BIOPHYSICS E. J. CASEY A* BIOPHYSICS Concepts and Mechanisms REINHOLD BOOKS IN THE BIOLOGICAL SCIENCES Consulting Editor PROFESSOR PETER GRAY Department oj Biological Sciences 1 niversity of Pittsburgh Pittsburgh, Pennsylvania CONSULTING EDITOR'S STATEMENT It is unfortunate that many students of biology regard biophysics as an esoteric and "•difficult" subject. The introduction of Professor Casey's "Biophysics: Concepts and Mechanisms' 1 to the Reinhold Books in the Biological Sciences should do much to dispel this view. Certainly, if every premedical student had a course in biophysics — and certainly no better book than Casey's exists for that purpose today — he would find his subsequent struggles with physiology enormously simplified. This is not to suggest that Professor Casey either dilutes or oversimplifies his subject. The simplicity of this book lies in the transcendent clarity and utter logic of the presenta- tion. A brief introduction to the necessary mathematics starts the book. This leads to a discussion of the physical forces exemplified in man, of mat- ter waves, electromagnetic radiations, and radioactivity as they apply to biological research. The author then passes to big molecules, and through them to an introduction to bioenergetics and the speed of biological proc- esses. The chapter on biophysical studies on nerve and muscle that follows draws point to all that has come before. The chapters on ionizing radiations and biophysical control excellently round out the broad scope of the book. All this, it must again be emphasized, is couched in language intelligible to any interested science major. I feel confident that the physicist, clinician, and biologist will find this book an ideal synthesis of an exciting interdis- ciplinary science. Peter Gray Pittsburgh, Pennsylvania October, 1962 c ■ BIOPHYSICS Concepts and Mechanisms \ E. J. CASEY : University of Ottawa Head, Power Sources Section Defence Research Chemical Laboratories Ottawa, Canada ^ REINHOLD PUBLISHING CORPORATION, NEW YORK Chapman & Hall, Ltd., London Copyright © 1962 by Reinhold Publishing Corporation All rights reserved Library of Congress Catalog Card Number 62-21000 Printed in the I 'mted States of America TO MY WIFE, MARY MY PARENTS and MY CHILDREN Preface This book is primarily intended to provide the student of biological sciences or of medicine with a substantial introduction into Biophysics. The subject matter, discussed in the Introduction, has been carefully chosen during ten years of teaching the subject. During this time the author has watched, in the literature, the subject begin to crystallize out from a rather nebulous mass of ideas and practices; and at the same time he has been able to observe what the students of this discipline require. Therefore, the book has been written with the needs of both student and teacher in mind, with the hope that this presentation of the choice of subject matter and the method of presenting it will be useful to others. Three objectives have been kept in mind in the presentation: (1 ) to build up from the easy to the difficult; (2) to make the presentation interesting; and (3) to unify it. Accordingly, the book generally increases in difficulty from an oriented review with pertinent examples in the first part, through more difficult material in the middle and later parts. Occasional relaxations, which reduce the information rate and afford occasions for exemplification with biological material, are included. A rather vigorous insistence on dimensional analysis has been hidden in the presentation, in the attempt to make the concepts and definitions precise. Following early definition, different units and methods of expressing them are used, so that the reader will not be awed by them when he studies further elsewhere. Wherever possible, recent work is introduced. Since the name "Biophysics" means so many different things to so many different people, the big difficulty has been to decide what not to write. In the interests of a unified presentation within a two-semester book, the limits chosen were concepts and mechanisms, with a minimizing of the method- ology which has already been treated in elegant fashion by others. There are some novel features about this book. The author has found them useful in his classes and would be pleased to receive the reader's opinions. Although bioenergetics in the broad sense of the term permeates the major part of the book from Chapter 2 through Chapter 9, it reaches its peak of interest in Chapter 7 in a conceptual presentation where the IX x PREFACE rigor of thermodynamics is sacrificed in favor of the development of a useful impression containing the necessary relationships: and these are illustrated. The electromagnetic spectrum (Chapter 4) and the matter wave spectrum (Chapter 3) are both surveyed, and stress is placed on those fractions which interact with (exchange energy with) biological material. The treatment of the effects of ionizing radiations (Chapter 9) surveys the hierarchy of struc- tures, from effects on simple molecules rrght up the scale to man. The unified treatment of speeds (Chapter 8) attempts to show similarities and differences of mechanisms among all rate processes: chemical reactions (catalyzed), fluid flow, diffusions, and electrical and heat conductance. The apparatus of physical control is described in Chapter 10; and in Chap- ter 11 the bases of control biophysics are introduced in terms which attempt to span the bridge between computer technology and brain mechanisms. The author has not hesitated to introduce a difficult concept if it would later serve a useful purpose, but has tried to get the reader through it in a simple manner. Because the scope is so broad, depth in every part of the subject could not be achieved in a book of this size. However, the bibliography is substantial, and further reading is explicitly suggested in those cases where the proper direction is not obvious. The chief inspiration for this work was the late Dr. Jean Ettori, Associate Professor at the Sorbonne and Professor of Biochemistry at the University of Ottawa. Known to his students as "the man who always had time," he died a hapless victim of cancer in 1961, at the age of 56. This man, who had gifts of vision in the biosciences as well as deep humility and love for his students, introduced the author to this subject and emphasized the need for what he called a "psychological presentation." The following colleagues, all specialists in their own right — in chemistry, physics, or the biosciences — read parts of early drafts of the manuscript and made many helpful suggestions: Dr. C. E. Hubley, Prof. A. W. Lawson, Prof. L. L. Langley, Dr. J. F. Scaife, Prof. M. F. Ryan, Dr. S. T. Bayley, Mr. G. D. Kaye, Mr. G. T. Eake, and Dr. G. W. Mainwood. Several other close colleagues helped by catching flaws in the proof. Mrs. Lydia (Mion) Labelle and Miss Nadine Sears struggled through the typing of a hand-written manuscript, Miss Sears in the important middle and late stages, and produced something which Mrs. Dorothy Donath of Reinhold could further mold into a finished text. The perceptive Miss Rosemary Maxwell turned out the best of the line drawings, and these in turn illustrate her talent. The author has had the encouragement of Dr. J. J. Lussier, Dean of the Faculty of Medicine, University of Ottawa, and of Dr. H. Sheffer, Chief PREFACE xi Superintendent of the Defence Research Chemical Laboratories, Ottawa, where the author carries on a research program in the interests of National Defence. E.J. Casey Ottawa, Canada October, 1962 Contents PREFACE ix INTRODUCTION 1 Scope 1 Subject Matter — a classification 3 Method of Presentation 3 1. THE SYSTEMS CONCEPT AND TEN USEFUL PILLARS OF MATHEMAT- ICAL EXPRESSION 6 The Systems Concept: introduced in general terms 6 The Ten Pillars: variable, function, limits, increments, instanta- neous rate of change; the differential and integral calculus; distribution of observations; expression of deviations; in- dices and logarithms; infinite series 8 2. SOME PHYSICAL FORCES EXEMPLIFIED IN MAN 26 Mechanical Forces: Newton's laws; units; levers; compressed gas 27 Osmotic Force: properties; water balance 35 Electrical Forces: bioelectrics; colloids; intermolecular forces; hydrogen bond 38 Generalized Force 44 3. MATTER WAVES; SOUND AND ULTRASOUND 47 Properties of Matter Waves: definition and illustration; absorp- tion 48 Sensitivity of a Detector and the Weber-Fechner Law 54 The Body's Detectors of Matter Waves: ear; mechanoreceptors 56 Speech 59 Noise 59 Physiological Effects of Intense Matter Waves: applications; therapy; neurosonic surgery 60 XIII CONTENTS ELECTROMAGNETIC RADIATIONS AND MATTER 67 The Structure of Matter: elementary particles; atomic structure; the nucleus; molecular structure and binding 68 Electromagnetic Radiation: nature; spectrum; absorption 76 Some Interactions of Electromagnetic Radiations and Living Matter: warming (infrared); visible (twilight and color vision); photochemical (ultraviolet); ionizing (X and gamma) 82 Microscopy: optical microscope (interference and phase con- trast); electron microscope 95 5. RADIOACTIVITY; BIOLOGICAL TRACERS 102 Ionization and Detection: positive ions; electrons; gamma rays; neutrons 104 Disintegration (Decay): half-life; energy distribution; decay products 112 Penetration of the Rays into Tissue 116 Uses as Biological Tracers: of molecular reactions; of fluid flow; in metabolic studies; radioactive mapping 1 17 BIG MOLECULES— STRUCTURE OF MACROMOLECULES AND LIVING MEMBRANES 125 Structure: crystalline macromolecules; dissolved macromole- cules (static and dynamic methods); living membranes 126 Isomers and Multiplets: electron transitions and triplet states 143 Replication and Code-Scripts: DNA and RNA; coding theory 147 Mutations and Molecular Diseases: hemoglobins; others 156 A CONCEPTUAL INTRODUCTION TO BIOENERGETICS 161 Laws (3) of Thermodynamics: statements; heat content of foods; free energy; entropy 163 The Drive Toward Equilibrium: free energy released; role of adenosine triphosphate, the mobile power supply 175 Redox Systems; Electron Transfer Processes: Nernst equation; indicators and mediators 179 Measurement of A H, A F, and T A S 184 Concentration Cells; Membrane Potentials 185 Negative Entropy Change in Living Systems 187 CONTENTS xv 8. SPEEDS OF SOME PROCESSES IN BIOLOGICAL SYSTEMS 192 General Principles: equilibrium us steady-state; rate-control- ling steps 193 Chemical Reaction Rates: effects of concentration and tempera- ture; the specific rate constant; catalysis by enzymes 195 Diffusion; Osmosis: diffusion coefficient; permeability con- stant 207 Fluid Flow: fluidity; laminar and turbulent flow; properties of plasma and of blood 212 Electrical Conductance: specific conductance; volume conduc- tor; EEG and EKG 219 Heat Conduction: heat production; heat loss 224 Formal Similarity and Integration of Five Rate Processes 230 Weightlessness 231 9. BIOLOGICAL EFFECTS OF IONIZING RADIATIONS 234 Dosimetry: dose units and measurement 236 Primary Effects: direct vs indirect; on molecules; oxygen effect 241 Biophysical Effects: coagulation; modification of transport prop- erties 245 Physiological Effects: sensitivity of cells; microirradiation of cells; irradiation of organs and tissues 247 Effects of Whole-Body Irradiation: present state of knowledge; therapy 254 10. BIOPHYSICAL STUDIES ON NERVE AND MUSCLE 262 Transient Bioelectrics in Nerve: historical review; tracer and voltage clamp techniques; cable and permeability theories; in central nervous system 262 Molecular Basis of Muscle Contraction: damped helical spring; energetics; structure; molecular kinetics of contraction 277 Effects of Environment on Control 290 11. THE LANGUAGE AND CONCEPTS OF CONTROL 295 The Systems Concept Redefined: information; entropy; measure- ment and noise; feedback; memory; implementation; control 296 Analogies: digital nature of nerve propagation; digital and analog computers 305 XVI CONTENTS The Computer in Biological Research: a study on the kinetics of iron metabolism 309 EPILOGUE— A PERSPECTIVE 315 TABLES OF COMMON LOGARITHMS AND EXPONENTIAL FUNC- TIONS 317 LIST OF SYMBOLS 319 INDEX 321 Introduction SCOPE Biophysics is today the youngest daughter of General Physiology, a sister to Biochemistry and Pharmacology. The subject matter is not yet very well defined, as the introduction to almost any of the recent essays on the subject quickly attests. Although the basic skeleton is clear enough — it being the engineering physicist's concept of a "system" suitably molded to describe the living thing — it may be many years before the dust has settled on dis- cussions of what appendages are proper to the skeletal framework of the subject. Consider some of the pertinent disciplines in terms of Table 1. Biochem- istry and biophysics attempt to describe and interpret the chemical and physical processes of biological materials in terms of the principles of or- ganic chemistry, physical chemistry, and physics. Biophysics is concerned with questions about the physics of biological systems. It has the advantages of less complexity and more certainty than the biological subjects, but has the disadvantage of being limited to only specific aspects of the whole living system. For the human being, biophysics can be thought of as providing a description of his whole physical system from the particular view of physics. For medical research, for the highest forms of medical specialization, and for the general medical practitioner of the years to come, the requirement seems inevitably to be a strong background and experience in the medical arts, coupled with a thorough grounding in the scientific knowledge of medi- cine and the scientific approach to it. The same is true of the biosciences. The scope of biophysics today is rather broad, if judged by the attitudes of authors of papers in several of the current journals, and in various essays. Yet the master, A. V. Hill, a Nobel prize winner who published his first paper in 1910 and is still active in research and physiology, has cautioned that the use of physical techniques or ideas alone for investigation of bio- logical problems does not of itself make biophysics. He defines the subject as: "the study of biological function, organization, and structure by physical and physiochemical ideas and methods," and then hastens to emphasize that he has put ideas first. He further expands* and drives home the key point as follows: *From "Lectures on the Scientific Basis of Medicine," Vol. 4, Athlone Press, London, 1954-1955; reprinted inSdence, 124, 1233 (1956). 1 INTRODUCTION There are people to whom physical intuitions come naturally, who can state a problem in physical terms, who can recognize physical relations when they turn up, who can express results in physical terms. These intellectual qualities more than any special facility with physical instruments and methods, are essential .... Equally essential, however, are the corresponding qualities, intuitions and experi- ence of the biologist .... The chief concern in the development of biophysics is that those [experimental] skills should be acquired by people who start with the right intellectual approach, both physical and biological. On the question of scope of medical biophysics, Hill says: ... If biophysics is to make its contribution to medicine, it is necessary that most physicians should have some idea at least of what it is about, while some physicians should have a pretty good idea. The ideas and methods of physics and physical chemistry are being applied today and will increasingly be applied, not only directly to physical medicine and radiology, but to neurology, to the study of circulation, of respiration and excretion, and of the adjustment of the body to abnormal conditions of life and work. At longer range, moreover, they will be aimed at the fundamental problems of minute structure and organization, of the physical basis of growth and inheritance, of the ordered and organized sequence of chemical reactions in vital processes, of the means by which energy is supplied and directed to vital ends. TABLE 1. Disciplines Surrounding Biophysics o u CL >. > a. .a. | uUl V TJ -a «-> en >> m "3d a C O '35 £ re l— ' v s_ plotted against x is a straight line of the form y = mx + b, with b = 0. The ideal gas law, PV = nRT, again can be used as a pertinent example. (b) y = mx 2 -f b is a parabolic rational function. In the case of the area of a spherical cell, the value, A, increases faster than that of the radius, r, so that the plot of A( =y) vs r( =x) sweeps up rapidly in a curve toward higher values of A, as r is increased. (c) N/N = e~ kl is an exponential rational function, in this case a decay (minus sign) or lessening, as time / increases, of the fraction N/N 0i where JV is the value of JV when t = 0; and A; is a proportionality con- stant. This function has less curvature than the parabolic. Radioac- tive decay is an example. The constant, k, can itself be negative. The weight of a growing baby is an example. (c') y = log x is a cousin of (c), called the logarithmic function. It has the same curvature as (c) but a different node. An example is the voltage across the living cell's wall, a voltage which is dependent upon ratio of salt concentrations inside and outside the cell. (d) y = k sin / is aperiodic function. The familiar sine wave of alternating current, the volume of the lungs as a function of time, and the pres- sure in the auricle of the heart as a function of time, are all examples. Figure 1-2 illustrates the four functional relationships. THE TEN PILLARS 11 These functions are all continuous; that is, at no point does the slope change suddenly from one value to another. It is probable that there are no discon- tinuous functions in nature, although the change in slope may be so sharp as to seem discontinuous in the first and cursory observation. Thus, phe- nomena involving the interface or juncture of two phases, as for example at the cell wall, are examples of rapidly changing continuous functions which at first sight appear to be discontinuous. 3. Limits If a variable, changing in accordance with some assigned law, can be made to approach a fixed constant value as nearly as we wish without ever actually becoming equal to it, the constant is called the limiting value or limit of the variable under these circumstances. A circus abounds with examples in which exceeding a limit in either dis- tance or time would mean a severe penalty. Consider the "hell drivers" who ride motorcycles inside a 40-ft cylinder, approaching the top — the limiting height — as closely as they dare, yet never suffering the disaster of actually reaching it. In other words, if y = /(*), and if, as x approaches a, y ap- proaches some value, b, then b is said to be the limit of/(x) when x equals a. In shorthand, for the functional relationship y = f(x), if x — * a as y — * b, then Lim f(x) = b x — '0 It is often useful to approach a limiting value and study its properties without having to suffer the embarassment sometimes associated with the limit itself. This concept was introduced by Leibnitz 300 years ago. 4. Increments A small fraction of any quantity under observation is called an increment. Increment is thus exactly translated as "a little bit of." It is given a symbol, the Greek letter delta, A. As the variable, x, increases (Figure 1-3) from zero to high values, that amount of x between A and B (i.e., x 2 — x x ) is "a little bit of" x, and is written in shorthand: \x. !— *f i i i r A P B 40mph Figure 1-3. Increments of Distance and Time, Ax and Af, used in defining velocity, Ax/Af, abouf point P, or dx/dr ai point P. 12 THE SYSTEMS CONCEPT Increments may be as large or as small as we like. If we reduce the dis- tance between A and B, the value of Ax is reduced; this can continue until Ax is infinitesimally small (so small that we cannot think of anything smaller). Infinitely small increments are called infinitesimals, and are written in shorthand with the Arabic letter "d", i.e., d.v. Combining the ideas of Sections 3 and 4, it is seen that as A and B ap- proach P, Ax gets smaller and smaller until, at the limit, Ax — ► dx, and it can be made infinitely small. This means that if we view the point, P, from B, we can move B in on P as closely as we please — in fact to an infinitely small distance away — and observe Pfrom as closely as we please. At the limit we observe Pfrom an infinitely small distance away, i.e., as A.v — > 0. With the concepts of increments and limits we have implicitly intro- duced the concept of continuous number, as opposed to the discrete number which is familiar to us in our unitary, decimal, and fraction systems. Con- tinuous number admits of the possibility of continuous variation of x be- tween A and B; the number of steps can be infinite. Continuous number is involved when a car accelerates from to 40 mph: the car passes through every conceivable velocity between and 40, and not in the discrete jumps which our decimal and/or fraction systems would describe. At best, these latter are but very useful approximations, and can be considered as con- venient, regular stop-off points, or stations, along the path of continuously increasing number. 5. Instantaneous Rate of Change Any living being is a complex system of interrelated physical and chemi- cal processes. Each of these processes in the "well" being is characterized by a particularly critical rate (speed or velocity) which enables it to fit into the complex system without either being too slow and holding all the other subsequent processes back, or too fast and allowing a runaway of certain subsequent processes. The study of the factors affecting the rates of processes is called "kinetics," and is discussed in detail for some biological processes, in Chapter 8. Average rate or speed, over some time interval, is often useful; but it is the instantaneous rate, or the speed at any instant, that is most useful for an understanding of these complex, interrelated reactions. If j; = f(x) and the function is continuous, we may be interested in how- fast y changes at any value of .v. In a diffusion process, for example, y would be a concentration and v the time. The question is: How much is the concen- tration in some particular volume changing per second at some particular second in time? The following three examples, one experimental, one graph- ical, and one analytical, illustrate the use of limits and increments to de- scribe this situation. THE TEN PILLARS 13 (1) Experimental : To measure the instantaneous velocity of an automobile (refer again to Figure 1-3) requires measurements of distance and time be- tween two stations, A and B. Two observers with stop watches and a tape measure can easily do this. They measure a value of Ax/ At, which is the in- crement of distance covered in an increment of time. But the car is acceler- ating between A and B, and hence Ax/ At is only an average value between A and B, and may be quite different from the velocity as the car passes P. Better values can be obtained the closer the observers are to P, but of course no value can be obtained if both observers are at P because Ax = and At = 0, and 0/0 is indeterminate, or can have any value from — « to + °°. The best value is obtained by taking observations at several values of A and B, at smaller and smaller values of Ax, until a good extrapolation to Ax = can be made. Hence the limit of Ax /At as At approaches zero is the instantaneous velocity at the point, P. In shorthand notation, the instan- taneous velocity at PisLim Ax/ At. A/— This symbolic description is further simplified by use of the infinitesimal symbols: Lim Ax/ At = dx/dt. Conversely the previous statement is actu- ally the definition of dx/dt. In other words, dx/dt is the instantaneous rate of change of x as t changes. A very simple experimental check on the method is to ride in a car and note the speedometer reading at point P. Both of these methods of determining instantaneous rate are exemplified in biological processes. (2) Graphical: A graph of the function which expresses the volume of the spherical cell, V = 4/37rr 3 , is shown in Figure 1-4. The question arises: How fast does the volume of the cell change with change in radius at a par- ticular value of the radius, r,? In other words, how "steep" is the slope of the curve, V vs r, at the point, r,? Slope or gradient is defined by surveyors as "rise"/"run," where "rise" is the vertical height from the base to the top and "run" is the level, or hori- zontal distance from the foot of the hill to the top. The ratio "rise/run" de- fines the value (trigonometric function) of the tangent of the angle enclosed by the level direction and the direction toward the top. The same is true in analytic geometry, the slope of the straight line join- ing P and P' being given by the ratio of the distances between P and P' as measured along the ordinate and along the abscissa. For example, slope V 2 - F, = AV/Ar. r~, — What we want to know is the value of the slope of the straight line which cuts the curve, V vs r, only once and at point P, that is, the slope of the tangent (geometrical figure) at P. This will give the instantaneous rate of change of Fas r changes, at P, or d V/dr at r,. 14 THE SYSTEMS CONCEPT RADIUS, r Figure 1-4. Volume of a Spherical Cell as a Function of its Radius. Determination of rate of change of V as r changes, i.e., dV/dr. This case is now similar to (1) and need not be discussed in detail. A point, P', is chosen; a straight line joining P and P' is drawn, and the value of A V/Ar determined from the graph. At successive points closer and closer to .Pthe same thing is done, until it is more or less evident what will be the limiting value of A V/ A r as A r approaches zero. Once again, Lim A V/Ar = d V/dr, the slope at P. It turns out that for this case d V/dr = Airr 2 . (3) Analytical: A simple example* will illustrate one way in which this can be done algebraically. The law established by Galileo at Padua governing the free fall of a body (Figure 1-5) toward earth, is expressed as S = 1/2 gt 2 , where S is the dis- tance fallen, t is the time of fall, and g is the value of acceleration due to gravity (32 ft per sec per sec.) This example is chosen not because of its specific relation to medical physics but because of its simplicity as an illus- tration of the algebraic determination of instantaneous rate of change by means of the method of increments. The experimental and graphical ex- amples, (1) and (2), are limited in that an extrapolation of incremental pro- portions is always necessary. In the algebraic method this is not necessary, but the limit still can be examined from as close in as it is possible to imagine. *As an alternative one could have considered a child blowing up a balloon, and asked the question: How fast does the area of the balloon change as the radius changes? The area is given by A --= 4irr , also a parabolic function. Less easily conceived examples appear later. THE TEN PILLARS 15 The question is: What is the velocity of the falling body at the instant it passes the point, S ? At S, S = 1/2 gt 2 -d-1) At 5 + AS, S + AS = 1/2 g(t + A/) 2 . Multiplying out the square, S + AS = 1/2 gt 2 + gtAt + 1/2 gAt 2 - -(1-2) Between the two points, then, the value for AS is given by Eq. (1-2)- Eq. (1-1): AS = gtAt + 1/2 gAt 2 The average rate, over a small increment of time is: AS/At = gt + 1/2 gAt Hence, the instantaneous rate is: dS/dt = Lim AS/At = gt + 1/2 g x = gt A/— That is, the instantaneous rate of change of distance with time (or velocity) at the point, S, is: dS/dt = gt (1-3) For example, 5 sec after free fall starts, Eq. (1-1) says that the distance fallen is 400 ft; and Eq. (1-3) says that the velocity as it passes the 400- ft mark is 160 ft per sec. Maximum and minimum values of functions with changing slope and curvature must be given by the values of the function for which the instan- taneous rate of change, or slope, is zero. This can be visualized in the periodic function of Figure 1-2, for example. 6. The Differential and Integral Calculus It has been seen that, given the explicit form of the "mother" function, it is possible by the method of increments to determine the explicit form of the s ■ t ^ t + ^t Figure 1-5. The Falling Body. 16 THE SYSTEMS CONCEPT expression which describes the instantaneous rate of change-the "daugh- ter " or derived, function. A system of "operations" has also been devel- oped by which the same thing is accomplished. In this sense d/dx is an "operator," operating on y in a specific manner which accomplishes the same result as the method of increments gave us in Example (3) . Conversely, if the rate of change is given (most often directly from the experiment), it is possible from the daughter equation to reverse the method of increments, and establish and examine the mother equation (Figure 1-6). The process is simply to sum the increments, under special conditions, when they are infinitesimally small. A system of operations has also been worked out for this process. The operator is symbolized as an elongated S , called the "integral sign," f, contrasted against the operator, "d", for the inverse process. .-. ^Pj^ e_ntio t_i £n rate of change Figure 1-6. Definition of Differentiation and of Integration. Described in the previous Sections 1 to 5 are the basic ideas of the calcu- lus The process of finding from the mother function, F{x), the daughter function, F'(x), which expresses rate of change, is called differentiation, or obtaining the derivative or derived function; the reverse process of summation of an infinite number of values of the derived function, F'(x), to give the mother function, F(x), is called integration or obtaining the integral. Two more definitions in shorthand will prove to be useful, the second order derivative and the partial derivative. Both are actually quite simple concepts. We often run into a situation in which we wish to express how fast the speed is changing. (Consider the automobile example, given in Section 4, in which we are now interested in acceleration.) Since speed is dS/dt, the rate of change of speed is d/d«dS/dt), which is abbreviated d 2 S/dt 2 with the operator "d," in the numerator squared and the whole differential in the denominator squared. It is obvious that the rate of change of acceleration would be expressed as d'S/dt\ and that higher orders exist, although they are not of common interest to us here. THE TEN PILLARS 17 Sometimes one or more independent variables (y, z) are kept as constants of the system while another (x) is varied. The rate of change of the dependent variable, 0, as x changes, is expressed as an incomplete or partial derivative. To emphasize the partial character, a rounded operator, d, is used; and the constants of the system are stated as subscripts outside parentheses which enclose the partial derivative. Thus: (d(f>/dx) y>i expresses the rate at which changes as x is changed, when y and z are kept constant. The second-order partial derivative, the "acceleration," is expressed as before: (d 2 /dx 2 ) M This notation is used in all heat and mass transfer- considerations. For instance, note the Haldane quotation which introduced this chapter. At this stage of development of biophysics (1962), the terminology of the calculus is being used in published work, hence the need for introduction to the bases and terminology of the subject. But explicit descriptions of most biophysical phenomena are very rare; hence there seems to be no need to in- troduce the operational calculus into an introductory book on biophysics at this time. Therefore no attempt has been made to display the actual opera- tions by which either differentiation or integration is accomplished. Opera- tional calculus is treated in detail in many standard textbooks. 7. Distribution of Observations A great many biological phenomena lend themselves to statistical meth- ods of expression, i.e., age, height, weight, bloodcount, sugar analysis, etc. This is so true that the "average value" over a large number is considered the "normal" value, describing the "normal man." Hence it is instructive to examine some of the methods of statistical expression, and to discuss their reliability. Statistics has come a long way since the publication in 1662 of John Graunt's "Natural and Political Observations Made upon the Bills of Mortality," a study based on the records kept during the Black Plague in London; and since Sir Edmund Halley (of "Comet" fame) wrote his basic paper on life insurance, which appeared 30 years later. In the 20th century statistical methods have penetrated nearly every field of learning in which numerical measurement is possible. Moroney's book 4 gives a delightful in- troduction to the subject. First of all, there are two factors which will result in a distribution in a number of observations. One is errors in measurement; the other is a real 18 THE SYSTEMS CONCEPT distribution in what is being measured. Measuring the length of a room with a 12-in. ruler will result in a fairly wide error, and although the mean value of a number of observations should be close to fact, there may be a large uncertainty in an individual measurement. Besides such random errors, there may exist also constant errors which are sometimes very important but too rarely recognized. Suppose the ruler has been made 1/16 in. too short at the factory. If the room were 32 ft long, in addition to the random errors, every measurement would have been 2 in. short: even the mean value cannot be trusted in the presence of a constant error! It is revealing to read the temperatures on several of the thermometers in the laboratory thermometer drawer! Constant errors and the need for calibration become quite obvious. Even under the most carefully controlled experimental conditions, unknown constant errors creep in. In addition, personal bias is always with us, in reality if not in principle. The variation in the quantity being measured is often called "biological variance." Consider the height of 80 people at a lecture — it usually has a distribution from about 5 ft, in. to 6 ft, 3 in., with the average approxi- mately 5 ft, 7 in. Deviations from 5 ft, 7 in., however, could hardly be con- sidered as errors or abnormalities! Constant errors are deadly and can result in gross misinterpretations. Analytical chemistry done without proper calibrations is an example. It has been shown to be prevalent even in routine analyses done day in and day out in the hospitals, with large variations in mean values being reported between them — each hospital apparently having its own constant errors! This is embarrassing, but it is a fact. Under these conditions, diagnoses made with reference to some published work from another hospital could easily be wrong. It is necessary continually to be on the alert against con- stant errors, or "biased [not personal] observations," as they are sometimes called. Random errors and natural distribution in the variable measured can both be treated with statistical methods. The most reliable methods, and in fact the only reliable method in constant use, presuppose that the observa- tions distribute themselves about a mean or average value such that the density of points is greatest at the mean and progressively less and less as the deviation from the mean becomes larger. That is, it presumes a "normal" distribution in the observations. Figure 1-7 shows the normal distribution curve. It can be interpreted two ways: (1) P represents the number of observations, N, which are Ax units less than the mean; (2) P represents the probability that any measurement now being made will have a deviation less than Ax from the mean. THE TEN PILLARS 19 It is axiomatic that any expression of confidence made in terms of normal distribution, presupposes normal distribution; and that any such expression concerning a distribution which is not normal is not only unwarranted, but also useless, and may be quite misleading. There are statistical methods for handling non-normal data, but they are not simple and are seldom used correctly. Mainland's book 3 goes into some of these, using examples of medical interest. -^x + ^y. DEVIATION FROM MEAN VALUE Figure 1-7. Normal Distribution of Observations. Solid Curve: Area under curve be- tween -a and +a includes 68 per cent of observations; between —2a and +2o, 94 per cent; and between -3a and +3o, greater than 99 per cent. Blocks: Typical Observa- tions of Heights of Thirty People at a Lecture. 8. Expressions of Deviations The most common method of expressing a number of observations, x, of the same phenomenon is by the common average, or arithmetic mean, x. There are others, such as the median and the mode, which have some use in nearly normal distributions, but only the mean will be considered. Deviations Ax from the mean can easily be computed by subtraction, and then averaged, the result being expressed as the mean deviation Ax from the mean x. A very common method of expressing the distribution is by the standard deviation, a, defined as the square root of the average of the deviations squared: a = y/Ax 2 , or a = y^Ax 2 /n 20 THE SYSTEMS CONCEPT Bessel's correction is introduced if the number, n, of samples is small (< 30); then a = j/£ Ax 2 /(n - 1) The most probable deviation, r, is that value of the deviation such that one- half the observations lies between the limits ±r. The relative deviation, usually expressed as a per cent, is the fraction which the deviation is of the observed mean value, i.e., Ax/x. Each of these has several names. In the case of random errors, "devia- tion" should read "error," of course; Ax is often called the absolute error of the measurement. Relative error is sometimes called per cent error or proportional error. These are discussed in detail, and examples are given, in Mainland's book. Superposition of Errors. In the determination of a quantity, A, af(x, y, z) which requires measurement of x, y, and z, each with an absolute error, the errors must be superimposed one upon the other, or added; the reliability of the value obtained for A is no better than the sum of the errors in x, y and z- That is, the relative error in A is the sum of the relative errors in the meas- urements oix,y, andz- 9. Indices and Logarithms In arithmetic the ancient Greeks devised and used a notation, now called that of indices, to express in shorthand the number of times a number is to be multiplied by itself. Thus, "2 multiplied by itself 5 times" (i.e., 2 x 2 x 2x2x2) = 32. This is written in shorthand as 2 5 = 32. The index, 5, is placed as a superscript to the base number 2. A number of laws of indices can be shown to exist for the manipulation of such numbers. These laws were observed for cases in which the indices are whole numbers. Now there is no reason to suppose that the rules would be different for fractional indices, although to multiply 2 by itself 5 1/2 times would really be tricky! Nevertheless, the rules are assumed to apply to fractional indices, as well as to whole-number ones, and further also to algebraic, unknown indices. In general, the laws of indices are as follows: (\)a m = axaxaxaxa m times (2) a m a" = a m+n (3) a m /a n = a m -" if m > n 1 .. or a m /a" = it n > m THE TEN PILLARS 21 (4) (a m ) n = a mn (5) (ab) m = a m b m (6) {a/b) m = a m /b m Fractional indices are called roots. Thus, a i = y/a, the square root of a; and in general a'/"' = m \/~a, the m lh root of a. (7) a" = 1 (8) a~" = 1/a" (9) a" = °o (10) a~' = 1/a" = Logarithms Let .4 = a". The index x, which tells how many times the base number a must be multiplied by itself to give A, is defined as the logarithm of A to the base a. In shorthand this statement is given by x = lbg a A, where "to the base a" appears as a subscript to the abbreviated "logarithm." Logarithms are indices and must obey the ten Laws of Indices, just as any other. For example: log AB = log A + log B log A/B = log A - log£ log A m = m log ,4 A change of base from base a to base b turns out to be analogous simply to a change of variable. In other words the logarithm to the base, a, is re- lated to the logarithm to the base, b, by a constant, \o% b a. One is a linear function of the other. This can be shown as follows. Suppose A = a" and A = b y , so that a x = b y . Then log a A = log a b>, or x = y log a b. There are two systems of logarithms in daily use in biophysics, as in all other science and technology: (a) Common logarithms, to the base \0(y = \0" for example), used to simplify the manipulations of multiplication and division, based on rules (2) and (3). The abbreviation is log, or log l0 . (b) Natural logarithms, to the base e (y = e x for example), where e = 2.71828. . . . The base, e, and the functional relationship,}' = e", occur over and over again in man's description of nature, and therefore will be illustrated further. The abbreviation is In, or log f . 22 THE SYSTEMS CONCEPT Conversion, as described above, is accomplished as follows: log A = In A 2.303 where 2.303 = log, 10. 10. Infinite Series,- y = y e ox A series is any group of numbers, arithmetically related, which differ from each other in some regular and explicit manner. Thus 1+2 + 3 + 4 + 5 + n is a series. This particular series is divergent, since the larger the n chosen, the greater the sum becomes. There are other series which are convergent, whose value approaches a limit as the number of terms is increased toward infinity. One such convergent series is x x 2 x 3 x 4 1 + — + + '■ + — + 1 2x1 3x2x1 4x3x2x1 This series, for a value of x = 1, simplifies to 1 l 2 l 3 l 4 1 + — + + + + 1 2x1 3x2x1 4x3x2x1 which converges to the numerical value 2.71828 .... as more and more higher index terms are added. In shorthand e x is written for the first, and e x or e for the second series. Thus X X L X 3 X* and e x = 1 + — + — + ■ + ■ + 1 2x1 3x2x1 4x3x2x1 1 l 2 l 3 e = 1 + — + — + + = 2.71828 1 2x1 3x2x1 More generally, when x is preceded by a constant, k, kx is substituted for x : , , kx (kx) 2 (kxy (kxy e** = 1 + h - — — + — + — + 1 2x1 3x2x1 4x3x2x1 The constant, k, simply tells how slowly the series converges for any particular value of x : the greater the value of k the greater the number of terms which will be necessary to define e kx to a chosen number of significant figures. Now, when x is the variable, and k constant, we can call its evaluation proportional to jv and write or y a e kx (1-4) THE TEN PILLARS 23 The series typified by e kx is the only functional relationship in all of mathe- matics for which its instantaneous rate of change at a value of x is exactly proportional to itself. That is, it is the only function for which both y a e kx (1-4) and dy/d.v « e kx (or « y) (1-5) are true. For completeness, if the proportionality constant in Eq. (1-4) is intro- duced, y =y ^ ---(1-4') and dy/dx = ky e kx ,__(l-5') or dy/dx = ky This, however, explains the importance of e x in mathematics. The im- portance in biophysics is that a great many naturally occurring phenomena are observed to behave according to Eq. (1-5'): many chemical reactions, growth, diffusion processes, radioactive decay, radiation absorption phe- nomena, etc. (Figure 1-8). TIME Figure 1-8. Two Exponential Relationships: Growth (positive k), and Decay (negative k). For example, let y be the number of atoms of a given sample which give out a radioactive emanation (alpha, beta, or gamma ray), and x be the time. Eq. (1-4') says that the rate of emanation is always proportional to the num- ber of atoms which are left and are capable of disintegrating, a statement 24 THE SYSTEMS CONCEPT which, if reflected upon, will become quite obvious because it is not only a "natural" law, an observed law of Nature, but also a logical deduction. In our examples, most commonly a decay is involved, in this case the decay of a concentration. Thus k is a negative number. If the minus sign is taken out of the k and k replaced by — X, the expression becomes N = N e~ Xl , sometimes written N = N exp(-Xt), for radioactive decay, where JV is the number of particles present when t = 0. Figure 1-8 shows the shape of the exponential curve for positive k values (growth), and for negative k values (decay). Note that the former increases to infinity, unless checked by the onset of some other law; and that the latter decays toward zero, reaching zero only after an infinitely long time, although it may be below the lowest measureable value within a very short time. The larger the value of k, the faster the growth curve sweeps upwards, and the sooner the decay curve approaches zero. PROBLEMS 1-1 : (a) If a student must pass biochemistry, and John is a student, then . . . ? (b) If y = 2x and Z = y, then what functional relationship exists between Z and*? (c) Uy =/,(*) and £ = f 2 (x); and f 2 (x) = /, (x) -f 3 (x), then what is the rela- tionship between x andy? (d) If A °c B, and B °c C, what is the relationship between A and C? (e) If the weight of a given volume of gas is proportional to density, and if the density is proportional to its pressure, then what is the relationship between weight of a given volume and its pressure? 1-2: Choose at random, alphabetically for example, the heights in inches of 25 students. (a) Is the distribution normal? Was the sample biased? (b) What are the average deviation, Ax, and the standard deviation, a? (c ) What fraction of the sample falls within the mean deviation from the mean? (d) What fraction of the sample falls within one standard deviation from the mean? If the distribution had been normal, what would have been the fraction? (e) What fractions of the sample fall with ±2 a and ±3 a? If the distribution had been normal, what would have been the fractions? 1-3: Make a table showing how the distance fallen, the speed, and the acceleration of a parachutist change in the first 5 sec before the chute opens. (Make the calculations for each second.) Suppose he hits the earth at a velocity of 120 ft per sec without the chute opening. From what height did he jump? 1-4: The decay of Sr 90 follows the exponential law N = JV e~ Xl , where N is the concentration of radiating material at any time, t; N Q is the concentration at some arbitrary zero of time; and X is the decay constant of Sr 90 , namely 0.028 years" 1 (i.e., 0.028 is the fraction lost per year). REFERENCES 25 (a) Make a table showing values of -Xl, e~ Xl , and N e~ Xt for various values of / (years), assuming that N = 100% at/ = 0. (b) From the results, make a plot of JV vs t, and estimate the half-life (the time, r, in years, when N = 50% of A ). (c) Sketch decay curves for P 32 (t = 14.3 days), I' 31 (8 days), C' 4 (5100 years), Co 60 (5.3 years), Po 210 (138 days), and Ra 226 (1620 years), all on the same graph. Compare them. REFERENCES 1. Petrie, P. A., et al., "Algebra — a Senior Course (for High Schools)," The Copp Clark Publishing Co. Ltd., Toronto, 1960. (See p. 314jffor discussion on incre- ments.) 2. Thompson, Silvanus P., "Calculus Made Easy (Being a Very Simplest Introduc- tion to Those Beautiful Methods of Reconing which are Generally called by the Terrifying Names of the Differential and Integral Calculus)," 3rd ed., MacMillan& Co. Ltd., London, 1948. 3. Mainland, D., "Elementary Medical Statistics," W. B. Saunders Co., Philadel- phia, Pa., 1952. 4. Moroney, M. J., "Facts from Figures," 3rd ed., Penguin Books Ltd., Toronto, 1956. CHAPTER 2 Some Physical Forces Exemplified in Man (Mechanical; Osmotic; Electrical) All physical reality is a manifestation of what force does. On the ques- tion of what force is, science can do no better than to call it by other names. (Truth is a virtue, however inconvenient.) INTRODUCTION Force and energy, along with optics and acoustics, are the concerns of classical medical physics, and some of the principles have been understood for well over a hundred years. In this chapter the nature and the units of force are reviewed, and the relationship between force and energy discussed. The transfer of energy is reserved for Chapter 7. The living system is in a state of continual exchange of force and energy with the environment. What is force? According to Newton (1687), it is vis impressa, an influence, measurable in both intensity and direction, operating on a body in such a manner as to produce an alteration of its state of rest or motion. Generically, force is the cause of a physical phenomenon. It is measured by its effect. Further penetration of the nature of force seems destined to remain a philosophical question, because the range of experi- ment stops at measurement of the effects. By experiment it is possible to measure the effect of different forces on the same object, and devise a system of interconversion factors by which one kind of force is related to another (for example, mechanical to osmotic). Ef- 26 MECHANICAL FORCES 27 forts to penetrate the generic nature of the "force field" — to develop a uni- fied theory — received much impetus, without much success, during the life of Albert Einstein, but one notices now that efforts at unification are falling off as theorists drift into other problems. Hence the question most funda- mental to all science, biophysics included, viz: "What is force?", seems destined to remain unanswered for a long time yet. It is a more fundamental question even than "What is life?", for life is only one manifestation of force! MECHANICAL FORCES Newton's Three Laws of Motion These three laws are the basic description of mechanical systems. From the simple statements can be inferred many properties of mass and inertia. First Law: A body at rest tends to stay at rest, and a body in motion tends to continue moving in a straight line unless the body is acted upon by some unbalanced force (F). The property of the body by virtue of which this is true is given the name inertia. The measure of amount of inertia is called the mass (m). Second Law: A body acted on by an unbalanced force will accelerate in the direction of the force; the acceleration (a) is directly proportional to the un- balanced force and inversely proportional to the mass of the body. This second law describes the familiar experimentally derived relationship F a ma, or F = kma. If the dimensions of F are suitably defined, this be- comes F = ma. The need to choose the dimensions in this manner results from the fact, discussed earlier, that we really do not know what the nature of force is, but rather do we know only its effects. This is certainly true of the common forces of gravitation, electrostatics, and magnetism. Yet fric- tional force we are able to relate to physical interference of microrough- nesses and physical attraction of two surfaces — and thus have some idea of what this force is. The force exerted by the finger to push the pencil, or the force exerted by the thumb on a hypodermic needle drive home to us a meaning of mechanical force based on its effects. Third Law: For every physical action there exists an equal and opposite reaction. The recoil of a rifle as the bullet is ejected, and the swinging arms which help man to maintain his balance while walking briskly, are examples. Careful consideration of the statements themselves will enable the reader to appreciate the far-reaching consequences of these laws, consequences which range from suspension bridges to the molecular interactions of bio- chemistry, from the effects of high centrifugal forces on the pilot of a high- speed aircraft to the simple levers of which the human body in motion is a remarkably complex, though well coordinated, example. 28 SOME PHYSICAL FORCES EXEMPLIFIED IN MAN Units and Dimensions It is useful now to introduce definitions of certain quantities in mechanics. By the first law, a force is defined as anything which changes the state of rest or of motion in matter. The basic unit of force, in the centimeter-gram- second system, is called the dyne. This is the force which will produce an ac- celeration of 1 cm per sec each sec (1 cm sec -2 ) on a mass of 1 gram (1 g). All other forces (electrical, etc.) can be related by suitable experiments to this fundamental quantity of motion. Force gives to mass an energy, a capability of doing work. In the system of mechanics, the amount of energy acquired by a mass under the influence of a force depends upon how long or over what distance the force acts. The energy imparted to 1 g of mass by a force sufficient to give the mass an ac- celeration of 1 cm sec -2 within the distance 1 cm, is called 1 erg. One erg = 1 dyne cm. This is an inconveniently small unit of energy, and a quantity often million (10 7 ) ergs has been defined as 1 joule (1 jou). By contrast with this definition of energy units in the mechanical system, the unit of heat energy, the small calorie, has been defined as the amount of energy which it takes to raise the temperature of 1 g of water 1° C, between 4.5 and 5.5° C, where water is the most dense.