ELEMENTARY

SELECTED TOPICS

H. T. EPSTEIN

PRINCIPLES OF BIOLOGY SERIES

ADDISON-WESLEY PUBLISHING COMPANY

ELEMENTARY BIOPHYSICS

SELECTED TOPICS

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HERMAN T. EPSTEIN, Brandeis University

ADDISON-WESLEY PUBLISHING COMPANY, INC.

READING, MASSACHUSETTS • PALO ALTO ■ LONDON

This book is in the

ADDISON-WESLEY

SERIES IN

THE LIFE SCIENCES

Consulting Editors:

EDWARD HERBERT

Massachusetts Institute of Technology

HERMAN LEWIS

National Science Foundation

Copyright © 1963

ADDISON-WESLEY PUBLISHING COMPANY, INC.

Printed in the United States of America

All rights reserved. This book, or parts thereof,
may not be reproduced in any form
without written permission of the publisher.

Library of Congress Catalog No. 63— 1 6566

For Becky, Karen, Erika and David

Preface

This monograph has an explicit purpose: to facilitate the introduction
of biophysics into the elementary biology course.

The reason for wanting to introduce biophysics is that the obvious
trend in biology is toward physical and chemical biology. By the time
the present generation of students has reached scientific maturity, twenty
years will have passed, and our students will be scientifically obsolete
unless they have been taught to think in physico-chemical terms from
their earliest courses.

At present, practically all biophysics courses are given on the senior-
graduate level. In teaching my own course on that level, I have been
struck many times by the fact that a substantial fraction of the topics in
biophysics utilizes only elementary mathematics, physics, and chemistry.
There arose the possibility of designing an introductory biophysics course
using only those elementary courses as prerequisites. However, it is un-
likely at present that any department, including my own, would install
such a course, because of the pressure of covering the broad scope of
science that is included in present-day biology. Another means of
achieving the goal is to give, say, a three- or four-week introduction to
biophysics within the introductory course as just another of the facets of
biology that are surveyed. The economy of paperback monographs has
permitted the writing of a short introduction to biophysics without
writing an entire textbook.

The majority of introductory biology courses appears to be given in
the freshman year. Thus, even by placing the "month of biophysics" at
the end of the course, the students will have had only the year of biology
and probably a year of chemistry or physics, along with high school
mathematics and science. Accordingly, one of the severe problems has
been to choose topics suitable for a very elementary presentation. There-
fore there have been important omissions, aside from the fact that this
short coverage is intended for beginning students. For example, there is
no discussion of membrane properties, nerve physiology, enzyme kinetics,
information theory, etc. Some items are omitted for the reason that the
complexity is too great, others because I judge them not sufficiently
developed at the present time, and still others because a monograph is
necessarily rather short. Even though I wanted to offer teachers a selec-
tion among possible topics, some have been omitted because they are
likely to be covered in other introductory courses. What remains, then,
is a selection that seems reasonable to me.

iv PREFACE

The chief writing problem has been to achieve simplification without
oversimplification so that statements have a reasonable range of validity
rather than no validity at all. There are tremendous simplifications in
the topics presented. Some simplifications shock me, let alone others
who would have simplified in an entirely different way. I offer no apology
for what has been done, for my judgment is that I have chose a reason-
ably good way of getting ideas and attitudes across to beginning students.
I would prefer this way to an appreciably more involved one required
for more accurate presentations.

The quantitative, experimental point of view of the physicist and
chemist has been, in my judgment, their most important contribution to
biology. I have presented topics which seem to me to communicate this
quantitative, experimental point of view. Biometry has been included
because I think that an introduction is desirable early in the game and
that, for most purposes, no more than an introduction is needed by most
students. To say this bluntly, I think that statistical methods are highly
overrated, and too often serve as excuses for not doing better thought-out
experiments. To be fair, I must also state that professional statisticians
know the uses and limitations of their subject; it is the statistically ill-
informed scientist who tends to overrate and misuse statistics. Therefore
I have presented the topic from the point of view of the problems faced
in evaluating measurements, rather than as an exercise in application of
mathematics to experimentally obtained numbers.

The chapter on physical forces and chemical bonds is placed next
because I believe that the molecular biology of today is only a pale be-
ginning of molecular studies in biology. Molecular differentiation, mo-
lecular neurophysiology, etc. are on the horizon.

The physics of vision, hearing, and muscles is going to develop
increasingly, and in 20 years may well be the first real meeting place of
biology and psychology.

Biophysical methods will continue to serve as basic equipment in the
experimental baggage of biological scientists. An introductory survey
has been written for those who would like just a quick look at the
methods. For those who want more depth, I have chosen to illustrate
with the ones which seem to me most likely to be generally applicable.

Finally, I would like to express my gratitude to the National Science
Foundation for their award of a Senior Faculty Fellowship during the
year 1959-60. This fellowship gave me the leisure to think about bio-
physics, and I developed the point of view which has now led to this
monograph.

Waltham, Mass. H. T. K.

June 106.)

Contents

CHAPTER 1 THE MATHEMATICAL TREATMENT OF DATA

Introduction 1

Measurements and their variation 1

The asymmetrical distribution 8

Computation of errors in compound quantities .... 10

Applications of statistical analysis 12

1. The ratio of a given deviation to the standard deviation 13

2. The ratio of one variance to another variance ... 13

3. The chi-squared test 16

Treatment of experimental data 20

1. Data-fitting using the least-squares principle ... 20

2. Correlation 22

CHAPTER 2 PHYSICAL FORCES AND CHEMICAL BONDS

Introduction 25

Physical forces 25

Chemical bonds 27

Molecules in solution 33

Bonds and energy exchange 35

Energy and biological processes 37

CHAPTER 3 PHYSICAL ASPECTS OF VISION

Sensitivity of the eye 39

Structure of the retina 44

CHAPTER 4 PHYSICAL ASPECTS OF HEARING

Introduction 46

Models for specific nerves for specific frequencies .... 47

CHAPTER 5 LIGHT ABSORPTION EFFECTS

Introduction . 52

1. Light absorption 55

2. Birefringence 58

3. Dichroism 60

4. Action spectra 61

CHAPTER 6 PHYSICAL ASPECTS OF MUSCLE

Introduction 66

1. The over-all action 66

2. Muscle structure 71

V

vi CONTENTS

CHAPTER 7 METHODS FOR DETERMINING MOLECULAR SIZE AND

SHAPE

Introduction

. 75

1. A survey

75

(a) Kinetic methods

75

(b) Static methods

78

2. Diffusion

80

3. Centrifugation

83

An example of the use of centrifugation in biology

87

The mathematics of the sedimentation velocity method

90

4. X-ray diffraction

92

CHAPTER 8 ISOTOPE METHODS

Introduction 99

1. Tracer experiments 101

2. Localization experiments 102

3. Studies utilizing the disintegration itself 103

CHAPTER 9 RADIOBIOLOGY

Introduction 106

The target theory 108

Multitarget analysis 113

Biological effects of radiation 117

1. Mutation 117

2. Ultraviolet light inactivation and reactivation . . . 118

REFERENCES 119

INDEX 121

CHAPTER

1

The Mathematical
Treatment of Data

INTRODUCTION

With a given set of data, the problem is always to extract the maximum
amount of information. This in no sense relieves the investigator of the
problem of designing better and more extensive and inclusive experi-
ments. But the existing data surely deserve to be analyzed as fully as
warranted. The word warranted is the problem word, and it is to this
problem that the mathematical analysis of data is directed. We would
like to know the information that can be gotten from some data includ-
ing the most likely values of the various quantities measured and,
equally importantly, the probable uncertainty of the values thus ob-
tained.

A measurement of any property of a system is, by itself, almost with-
out significance, because we do not know the uncertainty in the measure-
ment. A statement as extreme as this needs some justification. We feel
intuitively that the purpose of a measurement is to know something
accurate about a property. If we are greatly uncertain about the result,
then we have learned little. For we then have to say that the measure-
ment might be some particular number but that it also might be, say,
1000 times greater or less than that number. Thus, not even knowing
the uncertainty in the measurement is equivalent to changing the number
1000 to any number you may wish to insert. The whole purpose of
statistical analysis is to show us how to maximize the relevance of the
measurements we make.

MEASUREMENTS AND THEIR VARIATION

As a simple first example, let us take a series of n individual measure-
ments of something: x 1 ,x 2 , . . . , x n . Why do we take more than one
measurement? What have we gained by taking, as we usually do, the
arithmetic average of these n measurements?

If we consider that the sources of uncertainties in measurements act
randomly, then in a set of measurements we are as likely to get an
individual result higher than the "true" value as to get one equally much
lower than the "true" value. Indeed, unless measurements somehow

2 THE MATHEMATICAL TREATMENT OF DATA

cluster around the "true" value, measurements will avail us nothing.
Thus we are led to the assumption that the value around which the results
cluster is near to the "true" value; indeed, if we think about the problem
carefully, we come to the realization that we have no way of ever know-
ing the "true" value. So, we define the "true" value as the value around
which measurements do, in fact, cluster. Of course, we now need to
specify a means of extracting the true value (we henceforth omit the
quotation marks, which, nevertheless, are still there so far as knowing
the "truth" is concerned). If we could make an infinite number of
measurements, the value which recurred most frequently would be taken
as the true value. What do we do about the fact that we normally take
only a few measurements and essentially never take a number even
remotely resembling a very large number? How do we extract an esti-
mate of the true value from a small, finite set of n measurements? The
answer involves an analysis of kinds of errors.

In principle there are two kinds of errors in measurements. First, there
are the so-called systematic errors — such as those resulting from the use
of an inaccurate scale. This kind of error source is eliminated only
through painstaking examination of the measurements themselves and
analysis of the machines and procedures involved. We have no way of
being sure that all such errors have been eliminated ; we hope for the best.

The second kind is the chance error with which biometry is concerned.
We assume that these chance errors are just as likely to yield measure-
ments that are higher than the true value, as they are to yield measure-
ments that are lower than the true value. On this assumption, the devia-
tions, d i (=x i —a), of the individual measurements, x h from the true
value, a, should total zero. This condition can be stated algebraically
and we now show how this enables us to determine the value of a from
a given set of measurements.

J] di = d x + d 2 + d 3 + • • ' + d n = Yl ( x i ~ a )

i i

= xi + -r 2 + x 3 + • • • + x n — na.
Setting the sum of the deviations equal to zero gives

= x-i + x 2 + x 3 + • • • + x n — na
or

Xi + X 2 + X 3 + • • • -f X n Hi Xi

a =

n n

This condition leads directly to the choice of the arithmetic mean as the
desired average. With this choice, the average deviation is zero, since
there must be as much positive deviation as negative deviation. To get

MEASUREMENTS AND THEIR VARIATION 3

information about the deviations, we cannot use the average deviation,
since a set of measurements with large deviations from the average will
appear equivalent to a set with small deviations, in that both have an
average deviation of zero. It is plausible that we are really interested in
the absolute values of the deviations as a measure of the accuracy and
reproducibility of our measurements. Now, it is not simple mathemati-
cally to work with absolute values. Therefore statisticians have used the
least complicated way which is almost equivalent to using the absolute
values: they use the squares of the values of the deviations of the indi-
vidual measurements from the average. Since these are always positive,
the sum of these squared deviations is also always positive, and the
quality of the set of measurements can be inferred from the smallness of
the sum of the squared deviations.

Now, however, we must ask what average must be taken to minimize
the sum of the squared deviations; this value of a will be the one around
which the individual measurements cluster according to our proposed
criterion of least-squared deviations.

For those who cannot use the differential calculus, the result is given
a few lines below. For those who can use the calculus, the derivation
follows.

We wish to minimize the expression for the total variation:

2 d 2 i = J2 ( x i ~ «) 2 = (*i - «) 2 + ( x 2 - a) 2 + • • • + (x» - a) 2 .

i i

To minimize this expression we differentiate with respect to a, keeping
everything else constant, and set the result equal to zero:

2(.n - o)(-l) + 2(x 2 - o)(-l) + • • • + 2{x n - o)(-l)

= 2£(s< - o)(-l) = 0.

i

Dividing by 2 and collecting terms, we find

— Oi + x 2 H + x n ) + na = 0.

Thus, finally, solving for a, we obtain

•>'l + X 2 + • • • + Xn 2~2iX{

a =

n n

We have thereby shown that the value of a which minimizes the sum
of the squared deviations is again just the arithmetic mean.

We need to introduce a few new terms at this point. First, V, the
average variation is simply the total variation divided by the number of
independent measurements taken:

v _ . E i di 2 _ £ f (xi — a) 2
n n

4 THE MATHEMATICAL TREATMENT OF DATA

This average variation is called the variance, and we shall use the
term, even though it seems unnecessary to lose the directly given de-
scription inherent in the longer name.

Now, we could specify the results of our set of n measurements by
saying they yield the estimate of the true value to be a ± V. This seems
somewhat silly, for it would mean, for instance, that we would give the
length of something as 5.7 cm ± 2.3 cm 2 , and it is silly to talk of squared
centimeters as expressing the variation in the measurement of centi-
meters. It would appear more reasonable to take the square root of the
variance, so that the average and its variation would then have the same
dimensions. Thus, in the illustration above, we would say 5.7 cm ±1.5
cm, and this statement has a sensible ring to it.

Adopting this change, we need to name the new quantity — the square
root of the variance — and it is called the standard deviation. It is usually
denoted by s, or by the equivalent Greek letter cr (sigma). So the results
of a set of measurements are summarized as a ± s, where

a = >

n

and

s =

\ Li U'j — a )'

n

Now that we have this formidable looking apparatus, we have to ask
what it means. Of what use is the averaging procedure? What signifi-
cance attaches to this standard deviation? The procedures of statistics
supply answers to these questions, and the answer is based upon finding
that sets of n measurements do not exhibit many different kinds of
clustering around the average a but that, surprisingly, only two kinds of
clustering are actually obtained. To be concrete, if we plot n it the num-
ber of times we get the measurement x i} we obtain only two different
curves, as sketched in Fig. 1. In practice, we divide n t by the total
number of measurements to obtain the fraction of measurements which
yield the result x t . This fraction is called the frequency, f i} of a par-
ticular measurement, and it is / { which is indicated in the figure.

Note that these curves differ in that one is symmetrical, the other
asymmetrical. Now, it has been possible to derive mathematical ex-
pressions for frequency curves under assumptions to be set forth later.
It turns out that there are basically two types of curves, and the two
types, when graphed, look very much like the two empirically obtained
curves. There is a famous quotation to the effect that mathematicians
believe their theoretical curves are relevant because they have been
shown to exist in nature, while experimental scientists believe the experi-

MEASUREMENTS AND THEIR VARIATION

Xi

xi = the measured value of some property

/i = the frequency with which the value Xi
is obtained in the set of measurements

Fig. 1. A schematic representation of the two kinds of sets of measurements
usually obtained experimentally.

mental curves are reasonable because their relevance has been proven
by the mathematicians. In fact, of course, there is an inseparable mutual
interaction between the two approaches.

To derive the symmetrical curve, the mathematicians assume that
there exists a large number of small, randomly occurring, positive and
negative contributions of error to the individual measurements. Thereby
they derive the following expression for the frequency, f u of deviations
from the average value :

fi = "4= e~ (Xi

-a) 2 /2s 2

Here the pi and 2's result from the requirement that the individual
fractional frequencies add up to unity. That means that the fractions of
cases with all possible deviations from the average must includ 3 all cases,
so that the sum is unity (or 100%, if expressed in percentage terms).

This distribution of deviation frequencies is called, variously, the
Normal Error Curve, the Normal Distribution, or the Gaussian Distribu-
tion, after the great 19th-century scientist Gauss. The bell-shaped distri-
bution is plotted in Fig. 2, which also indicates the frequencies for
deviations equal to zero, ±s, ±2s, and ±3s.

We can now begin to answer some of the many questions we have
been accumulating. First, what fraction of the deviations lies within
±s of the average value a? This amounts to finding the fraction of cases
in the shaded area of Fig. 3. (For those who understand the calculus,
we can say that the fraction is found mathematically by integrating the
Gaussian Distribution from a — s to a + s.) The result is that about
% of the deviations fall in this range; conversely, of course, % fall out-
side this range. This means that if the average value of a measurement is
a, then by chance alone one will have a measurement deviating from a by

THE MATHEMATICAL TREATMENT OF DATA

a — 3s a — 2s a

a + 2s a + 3s

.r = the measured value of some property

/=the frequency with which the value
x is obtained in a set of measurements

Fig. 2. Gaussian distribution of measurements, showing the average value a
and the frequency of measurements equaling a or deviating from a by one, two,
or three standard deviations.

Fig. 3. The shaded area is proportional to the fraction of all measurements
lying between a — s and a + s.

at least ±s in one of every three measurements. By calculating in an
entirely similar way, the following table is obtained:

Occurs in fraction

A deviation of:

of measurements

±s or more

1/3

±2s or more

1/20

±2. 3s or more

1/100

±3.s or more

1/400

Note that a deviation in only one direction will occur in half the given
fraction of measurements. That is, for example, a deviation of 3s or
more will occur, by chance alone, once in 800 measurements. Similarly,

MEASUREMENTS AND THEIR VARIATION 7

a deviation of —2s or more (in the negative direction) will occur once in
40 measurements.

This table will provide the basis for much of the analysis we will
present. With its presentation, we have answered one of the two ques-
tions we set out to answer about the standard deviation: what is its
significance?

The other question concerns the utility of the averaging procedure
employed in obtaining the arithmetic average. We already know that
the clustering of measurements around this value is greatest, but still
more can be said. If we ask the mathematicians to compute the cluster-
ing of the averages themselves, we are asking what would be the distri-
bution of averages if we made a set of n measurements many, many
times. Another way of asking the same question is to ask for the standard
deviation of the average itself. The mathematician's reply to the latter
question is

V

or

n

V

V = —
n

That is, the average variation, or variance, of the average measurement
is only one-nth of the variance of the individual measurements. This is
the result we want. It says that the reason for taking the average is that
the average is roughly -\Jn times more likely to be the true value than is
any individual measurement. s a is called the standard error by statis-
ticians.

X or a

Fig. 4. A schematic representation of a set of individual measurements, x,
and of a set of averages, a, obtained by making many repeated sets of measure-
ments. The relative narrowness of the curve for a shows that the average ob-
tained in any single set of measurements is very likely to be close to the "true"
value.

Still another way of indicating the same information is shown in the
schematic drawing in Fig. 4, which shows the distribution of individual
measurements and also the distribution of averages. The latter is so
narrow that it is very unlikely that we will ever get a much different
result, even by chance.

8 THE MATHEMATICAL TREATMENT OF DATA

One problem with our analysis thus far is that we have asked the
mathematicians to tell us the distributions when we have many, many
sets of measurements. Suppose we don't have so many measurements.
After all, who feels like making more than, say, a half dozen, or maybe
ten, measurements, let alone an essentially infinite number? The answer
to our suppositional question is a surprisingly simple one. The mathe-
maticians tell us that the difference comes in two places: in computing
the standard deviation, and in computing the percentages given in our
reference table above. The computation of s changes only by substi-
tuting n — 1 for n. This has the effect of increasing s, and is increasingly
important the smaller n is. For instance, if we make only two measure-
ments, the variance would double as a result of this correction. Next, if
we ask what the table percentages become, we have to realize that the
answer will depend on n — there will be a different answer for each value
of n. Thus we would need not just the simple table above, but a set of
tables. We will not present such a set, but tell you that you can find
them in any standard statistics book. But, in the spirit of the present
exposition of the subject, we can tell you that these tables aren't really
so necessary. In practice, if n is as large as 10, one makes little error in
using our own little reference table; if n is 20 or more, there is really no
error in our table. So, just by modifying the expression for s to

Kxj - - a
'' \ n - 1

we obtain most of the correction needed because of the small samples.

Up to this point, we have been dealing with the symmetrical curve for
the distribution of deviations from the average. We have still to discuss
the asymmetrical distribution before giving some applications of the
analytical procedures which have been developed.