* (As the temperature is lowered, water molecules begin to line up in "anticipation" of freezing, and the volume increases; as the temperature is raised, increased thermal energy tends to drive the molecules apart, and the volume also increases). Experi- mentally, by transformation of mechanical motion into heat in a water calorimeter, 1 cal has been found to equal 4.18 jou. One thousand cal, or 1 kilocalorie (1 kcal), has been defined 1 Cal, or large calorie. This is the unit used to describe the energy available from different foods. Power is the rate at which energy is expended; that is, energy expended per unit time. The basic unit of power is the joule per second, called the watt (w). One-thousand watts is 1 kilowatt (1 kw). One horsepower (1 hp) is equivalent to 746 w or 3/4 kw. Entergy exists in two general forms, kinetic and potential. Kinetic energy is that possessed by mass in motion. In mechanics potential energy is that possessed by a mass because of its position. In other disciplines potential energy assumes different forms: the energy stored in chemicals, or that stored in extended muscle, or in an electrostatic charge separation across a cell membrane, could be released to do useful work or provide heat. Heat energy is all kinetic energy. It is the total energy of motion of all the molecules in the body under consideration. Temperature is an indicator of the amount of heat in a body, and can be considered to be the "force-like" *The amount of heat required to raise 1 g of a substance 1°C is called the specific heat, c. It can be measured under constant pressure (c p ) or under constant volume (cy ). MECHANICAL FORCES 29 factor of heat energy. The accompanying capacitive factor in effect sums up the energies which can go into all the vibrations, rotations, and translations of each molecule. This capacitive factor is called entropy, S. Heat energy is therefore given as the product TS, and 5* must have the units calories per degree, since the product must be simply calories. Heat energy was chosen over electrical, mechanical, or other forms for no other reason than that it is so common. All forms of energy can be factored into two parts, a potential part and a capacitative part: thus in addition to heat energy, we have force times distance for mechanical energy; voltage times charge for electrical energy; pressure times volume for the mechanical energy contained by a compressed gas; chemical potential times number of moles for chemical energy. Energy and its factors will be considered more fully in Chapter 7. Kinetic energy of mass in motion is given by force x distance, which has the dimensions (g cm/sec 2 ) cm, or g cm 2 /sec 2 . Kinetic energy of motion is also given by the familiar 1/2 mv 2 , with the same dimensions. Another familiar property of mass in motion is the momentum, M, defined as mv. Hence KE = 1/2 Mv. Some of these quantities can be illustrated by the example of a 200-lb** football player running at full speed with the ball. His potential energy in the form of food has been reprocessed into glycogen, etc., and stored as po- tential energy. That part ready for rapid conversion is available in the form of the mobile chemical adenosine triphosphate (ATP), whose role as a mo- bile power supply is wondrously general throughout the living system. Dur- ing the motion this chemical energy is being transformed, at least in part, to the mechanical kinetic energy of motion. His KE amounts (speed 100 yds in 12 sec; 1 lb = 454 g) to about 26,000,000,000 (or 26 x 10") ergs, or 2600 jou, about 550 small calories. If he is stopped completely within 1 sec by collision, he will have transferred energy at an average rate during that second of 2600 jou per sec, 2600 w, or just over 3 hp. If that energy all went into heat, it could vaporize about 1 g of water. On the other hand this energy could have been transformed into electricity, and the power delivered could have lighted twenty-four 100-w light bulbs to full brilliance for a sec- ond! A further insight into the power expended in such collisions can be gained if it is remembered that the bulk of the energy is transferred in about 1/10 sec of contact, during which time the power is about 30 hp! It is obvious that, in spite of the delights attached to such athletic pursuits, from the point of view of pure physics alone, they are sheer waste of energy and power which could be used more efficiently to do other tasks. In fact even ** Weight, a force. Since F = ma: 1 lb force = 1 lb mass x 32 ft/ sec 2 , and 980 dynes force = 1 g force = 1 g mass x 980 cm/ sec 2 . (1 lb force is the force of attraction between the earth and 454 g mass.) 30 SOME PHYSICAL FORCES EXEMPLIFIED IN MAN at its slowest, when no work is being done, basal metabolism amounts to about 0.1 hp. The human machine needs a minimum of 0.1 hp to keep it alive, and can put out continuously a maximum of about 0.01 hp of useful mechanical work, with occasional surges to several horsepower. The football player's momentum just before collision was (200/32) x (300/12) = 154 lb sec. If this were transferred in 0.1 sec during collision, the impressed force, defined as rate of change of momentum, dM/dt, was 154/0.1 = 1540 lbs. This can be expressed as a "shock" (force per unit mass) of about 7.7 g, where g is the acceleration of all bodies due to gravita- tional attraction to the earth (32 ft/sec 2 , or 980 cm/sec 2 ). The value 7.7 g is obtained directly from the second law, viz J7I 154 ° 77 a = t m = = 7.7 g 200/g By contrast, and as further illustration, the passengers on a modern com- mercial jet line experience about 2 g during take-off. The jet pilots for fighter aircraft and the astronauts have been tested up to 18 g. The famous right hand of boxer Joe Louis was said to impart up to 40 g to a stationary and nonelastic target. A laboratory centrifuge will provide a centrifugal acceleration of some thousands ofg; and the ultracentrifuge used in sedi- mentation experiments in which molecular weights of large molecules are obtained, develops up to 100,000 g. Centrifugal motion is convenient for varying at will the inertial mass of a body: e.g., in the human centrifuges in space-research laboratories. As a machine, man is very versatile. However, he is quite inefficient be- cause of the continuous power being expended to keep him alive when he is not "in use." His highest purely physical role is that of a computer. Two forces will now be considered: a mechanical force as applied to a lever, and the mechanical force of a compressed gas. The Lever A lever is one of a great number of machines — devices for doing work. This particular device permits mechanical energy to be factored into such values of force and distance that some desired mechanical result can be ac- complished. The lever does not create energy, of course, but simply makes the energy more available to do the particular job at hand. The familiar example of the crowbar to dislodge a large stone, using a log as a pry, is an example. In this case a relatively small force applied over a relatively large distance at the hands is transformed into a relatively large force applied over a relatively small distance at the stone. The mechanical advantage is the ratio of the two forces; it is inversely proportional to the ratio of the two dis- tances since F i d ] must equal F 2 d 2 . MECHANICAL FORCES 31 The three classes of levers, expressed in terms of the relative positions of applied force, F a , resultant force, F r , and fulcrum, with directions denoted by the arrows, are given in a classical example in Figure 2-1. 2nd class weight of jaw weight of body weight of head Figure 2-1. First-, Second-, and Third-Class Levers. The muscular-skeletal system of the human body is a complex system of levers. The majority of these are third-class levers. A runner on tiptoe has a second-class lever in his foot: the ball is the fulcrum, F a is at the heel, ap- plied by Achilles' tendon and the calf muscle, and F T is exerted near the instep. The jaw, the forearm, and the fingers of the hand are all third-class levers. However Jiu-jitsu is a study in first-class levers, and the arm and leg locks used in wrestling are almost invariably first-class levers. While doing push-ups the body is operating as a second-class lever. The pump of an old- fashioned well and a wheelbarrow are second-class levers, and there are countless other examples of each among man's tools. Simple levers were man's first machines. Compressed Gas Pressure is mechanical force per unit area (Figure 2-2). Atmospheric pressure is simply the weight force of a column of air 1 cm 2 in area and of a height, h, equal to the effective height of air above the earth. From basic definitions P = p gh, where p is the average density over the height, h. The units of pressure are dynes cm -2 , and of g, cm sec -2 . However, it is common practice, where differences or ratios of pressure are involved, to ignore the factor, g, which is constant at any particular spot on the earth's surface. The weight of the column of air is about 1 ,050 g or 1 5 lb above 1 in. 2 The common unit is 15 lb (force) per sq in. (15 psi) = 1 at- mosphere (1 atm) at sea level. 32 SOME PHYSICAL FORCES EXEMPLIFIED IN MAN f .. p t + p 2 + p 3+ p 4 (pressure = force per unit area) Figure 2-2. Pressure and Force. It has been found that 15 psi can support a column of mercury about 30 in. (76 cm or 760 mm) high. That is, if a glass tube of any diameter (the larger the cross-sectional area the larger the force, since the pressure is 15 psi) is mounted vertically in a pool of mercury, and if the air in the tube above the mercury is exhausted substantially to zero pressure, the air pres- sure on the outside of the pool will force the mercury up the tube to a height of about 30 in. above the level in the pool. If the supporting pressure (dif- ference between air pressure on the mercury in the pool, open to air, and on the mercury in the column) is less than 15 psi, the height of the column is correspondingly less. Atmospheric pressure varies with the weather, from about 29 to 31 in. of mercury between very stormy, low-pressure weather and fine, high-pressure weather. Living systems operate under this continuous pressure of 15 psi, but do not collapse for two reasons. Firstly tissue is about 80 per cent water by weight, and water is nearly incompressible. Secondly, air can pass fairly freely into those interior parts which are not solid or liquid, and the internal gas pressure is about the same as the external. A large reduction in pressure (e.g., 12 psi) over a small area of the skin surface can be tolerated for some minutes without ill effects. On the other hand, pressure-increases up to 327 psi at a new record depth in water of 726 ft were recently tolerated. The cur- rent skin-diving record is 378 ft, where the total pressure, P, is of the order of 12atm! The total pressure (psi) is given by: P = P atm + 0.43 D where P alm = 14.8 psi, D is the depth in feet, and 0.43 is the weight, in pounds, of a column of water 1 in. 2 in area and 1 ft high. At the record skin- diving depth, the total force on the body (20 sq ft) is about 270 tons! The troubles start when pressure changes occur rapidly, such as during collisions or impact. Consider the skin diver equilibrated 200 ft below the MECHANICAL FORCES 33 surface of the water. An extra amount of nitrogen will be dissolved in all the body fluids, including the blood stream. Henry's law describes how the amount of gas dissolved, w, increases linearly as pressure increases: i.e., w = HP, where H is the proportionality (Henry's) constant. This expresses the condition of the diver at equilibrium with his environment. If now, sud- denly, he rises to the surface, the nitrogen which has diffused into the blood stream is not able to diffuse out fast enough, and will come out of solution in the form of small gas bubbles, which rapidly coalesce to form larger ones. Under the conditions described, the bubbles so formed would be easily large enough to form "air locks" and prevent the flow through the blood capil- laries. This illustration simply shows the physical facts of the condition known as "bends": circulation ceases, waste products of muscle activity ac- cumulate, muscles cannot be reactivated; excruciating pain, paralysis, and death can result. The only treatment is to increase the pressure in a pressure tank in the hope that the nitrogen bubbles will redissolve. A second problem, and often a more important one, illustrates another physical point. It is a fact that sometimes during fear the individual will hold his breath tightly as he pops to the surface from a considerable depth: since the opening at the epiglottis is small, only a small force by the muscles is necessary to apply the considerable pressure needed to keep this valve closed. Up from even 25 ft, for instance, the external pressure has dropped from 30 psi to 15, and if the extra gas is not exhaled, the excess pressure is a full atmosphere on the delicate walls of the lungs. Punctures, called air embolism, can occur, and cause a condition not unlike pneumonia, in which air-CO, exchange on the lung walls is retarded. The results are similar in the case of a high-flying airman if he is ejected from the aircraft and is unprotected by a pressurized flying suit; or in the case of a space traveller whose pressurizing equipment fails. In these cases, in which the pressure is suddenly reduced from about 1 atm to (say) 0.01 atm, a second, more serious factor is introduced in addition to the first: the body fluids boil at pressures below about 25 mm Hg at 37°C. Facts which the anesthetist should know about gases are expounded and illustrated beautifully by Macintosh et al. 5 ; and aside from the ideal gas law, Henry's law, and recollections about thermal conductivity and resistance to flow through tubes — properties which are discussed briefly later — no further discussions on gases are presented in this book. The reader will have erred if he fails to consult Macintosh at this level of study. Some Important Mechanical Properties If a mechanical pressure (dynes cm -2 ) produces deformation, the pressure is called a stress. The amount of deformation, e.g., deformed length divided by the original, unstressed length, is called the strain. 34 SOME PHYSICAL FORCES EXEMPLIFIED IN MAN Elasticity is the property by virtue of which a body resists and recovers from deformation produced by a force. If the elongation, s, is produced by a weight of mass, m, in a sample with cross-sectional area, A, and length, /, the modulus (Young's) for stretching is given by stress mg /s mgl strain A/ I As which has dimensions of a pressure, m is high for materials difficult to stretch. The smallest value of the stress which produces a permanent alteration is called the elastic limit. Concussions, fractures, torn ligaments, and even bruises are examples of tissues having been forced beyond their elastic limit, usually during impact. Impact resistance, or hardness, can only be measured relatively. It usually is done by dropping a hard steel sphere, or pointed instrument, on the ma- terial, then reading either the diameter of the deformation caused by the sphere, or the depth of penetration of the pointed instrument. Bone, teeth, and nail have yielded useful values for impact resistance. Impulse is the product of pressure (stress) and time of application (con- sideration of the second law will show that impulse is also equal to momen- tum transferred per unit area). This is the physical description of the im- pact. Impulse measurements during impact applied directly to the brains of animals show that impulses composed of pressures of 30 to 90 psi acting for 1 millisecond (1 msec) or more cause physiological concussion (defined here as an immediate posttraumatic unconsciousness). Further, the impulse necessary to cause such damage increases rapidly with decreasing stress or pressure. There is a minimum time of application, of course, below which no damage is done. Analysis of stress-strain patterns in the human being has been going on for many years, especially studies on bones in relation to how bones are formed, grow, and are broken; and on lumbar intervertebral discs. Strain in a bone is most accurately measured by an electric wire strain gauge; the electrical resistance of the wire changes with stress. By transverse loading of a femur, for instance, with stresses of ~1 ton/in. 2 , strains of the order of only 0.0001 in. /in. are found. The bone is remarkably rigid. On the other hand, the discs are relatively easily strained, as they must be if they are to do their job during spinal maneuvers. Strains per disc are of the order of 0.02 in. On Hydro- (or Hemo-) Statics It was indicated on page 30 that the gravitational force of attraction of a body to the earth is given by m g, where m is the mass in grams, and g is the acceleration (cm/sec 2 ) or the force by which 1 gram mass is attracted to THE OSMOTIC FORCE 35 the earth at sea level (980 dynes/g). Our goal now will be to show what problem is introduced by the simple facts that man's head is 6 ft away from his feet and he walks upright. Two fluids circulate independently through the body: blood and lymph. Both move via a canal system. The former is a closed system driven by a pump; the latter is driven by muscle movement along the canals. Because a column of air 6 ft high, of 1 in. 2 cross-section, has negligible weight, there is no difference in the weight force of air at the head and feet. However, the weight of a column of water (or blood) of the same dimen- sions is 2.8 lb, quite an appreciable fraction (12 per cent) of 14.8 psi of at- mospheric pressure. In terms of the mercury manometer (1 atm, 14.8 psi, supports a column of mercury 30 in., or 760 mm high, remember?) this extra pressure at the feet due to the weight of the blood is 120 mm over 760. Hence the pump must force blood along against a 120-mm back-pressure. Add to this a small resistance to flow, mostly in the large arteries and veins in which the total area of flow is relatively small and the flow rate high. The heart is a pulse pump. It distends, collecting a volume of blood freshly oxygenated in the lungs, closes its inlet valves, and contracts, forcing the blood out through the aorta. The aorta, like the rest of the circulating system, has elastic walls, which, in turn, distend under the hydraulic force impressed by the contracting heart muscles. The pressure-rise in the aorta, for a rather typical stroke-volume of 30 cc, may vary from 30 to 150 mm Hg pressure depending upon the reaction of the walls of the arterial system to the pulse and the physical position of the person. In the highly elastic walls of the young and healthy the value will be small; as the tissues become harder with age, or disease, it will rise. The maximum value is called the systolic pressure, and is due directly to the factors outlined. It is usually of the order of 120 mm. The minimum value — reached after the walls of the aorta, distended by the stroke from the heart, have relaxed to the original diameter, having forced the blood along the artery-capillary system — is called the diastolic pressure. Typically in a healthy, adult male it is ~80 mm Hg. The mean value is about 100. The pulse period is about 1 sec. Because the veins in the legs are more easily distended than the arteries, most of the venous blood is stored there and re- called when needed. The center of gravity is thus lowered, and storage re- quires less work. THE OSMOTIC FORCE What Is It? One of the most important forces at work in the living system is the os- motic (literally, Greek: "impulse") one. It is the force which drives the dif- fusion of water, nothing more, and is a property of a solution just as are 36 SOME PHYSICAL FORCES EXEMPLIFIED IN MAN freezing point, vapor pressure, and boiling point. All of these properties have a value which depends only upon the number of solute particles present in the solution. Thus, pure water has no osmotic pressure; and the greater the concentration (c) of alcohol, for instance, dissolved in water, the greater the osmotic pressure. In fact the osmotic pressure, ir, varies directly as the concentration (number of moles, n, per volume, V): 7T = —RT = cRT V where R is the universal constant and Tthe absolute temperature. Note the analogy with the ideal gas law: PV = nRT Hence the former could be considered to be an ideal solution law. Naturally, the higher the concentration, c, of solute the faster will such a solution diffuse into pure water. However, conversely, the lower the solute concentration the higher is the water concentration, until in the limit, the solution is pure water. Since the laws of diffusion are just the same for water as for any solute, water will diffuse from the solution of higher water con- centration to that of lower water concentration; that is, it will diffuse from the solution of lower salt concentration to the solution of higher salt concen- tration, or, in other words, from the solution of low osmotic pressure to that with high osmotic pressure (see Figure 2-3(a)). It will diffuse from pure water into any solution. The diffusion of water is called osmosis. The direc- ts (b) high TT O solute ' low TT Olo , X>o °I°Q o oU ola solvent O 1° ° .net solven t flow O >L o ° r\ ^■membrane _ O 0:6! o hydro- static pressure highTT (ii) lowTI ^^stretched ibrane size erythrocyte membrane Figure 2-3. Water Balance, (a) High and low osmotic pressures; (b) osmotic pressure difference balanced by applied mechanical pressure; (i) hydrostatic, (ii) elastic, restor- ing pressures. THE OSMOTIC FORCE 37 tion of osmosis is determined by the osmotic pressure difference between the two solutions in contact, but otherwise there is no relationship between osmosis and osmotic pressure. The osmotic pressure can be measured by determining the mechanical pressure which must be applied to the solution of high osmotic pressure so that osmosis ceases. The mechanical pressure might be a hydrostatic one (Figure 2-3 (b) i), an elastic restoring force per unit area (Figure 2-3 (b) ii), or some other. Water Balance In the body (mostly water) the balance among tissues is maintained by a curious assortment of mechanical and osmotic forces, dictated in large part by the physical characteristics of membranes which separate the fluids. All living membranes pass water with ease. It is the solute content which deter- mines the osmotic pressure difference between the two solutions separated by the membrane, and this is determined in part by the membrane itself. Some membranes pass everything— water, salts, molecules — excluding col- loids and larger particles; the large intestine is an example. Membranes in the kidney pass water, salts, and many small molecules readily and rapidly. The membrane which forms the cell wall of the red blood cell passes water and salts, and some small molecules readily. Nerve cell membrane passes water and Cl~ readily, but balks at most molecules (its metabolic rate is low), and lets K + and Na + through only with difficulty. Since those species which can pass freely equalize their concentrations on opposite sides, only those which are restricted from passage can give rise to a difference in osmotic pressure. In the erythrocytes, water balance is thus controlled by the difference in soluble protein content between the cellular fluid and the plasma. Since the concentration is slightly greater inside than outside the cell, water runs in. As the cell walls become stretched, the re- storing pressure (the wall is elastic, like a balloon) applies a mechanical pressure on the liquid. An equilibrium is reached at which 7T, = 7T + P R where the 7r's are osmotic pressures inside and outside the cell, and P R is the restoring pressure of the walls of the distended cell. Table 2-1 gives a quantitative illustration of this important point. When membranes are ill-formed and cannot discriminate as they should, or when metabolic processes produce impenetrable species such as a protein whose concentration is different from the normal, the osmotic pressure dif- ference, 7r, — 7r , is not the same, and the powerful osmotic force differs from what it should be. The small mechanical compensation mechanisms (such as the restoring force in the erythrocyte wall) become strained, and edema 38 SOME PHYSICAL FORCES EXEMPLIFIED IN MAN may result. These facts are the physical basis of the salt-free diets and other chemical attempts to control water balance. TABLE 2-1. The Be jlance Between Osmotic Pre ssure Difference and Restoring Pressu re in Cell Walls. Ion content of blood plasma (meq/1) : Na + 138 ci- 105 K+ 4.5 HC0 3 - 25 Ca++ 5.2 protein 16 Total: 149.7 Mg ++ 2.0 po 4 - 3 2.2 /. 7r = 7.4 atm so 4 = 0.5 remainder 1.0 Ion content of i red blood cells (meq/1) : Na + 16 ci- 55 K + 96 HCO3- 15 Total: 117 Ca ++ 0.5 other ions 47 .:. w t = 5.7 atm Mg ++ 4.6 P R = 7T - IT, = 1.7 atm (25.5 psi) exerted by stretched walls of cell. If cell radius is 10m (10 " 3 cm), total force exerted by stretched cell wall is only 0.00005 lb. ELECTRICAL FORCES Electrostatic Force Like the gravitational and osmotic forces, we know little about the nature of electrical and magnetic forces either, but we can go a long way by study- ing and applying their effects. The basic concept of electrostatics is that of the potential, ^ (psi), at a point. The potential is defined as the work required (hence it is an energy) to bring one positive charge from an infinite distance and place it at the point or position in question. The unit of potential is, therefore, joules/ coulomb. Potential itself is impossible to measure, but differences in potential can be measured very accurately by the work they can do in the field or volume of space in which they exist — work of repulsion of pith balls, for example, or the work involved in deflecting the needle of a voltmeter or driving electric charges through some closed circuit. The potential difference, ^ 2 - ^ between two points is usually called " Fjou/cou, or volts." The term "charge" should be amplified. It is the quantity or amount of electricity in a bundle — whatever electricity is. We know there are, for- mally, two kinds of electrical charge; they are called positive and negative. ELECTRICAL FORCES 39 Positives repel; negatives repel; but positive attracts negative. Coulomb ob- served that the force of repulsion of like charges increases as the size of each, and decreases as the square of the distance. Thus F = ed 2 where Fis the force in dynes, <7, and q 2 are the charges in coulombs, d is the distance in centimeters, and e is the proportionality constant, called the di- electric constant (Figure 2-4). Unit charge is formally defined through Coulomb's facts: when two like charges are 1 cm apart and repel each other with a force of 1 dyne, each carries unit charge. DISTANCE x Figure 2-4. Interaction of Electrical Charges: (a) Coulomb's case; (b) field strength. Bioelectric Potentials At the microscopic level the most important potential differences in the living system arise from concentration differences (why they do will be seen later), and these occur almost without exception across living membranes. For example, in heart muscle cell the potential difference or voltage between the inside and outside of the cell, across the cell membrane, is about 85 mv, on the average, and cycles above and below this, as the heart beats. 40 SOME PHYSICAL FORCES EXEMPLIFIED IN MAN The electric field strength (see Figure 2-4) is denned as the voltage gradient, X), dV/dx, i.e., the voltage change per centimeter of effective thickness of membrane across which the force acts. In cells it has been variously esti- mated that the effective part of the membrane is only about 100 angstroms (100 A), 100 x 10 -8 cm, thick. The field strength across the membrane is therefore a phenomenal 85,000 v/cm, or over 200,000 v/in.! Electric field strength enters many phases of biophysics, and will appear often throughout this book, e.g., whenever membranes or bioelectric phe- nomena, such as those which give rise to the electrocardiogram and en- cephalogram, are introduced. The voltage gradient, TJ, (i.e., electric field strength) is the force which causes charge to flow — for positive charges, in the direction from higher to lower potential. The rate at which they flow (the current, i) is proportional to the forced. Thus l oc V Since the potential difference acts over the same path as the charges flow, the path length can be taken into the proportionality constant, and the result becomes i = AT amperes where K is the current if the impressed voltage is 1 v. This is Ohm's law. Transfer of charge is discussed further in Chapter 8. Colloids At the microscopic level the most important electrostatic forces are those which help to stabilize colloids. Colloids are suspensions of liquid or solid particles in a liquid medium (water, in our case). The particles are of the order of microns (1/i = 10 -4 cm) in diameter, and may be single macro- molecules, heavily hydrated, or collections or agglomerates of molecules. Characteristically, stable colloid particles (which do not agglutinate or precipitate) have excess like charge, and so repel each other. The repulsion promotes stability. The excess charge usually arises ultimately from the fact that the agglomerate contains acidic and basic chemical groups (e.g., — COO - , — NH 3 + , — P0 4 = ) whose extent of ionization at the tissue pH (~7) depends upon electrostatic interactions with other chemical groups nearby in the molecule. Since these interactions will differ from molecule to molecule, a chemical change in the colloid, an increased salt concentra- tion, or a shift in pH can weaken electrostatic repulsion and coagulate the colloid .... This is considered by some to be the mechanism by which anti- bodies work, and to be the reason why the blood groups are incompatible. ELECTRICAL FORCES 41 Iniermolecular Forces At the molecular level electrostatic interactions occur of such a profound nature that they are reflected all the way up to the physiology of the system. In this group we discuss not only charge-charge (ion-ion) forces, but also those arising from interactions involving dipoles, and even induced dipoles. With these concepts, along with that of electron dispersion in an atom-atom bond, we can then describe not only the "Coulombic forces" but also the so- called "London-van der Waals forces" operating between big molecules such as lipoproteins; and finally, with the concept of proton (H + ) exchange between neighboring groups (two oxygens, for example), we can describe the extremely important "hydrogen bond." For reasons which are reviewed in Chapter 4, in a molecule which is not symmetric, such as CO, one end accumulates more of the electronic charge than the other. In CO, the oxygen atom has the extra bit of negative charge, and the carbon is left slightly positive, by difference. The molecule has within it a permanent charge separation, and is called a permanent dipole. This and its weaker sister, the induced dipole, are shown in Figure 2-5. 8* OS- S' permanent dipole i p v i 0^ ^Osi.*NH 3 \ 8* -^B- induced dipole Figure 2-5. Electrostatic Charges in Molecules. Water is a permanent dipole, its hydrogen ends being positive to the nega- tive oxygen. The — CONH — linkage between amino acids in proteins is also a permanent dipole, as are the — COOH groups of organic acids, and many others. Although these are small charges, Coulomb's law applies to them, and fairly strong electrostatic forces can exist, firstly between permanent charges and permanent dipoles, and secondly between one permanent dipole and 42 SOME PHYSICAL FORCES EXEMPLIFIED IN MAN another. Water molecules attract each other, dipole to dipole, and give to bulk water a structure of oriented dipoles. Ions attract one end of the dipole and repel the other, and the result is an array of water dipoles oriented radially outwards from a central ion. The dipoles on large molecules can be hydrated by attraction to water molecules. Big molecules can be attracted to each other, or indeed have one part folded back and attracted to another part where two dipoles fall in close proximity, or where one dipole falls close to a charged group. Thus the dipolar character helps to determine not only composition but also structure. Still weaker forces exist between induced dipoles. Even if the molecule is symmetrical about an atom, a strong positive or negative charge can some- times induce the molecule's electrons to move a bit, so that the charge dis- tribution becomes distorted. Such induced charge separation is called an induced dipole. Interactions between the mutually induced dipoles of two molecules in close proximity are called the van der Waals forces. Further, it is postulated that the electron cloud of a molecule is in continuous motion, continually varying both the size and direction of its dipole. It induces a further dipole in its neighbor, and the new "dynamic" dipole interacts with the old static one in a manner which seems to confer an extra stability on the intermolecular "bond." The extra force of attraction is called the "dis- persion force," first postulated by London in 1930. Since one occurs when- ever the other does, today the mutually induced dipole and dispersion forces of attraction are referred to as the London-van der Waals forces. They are very weak by comparison with Coulombic forces, principally because the charges are not only small but deformable. However, in the absence of charged groups and when two molecules can come into close proximity (< 5 A) at a great many places over a fairly long distance (~15 carbon atoms in each molecular chain), considerable binding between the two has been shown to be accountable on the basis of London-van der Waals forces. Such is the case in lipoproteins in which a long hydrocarbon (and therefore with no polar groups and no permanent dipoles) chain becomes and remains intimately bonded to a polyamino acid or protein molecule. The strength and the sensitivity of this bond to interatomic spacings have been very evident in recent studies of lipoproteins in nerve cell membranes of the cen- tral nervous system. For example, one form of encephalitis is currently thought to be due to a change in binding which occurs as a result of inac- curate protein synthesis and poor binding to its lipid. Whereas Coulombic forces are fairly long-range forces (al/a' 2 ) the London-van der Waals forces are very short-range ( « \/d 7 ) but become im- portant when the particles approach very close to one another (see Table 2-2). ELECTRICAL FORCES 43 TABLE 2-2. Dependence of Force and Energy of Attraction Upon Distance Between Particles Name Interaction Force Energy Proportional to Coulombic London-van der Waals London-van der Waals in long-chain molecular associations ion-ion \/d 2 \/d ion-dipole \/d 5 \/d* dipole-dipole \/d 7 l/d 6 dipole-induced dipole or induced dipole-induced dipole }/d 7 \/d 6 (as above) \/d 6 \/d 5 The Hydrogen Bond In the covalent bond two atoms are said to be held together by "shared pairs" of electrons, and the postulate that the electron of a pair can spend part of its time around each atom is thought to confer extra stability on the bond. This is the process known as "exchange." In a similar manner the hydrogen ion of an — OH group, if it finds itself in the vicinity of a second, somewhat negative oxygen, halogen, or nitrogen group may, by thermal agi- tation jump the gap to this second group. Ideally it may continuously os- cillate between the two, and on the average assume a position half-way be- tween them. When this occurs, the strong positive charge is equidistant from two negative charges, is attracted to them both, and so forms a bridge — a weak bond. This is the currently fashionable "hydrogen bond" (Fig- ure 2-6). It is very versatile in the sense that, in tissues especially, which are 80 per cent water, it can be credited with much of the secondary structure ,a^ v ■0^ About 5 kcol needed to break I mole of hydrogen bonds I Figure 2-6. Hydrogen Bond — a Shared Proton. 44 SOME PHYSICAL FORCES EXEMPLIFIED IN MAN of big molecules — for instance, for the paracrystallinity of the regular molec- ular arrays so common in tissue, such as in muscle fiber and in the aqueous humor of the lens of the eye. Electromagnetic Force Although we live in the magnetic field of the earth, no information exists on the response of a man to large changes in magnetic-field strength. To small changes there is no response, as far as is known. Many molecular ef- fects are known, however, of which the recent exploitation of the so-called nuclear magnetic resonance phenomena, in which the location of a hydrogen atom in a molecule and the arrangement of atoms in molecular complexes can be learned, are exciting examples. However, on biological systems the effects of magnetic fields are yet poorly understood. Small animals placed in fairly strong magnetic fields of ~4000 gauss (at $2/gauss, ' 1 lb> of electromagnet/ gauss) show inability to repro- duce. Cell division and growth are inhibited. Interference with the collec- tion of the mitotic apparatus in preparation for cell division is implicated. In this respect the effect of a magnetic field is similar to the effects of X or gamma rays. The effects of electromagnetic forces — oscillating forces of unknown na- ture, which interact with both electric charges and magnetic poles, and with other electromagnetic forces — are better understood and are most important in the living system. In fact, the more the question is studied, the more it is realized in how many aspects of inanimate as well as animate subjects, electromagnetic forces play an important part. Usually electromagnetic phenomena are described by their interaction energy, rather than force; this expedient enables us to by-pass their nature, and concentrate upon their effects. An "oscillating potential" permeates electromagnetic energy. It is a periodic function of time (see Figure 1-2). The amount of energy in a packet depends only upon its number of cycles per second. Because of their importance, Chapter 4 is devoted almost completely to electromagnetic matters. Yet will all this preoccupation with force, the physicist still is unable to cope with some really big ones, such as political "forces," and economic "pressures." In "The Razor's Edge" (1944), W. Somerset Maugham con- cludes: "Goodness is the greatest 'force' in the world!". . . . Unfortunately, we cannot measure it. GENERALIZED FORCE Although temperature is not usually thought of as a force, it is the driving force for heat-energy flow. Discussion on driving forces for several processes which occur in the living system is contained in Chapter 7. PROBLEMS 45 All forces are, quite literally, ''factors of energy." Thus, a generalized driving force times a quantity yields energy. Some examples are: Mechanical force x distance = mechanical energy or work Gas pressure x volume of gas = mechanical energy or work Osmotic pressure x molar volume = osmotic energy or work Electrical potential x charge = electrical energy or work Temperature x entropy = heat energy or work Chemical potential x concentration = chemical energy or work The inherent difficulties of considering both temperature in "degrees" (fractions of a length of a liquid metal along a tube!) and chemical poten- tial (actually an energy per unit concentration) as "forces," are expounded further in Chapter 7. What happens to a biological system when the force responsible for the acceleration due to gravity (g) is removed — that is, becomes weightless — is critically important to future space travel. The meager information on the few human beings who have so far orbited the earth is reviewed in Chapter 8. PROBLEMS 2-1 : A 200-lb football player is running full speed at a rate of 100 yd in 12 sec. Cal- culate his kinetic energy in ergs; in joules; in calories; in Calories or kilocalories. If he were stopped completely in 1 sec, what power would he deliver during that 1 sec (in watts; in horsepower; in Cal/hr)? Compare this with the basal metabolic rate of 0.1 hp, or 60 Cal/hr (1 lb = 454 g; 1 cal = 4.18 jou; 1 hp = 746 w; 1 jou/sec = 1 w). 2-2: Values of the solubility of nitrogen and oxygen in water are 0.001 50 and 0.00332 g of gas at 1 atm/100 g water, respectively. Approximately how many cubic centimeters of each gas are contained dissolved in the body fluids (200 lb, 80 per cent water) under 1 atm of air (20 per cent oxygen, 80 per cent nitrogen)? Neglect the fact that the solubility of gases is less in salt solutions than in pure water. An anethetist may use a mixture up to 90 per cent oxygen, but he always re- tains about 5 per cent C0 2 in the inhaled gas. Why? 2-3: Assuming the total area of the adult human body to be 1 sq yd, calculate the total force due to the atmosphere (pressure 14.7 lb/in. 2 ) on the body. In dynes; in tons force. Calculate the total force on a skin diver at a depth of 450 ft. Why is he not crushed? What precautions must he take while coming up to the surface? Why? 2-4: Make two tables showing forces of repulsion — in dynes, of two like unit charges, each with 3 x 10" 10 electrostatic units of charge — at distances 0.1, 1, 2, 5, and 25 A apart; one table for a medium of air or a vacuum (dielectric con- stant = 1), and the other for an aqueous solution (dielectric constant = 72). Plot the numbers, force vs distance, for each case. 46 SOME PHYSICAL FORCES EXEMPLIFIED IN MAN REFERENCES 1. Harrington, E. L., "General College Physics, " D. Van Nostrand Co., Inc., New York, N. Y., 1952. 2. Randall, J. T., "Elements of Biophysics," the Year Book Publ., Inc., Chicago, 111., 1958. 3. Glasser, O., Ed., "Medical Physics," Vol. Ill, Year Book Publ., Inc., Chicago, 111., 1960; papers by Carter, Featherstone, Lipson, et al. 4. Moore, W.J., "Physical Chemistry," Prentice-Hall, Inc., New York, N. Y., 1950. 5. Macintosh, Sir R., Mushin, W. W., and Epstein, H. G., "Physics for the Anesthetist," 2nd ed., Chas. C Thomas Publ. Co., Springfield, 111., 1960. 6. Robbins, S. L., "Textbook of Pathology With Clinical Applications," W. B. Saunders Co., Philadelphia, Pa., 1957. 7. Wolf, A. V., "Body Water, " Sci. Amer., 199, 125 (1958). CHAPTER 3 Matter Waves: Sound and Ultrasound (On Music and Noise "from CtoC," On Speech and Some Therapy) According to Sir Richard Paget, human speech began by the performance of sequences of simple pantomimic gestures of the tongue, lips, etc. . . . Consider the word "hither. " The tongue makes the same beckoning gesture, while [one is] speaking this word, as is made with the hand. (H. Fletcher. 3 ) INTRODUCTION Our senses of touch and hearing reveal an environment which contains a bewildering array of matter waves: the breeze; falling raindrops; noise, speech, and music; earth tremors, shock, or blast waves; the vibrations en- countered when riding a horse, or when operating a jack-hammer. Bees and some other insects, and bats too, send and receive, and are guided in flight by very high-frequency matter waves. Thus waves in matter have a great spectrum of manifestations, uses, and effects. It is the purpose of this chapter to illustrate them, for matter waves and electromagnetic radiations together comprise the most important method of man's continuous exchange of force and energy with his environ- ment. The latter are introduced in Chapter 4. They are fundamentally very different from matter waves, although often confused with them. In 47 48 MATTER WAVES: SOUND AND ULTRASOUND matter waves the medium itself — solid, liquid, or gas — moves back and forth. PROPERTIES OF MATTER WAVES Definition Matter waves are of two types, which differ only in the direction of the vibration relative to the direction of propagation. In transverse waves the vibration is perpendicular to the direction of propagation (a plucked violin string, for example). In longitudinal waves the vibration is parallel to the direction of propagation (the pressure waves from a blast, or in front of a piston, for example). Most of the matter waves which are of interest here are, like water waves, a combination of both. The two basic properties are the pressure (force/unit area) of the wave and its rate of change with time. The former is usually called the ampli- tude, \p (dynes/cm 2 ). The latter is usually expressed as the number of times the value of \p cycles back and forth per second, i.e., as the frequency (cycles/sec). All matter waves, no matter what the shape, can be expressed as a super- position of simple, sinusoidal waves, of the type discussed in Chapter 1. There are traveling waves and standing waves (Figure 3-1 (a) and (b)). A biost, shock, water waves auditory region (sound) ultrasonic region I I I '(b) I 20 21,000 .000,000 CYCLES PER SECOND Figure 3-1. (a) Traveling Wave Such as Sound in Air; Standing Wave Such as On a Vibrating Violin String; (b) Range of Matter Waves. PROPERTIES OF MATTER WAVES 49 sound wave moving through air travels from its source and imparts an energy to the receiver. This energy is primarily in the direction of propaga- tion, but with scattering some of it becomes transverse. By contrast, the standing wave can impart no longitudinal energy — it has none. But it can impart transverse energy to the medium. The generation of the sound by the vibrating violin string is an example. The intensity, /, of the matter wave is the power delivered by it per unit area. In. other words, / is the rate at which the wave expends energy. All traveling waves move at a certain velocity, v (cm/sec). Hence the product of amplitude (a pressure) times distance is the energy expended per unit area: w = \p d (dynes/cm 2 x cm = ergs/cm 2 ) The product of amplitude and velocity is the power expended per unit area: I = \p v (dynes/cm 2 x cm/ sec = ergs/cm 2 sec) The intensity or power expended per unit area by the traveling wave, is highest for those media having molecules with the greatest number of de- grees of freedom in which energy can be stored — gases for example. Both the range and speed of sound are highest in solids, somewhat less in liquids, far less in gases. However, for any medium of constant density, p, the ve- locity has a fixed value. This fact results in another useful relationship, that between amplitude (pressure) and intensity (power): / = Vlvp which says simply that power delivered per unit area to any medium is pro- portional to the pressure squared, if velocity and density are held constant.* This (/ cc \^ 2 ) is a very useful rule-of-thumb, applicable, it turns out, to all field phenomena. Useful also is the fact that, although low-frequency waves are easily re- flected and diffracted by air and hence are nondirectional (or will go around corners), high-frequency waves are only slightly scattered by air. Therefore, the latter can be beamed in a preferred direction from a source, and even focused on a particular spot by proper (saucer-like) design of the vibrating source. *Dimensions: 3 sec cm = ergs/cm sec (Work it through.) 50 MATTER WAVES: SOUND AND ULTRASOUND Illustrations Frequency Matter waves have a broad range of frequency, from zero up to the current practical upper limit of about 1,000,000 cycles per sec (cps) in use in some ultrasonic-therapy and submarine-detection studies (Figure 3-1 (c)). The human ear is most sensitive from ^50 to ^10,000 cps; the range of man's ear, however, may be from 20 to 21,000 cps. This, then, is the auditory or sound range. Speech requires 60 to 500 cps. The piano ranges from 27.2 to 4138.4 cps. The great basso profondo, Italo Tajo, could reach a minimum of ~60 cps; the diminutive coloratura soprano, Lili Pons, could hit 1300 cps on a good day. Of course, these are the basic frequencies, and it is understood that a basic frequency generated by any physical vibrator will contain over- tones, or harmonics, which are multiples (2x, 4x, even 8x) of the basic frequency. The quality of the tone is determined by the sum of all the com- ponents: the basic frequency plus its harmonics. Training and youth combine to produce a receiver which can hear low- power sound up to 12,000 cps. Some musicians can detect overtones from their instruments up to 14,000 cps, but these are few. Most of us can detect frequencies up to 18,000 from a signal generator, if the signal is intense enough, and the odd person can detect up to 21,000 cps. Dogs do it with ease. Porpoises have a phenomenal sonic system in their heads which can sweep frequencies repetitively from a few cycles to many thousands of cycles — both send and receive. Below and overlapping the auditory range for man is the range (0 to 50 cps) of blast and shock waves, earth tremors, water waves, and the like. The masseur will use vibrations 1 to 50 cps; a ship will roll at 0.1 cps. An air hammer operates at ~ 15 cps, and we hear the overtones. Above the range of sound, from 20,000 up to > 1,000,000, lies the im- portant range of ultrasound, and the science and technology known as ultrasonics. Velocity The speed of matter waves depends sharply upon the medium, and in the case of a gas, its temperature and pressure. For instance, in air at 0°C and 1 atm pressure the speed is 331 meters/sec (mps) (730 miles/hr). In water and soft tissue it is 4 1 2 times higher than in air, and in solids it goes up to 5000 mps. The velocity of sound through fat is 1440, through muscle 1570, and through bone 3360 mps. Velocity is independent of frequency; and it is probably just as well, other- wise the low tones of the organ might reach our ears later than the high tones of the same chord! PROPERTIES OF MATTER WAVES 51 Amplitude and Intensity There is a minimum pressure and power of matter waves below which the ear cannot detect the wave. This value is about 0.0002 dynes/cm 2 , an ex- tremely small value because the ear is very sensitive. The corresponding power or intensity limit is ~10 9 ergs/cm 2 sec, i.e., ^lO" 16 w/cm 2 ! This value places its sensitivity very close to the threshold of the power in heat motion, and thus very close to the minimum background agitation of matter in our environment. The maximum amplitude the eardrum can stand, with- out certain irreparable damage resulting, is ~200 dynes/cm 2 . Therefore, the range of sensitivity of the ear is phenomenally high, one to a million. It is the most sensitive at 1,000 cps. The sense of touch, particularly on the fingers and tongue, is not nearly so sensitive, but responds down to much lower frequencies. To our knowledge, man has no detection apparatus for frequencies above about 20,000 cps. However, there is some evidence that ultrasound can penetrate to the brain and cause psychological aberrations, which may or may not be a result of organic damage. One of the most convenient ways of generating matter waves of controlled frequency is by means of the vibrating crystal. Certain crystals are piezo- electric — that is, they expand or contract if an electric voltage is applied to contacts with two different crystal faces (Figure 3-2). The amount of the (o) £l_ + V volts crystol (b) applied voltage, V (c) time radiating, vibrati ng surfoce -target beamed ultrasound crystals Figure 3-2. About Piezoelectric Crystals: (a) Voltage difference is applied between two opposite faces, (b) The length changes as the applied voltage is changed, (c) Varying volt- age, V, gives varying length, y. (d) Concave radiator concentrates matter waves on a target. 