THE ASYMMETRICAL DISTRIBUTION

The Gaussian Distribution of deviations dealt with the case of a large
number of small sources of error. The asymmetrical distribution deals
with the case of having basically two alternative situations, one of which
is very unlikely. Indeed, the one which is unlikely is so unlikely that it
would be irrelevant if it were not for the fact that there were so many
cases in question that the total possibility is not negligible. Take, for
example, the case of distributing samples, each 1/100 ml, of a bacterial
suspension containing 10,000,000 cells per milliliter. The probability of
finding any individual bacterium in a particular sample is very small:
1/100 of 10,000,000, or 1/100,000. Yet, because of the large number of
bacteria in the sample, the probability of finding some bacteria in the

THE ASYMMETRICAL DISTRIBUTION 9

sample is actually reasonably large. This situation has been analyzed and
the distribution which governs it has been given the name of the French
mathematician Poisson. The Poisson Distribution is

— a n
r> i \ e a

n

where e is the base of natural logarithms, and equals 2.71828. This
represents the probability of finding precisely n of something if the
average is a. If we were talking the language of deviations, we would
be talking about the quantity n — a. Here, however, the least we can
have is a equal to zero, so that we do not have very many negative
deviations: if the average were 2, then we would have negative deviations
of only —1 and —2. Thus this distribution is basically asymmetrical.

As an illustration of the use of this distribution, suppose we distribute
100 spores into 100 test tubes at random. The probability of finding any
given spore in any given test tube has an average value of 1/100, but
the probability of finding some spore is quite appreciable. Indeed, the
average number of spores per tube is 100/100, or 1. The Poisson formula
says that if the average is 1, then, for example, the probability of finding
precisely 2 spores in a tube is

P 2 (D = 6 ' 2 * ^ = 0.18,

as we can determine from tables or from direct computation using the
numerical value of e.

One of the most important uses of the formula has to do with finding
the zero class— the probability of finding none if the average is a. We
know intuitively that if we tossed 100 spores randomly into 100 test tubes,
there would be a substantial number of tubes having no spores. The
formula gives the zero class as

Po(a) = e~ a .

In the illustration above, e" 1 equals 0.37. In 377c of the tubes there will
be no spores at all.

The zero class formula is just as useful in the inverse way. If we
measure the zero class, we can use the formula to obtain the average. If,
for example, we found 2% of the test tubes without spores, we would solve
the equation

0.02 = e~ a .
A look at some tables gives the result for a as a little less than 4 spores

10 THE MATHEMATICAL TREATMENT OF DATA

per cell. The following table for the zero class may be of some use:

a = 1 2 3 4 5

p (a) = 0.37 0.14 0.05 0.018 0.0067

The Poisson distribution turns out to have another interesting and use-
ful property. If the standard deviation of things satisfying this distribu-
tion is calculated, it is found that

s 2 = a.

This remarkable result permits obtaining the standard deviation from
the average alone. Since the average can be obtained from the zero class,
it becomes an exceedingly simple matter to obtain the standard deviation.
In the illustration above, where there were 2% of the tubes unoccupied by
spores, we not only know from the little table that the average occupancy
is about 4, but also that the standard deviation is close to 2.

This property is frequently used in the design of experiments. Suppose
we wish to determine the number of spores to an accuracy of, say 5%.
We have only to solve the relation

* = ^ = 0.05.
a a

The result is a equals about 400. So we know that we must count until a
total of four hundred spores have been tallied in order to know the num-
ber 400 to an uncertainty of 5%.

COMPUTATION OF ERRORS IN COMPOUND QUANTITIES

It frequently happens that the desired quantity cannot be measured
directly, but can be determined from two or more other measurements.
For example, the molecular weight of a particle can be determined by a
combination of sedimentation and diffusion data according to the follow-
ing formula:

s RT

M

D \ (di/d p )

where M is the molecular weight of the particle of density d p , suspended
in a liquid of density d l} at a temperature T, and R is the gas constant.
Respectively, 1) and s are the coefficients of diffusion and sedimentation,
and are obtained experimentally. For our purposes, the desired quantity,
M, is given in terms of the complex ratio of 5 measured quantities. If
each of these has its own standard deviation, what is the standard devia-
tion of M?

COMPUTATION OF ERRORS IN COMPOUND QUANTITIES 1 1

We consider the problem in two simplified special cases. First, the case
where some quantity M equals the product of two other measurements:

M = AB.

Then, using s M ,s A , and s B for the standard deviations of the three quan-
tities, we have

M ± s M = (A ± s A )(B ± s B ) = AB ± Bs A ± As B ± s A s B

or, neglecting the term s A s B as being usually much smaller than the terms
retained,

sm = ±Bs A ± Asb-

Dividing by M and its equivalent AB yields

sm_ _ sa sb

M A B '

This tells us that the fractional uncertainty in M is compounded of the
fractional uncertainties in the two factors A and B. Because of the plus
or minus signs, we don't know what to do at this point. Statisticians have
been able to show that the most likely value in this case is given by the
square root of the sum of the squares of the fractional uncertainties in
A and B:

M

JW + m

It is a relatively simple matter to extend this to the product of more than
two factors. And since a quotient A/C is equivalent to a product A(l/C),
an entirely similar formula holds for quotients. The general results can
then be indicated as

A ■ B

M

sm
M

C

=VG) 2 +Gf) 2 +0r

The second part of the problem is the case where M equals the sum
(or difference) of two quantities:

M = A + B,
M ± s M = A ± s A + B ± s B ,

sm = ±s A ± s B .

Thus, when the relationship is addition (or subtraction), the actual un-
certainty, not the fractional uncertainty, is compounded of the actual

12 THE MATHEMATICAL TREATMENT OF DATA

uncertainties of the factors making up M. As before, the most likely
value is the root-mean-square one:

S M = Vs 2 A + S%

Now, if one has a more complicated relationship, such as
the standard deviation may be written as

SM //^i\ 2 /^B\ 2 / Sc+D \2

m~ \ika) + u; + \C + Dj '

As we have just shown,

S C+ D = s c + s D .

Thus, finally,

SB\2 , (Sc + Sd)

SM / /SA\ 2 , /Sb\ 2 {S C j- Sp)

M SKA) ^ KB) f (C+2)) 2

APPLICATIONS OF STATISTICAL ANALYSIS

The general idea in these analyses is to compute the probability that a
particular situation could have arisen by chance. If it could have oc-
curred by chance once in a million times, everyone would agree that
chance is not the explanation; there must be an underlying mechanism
for the deviation. Suppose, however, that by chance alone the situation
could have arisen once in 10 times — what this means is that a deviation
this great or greater will occur that often. Do we then say that a 1/10
chance is too unlikely, or that it is so likely to happen that the deviation
is not to be regarded as significant?

Statisticians have found through experience that an event that can
occur once in 10 times by chance alone is not sufficiently unlikely to
warrant undue interest. They have arrived at the following statements:

A probability of 0.05 is significant.

A probability of 0.01 is highly significant.

In the applications we shall present, these two levels of significance
will be utilized repeatedly.

There are three main kinds of statistical computations that we shall
set forth. These involve:

1. the ratio of a given deviation to the standard deviation;

2. the ratio of one variance to another variance;

3. the chi-squared test.

APPLICATIONS OF STATISTICAL ANALYSIS 13

1 . The ratio of a given deviation to the standard deviation

As an illustration, suppose we are studying the effect of adding various
biochemicals to a nutrient medium and find that, after a given time, there
is an average of 183 organisms per test tube, with a standard deviation
of 9 organisms. One test tube (that has one particular biochemical addi-
tive) contains 212 organisms. Is this increase significant?

The deviation is 29 organisms, which is slightly more than 3s. From
our previous reference table we find that a deviation of this much or more
will occur by chance alone about one time in 800. Therefore we say
that the deviation is highly significant and it is therefore highly likely
that the particular biochemical had a real, positive effect.

If there had been another tube with 198 organisms, the deviation would
be 15 organisms, and the ratio of this deviation to the standard deviation
is almost 2. Thus the probability of this happening by chance alone is
almost 0.025, and is therefore considered almost significant. Thus we
don't quite know whether to consider that the particular additive had a
real effect. We have to do more or better experiments to decide.

The ratio of the observed deviation to the standard deviation should
really be evaluated by using tables which depend on the number of
independent measurements, as explained previously. This computation
was done many years ago by a man publishing under the pseudonym
Student. He called the ratio t, and you will therefore find this test in the
textbooks under the name Student's t test.

The evaluation of n is not entirely trivial. When we choose a par-
ticular average a, the variability in the set of measurements is reduced
by the choice. Indeed, given n — 1 measurements and a, one can recon-
struct the nth measurement. Thus, knowing a, there are only n — 1
independent measurements. Similarly, every time we fix a parameter
like a or s or the total number of things counted, we impose a condition
which allows us to reconstruct the entire series of measurements with one
fewer measurement.

In a way entirely similar to that for individual measurements, we can
compare deviations of averages with standard deviations of averages.
If two laboratories measure the concentration of viruses in a vaccine
preparation, the deviation between their average values should be com-
pared with the standard deviation of the averages (the standard error)
to see whether there is agreement or systematic deviation between their
results.

2. The ratio of one variance to another variance

It happens with surprising frequency that there is need to compare
two variances. The analysis of the problem ends up with an estimate
of whether the ratio of the two variances is sufficiently different from

14

THE MATHEMATICAL TREATMENT OF DATA

Dilute starting culture
of bacteria

Growth period of several hours

Concentrated final culture

Concentrated final

culture distributed

1 ml per tube into

many tubes

Fig. 5. Bacteria are grown for a given period of time from a dilute starting
culture to a concentrated final culture. This final culture is distributed in 1 ml
aliquots into many tubes. These tubes are then sampled to determine the num-
ber of mutant bacteria in each tube.

unity to warrant the deduction that there are different mechanisms in-
volved in producing the fluctuations which result in the deviations. As
with the deviation ratio test, the actual conclusion is that by chance
alone a ratio this large or larger will occur in a certain fraction of cases
with the given number of independent measurements in each of the vari-
ances. And, once again, the question of significance is decided on the
basis of one chance in 20 being significant and one chance in 100 being
highly significant.

As an example, we take an investigation which was historically a
turning point in the understanding of the mechanism of mutations. The
appearance of a mutant cell is detected by placing the cells in an environ-
ment in which the mutant property can be expressed. An illustration from
bacterial genetics would be the appearance of resistance to a given anti-

APPLICATIONS OF STATISTICAL ANALYSIS

15

Dilute starting culture
of bacteria

Dilute starting culture is
distributed 1 ml per tube
into many tubes

IP

ip

'

W

III

Growth period of
several hours

I

Concentrated final cultures
of bacteria, each tube
grown separately from the
dilute starting culture

Fig. 6. This experiment differs from that of Fig. 5 in that the bacteria are
distributed into separate tubes before the growth occurs. The final tubes are
then sampled to determine the number of mutant bacteria in each tube.

biotic. One asks whether the testing with the drug induced the resistance
in a small fraction of the cells or whether the property had occurred
previously, by chance (spontaneously) , in that small fraction of the cells.
Thus the question was posed as to whether mutations are spontaneous or
induced. The answer to the question involved an anlysis of variance.

Consider, as in Fig. 5, a flask containing 100 ml of medium, inoculated
with some cells which then grow up to some final concentration. If
the 100 ml are distributed in 100 tubes and analyzed for the number of
mutants per tube, we will find an average value for the mutant concen-
tration and a variance calculated from the deviations from the average.

Next, consider a parallel experiment in which the 100 ml are distributed
into the 100 tubes as soon as the flask is inoculated. The bacteria then
grow up to the same final concentration as before (see Fig. 6). These
100 tubes are now analyzed for the number of mutants per tube and,
as before, an average and a variance are obtained.

If the mutation mechanism is by induction of drug resistance, then
the results of the two experiments should be entirely similar, since the
drug is acting on the same number of cells in the same number of tubes.

16 THE MATHEMATICAL TREATMENT OF DATA

If the mutants arise spontaneously, we will expect the same average
number of mutants, since the same total number of cells is being tested
in each case. But in the second experiment the deviations should be
greater, since in some tubes a mutant may arise early during growth
and give rise to a large percentage of mutant cells; alternatively, in some
tubes mutants may chance to arise very late, so that the percentage of
mutant cells in these tubes will be very small. Therefore there is a clear
difference in prediction by the two models. The induction model predicts
the same variance in both experiments; the spontaneous arisal model
predicts a variance in the second experiment which is much greater than
the variance in the first experiment.

In several control experiments, in which the number of resistant
bacteria was determined in different samples from the same culture flask,
it was found that the mean number of mutants equaled the variance with
reasonable fidelity. For example, where the mean was 3.3, the variance
was 3.8; where the mean was 51.4, the variance was 27. Now, in the
second part of the experiment, the number of resistant bacteria was
determined in the set of similar samples from different culture tubes.
Here the means were very different from the variances. For example,
where the mean was 3.8, the variance was 40.8; where the mean was 48.2,
the variance was 1171.

Using tables of the variance ratio (with the proper number of inde-
pendent measurements for each variance), it was shown that the experi-
mentally found ratio was far outside any chance fluctuation of the ratio
away from unity. Thus the mechanism of spontaneous, noninduced mu-
tation was first demonstrated.

3. The chi-squared test

Chi squared (x 2 ) is defined as

,2 T,i 0* — a) 2

X"

s 2

'th

Now, for a largo number, n, of independent measurements, this ratio
should be approximately n, since the numerator is n times the experi-
mental value of the variance and the denominator is the theoretical value
of the variance obtained by using an essentially infinite number of meas-
urements. So the tables of chi squared show the number of independent
measurements as a parameter. Up to this point, chi squared is just an-
other variance ratio test, with the denominator being a theoretical vari-
ance (approximated by taking an inordinately large number of measure-
ments or estimated by theoretical methods beyond the scope of our
treatment).

APPLICATIONS OF STATISTICAL ANALYSIS

17

The chief use of chi squared is in determining the ratio for a whole set
of different points with different theoretical values. For instance, if we
wish to compare an experimental set of values with any theoretical
curve, we use chi squared. In this instance, the variance in the denom-
inator is replaced by the mean value predicted by the curve, since the
Poisson Distribution afforded us the insight that the mean is equal to
the variance. The numerator remains n times the experimental variance
even though in application of chi squared there is usually only one
measurement made at each point, so that, usually, n = 1. Next, we
sum the expression for all the points of the curve at which experimental
values were taken. This sum over the various points of the curve is
indicated by a subscript p to distinguish it from the sum of deviations at
individual points. This sum of chi squareds is then written as

= E

(x p — a p y

Since, as we have remarked, the means are good estimates of the theo-
retical values at each point, the expression is more usually written as

= E

(*

exp

•r t h) :

Zth

This is the formula given in most textbooks.

Since the value of chi squared at each point should be unity, on the
average, the sum should total simply to the number of independent
measurements. Indeed, if we look at tables for chi squared, we find that
the values are very closely equal to the number of independent measure-
ments at a probability of occurrence equal to Va- The following table
will assist us in demonstrating the use of chi squared.

X 2

n

v -- 1/3

p = 0.05

p = 0.01

1

1.09

3.8

6.6

2

2.2

6.0

9.2

3

3.3

7.8

11.3

4

4.3

9.5

13.3

5

5.4

11.7

15.1

6

6.5

12.6

16.8

10

10.0

18.3

23.2

20

20.5

31.4

37.6

As an illustration, we compare some experimental data with a straight-
line formula to find whether it can represent the data adequately.

18

THE MATHEMATICAL TREATMENT OF DATA

Fig. 7. A plot of some experimental data and a line which is to be tested by
the chi-squared criterion for its goodness of fit to the data.

The plot of the data and the theoretical curve are shown in Fig. 7.
The theoretical curve was derived by methods to be discussed in a later
section. The equation is y th = —0.1 + 1.3.r. The values of y exp and y th
at the experimental values of x are

x = 1 2 3 4 5
2/th = 1.2 2.5 3.8 5.1 6.4
z/ exp =21547

The deviations are, for the straight line,

(2 - 1.2), (1 - 2.5), (5 - - 3.8), (4 - - 5.1), and (7 - - 6.4).
Their chi squared is then given by

X'

0.64 . 2.25 . 1.44

1.21 0.36

1.2

= 0.533

= 2.613

2.5 3.8 ' 5.1 6.4

0.900 + 0.379 + 0.237 + 0.564

There are five measurements, and we have chosen the two parameters
of the straight line (the intercept on the y-axis and the slope of the line)
to make a best fit, so there arc only three independent measurements.

Looking at the chi-squared table, we see that for three independent
measurements we could get a chi squared of 3.3 or more in one of each

APPLICATIONS OF STATISTICAL ANALYSIS 19

three sets of measurements we might make. Therefore the deviations
from the straight line are not significantly different from what would be
expected by chance alone. Thus the straight line is an adequate repre-
sentation of the data.

In genetics, chi squared is usually used as in the following example.
The problem is whether there is a correlation between two properties:
hair color and eye color. It was found that, of 80 individuals, there
were 36 with blue eyes and light hair, 11 with brown eyes and light hair,
9 with blue eyes and dark hair, and 24 with brown eyes and dark hair.
Are the deviations from chance great enough to warrant the conclusion
that hair color and eye color tend to vary together?

Note that there are 47 light-haired individuals (and therefore 33 with
dark hair). There are 45 blue-eyed people (and therefore 35 with brown
eyes). If the light-haired people had the same proportion of blue and
brown eyes as the entire population studied, then 45/80 of them would
have blue eyes. This is 45/80 X 47 or 26.43 people.

It is not necessary to compute any of the other classes of individuals
because there must be a total of 47 light-haired people, so that theo-
retically there would be 47 — 26.43 = 20.57 light-haired people with
brown eyes if the eye color distributes as for the entire group of people.
Further, since there are 45 blue-eyed people in all, and 26.43 have light
hair, the remaining 45 — 26.43 = 18.57 people should have dark hair.
Finally, the number of dark-haired, brown-eyed people should be 14.43.

Chi squared is then computed as the sum of the four terms:

x

(36 - - 26.43) 2 (11 - 20.57) 2 (9 - 18.57) 2 (24 - 14.43) 2
26.43 20.57 + 18.57 14.43

= 19.19.

The number of independent measurements is unity. We saw this
directly, since as soon as we had computed one theoretical number (26.43
blue-eyed, light-haired people), the other three numbers were fixed. We
can also fairly readily see this by counting the number of quantities we
have fixed, since we fixed the total number of people to be 80, the total
number of blue-eyed people to be 45 and the total number of light-haired
people to be 47. Thus we have only one of the four measurements which
can be arbitrarily made.

Looking at our table, we see that for n = 1, the probability of getting
a value of 6.6 for chi squared is already 0.01. To get a value of 19.19
would correspond to a probability much smaller than this. Therefore
the deviations from theoretical values can by no means be accounted for
by chance fluctuations. Thus we conclude that the hypothesis must be
that eye color and hair color do not vary randomly, but are associated.

20 THE MATHEMATICAL TREATMENT OF DATA

TREATMENT OF EXPERIMENTAL DATA

1 . Data-fitting using the least-squares principle

The least-squares principle was used to obtain an index of the varia-
tion in data. It has other uses, one of which is the subject of this sec-
tion. The problem of finding a theoretical expression to fit data utilizes
this principle to choose the parameters of the curve so as to minimize
the variation around the curve. We illustrate the approach by finding the
expressions for the parameters of a straight line which is to be fitted
to the data.

The equation of a straight line is

y = a + bx.

Consider the difference between the theoretically and the experimentally
determined y:

2/th — */ex P = a + bx — y QXp .

These differences are sketched in Fig. 8.

Our principle of least squares tells us to square this difference, add all
the squared differences for all the points, and then to choose a and b so
as to minimize the sum. For those with mathematical facility, the deri-
vation of a and 6 will be presented. For other readers, the results are
given below.

!lth = (i+bx

Fig. 8. A plot of some experimental data relating y and x and a theoretical
line whose slope and intercept are to be chosen to minimize the total deviation
of the theoretical values of y from the experimental values of y. The short
vertical lines are the deviations themselves: ?/ th — ?/ exp .