52 MATTER WAVES: SOUND AND ULTRASOUND expansion or contraction increases with increasing applied voltage. Quartz and barium titanate are currently in wide use. If the applied voltage is varied, the crystal shape varies accordingly, or vibrates, and the matter wave so established is transmitted by contact with the medium. The ampli- tude of the vibration is higher the higher the vibrating voltage applied. The frequency of vibration follows that of the electrical signal, if the crystal is not too big. Figure 3-2 illustrates these points. Apparatus with output which ranges from a few to a million cycles per second, and from next to nothing up to a few hundred watts per square centimeter of crystal, has been built and used. Constructed with a concave radiating surface (Figure 3-2 (d)), an array of piezoelectric crystals, if properly oriented, can be made to focus an intense beam of matter waves at a point a few centimeters from the radiating sur- face. For example, in recent therapeutic work beams of 1 Mc (1,000,000 cps) were focused on a small target, and delivered energy at a rate (inten- sity) of 8 kw/cm 2 of cross-section of the target ! Absorption If waves are diverging, or being dissipated or scattered, the important gen- eral rule, called the "inverse square law," is obeyed. It says simply that the intensity, /, decreases as the distance from the source gets larger, in such a manner that if, for example, the distance between source and receiver is doubled, the intensity at the receiver falls to only one quarter. Quantita- tively, I(x) oc \/x 2 where I(x) is the intensity at any distance, x, away from the source. See Figure 3-3. If a parallel beam of matter waves is absorbed by the medium, the rate of absorption at a point is proportional to the intensity at that point; or dl/dx = -kl which integrates (see Chapter 1) to / = I e-* if / is the value of / where x = 0. For the case in which the waves are diverging and also being absorbed, a linear combination of the inverse square law and the absorption law applies. The energy absorbed from the matter-wave beam by the medium contri- butes to the thermal motion of the molecules of the medium. The absorp- tion coefficient, k, is intimately related to several physical properties of the medium. PROPERTIES OF MATTER WAVES 53 Figure 3-3. Inverse Square Law. Radiation from source S diverges. Intensity (w/cm 2 ) at distance, d, is four times the intensity at 2d because the same radiation is spread through four times the area by the time it reaches 2d. However, there are two principal mechanisms of absorption of matter waves by tissue: (a) Fnctional resistance: The momentum of the propagation, which is directional (Fig. 3-1 (a)), is passed to the molecules of the tissue, which be- come momentarily polarized by the pulse of pressure. The directed energy thus received quickly decays into random, nondirectional molecular motion. This mechanism can be called "molecular absorption." It is important at medium and high frequencies. (b) Elastic reactance of the bulk tissue: Absorption occurs by movement of the bulk material; mass is displaced, and macro-oscillations result in sym- pathy with the impinging, oscillating pressure. Because the tissue is not perfectly elastic (i.e., the molecules will realign themselves so that they won't be polarized), the absorbed energy quickly dissipates in front of the pressure pulse as molecular motion or heat. This is the only method by which energy is absorbed at low frequencies — during earth tremors, train rumble, or massage, for example. This mechanism can be called "elastic absorption." Reflection, due to the inertia of the tissue (its tendency to remain at rest unless forced to do otherwise — Newton's first law of motion), occurs at 54 MATTER WAVES: SOUND AND ULTRASOUND high frequencies for soft tissue and even at low frequencies for dense tissue such as bone. Truly elastic tissues simply reflect incident matter waves. The absorption coefficient for molecular absorption (k) is well known for air and water: 3vp -> c _p °v jc ^ P ^ V where /is the frequency (cps) of the impinging wave, v the velocity (cm/sec), p the density (g/cm 3 ), rj the viscosity (dyne sec/cm 2 ), A" 7 the heat conductiv- ity (cal/sec deg cm), and the c's are the specific heats (cal/deg g) at constant pressure, P, and constant volume, V. Hence the energy absorbed per centi- meter of penetration of the impinging wave increases linearly with the vis- cosity or "stickiness" of the medium and with its thermal conductivity; in- creases very rapidly with increasing frequency; but decreases with increas- ing density. For water, which is a sufficiently good approximation to soft tissue for present purposes, k/f 2 = 8.5 x 10" 17 sec 2 /cm. For air the value is 1000 times higher, because although rj is 50 times smaller for air than for water, v is 4^2 times smaller and p is 1000 times smaller. For liquids only the first term (the frictional or viscous one) is important; for gases both are im- portant. Therefore it is useful to aerate a tissue before sonic therapy is ap- plied, because absorption is higher. Since reflection increases with increasing frequency, the method of appli- cation is important. In the absence of reflection, the above expressions describe the situation well. Direct application of the vibrator to the tissue assures this. However, if the sound is beamed through air, the situation is quite different: reflection occurs. Quantitative studies on tissues are only recent. The general rule which has emerged is as follows: Beamed through air, sound of high frequency suf- fers little absorption, and little damage results. The depth of penetration increases with increasing frequency. Most (>95 per cent) of the incident energy passes right through, or is reflected. Some of Von Gierke's figures (1950) are: 5 to 6 per cent absorbed at 100 cps; 0.2 to 4 per cent absorbed at 1000 cps; and <0.4 per cent absorbed at 10 kc. Beamed through liquid or solid, ultrasonic radiation is easily controlled and its absorption pre- dicted. More will be said about this later, in the section on therapy. SENSITIVITY OF A DETECTOR, AND THE WEBER-FECHNER LAW It is a fact that whether or not a receiver will detect a signal depends upon how much the signal differs from the background noise. The dependence is SENSITIVITY OF A DETECTOR, AND THE WEBER-FECHNER LAW 55 not a simple proportionality, but rather a logarithmic one. Thus, the sensa- tion, or loudness, L, is given by L oc log///° where 1° is background intensity, and / is the intensity, over background, of the signal to be detected. This is the basic form of the Weber-Fechner law. It has many manifestations. For instance, if there are two signals equally strong, with different backgrounds, the resolution of (difference in loudness), L 2 - L, , is related to the ratio of the intensities of the two backgrounds, 1° and I 2 °, as follows: L 2 — L { oc log I°/I° This is a law which has rather wide application, not only in the psycho- logical sensations but in detection of electromagnetic waves of many fre- quency ranges, from the radio to the infrared. Therefore its implications should be very thoroughly contemplated. Because of this logarithmic law, it is convenient to express power ratios by a logarithmic unit, so that sensation becomes approximately linearly pro- portional to this unit. The unit is called the ^bel," (b) and is equal to the logarithm of the ratio of two sound intensities if they are in a ratio of 10 : 1. The number of bels then is given by b = log 1/1° For sound, the value 1° is arbitrarily chosen to be the lowest one which a human ear can detect (10 -16 w/cm 2 ; or, in pressure units, 0.0002 dynes/cm 2 , since the same conversion factor applies to numerator and denominator). The bel unit is too large for convenience, and the decibel, one tenth of a bel, has received wider use. Therefore, the number of decibels is: db = io log i/r Another form of the Weber-Fechner law, then, is L « db It holds true for all sensory receptors. Some minimum discernible relative changes,** (/, - 7°)//° (where I, is threshold intensity), which man can detect are: Brightness of light: 1 per cent Lengths of lines: 2 per cent Feeling of weight: 10 per cent Loudness of sound: 30 per cent ** Remember relative error, defined in Chapter 1 ? 56 MATTER WAVES: SOUND AND ULTRASOUND Sensitivity, S, of a detector, or discernment per decibel of signal over back- ground, is defined as s = log r/M t where A I, = I, - 1° . Sensitivity is higher the smaller is the value of A/,. Usually when 6" is determined at different values of an independent variable, the result is expressed as the sensitivity relative to the maximum value taken as unity (S/S max ). The sensitivity of the ear is so expressed in Figure 3-4. moximum sensitivity 0.01 10 100 1,000 FREQUENCY (cycles per second; 10,000 100,000 Figure 3-4. Sensitivity of Human Ear at Different Frequencies of Sound Waves. The indi vidual's sensitivity curve may differ markedly from this average curve. THE BODY'S DETECTORS OF MATTER WAVES Introduction In this section are given an outline of the structure of the ear and a de- scription of the mechanism of the sense of touch. This sketch is meant to show the important general features, but does not penetrate into either the depths of the mechanism nor the psychology of the resulting sensations such as loudness and pitch. A very well written and concise display of the bio- physics of hearing is found in the book by Stacy et a/. 6 An up-to-date survey of the physiology of hearing is given by Whitfield, 7 and a masterful discus- sion of biological transducers (converters of mechanical to electrical stimuli) was recently given by Gray. 8 To delve deeply into this aspect of the subject is, unfortunately, beyond our scope, although it is currently a very active part of biophysical research. THE BODY'S DETECTORS OF MATTER WAVES 57 Notes on the Ear The structure of the ear can be pictured, in simplest terms, as consisting of three main parts: the pinna (lobe) and external canal, the middle ear, and the cochlea. The canal and the middle ear are separated by the tym- panic membrane (ear drum) which covers and protects the latter. The middle-ear cavity contains a system of three bony levers, the ossicles (the malleus, incus, and stapes) whose main job seems to be to act as a matching device transmitting matter vibrations between the two fluids: the air outside in the external canal, and the perilymph inside the cochlea. The cochlea is a spiral canal within the bone of the skull. It is divided axially into three channels by membranous partitions. Into one of these, the scala vestibuli, is inserted the end of the stapes; this chamber, then, receives directly the transmitted vibrations. Through the membranes, vibrations are passed laterally into the other two canals, the scala media and the scala tympani. These two are separated by the basilar membrane, which receives the end- ings of the auditory nerve, and the cells of which are the transducers that convert the mechanical energy of vibration into the electrical energy trans- mitted along the nerve. Most recent work has been aimed at the mechanism of action of the region of the basilar membrane, the transducer. Some of the cells on the membrane have hair-like processes projecting from their upper ends and attached to the overhanging, tectorial membrane. Relative move- ment between the tectorial and basilar membranes distorts the cells of both. Note Figure 3-5. The analogy with piezoelectric crystals is usefully drawn at this point: distortion of the shape of the transducer in both cases leads to change in the potential difference between two points on the surface of the transducer — in one case the surface potential of the crystal, in the other case the membrane potential of the cell. An accumulation of evidence now exists — Von Bekesy 13 received the 1961 Nobel Prize in Physiology and Medicine for this work, done at Harvard — that a traveling wave passes along the basilar membrane during excitation. The position at which the wave achieves its highest amplitude (think of the whip) is dependent upon the frequency of the wave being detected. Therefore, nerve signals from different tones arise at different spots, each spot associated with specific nerve endings. At low frequencies the whole basilar membrane vibrates in sympathy with the incoming matter wave. The question of membrane potential change will be considered in Chap- ters 7 and 10, in reference to erythrocytes and nerve cells, upon which voltages have been directly measured in vivo. Deformations in the structure, or failure of the ear to respond to matter waves, is the subject matter of the otologist. Corrections are applied some- 58 MATTER WAVES: SOUND AND ULTRASOUND times simply by amplification of the signal reaching the tympanic mem- brane, sometimes, although less commonly, directly to the cochlea by stimu- lation of the bone structure which surrounds it. Surgery is often necessary to free the "frozen" lever system. Reissner s membrane bone auditory ner ve tec torial membrane transducer cells basilar membrane COCHLEA non-elastic oining f i ber s\ auditoi y nerve end ings Figure 3-5. Schematic Drawing of Cross-section of the Cochlea, the Inner Ear. The three scalae are separated by deformable membranes. The transducers are fastened to the tectorial membrane by fibers. Relative motion between the tectorial membrane and the basilar membrane causes stretching of the transducer cells, resulting in change in membrane permeability, and therefore ionic composition and membrane potential. This change activates the nerve endings attached to the cells, and the impulse is carried down the auditory nerve to the brain. The Sense of Touch And Other Mechanoreceptors A magnificent array of mechanoreceptors (as well as photo-, chemo-, and thermal receptors) is displayed by the human body. These bring in informa- tion from the environment, and then provide a feedback of information con- cerning an action taken. The most sensitive transducers, other than those in the ear, are found on the tip of the tongue and on the tips of the fingers, although mechanoreceptors are located all over the body, so closely spaced that no pressure change on the surface, above some threshold value, goes undetected. They all have three parts in common: (1) a mechanism for transmitting a pressure change to the receptor cell; (2) the deformable receptor cell, the deformation of which (apparently) changes its cell membrane potential at a point intimately associated with (3) a specialized ending of a nerve cell's SPEECH 59 axon. Speculations are rampant on the mechanism of this transposition. Transduction through changing electrical potentials across the receptor cell wall is currently a very popular generalization; but reliable details of mech- anism, unfortunately, are too few. SPEECH Three resonators, or vibrating cavities, are responsible for the organized noise which we call speech. They are (1) the vocal chords, which close the exit used by air exhaled from the lungs; (2) the throat and the mouth; and (3) the nasal cavity. The vocal chords, the tongue, and the lips control the changes in vibration which are induced in the exhaling air stream and which are the sounds of speech. The combination of these three moving parts, each of which can take several different shapes, gives remarkable versatility in the production of sound. The fundamental sounds of speech are divided into six classes: pure vowels, diphthongs, transitionals, semivowels, fricative consonants, and stop consonants. The subject of phonetics is well known, is heavily illustrated in any good dictionary, and needs no review here. Amplitude and intensity are controlled mainly by the rate of expulsion of air, although secondary resonators such as the head and the chest play a small role. Speech sounds have been analyzed on many people by the Bell Telephone Laboratories, for obvious reasons. Some of the results are contained in the classic book by Fletcher. 3 For instance "oo" as in "pool" spoken by men (by women) has a mean fundamental frequency of 140 cps (270 cps), a mean low frequency of 411 (581 for women), scattered high frequency of 3700 (4412 for women). All speech sounds have been carefully recorded and ana- lyzed, and the sounds of the "average man" used for microphone design. The fundamental speech sounds have a power. When one talks as loudly as possible without shouting, the average speech power is about 1000 micro- watts (1 nw = 0.000001 w) at the source. When one talks in as weak a voice as possible, without whispering, it drops to 0.1 fiw. A very soft whisper has a power of about 0.001 ^w. Very loud speech is ~20 db over average speech power; a soft whisper is ~40 db under average. NOISE High-intensity noise has become one of the most disturbing problems of the modern way of life. Noise is usually defined as any unwanted sound, and hence the classification is highly subjective. High-intensity noise is usually defined as any unwanted sound greater than 85 db (see Table 3-1). 60 MATTER WAVES: SOUND AND ULTRASOUND Noise has many components — matter waves of many frequencies. The "buzz" from speech in a crowded room will center in the range 300 to 6000 cps. The noise generated by a wood planer has most of its energy between 200 and 2000 cps, while a power saw will emit noise from 50 to 6000 cps. Only low-pitched or high-pitched voices can be clearly understood. This is the crux of the problem facing communication engineers and otologists alike: to provide a sufficient sound intensity level (over background noise) to the middle ear. This question is considered in more general terms in Chapter 11. TABLE 3-1. Some Sources of Noise* Location Power (w/cm 2 ) Sound Power Level** (db) 50-hp siren 10" 2 140 (100 ft away) Submarine engine room io- 5 110 (full speed) Factories IO" 4 to IO" 8 76 to 128 Woodworking plants 10~ 4 to IO" 8 80 to 114 Subway car IO" 7 to IO" 8 80 to 90 Loud radio (2 ft away) Speech at 2 ft Speech at 1 2 ft Private office IO" 8 [ I0" 12 tol0- 8 | IO"' 2 80 60 normal, 77 shouting 43 normal, 61 shouting 40 Average home io- 13 30 Library 10 -h 20 "Silence" IO" 16 * After Neeley, K. K., "Noise — Some Implications for Aviation," Caw. Aeronaut. J., 3,312 (1957). ** Referred to 10 -16 w/cm 2 , the threshold of hearing. Exposure of man to high-intensity noise has several effects: change in hearing acuity, and mechanical or pathological damage to the cochlea; tem- porary blindness (>140 db); changes in ability to perform skilled and un- skilled tasks; feelings of fear, annoyance, dissatisfaction, and nausea. Dis- cussion of some of these effects follows in the next section. PHYSIOLOGICAL EFFECTS OF INTENSE MATTER WAVES The physicochemical basis of the physiological damage is fairly well understood. Five facts are important to the discussion: (1) During the absorption of matter waves, a front of high pressure pre- cedes a front of reduced pressure through the tissue. There is therefore a differential pressure, or a pressure gradient, along the tissue which stretches and compresses it in sympathy with the incoming wave. If the amplitude is PHYSIOLOGICAL EFFECTS OF INTENSE MATTER WAVES 61 such that the elastic limit is exceeded, tearing can result. Thus 160 db will rupture the eardrum itself, probably the toughest part of the soft tissue of the whole organ! (2) At high frequencies, the compression occurs so fast that energy is passed from the matter wave to the recipient molecules so rapidly that it has no time to disperse through molecular vibrations. The molecule be- comes phenomenally "hot" or energetic, and may fly apart. Thus chemical bonds are broken (Figure 3-6 (a)). Water is decomposed to H 2 and H 2 2 . gas or steam irradiator metal pan liquid making contact with brain through hole in skull. (a) (b) Figure 3-6. (a) Cavitation and Production of Broken Water Molecules by Ultra- sound. The OH fragment is a rapidly effective oxidizing agent, (b) Irradiation of a Small Locale in the Brain. (Success with Parkinson's disease reported.) (3) During rarefaction (low-pressure part of the wave), any dissolved gas in the tissue may coalesce into bubbles; and in fact bubbles containing only water vapor may form, breaking molecular bonds as they form, and breaking more bonds as they collapse and release their high surface energy. This is called cavitation. It occurs in water at power levels as low as 140 db. This critical power level decreases with increasing frequency. (4) With the breaking of bonds, free radicals are produced, which, for reasons to be discussed in Chapter 4, cause a (net) oxidation reaction to occur in most aqueous solutions. Three watts of power introduced at 500,000 cps, for example, will cause oxidation. (5) Because of general absorption of energy within the volume irradiated with matter waves, a general temperature rise occurs. This upsets the metabolism of the tissue in a manner discussed later in Chapter 8. Irradia- tion by 1 megacycle (Mc) at a power of 50 w/cm 2 , for example, raises the temperature of water from 20 to 50° C in a few minutes. Some specific observations of effects of sound waves on man are given in Table 3-2. For obvious reasons, experiments using high-power sound are carefully and selectively done on man. However, an accumulation of experience is 62 MATTER WAVES: SOUND AND ULTRASOUND being gained on animals, principally guinea pigs, rats, and mice. The in- vestigations have not been extensive enough to denote anything other than generalities. However, at 165 db, 500 to 400,000 cps, on guinea pigs, pathological changes occur in both the inner and middle ear; lesions appear in the organ of Corti, and it is ruptured from the basilar membrane. Hemor- rhages start where the malleus meets with the eardrum. Convulsions often result. The skin becomes blistered and reddened. Death is hastened by the damage. TABLE 3-2. Effects of High-Intensity Sound on Man* Frequency (cps) Level (db) Effect stimulation of receptors in skin mild warming of body surfaces nausea, vomiting, dizziness; interference with touch and muscle sense significant changes in pulse rate pain in middle ear changes in muscle tone; increase in tendon reflexes; incoordination minor permanent damage if prolonged major permanent damage in short time vibration of muscles in arms and legs resonance in mouth, nasal cavities, and sinuses ♦Collected by Neeley, K. K., "Noise — Some Implications for Aviation," Can. Aeronaut. J., 3, 312 (1957). SONIC AND ULTRASONIC THERAPY Certain uses have already been demonstrated; others await discovery, for the technique is very new to medicine. The following applications are al- ready well known in principle, and are now being introduced in practice very cautiously — for the early 1950's saw the period of novelty wax strong, and then wane into a hard reappraisal in the mid-50's; and one now observes the gradual emergence of the place of vibrations in the medical arsenal. Details can be found in the reviews of two masters of the subject, R. F. Herrick 10 and W.J. Fry 1 and in the book edited by E. Kelly. 2 Present Applications (1) Subcutaneous lesions can be located by ultra high-frequency matter waves. They focus well at 1 Mc, and penetrate to a useful depth. The depth of penetration is a function of the power of the source. Since reflection of matter waves is greater the higher the density of the medium, tumors can be distinguished from normal tissue at a location deep below the surface. 100 110 2000 to 2500 >150 Jet engine 130 to 155 100 to 10,000 105 140 130 to 140 ~160 ~190 50 ~120 700 to 1500 130 SONIC AND ULTRASONIC THERAPY 63 (2) Based on the same principle, the rate of blood flow through the ar- terial system can now be measured by reflected ultrasound, in a nondestruc- tive experiment in which all instrumentation is external to the body. (3) Dentists have begun to apply sound to the ears of patients during drilling, because it has been found that the brain cannot perceive pain from the teeth and sound from the ear at the same time. The sound in this case acts as a local anesthetic. (4) "Rapid massage" heat therapy is now quite common, with an assort- ment of low-frequency vibrator pads and belts available, and experimental models operating in the 12,000 to 50,000 cps region. For deep "massage" higher frequency ultrasound is used; it has the added advantage of comfort from noise. (5) Certain skin diseases can be treated with beamed and focused ultra- sound. Thus viruses are destroyed (literally shaken into little bits!) by ultrasound, and a future in sterilization seems assured. In this application its competitor is soft X rays. (6) "Neurosonic surgery" is now well advanced on animals, and has re- ceived some experimental evaluation on humans. The most spectacular suc- cess so far has been achieved in treatment of Parkinson's disease, the shaking palsy. Because of its future importance,*** some details will now be given. "Neurosonic Surgery" The ultrasonic radiation reaches the brain through a hole cut in the skull, and the matter waves are beamed and focused on that part, deep in the brain, in which involuntary movements are controlled (Figure 3-6 (b)). The energy dissipated by the beam is concentrated at the focus of the beam, and gently destroys the metabolic activity at the site (the substantia nigra). The method, when used carefully, has the advantage over all others that it pro- duces lesions at the focus of the ultrasonic energy without interfering with the normal blood flow from one part of the brain to another through the region irradiated. Of course this is a great advantage from the medical point of view. The techniques were worked out first on hundreds of cats and monkeys, and are now very cautiously being applied to man. Functional disruption of nervous conduction occurs within a few seconds of exposure to ultrasound of sufficient dosage to produce lesions: 980,000 cps, 1.8- to 3-sec duration, and particle velocity amplitude of 350 cm/sec, from a generator with the capability of 20 to 1000 w/cm 2 . From the therapeutic viewpoint it has been found possible to irradiate simultaneously the four small parts of the brain which are active with respect to Parkinsonism in the four limbs. ***In spite of the fact that Parkinsonism may be dying out. Thus the average age of these patients is steadily increasing, in North America, a trend which, if it continues, would indicate that the disease may have died out naturally by 1985. 64 MATTER WAVES: SOUND AND ULTRASOUND Other conditions reported treated successfully by this method at this date include a case of cerebral palsy and one of phantom limb pain. The prin- ciple is simple enough: to produce lesions, without excessive damage, at the tiny spots in the brain which control the function which appears disordered. Conversely, using this tool to inhibit temporarily the various functions con- trolled by the brain, one not only can obtain a micromap, in three dimen- sions, of the control sites, but learn something of the mechanism of control as well. The facts of microirradiation and selective absorption and damage, augur well for the future of "neurosonic therapy" as a strong competitor to the mechanical, electrical, and chemical techniques now in use in brain dis- orders. Figure 3-7. Equipment for Clinical Ultrasonic Irradiation of a Patient with a Hyper- kinetic Mental Disorder. Upper right and insert: The multibeam irradiator itself. (Cour- tesy of W. J. Fry, University of Illinois Biophysics Research Laboratory.) The Dunn-Fry Law As the quotation from Lord Kelvin (Chapter 1) said, it is always com- forting to be able to state quantitatively an important fact. On animals it has been found that the time, t, of irradiation to a chosen physiological state — in this case to paralysis of the hind legs of young mice — is related to the intensity, / (power), of the irradiating ultrasound (982 kc/sec, hydrostatic CONCLUSION 65 Q UJ or 300 250 200 co ~ -?- E UJ o h- V z •♦— o o * -z. *— o CO III < z> cr CO t- CO *- 100- =5 h- 150 IRRADIATION TIME t (seconds) Figure 3-8. Threshold Energy for Paralysis as a Function of Ultrasonic Intensity, curve shows data of W. J. Fry and F. Dunn, 1956. Broken curve shows how the th is much higher than expected at very short irradiation times. Solid reshold pressure 1 atm, starting temperature 10° C) by the simple expression t oc i/vTT the Dunn-Fry law, which says simply that the time to paralysis is shorter the higher the intensity; but that the damage occurs relatively more slowly for large intensities than for small intensities. This is one of the best rules-of-thumb so far worked out in biophysics of ultrasound therapy. It remains to be seen whether it is of general applica- bility. Intuitively one would think it should be. In any case it might be well to state the following memory aid: Probably because of general heating and of molecular excitation induced by absorbed ultrasound, metabolic, physio- logic, and histologic changes occur in tissues. In otner words, tissues Fry until Dunn! CONCLUSION "Like some other agents which have been introduced into the arma- mentarium of clinical medicine, medical ultrasonics passed through the early stages of enthusiasm, followed by a reactionary stage of pessimism, before it achieved the stature presently accorded it. Currently there are promising developments and interesting applications of ultrasound for medical diag- nosis, for therapy, and for biologic measurement." (J. F. Herrick. 12 ) The next ten years should be interesting ones from this point of view. 66 MATTER WAVES: SOUND AND ULTRASOUND PROBLEMS 3- 1 : Express in decibels the sound which delivers 1 50 times the power of background noise. 3-2: (a) Calculate the value of the absorption coefficient of sound in tissue at 50; 1000; 10,000; and 500,000 cycles per second (cps). (b) Make a plot of intensity vs depth in tissue for each frequency. 3-3: How would you employ the inverse square law to "protect" yourself from an intense source of noise? Suppose you wanted to reduce the noise level by a factor often. What could you learn about this problem from a = f(rj) as these terms are defined in the text? 3-4: Two signals enter your ear: one at 500 cps, with intensities / and 7° equal to 10~ I2 and 10" l5 w/cm 2 , respectively; and the other at 6000 cps with intensities /and 1° equal to 10 14 and 10~ 16 w/cm 2 . Which will seem the louder? REFERENCES 1. Fry, W. J., Adv. in Biol, and Med. Phys., 6,281 (1959): a review, illustrated. 2. Kelly, E., Ed., "Ultrasound in Biology and Medicine," Amer. Inst, of Biol. Sciences, Washington, D. C, 1957. 3. Fletcher, H., "Speech and Hearing," D. Van Nostrand Co., Inc., New York, N.Y., 1946. 4. Ruch, T. C. and Fulton, J. F., Eds., "Medical Physiology and Biophysics," W. B. Saunders Co., Philadelphia, Pa., 1960. 5. Herzfeld, K. F. and Litovitz, T. A., "Absorption and Dispersion of Ultrasonic Waves," Academic Press, New York, N. Y., 1959. 6. Stacy, R. W., Williams, D. T., Worden, R. E., and McMorris, R. O., "Essen- tials of Biological and Medical Physics," McGraw-Hill Book Co., Inc., New York, N. Y., 1955. 7. Whitfield, I. C, "The Physiology of Hearing," in Progr. in Biophysics, 8, 1 (1957); a review. 8. Gray, J. A. B., "Mechanical into Electrical Energy in Certain Mechano- Receptors," Progr. in Biophysics, 9, 285 (1959); a review. 9. Neely, K. K., "Noise — Some Implications for Aviation," Can. Aeronaut. J., 3, 312(1957). 10. Herrick,J. F. and Anderson, J. A., "Circulatory System: Methods — Ultrasonic Flow Meter," in "Medical Physics," Vol. Ill, O. Glasser, Ed., Yearbook Publ., Inc., Chicago, 111., 1960, p. 181. 11. Gardner, W. H., "Speech Pathology," ibid., p. 637. 12. Herrick, J. F., Proc. Inst. Radio Engineers, Nov., 1959, p. 1957. 13. Von Bekesy, G., "The Ear,"5W. Amer., Aug., 1957; a review. CHAPTER 4 Electromagnetic Radiations and Matter The next thing is striking: through the black carton container, which lets through no visible or ultraviolet rays of the sun, nor the electric arc light, an agent (X) goes through which has the property that it can produce a vivid fluorescence .... We soon found that the agent penetrates all bodies, but to a very different degree. (W. C. Roentgen, Annalen der Physik und Chemie, 64, 1 (1898).) INTRODUCTION Within fifteen years, just before the turn of the century, complacent classi- cal physics received three rude shocks. The first was Julius Plucker's de- scription (circa 1890) of the electrical discharges which take place in gases under low pressure and high voltage (the embryo of the "neon" sign). The second was Henri Becquerel's discovery of natural radioactivity in 1895; and the third was Wilhelm Roentgen's discovery of X rays, reported in 1898. In the years since then, the three discoveries have collectively engendered in- tense investigation of: (1) the structure of molecules, atoms and nuclei; (2) arrangements of molecules in crystals and other, less well-defined molec- ular arrays; (3) the electromagnetic spectrum, from X rays through visible to infrared radiation; and (4) the interactions — and in fact interconversion! — of electromagnetic energy and matter. In this chapter a review is given of those facts and theories which are useful to an understanding of the bio- physics of the interactions of electromagnetic radiation and living matter. 67 68 ELECTROMAGNETIC RADIATIONS AND MATTER THE STRUCTURE OF MATTER The Elementary Particles and Atomic Architecture Some of the key experimental facts accumulated within a few years of 1900 illustrate the bases upon which our knowledge of structure depends. Roentgen found that his unknown, or "X," rays would cause fluorescence in zinc sulfide and barium platinocyanide; and further that they would ionize gases and darken a photographic plate. They were therefore easily detected by an electroscope, or by an increase in current through a gaseous discharge tube, or by photographic techniques. He studied penetration through paper, wood, and metals, and showed that difference in penetration is one of degree rather than of kind (cf. the quotation which opened this Chapter.) A fluorescent screen on each end of a cylindrical gaseous discharge tube showed that particles, presumably charged, pass between the electrodes in each direction. By placing metal shields between positive and negative elec- trodes, and by impressing a voltage between horizontal plates placed with their plane parallel to the direction of flow, it was shown that the rays com- ing from the positive electrode bend toward the negative horizontal plate, and are therefore positively charged; and likewise the rays from the negative plate bend toward the positive plate, and are therefore negative. The nega- tive particles were called cathode rays, and positives canal rays. In 1897, J. J. Thomson (not William Thomson, Lord Kelvin) measured the deviation of the (negative) cathode rays in an electric and magnetic field, and obtained a value for the quotient of the charge to mass, i.e., e/m. This value was found to be the same (1.757 x 10 H cou/g) no matter what ma- terials were used. Cathode rays were therefore recognized as elementary particles of matter, and were called electrons. The (positive) canal rays, how- ever, were found to be different for different materials. By an ingenious experiment in late 1897, Milliken was able to obtain an independent measure of e, the charge on the electron. One or two electrons were trapped on atomized oil particles, and the electrical force necessary to prevent each oil particle from falling under the influence of gravity was measured. Since the size of the particle could be determined from the rate of free fall, the charge absorbed by the particle could be evaluated. The smallest value obtained, 4.78 x 10~ 10 electrostatic units (1.600 x 10~ 19 cou), corresponded to one electron absorbed. From Thomson's value of e/m, the mass could then be determined as 9 x 10" 28 g. This was an astounding achievement, the fact that exact meas- urement of this mass was possible by these means, whereas the most sensi- tive chemical balance weighs to only approximately 10" 6 g! For the canal rays, e/m for H + was found to be 1820 times smaller than for the electron. Faraday in 1830 had shown by electrolysis that the charge THE STRUCTURE OF MATTER 69 on the hydrogen ion was equal and opposite to that on the electron (being simply the absence of an electron), and hence the mass of the H + was deter- mined to be 1820 times the mass of the electron, i.e., approximately 2 x 10" 24 g. In 1896 Becquerel reported that he had accidentally discovered a pene- trating emanation from uranium salts. Thus, his photographic plates, kept in a drawer, with a key in the drawer above, became exposed with the im- print of the key in the presence of some phosphorescent minerals — notably salts of uranium — lying on the top of the bench. These emanations were also found to ionize gases. The Curies, in 1898, extracted a concentrate from pitchblende which had high emissive power, and named it radium (hence the terms "■radium-active" or "radioactive" elements, and "radioactive emana- tion"). They measured the strength of the emission by means of an electroscope. This instrument is essentially a vertical metal rod with a thin gold leaf at- tached to it by one end. If the electroscope is charged, the free end of the gold leaf is held out from the main shaft by repulsion of the like electro- static charges. It falls to the shaft in the presence of ionizing radiation, at a rate which increases with the strength of the emitter, because the electro- static charge on the metal is neutralized by charged particles formed during the absorption of radiation. Today ionization chambers based on this prin- ciple have wide use: a burst of current due to ionizing radiation is ampli- fied and recorded. One pulse of current occurs for each bundle of emanation absorbed. Ionization chambers are discussed in Chapter 5. In an experiment whose origin is obscure but which was refined and ex- panded by Rutherford (see Figure 4-1), three fractions emanating from a radioactive source such as radium were separated, and called alpha (a), beta (/3), and gamma (7) rays. It was found that alpha rays are positively charged and are much heavier than the betas. They are completely stopped by thin paper or a few milli- shields rodioactive source Figure 4-1 . Rutherford's Separation of Alpha, Beta, and Gamma Rays, by Means of an Electric Field Applied Between the Deflecting Plates. Tube is evacuated. 70 ELECTROMAGNETIC RADIATIONS AND MATTER meters of air, and lose one half their intensity if directed through 0.005 mm aluminum foil. By contrast, the beta rays are negatively charged, only weakly ionize gases, can travel many centimeters through air, and lose one half their intensity only if passed through 0.5 mm of aluminum sheet. The gamma ray has no charge. It strongly ionizes gases and penetrates up to 4 in. of lead. Careful determination of e/m showed the beta rays to be fast electrons, traveling at speeds up to 0.99 times the velocity of light (3 x 10 10 cm/sec). Similar experiments, and actual collection of alpha rays in a lead box, showed that the alphas are helium ions, He ++ . Experiments on penetration and analogous properties indicated that the gammas are simply electromag- netic waves like light, except of very short wavelength, shorter (or "harder") and more energetic than X rays. Rutherford's famous scattering experiments, performed about 1911, dis- closed the inner structure of the atom. Alpha rays were used as the bullets and metal foil as the target (Figure 4-2). He surrounded the target with a photographic plate nucleus paths of 1 \ ©/ ulpliu * w particles *■ scattered alphas -^^ 'atom of Ni © Ni foil Figure 4-2. Scattering of Alpha Rays by Nickel Nuclei. Definite scattering angles and even back-scatter were observed. See text. cylindrical photographic plate, and observed, in addition to dark spots re- sulting from direct penetration through the foil, dark spots at certain char- acteristic angles of scatter. Most important, though, was the observation of ia^-scattering, in which the incident radiation was reflected almost straight back, like a ball bouncing off a wall. In his own words, in a lecture delivered at Cambridge many years later, in 1936, Rutherford said: On consideration, I realized that this scattering backwards must be the result of a single collision; and when I made calculations I saw it was impossible to get anything of that order of magnitude unless one took a system in which the greater part of the mass of the atom was concentrated in a minute nucleus .... The back-scatter requires such energy that the alphas must penetrate to within 1/10,000 of the center of the positive charge in the atom; this means that the positive charge is centered in a nucleus of diameter 1/10,000 that of THE STRUCTURE OF MATTER 71 the whole atom. The atomic diameter calculated from Avogadro's number (6 x 10 23 atoms per gram atomic weight) and the density of, say, nickel (8.9 g/cc) is found to be approximately 10~ 8 cm (1 A). Therefore the diam- eter of the nucleus is approximately 10 l2 cm. Of primary importance to an understanding of penetration of energetic radiation into tissue was the deduction: the total positive charge is centered at the nucleus, which con- tains also most of the weight of the atom. The negative charge, equal in magnitude to the positive but of negligible weight, is in the orbital electrons. Atomic theory then developed rapidly, between 1910 and 1925. Max Planck suggested that light is emitted and absorbed in bundles of energy (quanta); and Niels Bohr postulated that the electrons are held in definite orbits or levels around the nucleus, bound to the nucleus by positive-negative attraction, yet held from each other by negative-negative repulsion, thus pre- serving a definite diameter for the whole atom. It was in 1926 that Erwin Schroedinger proposed an expression relating energy to radius, which for the first time gave these qualitative ideas quan- titative expression. It describes a model of the atom in which the electrons exist in a series of levels or orbitals, given the names K, L, M, etc., the K-shell being next to the nucleus. Figure 4-3 illustrates the spherical and Figure 4-3. Sommerfeld's Atom with Elliptical (p) and Spherical (s) Orbitals. Three p's are at right angles to one another. Each orbital can hold two electrons, whether both from the one atom or a "shared pair" in a bond. As drawn, this "atom" could accommodate 2 electrons in the K shell (Is) and 8 in the L shell (2-level). Thus it represents atoms from hydrogen (1 elec- tron) up to neon (10 electrons). The 3s, 3p, etc., orbitals, only slightly larger, and not shown, accommodate orbital electrons of elements higher in the periodic table. 72 ELECTROMAGNETIC RADIATIONS AND MATTER ellipsoidal orbitals first envisioned by Sommerfeld and described by Schroedinger. Each orbital can accommodate two electrons only, according to Wolfgang Pauli's "exclusion principle." The quantitative theory has now been tested experimentally for 36 years, by observation of the "light" emitted by excited atoms, and it describes, with the most beautiful precision known in science today, the observed results (more about this later). The in- ference is that Bohr's guess was right. But nobody knows why! Werner Heisenberg's introduction of the "uncertainty principle," and later his new formulation, called wave mechanics, in which all the elementary particles (and hence all matter) are considered to follow the undulations of electromagnetic waves, have only served to strengthen the grasp that this particular atomic model, or theory, has on science. The model discloses that there are sublevels in which an electron may find itself within the electron cloud: the s, p, d, and / levels,* or orbitals, as they are called (Figure 4-4). In each of these the electron is confined within a certain spherical or cigar-shaped volume about the nucleus. The orbitals of the outermost electrons of the atom overlap with those of the neighboring atom, and form a "bond." p-orbitol s-orbito schematic de Broglie s standing waves Figure 4-4. Schematic (exaggerated and distorted) s and p Orbitals with de Broglie's "Pilot Waves," Which are Thought to Guide the Electrons in Their Orbits. Working from the inside to the outside, we discuss interatomic binding after a section in which we focus attention on the hard, heavy, positive core of the atom, the nucleus, knowledge about which is so important to the understanding of radioactivity and its biological effects. *For sharp, principal, diffuse, and fundamental: descriptive codings used by spectroscopists to describe spectral lines. THE STRUCTURE OF MATTER 73 The Atomic Nucleus Since World War II much research has centered on the forces which hold the nucleus together. The nucleus carries all the positive charge and most of the mass of the atom. As a result of bombardment experiments (Fig- ure 4-2), especially on light nuclei, by 1930 it was known to be composed of two main particles, protons, p, (H + ) or bare hydrogen nuclei, and neutrons, n, particles of the same weight as protons, but with no charge. Moseley showed in the year 1914 the correlation between atomic number and positive charge on the nucleus; and isolation and identification of isotopes (same atomic number, different atomic weight — i.e., more or fewer neutrons) fol- lowed at a fast pace, until today more than 600 isotopes of the 108 elements are known. Some nuclei are stable, but some are unstable, and fly apart spontaneously into fragments. These are the radioactive isotopes. Some un- stable isotopes do not exist in nature, but can be produced artificially by nuclear bombardment (by n, p, etc) techniques. They are called artificially- radioactive isotopes. Experimental bombardment of the nucleus and examination of the prod- ucts by cloud chamber, ionization chamber, energy-balance studies, photo- graphic, and other techniques has disclosed about 20 new particles. First came the neutrino and the positive electron, or positron, then a number of new particles, at first all called mesons. Named after the great theoreticians, Bose and Fermi, these are now classified into: Bosons (spin = 1) (a) pions, or light mesons (t° : 264.2; tt ± : 273.2) (b) A;aons, or heavy mesons (k°: 965; k ± : 966.5) Fermions (spin = 1/2) (a) leptons, or light particles {n*: 206.77; e*: 1; neutrino) (b) barions, or hyperons and nucleons (Xi*: 2585; 2*: 2330; A°:2182; p*: 1836; n°: 1837) The mass (in multiples of the electron mass) and charge (°, + , or " super- scripts) of these particles (ir, k, Xi, p, etc.) are given in parentheses. The bosons exist in the nucleus and contribute to its phenomenal binding energy. Isolated, all but the electron, proton, and neutrino are unstable. However, the neutron persists for about 20 minutes on the average. The others last only 10- 6 tol0- 10 sec. Of some particular interest may be the muon (p*), well established as a cosmic-ray product in the atmosphere in which we live. It is ultimately pro- duced by the impact of a cosmic ray proton and an atomic nucleus in the upper atmosphere. A 7r-meson is first produced, which in turn decays 74 ELECTROMAGNETIC RADIATIONS AND MATTER rapidly into the muon plus a gamma ray. The muon disintegrates into a fast, ionizing electron and two more gamma rays, at sea level. The atom and its nucleus were recently detailed in delightful form by Gamov 10 , in a little book highly recommended for its simple, colorful de- scriptions of very complex phenomena. Molecular Structure and Binding It is the outer, or valence, electrons of the electron cloud which are evi- dently involved in binding atom to atom (Figure 4-4). Two distinct cases, and one intermediate case, have been studied thoroughly. First, the valence electron in "atom 1" can jump into an empty orbital of "atom 2," leaving atom 1 positive and making atom 2 negative. Strong electrostatic binding exists (Coulomb's law) because the charge separation is small. This is the case in all salts, both inorganic and organic. The bond is called ionic. Secondly, the electron from atom 1 can simply exchange, or be "shared" with that of atom 2. For instance if each of the two valence electrons is in an s (spherical) orbital, and the orbitals can overlap so that exchange or sharing takes place, a "sigma" bond is formed. If both are in p ("probing") orbitals (cigar-shaped), and if they overlap, a so-called pi (7r) bond is formed (Figure 4-5). Indeed combinations of s and p, called "hybrids," are pos- sible. For example each of the four bonds made by a carbon atom is a hy- brid of one s and three/? valence electrons — imagine, in Figure 4-4, the 2s and 2p electron orbitals as distorted; it is a mixture, called an sp^ hybrid. The four are directed tetrahedrally from each other, like four long noses, each to form a bond (i.e., to share a pair of electrons) with a neighboring atom. In the case of water, each of the p orbitals of oxygen overlaps with s of hydrogen to form a bent (109°) molecule. The bond is called covalent. TT-bond electrons closed loop (b) TT bond electrons, open path, \ m obi le carbon atoms Figure 4-5. Diagrams of Overlapping it Bonds: (a) A closed loop to form a dough- nut of negative charge above the plane of a benzene ring; (b) on a protein with open and ringed molecular structures, in which 7r-bond electrons are somewhat mobile and can transfer charge from one end of the molecule to the other, if forced. THE STRUCTURE OF MATTER 75 In between the ionic and covalent bond is the dative bond, in which the electron of atom 1 is partially given over to atom 2, although exchange and overlap still occur. Organic-phosphorus molecules are an important ex- ample (ATP, for instance, the "mobile power supply" in the living system). The oxygens of the phosphate assume a definite negative charge because of dative bonding. Of special importance is the w bond, formed by the overlap of two p orbi- tals ("probosci"). It often forms the second bond in the "double bond" of conjugated organic molecules, and restricts the relative rotation of atoms 1 and 2 if joined by the it. But the most important property of the it bond is its position, directed parallel to, but not coaxial with, the atom — atom axis (Figure 5 (b)). Although it helps to bind atom 1 to atom 2, it is an ac- cumulation of negative charge outside the volume containing the two atoms. It therefore can form weak bonds (complexes) with positive ends of other molecules in the vicinity; but, most important, it can exchange electrons with other it bonds close by, and hence provide a pathway by which elec- trons can run along a molecule from a point of excess negative charge to a point of deficiency of charge. Hence some organic molecules in tissues are electronic conductors, a fact which only recently has been appreciated with respect to nerve conduction and photosynthesis. (This very important topic is pursued in Chapter 6.) Further, the possibility of different electronic states in molecules, with different types of bonds, has profound ramifications in interactions of the molecule (and the tissue of which it forms a part) with electromagnetic radiations. These very important topics are also discussed in Chapter 6. It is obvious that the elementary particles are the building blocks of the living stuff. From the molecular point of view, however, it is not at all clear where the line is to be drawn between the living and nonliving. Usually the attributes of growth and reproduction are used to classify the living. Yet, in a supersaturated solution, copper sulfate crystals will "grow," layer upon layer; and if the temperature is allowed to fluctuate up and down with a frequency of one or two cycles per day, they will "reproduce" themselves, by "seeding," in the form of many crystallites on the walls of the container. In- deed, Teilhard de Chardin, in 1945, proposed that all the elementary par- ticles of matter are living, that they have the potency to do the things which living things can do, but that this potency is, to us, masked behind the gross behavior of large numbers. The gross behavior — statistical behavior — is all that our experimental techniques can today perceive in inanimate na- ture. Our techniques can examine the highlv organized individual man in which ~10 28 particles are organized and controlled from within, although this inner FORCE is not amenable to physical examination as we know it today. From the point of view of elementary particles, the only difference be- tween living and nonliving matter is one of organization. 