TREATMENT OF EXPERIMENTAL DATA 21
The mathematical problem is to minimize

E(2/ — 2/exp) 2 = E(« + bx — 2/exp) 2 -

We take the derivative with respect to a and b in turn and set the deriva-
tive equal to zero, obtaining

= E2(a + bx — i/exp),

= L2(a + bx — y exp ) (x).

Factoring out the 2's, and multiplying as indicated in the second equation,
we obtain

E<* + bEz — HVexp = = na + 6(2» — El/exp,

Lax + bY,x 2 — E^exp = = a(E-*0 + 6(2> 2 ) — E^exp-

We can solve these two equations for a and b to give us the values of
a and b which minimize the deviations from the theoretical line:

(Es 2 )(E2/exp) — (EzyexpXEs)

n(E* 2 ) - (Ex) 2

b _ n(£,xy exp ) — (E-g)(E^/exp)
"(E* 2 ) - (E*) 2

To show the use of these expressions, and also to demonstrate that
the expressions are much more fearful-looking than actually harmful,
we illustrate by obtaining the straight line used in the previous section
on chi squared. Because we will need the sums, we tabulate the data
again, along with the operations needed for this application.

Ex = 1+2 + 3 + 4 + 5= 15

E2/exp = 19

E^ 2 = 55

E^exp = 70

We substitute the various numbers in the expressions for a and b,
obtaining

(55)(19) - (70)(15) ^5
(5) (55) - (15) 2 50

(5) (70) - (15) (19) 65
5(55) - (15) 2 " 50 " '

The least-squares straight line for these data then is

y = -0.1 + 1.3a;.

X = 1

2

3

4

5

2/exp == ^

1

5

4

7

x 2 = 1

4

9

16

25

.VexD — "

2

15

16

35

22 THE MATHEMATICAL TREATMENT OF DATA

If we had wanted to fit a more complicated curve, the procedure would
have been entirely similar. For example, suppose we had decided to try
something quite complicated, such as

y = a sin bx + c log x.

Aside from the purely mechanical problem of carrying out the calculus
operations, the procedure is entirely similar. We find expressions for
a, b, and c so as to minimize the sum of the squares of the deviations
from this expression, thereby obtaining three equations for the three
parameters. We could, in fact, go as high as five parameters with these
data, and then we would be able to fit the data exactly at the experi-
mental points, although the curve might fluctuate wildly between those
points.

There are other methods of obtaining curves to fit data, but the curves
obtained by the least-squares principle are the generally accepted ones.
Further, there are many subtleties in the least-squares approach which
the student can find in the standard texts.

2. Correlation

We can say a few things about the concept of correlation. Here we
ask the question of the extent to which variations in one variable are
accompanied by a parallel variation in another variable. It must be
strongly emphasized that there is not the slightest implication of a direct
connection between the two variables. The pedagogically favored illus-
tration of this point is the finding of an almost perfect proportionality
between teachers' salaries and liquor consumption. The pedagogue then
declares that it is manifestly ridiculous to deduce the validity of the
causative relation — the teachers are drinking all that liquor. If we study
the details of the situation, we discover that teachers' salaries rise during
prosperous times and during such times the entire population has extra
money, some of which is spent on liquor.

In the way in which we will introduce correlation, we look at the extent
to which points cluster about the presumed line expressing the possible
correlation, and compare it with the clustering around the mean value
of all the points. Consider Fig. 9.

If we call y the average value of //, then the variation around the
average is £(// y) 2 .

The variation around a proposed line (to be found theoretically) is

H(>J - - 2/th) 2 .

TREATMENT OF EXPERIMENTAL DATA

23

Fig. 9. A plot of some experimental data (o) ; their deviations (---) from
the mean, y; and their deviations (— -) from a line proposed to fit the data.
The extent to which x and y are said to be correlated is given by the extent
to which the line is a better representation of the data than is the value y. The
criterion for better representation is the ratio of the sum of the squares of the
deviation from the line to the sum of the squares of the deviations from the
mean, y. The quantitative use of the criterion is given in the text.

Consider the ratio of these two variations:

E(y — Vth) 2

Z(y - y) 2

If there were a perfect correlation between y and x — that is to say, if
the line fit the data perfectly — the numerator would be zero. If the line
is less than a perfect fit, the numerator becomes increasingly large until,
finally, if there is absolutely no connection between y and x, there will
be as much variation around the line as around the mean value of y.
Then the ratio will be unity. Thus the values of this ratio vary from
zero for a perfect correlation to unity for completely unrelated things.
Since we wish to speak of the extent to which things are "co-related," the
statisticians have chosen to use one minus this ratio as the index of
correlation, R. Then perfect relation is unity and complete lack of
relation is zero.

R

1

E(y - V) 2

There is one more complication to be taken care of. The way we
have chosen the relationship index, it is always positive. This presents
a problem, because it is possible for things to be negatively connected, in
that an increase in one variable correlates with a decrease in the other.

24 THE MATHEMATICAL TREATMENT OF DATA

This problem has been solved by using the square root of the relationship
index with the proper sign. That is,

r = ±\/R ,

where r is what is technically defined as the correlation coefficient. It
can be shown algebraically that this definition of r is entirely equivalent
to the one used in standard texts:

£(z — x)(y — y)

VZ(x -- x) 2 -Z(y -- v) 2

Here the numerator is to be understood as showing the extent to which x
and y vary together. For a large number of pairs of values of x and y, if
they were unrelated, then for each particular plus value of (y — y), there
would be both a positive and a negative value of (x — x) , so that in the
sum these terms would cancel out. If there were actually a correlation,
then when the x difference was positive the y difference would also tend
to be positive and there would be no cancellation. The purpose of the
denominator is to make the coefficient dimensionless and to give it the
values zero and unity for, respectively, unrelated and perfectly related
variables. To prove the equality of the two definitions, use must be
made of the least-squares line relating y and x.

By using our definition based on clustering, those of us with mathe-
matical facility may use any type of theoretical expression for y and
can see how to compute the correlation even for curved lines. The rest
of us have to take the expressions for r as given.

A point which is very much understressed in many standard treat-
ments is that the correlation index — the square of the correlation coeffi-
cient — is the direct measure of the extent to which variables are cor-
related. Thus, a correlation coefficient of 0.6 looks respectably large, but
we note that its square is 0.36. This tells us that only about % of the
variation in y is correlated with variations in x, the remainder being
associated with factors which do not affect x. From this point of view
we may say that one-third of the factors determining x and y are shared,
the others being different for the two variables.

CHAPTER

2

Physical Forces
and Chemical Bonds

INTRODUCTION

As the current pace of advances in molecular biology and biochemistry
accelerates, a detailed understanding of the structure, interactions, and
functions of biologically important molecules becomes increasingly neces-
sary. This means that the forces involved in chemical reactions must
be increasingly taken into account, and this requires detailed information
about the nature of the various chemical bonds. Consequently, in this
chapter we present the basic physical forces and try to give an idea
of how they operate to create the various chemical bonds. Funda-
mentally, the study of such matters involves comprehension of the com-
plex mathematical apparatus of modern theoretical physics and theo-
retical chemistry, and it is impossible to get around this requirement.
Nevertheless, almost all scientists have geometrical and mechanical
images and models in their minds which guide them in their use of such
a mathematical apparatus to the point that these images and models
serve as the bases of their "intuition" when considering new problems.
These images and models are admittedly limited in scope and accuracy,
and each scientist may have his own peculiar set. This writer believes
that it is useful for beginning students to acquire some intuitive feelings
about such matters even before they acquire the detailed apparatus for
understanding the matters. Indeed, in some respects it is only to the
extent that today's complexity becomes incorporated into the intellectual
equipment of the youth that youth has the capacity to transcend the
elders. In this spirit, then, is the following discussion of forces and
bonds presented.

PHYSICAL FORCES

The difference between Newtonian and quantum physics is of a nature
which far transcends their procedural differences. But, for our purposes,
we find it useful at this point to look at an aspect of the procedural dif-
ferences: the emphasis of Newtonian physics on the concept of force; and
the emphasis of quantum physics on the concept of energy. Indeed, it is
in great part this shift which permitted physics to provide the basis for

25

26 PHYSICAL FORCES AND CHEMICAL BONDS

its deep contributions to chemistry, which has always worked on the
level of energy.

Newton started from the Galilean principle of inertia (known also as
Newton's first law of motion) which asserts that bodies move with con-
stant velocity unless a force acts on them. Thus, the observation of a
nonconstant velocity (an acceleration) is taken as evidence for the
existence of a force on the body. Then Newton's second law provides a
recipe for quantitative work by asserting that the acceleration produced
by the force is directly proportional to the force. It is also inversely
proportional to the mass of the body. These two assertions are combined
in the usual statement of the law:

/
a = — •
m

When an acceleration is then observed, it is studied in as many dif-
ferent situations as needed to give us an idea of its general behavior:
its dependence on time, distance between bodies, dependence on masses,
velocities, etc. By such a study, Newton was able to deduce the expres-
sion for the first force ever studied as such: the gravitational force. He

deduced its formula as

mim 2

./grav "

Gr' 2

where ?% and m 2 are the masses of the two bodies exerting gravitational
forces on each other, r is their separation, and G is the gravitational
coefficient which puts / in the proper units.

The procedure for using this result in a case where a gravitational
force is known to be acting is to insert the expression for /, with the
proper values of the masses, into the first equation, solving for the ac-
celeration, a. From a, by standard mathematical procedures, the position
and velocity of the body in question can be determined at all times. For
physics, the problem is solved.

Are there any other forces in nature? How could we recognize them?
In a situation in which we can show the gravitational force not to be
effective, the observation of an acceleration is evidence for a new force.

Peasants from the province of Magnesia (Greece) long ago observed
that some stones could attract or repel other stones, and two aspects of
the situation permit us to deduce the existence of a new force. First,
there is both an attraction and a repulsion, whereas with gravitation
there is only attraction. Second, these stones exert their effect along a
table top, at right angles to the predominant gravitational attraction of
the earth. Thus the existence of what we call magnetic force was deduced.