76 ELECTROMAGNETIC RADIATIONS AND MATTER ELECTROMAGNETIC RADIATION; NATURE AND SPECTRUM The electron clouds of atoms and molecules can be excited by various methods — by heat, bombardment by some charged particle, and by absorp- tion of incoming radiations. A simple example is the flame test for sodium: if a sodium salt is heated in a flame, it glows with a characteristic yellow glow. It is not burning (i.e., being oxidized by oxygen). Rather, the valence (outermost) electron gets excited (accepts energy) and "jumps" to a higher- energy orbital, from a 3s to a 3/?. Imagine the next set of orbitals around the nucleus in Figure 4-3. Its lifetime there is short, however, and it falls back to the original state ("ground state"), and emits the extra energy as electromagnetic radiation {light in this case) of such a wavelength (5893 A) that it excites the cone cells on the retina of the eye. Biology is entering its electromagnetic age. Many parts of the electromag- netic spectrum are beginning to be used for diagnosis and therapy, as well as for studies which are leading to a better understanding of the roles of each of the parts in the systematized whole. Nature of Electromagnetic Radiation The exact nature of electromagnetic (em) radiation is unknown. What is known is that the wave has two component parts, an electric part and a mag- netic part, moving in phase, but in direction 90° from each other — much like two vibrating strings, one going up and down while the other goes back and forth — superimposed on each other. Each oscillates about an average value (zero) at a frequency which depends upon electronic vibrations in the source. The em waves travel in a straight line, and have energies inversely proportional to the wavelength, or directly proportional to the frequency (number of cycles per second). The wave carries no net electrical charge, and no net magnetic moment, but because of the components which can in- terfere or react with electric or magnetic fields, it can lose or gain energy (i.e., change frequency). All em waves travel at the velocity of "light." They have both wave properties (such as the capability of being reflected or diffracted) and particle properties (such as delivering their energy in bundles or quanta.). The unit bundle of electromagnetic energy is called the photon. Undulations in the electromagnetic field are described by the celebrated Maxwell equations (1873). Electromagnetic radiations vary only in frequency, and through this, in energy. Therefore their use requires handling the energy contained in the radiation. For example, we know how to handle light with mirrors, lenses, microscopes, and prisms, and to detect it by photographic plates, photo- electric cells, the eye, etc. Handling, or making it serve a useful purpose, is simply a question of using equipment which does not absorb the light. Detec- ELECTROMAGNETIC RADIATION; NATURE AND SPECTRUM 77 tion is simply a question of providing a medium which can absorb the light, or a medium with which the light can interact and be partially absorbed, to appear as another, more familiar form of energy. Electromagnetic radiation propagates with undiminished energy through a vacuum, always at the speed of light no matter what the frequency. The Electromagnetic Spectrum — A Survey Table 4-1 gives some properties of interest for the whole spectrum of elec- tromagnetic radiations. Since the em radiation has both wave and particle properties, the wavelength range of the different sections is given, and the energy associated with an excitation in each section is given in electron volts (1 electron volt/molecule = 22,000 cal/mole). Common means of detecting and of handling the radiations are noted; and what happens during absorp- tion is indicated. If one expects to gain insight into the interactions of electromagnetic radiations and matter, one must study the two Tables, 4-1 and 4-2, ex- haustively. There is no easier way. One will find, for example, from in- spection of the dimensions of the wavelength, A, and frequency, v, that they are related through the velocity, c, which for all electromagnetic radia- tions in vacuum, no matter what the wave length, is 3 x 10 10 cm/sec (186,000 miles/sec). Thus v = 3 x 10 !0 /A cycles/sec Table 4-2 indicates some of the effects of the interaction of various "cuts" of the spectrum with matter. It is certainly true that radiation of short wave length (high frequency) carries more energy, is more penetrating, and can do more damage than that of long wave length. Thus, at wavelengths from 20,000 to 500,000 A, the radiation simply tickles the molecules into a rota- tional and vibrational frenzy (high heat energy;. Radiation of 4000 to 7800 A excites electrons in the pigment molecules of the retina of the eye, and is visible. (Maximum sensitivity of the eye is at about 6000 A.) Radia- tion of wavelength 2000 to 4000 A (ultraviolet) excites even the bonding elec- trons in a molecule, and so loosens up a bond that chemical reactions may take place which otherwise could not. Wavelengths below 2000 A, in the hard or vacuum ultraviolet, actually drive electrons out of a molecule, or ionize it; and as the wavelength gets shorter, and the radiation "harder," more and more ions are formed in the wake of the incoming radiation. In the X-ray region (X = 1 A) the electrons of even the K shell of the atom, the most tightly bound ones, can be excited or ejected; and in the gamma region (~0.01 A), even the nucleus can be penetrated by the radiation, al- though electrons in the atomic cloud are a more probable target. 78 E c tt cu cu Q CU C bo co a: -a "o 53 J2 — ■ co o be «u ^ CO " IU E £\S CJ — <" -rl -5 c u CO CO U bo _c to t- bc T3 • o >, _Q CO j3 * C • C CO 3 CO £.2E 3 _: 3 JZ - CU JO c e £0 CU JO 3 _o 6 to cu 0"> O S a- -t cu cO -C cu C _o to N '5 o i u x 9- CL CO _ bD cu cu o o u3 3- u o cu § 2 cu 3 CU CU JZ CU -3 cu .3-. o S3 - ? c E «-> o ci — .- o u rl ^ u ° " -£r ° « .S T3 « cu O ' cu CO s- S g -o -a cu c C •- _Q C CO cu CO 3 3. 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O o u E o E 3 i_ ^« (J CU a to c a> o E o iu (U CU £ 0) 4) cu <3> 4? 3 C « cr o i) « cc - - 2r cu o> c o c cu D i- cu 3 O 3 O O [_ Ou in JO O CU 3 CO "3 3 O CU cu CO o O =o O o X ro — O O O OJ O ) * 2 2 o © o < - X X (V) ,- c^ <^> I © O © o o to cu cu ^ CO ^-v © M U O CU ^ r— E6o CU o un © LT) CU >- CU 3 •< CU u CO a. © o © © ••< »< •< o o d o © oo © o © © o © o oo o © 00 o © © © © © 8° 8 N - E cu © © LTl O © O 00 LT) r» cm '-"" 2 2 •< o o © © © © ©~ o en £ cu o o o E cu © o © © o © o CO c o o ^5 o -a CO -o CO CO < CU E CO O U CO E E CO O X CO u CO cu cu > cu cu 3 cO CO ^ »— i 3 O XJ u M >-, u CO in > CO CU Z U CO cu cr cu be CO D -3 -3 CO CO t_ 5_ t_t cr CO CU CO Ih CU -o JZ cO bO o CQ 79 TABLE 4-2. The Electromagnetic Spectrum — Absorption Radiation Source Absorbed by Effects of Absorption Cosmic nuclear reactions nucleus; electron artificial radioac- on sun cloud of atoms tivity, fission, ex- and molecules citation, ioni- zation Gamma radioactive elements nucleus; electron artificial radioac- cloud tivity, excitation, ionization X metals hit by high- electron cloud excit. or eject, of K- speed electrons shell electrons Vacuum UV sun; atoms hit by electron cloud excit. or eject, of L- med. speed elec- or M-shell elec- trons trons Far and near UV gas discharge tubes; electron cloud excit. of sub-shell sun* and valence electrons Visible sun; thermally ex- electron cloud excit. of valence cited atoms electrons Near infrared red-hot bodies (e.g., vibrating perma- increased kinetic fireplace); sun nent dipoles in energy of vibrat. molecules (incr. temp.) Far infrared red-hot carbon; sun rotat. and vibr. incr. kinet. energy perm, dipoles of of rotat. and molecules vibr. (incr. temp.) Microwave klystron radio tubes rotation of perm. incr. kinet. energy (radar) dipoles of rotat. (incr. temp.) Ultra high-freq. tubes and tuned reradiated by con- unknown; interac- radio circuit ductors (metals, the body, etc.) tion with nerve? High-freq. radio tubes or transistors reradiated by con- unknown and tuned circuit ductors (metals, the body, etc.) Broadcast tubes or transistors reradiated by con- unknown and tuned circuit ductors (metals, the body, etc.) * Estimates of the internal temperature of the sun go as high as a million degrees K. Spectroscopic meas- urements give the temperature of the incandescent gases surrounding the sun to be about 6000° K. A black body at 6000°K radiates some energy at nearly all wavelengths, but the maximum energy is radiated at about 5000 A, right in the middle of the range of wavelengths visible to man. This is no coincidence, of course, for man's senses are adapted to his environment. After absorption of the damaging short-wavelength ionizing radiation by the upper atmosphere, the total energy reaching the surface of the earth on a clear day is ~ 1.25 cal/min cm 1 . However, above the a phere space travelers will have to be protected against the small amounts of ionizing radiation which extend right down to wavelengths in the X-ray region. The most prominent of these is the strong emission of excited hydrogen atoms, the"Lyman-alpha" line, at a wavelength of 1215 A. 80 ELECTROMAGNETIC RADIATIONS AND MATTER Quantitative expression of these ideas followed Planck, who, in 1901, pro- posed that the energy, e, contained per photon in incoming electromagnetic radiation is proportional to the frequency, v, of the radiation. Thus e = hv where h is the proportionality (Planck's) constant, equal to 6.62 x 10" 27 erg sec/photon (1 electron volt, ev, = 1.6 x 10 12 ergs). Let w } ,w 2 , and w 3 be the energies of binding of different atomic or molec- ular orbital states of the electron to the nucleus, and accept Bohr's as- sumption. If e = w } , w 2 , or w 3 , absorption of the incoming radiation will easily occur,accompanied by excitation of the electron from its "ground state," or orbital of lowest energy, to an excited state. If e ^ w y , w 2 , or w 3 , then absorption does not readily occur, although in favorable cases w x can be taken from a larger e, the electron excited to state 1, and the radiation pass on with reduced energy (e - w, = hv 2 ) and lower frequency (longer wave- length). This is one aspect of the famous "Compton scattering." If f is greater than some critical value, w, the ionization energy, the elec- tron can be ejected completely from the atom or molecule, and may have any kinetic energy up to and including e — w. Since the electron has a mass of 9 x 10 _28 g, the kinetic energy (1/2 mv 2 ) is less than, or equal to, e — w. Now a negative particle of velocity v, just like any other member of the elec- tron cloud about a molecule, but moving with high velocity, is a very good ionizer itself. Hence the ionization process continues along a track through the tissue until all the incoming energy, e, has been dissipated either as heat or in producing ions. The Laws of Absorption In the tables of properties of em radiations, the bases of the techniques for handling them were implied. What happens when absorption takes place was also indicated. We consider now the extent of absorption, and its con- verse, the depth of penetration. In brief and in summary, absorption of electromagnetic radiations is governed only by the laws of chance. The chance that a photon will be ab- sorbed depends only upon the number of target electrons and nuclei in its path. From the fact that the higher energy (shorter wave length) radiations penetrate deeper into any given material, it is inferred that they are more difficult to capture — have a "smaller capture cross-sectional area." Con- versely, the denser the target material the greater is the number of potential targets per centimeter of the photon's path, and hence the greater is the ab- sorption per unit length of path. These ideas are expressed quantitatively in Lambert's law. The rate of ELECTROMAGNETIC RADIATION; NATURE AND SPECTRUM 81 absorption is directly proportional to the amount to be absorbed; or -dl/dx = k'l where x is thickness and / is intensity, or number of photons passing 1 cm 2 per sec. This is one of the natural functions (Chapter 1) for which / is ex- pressed explicitly as / = /, -k'x where I is the intensity when x = 0, just as the radiation enters the ab- sorbent; k' is a constant, characteristic of the absorbent (larger, the better the absorption capacity of the medium), called the absorption coefficient. The plot of / vs x is shown in Figure 1-2 (c). Since In I /I = k'x, conversion to common logarithms by dividing by 2.303 gives log I /I = kx, where k' = 2.303 k, and k is called the "extinction coefficient." Lambert's law is applicable over the whole electromagnetic spectrum, and, you will remember from Chapter 3, is useful also to describe the ab- sorption of matter waves. It is an obvious but very important point that the extinction coefficient of a substance will be different at different wave- lengths. From the far infrared, through to the near ultraviolet, the extinc- tion coefficient is large only for particular wavelengths. Such specificity is a property of molecular absorption. If these molecules are suspended or dis- solved in a medium, k will be directly proportional to the concentration, c (Beer's law). Thus k can now be factored into ac, where a is called the molec- ular extinction coefficient. Formally then: log I /I = acx (Beer-Lambert law) The specificity for absorption of selected wavelengths disappears from the far ultraviolet through to gamma radiation — continuous absorption occurs accompanied by ionization — and the extinction coefficient decreases more or less linearly with decreasing wavelength (i.e., with increasing energy /photon). Thus ultraviolet light penetrates only a small fraction of an inch of tissue; and the k for tissue for near ultraviolet is very large. By contrast, soft X rays penetrate tissue with only a small amount of absorption per cm; and k is smaller. However, each photon of X rays absorbed carries roughly 1000 times more energy than each photon of near ultraviolet, and therefore only 1/1000 as much absorption is required to do the same damage. It is seen then that the important quantity is the energy absorbed per unit volume, because this determines the subsequent effect: warming of tissue, triggering of the optic nerve fiber, providing the energy for photochemical synthetic processes, or ionization and rupture of molecular bonds. 82 ELECTROMAGNETIC RADIATIONS AND MATTER The molecular extinction coefficient is strongly dependent upon wave- length, as we shall soon see. The optical transmission is defined as 100 7// per cent. The optical density, often used, is defined as log (I /I), and increases linearly as concentration of absorber is increased. SOME INTERACTIONS OF ELECTROMAGNETIC RADIATIONS AND LIVING MATTER The parts of the spectrum which are of biophysical importance can be conveniently classified under four main titles: the warming region, the visible region, the photochemical region, and the ionizing region. Each of these is illus- trated below. Enough of the principles are given to introduce infrared and ultraviolet therapy. The visible region is considered in more detail, for obvious reasons. X and gamma rays, and hard ultraviolet too, are intro- duced here in principle only. Detection and absorption are discussed in Chapter 5, and Chapter 9 deals exclusively with biological effects of all the ionizing radiations. The Warming Radiations (Infrared) Electromagnetic radiation in the infrared range is always associated with heat energy of those molecules which contain permanent dipoles. Its ab- sorption results in increased rotations and vibrations, and therefore in in- creased temperature. Infrared radiations are then logically called "heat rays. The penetration into tissue is appreciable, although the extinction coeffi- cient is large. The warming effect of absorption by the very outer layers of the skin can be felt beneath the surface because of the poor but substantial heat conduction of the tissue. Infrared-lamp therapy is based on this prin- ciple. Since the tissue is 85 per cent water, the strongest absorption would be expected to occur particularly near the strong water-absorption wave- lengths: (1) vibrations at 28,200 and 63,000 A, (2) rotations from 500,000 to 1,200,000 A, as well as (3) some absorption by mixed vibrations and ro- tations at nearly all wavelengths greater than about 8000 A. Intense infra- red electromagnetic radiation, when absorbed by tissue, causes gas and steam pockets which lead to lesions and blisters. Infrared Spectra The wavelengths absorbed often provide clues as to what rotation or vibration is absorbing the incoming radiation. In the instrument called the spectrometer a small slit of light from a continuously burning carbon arc — a good source of infrared radiation — passes through the absorbent and then on through a triangularly shaped crystal (prism) of KC1 or KBr; the trans- mitted radiation is broken up — the longer wavelengths will be bent sharply SOME INTERACTIONS WITH LIVING MATTER 83 within the crystal, the shorter wavelengths less so — and the image of the slit will appear as darkening on a photographic plate, at positions proper to the wavelengths entering the slit. Thus the absorption bands of water corre- O spond to O — H stretching vibrations and to H H bending vibrations. This is true for any absorber with rotating or vibrating dipoles. Many thousands of spectra have been determined, principally in organic mole- cules, for purposes of learning what polar groups there are in the molecule, or for identification of a particular substance in a mixture. Continuous use is now being made of this technique in investigation and control of barbitu- ates and narcotics, for example. Each material has a characteristic spec- trum (plot of absorption vs wavelength), easily reproduced, in many cases easily identified. Figure 4-6 shows two examples, and gives an indication at the bottom of what rotations and vibrations within the molecule may be responsible for each absorption peak (pointing down). Visible Radiations This region is noteworthy for the sole reason that the animal body is equipped with a very sensitive set of living cells which can detect wave- lengths of 4000 to 7800 A coming in from excited molecules in the environ- ment. Molecules in the environment are excited by radiation which pours in from the sun at all frequencies proper to a hot body. The reradiated energy from the excited molecules of a tree, for example, outlines its shape; the exact composition of the reradiated energy defines its brightness and what we perceive as its color. The eye is a device by which the energy of an electromagnetic radiation pattern is converted into the energy associated with the various nerve im- pulses which can traverse the optic nerve to part of the brain. It is a trans- ducer in the sense that it provides a mechanism by which electromagnetic radiation of wavelengths in the critical range can be received, focused, sorted out, and then converted into the chemical, thermal, and electrical energy which is necessary to trigger nerve propagation. In general, the energy car- ried by a nerve impulse is much greater than that of the light photons which trigger the propagation. This subject is considered in Chapter 10, and we confine ourselves here to what takes place before the nerve is triggered. Architecture of the Eye Figure 4-7 is a simplified sketch of the basic parts of the eye. It illustrates principally the roles of the lens, the retina, and the optic nerve. Light of intensity I Q ergs/cm 2 from a light source falls on the cornea. About 96 per cent passes on through the lens, and about 4 per cent is reflected. The cornea, the aqueous humor, the lens, and the vitreous humor are essentially 84 -• \- *: o 1 * °5 u a> c - o •55 c cr < 0) o m ac X to 'u »- z "E 2 CO E 2 O 2 1 1 < £ _ O O o o E _J £ £ C O co 1/1 o 11 o o z LU CD X o ■5 te ^0 £ > t E - < 1 ■c ° « a S- » CO •£ 3 3 ec 7 2 2 E t- *" CO X — 3 U o g 5. 't in 1 c a. >- Q. o Z X »- O t 8.3 CO o -2 c I 3 O I V \£ LU o 4> Q. S <-> q o ; JD O - • O >. lO • ~ «> ,V 3 3 X U 3 ~ 01 °-== in cJ IE £ SOME INTERACTIONS WITH LIVING MATTER 85 point light source (the object) ornea aqueous humor ciliary muscles optic nerve Figure 4-7. Architecture of the Left Eye, Viewed from Above. liquid crystal materials and are, of course, transparent. About 48 per cent of 7 reaches the retina. The iris acts as would the diaphragm of a camera, controlling the area of the pupil, and hence the total energy admitted. The incoming light, which is usually divergent from the source, is focused on the retina by the lens. The distance, q, between the lens and the image (of the light-source) on the retina is constant, but the lens-to-object distance, p, may vary widely from about 4 in. to a mile. To be versatile, then the focal length, /, defined as _L _L J_ / ~ P + q must be adjustable if objects at different distances are to have sharp images on the retina. Now the focal length depends upon the geometry of the lens: a thick lens will have a short focal length, and a thin lens a long focal length. Because the lens is a liquid crystal much like jelly, its shape can be changed by the tension exerted by the ciliary muscles. This tension is in turn con- trolled by a nervous signal fed back from the retina, the cells of which esti- mate the sharpness of the image. This process is known as accommodation. Photosensitive Cells The focused light falls on two types of cells on the retina, rod cells and cone cells, named because of their shape. The rod cells (scotopic vision) are the more sensitive to light, and distinguish for us light from dark when the intensity is very low (twilight vision). On the other hand the cone cells (photopic vision) are less sensitive, can resolve large amounts of light into its components, and therefore detect details of the image, such as shape and color. The photosensitive cells are present in large numbers, estimated at 126,000 cells/mm 2 . Most of the cone cells are clustered close together about 86 ELECTROMAGNETIC RADIATIONS AND MATTER a center called the fovea centralis. The distribution of rod cells is different (Figure 4-8) — practically none at the fovea, but otherwise distributed in great numbers over the whole area of the retina. * eg 2 2E a. UJ Cl tf) o z < O I CO o o ce u. O 160 - 120 - 80 - 40 HORIZONTAL ANGLE (DEGREES) FROM FOVEA CENTRALIS Figure 4-8. The amount of rhodopsin and the number of rods per unit area have a similar dependence on angle bounded by the incoming light and the central meridian in which incoming light falls directly on the fovea. The optic disc, where the optic nerve enters, is about 16° to the nasal side, and there- fore a blind spot exists there. (Locate the blind spot in your right eye by first focusing the eye on the black dot, then turning the eye 16° to the left — i.e., about 4 in. if the dot is 10 in. from the eye.) (After Rushton.') A brief discussion is now given of those molecules, known as pigments, which are not only the absorbers of the incoming radiation but also the transducers, the "machines" by which the incoming energy is trapped and "led across" into another form, not heat, which can trigger the optic nerve. Actually there are two separate subjects to discuss: twilight vision and color vision. Although much has been learned by direct experiment on animals, Rushton 1 complained in his recent review: "Measurements upon human pig- ments have only just begun, and it is to be hoped that far better experiments will be made." We give here a summary of the present understanding of this SOME INTERACTIONS WITH LIVING MATTER 87 important vital process, keeping pretty close to the facts, by-passing the theories. Twilight Vision As mentioned above, cells of two general shapes are found on the retina, rod and cone, the rod cells being responsible for the very sensitive detection of light from dark when it is almost dark. These cells distinguish the shape of the object, and although this is their primary role, they also permit us to dis- tinguish colors. The pigment responsible for twilight vision is a molecule called rhodopsin, the classical "visual purple." It is a condensation product of the carotenoid, retinene, and a protein called opsin. Retinene is a 20-carbon, ringed com- pound, the aldehyde of vitamin A, and its structure is well known. How- ever, not very much is known about opsin. Another opsin has been identi- fied, attached to retinene in the pigment todopsin. Further, an isomer of retinene has been combined with the original opsin, and cyanopsin formed. However, only rhodopsin is active in twilight vision. The extinction coefficient of rhodopsin, extracted in bile solution or in digitonin, has a maximum value at 5000 A. It drops off rapidly at both higher and lower wavelengths. Thus at 5500 A it is already down to about 25 per cent of the maximum, and at 5800 A is nearly zero; while at 4000 A it is also 25 per cent of the maximum value, but then remains about the same to wavelengths below those detected by the eye (smaller than 4000 A). The Beer-Lambert law is obeyed exactly for weak solutions of rhodopsin. Further, Figure 4-9 shows that the sensitivity of the human eye is deter- mined directly by the absorption of light by rhodopsin. To man's eye rhodopsin has a rose color; it absorbs strongly in the green (5000 to 5800 A) and yellow (5800 to 6000 A) regions and to a lesser extent in the blue (4200 to 5000 A), and reflects all the rest; it is this reflected light which falls on man's eye as he looks at the pigment, whether on the retina through an ophthalmoscope, or in solution. This is why it is "colored" rose. It follows from the preceding paragraph that the fewest number of pho- tons which will trigger the nerve will be those of wavelength 5000 A, for it is here that the extinction coefficient is greatest. Incidentally, the unit of light energy falling on the retina is the troland. At this wavelength it amounts to about 100 quanta falling on a rod per second. However, the rhodopsin of a rod is half-bleached by about 0.03 trolands, or 3 quanta per rod. It happens that 1 troland is the retinal illumination when 0.1 millilambert (mL) is viewed through a pupil 2 mm in diameter; and 0.1 mL is the brightness of a white screen illuminated by 1 candle at a distance of 1 m. Rhodopsin is "bleached" by white light. Its color fades rapidly through 88 ELECTROMAGNETIC RADIATIONS AND MATTER yellowish to clear. In the dark, in vivo, the color is restored. The process can be summarized as follows: photons + rhodopsin k A, (bleaching) k 2 bleached vitamin A -f energy (to nerve endings) + retinene + energy (regeneration) The scheme above indicates that the greater the intensity of the incoming light, the more will the rhodopsin be bleached. In twilight most of the pig- ment exists as rhodopsin, and the sensitivity is greatest. In daylight, most of it will be bleached, and the sensitivity least. "Dark-adaptation" is very familiar to us all; it is slow because the speed of regeneration of rhodopsin is o in z UJ Q < a. o 0.7 - 0.6 - 0.5 - 0.4 - 0.3 - 0.2 - 0. I - 250 300 350 400 450 500 550 3000 4000 5000 600 m M o 6000 A WAVELENGTH Figure 4-9. The spectrum of human scotopic (twilight) vision sensitivity (crosses), and the absorption spectrum of rhodopsin (solid curve) are the same. (After Rushton. 1 ) SOME INTERACTIONS WITH LIVING MATTER 89 slow. The reader is invited to contemplate the expression of the Weber- Fechner law in this organ: 5" oc log r/M, It says that the sensitivity, S, increases as the difference between the threshold intensity and that of the background decreases. This photochemical description of twilight vision, although satisfactory in general, apparently needs revision, for serious troubles arise when quantita- tive description is attempted. It now seems likely that individual pigment molecules are attached to individual nerve endings, and the excitation of just one pigment molecule by incoming radiation is sufficient to trigger the nerve. Thus, although it takes upwards of half an hour for dark adaptation to oc- cur—that is, for the bulk rhodopsin to be regenerated in man after a bleach- ing — the minimum time during which the eye can recover enough from a flash to see another flash is about 0.01 sec. Color Vision The cone cells somehow distinguish between wavelengths, and thus dis- tinguish colors. The Young-Helmholtz theory, usually accepted, and now nearly 100 years old, suggested that three color-sensitive pigments exist, each one sensitive to one of the basic colors: red (6200 to 7800 A), green (5000 to 5800 A) and blue (4200 to 5000 A); and that various intensities mix to give the colors and qualities commonly referred to as hue, brightness, etc. The Young-Helmholtz theory is based on the experimental fact that by a proper mixture of red, blue, and green light in an object, any shade of color can be matched. The theory is that the three pigments absorb definite frac- tions of the visible spectrum and overlap one another, and that the optic nerve can receive and transmit signals which correspond to any and all wavelengths of the spectrum. Apparently this theory now requires major modification as a result of the very recent (1959) work of E. H. Land. 7 In some remarkable experiments he has shown in effect that the full range of colors can be recorded by the brain provided only that the proper mixtures of intensities of two wavelengths (one greater than, and one less than, 5880 A (yellow)), fall on the retina! It seems that the information about colors other than the two incoming wavelengths is developed in the retina. The possibility that the pigment molecules are in intimate contact in the cone cells, and dis- tribute the excitation energy among themselves in a manner controlled by the intensity pattern of the incoming light, immediately suggests itself. But more work is clearly needed following this surprising turn. Another recent surprise is that some evidence has been turned up that other molecules in the neurones, in the nerve pathway itself, contribute to the color perceived in human vision. 90 ELECTROMAGNETIC RADIATIONS AND MATTER In spite of the credence placed in the Young-Helmholtz three-pigment theory of color vision, there is no direct evidence that three pigments exist in the cones. There is direct experimental evidence for two, however; this will now be recalled. Protanopes (color-blind people) cannot distinguish green from red. By measurement of the intensity of the light reflected from the retina as a function of incident wavelength on protanopes, it has been shown that a definite absorption by a pigment, given the name "chlorolabe," takes place with maximum at about 5400 A. Now the protanope can see green, but not red. This fact means that a sec- ond pigment, given the name "erythrolabe," is missing in the protanope. Difference spectra (unreliable) of two pigments in the normal fovea (collec- tion of cone cells) show that the maximum absorption of the second, or miss- ing, pigment is about 6000 A. Thus there is good knowledge of one pig- ment, the chlorolabe, and knowledge of the existence of a second, erythro- labe. There is no experimental knowledge of a third in cones. But, of course, Land's new work indicates that only two are really necessary, one sensitive above and one sensitive below 5800 A. The two pigments discussed have these qualifications. Recall that the optical density maximum for rhodopsin is at 5000 A. What the relation is between the excited pigment molecule and the color perceived is poorly known. Experimental approaches include that of meas- uring the electrical signals in the optic nerve (the electroretinogram, ERG) during stimulation by light, the reflection densitometry experiments men- tioned just above, studies of the rates of bleaching and recovery (adapta- tion), visual acuity, color perception, and Land's new work. However, since the excitation energy for electrons in large molecules is so dependent upon structure, it would not be surprising if rhodopsin, chlorolabe, and erythro- labe turn out to be very similar in composition. The answer will lie in knowledge of the structure of these molecules. Incidentally, an important new fact, bearing upon acuity especially, is that the eyeball is never still, but rather is in a state of small, almost im- perceptible oscillations, such that the incoming light falls on a spot on the retina for only a few microseconds before it is deflected away. If the eyeball is fixed relative to the light source, color vision disappears. Physical Defects of the Eye If the lens is too thick or the eyeball elongated (myopia), the ciliary muscles are not able to make sufficient adjustment of the focal length to permit distant objects to be focused on the retina. The phenomenon is known as nearsightedness, and can be corrected with the aid of glasses with a concave lens of the proper focal length. If the length of the eyeball is too SOME INTERACTIONS WITH LIVING MATTER 91 small, the condition is called hypermetropia, and can be corrected with a con- vex lens of proper focal length. The lens of the eye often does not have the same curvature over all its sur- face, and light passing through the area of improper curvature will not be properly focused on the retina. The lens of such an eye is said to be astig- matic. A properly ground astigmatic glass lens can compensate. Sometimes translucent or opaque tissue grows in or on the liquid crystal material of the lens and absorbs the incoming light before it reaches the retina. Such tissues are generally termed cataracts. Some can be removed by surgery; some are too extensive. Depth Perception Two detectors in different locations can inherently provide more informa- tion than one; and if relative information is recorded and interpreted from the two signals, more information is available from the two detectors than if each were interpreted separately. This is the reason sensory organs come in pairs. Typical of the relative information obtainable from two stations, in general, are direction and distance, or depth. Sound can be reflected, and hence the directional information provided by two stations is important. Light travels in a straight line to the eye, and therefore directional informa- tion is not important. However, the information derivable about distance or depth is important when we attempt to compare distances or develop a per- spective view. Ideally the eyes may each be rotated about 50° from a central line of vision. The two have to be in focus at the same time, on a near or a far object, and this requires a facility of minor individual adjustment. If the eyes cannot be made to focus (crossed eyes), sufficient correction can some- times be made with a suitable set of glass lenses, but often the cross must be corrected by shortening the lateral muscles or by suitable exercises designed to strengthen them. Photochemical Radiations (Ultraviolet) Photosynthesis Subshell electrons are excited by the ultraviolet. The absorbed energy may be passed off to the vibrations or rotations of nearby molecules and ap- pear as heat energy; it may be re-emitted as ultraviolet; or it may excite the molecule and make it more susceptible to chemical attack by neighbor- ing molecules. Thus in the last case the ultraviolet may provide some or all of the activation energy needed for reaction to occur, and thereby increase the rate of reaction (treated later in Chapter 8). In fact, the photochemical mechanism is sometimes the only mechanism by which certain reactions can take place at a reasonable speed at biological temperature. 92 ELECTROMAGNETIC RADIATIONS AND MATTER Because they carry more energy than photons in the visible region, the photons in the ultraviolet region are less likely to be absorbed. They pene- trate deeper into the absorbent and excite molecules at the point at which they are finally caught. Of all the synthetic biological reactions whose rate is sensitive to ultra- violet light, probably the photosynthesis of simple organic sugars from C0 2 and 2 in plant leaves is the best understood; and yet the understanding of this basic process is not completely satisfactory. Of course if it were, we should be able to reproduce the syntheses in a test tube; but we cannot. More important to present considerations is our knowledge of photo- catalyzed syntheses of the vitamins from basic components. Some of the vitamins have been purified, crystallized, and synthesized, and hence their chemical composition and structure are known. Consider the antirickets vitamin D 2 (calciferol) for instance. Its structure is well known: two six- membered rings and a five-membered ring attached to an unsaturated aliphatic side chain of six carbon atoms, with a molecular weight of 393. This molecule is formed through the absorption of ultraviolet radiation of 2500 to 3000 A by ergosterol, a sterol molecule whose structure also is well known. The synthesis occurs in at least two steps. The absorption is con- sidered to take place at a carbon-carbon double bond, and the absorbed energy to go into excitation of the t electrons which form the bond. The opening of a benzene-like ring follows, and further rearrangements of the atoms and bonds give the biochemically active vitamin B 2 structure. The re- action will not occur at all unless photolyzed. This synthesis takes place in the human body at a location to which both the molecular components and ultraviolet radiation are accessible: that is, just beneath the surface of the skin in the living tissue serviced by the blood capillaries. Thus the principle upon which ultraviolet therapy is based, and the advantages of moderate exposure to sunlight, both become apparent. Phototherapy Prolonged sun bathing can damage skin pigments and can cause ery- thema. For instance, on the average it takes only 20 microwatts (/xw) of ultraviolet of wavelength 2537 A (from a mercury vapor lamp) falling upon the skin for 15 min to produce erythema. It is fortunate that the very in- tense ultraviolet radiation from the sun is attenuated (scattered, absorbed, converted into radiation of longer wavelength) by the ozone and nitrogen compounds in the upper atmosphere. Ultraviolet radiation would be a prob- lem in space travel if it were not so readily reflected by metallic surfaces. The effects on the eye are well known and have been implied in the discus- sion of the chemistry of the eye: the higher-energy photons of the ultraviolet in falling on the retina can keep the rod and cone cells devoid of rhodopsin SOME INTERACTIONS WITH LIVING MATTER 93 and damage the color pigment molecules. Snow-blindness and "whiteouts" are the result. Further, ultraviolet has been attributed in some cases to promoting the growth of cataracts and photothalamia, or inflammation of the cornea. However, ordinary window glass absorbs all the dangerous ultraviolet, and colored inorganic materials can be added to filter out (or absorb) any undesired range of wave lengths. Therefore, protection is no problem, if properly sought. Ultraviolet light has a lethal effect on primitive animal and plant life. This fact is used to good advantage in destroying the bacteria, eschenchia coli and bacteria coli, in foods or in our water supply. Each of these is killed by about 14 x 10" 6 ergs per bacterium. Among the abnormalities successfully treated with ultraviolet light are conjunctivitis, fibrosis, acne, and surface in- fections of various kinds. Certain heavy metals (calcium, gold, silver, etc.) and certain highly absorptive molecules (methylene blue, quinine, etc.) sometimes increase the therapeutic value of the ultraviolet irradiation. The shortest-wave, vacuum-ultraviolet radiation overlaps the X-ray re- gion. The principle difference between the two regions in the present classi- fication is whether ionization and bond rupture is the exception (ultraviolet) or the rule (X and gamma). The vacuum-ultraviolet will be discussed im- plicitly in the next section, for the differences between it and the X ray are of degree rather than of kind. Ionizing Radiations (Mainly X and Gamma) Principles The only distinction between the radiations more and less energetic than that with a wavelength about 2000 A is one of excitation vs ionization. That is, at wavelength X greater than about 2000 A, excitation of electrons of the electron cloud takes place as the rule, and ionization takes place only in special circumstances; while at X less than about 2000 A the electrons can be knocked right out of the atom by the absorbed photon. As X decreases, the loosely held orbital electrons are the first to go, followed by the subshell elec- trons, and as X — » 1 A (X-ray region) the tightly bound K-shell electrons can be ejected. A simple calculation will make this important point clear. It takes an in- put, w, of ~230 kcal to make 1 mole of ions out of 1 mole of atoms, i.e., 10 ev to make an ion out of an atom. (This is the energy carried by each photon of em radiation of wavelength 1200 A.) Now the gamma radiation of the radioactive isotope of cobalt of atomic weight 60 (referred to the hy- drogen atom as 1), Co 60 , used in deep radiation therapy for cancer, has an energy of about one million electron volts (1 mev/photon). Therefore, each photon would leave a wake of about 1 6 / 1 = 10 5 pairs of ions (or molecules which have been ionized) before it loses all its energy. 94 ELECTROMAGNETIC RADIATIONS AND MATTER The electrons lost may have been valence, or bonding, electrons — active in holding the molecule together. In covalent bonding two paired electrons form the bond between carbon atoms, as in a sugar molecule for example. Ionization weakens the bond and perhaps breaks it; in any case the unpaired electron left is chemically very reactive and will make a new bond at any time or place. Cross-bonding of molecules, the synthesis of new molecules, polymerization of old ones, etc., all can occur. It is not hard to envisage how such reactions could adversely affect the tightly geared steady-state of normal living tissue. It is convenient to reserve further discussion of the effects of ionizing radia- tions until the principles of radioactivity have been outlined. The radioac- tive emanations, alpha, beta, and the nucleons, are ionizing radiations, as are gamma and X, and the effects of all are conveniently discussed together. Diagnosis by X Rays The absorption of electromagnetic radiation increases with increasing density of the absorbent. Differentiation of diseased tissue from normal is based on this fact. The higher the speed of the electrons which impinge on the target metal, the harder the X rays so produced. Machines avail- able today produce X rays from electrons which have been accelerated by thousands to millions of volts. In general, the greater the voltage, the greater the energy of the X-ray photons, and the greater their penetrating power. For example, at 40,000 v (i.e., 40 kilovolt potential (kvp), in radiation terminology) almost any tissue will stop some of the X radiation and cast a shadow on the fluorescent screen or photographic plate behind it. At 80 to 100 kvp, commonly used in medical diagnosis, the radiograph displays shadows which differentiate fat and other soft tissues from air space and from bone. Whenever it is possible to insert molecules containing heavy metal atoms into a region of interest, differentiation of tissues in the region is enhanced (Figure 4-10). Thus barium sulfate solution is commonly administered as an enema so that the lower part of the intestines may be examined (by X radiation). Iodine in a variety of compounds is also widely used to increase differentiation. For instance, in iodophthalien it is preferentially taken up by the liver and stored in the gall bladder; thus gallstones, if present, are easily seen. Similarly, the kidneys, uterus, blood vessels, and even the heart can be made visible to X-radiography (see Figure 4-10 (b), for example). Location of broken bones, of swallowed pins, of stomach ulcers and of tumors is routine. The use of X rays for diagnosis introduces the serious question of the ex- tent of the damage done by the rays absorbed. A complete fluoroscopic gastrointestinal examination with barium sulfate can be done by a competent physician with the dose to the region irradiated not exceeding 20 rads (the MICROSCOPY 95 impinging x-rays 80 kvp 1.32 1.35 + + Co 20 5.0 0.99 Four targets or absorbers 1.54 transmitted x- rays photographic plates Figure 4- 10a. Absorption of X Rays by Atoms. Energy of the incoming wave is trans- ferred to the electron cloud. Absorption is proportional to electron density, electrons per cubic A (bold numbers inside). Number of electrons (i.e., atomic number - valence) and atomic weight are given, as is atomic radius (at 7 o'clock). Note shift of both ampli- tude (number of photons per sec) and frequency (energy per photon). unit is defined later — only relative numbers are of interest now), although electronic intensification of the image now permits one to reduce this dose by a factor of ten. Although immediately measurable damage appears only if the dose is hundreds of times higher, more subtle effects, such as malignant growths, may show up years or even generations later if the greatest caution is not exercised. The effects of absorbed radiation dose can be cumulative. These questions are considered in more detail under "Therapy" in Chap- ter 9. MICROSCOPY A microscope is a device which throws a large image of a small object on the retina of the eye. It does this by passing definitive light through a sys- tem of lenses. A few useful notes are now given on the two most common types. All the necessary details are set out in a very useful, practical manner in the little book by Martin and Johnson entitled: "Practical Microscopy," 8 and in literature happily supplied by the optical companies. 96 ELECTROMAGNETIC RADIATIONS AND MATTER Figure 4-10b(i). Absorption of X Rays by Tissues. Abdomen with Barium Sulfate in the Colon. Note the differences in absorption of X rays by skeleton (vertebrae, sacrum, ribs, etc.), soft tissue (bottom edge of kidney, psoas muscle, liver), and gas pockets in stomach and colon. Low contrast film. Optical Microscope The small object to be viewed is illuminated either from above or below. In the former case reflected light, and in the latter case transmitted light, is allowed to pass through a convex objective lens of short focal length. In passing through the objective, the rays (visible region) are sharply bent, so MICROSCOPY 97 that a bright, but small image of the object exists within a few centimeters of the objective. About 10 cm away from the objective, and in line with the object, is the "eye-piece," or condenser, another convex lens with very short focal length, which throws an image of the objective's image on the retina if held about 2 cm away. Figure 4-10b(ii). Absorption of X Rays by Tissues (Continued). Ab- domen with Iodine Metabolized into the Kidneys. Note the difference be- tween the normal calyces of the kidney (white "horse," upper left) and the defective one (upper right). High contrast film. (Courtesy of A. F. Crook, Ontario Cancer Foundation.) Magnifications up to more than lOOOx are possible with the best instru- ments. The preparation of the lenses is the critical thing, for it is difficult and costly to grind a large lens which will not be astigmatic. If the lenses are perfect, the limit of resolution (the smallest distance by which two ob- jects can be separated and still be differentiated) is determined only by the wavelength of the light and the size of the aperture which admits the light. 98 ELECTROMAGNETIC RADIATIONS AND MATTER For white light, with an average wavelength about 5000 A and a numerical aperture of unity, the resolving power is 10,000 A, or 10" 4 cm, or 1 n. One can use monochromatic blue light to improve this somewhat; and the re- search use of ultraviolet (A = 2537 A from a mercury arc, for example) with fluorescent screens, is an attempt to push the resolution down to 0.1 ;u. In common practice, however, "good" microscopes used in schools and routine examination have a resolving power 5 to 20 /j.. The binocular microscope uses two microscopes in parallel, one for each eye. From this double input, one obtains depth perception. Phase-contrast and interference features have been superimposed on the simple microscope, broadening its versatility by improving the contrast be- tween different parts of the object under study. Contrast occurs in the normal microscope because of differences in density. In phase and interfer- ence microscopes, used when the density is about the same throughout (soft tissue is ~90 per cent water), advantage is taken of the facts that the speed of light through materials, which determines their refractive index, and the amount to which the plane of polarized light can be rotated, often differ if the molecular composition of the materials is different, even though their density is the same. To take advantage of these facts, two methods are available. Both present a highly contrasted image to the eye, one in inten- sity, one in color. The principles are really quite straightforward. The reader is referred to the trade literature for operating detail. Both are extensions of the normal bright-field transmission microscope; only the extensions will be noted here. In the phase microscope, an annular diaphragm is inserted in front of the con- denser lens and therefore before the light falls on the specimen, together with a phase plate composed of a thinly evaporated ring of dielectric on a background of thinly evaporated metal. Thus light passes at different speeds through different parts of the object to be viewed, and the emerging light waves are out of phase. At one point of emergence from the object the phase difference will be such that the waves cancel each other; at another they reinforce each other. The phase plate "fixes" these differences by retard- ing those which pass through the dielectric, and absorbing some of those which pass through the metal. Thus identification and analysis of the struc- ture of (unstained) living cells and tissues, the components of which are so similar in density that discrimination is impossible with the light microscope without killing and staining, is made possible. This instrument, invented by Zernicki in the Netherlands in 1932, is now an indispensible tool in clinical analyses — in bacteriological, histological, and, in particular, pathological studies of tumors and cancerous tissues. Note the contrast in Figure 4-11. The interference microscope is a polarizing microscope, adapted so that part of the light passes through the object and part around it, the two then being MICROSCOPY 99 Figure 4-11. Partially Crystalline Otoconiae (stones) of the Utricular Macula (bone) of the Organ of Balance in the Middle Ear: Sectioned, and in Negative Phase Contrast. Magnification 60 x . In addition to the sizes and shapes of the stones, note their darker center (glycoprotein) and the bright lamellar periph- ery (calcium carbonate). (Photograph courtesy of L F. Belanger, University of Ottawa Medical Faculty, and of J. Cytology and Cellular Comp.) . recombined to interfere constructively or destructively (as in the case of phase, above), and to present to the eye enhanced differences in density or color. Before the light passes through the specimen it is plane-polarized by passing through a crystal in which the light in all but one plane is absorbed. The emerging, polarized light is split into two beams whose polarized planes are rotated at right angles to each other after one has passed through a sec- ond crystal (birefringent). One beam then passes through the specimen, and the other around it. The one which passes through is rotated, absorbed, and retarded in different places to an amount depending upon the arrange- ments of the molecules ( — the term is "different optical paths"). The dis- torted light is then recombined with that by-passed, and their interference presents the image in different colors to the eye. If monochromatic light is used, the image appears in the form of differences in intensity; if white light is used, the image appears in the form of differences in color. Although it is not as sensitive as the phase microscope to differences in structure, the inter- ference microscope affords a wider field of view, can show subtle differences as shades in color, and has permitted (optical) determination of the amount of a particular absorbing material in the field of view. Since its inception, in 100 ELECTROMAGNETIC RADIATIONS AND MATTER the early 1950's, it has been used for quantitative studies of proteins in living muscle, growth rates of cells and parts of cells, and similar problems on living tissue which can be studied only with a nondestructive tool. Electron Microscope This development of the last twenty years has added a new dimension to the depth to which tissues can be viewed. After fixing and staining (e.g., permanganate, phosphotungstic acid, osmium oxide), a very thin cut to be examined is placed in high vacuum, and bombarded from below by electrons (from a hot filament) which have been accelerated through a small aperture. Some of the electrons hit dense parts of the object and are scattered and absorbed — the principle is the same as for X rays (Figure 4-10 (a)); others pass on through less dense parts and fall upon a fluorescent screen or photographic plate. Proper alignment permits, in today's machines, ampli- fications of 500 to 100,000 x , with resolution of a few angstroms. One instrument, which can be considered typical for biological work,** gives a 15- A resolving power; 600 to 120,000x magnification; and accelera- tion voltages of 100, 75, or 50 kv, to give electron beams of equivalent wave- lengths of 0.037, 0.043, and 0.054 A. The "lenses" are electric voltages be- tween charged plates. The amplification can be increased to over 1 ,000,000 x by photographing the screen, and enlarging the photograph. Others The ultraviolet microscope and fluorescence microscope have been used and improved since the early 1900's. They have some specialized uses in biological research. X-ray microscopy is useful when the sections to be studied are opaque to visible and ultraviolet light. For example, in histo- logical sections on bone, soft (~5 kvp) X rays are absorbed by the mineral component, passed by the organic component. Reflection microscopy, especially the slowly developing infrared reflection techniques, may find limited use in future studies on biological material. PROBLEMS 4-1 : Draw the shapes of sigma and pi bonds. 4-2: If all 10 28 atoms in a human being were lined up side by side, how long would be the line, in miles? 4-3: It costs an input of about 105 kcal/mole to pull the first hydrogen off a water molecule. "Light" of what wavelength will blast it off? (calculate it). 4-4: Sketch intensity vs distance for the penetration of electromagnetic radiation into tissue, presuming concentration of absorbent of 0. 1 moles/1 and molecular extinction coefficients of 0.1, 1.0, and 10.0. **The limitations should be realized: the tissue sample is dead, dry, and thin while being viewed in the electron microscope. REFERENCES 101 REFERENCES 1. Rushton, W. A. H., "Visual Pigments in Man and Animals and Their Relation to Seeing," Prog, in Bwphys., 9, 239 (1959). 2. Stacy, R. W., et al., "Essentials of Biological and Medical Physics," McGraw- Hill Book Co., Inc., New York, N. Y., 1955, p. 262. 3. Brindley, G. A., "Human Color Vision," in Prog, in Bwphys., 8,49 (1959). 4. Evans, R. M., "An Introduction to Color," John Wiley & Sons, Inc., New York, N.Y., 1948. 