Next, when a piece of fur is stroked with a piece of amber, both the
amber and the fur acquire the capacity to attract pieces of lint, paper,

CHEMICAL BONDS 27

and each other. Stones from Magnesia, however, have no effect. The
mutual gravitational forces of the amber and fur are inappreciable.
Therefore, there is a third force in Nature, called the electrical force.

Except for the nuclear force (which does not concern us here), these
are the only forces now known in Nature, and the explanations of natural
phenomena must necessarily involve these three forces in various com-
binations. With them, science has explained the consequences of a ball's
hitting a bat, light being absorbed by leaves, the trajectories of stars, and
the properties of molecules interacting with each other.

For completeness, it is worth mentioning that Einstein showed that the
electric and magnetic forces are really only two special aspects of a
single, unified, electromagnetic force.

There are other situations in which we speak of forces acting. For
example, when particles move through a solution, or an airplane moves
through the air, we speak of the drag or viscous forces acting on the
particle or airplane. In fact, on the molecular level there is no such thing
as a viscous force. There is something properly called viscous energy,
perhaps, but even this terminology would be unacceptable to some
scientists.

When a particle tries to move through a solution, it meets a resistance
deriving from the fact that in the liquid state the solvent molecules
attract each other strongly (generally by the van der Waals "force" to
be explained later). Thus it takes work for the particle to move between
the solvent molecules, since it must separate these molecules. Therefore,
when the particle has pushed the solvent molecules apart, and then
moves farther on, the molecules come back together with sufficient
energy (imparted by the particle and thus lost by that particle) that
they bounce off each other. This oscillational energy is communicated
by collision to the other molecules of the solvent, with the final result
that the average oscillation (thermal vibrational) energy is increased
somewhat. In this fashion, the energy required to move a particle through
the solvent produces heat, which our usual manner of speaking ascribes
to the work done against frictional forces.

Similar explanations are available for other apparent forces.

CHEMICAL BONDS

The accent on forces lasted more than 200 years, until quantum
physics shifted the emphasis to energies. In the meantime, chemists had
become increasingly occupied with energies, as thermodynamics developed
and was able to make assertions about energies involved in reactions
while saying nothing directly about forces. In this spirit, the chemists
talked about the energies of the bonds holding atoms and molecules to-

28 PHYSICAL FORCES AND CHEMICAL BONDS

gether, rather than about the forces. This has turned out to be more
than just a verbal difference, because all the bonds have turned out to be
electrical (or, rather, electromagnetic) in origin, so that the study of
chemistry using the force concept would have led to impossibly difficult
models.

The picture that we have of atoms is that of miniature solar systems
with a central positively charged nucleus and a number of planetary
electrons whose total charge is normally equal to the net positive charge
of the nucleus. If we consider a crystal of salt, sodium chloride, and ask
what holds it together, we have to know something of the conditions
within the crystal. If we toss some metallic sodium into water, the chem-
ists tell us that there will be a violent reaction, perhaps an explosion and
a fire. If we separately bubble some chlorine gas through water, some of
the gas seems to dissolve, the rest comes bubbling out. Breathing the
chlorine gas or drinking the chlorine water is harmful. Yet when we toss
some salt into water, nothing happens and no one is hurt by drinking
the water. Clearly, the sodium and chlorine cannot be in their native
states while in the salt crystal or in the dissolved state. X-ray analysis
of the salt crystal shows us that the centers of the sodium and chlorine
atoms are clearly separated from each other. The only difference in state
that we can imagine is that sodium and chlorine may have done some
swapping of their planetary electrons. Could such an apparently simple
swap have accomplished such profound changes in the properties of the
substances? The answer is a resounding yes, on purely experimental
grounds as sketched above.

In following up this kind of analysis, physics and chemistry have
shown that certain numbers of electrons are much more stable than all
others. Further, it was soon noticed that atoms which naturally had one
of these stable numbers were quite thoroughly inactive chemically. Thus,
the first attempt to explain chemistry from atomic properties utilized the
closeness of the atoms concerned to one of these stable states. In our
case, sodium turns out to have one electron too many; chlorine has one
too few. What could be more natural than for sodium to give its un-
wanted electron to the chlorine, thereby transforming both atoms into
stable configurations? If we make this hypothesis, we see that we have
simultaneously created a reason for the combination of sodium and
chlorine. For the former now has a net plus charge and the latter a net
minus charge. Electrically, the two atoms are simply pulled together.

Is it possible actually to explain the bonding by a simple electrostatic
interaction? The mutual energy of two charges depends in a very simple
way on the charges and their separation:

j? Q1Q2

CHEMICAL BONDS 29

(aX, ©\<S)/© S J®

^\. \ 30 / ^^

^^ /3o\ ^\

Na

Fig. 10. A two-dimensional illustration of the interaction of the central Na +
atom with its Cl~ neighbors. Most of the 180kcal/mol binding energy comes
from interactions with the four nearest CI - neighbors; each of these bonds is
shown to be about 30 kcal/mol, so that these four bonds add up to 120 kcal/
mol. The interactions with the eight next nearest CI - neighbors are much
weaker because the separations are much greater. In this illustration, each bond
is about 8 kcal/mol, so that these eight bonds total to 64 kcal/mol. In prac-
tice, the situation is really three-dimensional, and the repulsion of the nearest
Na + atoms must also be taken into account.

where E el is the electrical energy of the two charges of magnitudes q^
and q 2 separated by a distance r.

The value of r can be deduced in several ways (including x-ray
analysis) and the calculated mutual energy turns out to be 120 kcal/mol
(Fig. 10). The experimental binding energy is 180 kcal/mol. Thus we
see that the sodium chloride binding may be understood fairly well on
the basis of what is called a simple ionic bond between the two con-
stituent atoms. Electrical attraction of the ions with still other neighbor-
ing ions accounts for the other 60 kcal/mol.

Is it possible to explain all chemical bonds in these ionic terms? As we
have remarked above, if the electrical forces didn't exist, there would be
no atoms and no chemistry. But the simple picture of atoms giving or
receiving electrons and thereby being electrically bonded is too simple.
Consider the hydrogen molecule. It is composed of two originally neutral
hydrogen atoms. There is no reason to suspect that one of the atoms has
given its electron to the other. Perhaps it is reasonable to propose a
model in which the two atoms share each other's electron. The state of
atoms having two electrons had been found to be one of the stable states
described above. So, by sharing each other's electron, both hydrogen
atoms can attain this stable state. It is necessary to use quantum physics

30 PHYSICAL FORCES AND CHEMICAL BONDS

to do the calculation properly, but the sharing idea has proved to be a
good one, in that excellent agreement of theory and experiment is ob-
tained for the binding energy of the hydrogen molecule. Since the elec-
trons involved are the valence electrons of the atoms and since they are
shared in a cooperative way, the shared electron bond has been given
the name covalent bond.

For other molecular systems, an obvious extension of these notions has
proved useful. If the atoms share the electrons unequally — so that the
electrons spend somewhat more time with one atom than with another —
then there is a combination of sharing and ionic bonds. These situations
are usually spoken of as exhibiting some 'partial ionic bonding. To the
extent that the electrons are shared, we calculate the binding energy as
for a covalent bond. To the extent that electrons spend more time with
one atom than with another, the first is more negatively charged; this
requires that we calculate the electrical interaction as though each atom
had something less than a whole electron's worth of electric charge.

In this fashion, then, the electric bond between atoms has been de-
composed into ionic bonds, covalent bonds, and intermediate cases of
covalent bonds with partial ionic character. Do these models permit us to
picture everything that takes place in the whirling clouds of electrons?
Unfortunately, the answer is no. There are still three other bonds which
are thought of as involving quite different electron configurations. And
there is a slight extension of the covalent bond that we present first.

It is possible for electrons to be shared not just by pairs of atoms but
also by all the constituents of a piece of matter. Essentially, the elec-
trons, or rather, some of them, roam freely throughout the piece of
matter. This extreme case is that of metals; metals arc defined as sub-
stances in which electrons are completely free to move. We will not deal
further with this type of interaction.

The next effect to be set forth involves the situation in which alterna-
tive electron configurations are possible. We return to the hydrogen
molecule already mentioned as being bonded by a sharing principle which
we did not elucidate at all. If we now look a little more at this case,
we can find a feature which Newtonian physics would consider irrelevant.
At any one time, the electron which belonged to one of the atoms might
be near that nucleus while the electron which arrived with the other atom
would be near its nucleus. But the electrons might also be with the
opposite nuclei. Now, it would seem to make no difference which electron
is with which nucleus, since in any event only one electron will be at one
place at one time. But (simplifying the physics enormously) the pro-
cedures of quantum physics force us to count both possibilities, even
though our intuition tells us this is counting things twice. The exchange
of electrons shouldn't make any difference in Newton's physics, but it

CHEMICAL BONDS 31

makes a difference in quantum physics. The ultimate test rests on the
experimental value of the binding energy, and the result is unequivocally
in favor of quantum physics. Exchange energy does, indeed, exist.

H

C

/ \

HC CH

HC CH

\ /

C

H

Fig. 11. A schematic representation
of the benzene molecule: C 6 H 6 .

For biology, the exchange frequently appears in another form which
is called by another name: resonance. To illustrate this, consider the
benzene molecule shown in Fig. 11. Each carbon atom has four valence
electrons and therefore makes four covalent bonds. Each carbon atom is
bonded to one hydrogen atom. Thus, of the 24 bonding electrons initially
present, 18 are left for the intercarbon bonds, making three bonds per
atom. Figure 12 shows that there are two different ways of achieving
this bonding, both using the idea of forming double bonds between half
the pairs of atoms. As before, quantum physics tells us to count both
possibilities in computing the binding energy, and experiment confirms
this prediction. In this situation, there is a model for the exchange which
results in the name of "resonance" energy; the molecule is said to
resonate between the two forms. One can think that the electrons form-
ing the double bond spend half the time on one side of each carbon atom,
and half the time on the other side. By racing back and forth between
these two configurations, the electrons tend to pull the carbon atoms closer

H

l

H

l

1
C

• \

H— C C— H

1 II
H— C C— H

\ /
C

C

/ \
H— C C— H

II 1
H— C C— H

\ /

C

H H

Fig. 12. Two equivalent representations of the benzene molecule, with all
the bonds indicated. There are three bonds remaining per carbon atom after
the bond to hydrogen has been established. The three double bonds can be
placed in the two ways shown, and chemists speak of the structures as resonat-
ing from one form to the other.