5. Ruch, T. C. and Fulton, J. F., Eds., "Medical Physiology and Biophysics," W. B. Saunders Co., Philadelphia, Pa., 1960. 6. The Physics Staff, University of Pittsburgh, "Atomic Physics," 2nd ed., John Wiley & Sons, Inc., New York, N. Y., 1944. 7. Land, E. H., "Color Vision and the Natural Image," Proc. Nat. Acad. Set., 45, 115 (1959); Sri. Amer., 200,84 (1959). 8. Martin, L. C. and Johnson, B. K., "Practical Microscopy," 3rd ed., Blackie & Son, Ltd., London, 1958. 9. Shamos, M. H. and Murphy, G. M., "Recent Advances in Science," New York Univ. Press and Interscience Pubis., Inc., New York, N. Y., 1956. 10. Gamov, G., "The Atom and its Nucleus," Prentice Hall, Inc., New York, N. Y., 1961. 11. Richards, O. W., "Pioneer Phase and Interference Microscopes," N. T. State J. Med., 61,430 (1961). 12. Bennett, A. H., Jupnik, H., Osterberg, H., and Richards, O. W., "Phase Micro- scopy, "John Wiley & Sons, Inc., New York, N. Y., 1951. 13. Hale, A. J., "The Interference Microscope in Biological Research," Williams & WilkinsCo., Baltimore, Md., 1958. 14. Pritchard, R. M., "Stabilized Images on the Retina," Sri. Amer., 204, 72 (1961). 15. Hall, C. E., "Introduction to Electron Microscopy," McGraw-Hill Book Co., New York, N. Y., 1953. 16. Szent-Gyorgyi, A., "Introduction to a Sub-Molecular Biology," Academic Press, Inc., New York, N. Y., 1960. CHAPTER 5 Radioactivity; Biological Tracers Our sensory data, even with complex equipment, consists of flashes of light, of the rates of discharge of an electroscope, of audible clicks or totals from an automatic counter, of tracks of liquid particles in a small chamber, of the deposit of silver grains on a photographic film, of heat evolved, of certain color changes. From these simple observations scientists have al- ready created a complex and exciting description of particles far too small to be seen directly (Miner, Shackelton, and Watson. 3 ) INTRODUCTION Properties of the Emanations In 1897, we entered the golden age of nuclear physics. It was then that Becquerel, experimenting with pitchblende, which is fluorescent, acciden- tally discovered a new and exciting emanation from the material. The emanation was rather penetrating (through his desk-top), and darkened some photographic plates kept in a drawer below. The Curies extracted the element which gave rise to the activity — radium — and called the emanation "radium-activity/' from whdch we derive the modern name, radioactivity. Chapter 4 has already described how three components were isolated from one another by Rutherford, and named alpha, beta, and gamma rays. The relevant properties of each as determined from scattering experiments, etc., are gathered in Table 5- 1 . It is the penetrating properties of these radiations with which we are now primarily concerned. However, to understand penetrating properties of radiations from any radioactive source, we must first understand their origin (i.e., in the atomic nucleus) and their absorption, as well as the methods used to detect them, to identify them, and to measure their energy. 102 INTRODUCTION 103 TABLE 5-1. Physica Properties of Nuclear Particles Emanation Symbols Rest Mass (grams) Charge Nature v/c Source Alpha °.c§> 7.2 X 1(T 24 + 2 bare helium ions 0.001 to 0.1 unstable nuclei (u.n.) Beta ft • 9 X 1(T 28 ±1 electrons 0.1 to 0.9 u.n.; accel erators (ace.) Gamma 'Y f /w^ (not appli- cable) electromag- netic radia- tion 1.0 u.n. Proton P, o 1.8 x 1CT 24 + 1 bare hydrogen nucleus 0.01 to 0.2 u.n.; ace. Neutron n, O 1.8 x 10" 24 same, neutra- lized "fast," and "thermal" (slow) u.n. ; fission Deuteron d,OQ 3.6 x 10" 24 + 1 n + p u.n.; ace. Note: Charge is the number of units of 4.8 x 10~ electrostatic units (esu) of charge. Velocity is ;• ; and velocity of light, c, is 3 x 10 cm/sec. (The ratio v/c for protons in cosmic rays and in the Van Allen radiation belt above the earth's surface approaches 0.8 (or larger than that produced arti- ficially). The Nucleus As has already been seen (in Chapter 4), the size of the nucleus has been measured by means of scattering experiments and found to be 10" 12 cm, or about 10" 4 A. The nucleus carries all the positive charge and most of the weight of the atom. It is thus very dense.* The positive charge carried by such a dense particle is almost unimaginably high — for radium it is 88 times that of a hydrogen ion! — and it is therefore not surprising that the binding forces, whatever they may be, must be orders of magnitude stronger than those of the electron cloud of the atom; and even a minor reorganization or splitting must involve a mass-energy change. It is instructive for one to com- pare again (Table 4-1) the energy of visible light, ~ 1 electron volt/photon, with that of gamma rays, 1,000,000 electron volts/photon, which arise from nuclear rearrangements. ♦This can be illustrated by a calculation of the weight of a 1-cm cube of nuclei of nickel (Ni) atoms, for instance, it being presumed that the nuclei are close-packed, side by side. Sim e the diameter of each is ~10~ 12 cm, 10 12 nuclei side by side would be 1 cm long; and the cube would contain 10 36 nuclei. Each weighs 65 times as much as hydrogen, or 65 x : ! x 10~ 24 g. The weight of the 1-cc cube, then, is about 10 14 g. or approximately 100,000,000 tons! 104 RADIOACTIVITY; BIOLOGICAL TRACERS It is exactly this huge energy carried by the alpha or beta particle, or by the gamma photon (packet of light), which is responsible for its detection as well as the damage it does to the molecules of a tissue. Thus, as the emanation is absorbed by molecules of a gas, say, its energy is gradually dis- sipated by being passed over to the gas molecules; these in turn are at least excited, and many are ionized, a process which requires only a few electron- volts per molecule. The number of protons in the nucleus determines its positive charge, and hence its position in the periodic table. Protons plus neutrons determine the weight of the nucleus. There may be several numbers of neutrons which can combine with a given number of protons, and thus there can be several weights of the same element. These different weights of the same elements are called "isotopes" (iso topos — in the same place in the periodic table). Some isotopes are quite stable, some spontaneously disintegrate. For ex- ample, carbon with 6 protons in the nucleus, may have 4 to 9 neutrons in the nucleus, to form C 6 !0 , C 6 n , C 6 12 , C 6 13 , C 6 14 , C 6 15 . The isotope C 6 12 is the basic carbon in nature, and is quite stable, whereas C 6 H is a long-lived beta emitter also found in nature. The others are short-lived, and are made artificially by bombardment of nuclei by the "bullets" listed in Table 5-1 . IONIZATION AND DETECTION Ionization Positive Ions The mechanism by which ionization takes place in the path of each emanation is important to considerations of penetration. Each mechanism is different from the others because the emanations differ so remarkably. The alpha (He 2 4 ) ++ , the broton (H, 1 ) + , and the deuteron (H, 2 ) + are very small, but dense; the alpha carries the positive charge of two protons. Upon collision with electron clouds of a target material, it easily ionizes the atoms by pulling the negative electrons after it, wasting a small fraction of its kinetic energy in the process. Since it is likely to tear at least one electron out of every atom through which it passes, it leaves a very dense wake of ionization (Figure 5-1). The alpha of radium (Ra) has a kinetic energy of 4.8 x 10 6 electron volts, which means that it leaves a wake of about 140,000 ionized atoms. Thus in air it can travel a few inches; in metal it can pene- trate only about 0.0001 cm; and in fact can be stopped by a piece of paper! Although its path is short, the radiation damage or ionization along the path is intense. Actually, theory shows that the energy transferred per centimeter of path (called the linear energy transfer, LET) increases with increasing charge, q, and decreases with increasing velocity, v, as follows: LET cc q 2/ v 2 IONIZATION AND DETECTION 105 absorption by the nucleus decay of unstable nucleus photoelectric absorption Depth in Tissue Figure 5-1. Schematic Representation of Tracks of a Neutron (n), and of Alpha (a), Beta (/?) and Gamma (7) Rays in Tissue. Note that the density of ionization increases as energy is lost from the impinging ray. The alpha trail of ionization is dense, the beta trail is spotty, and the gamma and neutron trails are composed of spurs. From these considerations and the properties given in Table 5-1, one can understand that the differences among alphas, protons, and deuterons art- more those of degree than of kind. All are positive, heavy particles with high LET. Electrons The beta is a very small particle — a very fast electron. Its charge is either negative (as is the beta from P 32 ) or positive (as is the beta from P 30 ), al- though the negative is the more common among biologically interesting iso- topes. Because it is of light weight, with a mass only somewhat greater (rela- tively) than the mass of the electrons in the atom, a collision can result in energy transfer and a change in direction, similar to billiard balls in play. **] As a result the path traversed by the beta will be governed more or less by chance collision. It will have many changes of direction. Along the straight portions of the path, when the beta flies through the electron cloud of the 106 RADIOACTIVITY; BIOLOGICAL TRACERS atom, excitation can occur, accompanied by loss of speed, and hence loss of energy ( = 1/2 mv 2 ). The definite changes in direction result from collisions, and the energy and momentum transferred can cause ejection of the electron hit; i.e., ionization. When collisions are ''favorable," the trail of ionization, although sparse, may penetrate quite deeply into a tissue; but when unfavor- able, it will be very intense but very short. (See Figure 5-1.) Very fast electrons may penetrate the atom as far as the nucleus, and by interaction with the field of force about the nucleus lose energy, with the production of secondary X rays. These X rays are called bremstrahlung. Hence a hard beta source may produce a secondary radiation which is much more penetrating than the impinging betas. The initial velocities of betas from a source vary widely because the small neutral particle, the neutrino, is ejected from the nucleus along with the beta, and the energy of the disintegration is split between the two. It is the maximum energy of the betas which is usually given in tables of data. As a result of the energy distribution and the deflection of betas as they enter and lose energy in a target, the betas follow a nearly exponential law of penetration. Gamma Rays The gamma ray is electromagnetic radiation, like light, but of very short wavelength. Since it carries no charge, it is captured only by direct collision or wave-like interaction with a target: with the nucleus or the electrons of an atom. Some energy is transferred to the target electron and the gamma con- tinues on, usually in a modified direction, at reduced energy (e, = hv t ) where v, is frequency and h is Planck's constant. The recoil electrons are relatively slow, and are therefore good ionizers (see Figure 5-1). Just as in the case of X rays (see Figure 4-10 (a)), then, absorption of gammas arises from essentially two processes: (1) "pair production": strong interaction with the nucleus and production of a pair of electrons (e + and e~) — impor- tant in water only if energy of the y is above 3 Mev; and (2) "Compton ab- sorption": ejection of an electron at an angle, some of the energy of the gamma being lost, and the remainder ("Compton scattering") proceeding, usually in a changed direction, and always at lower frequency. The process (2) is repeated until, finally, the energy left from a succession of collisions is absorbed by the electron clouds of atoms (photoelectric absorption) and is ultimately dissipated as heat. At energies below about 0.2 Mev, elastic (Rayleigh) scattering reduces the absorption and increases the range of the gamma in water and soft tissues. Neutrons The neutron is as heavy as the proton, but carries no charge. Energy is lost only by collision with light nuclei, and hence it can penetrate as deeply IONIZATION AND DETECTION 107 as X rays. The nuclei set in motion by bombardment by fast neutrons (0.1 to 15 Mev) have a high LET and leave a wake of intense ionization. Slow (thermal) neutrons are ultimately captured by nuclei; the product is nor- mally unstable, and, for light atoms, usually emits a gamma ray. A good billiard player will attest that maximum energy transfer can take place be- tween two neutral "particles" if they have the same weight. Therefore, neutrons are slowed down, or "moderated" best by materials containing much hydrogen — water, paraffin, etc. Thus, penetration into these ma- terials is slight, or in other words, the absorption coefficient is high. Neutrons are by-products of nuclear fission, or of proton- or deuteron- bombardment of light nuclei; they have a half-life of the order of 20 min, can be quite destructive of living tissue, and are difficult to detect. The damage is caused by charged nuclei set in motion by the impact of the neutron, or from artificial radioactivity induced by capture of the neutron by the nucleus (Figure 5-1). Defection Ionizing radiation is detected by any one of four basic methods: (1) Exposure of a Photographic Plate: i.e., reduction of silver halides to silver along the path of the photon or particle. If the plate is placed in contact with a section of tissue containing a radioactive tracer, the plate will be exposed where the activity is. This method of mapping is now known as "autoradi- ography." Microradiography is another interesting technique in which a large shadow of a small object is allowed to fall on a photographic plate. This technique has been used for years with X rays as the source, and recently it has been demonstrated to be feasible and useful using alpha rays as the source. Figure 5-2 shows a micro X radiograph of a section of bone — the mineral content is clearly visible — and an alpha radiograph taken of the organic part after the mineral had been removed. (2) Ionization of a Gas Contained Between Two Electrodes: As the photon or particle passes through the gas it leaves a wake of ion-pairs. If there is no potential difference between the electrodes, the ions will recombine. If a potential difference is applied (Figure 5-3 (a)), each ion will migrate toward an oppositely charged plate. Those which reach the plate before recombin- ing will be discharged and produce pulses of current in the external circuit. The higher the potential difference, the less is the recombination. Thus at an electric field strength of about 10^/cm almost all the ions produced are "collected" at the electrodes. This is called the "saturation" condition, and most ionization chamber systems operate in this region. If the electric field strength is increased still further, the primary ions are given sufficient energy to produce secondary ionization of the s*as molecules, resulting in a multiplication of the original ionization. This is known as an 108 RADIOACTIVITY; BIOLOGICAL TRACERS i *• ** M- f (a) (b) Figure 5-2. Microradiography. (a) X-ray microradiograph (5 kvp) of a sec- tion of natural compact bone (tibia). Note the large (black) osteonic canals and the (light) mineralized regions. Magnification 500 x. (b) Alpharadio- graph (source 2 mc/cm 2 of Po 210 ) of a section of the same bone demineralized. Note regions of low-density (dark) and high-density (light) organic material. Magnification 150x. Together (a) and (b) demonstrate directly the regions of growth of young bone around the osteonic canal: tissue mostly organic, only lightly mineralized. (c) Alpha- radiograph showing filiform papillae (top) of the human tongue. Note the dense fibrous collagen core of the papillae and of the supporting base of the epithelium, and observe the low- density (black) mucous-forming cells at the bottom of the picture. (Courtesy of L. F. Belanger, University of Ottawa Medical Faculty, and D. H. Copp, University of British Columbia Medical School.) IONIZATION AND DETECTION 109 KZ3> ©-v © t (a) 0> ' phot 00 photons to • voltmeter Geiger threshold saturation (s) region Figure 5-3. Ionization Chamber: (a) sche- matic design — wire anode, A, and cylindrical cathode, K, filled with gas (e.g., Argon); (b) charge collected at A per pulse at different voltages. (See text.) "avalanche" process. The multiplication factor may be as high as 10 3 or 10 4 , so that the current pulse which is produced may be 10 3 or 10 4 times larger than the "saturation" pulse (Figure 5-3 (b)). Since the pulse size is propor- tional to the energy lost by the original photon or particle, a chamber oper- ated in this fashion is known as a "proportional" counter. At higher voltages, the multiplication factor for large pulses tends to be smaller than that for small pulses, and all pulses are multiplied to a constant size regardless of initial strength. The voltage at which this gaseous dis- charge starts to occur is known as the "Geiger threshold." Figure 5-3 shows an ion-chamber design from which the proportional counter and the Geiger counter may be developed. Figure 5-4 is a photo- graph of a typical unit. (3) Fluorescence Induced in Solids and Liquids: The light emitted after the ab- sorption of ionizing radiation by a fluorescent solid is reflected on to the 110 RADIOACTIVITY; BIOLOGICAL TRACERS Figure 5-4. Measurement of Radioactivity. Left: thin- walled Geiger tube for alpha- and gamma-ray detection. Right: a typical survey instrument with protected, detach- able detector tube. Typical ranges: to 0.25 mr/hr; to 2.5 mr/hr. photocathode of a photomultiplier tube, causing the ejection of more elec- trons. These are multiplied in number by an internal secondary-emission system to produce a measurable current pulse for each scintillation. A typi- cal arrangement is shown in Figure 5-5. Certain organic liquids also fluoresce, and very sensitive liquid counters have recently been developed. Each of the counters discussed in paragraphs (2) and (3) has specific uses, tor a radiation such as the l.Z-Mev gamma from Co 60 , for instance, the scintillation counter can have efficiencies as high as 15 per cent as compared to 1 per cent for a Geiger counter. Therefore, for medical tracer applica- tions of gamma in which the intensity is low, a scintillation counter would be preferred over a Geiger counter. However, if dosage is high, as it may be in radiation therapy, the extra sensitivity is not important. Figure 5-6 photons scintillation phosphor photocathode JT \ *fl X optical coupling to photocathode pre-amplifier T photomultiplier assembly pulse output to ammeter Figure 5-5. Schematic Drawing of Scintillation Counter. (See text.) IONIZATION AND DETECTION 111 shows a lead-collimated scintillation counter, useful, for instance, for ex- ploring the thyroid after radioactive iodine has been administered. External exploration of the organ for determination of size is known as scintography. Mechanical devices have been designed which control the exploration and print a map of the intensity of radiation from that area of the throat. (4) Chemical Reactions Induced in Aqueous Solutions: Water is broken up into H and OH, and these very reactive products undergo reactions with solutes to produce new chemicals. Oxidations or reductions, molecular rearrange- ments, polymerization of plastics, and corrosion of metals have all been used as detectors. Important quantitative aspects of absorption of ionizing radia- tion by aqueous tissues are developed in Chapter 9. Figure 5-6. Collimated Scintillation Counter. Top: disassembled to show photo- multiplier assembly. Bottom: assembled. With collimator (left) attached, the instrument can be used for scintography — tor detailed external mapping of the human body, above the liver for example, following internal administration of the appropriate radioactively-labeled chemicals. (Photographs courtesy of Burndepts Ltd., Erith, England.) 112 RADIOACTIVITY; BIOLOGICAL TRACERS DISINTEGRATION (DECAY) Rate of Decay; Half-Life We have no control over the disintegration of individual nuclei: if a nucleus is unstable, it will decay at a time which is completely unpre- dictable. However it is possible to describe and predict the fraction of a large number of unstable nuclei which will decay within a given period; that is, AN/ At is easily measured. In fact the number of nuclei (J\ ) which do decay within a given time is proportional to the number present which are able to decay. Thus AN/ At oc N or the instantaneous rate dN/dt ex TV- Insertion of the proportionately constant — A (called the "decay con- stant") gives -dN/dt = \N After the summation in the fashion indicated in Chapter 1, N = N e~ Xl where N is the number present at any arbitrarily chosen zero of time. This expression says simply that the number, N, of nuclei which are present at any time, t, is only a fraction of the number, N , which were present at zero time — the fraction being e~ Xt . Now, it is useful and instruc- tive to expand the fraction into the series it is, and write e~ Xi = l + + + l 2x1 3x2x1 A 2 / 2 A 3 ; 3 1 _ \ t + + The value of A differs for different radioactive elements. For Sr 90 the value has been measured to be 0.028 yr '. After five years, for example, .-* = 1 - (0.028 x 5) + (0 -° 28 X 5)2 - (°-° 28 X 5)3 + ^0.87 2 6 Therefore N = 0.87 N , or the fraction of N () left after five years is 87 per cent. Calculations for 10, 15, 25, 50 yr would span a time at which N is just 50 per cent of N . For Sr 90 this time is about 25 yr, and it is called the "half- life"— the time it takes active material to decay to 50 per cent of the original concentration, N . Half-life, r - In 2/A = 0.693/A. DISINTEGRATION (DECAY) 113 If two radioactive elements have been concentrated chemically to the same value of JV , the one with the shorter half-life decays faster, has greater "ac- tivity" (higher dN/dt) at time zero, or delivers more emanations per second to the tissue being irradiated. The unit of activity is the curie (c), that amount of radioactive material which provides 37 billion (i.e., 3.7 x 10 10 ) disintegrations each second. Thus 1 g of pure Ra 226 which gives off 4.8 Mev (average of 3) alphas, has a total activity of about 1 c. Sr 38 90 , which gives off only a 0.6 Mev beta, decays faster and is less dense than radium; 1 g of pure Sr 90 provides an activity of 147 c. However, since a pure radioactive substance is always contaminated by its daughter products, the activity per unit weight is deter- mined by the concentration of radioactive substance. Clearly 1 millicurie (mc) per gram might be usable in a medical application, whereas 1 mc per ton should be quite impractical. Specific activity is defined as the number of mc/g. Figure 5-7 shows decay schemes for several radioactive isotopes of use as tracers in diagnosis and as irradiation sources in therapy. Energy Distribution of the Emitted Rays Before we come to the question of depth of penetration and extent of ionization of the rays from a radioactive source, we must consider two more factors: the energy distribution (spectrum) of the rays from any given pure source, and the number and kind of products of disintegration. Both alphas and gammas are the result of a particular kind of fracture or rearrangement of unstable nuclei. One could consider the nuclei to be in excited states (think of an undulating water droplet), existing as such from the time of their formation (in the sun?) millions of years ago, and disinte- grating at a rate which we can measure but which we are not able to vary. Thus, although half the atoms of Ra 226 in a sample will undergo alpha decay in a definite and reproducible time, we do not understand why the disinte- gration of Ra 226 is always by loss of one alpha particle, a package of 2 pro- tons + 2 neutrons; and the most striking fact of all is that these alphas al- ways come off with the same velocity. The similarity of this quantum-like be- havior to the quantized absorption and radiation of light by the electron cloud of the atom, suggested to theoreticians that a Bohr-like model for the nucleus should be useful. Development of theory has proceeded along these lines, and has led at least to a quantitative description, if not an answer to the question "why?". The alpha or the gamma radiation from a single elemental source occurs at discrete energies — alphas of single velocity, gammas of single frequency (Figure 5-8). However, with the beta is expelled a neutrino, a tiny neutral particle of variable velocity; and therefore the beta radiation from a single elemental source has a distribution of energies — low, corresponding to a 114 No, 24 sodium-24 I. 39mev Mg 24 12 Co' 60 J 27~ cobalt-60 Ni 60_ '28 15 hrs K 42 (18%) (82%) 19 potossium-42 y 1.37 Co 42 20 /92.0 ,yl.53 12.5 hrs £3.6 y 2.76 . 59 (46%) (54%) Fe„ 1 1 4 5 days /30.46 yl 29 ^ rl.10 (43%) ^ (57%) Sr y 90_ 38 90 28yrs 32 0.54 15" 0.306 39 5.25yrs 7 I . I7mev Zr yl. 33mev 9o_ 40 phosphorus-32 j8 2.26 .32 14 2 days 0l.7lmev carbon-14 .14 6~ 4 1 C N H He — 5568yrs 0O.I55mev 3 i 12.26 y rs 00.0 1 8 me v tritium lod ine-131 (3%) (9%) (87%) 00.33 /30.6lmev (1%) y0.64 < ' 116 RADIOACTIVITY; BIOLOGICAL TRACERS "Daughter Products": Products of Radioactive Decay Any radioactive source, before being administered for any good reason, should be examined for the radioactivity and the chemical properties of its disintegration products. Refer to Figure 5-7. Thus, loss of an alpha means a shift downward of two places in the periodic table (e.g., radium — * radon); and loss of a beta means a shift upward of one place (e.g., iodine I 131 — > xenon 131 ), because these charged particles (electrons) are ejected from the nucleus, and it is the charge on the nucleus which determines the position of the element in the periodic table. Loss of a gamma results in no shift, but is simply a loss of energy during a nuclear reorganization. The daughter products often are unstable and give rise to further disin- tegration. Several steps may occur before a nucleus reaches a stable state. One of the simplest disintegrations is that of Na 24 , used in determining the role of sodium in a cell-membrane transfer. The scheme was seen depicted in Figure 5-7. The isotope Na 24 gives off a 1.39 mev beta to become excited Mg 24 (magnesium); but this in turn emits two hard gammas before reaching a stable product. The Ra 88 226 nucleus and its daughters produce a total of eight alphas, eight betas, and eight gammas before reaching the stable isotope Pb 82 206 (lead). Three isotopes of polonium (Po 84 ), two of bismuth (Bi 83 ), one of thallium (Tl 81 ) and three of lead take part in the disintegration scheme! Note that all the daughters except radon are solid elements. Although all have short half- lives, they take a fleeting part in the chemistry of the molecules in the vicin- ity in which they are formed. By interesting contrast with radium (Ra), Po 210 is a pure alpha emitter, and P 32 (phosphorus) is a pure beta emitter. I 131 and radio-gold, Au 198 , emit both betas and gammas. Decay schemes for some of these are given in Figure 5-7. PENETRATION OF THE RAYS INTO TISSUE It is preferable to discuss the penetration of the pure emanations and then to infer the effects of the mixed emission of mother and daughters. The alpha (and also the proton and deuteron) penetrates in a straight line until it is stopped (Figure 5-1), provided of course that it does not "hit" a nucleus (Figure 4-2). Because both the a and the target nucleus are so small, the likelihood of collision is small. Since alphas are monoenergetic from a source, all penetrate to about the same depth. Both beta-scattering and gamma-absorption are governed more or less by chance collisions in which energy is lost from the penetrating radiation. The intensity decays more or less exponentially with distance in each case (Figure 5-9). This is only true to a first approximation, however, because of scattering which is related to the geometry of the system. PENETRATION OF THE RAYS INTO TISSUE 117 2 2 c o c "*- !c O o o □ alphas (0 001 mm) (7 mm (I mm) gam mas or neu t rons (several feet) Depth in tissue Figure 5-9. Penetration of 1 Mev Alphas, Betas, Gammas, and Neutrons into tissue In simplest cases, the curve for gammas is truly exponential; that for betas has less curvature and reaches a maximum value, which is the depth of penetration of the fastest betas. Note that the area under each curve corre- sponds to 100 per cent of the impinging rays hitting the target. The depth of penetration is radically different for the three cases. TABLE 5-3. Ranges of Various Types of Radiation in Soft Tissue.* Range of Radiation in Material Usual Ionizing of Low Atomic Number Radiation Energy Range (mev) Particles in Tissue Actual Range in Air, NTP (cm) Equiv. Range in Watery Tissue (cm) Beta rays 0.015 to 5 electrons 0.1 to 1000 0.0001 to 1.0 Electron beams 2 to 10 electrons 300 to 8000 0.4 to 10 X rays and 0.01 electrons 230 0.23 Gamma 0.10 electrons 25,000 4.0 rays** 1.0 electrons 23,000 10 L 10 electrons 34,000 34 Fast neutrons** 0.1 to 10 protons many meters ~10 Slow less than 0.6-mev 0.8 (protons) 0.001 neutrons*** lOOev protons + 2.2-mev 400 (electrons) 0.5 gammas Proton beamsj 5 to 400 protons 30 to 80,000 0.035 to 80 Alpha raysf 5 to 10 alphas 4 to 14 0.003 to 0.01 *Krom "Radiological Health] [andbook," National Bureau of Standards Wa hington, D. C, I960 **Range for absorption of half the incident radiation. ***From "Safe Handling of Radioisotopes," Health Physics Addendum," < G Vppleton and I' N Krishnamoorthy. Eds., International Atomic Energy Agency, Vienna, I960 +From G. J. Hine and G. L. Brownell. "Radiation Dosimetry," Academii Press, [n< 1956 and W Whaling, "The Energy Loss of Charged Particles in Mattel " Handbuch der Physik, XXXI\ 118 RADIOACTIVITY; BIOLOGICAL TRACERS In review, the nature and properties of the four main types of emanation have been considered. Positive ions and electrons lose kinetic energy by charge interaction with the electron cloud of atoms in the path: the greater the electron density the greater the absorption. Gamma rays lose energy to the electron cloud principally by pair production or Compton scattering. A neutron must hit a nucleus to lose energy. When it does, either the nucleus (charged) recoils through the medium and ionizes as a positive ion, or the neutron is absorbed by the nucleus, usually to form an unstable iso- tope which decays with the expulsion of beta or gamma, proton or neutrons. The data of Table 5-3 illustrate these important principles. Note par- ticularly the variation of the range in tissue for radiations of different type and energy. Protons are the ionizing particles in tissue which is under fast- neutron irradiation because hydrogen of water is the most plentiful target in the tissue. . . . This table should be thoroughly studied and understood. USES AS BIOLOGICAL TRACERS One of the simplest, and yet one of the most intriguing applications of the properties of radioactive substances has been in their use as tracers. The age of the earth, the authenticity of oil paintings, the courses of water and wind currents, have been probed simply by analyzing for the pertinent radioactive isotope in the proper place in the proper manner. Three uses as tracers concern us here: (1) as an aid in determining the steps and paths by which molecular reactions occur, whether simple hydra- tions of ions, or the more complex syntheses and degradations of large bio- chemicals; (2) in plotting the course of fluid flow, through the blood capil- laries, across cell walls, etc.; (3) in plotting the time and space distribution of biologically active chemicals. Examples of each are now given to illustrate the principles. The book by Kamen, s now a classic in the subject of tracers, is highly recommended for further study. Tracers of Molecular Reactions The first use of isotopic tracers on a biological problem was reported by Hevesy in 1923; this was a study of lead metabolism in plants. When heavy water (D^O) became available in Urey's laboratory after the discovery of deuterium there in 1932, many biochemical problems were attacked: hydro- genations and dehydrogenations, cholesterol synthesis from smaller frag- ments, conversion of phenylalanine to tyrosine, etc. Then, by 1942, am- monium sulfate containing N 7 15 , instead of the more common N 7 14 , became available, and compounded the possibilities for biochemical investigations. Thanks to the nitrogen tracer, the fate of amino acids in protein synthesis could be followed. Probably the most important of all these investigations, from the point of view of biology, was the demonstration that protein mole- USES AS BIOLOGICAL TRACERS 119 cules are in a dynamic equilibrium with their environment: they are not fixed end-items, but rather they are continually breaking apart here and there, accepting new amino acids and rejecting old. The same thing has now been found in lipid and carbohydrate metabolic reactions. Thus a dy- namic steady-state must now be considered well established in the biochem- istry of life, even at the molecular level, a fact which could be established only by this unique tool, the isotopic tracer. To be useful as a tracer, the only requirement is that the isotope be present in an amount different from that occurring in nature. If the isotope is radioactive, its presence is easily detected by the ionization caused by its disintegration product. If it is not radioactive (deuterium, H, 2 , and nitro- gen-15, N 7 '\ are examples), it can be detected by two methods: (1) In the highly evacuated mass spectrograph, the atom is ionized by bombardment by electrons, and then, after the ion has been accelerated in an electric field to a prechosen velocity, it is allowed to enter the space between the poles of a strong magnet. It is deflected there by the magnetic field, by an amount determined by the weight of the flying particle: the heavier the particle the less the deflection. (2) By neutron activation: In some cases — N 7 1S is an ex- ample — the nonradioactive isotopic tracer can be made active by bombard- ment with thermal neutrons, and then its quantity measured as the radio- activity of the product, N 7 16 in this case, a hard beta and gamma emitter with a half-life of only a few seconds. Tracers of Fluid Flow The classical method of determining the flow pattern in the circulation system is to inject nitrous oxide, N 2 0, at one point and then sample at vari- ous times and places after the injection. The isotopic dilution technique, described under (1) and (2) above, has been used to map blood flow in the brain, advantage being taken of the fact that no new chemical reactions are introduced into the system in the ma- terials injected. During the past five years, the radioactive isotope method has also been applied to the very difficult problem of measuring the rate of flow of blood through various parts of the brain, and although these experiments have not been done as yet on man, the work (mainly on cats) is interesting and in- structive, and illustrates the power of the method. The chemically inactive, freely diffusible gas, CF 3 I 131 , has (5 and y emanations well-suited to the de- tection techniques already described. For example, ~300 microcuries (fie) are administered, either by injection into the blood stream in about 10 cc of salt solution, or inhaled from a prepared air mixture. The blood can be shunted through a glass tube from one part of an artery to another, and the activity of the shunted blood determined with a counter attached to the glass. 120 RADIOACTIVITY; BIOLOGICAL TRACERS Alternatively, autoradiographic techniques on deep-frozen sections of sacrificed animals can give quantitative information on blood flow at dif- ferent depths in the tissue and at different times. For example, through both superficial and deep cerebral structures the flow rate is about 1.2 cc/min per g of tissue — in all but the white matter, through which the rate of flow may be as low as 0.2. In the spinal cord the flow rate in the gray matter is 0.63 cc/min per g; in the white matter it is 0.14. Under light anesthesia these values are reduced about 25 per cent. All these values are given in terms of flow through 1 g of tissue, because there is just no good way to de- termine the number and dimensions of the blood capillaries in these tissues. Studies on Metabolism: Time and Space Distribution of Biologically Active Chemicals For information subsequently to be used in therapy of one sort or another, tracer studies on metabolism are probably the most important. Every tissue or organ has a definite turnover rate of its molecular components. Every substance which enters through the gastrointestinal tract or through the lungs into the blood stream, or is introduced directly into the body fluids through hypodermic needles, has one or more locations to which it goes, and a definite time (on the average) it stays there before being rejected in favor of new material. In practice, radioactive atoms are introduced into the molecules which compose the material to be studied. Where this material goes, and how long it stays there, as well as in what form it is rejected, can all be answered by proper use of isotopic dilution or radioactive labeling technique. For example, studies have been made on the metabolism of proteins, such as the rate of protein synthesis and nitrogen (N 15 ) transfer; on the intermediary carbohydrate metabolism (C 14 and P 32 ); on the intermediary metabolism of lipids — the pathways of fatty-acid oxida- tion and synthesis (H 3 ); on healing of bone fractures; on iodine metabolism (I 131 ) in the liver and in the thyroid; on turnover rate and growth rate of normal** and diseased tissue (C 14 , H 2 , O 18 , Fe 59 , Au 198 ); on the metabolism and turnover in teeth (P 32 ); and on blood circulation in the brain (I 131 )- In more detail: the metabolism of nitrogen in the living system has been studied by the introduction of N'Mabeled glycine or other amino acids, ammonia, or nitrates, into food. Measurement of the N 15 — by either activa- tion or mass spectrometry (since N 15 is a stable isotope) — as it appears in the urine, as well as analysis of the molecules in which the nitrogen is con- tained, has shown that the cellular proteins and their constituent amino acids are in a state of ceaseless movement and renewal. The proteins and amino **Other isotopes now in use in metabolic studies include: Cr , Na , S , CI , K , Ca , Mn 54 , Zn 65 , Br 82 , Rb 86 , I 128 . USES AS BIOLOGICAL TRACERS 121 acids are continually being degraded, and being replaced by syntheses. That the rates of breakdown and resynthesis are the same is attested by the fact that the concentrations are maintained constant during life. About 60 per cent of N ,5 -containing protein has been shown to appear as glycine in the urine within 24 hr after a high-protein diet has been eaten; about 80 per cent appears within 60 hr. Liver, plasma, and intestinal-wall proteins are re- generated much faster than those of muscle and connective tissue. The nitrogen that goes into ringed structures such as the porphyrins, which enter complexes with Fe +2 and Fe +3 to form the hemin of red-blood cells, turns over quite slowly: it takes 10 days for the hemin to be synthesized from isotopically tagged glycine, and then nearly 140 days before the deg- radation process (cell replacement in this case) reduces the concentration of tagged nitrogen to half the peak concentration (see Figure 5-10). indirect Time after oral administration Figure 5-10. Radioactivity in a Particular Vol- ume of Tissue as a Function of Time After Administration. Time and height of the maxi- mum depend upon location of the volume, upon what chemical compound is given, its normal biochemistry, where it was introduced (direct or indirect), and the half-life of the isotope. Other uses of radioactive tracers include the investigation of the effects of drugs and hormones on the turnover rate in particular tissues or organs. A subject of particular interest in recent years has been the role of insulin in the control of diabetes. In a diabetic, sugars are transported across the membrane and into the cell abnormally slowly, and they accumulate in the plasma, useless for supplying energy, via oxidation, inside the cell. Insulin, a medium-sized protein molecule whose structure has been well known since it was first synthesized in 1956, has been tagged with I 131 and introduced into the blood stream. Within minutes, more than a third accumulates in the liver and the kidneys. However, a fraction adsorbs in a nonspecific man- ner on all membranes accessible to blood plasma and intracellular fluids. Cell walls are no exception; and the adsorption of insulin has been associ- 122 RADIOACTIVITY; BIOLOGICAL TRACERS ated with an increase of the rate of sugar penetration (a process which itself has been followed by C 14 -tagged sugars). Whether the control exercised by insulin is simply by opening the access to pores, or whether it controls in a more subtle manner by increasing the activity of the enzyme (hexokinase) also thought to be adsorbed on the membrane, has not yet been settled. However, it can be seen that the use of radioactive tracers in such a phar- macological problem can make a valuable contribution to our knowledge of the processes involved. The pioneering work of Huff and Judd on the quantitative analysis of the time and space distributions of Fe 59 in blood plasma, will be discussed in Chapter 11 as a concrete example of how possible methods of action can be analyzed with a computer if it is fed reliable experimental measurements of where the Fe 59 goes and how long it stays there. We learn a little about what the iron does, and also something about just what processes are interfered with during blood diseases. Radioactive Mapping Administration of compounds of I 131 , followed bv external measurements of beta-ray intensity in the thyroid region of the neck, has been introduced in some centers as a replacement test for determining whether the thyroid is normal, over-, or under-active. A hyperactive thyroid may absorb up to 80 per cent of the tagged iodine; a hypoactive gland may absorb as little as 15 per cent before normal biochemical turnover elsewhere in the body re- duces the concentration via excretion. Mapping of the thyroid by I 131 scintography is common practice. Both the outline of the organ, and its turnover rate can be obtained from maps made at different time intervals after administration. The maximum activity of the emission is a direct measure of the uptake of iodine by the thyroid. The flow of fluids through various critical parts of the system can also be mapped satisfactorily by dissolving in the fluid a small amount of gas which contains a radioactive emitter, and mapping from the outside with a col- limated scintillation counter (Figure 5-6). Conclusion A great many elementary biochemical reactions are being studied via the tracer technique, and a few physical processes also. Some of these will be found mentioned as examples in different parts of this book. The techniques are reliable and extremely sensitive, and have the unique advantage that the introduction of the radioactive element can be done in such a manner as not to upset the chemistry or the physics of the process in vivo. Already in ex- tensive use in biological research — in his review Kuzin 12 was able to collect 358 references to new work published in 1959 alone!— now, led by successes REFERENCES 123 with I 131 and P 32 , radioactive tracer techniques have a wonderful future in medical diagnosis. As it does in so many subjects, the National Bureau of Standards, in Washington, periodically publishes reliable definitions of terms, values of universal and experimental constants, and tables and graphs of collated data on radiologically important parameters. The ''Radiological Health Hand Book" is indispensible to further study of this subject, as a quantitative sup- plement to the classic work of Kamen. 5 PROBLEMS 5- 1 : (a) What element is formed by the radioactive disintegration of: P~ £- 0' P 32 ^ Co 60 ^ p30 1+ 0- 8- Na 24 ^ Ra 226 -^ a 8* p o 210 _^ Na 22 % (b) Is the product radioactive too? 5-2: (a) Make a graph showing activity (counts per minute) against time, for up- take, utilization, and elimination of I !3! by the thyroid, (b) List five important reasons why I 131 is used in irradiation-therapy of goiter. 5-3: The 1.70 mev /3-ray of P 32 penetrates about 7 mm into tissue. The half-life is 14.3 days. A 1-millicurie (mc) source will deliver about 1 rad (radiation ab- sorbed dose) per minute. How long would it take for a 1 mc of NaHP0 4 , composed of P 32 , taken orally as a solution in water, to administer 6000 rads to an organ in which it concen- trates? REFERENCES 1. The Staff, Physics Dept., Univ. of Pittsburgh: "Atomic Physics," 2nd ed., John Wiley & Sons, Inc., New York, N. Y., 1944. 2. "Atomic Radiation (Theory, Biological Hazards, Safety Measures, Treatment of Injury)," RCA Service Co., Camden, N. J ., 1 959. 3. "Teaching with Radioisotopes," H. A. Miner, el al., Eds., U. S. Atomic Energy Commission, Washington, D. C, 1959. 4. Scientific American, issue on "Ionizing Radiations," Vol. 201, September, 1959: papers by S. Warren, p. 164, and R. L. Platzman, p. 74. 5. Kamen, M. D., "Tracer Techniques in Biology and Medicine," Academic Press, New York, N. Y., 1960. 124 RADIOACTIVITY; BIOLOGICAL TRACERS 6. Glasser, O., Ed., "Medical Physics, Vol. Ill," Year Book Publishing Co., Chicago, 111., 1960: several short articles, p. 302-364. See especially: "Locali- zation of Brain Tumors with /^-Emitting Isotopes," by Silverstone and Robertson. 7. Kity, S. S., Methods in Med. Res., 1, 204 (1948). 8. Munck, O. and Lassen, N. A., Circulation Research, 5, 163 (1951). 9. Clarke, H. T., Urey, H. C, and 16 others, "The Use of Isotopes in Biology and Medicine," in the Proceedings of a Symposium on the subject, The Univ. of Wisconsin Press, Madison, Wis., 1948. 10. Huff, R. L. and Judd, O. J., "Kinetics of Iron Metabolism," in Adv. in Biol, and Med. Phys., 4,223 (1956). 11. Freygang, W. H. and Sokoloff, L., "Quantitative Measurement of Regional Cir- culation in the Central Nervous System by the Use of Radioactive Inert Gas," Adv. in Biol, and Med. Phys., 6,263 (1958). 12. Kuzin, A. M., "The Application of Radioisotopes in Biology," Review Series, No. 7, International Atomic Energy Agency, Vienna, 1960. 13. "Scintography — A collection of Scintigrams Illustrating the Modern Medical Technique of in vivo Visualization of Radioisotope Distribution," R-C Scien- tific Co., Inc., Pasadena, Calif., 1955. 14. Cork, J. M., "Radioactivity and Nuclear Physics," 3rd ed., D. Van Nostrand, Inc., New York, N. Y., 1957. CHAPTER 6 Big Molecules (Structure of Macromolecules and Living Membranes; Isomers and Multiplets; Codes and Molecular Diseases) A score of diseases (including sickle cell anaemia and phenylketonuria) have so far been recognized as enzyme diseases, presumably resulting from the manufacture of abnormal molecules in place of active enzyme molecules. I think that it is not unlikely that there are hundreds or thousands of such diseases. I foresee the day when many of these diseases will be treated by the use of artificial enzymes .... When our understanding of enzyme activity becomes great enough, it will be possible to synthesize a catalyst, etc Thus did Linus Pauling emphasize to an international sym- posium of enzymologists in Chicago, in 1956, the relationship between the structure of the macromolecule and its chemical and physical roles in the living system. INTRODUCTION The structure of macromolecules and of arrays of them in living mem- branes and other tissues has occupied the attention of an important class of biophysicists for the past ten years. Using modern rapid-flow, quick-freeze- drying, and micromanipulation techniques, and armed with the phase and 125 126 BIG MOLECULES interference microscopes, the X-ray diffraction camera, and the elertron microscope — the last now in such an advanced stage of development that, in proper hands, it can resolve or "see" small particles just a few atomic diameters apart — researchers have been able to gain new insight into the actual shape of the molecule in the tissue, and even into the positions of atoms and groups of atoms within the molecule. Running concurrently with these physical researchers have been chemical studies which have finally solved the puzzle of the complete chemical composition of a few large, biologically important molecules. For example, although the hormone, insulin, has been known and used widely in the treat- ment of diabetes for nearly forty years, it was only in 1955 that Sanger and his colleagues at Cambridge were finally able to write down the complete structural formula. It contains 777 atoms! Since then, ribonuclease (RNAse), an enzyme containing 1876 atoms and which catalyzes the cleavage of ribonucleic acid, has also yielded the secret of its composition to the attack of persistent chemists. This completes the first big step toward knowing how this molecule works as a catalyst, although details of the struc- ture at and around the active site(s) are not yet known. This is the next big task, for if more than one of the chemical groups must exert their chemical effects on a specific part of the molecule whose hydrolysis is to be promoted, then their spatial arrangement must be very important. Not only must they be present, but they must be present at the proper positions in space if the catalytic activity of the site is to exist. In other words, if one of the players is out of position, the game is lost. Table 6-1 gives a spectrum of biologically important organic molecules, small and large — some containing a metallic oxidizable and reducible ion which enters the chemical reactions of the molecule. Although some details are given in the following sections, the discussion is just an indication of the scope of the subject. There are excellent reference sources: for example, the recent book of Tanford. 16 STRUCTURE Our purpose, first, will be to outline the structure of two big molecules of critical biological importance, myoglobin and hemoglobin, learned in the recent work of the schools of Kendrew and Perutz, respectively. The method used was X-ray crystallography, and although the chemical com- position has not yet been fully worked out for these two molecules, X-ray crystallographic studies have completely outlined the form of the molecule in the dry crystalline state. The second part of this section on structure is concerned with the cross- linked structure of liquid crystals, such as in the aqueous humor of the lens STRUCTURE 127 of the eye, anH of membranes — those of the erythrocyte cell wall which are relatively homogeneous, and those patchy, mosaic membranes exemplified by the wall of the small intestine. Crystalline Macromolecules Diffraction of X rays by the regular arrays of the electron clouds which surround the atoms or ions of a crystalline substance was introduced in Chapter 4. The X rays diffracted from a single crystal interfere with one another in a manner which is determined solely by the position and electron density of the target atoms in the crystal. If the diffracted rays are allowed to fall upon a photographic plate, from the position and darkness of the spots on the plate, one can (at least in principle) locate the position and electron density of the diffracting atoms in the crystal. The position of the spot tells the angle, 9, of constructive scatter of the X rays of wavelength, A; and the Bragg interference equation, nX = 2d sin 0, relates these values, the "order" of interference, n, and the wavelength, A, to the spacing, d, within the crystal responsible for the scatter. The blackness of the spot gives the amplitude. The superposition of those waves which give rise to the one which emerges from the crystal, however, must be inferred from positions of the atoms in the crystal. This is done by a trial-and-error mathematical method involving superposition of infinite series, a method which will not be described here. It was in 1951 that Pauling and Corey made the big break-through in our understanding of structure of proteins: they were able to determine from X-ray diffraction patterns that synthetic polypeptides formed of alpha amino acids all have a coiled, helical form. In other words, the back-bone of the polypeptide chain coils around and around, to form a cylindrically shaped molecular helix. This can be easily understood now, in retrospect, as follows. Since all the alpha amino acids have the structural formula H R I I N— C— COOH H H and since these condense through the — CONH-- linkage (Figure 6-1) in the form H R O ! H R O I llil I II • • • -N-C-C-T-N-C-C- • • • 1 2\ 3 ' 4 5 1 6 h ; h the atoms of the backbone of the chain, — N — C — C — , are repeated over and over again. 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U 2 X IU c be O p _c > _y < z O O u 3. a n > c r.. - u CJ - en u 3. O U w 3 3 u u 3 JOJ z 130 BIG MOLECULES carbon 6 falls almost directly above nitrogen /, and the two are hydrogen bonded about 1.5 A apart. The diameter of the helix so formed is about 8 A. The helix has an open core, about 2 A across,. and the R-groups, or side chains to the main structure, jut out radially from the central axis of the cylinder. Figure 6-1. The Planar -CONH — Linkage (boxed) Between Amino Acids in a Protein. Lengths in angstroms. Since the helical shape is a property of poly alpha amino acids, it was given the name "alpha-helix, " and it is now probably the most famous structure of macromolecular physical chemistry. Figure 6-2 is a drawing, similar to the original disclosure, which shows the main chain (bold bonds) and the posi- tions of attached groups ( — R); and which indicates the positions of the hy- drogen bonds, the "bones" which give the helix rigidity. It is now known to be the main structural component of «-keratin, hair, wool, nail, muscle, and connective tissue, etc. Recently it has been traced in muscle to the contractile enzyme, myosin itself. Because of the unique role of myosin, some of its physical and chemical properties are expounded in Chapter 10. One protein, of unquestioned importance, which has intrigued biological investigators for years, is hemoglobin, the "oxygen carrier" of the respira- tory enzyme system, first crystallized and purified by Hoppe-Seyler in 1862. However, with a molecular weight of 67,000, its amino acid sequence and the physical structure of the molecule have only slowly yielded to persistent STRUCTURE 131 Figure 6-2. Schematic Representation of the Alpha Helix of Protein. Three complete turns are shown. They start at the bottom C, wind out toward the reader through the next — N — C — , then back in through the plane of the paper, etc. (After Pauling and Corey, 1953.) 132 BIG MOLECULES investigation. The X-ray diffraction pattern of even single crystals was too formidable for analysis until M. F. Perutz, about 1950, began to substitute heavy metals ions such as Hg +2 at particular spots on the molecule and to analyze the effects of these strong X-ray scatterers on the spectrum. With this technique, now known as the "method of isomorphous replacement," it was possible by 1960 to show the surprising result that the protein of the molecule at 6 angstroms' resolution looks like several intertwined worms, with the heme groups attached — not a regular array at all. Studies con- tinue on the amino acid sequence, and on the analysis of the X-ray diffrac- tion pattern, in an effort to get even better resolution of the detailed struc- ture of the hemoglobin molecule. Inherently simpler, myoglobin (one Fe +2 ion only) has yielded not only to 6 A analysis (1956) but even to 1.5 A resolution (1958), work for which Kendrew and his team received a Nobel Prize in 1961. The main features of this molecule are depicted in the drawing shown in Figure 6-3. The a-helix (flat) heme group CH 2 < „ p CH,-/VV\ C-N N-C HC (Fe) CH X C-N N=C / I I \ HC-C C C C- CH ? X C C C / H | CH 3 CH 3 protein (ferrous ion) seqments R= -CH 2 CH 2 0H I 1 1 30 A Figure 6-3. Molecule of Myoglobin. (Drawn from the Model of Kendrew, 1958.) hydrogen-bon(d)ed, forms the framework of the worm-like segments, sudden turns in which are thought to be associated with the proline groups — an amino acid residue of odd structural configuration. The heme group sits ex- posed, with the iron ion ready for oxidation or reduction, or, preferably, simply complexing with 2 picked up from air. Although this is the configuration of crystalline myoglobin, the shape of the STRUCTURE 133 molecule dissolved in salty water may be quite different — for example, one can readily imagine the legs of this molecular octopus unfolding in the blood stream. Structural knowledge of many other big molecules is rapidly becoming available. This is a subject of intense interest. Straight chains and helices, some coiled into balls, some folded back and forth to form rods, others with randomly coiled shapes, are known or imagined. These forms are illustrated in Figure 6-4. 1 1 1 1 .)- (I . . . , , , , )- C ■ ' i i j random coil helix globe rod Figure 6-4. Some Molecular Shapes in Solution (schematic). Transitions one to another can be effected by change in pH, ionic composition, or temperature. Receiving much attention in the hands of F. O. Schmitt and the MIT School has been collagen, the structural component of connective tissue, tendon, skin, cartilage, etc. (Figure 6-5). Formed of three interwound molecular helices of protein, with molecular dimensions approximately 3000 A long x 30 A in diameter, it cross-links end to end to form fibers, and then side to side to form either sheets (two dimensions) or blocks (three di- mensions) of connective tissue with very varied physical properties: for ex- ample, tensile strength up to 100,000 lbs/in 2 , equivalent to that of a steel wire of the same dimensions! Now thought to be the basic information-carrier of the gene, and an ex- tremely important component of the nucleus of the cell, is desoxyribose- nucleic acid (DNA). At about 70 per cent relative humidity, it is an ex- tended, double-stranded helix, of molecular weight in the millions. Further discussion of the structure of DNA, and its sister nucleic acid, ribosenucleic acid (RNA), appears later in this chapter. Now, the backbone of the helices of DNA and RNA is ribose, a sugar, polymerized through phosphate groups. Polymerized sugars are the second major structural component of living tissue — cellulose and chitin are ex- amples. Hyaluronic acid and glycogen are polysaccharides which take an integral part in the biochemistry of life. Thus glycogen is the form in which sugar is stored as an energy reserve in the liver. Polysaccharides, like proteins, take many forms in tissue. One which seems to be unique is the pleated sheet of cellulose. 134 BIG MOLECULES € I ':••-. ' the angular ve- locity of the centrifuge (radians/sec). A more rapid method, used within the past few years, takes advantage of the fact that small volumes bounded by the top and the bottom of the tube reach equilibrium very rapidly; measurements of concentrations in these volumes can be made within a few hours, and an "average" molecular weight then evaluated. Direct Measurement of Size and Shape via the Electron Microscope. For those polymers whose shape and weight are the same, dry or wet, the direct meas- urement by the electron microscope is possible. A comparison of the results of different methods on the globular molecule icthyocol is shown in Figure 6-6. The nonequilibrium methods will now be outlined. Dynamic Methods These are based on four transport processes which are discussed as a group in more detail in Chapter 8. The following outline sumcies here: Diffusion under a concentration gradient and sedimentation under a centrif- ugal force can both be stated as the speed of the process under specific condi- tions, and these speeds expressed as D and s, respectively. An argument involving factional force offered by the water against movement of the macromolecules shows that the ratio of the two speeds, D/s, is related to the molecular weight, M, by (1 ~ P) D _ J_ RT s M ' an expression originally derived by Svedberg. Measurements of D and s, and of the densities of solid and solute permit evaluation of molecular weight. Intrinsic Viscosity. This property, /c as c — » (where rj is the Vo / viscosity of the solvent and r\ that of the solution), can be related to the vol- ume of the molecule and molecular weight by two expressions which in simplest form are: (1) \r\\ = 2.5 V for spheres (7000 V for a big, randomly coiled molecule such as DNA) — here Fis cc/g; and (2) [77] « M a where a is an empirical constant, usually 0.5 to 1 .0. Although measurement of viscosity is easy enough, the proportionality constants have an empirical character, and hence one always suspects the absolute values of size and shape so obtained. However, they are quite reli- STRUCTURE 139 ably indicative of change in molecular shape as environment is changed, and it is in this manner that they are usually used. Speed of rotation of a big molecule about an axis can be inferred by an opti- cal measurement calledy/oie» birefringence, and the result related to molecular weight. Both the optical technique involved and discussion of the propor- tionalities are beyond the scope of this outline, for they are very specialized. Proper use of the techniques outlined have shown many interesting prop- erties about certain biologically active molecules. Compare now the results of the dynamic methods with those of. static methods. Table 6-2 gives aver- age weight and dimensions of collagen, measured by five different methods, and of erythrocyte DNA by two methods. Our well-worked illustration, Fig- ure 6-6, shows the results of the direct measurements of size of dried ichthyocol** rods by electron microscope techniques as compared with the indirect measurements by light scattering, flow birefringence, and intrinsic viscosity methods. TABLE 6-2. Dimensions of Molecules of Collagen and DNA. Molecule Method Mol Wt Length Diameter Collagen* Osmotic pressure 310,000 — — Light scattering 345,000 3100A 13.0 A Intrinsic viscosity — 2970 13.6 Sedimentation and viscosity 300,000 — 12.8 Flow birefringence and viscosity 350,000 2900 13.5 DNA** Light scattering 4.7 to 6.2 million — — Sedimentation and viscosity 5.3 to 17.4 million 2030- 2350 A * The chief constituent of connective tissue (cartilage, tendon, etc.). (After Doty, Oncley, etal., Eds. (19 "'Extracted from human erythrocytes. (After Butler, el al. (I960).) These are particularly pleasing results, one result confirming the other. Such is often not the case for randomly coiled molecules for which the results of different methods may disagree violently with one another. Carbohy- drates are particularly perplexing from this viewpoint. Again, in solution DNA is a very large, randomly coiled molecule, an 'extended double- stranded helix, apt to polymerize and take any shape at all in response to its environment. Therefore the study of nucleic acid reproduction as a molec- ** As the name implies, ichthyocol is a collagen from fish, used as a food supplement, as are gelatin from animals and glutin from wheat. 140 BIG MOLECULES ular reaction, like reactions of other randomly coiled molecules in solution, is made just that much more difficult. Some very fine X-ray diffraction work has been done On crystalline DNA, but even in crystalline form it may as- sume several structural arrangements, depending upon the humidity. Molecular Structure of Living Membranes There are two main subjects of interest in membrane biophysics: the structure of the membrane, and its penetration by small and large molecules and ions. They are closely interrelated. Thus there exist, in the human body, membranes which have every degree of specialization — frorn the quite nonspecific mosaic membrane of the small intestine to the highly specific membrane of nerve cells which not only can distinguish sodium ion from potassium ion (a trick which analytical chemists have only recently learned to do) but even change the rate at which it lets them through! We confine ourselves here to considerations of structure only. Penetration is discussed in Chapter 10. From analytical and electron microscopic work, it has been found (Danielli and many others) over the past twenty-five years that most living mem- branes*** are laminar, composed of at least three, sometimes five, layers. The heart of the membrane is a bimolecular layer of lipid, flanked by thin layers of protein, or cellulose, or both (Figure 6-8 (a)). The cellulose, if pres- cellulose ond/or protein layers bimolecular fatty acid layer Figure 6-8. Schematic Representation of Layers in the Living Membrane. For many membranes the total thickness is about 75A. (a) Note the position of the defect or pore, (b) Plan view of lipid film. ent, seems to be there simply for structural reasons — to make the membrane mechanically strong. The protein layer can also provide strength. However, various metal ions and water form complexes with the protein, and some protein of most membranes has enzyme activity, a property which is cur- *** For example, the cell wall, the endoplasmic reticulum within the cell, etc. STRUCTURE 141 rently thought by some to be associated with control of the size of the holes through which penetration of ions and molecules occurs. Although the membrane may have a total thickness of hundreds of ang- stroms, the hydrophobic lipid layer, probably continuous, (and certainly the well-protected center layer), is estimated to be only 75 A thick. Figure 6-9 is an electron micrograph of two membranes touching each other, from which the 75 A figure can be directly measured. This is a pattern which has been found in practically all the living membranes so photographed. The membrane is not perfectly symmetrical, as different staining methods have shown; and in some cases — the erythrocyte wall, for example — there is definitely an assymetry. '-■•■>-,,* ■ --.■-■■ - . -">' - ^ Figure 6-9. Electron Micrograph of the Double Membrane of a Nerve. Osmic acid stains the outer protein layers (see also Figure 6-8), and scatters electrons (dark ridges), but does not absorb into the (light) lipid layer in between. Total distance across one membrane is about 75A. Magnification: 880,000 x. (Courtesy of J. D. Robertson, Harvard Medical School.) When ones tries to penetrate deeper into the structure of the membrane, one runs into singularly difficult problems. Although it must be made up of macromolecules of protein, cellulose, and lipid, those molecules probably are distorted and stretched, or cross-linked into a planar structure. Neither the structure nor the properties of degraded or dissolved membrane mole- cules would therefore be expected to reflect those of the living membrane by conventional techniques of analysis. And yet not only are the complete membrane structures too thin to be studied in bulk, but also they degenerate when dried for X-ray or electron-microscopic study. In other words, good techniques for studying living membranes in vivo are still needed. Certain very specialized membranes, such as those enclosing nerve and muscle cells, and the rod and cone cells of the retina, can be studied through examina- tion of the details of their specialty. For instance, much progress has recently been made in elucidating the structure of the mitochondrion mem- 142 BIG MOLECULES brane because of its unique function in electron transport in the step-wise oxidation of foods. But the general problem of direct knowledge of the struc- ture of living membranes probably awaits more knowledge of the structure of macromolecules in solution. Indirect methods — i.e., studies of penetration of the living membrane by water, ions, and molecules — are proving to be very helpful to studies of structure, because from such studies one can infer some properties of the membrane in vivo: pore size, for example. An estimate of pore size (length and area) requires at least two independent experimental measurements, because there are two-dimensional parameters, area and length, to be evalu- ated. Both the rate of diffusion of a substance down a concentration gradi- ent and the rate of flow of a fluid under a mechanical pressure, should be larger the larger the area of the hole in the membrane and the shorter its length. Although the rate of transport of water through the cell membrane of erythrocytes is very rapid, both rate of diffusion and rate of flow have re- cently been measured accurately enough to determine a value for average pore diameter in the erythrocyte wall in vivo. Diffusion rate of water was found by measuring the rate at which radioactively labeled water is picked up by the cells within a few milliseconds of being bathed in the labeled water. A fast-flow apparatus had to be used, the ingenious details of which are best described in the original papers. 8 Then the rate of flow into the cell was measured by suddenly changing the osmotic pressure (salt concentra- tion) outside the cell, and following the change in cell diameter by means of a light-scattering technique. From the results, an analysis gives about 7 A as the diameter of the pores in the erythrocyte wall. The beauty of this kind of experiment is that it is a measure of a physical property of the membrane while it is living and func- tioning normally. The limitation is that the analysis involves certain as- sumptions which may or may not turn out to be absolutely correct. In the next few years it will be supplemented by the so-called "differential osmotic pressure" approach of Staverman, in which pore size can be inferred by the "ieakiness" of the membrane to certain ions; and by the molecular- or ionic- sieve approach, in which a large number of ions of various sizes are tested for their penetration. The diameter of the largest one which can penetrate the membrane is the effective diameter of the pore. Further support for the pore theory comes from examination of mono- molecular layers of large fatty acids and lipids. The lipid is spread out on water in a pan with a moveable boom (the so-called "Langmuir trough"). The boom is then made to reduce the area which the spread lipid must cover, and the force required to move the boom is measured on a sensitive torsion balance. When the layer has closed in completely, the resistance to ISOMERS AND MULTIPLETS 143 movement of the boom increases sharply, and thus the continuous mono- molecular layer is formed. By means of electron microscopic examination it has been found that the molecules assume a two-dimensional crystal struc- ture, with many crystallites. Where these meet there is indication of defects or dislocations which could be the precursor of pores in the membrane- see Figure 6-8, (a) and (b). All these approaches presume that pores really exist, and ignore Beutner's old (1911) idea that the membrane's lipid layer is a continuous barrier through which ions and molecules penetrate by either chemical reaction or solution in the lipid layer. This idea still has much appeal, especially in view of what is now known about the changes in transport mechanisms through a film across which a large electrical voltage exists. Thus a typical membrane potential of 100 mv across a membrane whose thickness is 100 A, would exert an electrical field of 100,000 v per cm across the membrane, and nobody knows yet what that would do to a continuous lipid layer. Perhaps acidic and basic organic molecules are formed by electrical discharge, simi- lar to the reactions known in organic transformer oils, to give the layer more of an ionic character so that water and ions can more easily dissolve. Structure within the living membrane is a treacherous problem for study; but no problem is more intriguing, and none in biophysics more important. ISOMERS AND MULTIPLETS This section is concerned with (a) the stereoisomerism which is expected to occur in macro-organic molecules as well as in classical organic molecules; and with (b) excited states which one supposes to exist in macromolecules, by analogy with the properties of smaller ones. These subjects have a bearing on the physical structure of the molecules and their chemical reactivity; but the current practical interest is in their relationship to inherited characteris- tics, to disease, and to benign (passive) and malignant (invasive) tumors. Unfortunately this subject is, experimentally, still in its infancy, although the general principles had been discussed at some length by Delbriick and Schroedinger 6 by 1944. Since the principles are fairly straightforward, and the experimental work by contrast very complicated and as yet not too definitive, we outline first the principles, and relate them to a model, or working hypothesis. Isomers Stereoisomerism — the existence of two or more chemicals with the same composition and differing only in the arrangement of the atoms — has been known in organic chemistry for a hundred years. Such isomers are truly different compounds, having differing physical and chemical properties. The propyl alcohols will illustrate this basic point. Thus normal propyl 144 BIG MOLECULES alcohol has the following atomic arrangement: H H H I I I H— C— C— C— OH I I I H H H However, in isopropyl alcohol the OH group is attached to the central car- bon atom instead: H OHH I I I H— C— C— C— H I I I H H H "Normal" melts at -127° and boils at 98° C, while "iso" melts at -89° and boils at 82° C. Normal chlorinates slowly in PC1 3 , iso chlorinates rapidly. Not all isomers are so obvious. Consider adrenaline, which has the struc- tural formula HO HO < > — C HOH CH 2 NH CH 3 Two forms exist, which differ only in the arrangement of the groups of atoms attached to the tetrahedral carbon atom starred. The two forms differ in optical rotation. One is physiologically active; the other is not. As we proceed through the higher alcohols— for example, those with four carbon atoms or more and two OH groups— the stereoisomeric possibilities mount. In the sugars and celluloses in which rings of carbon atoms are linked to one another to form long chains, each carbon having an OH group, physical interference with free rotation about an interatomic bond adds further to the number of possibilities. In molecules of the size of nucleic acid molecules, the number of structurally different possibilities is enormous. Thus (the example is Schroedinger's) the two characters of the Morse code, dot and dash, can be arranged in groups of four-character letters in 30 different ways. If, however, we have a system of even five characters, and if five copies of each of the five characters are arranged into linear code- scripts of 25 characters, the total number of possible 25-character code- scripts is an astronomical 63 x 10 12 — that is, 63 million millions! Note that even though the total number of characters chosen to define uniquely the "isomer" is only 25, the number of possibilities is hard to envisage; and indeed this number does not count any arrangements with either side-chains or rings, and is limited even further in that it excludes anything but five ISOMERS AND MULTIPLETS 145 copies of each character to make up the 25! Of course, not just any arrange- ment of atoms gives a stable molecule; but on the other hand the number of chemical groups of which a macromolecule is composed ( — CH 2 , — NH, — CO, — C — S — , etc.) is certainly far more than five! .... One concludes that the number of stable isomers of a macromolecule must be huge, but at this stage of knowledge one really has no idea how many there are. Each must have a unique set of physical and chemical properties. Just as in the case of the simple alcohols, each must be a stable molecular entity. Excited States No molecule, even if anchored at some point, must be completely quiet if T > 0°K. Indeed, in an environment at 98°F (37°C) such a molecule, even if initially at rest, or quiet — i.e., in its vibrational and rotational "ground state" as it is called — will soon be buffeted into motion by neigh- boring molecules of gas, liquid, or solid, until its energy level or tempera- ture is, on the average, that of the environment. Heat energy enters the molecule as the energy of rotation or vibration if the molecule is anchored, and enters also as the kinetic energy of translation (linear motion) if the molecule is free. The vibrations and rotations may be thought of as standing or traveling matter waves moving across the molecule. Parts of the mole- cule can be fixed and immobile; other parts can be free. The distribution of energy within the molecule will be continuously changing. Macromolecules accept and give up energy to the surroundings in discrete bursts or bunches or quanta, if the quantum theory applies here as it is known to apply to 2- and 3-atom molecules. However, the energy differ- ences between mechanically excited states must be very small — so small that almost a continuous exchange of energy must be possible. The important point is that all of the configurations which result from heat exchange are configurations proper to one isomer; in principle the isomer may assume many shapes. Consider the random coil configuration of protein as an example. The one chemical entity may assume many shapes simply as a result of thermal exchange. Electronically excited states also exist but these are different. It was seen in Chapter 4 that electrons which make the bonds of molecules can absorb and re-emit electromagnetic radiation, and that some excited states can be reached by the absorption of such small amounts of energy that even local heat energy sometimes will do the trick. It is a general rule-of-thumb that whenever a bonding electron accepts energy of any kind and becomes itself "excited," the bond is weakened. Once weakened, it is more susceptible to thermal buffeting and to chemical attack. Its "defense" is to rid itself of the extra energy and get back into the bond; this it does by reradiation, or by transfer of energy into the mechanical motion of the molecule. 146 BIG MOLECULES The salient point is the following: If the extra energy in the molecule is large enough, quite by chance it may collect at a critical bond and loosen it sufficiently so that a rearrangement of groups within the molecule can occur, and thus produce a dif- ferent isomer. When this occurs in the DNA molecule of the gene, a mutation is the result. There are many other biological processes which seem to involve excited electronic states of molecules: oxidations seem to be in a class by themselves because of the number of reactions of molecule + 2 + hght which have been demonstrated. In some reactions light is absorbed, and then im- mediately (within 10~ 12 sec) re-emitted, at least in part (fluorescence); in others the absorbed energy is retained for some appreciable time, perhaps a few seconds (phosphorescence). However, the extra energy to excite elec- trons in a molecule may also be derived from chemical reactions in the metabolism, for there is plenty of it there! This obviously occurs in some bacteria (pseudomonas, vibrio, etc.), some crustaceans, the elaterid beetle, and the firefly, for these animals are chemiluminescent. That human beings are not luminescent may be a subtle reminder of two important facts: (a) in man the energy-producing metabolic reactions are more carefully delineated by enzymes, constrained to occur in many small steps, each one linked intimately with an energy-consuming metabolic process; and (b) there are electron and proton transfer reactions along large molecules, transfer mechanisms which can conduct the "energy" to where it can be used. In other words, in humans, because of the extra complexity of the system, the extra energy of excitation of molecules need not be radi- ated and lost; there is a mechanism provided by which it can be used. This can be illustrated further. Although most proteins in vitro have no phosphorescence at room temperature where molecular mechanical motion is relatively large, at low temperature (77° K) all the following proteins, plus at least 18 amino acids, show phosphorescence: fibrinogen, y 2 globulin, keratin, gelatin, zein, and bovine serum albumin, as well as egg albumin and silk fibroin. Aromatic rings with it (Pi) bonds in the molecules are a neces- sary condition for the phosphorescence. In some simple organic molecules (certain ketones, for example) the extra energy has been found to excite one of the unshared pair or nonbonding (n) electrons on the oxygen atom. Its excited position is one of the so-called 7r positions or orbitals of the molecule. The transition is called an "/? — ir" transition (Figure 6-10). The energy absorbed during an n - w transition is about 80 kcal/mole, and can be produced by ultraviolet light of wave length about 3000 A. The unshared pair of electrons form no bond, but they are paired in the sense that they have opposite "spins." The molecule which contains only paired electrons is said to be in a "singlet" 1 state (S = In + 1,' where n is REPLICATION AND CODE-SCRIPTS 147 jverlappin it electron above the plane of the atoms Structural r, formu la with conjugated double bonds q // \ // n-T excitation 2 s electrons an unshared pair (non bonding) Figure 6-10. The n-7r Electronic Transition (schematic). the number of unshared electrons). When excited, however, the promoted electron, now in a formerly empty ir orbital, is unpaired; S = 3, and the molecule is said to be in a "triplet 1 ' state. Triplet states are important be- cause they sometimes retain the extra energy, without radiating it, for rela- tively long periods of time. Thus molecules in the triplet state sometimes have time to collide with others which are similarly excited, and the total energy of the collision may be sufficient to cause the isomeric or mutation reaction. Based on the work of M. Kasha, Reid 10 has listed a few types of molecules (containing N, O, P, S) whose n — it transition and the subsequent triplet states probably are energy carriers in biological processes: Amides Aldehydes and ketones Amides Quinones Thioketones Pyridines Diazines and triazines Azo- and diazo-compounds Nitroso-compounds Pyrimidines possibly Carbonates Nitrates Nitro-compounds The mechanism of some isomeric reactions in which a triplet excited state is an intermediate is now fairly well understood. For large macromolecules, however, pertinent information remains for the future. Nevertheless the direction and importance of such work is now clear. REPLICATION AND CODE-SCRIPTS There are now four types of experiment which support the contention thai genetic information is carried by the nucleic acids, DNA and RNA. There is still little direct evidence from any species higher than virus or bacterium. 148 BIG MOLECULES The celebrated French work on the transplanting of DNA in ducks seems to open the doorway to studies on higher animals. The long extrapolation to humans may turn out to be correct, although it is certainly not yet justified, for this will take generations to prove. Bacterial transformation: If pure DNA, extracted from a suspension of bac- teria of one type (A) is added to a suspension of another type (B), the progeny of the thus-infected B type have characteristics of A. Virus reproduction: Bacteriophage T 2 , a virus, which can reproduce only after it has entered into a living bacterial cell, can be split — the protein part from the nucleic acid part (DNA). The DNA, shorn of its protein, can enter the bacterial cell and rapidly reproduce the intact T 2 phage particles again. Virus "synthesis'''': Tobacco mosaic virus can be split chemically into pro- tein + RNA. One can then reconstitute the virus, using protein of strain A and RNA of strain B. The progeny are of strain B only, having resnythesized their original protein. Genetic recombination of bacteria: In fertile strains of bacteria, in which DNA can be passed from the donor to recipient cells, the extent of the appearance of the characteristics of the donor in the progeny is proportional to the amount of DNA transferred. Some Properties of DNA and RNA These "nucleic" acids (found in the cell nucleus and in the cytoplasm) are substituted sugar molecules which are polymerized through phosphate linkages. In DNA the sugar is desoxyribose; in RNA it is ribose. Both have 5-carbon rings. The substituent groups on the sugar molecules are organic nitrogen bases. These are ringed compounds with two nitrogen atoms in the ring, and are four in number: adenine, guanine, cytosine, and thymine (in DNA) or uracil (in RNA). Linkages, etc., are shown in Table 6-3. From X-ray diffraction studies it is known that DNA is a helical molecule with 10 sugar residues per turn of the helix. In the "dry" (70 per cent RH) crystalline state two helices are found interlocked (Figure 6-11), each with its phosphate-sugar chain facing to the outside, and the purine and pyrimi- dine bases, hydrogen-bonded together, facing to the inside. f At cell division, the two interlocking helices separate, and each repro- duces, probably by a process analogous to crystal growth, as though each helix, separated, acts as a template or a die for the "casting" operation which forms the new molecule. That this occurs at mitosis, suggests that the helices are pulled apart by a force which exists only at mitosis. For instance if two ends, one from each helix, are attached to the membrane which encloses the nucleus, in the expansion before division (25 per cent by one measurement) the DNA helices could be pulled apart; then if each template reproduces its opposite by "condensation," two DNA molecules | A single-stranded DNA is known, in phage vJ > ++ I I I B.mo.oiM-Mg -.m B 305070 i oo (b) Figure 6-12. (b) Two Sedimentation Patterns (A and B) of the Ribosomes shown in (a). Note how the binding of these little particles is so dependent upon the medium. The numbers are the sedimentation rates (in svedberg units) of the different particles in the ultracentrifuge: the larger particles fall faster. (Photographs (a) and (b), courtesy of S. T. Bayley, National Research Council's Biophysics Section, Ottawa.) "Cogs" and "Cams" It is generally assumed that the code is contained in the arrangement of the four basic (2 pyridine and 2 pyrimidine) groups in the nucleic acid chain. There are at least 20 amino acids which must be distinguished. The smallest number of 4 basic groups which could be arranged in enough differ- ent ways to distinguish 20 amino acids is 3; and 3 in principle could dis- tinguish as many as 64 amino acids (4 1 ). Two suggestions have been made in which it is shown that, of the 64 pos- sible ways or arrangements, only about 20 are unique in a chain. One sug- gestion was made by Gamov, Rich, and Yeas in 1953, who postulated that the cyclic, helical structure of DNA would give rise to arrangements in which the 4 pyridine and pyrimidine bases jut out from the helix to form the 4 corners of a diamond on the external surface of the helix. Only 20 unique arrangements of the 4 bases could exist. Let us call this the cam theory 154 BIG MOLECULES partly because one thinks of a cylindrical cam with coding on its walls (Fig- ure 6-13), and partly because it is a degeneration of Gamovl The other suggestion, made by Crick, Orgel, and Griffiths in 1957, was that in a linear arrangement of only 4 characters, only about 20 unique groups of 3 could be written, provided that no character be counted as be- longing to more than one group of three — that is, if there is no overlap. We think here of a helical molecule with 20 arrangements of 3 bases which de- fine the code information along the chain. Partly because the process re- sembles the meshing of carefully fitted gears, and partly because of the initials of the inventors of the theory, let us call it the cog theory. Figure 6-13 is a schematic representation of the cam and the cog. triplet base code sugar ridge cam cog Figure 6-13. Cogs and Cams for Coding on DNA. Each spot represents a pyridine or pyrimidine base. Both theories have serious drawbacks, not yet resolved. In the Crick model, the amino acids in solution must "know" that they are forbidden to indulge in overlap; while in the, Gamov model a severe geometric restriction exists, viz., the DNA molecule (and hence the RNA whose shape is deter- mined by DNA) must always hold a very specific and rigid helical structure if the diamond arrays are to persist on the surface. However, successes in a flurry of investigation, genetic and biochemical, have engendered the belief that the basic facts of the amino-acid code car- ped by DNA may be completely known by 1963! There have been three recent remarkable disclosures. First, Nirenburg startled the International Biochemical Congress in Moscow in the Summer of 1961 by announcing that polyphenylalanine (a polypeptide) could be produced by adding poly- uridylic acid (i.e., an RNA, the pyrimidine bases of which are all uracil) to a cell-free solution of phenylalanine. This showed that a sequence of uracils (probably three of them) codes phenylalanine. Secondly, from elegant genetic studies, Crick et al. argued that: REPLICATION AND CODE-SCRIPTS 155 (a) A group of three bases (or, less likely, a multiple ol three l>ases) along the DNA helix codes one amino acid. (b) The sequence of bases is read from a fixed starting point along the helix. This determines what groups of three in sequence code an amino acid. (c) The triplets do not overlap each other. (d) Probably more than one triplet of bases will be found to code a single amino acid (that is, the code is "degenerate"!. Lastly, Ochoa et al., in March 1962, disclosed a three-base coding for each of the 20 amino acids, a code based on the increased rate of amino-acid uptake by E. coli protein to which had been added the polymerized bases of known composition. Other laboratories have been publishing partial codes also. Although they may be revised even before this book is printed. Table 6-4 lists tentative codings published by four different laboratories. Underlined are the codes in which the authors have expressed greatest confidence. TABLE 6-4. Triplet or Three-Base Codes for Each of the 20 Amino Acids of Proteins Symbol Tentative Codes ( 1 962) Amino Acid Ochoa Zubay Gamov Woese 23 e/a/. 20 et a/. 22 eta/." alanine ala UCG* UCG AAC UAG arginine arg UCG UGC AGG AGG asparagine asp N UAA -i UAG J UCA AGU GAU aspartic acid asp cysteine cys UUG ?CG glutamic acid gluN UAG -i UC( . ' UUA AUU UAU glutamine gluN glycine giy UGG UUG CUU GAG histidine his UAC UGU isoleucine ileu UUA UAC CAU leucine leu UUC UCU \GC UCG lysine lvs UAA UGA ccc CCG methionine met UAG UAU cuu proline pro ICC UCC ecu ccc serine ser UUC UGG < < ;u \\(, threonine thr UAC UAG ACU CAC tryptophane try UGG UAA tyrosine t vi- UUA 'AU UUU valine va 1 UUG UUG \.\u ( \<; phenylalanine pha UUU UUU GI I UUG I ' lll'.ic ll *l tnderlined < odes are C i \ tosine those thought l>\ the respe< live \ adenine authors to be vei ( , guanine ible I ' 156 BIG MOLECULES There are extensions and modifications of the cog and cam theories, and even other theories of the physical arrangements on DNA and RNA. The experimental problem is not made simpler by the fact that there are 20 x 19 x 18- •• = 2.3 x 10 17 different ways in which 20 different amino acids can be hooked together! Some "selection rules" must therefore follow from a code, for, as Gamov says "if one could spend only one second to check each assignment, one would have to work continuously for about five billion years, which is [estimated to be] the present age of our Universe! " Other experimental work has brightened the picture still further. For in- stance, only with a specific enzyme does an amino acid form a complex with ATP; polymerization and depolymerization occur in DNA and RNA; com- plex formation occurs between the low-molecular-weight, soluble (or "trans- fer") RNA and the DNA molecule; the helical shape of DNA is well estab- lished in moist air; and chemical analyses have been made of certain mole- cules. All these are experimental facts. There are many, many variables, better knowledge of which will clarify the theory. MUTATIONS AND MOLECULAR DISEASES The idea of "sick people from bad molecules" is not really new, although it certainly has been experimentally demonstrated in very convincing fashion and exploited heavily since 1948. While Washington was busy on the Delaware, Scheele in Germany showed that there is a good and bad form of adrenalin. By 1913, F. G. Hopkins was able to state with some bio- chemical authority: "Metabolic processes on which life depends consist in toto of a vast number of well-organized and interlocking enzymic reactions, interference with any one of which can product deleterious effects . . . ." The quotation from Pauling, with which this Chapter began, concerning the need for better understanding of macromolecules and catalysts, is the mod- ern approach to this question. We have seen that, because of structural and/or compositional changes in macromolecules, the following results may accrue: ( 1 ) Change in rate of chemical processes (2) Change in rate of physical processes (3) Introduction of new side reactions A simple example of (3), introduced before recorded history and persisting faithfully to our day, is offered in the different blood types in man: O, A, B, AB. These differ from each other only in that the colloidal-stabilizing mechanism of the macromolecules of the blood plasma is different: for if two of the types are mixed, they agglutinate or gel; the mixture becomes thick and refuses to flow. The physical nature of this subtle difference which makes them incompatible still escapes us. The production, by each indi- MUTATIONS AND MOLECULAR DISEASES 157 vidual, of antibodies (big molecules?) which are specific to that individual, and incompatible with those built by any other individual for the same pur- pose, is a well known phenomenon. Thus each individual has a specific bio- chemistry and a biophysics of his own, which becomes manifested in many ways. It is not surprising, then, that even small changes in structure or composition of certain large molecules can sometimes have disastrous results. A few examples will illustrate the point. No attempt is made to be ex- haustive. Lathe's thesis 1 reviews several other molecular diseases. Molecular Diseases There is both a broad, generic connotation and a rather restricted, spe- cialized one associated with the term "molecular diseases. " In the sense that all diseases involve molecules which are incompatible with the chem- istry or the physics of the system, all diseases are "molecular. " However, in the more restricted sense, the term has evolved to mean diseases caused by apparently small modifications of the chemical composition or the physi- cal structure of a particular molecule. The hemoglobin diseases, recognized only within the last decade, are now the classic example. Hemoglobins : There are at least ten known modifications of the hemo- globin molecule, each of which is associated with a pathologic condition. The normal molecule is characterized by certain values for sedimentation and diffusion constant (thence molecular wt.), electrophoretic mobility, elec- tric charge as a function of pH (determined by titration), solubility, ultra- violet absorption spectrum, etc. The most celebrated variant, S, which is found in erythrocytes from people with sickle-cell anemia, differs from the normal, A, principally in the manner in which it moves under the influence of an electric field: it moves faster, and at pH = 7, toward the cathode, whereas .4 is negatively charged at pH = 7 and moves toward the anode. Some of the pertinent characteristics of ten different forms of the hemo- globin molecule have been collected in Table 6-5. Although the differences were first observed clinically, and then correlated with differences in physi- cal properties, recent work has established that the differences arise because of different composition or arrangement in the amino-acid sequences of the protein. There are about 600 amino acids in the molecule. X-ray diffraction studies have shown that type A (normal adult human hemoglobin) mole- cules consist of four intertwined polypeptide chains. Two of these have a valine, then a leucine residue just prior to attachment to the nitrogen of the porphyrin (heme) group; two others have a valine, histidine, leucine sequence before attachment to the (iron-containing) porphyrin group. It is now known that modifications occur right at that point: a different sequence, or even different amino acids in the sequence, can occur. 158 55 c E _re c o O .^r ■£ en CU 3 en a 3 T3 O (L) bo 1 *— — ' en D tj D D (J o s-. bfi X CO o bo X CO "o en 15 _3 £ o en re "5 en CO C re XI CJ be re CU be C re X CJ £ 3 t- +-j CJ cu ,re re _q ^ o re E *" ~ P 1/5 £ u 5 ~~- re ^ t- ^" -H .S" ^ o o cu i_ X 4-J o en c CU O en re CU > '5 " r ri 2 ^ 3 S" (__ r- en -a 3 _C _c 3 * c £ 03 e £ re CJ en en IS en IS en rs en IS in c 'o o CU 3 3 CU C re o en 'o re o 'o re o CO re o _3 o 3 en c c (i 3 c e^-. 03 c n 5c en _4J en -a 'o re 'E re o c 'E re >- c 'E re c "re £ re ^T <-. 3 CD t/a <3 X) '<3 c re s o c o i— i o c c o a. , ii D II »-■ *— ' oo 1 — 1 oo -22 -c T 1 | | 1 I w u * , u O a C3 2 "-s >-od a «N Q CM a Elect phor Mobili P H = - 1 1 1 1 + CN ■ p c <= "O _ ,_ ^_^ en J .0 « 'c "C re C .2 "5 o c en O en CO "S re u-~ re t/1 CJ O E o bD '£ E '£ -♦— C CU CU CU Q. E c CO E 2 CJ en C re 2 2 C re IS CU c CJ c 3 _re cu C CJ 3 o c c re 1 '£ 1 O c o c O 3 17 - re re c- 'E CJ IT >> c t~* re CU ^-^ */i ■>> i_ > o re O re zz? E re T3 -a CU b CU cu eu O en ft CJ ■ CU X £ E E cs C V CU IS CU C re re CU jS CU CU c 2 c c c a. c 3 o CJ i re c c o o O z CO p D z I Z z PROBLEMS 159 There may be other modifications out farther in the protein, but this is not yet known. Likewise there may be many more modifications of hemo- globin than those listed. The work is really quite new. Unfortunately, prac- tically nothing is known of the shapes of these molecules — and won't be until more is known of their structure. Sufficient familiarity with the physio- logical reactions has been estimated to be about ten years away. The sickling of erythrocytes occurs when the hemoglobin-.^ is in an at- mosphere low in oxygen, and is a remarkable example of what "bad" mole- cules can do. It is now fairly well established that these bad molecules are so shaped that they can fit into each other and be piled up like a stack of saucers. In so piling, their strength is sufficient to push out the sides of the erythrocyte and cause it to buckle in the middle, i.e., to become sickle- shaped. On oxygenation, the stack collapses, presumably because the mo- lecular shapes are no longer so nicely complementary. Apparently the process resembles the growth of a crystal. The reader is asked to meditate on the known structure of myoglobin (Figure 6-3), and to study the pictures of Perutz et al. 24 on hemoglobin, before pressing further into this subject via Reference 25. Others. There are well over 20 diseases for which a "bad" molecule has been postulated as the cause. One other which is receiving considerable at- tention now is phenylketonuria. This is associated with mental deficiencies, and has been traced to the fact that one of the enzymes which catalyze the oxidation of phenylalanine through various steps toward pyruvic acid is not doing its job fast enough. Whether the offending enzyme molecule is not being synthesized, or has some physical deformity which renders it only partially active; or whether it or the substrate is not being transported fast enough to the place of catalysis, is not yet known. However, the result is ac- cumulation of phenylalanine in the blood stream, and interference with syn- thesis of nerve tissue. PROBLEMS 6-1: Erythrocyte DNA has a molecular weight of above five million. Calculate the diameter of the smallest sphere into which one molecule could be compressed. (Assume an average atomic weight of 12: it has Cs, N's, O's, H's, and a few P's and S's. Assume also that each atom occupies a cube 1 .2 A on each side.) If it were stretched out, the atoms end to end, what would he the total length of the chain? 6-2: Have you figured out how the two helical strands of DNA can unwind: for repli- cation, or for coding transfer- 1<\ A.' 6-3: Describe the four possibilities open to a big molecule in an electronically ex< ited state. 160 BIG MOLECULES REFERENCES 1 . Lathe, G. H., "Defective Molecules as a Cause of Diseases," Thesis, Leeds Univ. Press, Leeds, England, 1960. 2. Dixon, M. and Webb, E. C, "Enzymes," Academic Press, New York, N. Y., 1958. 3. Pauling, L., in "Enzymes: Units of Biological Structure and Function," edited by Gaebler, O. H., Academic Press, 1956. 4. Putman, F. W., Ed., "The Plasma Proteins, I: Isolation, Characterization and Function," Academic Press, 1960. 5. Oncley, J. L., et al., Eds., "Biophysical Science — A Study Program," John Wiley & Sons, Inc., New York, N. Y., 1959; papers by Kendrew, Doty, Rich, and many others. 6. Schroedinger, E., "What is Life?", Doubleday Anchor printing, 1956, of Cam- bridge Univ. Press book, 1944. 7. Butler, J. A. V., "Inside the Living Cell," Basic Books, Inc., New York, N. Y., 1959; Butler, J. A. V., etal., Proc. Royal Soc, A, 250, 1 (1960). 8. Solomon, A. K., Scientific American, 203, 146 (1960), and references. 9. Hoagland, M. B., Scientific American, 201, 55 (1959). 10. Reid, G, "Excited States in Chemistry and Biology," Butterworths Sci. Publ., 1957. 11. Gamov, G., Rich, A., and Yeas, M., Adv. Biol. Med. Pkys., 4,23 (1956). 12. Davson, H. and Danielli, J. F., "The Permeability of Natural Membranes," 2nd ed., Cambridge Univ. Press, 1952. 13. Shooter, K. V., "The Physical Chemistry of Desoxyribosenucleic Acid," Prog, in Biophysics and Biophysical Chem., 8,309 (1957). 14. Scientific Amer., Issue on "Giant Molecules," 197, No. 3, 1957; articles by Doty, Crick, Schmitt, Debye, and others. 15. St. Whitelock, O., Ed., "Cellular Biology, Nucleic Acids and Viruses," N. Y. Acad. Sci., 1957. 16. Tanford, C, "Physical Chemistry of Macromolecules," John Wiley & Sons, Inc., New York, N. Y., 1961. 17. "The Merck Index," 7th ed„ Merck & Co., Inc., Rahway, N. J., I960. 18. Bonnar, R. V., Dimbat, M., and Stross, F. FL, "Number Average Molecular Weights," Interscience Publishers Inc., New York, N. Y., 1958. 19. Crick, F. H. C, Barnett, L., Brenner, S., and Watts-Tobin, R. J., Nature, 192, 1227(1961). 20. Speyer, J. F., Lengyel, P., Basilio, C, and Ochoa, S., Proc. Nat. Acad. Sci., 48, 441 (1962). 21. Nirenberg, M. W., and Matthei, J. B., ibid., 47, 1588 (1961). 22. Zubay, G., and Quastler, H., ibid., 48,461 (1962). 23. Woese, C. G., Biophys. and Bwchem. Res. Com., 5,88 (1961). 24. Perutz, M. F., Rossman, M. G., Cullis, A. F., Muirhead, H., Will, G., and North, A. C. T, Nature, 185, 416 (1960). 25. Itano, H. A., Singer, S. J. and Robinson, E., in "Biochemistry of Human Genetics," G. E. W. Wolstenholme and C. M. O'Connor, Eds., Churchill Ltd., London, 1959; p. 96 ff. CHAPTER 7 A Conceptual Introduction to Bioenergetics Thermodynamics is a queer science. It is a system of logic based on three postulates which have never been proved or disproved. By clever juggling with symbols and ideas, it establishes relations between different forms of energy .... These are most interesting relations which allow us to peep behind the scenes of Nature's workshop .... Thermodynamics may yet tell us how Nature molds such complex phenomena as muscular contraction out of simpler reactions . (A. Szent-Gyorgyi. 