32 PHYSICAL FORCES AND CHEMICAL BONDS

o

Fig. 13. A particle is represented with its charges distributed so that the
negative charges tend to be concentrated near one end and the positive charges
near the other. The particle is thus equivalent electrically to the dipole indi-
cated on the right.

together than with single bonds. Indeed, the measured separation
(1.39 A) between the carbon atoms in benzene is actually intermediate
between the 1.54 A for a single bond and 1.34 A for a double bond be-
tween carbon atoms.

The next kind of bond to be discussed is that called variously the
molecular energy or the van der Waals energy. This is a bond between
electrically neutral atoms or molecules, or groups of atoms or molecules.
Electric neutrality does not necessarily imply zero electric charge ; it may
also arise from the equality of plus and minus charges. If the particle
we are now discussing happens to have some of the negative charges
separated from an equal amount of its positive charges, it is still neutral
as a whole, but forms what is called a dipole, as in Fig. 13. Here the
separation of the charges is indicated schematically by the circled plus
and minus signs. Such an entity, composed of plus and minus charges, is
called an electric dipole, by analogy with the similar situation with
magnetic poles. Now, dipoles can interact electrically, too, although the
attraction of one charge is offset to quite an extent by the repulsion
exerted by the other charge of the dipole. Thus dipole forces are usually
quite weak. We shall not enter into a discussion of these dipole forces
other than to say that there are several varieties of them, all contributing
to the van der Waals energy, and all decreasing much more rapidly with
particle separation than by the regular interaction described at the be-
ginning of this chapter. Because these dipoles are involved in the disper-
sion of light, the energy involved is also frequently called dispersion
energy.

The final bond to be discussed is the hydrogen bond. Consider, for the
moment, the OH - ion. We recall that oxygen picks up two electrons to
reach the nearest stable atomic configuration, and that hydrogen nor-

MOLECULES IN SOLUTION 33

mally gives up one electron. Thus the OH - ion should really be thought
of as = coupled to H+. This is an ionic bond. When we examine the
NH bond, a similar situation is found. For example, in ammonia, NH 3 ,
the nitrogen would be triply charged negatively to reach the stable con-
figuration. Now if the geometrical structures of molecules were such that
an NH 3 group should be near an oxygen atom (such as one forming part
of a C = group) one of the plus-charged hydrogens from the NH 3
group could oscillate back and forth between its group and the oxygen
atom, making essentially electrical bonds and thereby binding the two
groups together. Since the hydrogen nucleus is nearly 2000 times as
massive as the electron, the oscillations are much slower than for electron
bonds, and the resultant bond is correspondingly weak. But since large
molecules of biological importance contain many such groups, there can
be many hydrogen bonds which add up to a respectably large total
binding energy. For example, the nucleic acids' characteristic double-
stranded structure appears to be held together entirely by hydrogen
bonds.

To give an idea of the magnitudes of these bonds, the following table
shows typical values in each of the five categories of bonds:

ionic 100 kcal/mol

covalent 100 kcal/mol

partial ionic 50 kcal/mol

resonance 40 kcal/mol

hydrogen 2-10 kcal/mol

One final feature to be discussed is called saturation. In the ionic bond,
the ions interact electrically with all other electrically charged groupings
in their neighborhood; the ionic bond is therefore said to be unsaturated,
since other ions can join in. The covalent bond we have pictured is due to
a sharing of electrons, and the result would be that the shared electrons
would lie primarily between the two atoms. Therefore no other atoms
could normally become involved — without breaking this covalent bond.
Thus the covalent bond is said to be saturated, since it normally involves
just one pair of atoms. Such considerations become important in esti-
mating the effects of the entry of other substances in the neighborhood of
the existing bonded substances.

MOLECULES IN SOLUTION

Practically all the chemistry occurring in biological systems takes
place in solutions, almost all of which are water solutions. Thus it is im-
portant to look at the effects of water on the forces and bonds we have
outlined.

34 PHYSICAL FORCES AND CHEMICAL BONDS

Consider a charged particle, an ion, in a solution which also contains
many small ions (e.g., Na + , CI - , S0 4 , etc.). The minus-charged ion
on, say, a protein molecule will attract some of the cations in solution by
a simple electrostatic force. As shown in Fig. 14, these cations will tend
to cluster around the negatively charged protein, while the negative ions
will be repelled. The situation would look as in the figure except that
the plus ions also repel each other and attract the anions. Therefore the
real picture will have the cations pushed out quite a bit, intermingled to
some extent with the negative ions, so that it is only statistically that the
protein has more plus charges nearby than minus charges. This picture
has been worked out mathematically by Debye and Hiickel, and the
actual distribution of cations and anions around the protein can be
shown to be just the kind of swirling, interpenetrating clouds that our
simple approach yields. At some distance from the protein, of course,
the electric force of the protein is vanishingly small, so that for the
solution as a whole there are Debye-Huckel clouds around each charged
molecule.

+

Fig. 14. The clustering of plus and minus solution ions near the respective
minus and plus charged ends of a "dipolar" particle.

These ionic clouds affect reactions appreciably, for reacting molecules
must drag along their sycophantic clouds, thereby reducing their mobility
in solution, and therefore the reactivity. Also, because of the existence
of these clouds of ions, molecules held together by ionic bonds or hydrogen
bonds will have their binding appreciably reduced. It is possible that
this effect is important biologically, especially since hydrogen bonds have
been found to be responsible for the twisting and folding of such biolog-
ically important molecules as proteins and nucleic acids. Indeed, since the
latter are held together chiefly, if not entirely, by hydrogen bonds,
changes in the ionic concentration near nucleic acids could well be
associated with their ability to separate into component strands during
replication. The ease of separation of the strands has been verified as
being greatly augmented by lowering the salt concentration while heat-

BONDS AND ENERGY EXCHANGE 35

Fig. 15. A sketch of the water molecule and its electrically equivalent
dipole.

ing nucleic acid solutions; this effect is, however, primarily associated
with increase of the repulsion due to the negatively charged phosphate
groups.

A second aspect derives from the fact that the water molecules are
themselves dipoles; their positive and negative charges are permanently
separated from each other, so that the molecule as a whole is electrically
neutral, but still can effect electrical interactions. Thus water can act to
split molecules bonded appreciably by electric forces. As an example,
salt (sodium chloride) is found to be entirely separated into constituent
ions when salt is dissolved in water.

Some investigations have yielded results suggesting that, in water, the
hydrogen bond strength may be less than lOOOcal/mol. In any event,
the strengths of ionic and hydrogen bonds, due to electric forces, are
subject to important modifications when the molecules are dissolved in
water.

BONDS AND ENERGY EXCHANGE

Given the existence of the various chemical bonds, it remains to ex-
plain the energy exchanges involved in chemical reactions. In order to
stick two atoms together in, say, a covalent bonding, we have to bring
them close together. As long as they are many atomic diameters apart,
there is no force acting between the atoms, but as they approach each
other, the outer electrons of each exert a repulsive force on the other, so
that we have to do work to push them still closer together. It is not until
we have pushed them so close that an electron from one atom experiences
the attractive force of the nucleus of the other atom that the electron
swap which makes the two atoms exert a net attraction on each other can
take place. It turns out that the net attraction is not very great, but as
long as it exists, it holds the atoms together. If we now supply the
energy to move the electron back where it was in the first place, the
repulsion between the atoms is enough to force them apart and give them
an appreciable velocity. This velocity corresponds to an energy of
motion, which is the form in which we get back the energy we initially
expended in forcing the atoms together. Thus, as long as the atoms are
bound to each other, they have this latent energy. In this fashion, com-

36 PHYSICAL FORCES AND CHEMICAL BONDS

plcx molecules which contain great amounts of latent energy may be
synthesized.

Now, if such an energy-rich molecule collides, by Brownian motion,
with another molecule of the right kind (and evolution has so arranged
matters that there is a reasonable likelihood of a molecule of the right
kind being in the neighborhood ) , a rearrangement of both molecules may
occur. If the first molecule had a carbon-carbon bond, it could be broken
by interacting with a molecule of phosphoric acid so that one carbon
receives a hydrogen from the acid, and the other atom is coupled with
the remaining phosphate group. This carbon-phosphate bond also has
appreciable energy, derived in this ease from the carbon-carbon energy
which now produces very little of the high velocity of separation that a
simple splitting would have produced.

The way this could happen would be for the phosphoric acid to ap-
proach the covalent bond so closely that the electron orbits were thereby
disturbed. There would then be a certain probability that the covalent
bond would be established with the phosphate group by displacing a
hydrogen atom; the latter would then fall into place with the remaining
carbon atom. Schematically, the reaction may be written as follows:

c— c— c— c— c— c

+

H
O

I
HO— P— OH

II
O

1

C— C— C— H + C— C— C

I

o

I

HO— P— OH

II

The generalization then is that energy is exchanged by coupling a
dissociation reaction with an association reaction, with little energy loss
resulting. This coupling of reactions is a basic feature of biochemical
transformations, and in texts you find this written in general symbols as

A^ _^B

Here the symbolism means that the energy-releasing reaction A — » B is
coupled with the energy-requiring reaction C — > D.

ENERGY AND BIOLOGICAL PROCESSES

37

ENERGY AND BIOLOGICAL PROCESSES

Energy exchanges provide the key to understanding biological proc-
esses. Such a statement either must be obviously true (since the laws
of physics and chemistry must be obeyed) or else some vitalistic factors
must operate. Since none of the latter has been found necessary, as yet,
to explain biological processes, science continues to work along physico-
chemical lines.

The way in which energy is used in biological processes is of some
interest, and will be explained with the analogy of a waterfall, as in
Fig. 16.