7 ) INTRODUCTION Scope The manipulation of the energy available from many natural sources has been a problem of deep concern to man since the realization of the facts of motion. Then came the mastery of fire; the kinematic machine; the use of chemicals for ballistic purposes; and the water wheel for milling, and later for producing the most versatile energy form of them all: electricity. Our age is witnessing the development of the peaceful uses of atomic energy, the energy of nuclear reactions; and a slower but perhaps more far-reaching de- velopment of methods of trapping and storing the sun's radiation as heat, and chemical and electrical energy. Thermodynamics is the study of general principles which relate to trans- fer of energy from one form to another (Figure 7-1). By contrast with some of the more clearly understood systems, bioenergetics is still in its infancy, although biochemists have done much toward describing the energetics of 161 162 A CONCEPTUAL INTRODUCTION TO BIOENERGETICS (a) heat 100 % conversion (b) Figure 7-1. Energy Interconversion, (a); (b) Degradation of Different Forms of Energy into Heat Energy (the "Heat Death"). some pertinent chemical transformations, and physiologists have done some- thing toward relating chemical energy and work. The many relationships which must exist in living systems among mechanical, electrochemical, chemical, and heat energies are as yet poorly known. This chapter attempts to summarize the conclusions which arise from a generalized approach to energy transfer, and to indicate how far they can be carried into a descrip- tion of the living system. In this account, use will be made of three different types of symbols, small-case letters, capital letters, and script capital letters, which usually refer to 1 gram, to 1 mole, and to the whole system, respectively. The capi- tals and script capitals have the further property of being "variables of state" — being variables the value of which help to define the state or condi- tion of the system or subject, irrespective of past history. This will become more clearly understood as the subject is developed. Some Useful Definitions Energy (from the Greek word meaning "active in work") — usually defined as the potency for doing work. Remember the difficulties with definition raised in Chapter 2? kinetic Energy (KE) — energy of motion; energy contained within a bound- ary by virute of the motion of the parts contained therein. Potential Energy (PE) — literally "energy of position' 1 ; more generally energy stored in any metastable but convertible form. Heat Energy (HE or q) — in terms of the kinetic theory, identically equal to the kinetic energy of motion (rotations, vibrations, translations) of the com- ponent molecules. LAWS OF THERMODYNAMICS 163 Specify Heat (c) — the heat energy required to raise 1 g of a substance one degree in temperature. A particularly important specific heat is that of water, by which the unit of heat energy is defined: One calorie is the amount of heat energy required to raise 1 g of pure water 1°C, from 3.5 to 4.5°C (where it is the most dense) at 1 atm pressure. Heat Capacity {(.') — the heat energy required to raise 1 molecular vvt of substance 1 deg in temperature. The units of specific heat, c, are cal/deg Cent, g; and of heat capacity, C, are cal/deg Cent. mole. Other forms of energy to be discussed are mechanical, electrical, gravita- tional, chemical, nuclear, etc. Energetics or thermodynamics is the study of interconversion of these. In biological systems the subject is usefully- called bioenergetics. That part of the subject dealing with electromagnetic and matter waves was considered in Chapters 3 and 4, and is expanded in Chapter 9. LAWS OF THERMODYNAMICS Statements of the Three Laws There are three general principles which summarize human experience with energy interconversion. They are negative laws in the sense that they cannot be proved always to hold, but nevertheless never have been known to be violated. The First Law: The first law states simply that energy can be transformed from one form to another but cannot be created or destroyed. After the equivalence of matter and energy were recognized (and proved in nuclear reactions), the law was generalized still further to read: "mass-energy" in- stead of "energy." The Law stands as written, needing no extension, for all cases in which any form of energy is converted into heat: 100 per cent conversion can al- ways be realized. In Figure 7-1 (b) each of the arrows originates in a form of energy other than heat. The Second Law: For any machine which converts heat into mechanical work, chemical into electrical energy, or the like, it is a universal experience that only a fraction can be converted; the rest remains unavailable and un- converted. There is thus an amount of unavailable energy as well as available energy from the conversion. The unavailable, it would be logical to assume, is the heat energy which must remain in the molecules of which the final state (i.e., the product) is composed. The Third Law: At 0°K (-273.16°C), the absolute zero of temperature, at which all molecular motion has ceased, matter should be in a state of perfect order, the molecules being perfectly aligned or oriented, and per- fectly quiet. This law is concerned with the absolute heat energy contained in molecules at any temperature. Although our present interest is in changes 164 A CONCEPTUAL INTRODUCTION TO BIOENERGETICS from one state to another, rather than absolute quantities in any state, the absolute quantities disclosed via the Third Law permit easy evaluation of the changes. More Detailed Consideration of the First Law. Enthalpy or Heat Content The internal energy of a body is defined as the sum total of all the kinetic and potential energy contained within the body. When expressed per gram molecular weight it is given the symbol U cal/mole, and is a "state vari- able," that is, one whose value depends only upon the temperature, pres- sure, and composition, irrespective of how it arrived at this condition. Heat energy, (that contained in the motion of the molecules), potential energy of the electron cloud of the atom, and the binding energy of the nucleus all contribute to the internal energy. If a transformation takes place in one molecular weight of a substance, two things in general can occur: energy can be taken in by the substance, and work can be done. If an amount of energy, q, is taken in, and an amount of work, w, is done, the difference, q — w, must be the increase in energy of the substance during the process; this difference must be stored as internal energy, and hence the change in internal energy is: AU = q - w where AU U 2 - U x or ^final ' ' ^initial Now AU = q — w is the concise, algebraic statement of the First Law. The concepts are illustrated in Figure 7-2. >> a UJ (a) environment final AU (b) initial State Figure 7-2. The First Law of Thermodynamics: (a) a state diagram showing internal energy change, A il, during a process; (b) the process: heat taken in, q, and work done, w. LAWS OF THERMODYNAMICS 165 One could generalize to complex, nonmolar quantities of varied composi- tion; the law would still be conceptually the same: AU = q - w More will be said about this generalization later. The first law can be extended into a more useful form for processes taking place at constant pressure. Since any substance, this book, for example, has an individual and independent existence in space, and since it occupies a certain volume and has an area upon which the air pressure (i.e., weight of the column of air above it) is 15 lb/sq in., the book does not have as much internal energy as it would have if it were in a vacuum, because it already has done a considerable amount of work against atmospheric pressure. That is, it has already expended enough energy (or "work of expansion"), W, to roll back the atmosphere and create a hole or vacuum in which it can exist. Hence the internal energy U = KE + PE - W The work of expansion, W, can be easily evaluated. Consider the cylinder with frictionless piston of area, A, enclosing a volume of gas, V. From the definition of work: Work = force x distance = PA x AV/A = PA V = P( V 2 - V, ) Since we are considering an initial state, V v of zero volume, in general W = PV. Substituting, U = KE + PE - PV = H - PV where H is the internal energy contained per mole in a vacuum (when P = 0). The quantity, H, is called heat content, or preferably enthalpy because really potential energy as well as heat kinetic energy is included. A little thought about the definition will lead one to the conclusion that H should be a very useful quantity for comparison purposes because its value is independent of any volume change which may accompany a transforma- tion or process. Further, for the case of chemical reactions, AH = H 2 - //, (note the parallel with A U) must be identical with q, the heat taken in dur- ing the process for the case in which the only work done is that of expansion; i.e., q = AH. Many biological processes occur in solution, with no appreci- able change in volume, and in these cases AU = AH. 166 A CONCEPTUAL INTRODUCTION TO BIOENERGETICS Now AH = q may be positive or negative depending upon which is larger, the enthalpy of the final or of the initial state. The former characterizes an endothermic reaction; the later an exothermic reaction. As a general rule anabolic reactions are endothermic; catabolic reactions are exothermic. More specifically, the synthesis of proteins in the metabolism of the living svstem is endothermic; the combustion of glycogen and other food stores is exothermic. For chemical reactions the value of q or AH, the "heat of reaction," can be measured calorimetrically. and quite accurate values obtained. For in- stance, for the simplest reaction H 2 + 1/2 O, = H 2 the heat of reaction &H = #final - ^initial = H( 1 mole FTO) - H( 1 mole H 2 + 1/2 mole 2 ) and although the absolute value of the enthalpy (or internal energy) for neither reactants nor product is known (Who knows how to determine the sum of all the potential energies in the nucleus, for example?), the difference, AH, can be obtained with great precision: —57,798 cal/mole at 25°C, the minus sign indicating that the reaction is exothermic. An especially useful heat reaction is the heat of formation, AH., the enthalpy change which occurs during the reaction by which the molecule of interest is formed from its elements. Actually the example above was a formation reaction. Another now follows: 6CW + 6H 2 (£) + 3 2 (£) = C 6 H,,0 6 (glucose) AH f = -279,800 cal/mole From a table of heats of formation, heats of reaction can be computed as AH = (A//,) producls - (AH f ) reactants The heat of combustion or burning of glucose could be computed, from heats of formation, from the following reaction: C 6 H 12 6 + 6 = A5' + q' hm + q' a + 7-AS then -3000 = AS' - 1400 + q\. x + 7~AS If the food taken in and burned was glucose, for example, XAS can be evaluated as follows. A A JC of -3000 Cal arises from 4.5 moles of glucose (Table 7-3), and therefore TA$ = 310degK x 4.5 moles x 514 cal/deg mole = 700 Cal Our problem then reduces to q ' cx -f AS' = -2300 Cal. The value of total rate of heat loss has been measured for man in many aspects (look ahead to Table 8-1 1), and in an average day q' ex can be at least as large as the basal metabolic heat loss, and usually runs in excess of 2000 Cal. Therefore -AS' will be less than 300 Cal. The external work AS' can be roughly estimated, especially for an unskilled laborer. Suppose he is required to dig a hole 8 ft square and 4 ft deep; the work of lifting alone is about 30 Cal, and this represents at most a third of the total work expended in loosening, picking, and lifting operations associated with the job. Loco- motion, eating, and the other daily external expenditures probably account for the rest of the 300 Cal of external work. An estimate of the internal work done per day can also be obtained. In our example above, the total free energy available was 3700 Cal (3000 + 700). If the efficiency, S , was 37 per cent, then AS' + AS' int = 1370 Of this, about 300 Cal was external work, A^', as we saw above; and there- fore the internal work, A L J' int , which kept the metabolic process running, was about 1170 Cal, 34 per cent of the metabolic heat loss, q' . The reader is invited to consider other aspects of man's life and work from this point of view: to put other estimated values into the Second Law and juggle them about, hence to become familiar with both the clarity of concept and the difficulty of successful detailed application at the present state of knowledge. THE DRIVE TOWARD EQUILIBRIUM 175 THE DRIVE TOWARD EQUILIBRIUM The Driving Force It is a familiar fact that if two mechanical forces of difFerent magnitude oppose each other at a point, the resulting movement will be in the direction of the larger force. Similarly, it seems almost axiomatic that if two systems of different free energy. F, are made to oppose each other, provided they are able to interact, the interaction will proceed in the direction of the larger. For chemical reactions, if the free energies of formation for reactants and products are known, then the free energy of reaction. AF, is simply the dif- ference between the two. This value, AF, represents the maximum amount of work available from the reaction of 1 mole of reactant into product. Since AF = F finaj - /'„„,,,,, a negative value of AF means that the reaction will proceed spontaneously from reactants to products. Such a reaction is said to be exergonic. If (see Figure 7-3) AF is positive, free energy must be supplied from the outside — another reaction perhaps — before reactants will go into products; the reaction is said to be endergonic. The analogy with exothermic (negative AH) and endothermic (positive AH), introduced earlier, is obvious. State Figure 7-3. Free Energy of Initial and Final States. For exergonic (free energy-producing) processes, AF (= F fin — F in ) is negative; for endergonic (free energy-consuming) processes, AF is positive. The energy-producing reactions in the living system are numerous. Nearly all the primary sources are the combustion of food products. By suitable carriers the free energy required by the endergonic syntheses of anabolism is trapped and carried through the blood stream to the locations at which the synthetic processes take place. Naturally, free energy is not a driving force, although it is often considered as such. Nor is the partial molal free energy, (dF/dn) T

r ^ r 0— p--o-p-o-P'-o--ch. o n- :h / N w O" o o c H (L) H H C J .'/I C--C H I I OH OH HC I NH It enters many chemical reactions in the living cell, coupling, in some un- known manner, in such a way that the free energy of hydrolysis (splitting off the terminal phosphate group at L), or dephosphorylation as it is often called, —7.7 kcal/mole, is passed to the reaction to which it is coupled. For example, adsorbed on the enzyme myosin in muscle, the molecule hy- drolyzes, and the free energy appears as the mechanical work of contraction of the muscle; coupled with RNA it supplies energy for protein synthesis. Its hydrolysis products are adenosine diphosphate (ADP) and phosphate ion(P). To become rephosphorylated, as it must, it is carried to the "energy fac- tory" of the cell, the mitochondrion (there are 50 to 5000 of these little double-membraned, 2- to 5-micron bodies per cell), and there the ADP and P are coupled with some step of the respiratory enzyme's oxidation of glucose by 2 , receiving the 7.7 kcal of free energy needed to force the ex- pulsion of water and the regeneration of ATP. In plants, the recoupling can occur photochemically through chlorophyll and its enzyme system. The re- action can be represented as: "discharging" ATP + H 2 , ADP + P "charging" 178 A CONCEPTUAL INTRODUCTION TO BIOENERGETICS and it is reversible. Left to right, it couples in wherever free energy is needed throughout the living system. Right to left it becomes "charged back up," ready to supply energy at another site. Now the living system is not wasteful of free energy without a good pur- pose, such as to keep the system warm in a cold environment. Thus most endergonic processes occur in steps of about 8 kcal/mole, or slightly less, making full use of the free energy of the hydrolysis reaction. Likewise the oxidation of foods also goes in steps of slightly more than 8 kcal/mole each, so that the charging reaction is also not wasteful. Indeed, the very complex sets of steps in the oxidation of carbohydrates, fats, and proteins seem de- signed so that at several stages of each the ADP + P can couple in and be condensed into ATP. This is the principle of the Krebs (citric acid) cycle, for instance, in which it is estimated that 38 ATPs are reformed per mole- cule of glucose oxidized to C0 2 and H 2 0. This number permits an estimate of the efficiency of the recharge process to be made: 8 kcal/mole of ATP x 38 ATP's inn „ '- x 100 = 37 per cent 824 kcal/mole of glucose This efficiency is very respectable, especially since the reactions are going very fast. By contrast, a steam or diesel engine could probably do 20 to 30 per cent on glucose (for a short while!), and up to about 35 per cent on gasoline or oil; solar batteries can convert only about 10 per cent; and thermoelectric converters about 5 per cent from the fuel (including nuclear, or radioactive fuels). Other (like ATP/ADP) electrochemical devices — eg. batteries and fuel cells — are able to give very high efficiences (>80 per cent) if operated slowly, much less if required to operate very fast. A simple calculation (note the approximations) will emphasize the im- portant point of how efficient the human machine really is. Man's basal metabolic rate is about 70 kcal/hr. If this is all expended through ATP, the turnover (charge-recharge) rate is 70/8 ~ 9 moles ATP/hr. If we assume that a 150-lb man of density about 1 g/cc contains on the average 10~ 4 moles ATP per liter, the turnover time for ATP is: 150 1b x454g/lb x 1 1/1000 g x 10~ 4 moles/1 „- — 2 — x 3600 sec/hr ~ 30 sec 9 moles/hr That is, each ATP molecule in the body is hydrolyzed and reformed about once every 30 sec! At this speed of discharge and charge, a man-made bat- tery would have an efficiency well below 1 per cent. Indeed, it would burn up in the attempt! Hence 37 per cent in the living system is truly remark- ^ , , , , • , / c 70 kcal/hr able. To supply the basal energy, it burns the equivalent ot ~ 1 / g ~4 kcal/g glucose each hr, 24 hr a day. REDOX SYSTEMS; ELECTRON TRANSFER PROCESSES 179 The ATP-ADP system is one of a class of oxidation-reduction (redox) or electron-transfer systems operating in the living being. There are many others. REDOX SYSTEMS; ELECTRON TRANSFER PROCESSES Equivalence of Electrical and Chemical Energy Oxidation-reduction reactions have very wide exemplification in living systems: They bring about energy-producing oxidations of food; electro- chemical reactions in the brain and nerve; hydrogenation of oils and dehy- drogenation of fats and sugars, etc. Some are simple electron-transfer re- actions, the reaction Fe+ 2 __» p e +3 + g - for example. The free energy of this /W/-reaction (There must be a place for the electron to go!) can be trapped as un-neutralized electrons — i.e., as electrical energy. In fact if a metallic or molecular electron-acceptor is present at the site, such as H + + e~ — 1/2 H 2 the chemical free energy of the total reaction 1/2 FT + Fe +3 — H + + Fe +2 can be drained off as electrical energy. This transformation is almost re- versible (and therefore highly efficient), even at fairly high speed. The free energy of oxidation of foodstuffs is guided by a series of redox enzymes through a particular reaction scheme, in which each step of the process is a fairly efficient redox process. Most of the free energy of each step is trapped as an electron per molecule, and then passed on at the site where it can be used. Equivalence of electrical and chemical energy is a requirement of the First Law. Thus AF calories/ mole of reaction must be equal to the electrical energy derived per mole of reaction. Now Faraday showed about 1830 that 96,500 coulombs (amperes x seconds) are required to oxidize or reduce one equivalent weight of redox substance; and one equivalent weight is defined as the weight which will transfer one electron per molecule. Hence if the number of electrons transferred per mole, or the number of equivalents per mole, is n, and if 96,500 cou/equiv is abbreviated to F, then the product nF is the number of coulombs required to oxidize or reduce 1 mole. But elec- trical energy in joules is volts x coulombs. Therefore -AF = nF E What voltage is E? It is the voltage measured between the hydrogen end 180 A CONCEPTUAL INTRODUCTION TO BIOENERGETICS and the ferrous-ferric end of the reaction cell. To make this measurement, and thereby to measure AF, one might simply bubble hydrogen over a piece of platinum (the metallic contact) in \N-ac\d soution; and attach the plati- num through a voltmeter to another platinum piece sitting in equimolar fer- rous and ferric salt solution. The two solutions must be connected if the circuit is to be complete. The value measured in this case is 0.77 v, con- sistent with a free energy of reaction of about 40 kcal per mole of hydrogen consumed. The ferric end is positive to the voltmeter, the hydrogen negative. The concentrations may not be as stated, however, and we would expect, and indeed find, that the voltage measured would then differ from 0.77. The conditions specified in our example are arbitrarily chosen "standard state conditions": unit (1) activities of reactants and products, 1 atm pressure, 25° C; and reversibility. We have already seen what a deviation from unit activity ratio will do to AF. Purely as a matter of convenience and of convention, since the absolute value of no redox system is known, the normal hydrogen electrode (NHE) (1 atm pressure, normal acid, and H 2 on platinum) has been chosen as the standard reference, and defined as zero volts. All other redox systems are referred to this standard. In fact a table has been drawn up of known standard redox potentials, F°'s, and is called the electromotive series. However, a special table has been drawn up for biological redox systems. It differs from the standard F°'s, referred to the NHE, in two ways: all the redox reactions are measured against hydrogen at pH = 7, not zero; and since the effective concentrations or activities are not usually known for bio- logical molecules, measured concentrations are used instead; and the tabu- lated values, E ml , refer to equal concentrations (midpoint, m) of oxidized and reduced form (i.e., material 50 per cent oxidized). Table 7-4 lists some of these. A very complete discussion of biological redox systems is given in the remarkable book of W. Mansfield Clark, 2 who has spent a lifetime making a systematic study of, and attempting to organize our knowledge of this subject. Free Energy and Concentration. The Nernst Equation The free energy of reaction, and hence the emf, F, of reaction, varies with the concentrations, as is evident from the relation between AF and K given above. Insertion of nFE° for -AF°, and nFE for -AF, and rearrange- ment gives the famous expression of the emf as a function of concentrations, introduced just before the turn of the century by Walther Nernst: DT E = F°- —\ n (a m /a T J nb REDOX SYSTEMS; ELECTRON TRANSFER PROCESSES 181 TABLE 7-4. Redox Potentials f m7 of Some Important Biochemical Reactions. Steady-state Redox Process Ki Redox Catalyst Hydroxide ions - oxygen + 0.80 + 0.35 Ferrous - ferric + 0.29 cytochrome A + 0.25 cytochrome C + 0.14 hemoglobin Succinate - fumarate 0.00 -0.04 cytochrome B Alanine - ammon. pyruvate -0.05 -0.06 flavoprotein Malate - oxalo acetate -0.10 Lactate - pyruvate -0.18 riboflavin Ethyl alcohol - acetaldehyde -0.20 Hydroxy butyrate - acetoacetate -0.28 -0.32 DPN (diphosphopyridine nucleotide) -0.35 glutathione (estimated) Cystine- cysteine -0.39 Hydrogen - hydrogen ions -0.42 Pyruvate - carbonate + acetyl pH -0.48 Acetaldehyde - acetate -0.60 Note: At pH 7, and at 50 percent oxidation, measured against the normal hydrogen electrode. Values given are approximate. Complete data on these and many other biological redox systems are given by Clark. 2 E° is the value when the ratio of activities of oxidized and reduced species is unity (In 1 = 0), and the second term is the correction for any ratio not equal to unity. Usually T is 37° C (310°K); R is always 8.3 jou/deg mole, F is always 96,500 cou/equiv; and In x = 2.303 log x. Insertion of these numbers gives the common form of the Nernst Equation „ „ (1060 , E = £° log (a ox /a red ) For the simplest case, H 2 = 2H + + 2e the a red = 1, being an element; n = 2; and since pH = -log (a H .), and E° = by definition, the emf of the hydrogen electrode, referred to the NHE, as a function of pH is: E = -0.06 x pH volts 182 A CONCEPTUAL INTRODUCTION TO BIOENERGETICS Plots of E vs a H + and of E vs pH are shown in Figure 7-5. It can be seen that at the physiological pH of 7, E ml on the \HE scale is —0.42 v. 0.0 0,4 2 -0.82 -m 7 (all reduced) 100% (all oxidized) % oxidized (b) Figure 7-5. Reversible Potential of an Oxidation-Reduction Reaction: (a) as a function of pH,onthe normal hydrogen electrode (NHE) scale; (b) as a function of per cent oxidation. Definition of E m7 : potential (on the NHE scale) when pH = 7 and when the redox system is 50 per cent oxidized. As a further clarification and as a summary, Figure 7-6 shows schema- tically the relation between the NHE scale of £"°'s (pH = 0), to which AE values have been traditionally related through —AE = nEE, and the physio- logical scale, E m7 (pH = 7). The latter is now commonly used as a relative measure of free energy changes in biological reactions. The values in Table 7-5 have been measured simply by putting a platinum wire into a mixture of equal concentrations of sodium succinate and sodium fumarate at pH 7, containing an enzyme and a mediator (discussed later), and measuring its voltage against a hydrogen electrode in the same solution. Such measured values can be used to predict the direction of reaction, or as a basis for com- parison, but not for the determination of AE, because the effective con- centrations (activities) are not known. It is well to be clear on this limitation of the £" -, listing;. Difficulty often arises in this subject because of notation. Different authors use different subscripts and superscripts. In this book we have de- fined, and use, only E, E°, and E m7 . One should be aware of the variations which one may find. Further, one should understand clearly that the values given in the table for intermediary processes of oxidation are midpoint values; that although these redox systems are generally poised at their most stable point (Figure 7-5), a tight control must be kept by the living system at all times on the concentration of oxidized and reduced states of each system; REDOX SYSTEMS; ELECTRON TRANSFER PROCESSES 183 that too much variation could cause a normally proceeding reaction actually to go backwards ! A special application of the Nernst Equation is discussed under concen- tration cells. + 1.22 + 0.80 -0.42 pH = pH=7 Figure 7-6. £ m7 's (center vertical line), and Their Relation to the Corresponding P's. (See text and Table 7-4.) Balky Redox Reactions There are three tricks provided by nature to promote electron exchange in oxidation-reduction reactions. The first is catalysis : providing a surface or a site on which the exchange can rapidly take place. For example, electrons exchange immeasurably slowly between H 2 and H + in solution, but if a sur- face such as finely divided platinum metal is added, electron exchange is rapid, and the potential readily manifested. The second trick is the use of an indicator redox system. If one wishes to know the redox potential of a solution in which the electron transfer is slow or sluggish, one can add a very small amount of an entirely foreign redox system, which exchanges electrons rapidly with the system of interest, and which is either itself highly colored or exchanges rapidly at a metal elec- trode. In the first case the depth of color of the resulting solution can be related to the redox potential; and in the second case the potential can be read directly against a reference electrode. Methylene blue, a colored redox dye, is one of a class of dyes commonly used for this purpose, while the addi- tion of a small amount of potassium iodide often will permit direct measure- ment of the redox potential of the solution against some suitable reference 184 A CONCEPTUAL INTRODUCTION TO BIOENERGETICS electrode. If the redox indicator (KI, for example) is present to an amount much less than the redox systems in the solution to which it is added, it can exchange electrons (KI — * I 2 ) until its potential (determined by a x /a KX ) is the same as that of the solution. The third trick is really a combination of the first two. If a solution con- tains two reactants, such as glucose and oxygen, which can react together spontaneously (negative AF), the reaction will be extremely slow unless the solution contains mediators. Consider one step in the over-all process, for example succinate added to pyruvate in a test tube. Although these two ions can exchange electrons (and hydrogen atoms), with the liberation of free energy, they don V unless a redox system such as cytochrome-C is present as a mediator. Its job is to couple with succinate and reduce it to fumarate, then (itself now oxidized) to oxidize pyruvate. In other words it provides a path by which the over-all reaction can go in two steps, via the mediator, whereas it could not go at all in one. The whole respiratory enzyme sys- tem is a system of mediators, permitting the complete, controlled oxidation of glucose by oxygen to go in discrete - steps, the free energy of each step being thus made readily available to recharge ATP, for example, and there- fore to be usable elsewhere in the system. There seem to be no generic differences among electrochemical catalysts, redox indicators, and mediators. The name used depends upon one's point of view. Indeed, in his classical work on the succinate-fumarate system, Lehman (1930) called succinic dehydrogenase the catalyst and methylene blue the mediator. MEASUREMENT OF AH, AF, AND TAS The simplest way to measure all three energies is in an electrochemical redox cell, described in the previous section, if indeed the reaction is an oxidation-reduction reaction. Thus AF is directly related to the voltage on the NHE scale by -AF = nFE, and A S is directly related to the rate of change of AF with temperature through the relationships' ^1=-A<>; and AS = nF d -^- dT dT Since AFand AS can be so determined, AH can be obtained from the Sec- ond Law: AH = AF + TAS However, AH, the heat of reaction, is itself hard not to measure! If no work at all is extracted in a calorimeter experiment, as a process is allowed to go spontaneously to equilibrium, all the free energy is wasted away into heat, and A His the quantity of heat measured in the experiment. CONCENTRATION CELLS; MEMBRANE POTENTIALS 185 Measurement of the equilibrium constant, in the usual manner, gives a measure of AF, since -AF = RT\n K eq Further, d In K eq AH dT " ~RT~ 2 and therefore measurement of the equilibrium constant at several tempera- tures allows evaluation of A //by an alternative method. The Third Law, stated early in this chapter, provides another avenue for the determination of the thermodynamic energies. The law says that the entropy of all elements in their stable states (viz., S °) is zero at absolute zero temperature (where all molecular motion ceases). Thus the entropy of all pure substances at 0°K is also zero. Further, the entropy at the normal body temperature of 37°C is the sum of all the little ways heat energy can be stored by the material; and it can be evaluated from the heat capacity, C. , of the substance measured at different temperatures from 37° C down to abso- lute zero. Within the past 25 years, literally thousands of "third-law en- tropies" have been so evaluated. Table 7-5 lists some of these values for biologically important molecules. Then, as Szent-Gyorgyi, 13 the energetic contemporary physiologist, so aptly stated in the quotation which opened TABLE 7-5. Some Free Energies of Formation and Third Law Entropies. -Af f ° (Cal/mole) S Q (cal/deg mole) H 2 0(1) 56.7 16.75 H 2 0(g) 54.7 45.13 NaCl(s) 91.7 C 2 H 5 OH 40.2 38.4 C 12 H 22 O n (sucrose) 371.6 C0 2 (g) 51.08 HAc 94.5 38.0 this chapter, a large, formal system of very useful numbers has been calcu- lated and tabulated from known experimental results. The National Bureau of Standards, Washington, D. C, has published handbooks of useful data. Tables 7-1 and 7-3, as well as 7-5, present very carefully selected samples, of biological and medical interest. CONCENTRATION CELLS; MEMBRANE POTENTIALS If two vessels containing different concentrations (two glass vessels con- taining 2 at different pressures joined by a closed stopcock; or two salt 186 A CONCEPTUAL INTRODUCTION TO BIOENERGETICS solutions of different concentrations separated by a suitable membrane. Fig- ure 7-7) are allowed to interact, the difference in free enegery, AF, can be manifested by transport or movement of molecules or ions. By a rather neat argument involving the dependence of electrical potential upon concentra- tion of ions, it can be shown that the A/ 7 can also be manifested as a poten- tial difference in such a system. With suitable electrodes the value can be measured. A form of the Nernst equation relates the emf of this concentra- tion cell to the ratio of the salt activities. Thus 0.060 log (a, /a,; This equation shows the relationship between the potential and the activity ratio for condition of no transport across the interface. For example, for a cell composed of IN - NaCl:0.1A r - NaCl, in which the activity ratio is about 10, the value of £ conc = 0.060 v ( = 60 mv). I I salt in J water) membrane dif f use interface a i greater than o 2 Figure 7-7. Concentration Cell (left); with Transport (right). If flow or transport of ions or water occurs, and it usually does to some extent across living membranes, the value observed, E, differs from E by a "diffusion potential," E m , which can be approximated by either the Hen- derson (1911) or Planck (1915) equations, and measured, approximately, under certain rigorous experimental conditions. Thus, E = £ 'diff Values 50 to 100 mv are found routinely in living systems, across the mem- branes of nerve cells and red blood cells, for example (see Table 7-6). These values are due principally to potassium chloride concentration differences across the membranes. It is interesting to note that in the electric eel, simi- lar cells are arranged in series, and potential differences of 200 to 1000 v are usually observed! In nerve, the stationary values of about 80 mv are modi- fied rapidly with passage of a stimulus, due to a change in permeability. NEGATIVE ENTROPY CHANGE IN LIVING SYSTEMS 187 TABLE 7-6. Membrane Potentials, E, Observed, and Calculated from Measured Concentration Ratios Across Cell Walls. E (millivolts) System KCI cone / KCI cone inside / outside Observed Calc by Nernst Eq. Loligo (squid) nerve axon 19 1 50 to 60 74 Sepia (cuttlefish) axon 21 1 62 77 Carcinus nerve cell 34 1 82 89 Frog muscle cell 48 1 88 98 Human muscle cell 50 1 85 to 100 99 Actually, any activity difference between two solutions separated by a mem- brane is a sufficient condition for a membrane potential to exist. Three cases will give rise to a potential difference: ( 1 ) Two concentrations of the same salt (restricted flow). (2) The same (or different) concentrations of two different salts. Even though the concentrations are the same, the effective concentrations or activities differ because of different interactions with the solvent and with each other. (3) Free flow through the membrane, except for one macromolecular ion. This is a rather famous equilibrium, exemplified across living cell walls, and described quantitatively by Donnan. To sort out these possibilities on living membranes is one of the hardest tasks in biophysics today. The subject will be considered one step further: the time-variation of the potential across nerve-cell membrane (Chapter 10). NEGATIVE ENTROPY CHANGE IN LIVING SYSTEMS The concept and the quantity entropy has been very carefully introduced in a simple manner, as a specific heat — a very special specific heat, to be sure — and this idea of entropy is sufficient for many considerations. But the implications are more far-reaching than at first suspected. Thus, an increase in entropy during the course of a reaction was described as meaning that the modes of rotation, etc., of the products were more numerous than those of the reactants. This interpretation means that the amount of complexity in the system has increased with reaction, and could be rather loosely ex- tended to mean that the amount of disorder in the system has increased. Thus the extra heat, q', lost during a process done in a nonreversible manner con- tributes quantitatively to the disorder of the system and its environment. The idea of entropy being associated with disorder or randomness can be introduced systematically and logically through statistics. Briefly, the 188 A CONCEPTUAL INTRODUCTION TO BIOENERGETICS method takes the following form: The properties of a quantity, In 12, are considered in some detail, and it is shown that In 12 has the two fundamental characteristics of thermodynamic entropy: (1) that In 12 for two or more in- dependent systems is the sum of the In 12's for all the individual systems — that is, that In 12 is an extensive property dependent upon quantity; and (2) that In 12 increases for all spontaneous changes occurring in a system for which the quantity of material and the energy are held constant. Both these properties have been introduced earlier, although not in just this form. The proportionality constant, R (cal/deg mole), then is introduced to relate S and In 12: S = R In 12 In this development 12 is a pure number, the number of ways in which the particles or parts of the system can be arranged (organized or disorganized). For one of a pair of playing dice the number is 6 (six sides). For a mole, which contains 6 x 10 23 molecules, this number, 12, could be counted out, if we were clever and patient enough! However, approximations can be made through the methods of statistics which give closely enough the num- ber of ways the particles can be arranged. Hence the expression above means that the entropy, S, of a system increases as the number of ways in which the system can be arranged increases. The greater the chaos or dis- order, the greater the number of ways; and the greater the entropy of the system. It has already been shown that all naturally occurring processes, which occur irreversibly, make a positive contribution to the entropy and hence the heat energy of the universe. If there are no violations of the Second Law elsewhere in the universe, the available energy is decreasing all the time, and the universe is approaching the ominous "heat death" or "entropic death," in which the free energy will have reached zero and the entropy a maximum or upper limit. We have then the two interesting possiblities: a one-step creation during which the whole was wound up, from which condi- tion it has been slowly running down ever since; or the continuous violation of the Second Law is occurring somewhere in the universe. An interesting question, then, is: Is continuous creation occurring within the living thing? Hence, one of the more important aspects of this study of entropy changes centers on the fact that, although the net result of any physical process must be (Second Law) a positive entropy contribution to the universe, there are some processes in which the entropy definitely decreases within a limited space; and it is not very obvious where the overriding increase, if any, occurs to the universe. The process referred to is the creation of the living thing (Figure 7-8), which, although very complex, is certainly not disordered. In fact it is much more highly ordered than the components from which it is NEGATIVE ENTROPY CHANGE IN LIVING SYSTEMS 189 made. Growth of the living system, controlled from the outset by a molecule such as DNA (desoxyribonucleic acid), must be one of the great "consumers of entropy" or "producers of negative entropy" Is it in the growth of an ever-increasing number of living individuals that we find our continuous creation? .... Although during death and decay the order of life is gradually replaced by disorder, the quantity of physical order existing at any one time seems to be increasing each generation, and higher social and economic order runs parallel with the higher physical order of a larger population. Expanding Universe (entropy increasing) r ii i i \ i ' — — ~z LJ 1 1 Protein Molecule (very complicated but highly ordered) Figure 7-8. Entropy Changes. Growing Li ving Thing (entropy decreasing) Some attempts have been made go give quantitative expression to these ideas. Most of these attempts since 1930 involve the concept of the "steady- state," which is treated in the next chapter; but even these attempts do not permit the use of numerical examples, and although inherently very interest- ing, cannot be treated quantitatively in this book. On the other hand, per- haps Teilhard de Chardin was right when he suggested that, taken as a whole, the universe is evolving toward a single, highly organized arrange- ment in which all the ("living") elementary particles of matter have achieved their ultimate state of development; that as living systems organize them- selves more and more, over many more thousands of years, the statistical expression of behavior in terms of the average of random motion of many subparticles, will gradually give way to expressive dominance by the grand ensemble of organized living things. Unfortunately we simply have no way at all of evaluating the sociological and economic interaction energies, nor indeed the psychological, spiritual and moral energies of our own minds. Armed with the background presented in Chapters 4 to 7, the reader will now want to push on more deeply into certain aspects of energy transfer in 190 A CONCEPTUAL INTRODUCTION TO BIOENERGETICS living systems. It is recommended that he take the appetizers, References 13 and 14, before he starts the full courses offered by References 2, 6, 10, or 15. PROBLEMS 7-1 : If a man submits to a diet of 2500 Cal/day, and expends energy in all forms to a total of 3000 Cal/day, what is the change in internal energy per day? If the energy lost was stored as sucrose (390 Cal/100 g), how many days should it take to lose 1 lb? (Ignore water loss for this problem.) 7-2: (a) From the following heats of formation at 25° C, compute the heat of com- bustion (i.e., the "fuel value") of d-glucose. Give the answer in Cal/mole and Cal/gram. \H f All elements (Na,0,, etc.) CO, -94.4 Cal/mole HX> -64.4 C 6 H 12 6 (d-glucose) -279.8 (b) Given the heat of combustion of sucrose to CO, and H 2 to be 1349 Cal/mole, compute the heat of formation from the elements. 7-3: (a) From the values given for AH and AFfor any two reactants tabulated in the text, calculate the entropy change per mole, (b) For each of these two cases, calculate the standard emf of the reaction. Are these values for pH = 7? 7-4: Given the fact that the standard emf 's for the redox systems methylene blue and maleate-succinate are respectively 0.05v and 0. 1 v, at the physiological pH of 7, calculate the standard free energy of reaction (at pH = 0). {Note how important it is to define the pH, or alternatively that the living system keep its pH con- stant.) 7-5: (a) Using the Nernst equation, plot E as afn of pH for: (i) 1/2 H 2 — H + + e" -0.42 v (ii) succinate — * fumarate 4- 2H + 2e~ -0.00 v (iii) 4 OH" — O, + 4e" + 2H 2 +0.80 v (iv) Cu — Cu ++ + 2e"atpH = 7. +0.36 v (b) If E Q = 0.50 v and n = 2, plot E as fn of per cent oxidation from to 100 per cent. REFERENCES 1. Clark, VV. M., "Topics in Physical Chemistry," 2nd ed., The Williams and WilkinsCo., Baltimore, Md., 1952. 2. Clark, W. M., "Oxidation-Reduction Potentials of Organic Systems," The Williams and Wilkins Co., Baltimore, Md., 1960. 3. Fruton,J. S., and Simmonds, S., "General Biochemistry," John Wiley & Sons, Inc., New York, N. Y., 1953. 4. Glasstone, S., "Thermodynamics for Chemists," D. Van Nostrand Co., Inc., New York, N.Y., 1947. REFERENCES 191 5. Kaplan, N. O., in "The Enzymes — Chemistry and Mechanism of Action.'' J. A. B. Sumner and K. Myrback, Eds., Vol. II. Pan 1, Acad. Press Inc.. New York, N. Y., 1951. 6. Sodeman, VV. A., Ed., "Pathologic Physiology: Mechanisms of Disease," 2nd ed., W. B. Saunders Co., Philadelphia, Pa., 1956. 7. Szent-Gyorgyi, A., "Thermodynamics and Muscle," in "Modern Trends in Physiology and Biochemistry," E. S. G. Barron, Ed.. Acad. Press Inc., New York, N.Y., 1952, p. 377. 8. Teilhand de Chardin, P., "The Phenomenon of Man." Harper & Bros. London, 1955. 9. Wilkie, D. R., "Thermodynamics and the Interpretation of Biological Heat Measurements," Prog, in Biophys., 10, 259 (1960). 10. Augenstine, L. C, Ed., "Bioenergetics," Acad. Press, New York, N. Y., 1960: dealing mainly with energy absorbed from radiations. 11. West, E. S., "Textbook of Biophysical Chemistry," The Macmillan Co.. New York, N. Y., 1960: good discussion on energy of metabolism, with worked examples, p. 386, eg. 12. George, P. and Rutman, R. J., "The 'High Energy Phosphate Bond' Concept." Prog, in Biophys., 10, 1, 1960. 13. Szent-Gybrgyi, A., "Bioenergetics," Academic Press, New York, N. Y., 1958. 14. Lehninger, A., "How Cells* Transform Energy," Scientific American. 205, 62 (1961). 15. Oncley,J. L.,eial., Eds., "Biophysical Science — A Study Program," John Wiley & Sons, Inc., New York, N. Y., 1959: papers by Lehninger, Calvin, and others. 16. Lewis, G. N., and Randall, M., "Thermodynamics," revised by K. S. Pitzer and L. Brewer, McGraw-Hill Book Co., Inc., New York, N. Y., 1961. CHAPTER 8 Speeds of Some Processes in Biological Systems The ultimate goal of biophysical kinetics is the understanding of that remarkable integration of heat, mass, and work transfer by chemicals which maintains so reliably the steady-state condition in every spot in the living system. INTRODUCTION Biophysical kinetics is the study of the rate or speed at which chemical reactions or physical processes take place. Factors which influence the speed are elucidated in detail, when possible, by experimental methods, and are then analyzed in terms of the actions of the molecules which give the over- all result. It is the study of mechanism of reaction, and of molecular mech- anism in particular. Kinetics is formally defined as "that branch of dynamics which treats changes in motion produced by forces." It is the purpose of the subject to define and interpret these forces, which may be functions of temperature, pressure, molecular interactions, concentration gradients, electrical poten- tial, etc. Within the broad field of kinetics there are two main subjects which are of interest in biology: ( 1 ) Kinetics of chemical reactions in solution. (2) Kinetics of physical process such as diffusion, fluid flow, transport of electrical charge, and heat conduction. The basic principles of the main subject are sketched, and then each of the subjects of particular interest is considered. Since chemical re- 192 GENERAL PRINCIPLES 193 actions are covered more or less comprehensively in textbooks in biochemis- try, and since physical processes are very numerous in the living animal but usually receive very little attention from the kinetic point of view, most of the effort is put on the kinetics of physical processes. The presentation em- phasizes the formal similarity of all these processes, and the fact that there are many common factors upon which the rates depend. Unfortunately we do not know enough at this time to achieve very much of the ultimate goal mentioned in the Foreword. GENERAL PRINCIPLES Rate-Controlling Step If any physical or chemical process goes from initial state to final state through a series of intermediate steps, usually one of those steps is inherently slower than the others and controls the rate of the over-all process. For example, a bucket brigade passing pails of water hand to hand from the river to the burning house can transport water no faster than the little old lady who forms the slowest link. The principle is true for chemical and physical processes as well. In most processes in which we are interested, the over-all process involves physical transport as well as chemical reaction. One of the physical steps or one of the chemical steps may be rate-determining. A measurement of the over-all rate or speed is always a measure of the speed of the slowest step. Consider the chain of events: If the reaction B — * C is the slowest, then the over-all rate is the rate of B — C. (As an exercise, apply this principle to the over-all event of free air becoming dissolved in the blood stream. What would you expect to be the slowest step?). Equilibrium If a process can proceed forward or backward, starting as either reactants or products and produce products or reactants, respectively, the process will move spontaneously (although perhaps slowly) in a direction toward mini- mum free energy for the over-all reaction materials: The reaction will "stop" when the concentrations are such that the work the reactants can do equals the work the products can do, and then apparently the reactions in both directions cease. The materials have then reached thermodynamic equi- librium. The rate of the forward reaction will depend upon the inherent attraction the reactants have for each other, and upon the concentrations of the reac- 194 SPEEDS OF SOME PROCESSES IN BIOLOGICAL SYSTEMS tants. The same is true of the reverse reaction. Thus, if aA + bB^ cC + dD where k ] is the measure of inherent attraction A and B have for each other, the over-all rate of reaction in the forward direction of a moles of A with b moles of B (i.e., i\ = -d[A]/dt, or -d[B]/dt, where [ ] denotes concen- tration), is: ,, = k,[A][A] ••• x [B][B] ••• = k,[A]"[Bf Similarly v 2 = k 2 [C]'[DY This first principle, that of mass action in reaction kinetics, was demon- strated quantitatively by Wilhelmy in 1850. At equilibrium the over-all reaction ceases. Therefore i 1 , = v 2 at equi- librium: k.imBY = k 2 [C\[DY [CYiDY = k L = K [A]"[BY " k 2 ' eq where K is the equilibrium constant. This form of the Law of Mass Action was stated thus by Guldberg and Waage in 1863. For any reaction -AF° = RT\n K eq , which states that the free energy change per mole (eg., refer to sucrose oxidation) is a measure of the position of equilibrium. Steady State Consider again the consecutive process discussed above and consider specifically the case in which the supply of A is unlimited, so that the con- centration of A, [A], never changes. If£, > k 2 , A will be converted into B faster than B will be removed into C, and B will accumulate. Since the rate of the reaction B — ► C is v 2 = k 2 [B] as we saw above, as B accumulates, v 2 increases until it reaches the value of v { . At this point B will have reached its steady-state concentration be- cause the concentration B neither increases nor decreases further. The same is true of the other steps. In the steady-state then »1 = V 2 = V i = V 4 or k t [A] = k 2 [B) = k,[C] = k 4 [D] ON CHEMICAL REACTION RATES; ENZYMES 195 Since the specific rates are all different, the steady-state concentrations are different; but if the process is in the steady-state condition, the concentra- tions are constant. If the back reactions proceed at a measurable rate, the situation is more complicated, but the principles are the same. When you hear the word "equilibrium" used, then think: Which is meant, true equilibrium or steady-state? In the latter case, continuous processing occurs; in the former no net reaction occurs. Figure 8-1 illustrates this difference. source (lake) tumbling stream Equilibrium Steady State Figure 8-1. Equilibrium and Steady State. ON CHEMICAL REACTION RATES; ENZYMES Concentration and Temperature The law of mass action has already been outlined under the discussion of the approach of a system toward true equilibrium. The rate is always pro- portional to some power of the concentration of reactants, and this index is called the "order" of a reaction. There are really two orders obtainable from experiments, one with respect to time, and the other with respect to concentration. These will have the same value if the reaction is a simple one in which the slowest step is the first step, the one which involves reactant concentrations explicitly. If some other step than the primary one is rate-determining, or if products interfere with or inhibit the reaction, the power, a, of the concentration, [A], which describes best the over-all rate may be different from that which describes the initial rate. Complicated cases are not considered here. Some of the simpler cases are collected in Table 8-1, which shows the rate equation and the expression and dimensions of the proportionality constant, k, called the specific rate con- stant, when a = 0, 1/2, 1, and 2. In Table 8-2 are collected values of the specific rate constant for some first and second-order reactions. 196 SPEEDS OF SOME PROCESSES IN BIOLOGICAL SYSTEMS TABLE 8- 1 . Summary of Rate Equations for Some Chemical Reactions. Order* Rate** Equation Expression of Specific Rate Constant Units of Specific Rate Constant v = k k c t moles/ liter sec i/2 v = k(c - c)* k 2 moles'/liter' sec i v = k(c — c) k 1 In °° _, t c - c 2 v = k(c - c) 2 k 1 c liters/ mole sec 1 %(cq - c) Reactions are of the general form: Products ( f o - c ) wherec is initial concentration of A, andf is the amount of some product formed at any time, t. *The index of the concentration in the rate or velocity equation. **Velocityi> = dcjdl. TABLE 8-2. Table of Specific Rate Constants, k Process Order Specific Rate Constant (25° C) Mutorotation of glucose 1 1.03 x 10- 4 sec-' Myosin-catalyzed hydrolysis of ATP 1 3.0 x 10- 4 sec-' Decomposition of N 2 O s 1 3.4 x 10- 5 sec-' Pepsin-catalyzed hydrolysis of a di- aminoacid substrate 1 2.0 x 10- 7 sec-' Decay of Sr 90 1 8.9 x I0~ l0 sec- 1 Pyridine + ethyl iodate — ► N(C 2 H 5 ) 4 I 2 1.25 x 10" 4 liters mole" 1 sec -1 Thermal decomposition of HI (gas) 2 5 x 10~ 4 liters mole -1 sec -1 The rate of every individual chemical reaction or physical process in- creases with increasing temperature, i.e., with increasing kinetic energy in the molecules. This is true without exception. However, in some physio- logical processes an increase in the temperature permits certain side reac- tions to occur, which so interfere with the chain of events that the rate of the over-all process decreases with increasing temperature. For a great many chemical reactions it is found experimentally that the rate of reaction just about' doubles for every 10 Centigrade degrees of rise in temperature. For most physical processes involving mass transfer, the rate goes up from 1.1 to 1.4 times in a 10-degree rise. There are many exceptions ON CHEMICAL REACTION RATES; ENZYMES 197 to these rules of thumb, of course: for example, certain free radical recom- binations have no temperature coefficient of rate; and by contrast the rate of inactivation of enzymes by heat, and of the denaturation of proteins, can in- crease by 1000 times over a 10-degree rise! The last column of Table 8-3 illustrates this point quantitatively. TABLE 8-3. Dependence of Rates or Speeds of Various Processes Dn Temperat jre* Process Activation Energy, E* Rate at 37°/Rate at 27° C Free radical combination 1.0 Free radical + molecule — ► products to 0.3 1.0 to 1.01 Transport in water solutions (diffusion, viscous flow, ion mobility) 1.0 to 5.0 1.06 to 1.28 Transport in fat and lipid (diffusion, osmosis) 8 to 15 1.5 to 2.2 Molecule + molecule — ► products (hydrolyses, neutralizations, rear- rangements and condensations) 10 to 30 1.8 to 5.0 (a) uncatalyzed 15 to 30 2.2 to 5.0 (b) catalyzed 10 to 20 1.8 to 3.0 Denaturation of proteins and inactivation of enzymes 30 to 150 3.0 to 3000 'Different processes of the same general type may have different activation energies Therefore both A* and the ratio of rates are given as a range of values. Units of E* : kcal/mole. In general this dependence upon temperature is understandable in terms of the postulates of the kinetic theory of matter. Molecules are presumed to be in a state of continuous motion and have a heat content (H) which de- pends upon the number of (degrees of freedom of) rotations, vibrations, etc. It is axiomatic that in such a case of random motion not all molecules will contain exactly the same kinetic energy at any one instant. In fact, it is in- herent in the kinetic postulates that the energy distribution must be of the form shown in Figure 8-2. The average heat energy, Q^ v , per mole of material is 1/2 RT (300 cal) for each translational degree of freedom, RT (600 cal) for each vibrational degree of freedom, and 1/2 RT [or each rotational degree of freedom. For a diatomic gas at 27°, then, with one degree of vibrational freedom, two of ro- tational, and three of translational, the average heat energy, Q^ v , is 2100 cal per mole of gas. In any collision of reactant molecules which is to result in reaction, a mini- mum or threshold energy must be involved in the collision, or else the mole- cules will simply bounce off each other. Let this threshold energy be E*. A few molecules will have the excess energy sufficient to react; not every col- 198 SPEEDS OF SOME PROCESSES IN BIOLOGICAL SYSTEMS threshold energy E Energy E Figure 8-2. Maxwellian Distribution of Energies in Molecules. lision need be a fruitful one. At a higher temperature, T 2 , more molecules have the necessary threshold energy to react, and therefore the rate is faster. Experimentally, Svante Arrhenius, about 1889, observed that the rate in- creased exponentially with the temperature. Since in solutions, the con- centrations do not vary appreciably with the temperature, the temperature- dependence is practically all in the rate constant, k. Thus k = Ae- E *' RT where A is a constant in moles per liter per second, E* is the threshold energy in calories per mole, R the gas constant (1.987 cal per degree per mole), T the temperature in degrees K, and "e" is 2.71828, the base of natural logarithms. Taking logarithms of both sides In A = In A - E*/RT or, changing to the base 10, the more familiar system: log k = log A - E*/2.303RT Hence a graphical plot of experimental results of rate measurements at dif- ferent temperatures plotted as log v vs \/T has a slope of -E*/2303 R; and, since R is known, the value of the threshold energy, E*, can be determined (see Figure 8-3). Table 8-3 gives values of E* for different kinds of processes. E* is often called energy of activation as well as threshold energy, and the measured value can often aid in the characterization of the rate-determining step of a process. ON CHEMICAL REACTION RATES, ENZYMES 199 slope A log v A l/T 2.303 R ( T in degrees Kelvin ) Figure 8-3. Arrhenius Plot of Log Rate vs l/T; Determination of Activation Energy. Referring back to Figure 7-3 which describes a process proceeding from an initial state to a final state, we know now from the preceding discussion that it must be modified with the insertion of an activation "hump" or bar- rier (see Figure 8-4). Thus E* is related to the extra heat content, A// } , the heat content change between initial state and "activated" state. activated complex c UJ State (a) | uncatalyzed State (b) Figure 8-4. Enthalpy (a) and Free Energy (b) of Components as They Pass from Initial to Final State Over the Activation Energy Barrier. Note the position of the activated complex, and the energetically easier path of the catalyzed reaction. 200 SPEEDS OF SOME PROCESSES IN BIOLOGICAL SYSTEMS More Factors of the Specific Rate Constant Various interpretations have been given to the pre-exponential term, A. The most successful has come from the theory of absolute reaction rates, which was pioneered by H. Eyring mainly in 1935, and expounded in detail in 1941 in the famous book by Glasstone, Laidler, and Eyring 6 , and since then in most books on physical biochemistry. Essentially the reacting molecules are pictured (refer to Figure 8-4) as proceeding through a state in which they are in a metastable state called the "activated complex," which is more or less in equilibrium with reactants in the initial state, 1. While in this complex, the molecules can either proceed to form product, the final state, or return to reactants, the initial state. If equilibrium can exist between reactants and complex, the thermody- namic functions can apply to this part of the reaction: thus H i - //, = A//*; S x - S x = AS*; and F* - F, = AF*. From statistical mechanical arguments the pre-exponential term by this theory reduces to" k T h where t is a "transmission coefficient," which expresses the fraction of com- plexes which proceed to products (often assumed to be 1.0); k g is the ideal gas constant per molecule (R/6 x 10 23 = 1.38 x 10" 16 erg per deg C per molecule), and h is Planck's constant (6.63 x 10~ 27 erg sec). The over-all rate, then, for a reaction such as that considered on p. 194, is: v = [A) a [B] b T^— g*sv* e-W/xr h It can be seen that a measured value of rate, v, at known concentrations of A and B, plus a measured value of the activation energy, AH*, permits the value of AS X to be obtained. Especially in biological processes has the evaluation of the entropy of ac- tivation been important. Remember, the entropy change tells us whether the heat capacity of the system has increased or decreased during the reac- tion, and since the heat energy contained within molecules increases with the complexity of the molecule, it is often possible to infer certain physical properties of the activated state, and hence of the molecular movements dur- ing reaction. This technique has proved useful in learning about the mech- anism of muscle contraction, for example, certain details of which are con- sidered in Chapter 10. In short, the rate of a process depends upon the concentrations and the temperature, and on the free energy change accompanying the formation of the activated complex from reactants. ON CHEMICAL REACTION RATES; ENZYMES 201 The role of a catalyst is to provide an alternate path which is energetically easier. Thus the catalyst, because of the energetic advantages it offers, acts as a guide-post to direct the reaction through preferred channels or path- ways (see Figure 8-4 (b)). This subject is now explored further. Catalyzed Reactions; Enzymes There are many chemical reactions and physical processes whose rate or pathway is controlled by one or more catalysts. Far surpassing all the rest in importance as biological catalysts are the enzymes. These are large pro- tein molecules, which are often bound with metallic ions and are always heavily hydrated. They have the special property that at some site(s) on the surface both the kinds of atoms and their arrangement are such that more or less specific adsorption of a "substrate" molecule can occur. The substrate molecule is the one which is to undergo hydrolysis, hydrogenation, trans- ammination, or some other reaction. In addition to the kind of atoms and their arrangement, a third essential requirement of the enzyme seems to be the presence, in the vicinity, of a large electric charge, usually in the form of a metallic ion such as Mg ++ , or a charged chemical group, such as -PCV 2 . The role of the charged group is to distort the electronic structure of the substrate molecule as it adsorbs on the enzyme, thus to make it energetically easier for the desired reaction to occur. The most easily measured manifestation of a catalyzed process is a lowered activation energy, E*. Some values are collected in Table 8-4. Note especially the numbers for the decomposition of H 2 2 . TABLE 8-4. Activation Enerc gies for Some Catalyzed Biological Rei actions. Reaction Catalyst £* (Cal/mole) Inversion of sucrose acid(H 3 + ) trypsin-kinase malt invertase 20.6 14.4 13.0 yeast invertase 11.5 Hydrolysis of ethyl but yrate acid(H 3 + ) pancreatic lipase 13.2 4.2 Decomposition of hydrogen peroxide (H 2 2 ) no catalyst platinum Fe + + 17to 18 11.5 10.1,8.5 liver catalase 5.5 Hydrolysis of urea acid(H 3 + ) urease 24.5 12.5 to 6.5 "The suffix "ase" denotes enzyme. 202 SPEEDS OF SOME PROCESSES IN BIOLOGICAL SYSTEMS A typical process can be illustrated as in Figure 8-5, and described as fol- lows, a hydrolysis serving as the example: First the molecule to be hy- drolyzed (the substrate molecule, S) bumps into the hydrated enzyme mole- cule, E; and if the collision occurs at the active site and is energetic enough, a slight bond will be made between the two, forming the enzyme-substrate complex, ES. The complex can then do one of two things: it can fall apart again (in which case we lose interest); or it can be "activated" — i.e., given excess energy by favorable collisions with its neighbors — and associate with a water molecule close by, to form the activated complex, ESK This in turn can either fall apart the way it was formed, or it can proceed on to split up in a new way — as reaction products — with the substrate molecule hydrolyzed and the enzyme ready to go again. The process is sketched at the top of Figure 8-5, the energy of the reac- tion path at the bottom, and the formal equation in the middle. For the purpose of formulation of the rate equations, the reaction can be written: E + S^ ES (formation of the M jchaelis complex, ES) 1 *-i ES ~^> products (activation and reaction to products) 2 where k t , k_ x , and k 2 are the specific rate constants for the respective steps. prod. 2 *• E + products Reoctonts Complex Activated complex State — Products Figure 8-5. Schematic Representation of Catalyzed Hydrolysis Reaction, Showing Formation and Activation of the Intermediate Complex, ES. ON CHEMICAL REACTION RATES; ENZYMES 203 Now this reaction can be very complicated, but for our purpose it will suf- fice to consider one (the simplest) set of conditions, and examine how the rate varies with changes in either enzyme or substrate concentration. First we assume that v_ } is much faster than v 2 , and therefore that reaction 1 is essentially at equilibrium, or that K = kjk , . The rate is then given by v 2 = k 2 [ES] where the square bracket again denotes concentration. Now the only prob- lem remaining is to compute [ES] from the equilibrium constant. If the initial concentrations of enzyme and substrate, respectively, are [£"] n and [S] , the concentrations of free E and S are given by [E] = [E] - [ES] and [S] = [S] - [ES] and therefore the equilibrium constant is given by K = \m Cq ([£]„ - [ES]) ([S] - [ES]) This becomes simpler if only the usual case is considered, namely that in which the substrate concentration is much higher than the enzyme concen- tration; for under this condition only a small fraction of the substrate mole- cules will ever be tied up as complexes ES because there are so few enzyme molecules with which the substrate can form a complex. Hence [S] - [ES] - [S} Rearrangement gives [ES] = K eq [E] [S] () [E] Q [S] i + /ysio K m + [s] if K m is defined as 1/A~, f] . This holds for any value of [S] at any time. Therefore the rate, v 2 , of the enzyme-catalyzed reaction (proportional to the concentration of complexes) is: k 2 [E] [S] v 2 = -d[S]/dt = K m + [S] This is the rather celebrated Michaelis-Menten Equation, and describes the rate as a function of initial substrate concentration under the particular con- ditions we assumed. A plot of v 2 vs [5"] is shown in Figure 8-6 for both high and low enzyme concentrations. The expression says that: (1) the rate is 204 SPEEDS OF SOME PROCESSES IN BIOLOGICAL SYSTEMS high [E] li > /a low [E] b c o (J c > K m , and K m + [S] Q ~ [S] , and the rate expression reduces to v 2 = k 2 [E] with rate independent of substrate concentration; but (note region b) if the substrate is in excess of enzyme, yet [S] << K r rate expression reduces to and K m + [S] « K m , the v 2 = [S] [E] t with rate increasing linearly with substrate concentration. It is clear then that the nature and the extent of the binding of the enzyme- substrate complex, ES (i.e., the value of K m ) is all-important: the bigger the Michaelis constant the smaller the extent of binding; and the weaker the binding, the slower the rate of hydrolysis. It is by good chance* that these E-S complexes generally absorb electro- magnetic radiation in the visible and near ultraviolet regions of the spec- trum. Hence their existence, as well as their K m , can be determined spectro- photometrically (by light absorption), and the value of K m compared with *This work was pioneered and developed to a highly specialized art by Britton Chance, of Yale University. ON CHEMICAL REACTION RATES; ENZYMES 205 that obtained by measuring rates at various concentrations of substrate and enzyme. The "catalyst law" (for enzymes, the Michaelis-Menten expression) rear- ranges to [S\ W£]„ - K eq from which it is seen that the slope of a plot ofv/[S] vs v gives K (= 1/A" m ) directly as the negative of the slope. Figure 8-7 is such a plot for the hy- drolysis of a particular dipeptide for which the stomach enzyme, pepsin, is a specific catalyst. The value of K m obtained is 0.0014 moles liter -1 . This result is typical. The inverse, the value of K at 25°, will usually be found to be between 100 and 600 liters/mole, which means that the substrate must be in excess 100- to 600-fold over the enzyme if the catalyst is to be more than 90 per cent complexed (i.e., "worked hard") at all times. There are cases (certain chymotrypsin-catalyzed reactions, for example) in which the binding of the complex is much stronger. By contrast, the myosin-adenosine triphosphate complex, formed during muscle contraction is relatively a very weak complex .... The value of K m is numerically equal to that value of the substrate concentration at which one-half the enzyme molecules are tied up as complexes. Electrical attractions and repulsions as well as the geometry of the molecules E and S determine the extent of K eq = 700 Figure 8-7. Determination of the Binding Constant of the Intermediate Complex in a Catalyzed Reaction (pepsin-catalyzed hydrolysis of carbobenzoxy-glutamyl-tyrosine ethyl ester, a dipeptide). Values plotted are those of initial rates found experimentally for six different initial concentrations of substrate. 206 SPEEDS OF SOME PROCESSES IN BIOLOGICAL SYSTEMS binding and hence the specificity of a particular catalyst for a particular reaction. Not all of the assumptions made nor the conditions assumed in the fore- going analysis are always met. Because equilibrium does not always exist in reaction 1, K m can better be expressed as (£_, + k 2 )/k ] , which of course reduces to l/K = k_ ] /k ] if k_ x >> k 2 , the case we have studied already. There are further complications, such as competition by two or more reac- tants for the one active site, which introduce more terms in the expression for v. Although these are of pragmatic interest in biological chemistry, further discussion here is beyond our scope — our purpose is simply to illus- trate complex formation and saturation of a catalyst. It should be remembered that the rate constant, k 2 , can be factored into k T h Because the change in entropy AS 1 accompanying activation gives an indi- cation of the change in the freedom of motion within the complex ES 1 , deter- minations of AS 1 and A// 1 have become very powerful tools for understand- ing the mechanism on a molecular scale. Some values are given in Table 8-5. A very interesting success story of this kind centers on myosin, the con- tractile substance in muscle and the catalyst for ATP hydrolysis. The "state of the art 1 ' is reviewed briefly in Chapter 10. TABLE 8-5. Kinetic Parameters for Some Enzyme-Catalyzed Reactions.* Enzyme Substrate (deg C) pH (moles ') (sec ] ) E 2 * ±S 2 * Pepsin carbobenzoxy-1-glu- tamyl-1-tyrosine 32 4.0 560 0.0014 20.2 4.6 a-Chymo- benzoyl-1-tyrosine trypsin ethyl ester 25 7.8 250 78 9.2 -21.4 Urease urea [CO(NH 2 ) 2 ] 21 7.1 250 20,000 9.7 - 7.2 Myosin adenosine triphos- phate (ATP) 25 7.0 79,000 104 13.0 - 8.0 \/K m . equilibrium constant for formation of ES complex (see Figure 8-5). k 2 : specific rate constant for unimolecular breakdown of the ES complex. £,* : energy of activation of ES complex. AS 2 *: entropy of activation of ES complex. Negative values are usually interpreted as evidence for the freeing of charged groups resulting in orientation of water molecules during activation. Values in the last three columns were taken at high substrate concentration and therefore refer to the activation of ES complex into product. *See the book bv Laidler 15 for collections of data. ON DIFFUSION; OSMOSIS 207 Generalization of Method Enzymes are not the only catalysts in the living system, of course. Sur- faces, acid (H + ), base (OH ), and metallic ions are all important catalysts. The general principles outlined above apply to these equally as well as to enzymes. The factoring method of analyzing rates — that of extracting from the proportionality constant one after another the variables and universal constants upon which the rate of a process depends — in some ways has reached its highest state of development in chemical kinetics; and it is scor- ing rather remarkable successes with some very complicated biochemical re- actions. Whether this method of analysis, which ultimately reduces to analysis of the intermolecular forces and molecular movements of a biologi- cal process, is properly termed "biophysical chemistry' 1 or "chemical bio- physics," is often uselessly debated. It is a matter of definition; and no definition has yet been generally accepted. We use this illustration of the factoring method not only to discuss the velocity of biochemical reactions in terms of molecular interactions, but also by analogy to discuss in the follow- ing sections the velocities of the physical processes of transport, namely dif- fusion, osmosis (a special case of diffusion), viscous flow, electrical con- ductivity of solutions and tissue, and heat conduction. ON DIFFUSION; OSMOSIS Diffusion may be defined as the movement, in a preferred direction, of one component relative to the other components, of a mixture or solution. The preferred direction is from the place of higher concentration to the place of lower concentration of diffusing substance. No flow of the whole fluid need occur — no turbulence, nor even convection; no gravitation, no electrical field is of importance to transport by pure diffusion. The fact that diffusion occurs is not surprising when one remembers that all molecules are in a state of continuous motion. The more molecules of type P there are present in a particular volume of solution, the greater the likelihood that some of these will gain enough excess energy to find their way out of this volume. Consider two unit volumes with a common face, one with concentration P in Q higher than the other (Figure 8-8). Because all molecules are in continuous motion (i.e., have kinetic or thermal energy), on the average more P molecules from volume 1 pass into volume 2 than the reverse. In fact, the greater the concentration difference (actually the gradi- ent dc/dx), the greater the speed at which they diffuse, other things being equal. Figure 7-7 was an earlier impression of this same idea. If, however, some sort of barrier to diffusion is placed between volumes 1 and 2, the rate at which P diffuses is slowed down; and the greater the thick- ness of this barrier the lower the rate becomes. To a first approximation, 208 SPEEDS OF SOME PROCESSES IN BIOLOGICAL SYSTEMS high 1 o w P "- P P in Figure 8-8. Illustration of Direction of Diffusion of P in a Mixture of P in Q. therefore, it is the rate of change of concentration, c, with distance, x, which determines the rate of diffusion. These intuitions were first set down and experimentally proven by the in- genious German anatomist, Adolf Fick, in 1855: j = DA dc_ dx (Fick's first law) where dcjdx is the instantaneous rate of change of concentration with dis- tance, called the concentration gradient, in moles per liter per cm; "j" is the flux (i.e., the flow rate, v) — the number of moles passing through a particu- lar area, A cm 2 , in 1 sec; and D is the proportionality constant, which con- tains all the other factors — some of them still unknown — upon which the rate of diffusion depends. Self-diffusion of water across an erthythrocyte membrane is an example. Absorption of gaseous 2 by the blood capillaries in the lung is another example: both the partial pressure of 2 and the con- centration in the circulating blood plasma are constant in time. Fick's first law is limited to the case in which concentrations do not change — the steady-state condition — and the source and the sink are infinite. However, there are many specific cases, particularly in the gastrointestinal tract and associated with assimilation of the degraded products of foods, in which the concentration gradient is not constant, the state is not steady. Any periodic or sporadic phenomenon which makes a sudden change in the rate of supply of reactants to a certain part of the living thing, will cause a deviation from the steady state. Thus in the volume in which the change occurs, the rate of change of concentration, dc/dt, is given by dc/dt = D d 2 c/dx (Fick's second law) Since d 2 c/dx 2 can be written — ( — , and since dcjdx is the concentration dx \dx) gradient, we see that the second law states that the rate at which the concen- ON DIFFUSION; OSMOSIS 209 tration changes within a volume is proportional to the rate of change of the concentration gradient at the boundaries of the volume. One simple example will be used to illustrate the problem described by Fick's second law. This will be done only qualitatively, for the detailed de- scription is too complicated to be practical here. Consider the red blood cell, with various components contained within, and separated from the medium by a membrane, the cell wall. There are fluids on both sides of the wall in osmotic equilibrium (see Chapter 2). This is a condition of no net change: potassium ion, at higher concentration inside the cell is being transported in both directions across the cell at equal rates; sodium ion, at higher concen- tration outside the cell, is being transported in by diffusion, out by "active transport," but both at the same rate so that there is no net change. Water moves across the membrane freely in both directions. (Recent radioactive tracer experiments using tritium have shown that complete exchange of water can occur in a few milliseconds.) If for some reason the "sodium pump," which provides the active transport, fails, then both K + and Na + will diffuse passively, each in the direction towards lower concentration (Figure 8-9). The rate of diffusion, expressed by the rate of change of con- centration, dc/dt, is given by the second law as D d 2 c/dx 2 . Solution of the equation for c, gives c as a function of t; or c = /(/). The form, /, can be worked out explicitly, provided certain other conditions are known. The result is approximately r K+ = c { + c 2 /y/T+t for the decay of the internal K + concentration and r Na+ = c[ — c' 2 /y/t + t for the buildup of internal Na + concentration to the concentrations of K + and Na + in the plasma in Time after failure (sec) Figure 8-9. Readjustment of Concentration of Na + and K + Inside the Erythrocyte Following Failure of the Sodium Pump — A Diffusion-Controlled Process. Final values, 1 38 and 1 6, are those of the plasma. 210 SPEEDS OF SOME PROCESSES IN BIOLOGICAL SYSTEMS which the cells are bathed. The inverse square root relationship occurs over and over again in diffusion-controlled processes. In Figure 8-9 are shown the initial concentrations (milliequivalents per liter) of Na + and K+ inside the cell (at / = 0), and their change toward the concentrations in the plasma (dotted lines) following failure of the sodium pump. Diffusion Coefficient D, and Permeability Constant P. Table 8-6 gives some representative values for the diffusion coefficient at 25°C in cm 2 sec -1 . The activation energy and the temperature coefficient of rate of diffusion in water solutions and in fat and lipid, were given in Table 8-3. TABLE 8-6. Some Diffusion Coefficients (D) (cm 2 sec ] ). Substance into Water (at 12° C) D x 10 5 Glycerine 0.42 MgS0 4 0.35 KC1 1.59 NaCl 1.09 Sugar 0.29 Urea 1.12 Just as the specific rate constant of a chemical reaction can be broken down into the factors upon which it depends, so also can the diffusion coef- ficient be factored. Diffusion is a "jump process," in which the movement of a species occurs by its being pushed from one position of rest to another as the result of favorable collisions with neighbors. The distance between successive positions of rest is called the jump distance, A. The activated complex in this case is pictured as being an intermediate position in which the jumping species is half way between rest sites and can go either way. Detailed analysis shows that D = t\ 2 JiL e -*Ft/RT h where A is the jump distance in cm, AF* is the free energy of activation (Fig- ure 8-4) for the "jumper," 7" is the absolute temperature (degrees Kelvin), k is the Boltzmann constant, and h is Planck's constant. The units of D are therefore cm 2 sec "'. Table 8-6 gives some values of D for different spe- cies diffusing into water. As in the case of chemical reactions, the term k g T/h is a constant at any temperature. The low diffusion constants (in molasses, in lipids, or in fats) and high values (in water or alcohol) are determined by the values of X, and ON DIFFUSION; OSMOSIS 211 by AF\ the energy of binding within the shroud of neighboring molecules through which the jumping species must penetrate if it is to move success- fully to the next position of rest. Two innovations have been introduced into discussions of diffusion in recent years, one for theoretical reasons and the other for practical reasons. Firstly, it is more proper to consider activities (effective concentrations) than measured concentrations, and more proper still to consider as the "force," the gradient of the chemical potential which drives the diffusion process; and therefore dc/dx is replaced by dn/dx, in the more esoteric discussions, if not in practice. Secondly, the thickness of the interface, at a cell wall for instance, is really a matter of definition rather than of position of chemicals. Who can say where the water phase stops and the heavily hydrated protein of the wall begins? Therefore dc/dx is hard to measure for living membranes, and re- course is made to a phenomenological trick: dx is taken into the diffusion constant, and the rate of flow is expressed as the difference between the flows in the two directions through the membrane. Thus j =(P,A Cl -(P 2 Ac 2 where 1 and 2 represent diffusions in the forward and back reactions, and c, and c 2 represent concentrations on the two sides; the (P's then have units cm sec -1 (velocity) and are called permeability constants. A few of these are collected in Table 8-7 for monovalent cations penetrating through living membranes. These permeability constants can be compared with values de- termined for synthetic interfaces also given in the table. TABLE 8-7. Some Permeability Constants ( § ) for Synthetic and Biological Membranes.* , _._ . Permeability Constant x 10 Intertace Diffusion ' _i. (cm sec ) K + into erythrocyte of: man 5.0 dog 1 .0 rat 10 KC1, KBr, KI into nitrobenzene 0.007, 0.075, 1 .4 Na + into erythrocyte of: rabbit 3.0 dog 0.5 Na + through frog skin 5.0 Na + (as iodide) into nitrobenzene 0.2 Alcohols into erythrocyte 10,000 to 100,000 Water into erythrocyte ~ 1 0,000 ♦Collected by J. T. Davies, J. Rhys. Coll. Chem., 54,185(1950). See also Ref. 17. For ionic flow the values in the table can be transformed very easily into electrical resistance units. Thus if the concentration of the salt at the mem- 212 SPEEDS OF SOME PROCESSES IN BIOLOGICAL SYSTEMS brane is 1 mole per liter, the values come out to 1000 to 50,000 ohms/cm 2 , in general agreement with values found by direct measurement for living membranes. The values determined depend on the permeability, discussed later. Osmosis Following the foregoing discussion, very little needs to be said about osmosis. It is simply the diffusion of water from the place of higher water concentration to the place of lower water concentration. More properly, it is the diffusion of water down an activity (effective concentration) gradient. The speed of the process is described by Fick's laws — the first for the steady state of constant concentrations, and the second for the unsteady state of changing concentrations. Osmotic pressure and water balance, both properties of the equilibrium state, were discussed in Chapter 2. As an anatomist, Fick naturally had an interest in these important proc- esses; but this interest must have been accompanied by a remarkable insight. ON FLUID FLOW; BLOOD Poiseuille's Law Holding a special place among the kinetic processes of importance in biology is the transport of fluids, both gases and liquids, along tubes and in and out of storage chambers. One need mention only the circulation of blood and the respiration of air as examples. The first striking fact is the flow itself: it takes place (almost) no matter how small the applied mechanical force; and the rate of flow increases line- arly with increasing driving force. Flow is opposed by frictional forces or "internal barriers" which the moving fluid must surmount — the smaller the internal barriers the faster the flow resulting from a given applied force. Ideally at least, as was first stated by the French physicist, J. L. Poiseuille, in 1884, a liquid moves in a tube by the sliding of one imaginary layer of liquid over another. The surface layer moves very slowly, if at all, relative to the speed of layers far removed from the surface. The presumed velocity distribution is indicated by the lengths of the arrows in Figure 8-10. Figure 8-10. The Gliding Layers in Nonturbulent Fluid Flow. Length of the arrow is pro- portional to speed. ON FLUID FLOW; BLOOD 213 If P, and P 2 are the pressures measured at the points 1 and 2 in the tube, and R is the distance from the center bore of the tube, the driving force is given by ttR 2 (P x - P 2 ) The frictional force on the layer at distance R from the center is propor- tional to the area of the layer (2ttRI), and to the velocity difference between the layer we are considering and its nearest neighbors; in the limit this is dv/dR. After the two forces have been equated, integration (or summing all veloc- ities from that at the center of the tube to zero at the wall) gives P — P v = £j £2( r 2 _ R 2) 4/ where r is the radius of the tube. This expression gives the linear speed of the layer which is R cm from the center. is the proportionality constant, and is called the fluidity (the higher its value the higher the velocity). The total volume of fluid flowing per second through the tube is calculated by summing all the elemental volumes, 2irRdR, for which v is expressed. The result is the celebrated Poiseuille equation which' expresses rate of flow (cc/sec) of liquid through a tube of radius r and length / under an applied pressure difference of AP = P ] — P 2 : irr 4 dVldt = AP cc/sec 8/ If A P is given in dynes per cm 2 , r and / in cm, and the speed of flow in cc per sec, the fluidity, 0, must be cm per sec for a force gradient of 1 dyne per cm; i.e., has the dimensions: — /— — . It is the velocity of flow of a fluid sec/ cm under a unit force gradient. The case for gases is slightly more complicated because of the added fact that the volume depends strongly upon the pressure and the temperature. With the proper modifications the expression for rate of flow of gases approximates: Trr 4 P. 2 - P 2 dV/dt = ! 16/ - P n if P is the pressure at which the volume is measured. Fluidity, 0, and Viscosity, rj Table 8-8 gives values of the fluidities of various substances at different temperatures. Of the liquids, ether is the most fluid one listed; glycerine at 0°C is the least fluid — indeed at 0°C it is almost a glass! The fluidity of 214 SPEEDS OF SOME PROCESSES IN BIOLOGICAL SYSTEMS TABLE 8-8. Fluidities (0) poise 1 , or — cm /dyne sec/ cm Temp (°Q Hydrogen Air Ether Benzene Water Butyric Acid Castor Oil Glycerine -30 13040 6490 11980 5850 352 110 56 44 0.00041 0.000083 10 11760 5680 392 132 78 50 0.0013 0.0011 25 11300 5410 450 164 112 71 0.0036 0.005 37 10990 5290 500 196 144 85 0.01 0.02 50 5150 552 227 182 105 100 4440 352 gases decreases as the temperature is raised (see Chapter 2); but that of all liquids increases with increasing temperature. In liquids the higher the tem- perature the greater the number of particles which have the energy to over- come the internal barriers to flow; or in other words, the higher the tempera- ture the smaller the sticky frictional forces which must be overcome by the gliding laminae, and the faster the flow. The temperature coefficient of fluidity, factored as a specific rate constant, is given in the theory of rates as V In -SFt/RT V h.X e ASt/R e -AHt/RT where K is the volume of 1 mole of fluid, h is Planck's constant, N is number of molecules per mole, Avogadro's number, and AF X , AS\ and AH X refer to formation of 1 mole of activated complex in the glide plane as it slips from. one position of rest to the next .... The physical analogy between diffusion and flow thus is extended to the algebraic statement of the factors upon which they depend. The two processes can be directly compared in Tables 8-7 and 8-8. The experimental values of E* (related to AH 1 ) are usually the same for diffusion and flow (Table 8-3). This indicates the inherent similar- ity of the two processes. Indeed in diffusion the particles move individually at ran- dom from one position of rest to the next. In flow a plane of particles moves as a unit, and no relative motion occurs between members of a plane; adjacent planes glide past each other. The intermolecular forces which oppose dif- fusion are the same as those which oppose laminar flow. That is, the bar- riers to flow are the same, and hence £*'s are the same. The catalysts in this case are called surface-active agents. Washing detergents are good examples. The inverse of the fluidity, i.e., 1/0, is called the viscosity, usually ex- pressed by the symbol rj. Hence a high viscosity (cold molasses) means low ON FLUID FLOW; BLOOD 215 fluidity. Viscosity can be considered as the frictional force opposing the a x j- • dynes /cm . . , . . . . „ , , How. Its dimensions are — / , or dyne sec/cm , this unit is called the cm / sec poise, after Poiseuille. A very simple way to measure fluidity or viscosity is in the Ostwald vis- cometer. The capillary pipette is filled to a mark with fluid, and measure- ment made of the time it takes the fluid to run out of the pipette. This time is divided into the time taken by water, or some other fluid, to drain at the same temperature. The quotient is called the relative viscosity. A density cor- rection is necessary if the driving force (gravitational) is to be equal in the two cases. Solutions or suspensions (of molecules or particles respectively) in water usually increase the viscosity (decrease the fluidity). The fractional increase is (r;, — ri )/rj , where the subscripts s and refer to solution and pure water, respectively. But this value, often called 77', varies with the concentra- tion. It is convenient, then, to measure the 77' at several concentrations, and express each measurement in terms of unit concentration by dividing by the concentration at which the measurement was made. This number is called the specific viscosity. It is also concentration-dependent, because intermolec- ular interactions are higher at higher concentrations. It is useful, then, to extrapolate measurements of specific viscosity to infinite dilution (zero con- centration), for this value is the value of that part of the viscosity due to the suspension only, and unaffected by interactions which solute particles could have on each other. This value is called intrinsic viscosity, usually symbolized [77]. Values range from .02 for small- molecular- weight solutes to 20 for macromolecules, and to much higher values for suspensions of living cells. Turbulent Flow Laminar flow will exist in most fluids at low rates of flow. When the flow rate becomes high, the glide planes get off-track, and turbulence sets in. Small whirlpools and eddy currents are initiated, and the fluidity drops abruptly; therefore, if the rate of flow is to be maintained, higher driving force must be applied and more energy must be expended. Unless some result of particular value is derived from the turbulence (more rapid mixing of chemical reactants at a reaction site, for example), it is obviously waste- ful of energy. The circulatory system in man has certain features, such as flexible walls lined with hydrated protein "hairs," which help direct the fluid flow and damp out trends toward turbulence. The Reynolds number, Re, a dimensionless parameter of fluid flow, is defined as Re = 20 p vr 216 SPEEDS OF SOME PROCESSES IN BIOLOGICAL SYSTEMS where p is the density of the flowing fluid, the fluidity, v the velocity of flow, and r the tube radius. For homogeneous liquids flowing at constant velocity, it is general experience that the flow is laminar and turbulence can- not be maintained if Re < 2000. For blood, Re has been found to be 970 ± 80 over the pertinent range of flow rates and tube sizes. Therefore, laminar flow probably occurs in the blood vessels at all times, although turbulence may set in momentarily at the valves during the pumping action of the heart. Properties of Blood Plasma and Blood Previous discussion has implied that the fluidity (velocity per unit force gradient) is independent of speed of flow, v. Liquids for which this is true are called Newtonian liquids. Pure water is a good example. However, most real liquids are at least slightly "non-Newtonian" — that is, = f(v). One of the most complex examples of this behavior is blood — a suspension of cells in plasma, which itself is a water solution of salts and heavily hydrated macromolecules. Figure 8-11 is taken from results which show that the ease of pushing the fluid through a tube — in this case a glass one — decreases rapidly with intro- duction of macromolecules and cells into water. Thus the for plasma is about half that for water [144 — /— — -); and increasing amounts of red \ sec/ cm / blood cells reduce the fluidity still further. Yet, to a first approximation, > blood AP Figure 8-11. Rate of Flow of a Fluid Through a Tube as a Function of Driving Pressure. The slope is proportional to the fluidity, given in parentheses. The usual range of per- centage of total fluid volume filled with cells is shaded in. ON FLUID FLOW; BLOOD 217 'synthetic plasmo" o synthetic plasmo plus red blood cells AP (mmHgO/cm length) Figure 8-12. Fluidity (slope) of Synthetic Plasma to Which Different Volume Percent- ages of Cells Have Been Added. within the physiological range of operation both plasma and whole blood are essentially Newtonian; that is, their curves are linear; Poiseuille's law of laminar flow is obeyed. However, closer inspection of not only very low rates of flow but also very high rates reveals that the fluidities in these ranges are lower than in the intermediate range in Fig. 8-11: the fluidity is dependent upon flow rate in these regions. Thus at low flow rates an elasticity due to the formation of liquid crystals by hydrogen bonds makes flow more difficult and has to be broken down; at high flow rates turbulence sets in and makes flow more difficult. Figure 8-12 illustrates the first point. Notice how the fluidity (slope) changes with flow rate, when flow is slow. On the other hand, turbulence can actually be heard (or its effects can be heard) over the heart where very high flow rates accompany the high pressure part of the beat .... The de- pendence of viscosity (1/0) on tube radius (Figure 8-13), at first surprising, resolves to a question of the interruption of laminar flow when the diameter of the suspended particles (red blood cells) approaches the diameter of the tubes through which the suspension is flowing. This is the condition which exists in the blood capillaries — the process is more like an extrusion than a laminar flow. The velocity gradient across the tube is the cause of Bernoulli forces which not only make the cell spin, but also force it toward the center (the bore) of the tube. Further, the blood vessels are somewhat elastic and can increase their diameter under pressure. Thus the flow rate doubles for a 16 per cent increase in radius! This fact, plus the probably great differ- ence between the surface of glass tubes and the molecular-hair-lined** blood **These "molecular hairs" arc hydrated protein molecules, partly detached from the wall, and jutting out into the tube. 218 SPEEDS OF SOME PROCESSES IN BIOLOGICAL SYSTEMS 0.5 1.0 1.5 Tube radius (mm) ■»" Figure 8-13. Fluidity of Whole Blood vs Glass Tube Radius. vessels, makes the whole study very complicated, easily subject to gross mis- interpretation, and certainly needing more careful experimental definition. Circulation of Blood Description of the circulatory system is not our objective here. This was done in 1628 by, at the time, the radical physician, Sir William Harvey, whose description of experiments proving the continuous circulation of the blood — from the heart through the systemic arterial and venous systems, back to the heart, thence through the pulmonary arterial and venous sys- tems, and again back to the heart — is still one of the classics of clarity in medical literature. The pressure difference between aorta and vena cava across the pump, the heart, is about 100 mm Hg, or 0.13 atm. Along the large arteries and veins and in the main arterial and venous branches, the pressure gradient is small; but because these vessels are of large radius, the flow rate is rapid. The pressure gradient is at its peak along the capillaries and the arterioles; be- cause they have very small radii, the flow there is slowest — just where it should be the slowest— so that plenty of time exists for exchange to occur by diffusion through the walls of arterioles and capillaries. Figure 8-14 illus- ° O vena aorta cava capillaries aorta 1 vena R cava , 1 » capillaries Figure 8-14. Relative Areas and Pressure Drops in Different Parts of the Human Circulatory System. ON ELECTRICAL CONDUCTANCE; EEG AND EKG 219 trates this point, showing the pressure changes and the relative total cross- sectional areas. Two quantities can be measured, the flow rate (cc/sec), and the speed of flow (cm/sec). Measurements in the aorta show that enough blood flows past a flow-meter detector per second for one complete cycle to require 45 min. Insertion into the aorta of a bit of radioactive argon as an inert tracer, and measurement of how long it takes for the tracer to complete the ciruit, confirms this. Speed is less easily measured. One method is by tracer. The ultrasonic method (see Chapter 3) introduces no pathological changes, but needs calibration. ON ELECTRICAL CONDUCTANCE; EEG AND EKG The next rate process to be considered in this chapter is the movement of ions under the influence of an electrical field — in other words, the con- ductance of solutions of salts in water. This subject is basic to an under- standing of the gross current paths through the human body upon which are based the techniques of electrocardiography (EKG) and electroencepalog- raphy (EEG), and also basic to some of the transport processes driven by membrane potentials which are of importance in nerve conduction and elec- trical shock treatment. Towards the latter part of the last century the big-three "solution" pio- neers, Kohlrausch, Arrhenius, and Van't Hoff, showed that salts dissolve in water as ions. These are electrically charged and free to move about at random because of thermal energy, but subject to movement in a preferred direction under the force of an electrical voltage gradient. Positive ions are forced to the negative electrode, and negative ions to the positive electrode by the electrical field. The speed of movement, or mobility (centimeters per second under a voltage gradient of 1 v per cm) was understood quantitatively by 1923 (the work of Debye and Hiickel, Onsager, and later others) as being determined by the ease with which a charged ion, complete with "hangers- on" such as electrically charged ions and water molecules, can slip from hole to hole in the liquid. The process is very similar to diffusion, which was described earlier. The difference is that ions are charged and move under a voltage gradient, whereas the diffusing particle may or may not be charged and moves under a concentration gradient . If a potential difference exists for any reason be- tween two parts of an electrolyte, or is applied from the outside, ions move and current flows — in other words, charge is transferred. Hence this is just another transport process. Ohm's Law Concerning Current \{ n is the number of charge carriers per cc, w their average velocity under the impressed voltage, and q the electrical charge carried by each, then the 220 SPEEDS OF SOME PROCESSES IN BIOLOGICAL SYSTEMS amount of electrical charge passing per second through a plane of 1 cm 2 area, called the current 'density , I, is / = nwq If A' is the number of molecules per mole: n/N is the concentration, c, in moles/cc; and qN is the charge per mole. The charge required to oxidize or reduce 1 mole of anything is zF, where F is the charge (96,500 coulombs per equivalent weight) required to oxidize or reduce 1 g equivalent weight, and Z is the number of equivalent weights per mole (i.e., the number of electrons transferred in the redox reaction). This is Faraday's law. Summed (2) for all different ions, s, then I = FZc s WsZ s Since c s is moles/cc and w s is cm/sec, the current density has the dimensions: coulombs per cm 2 per sec, or amperes per cm 2 . Note that the current increases linearly with the concentration of charged particles, with their speed, and with the charge they carry. Specific Conductivity of a Solution, k This is defined as the current which passes for an impressed voltage gradi- ent, 13 , of 1 v/cm. That is, k = //I). This is a form of Ohm's law. Now although the dissociation of ions of a salt is usually complete, sometimes there is association and always there is hydration, and hence often the ef- fective "degree of dissociation," a, is less than 1. Introducing this concept gives FjLc sWl z s am p S k = a or ohm ' cm TJ volt/cm One more concept completes the picture. If the mobility, i± s , which is the speed under an impressed voltage gradient of 1 v/cm, is defined as w s /\), then * = Fj^c s n s z s a Note that this expression describes the rate of the electrical transport process. Thus k is the rate in amperes at which charge is transferred across 1-cm 2 area of electrolyte if the voltage gradient along the path is I v per cm. The value is proportional to the concentration. The proportionality constant factors into three constants (a, z, F) and the mobility, fi; and n is really the specific rate constant for the process. Therefore n plays the same role for conductance as does k for chemical reactions, D for diffusion, and 4> for fluid flow, respec- cm / v tively. The units of /x are / . Values of the mobilities of small ions sec/ cm ON ELECTRICAL CONDUCTANCE; EEG AND EKG 221 average about 0.001 (see Table 8-9) for the ions of tissue fluids. The con- ductance, k, then is easily computed from the above expression, since a « 1 for salts in tissue fluids. TABLE 8-9. Mobilities* (n) of Selected Ions in Aqueous Solutions at 27° C. IT 362 OH- 207 Na'+ 52 ci- 79 K + 77 I- 80 NH 4 + 76 N0 3 " 74 |Mg ++ 55 HCCV 46 2S0 4 = 83 Benzoate - 33 Blood Plasma Components: Albumins a-globulins /3-globulins Fibrinogen 7-globulins Erythrocytes 5.7 to 6.2 3.6 to 5.1 2.5 to 3.2 1.7to2.3 0.8tol.3 13 (buffered at pH 8.6) *Dimensions: cm / v sec/ cm x 10 s . For the small ions, the values refer to infinite dilution. From Ref. 20. However, just as diffusion and fluid flow are concentration-dependent, so is electrical conductivity; and it is useful to express conductance per equiv- alent weight. It is called equivalent conductance, A, and is given by pV ohm -1 cm -1 A = t / . m, /rf) describes the convection which carries the moisture away; and AP is the driving "force/" i.e., the difference in vapor pressure, P, of the liquid on the surface at skin temperature and that of water at the ambient temperature - the latter reduced by the relative humidity, RH. The important factor is the last one. Thus the liquid on the surface strives to set up an equilibrium pres- sure of vapor with the atmosphere which surrounds it, but never quite suc- ceeds, since the atmosphere is nearly always undersaturated (RH < 100 per cent). For example, if the skin temperature is 34°C (91°F) and the RH = 60 percent for an ambient of 20° C (68° F), quite common conditions, IP = P(34°) - 0.6P(20°) = 0.04 atm At very high temperatures {T a > 80°F) this method is the body's escape valve for excess heat. Each gram of water lost by vaporization removes 0.58 Cal from the skin. In the lungs, inhaled air becomes saturated and then is TABLE 8-11. Estimated Per Cent of Heat Loss, by Each of Four Principal Methods. Body's Heat Loss (Cal/hr) Per Cent of Skin Covered Per Cent Heat Loss by Activity Conduction and Convection Radiation Water Loss from Skin Respi- ration* Studying, fully clothed, 70° F 150 85 68 20 10 2 Studying, lightly clothed, 70° F 200 15 20 58 20 2 Resting for BMR test, 70° F 70 15 20 70 8 2 Running mile race, 60°F 1500 25 20 20 50 10 Sunbathing on beach, 90° F 350 15 10 8 80 2 Walking, heavily clothed, 0°F 350 95 50 8 2 40 *Assume 50 percent relative humidity. See Refs. 2 to 4, and 21 228 CD O- x CD ~o cd > o e CD u c 0) CD i_ n U_ _c u %_ i. cd O c ^ O c 0) o n t: n o Q. F D CD > X cd U_ >- O) o c o E c a) £ < to a> a> o 4- X i_ cd D i_ ._ a) t- -C to o — o <1) £ *_ n o ■ I (j> > o CN cd 00 LU ■•- CD o < t— c o u t£j 'u 0) Q. l/l ■D c o cd c 0) a c o o _0 CD > CJ C£: CD 'u o 8." CO a> o O ° D K oc Ik. 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'Z: ^ In CO CO 3 C ■«- ■ — U Z q_ CJ en O e > *J en u x cj X cj X en cj cu a. o u 230 SPEEDS OF SOME PROCESSES IN BIOLOGICAL SYSTEMS exhaled, with the same loss of heat per gram of water. Respiration then be- comes important, especially when the air is dry and/or cold. Urine and feces contribute a small fraction to daily heat loss. Heat Loss from the Body Under Various Conditions Table 8-1 1 illustrates that the escape valve for excess heat may be any one of the several methods of heat loss and will vary for different activities. The very important role of the skin as a heat insulator and as a water supplier to the surface, and the role of cover and clothing now become clear. To sum up: the maintenance of constant body temperature is a very re- markable example of the "steady state." In Chapter 7 we illustrated heat- producing reactions — chemical, physical, mechanical, etc. In this chapter we have discussed the rates of heat-producing reactions and the rate of heat loss. In the steady state there is continuous flow — and the rate of "waste" heat production is exactly balanced by the rate of heat loss, no matter what the ambient conditions. So it is with literally hundreds of processes in the living thing. FORMAL SIMILARITY AND INTEGRATION OF THE FIVE PROCESSES The method of presentation used in this chapter permits us to summarize in a table the factors upon which the rates of the five processes depend, and to note their similarities and differences. Since each of the processes was dis- cussed individually, no comment on Table 8- r 12 and its extension, Table 8-13, will be made now. other than to ask the reader to note that the classi- cal driving force and the role of the activated complex are both stated ex- plicity. The reader should consider these tables to be a memory aide, which, if understood, will give him a powerful grasp of the nature of each of these important processes occurring within the living system. In the living thing, these processes are not separate and distinct, isolated from one another. On the contrary, at every spot in the body probably three