If, as in the left sketch, the water falls straight down to the ground, it
bounces wildly because it has accumulated an enormous speed as it falls
from the top. The energy in the water at the top (its ability to do work)
is converted to the energy of motion of the water, and when the water
strikes the ground level, there is a glorious display of hydropyrotechnics,
after which the previously swift water molecules settle down to their new
(energy-wise) impoverished state of being like all other water molecules.
The energy of the waterfall has been dissipated into heat, for the water
at the bottom is now somewhat warmer than it was before.

In the right-hand sketch, the water falls only a small distance before
striking the vanes of a water wheel, which thereby is caused to turn. If
the water wheel is connected to something, it can turn that something

'■■■■■■ ''\viC3 \fc_a~-

T«..'V » 'I . ' I , J l | |ll ■■■! II II . I III . 1 . 1 * ),, * -

(a)

(b)

Fig. 16. In (a), the water falls freely to the bottom, bounces high because
of the large velocity each molecule acquires while falling, and finally settles
down to a quiet flow to the right. In (b), the water falls freely only to the first
water wheel. The small velocity acquired is imparted to the vanes of the wheel,
thereby turning the wheel and causing useful work to be done by whatever is
connected with the wheel. The water then falls freely again for a short distance,
picking up more velocity, which is imparted to the next wheel. Finally, the
water arrives at the bottom with very little energy to dissipate, so there is
little wild bouncing, and the water quickly settles down to its quiet flow to the
right.

38 PHYSICAL FORCES AND CHEMICAL BONDS

and cause it to do some mechanical work. Thus the water energy is
partly converted into usable mechanical energy. We say "partly" be-
cause it took some energy to turn the wheel itself, some energy is lost in
friction, and some in the bouncing off the wheel. Then the water falls a
little more before going through a second water wheel. The water must
fall somewhat, or it will have no motion to impart to the wheel ; each
time it hits a wheel, some of the energy is lost in the three processes
mentioned. Thus, in a series of interactions with a set of wheels, the
initial energy of the water can be fairly efficiently converted to usable
energy.

One way of storing the energy extracted by each water wheel is to
have it turn a magnet to create electric energy which could then, if
desired, be stored in a battery for later use.

The same process, in spirit, exists in biological events. If a complex
molecule exists, there is energy stored in its bonds, as described previ-
ously. The molecule exists because work was done in creating it, and
this energy is now to be extracted by taking the molecule apart. If the
molecule were taken apart all at once, the energy would be primarily
released as heat, for it is difficult to couple an explosion efficiently to an
energy storage device. Instead, the complex molecule is taken apart one
atom (or a very few atoms) at a time, and some of the energy released
is used to build up a small molecule. These small molecules play the role
of storage batteries, and may serve subsequently to provide the energy
for needs of the cell, such as movement, replication, etc.

In this stepwise fashion, the energy originally stored in the complex
molecules can be used, with reasonable efficiency, to create energy storage
molecules which then are of general use to effect the biochemical re-
actions. In practice, for example, the complex molecule could be a sugar,
and the small battery-like molecule could be ATP. The device of
storing energy in special molecules like ATP avoids the problem which
would otherwise arise in coupling energy-yielding dissociation of complex
molecules to energy-requiring reactions. There would otherwise be prob-
lems of different locales of the two processes, as well as problems in
timing.

CHAPTER

3

Physical Aspects
of Vision

Some of the physical aspects of vision are accessible to those who have
some knowledge of statistical analysis. There are many kinds of studies
that have been made, and we shall deal with several of these.

SENSITIVITY OF THE EYE

One important question about vision has to do with the number of
photons required for seeing. The question has to be refined, because an
immediate problem arises in deciding where these photons must be ab-
sorbed. Do they all have to be absorbed in the same rod (cones are
neglected, since we know they are used for high light-intensity vision) ?
Can a single rod effectively absorb more than one photon in any arbi-
trarily short time interval? Do rods or groups of rods cooperate in vision?
Do they cooperate in experiments to determine the minimum number of
photons that may be detected?

Studies of the sensitivity of the eye to light are carried out by ex-
posing the eye to flashes of dim light and measuring the fraction of times
(the probability) that the flash is seen. If light of any intensity could
be seen, the experiment would be uninteresting. But the results of these
experiments are more than normally interesting.

There are two ways of doing the experiment. The most direct way is
to shine progressively dimmer light flashes into the eye and to determine
the smallest intensity that is normally seen. The actual mechanics of
this experiment is of interest. The subject sits in a dark box with his
teeth gripping a plaster cast specially made to fit the subject's teeth.
This, plus some clamps on the subject's head, permits him to gaze
directly ahead at a very dim red light which also helps to fix the subject's
eyes. The rods are quite insensitive to red, so this light doesn't disturb
the experiment. Dark-adaptation was achieved by having the subject
sit in total darkness for 45 minutes. An aperture which can be illumi-
nated with light of arbitrary intensity and wavelength is situated at an
angle of about 20 degrees away from the line of sight. The size of the
aperture is chosen so that the image formed on the retina covers only

39

40 PHYSICAL ASPECTS OF VISION

about 500 rods. The amount of light needed for vision can be estimated
from the fact that stars of the 8th magnitude are visible, corresponding
to about 10 -9 erg/sec. In the laboratory setup, even less energy was
needed for detection.

The subjects were able to detect light of wavelength 510 mp whose
intensity corresponded to about 10~ 10 energy units (ergs) incident on
the eye. This corresponds to about 100 photons. This energy has to be
corrected for the various reflections and absorptions experienced by
photons before they reach the retina and are absorbed. At the cornea
about 5% is reflected, leaving 95 photons to continue. Half of these are
absorbed in the humors of the eye, leaving about 45 photons. Eighty
percent of these are absorbed in the nonsensitive portions of the retinal
structure, leaving about 10 photons to be absorbed for this barely de-
tectable flash.

In the actual experimental situation, it was found that light of a
particular intensity could always be seen. But when a slightly lower
intensity was used, the subject was always able to see the flash in a
certain percentage of cases. That is, for an intensity which can always
be seen, one cannot be found just slightly lower which is never seen.
Therefore, for each subject, that intensity which was seen 60% of the
time was arbitrarily taken as the threshold for that particular run.
There were variations from individual to individual, and even the same
person exhibited a day-to-day variation. In the experiments being dis-
cussed, seven subjects were studied, and the range for threshold seeing
was 2.1 to 5.7 X 10 -10 erg. One particular subject had a range of 4.83 to
5.68 X 10- 10 erg.

At a wavelength of 510 mfi, the energy of a quantum is 3.89 X 10 -12
erg. Thus, the actual range in photons is from

2.1 X 10- 10 5.7 X 10 -10

to

3.89 X IO-12 3.89 X IO-12

which is 54 to 158 photons. Since about 10% of the photons get through
to the retina, the number of photons needed for threshold sensitivity is
5 to 16.

We have 5 to 16 photons falling on 500 rods. On each rod, therefore,
there is an average (taking the 5-photon number for calculating pur-
poses) of 1 photon per 100 rods, or a chance of 0.01 that any one rod
received a photon. The chance that 2 photons will fall on the same rod is
then approximately 0.01 X 0.01 or 0.0001 — one chance in 10,000. Since
a total of only 5 photons reaches the rods, it is highly improbable that
two photons per rod are needed to produce a response. Thus, one photon
must react with one rod, affecting one molecule of the visual pigment,

SENSITIVITY OF THE EYE 41

and in turn this reaction initiates a sequence of other chemical reactions
leading finally to stimulation of a single nerve fiber. It must then be
true that the requirement for 5 photons is exerted at the level of the
nerves. The nerve excitation resulting from summing the effects of 5
individual nerve fibers is what is finally sufficient to trigger the optical
nerve into transmitting an impulse to the brain.

On quite general grounds, a result greater than one photon is to be
expected. If one photon were the threshold, then a single firing of a
nerve fiber would result in vision. A single nerve fiber can be excited
randomly by heat, fluctuations in chemical reactions in the retina, pres-
sure, etc., so that we should be seeing flashes all the time. Therefore, we
would guess that more than one photon would be needed, so as to reduce
the random triggering of vision by these agents when it is irrelevant to
the seeing processes. Of course, psychologists might maintain the en-
tirely possible position that the triggerings occur but are suppressed by
the mind so as to free the eye for purposeful vision.

A number such as 5 photons is interesting because the fluctuations in
this number (from the Poisson Distribution) yield a standard deviation
of \/5, which is about 2.3. Two standard deviations (2s) equal about 5.
Therefore, in some 5% of the cases where an average of 5 photons was
delivered to the retina, the fluctuations should be great enough to result
in no photons getting through and therefore the subject would not see
the flash that the experimenter thought he was giving.

This leads us to another way of approaching the visual threshold
problem. Consider that, for a subject to detect a flash, a threshold of n
photons must be equaled or exceeded on the retina in the area being
stimulated. When the number of photons arriving is less than n, the
subject will see no flash at all; when it is n or greater, the flash will be
seen each time. Thus, if this simple model were correct, the frequency of
seeing flashes of increasing intensity would rise abruptly from zero to
100% at the threshold value of n. However, there are two sources of
variation in such an experiment: biological variability and physical
variability. We can't do anything about biological variability, for the
term includes factors which we don't understand or cannot control. So
we focus attention on physical variability. In this situation it means
that when one shines n photons, on the average, on a retina, there will be
fluctuations of this number. Indeed, we know that the standard devia-
tion of the number is \/ n - That is, in one third of the cases, the number
will differ from n by \/n or more. Now, when the average is n, the frac-
tion of times there will be n or more is not unity, as in the simple non-
fluctuating example first mentioned in the paragraph, but is something
less than unity. For any given intensity of light, the flash will contain

42

PHYSICAL ASPECTS OF VISION

Simple model

Fluctuation model

Threshold intensity

/_

Intensity

Fig. 17. A schematic representation of the frequency with which a flash of
each intensity is seen, for two models. The simple model assumes that each flash
has the same number of photons. The fluctuation model takes into account the
fact that a given average number of photons (plotted on the abscissa) will have
a fluctuating number of photons in each individual flash, for experimental
reasons.

Intensity of the light flashes
(on a logarithmic scale)

Fig. IS. Calculated curves of the fraction of times a light flash will contain
at least n photons for the average intensity plotted on the abscissa.

a fluctuating number of photons, and only flashes containing at least the
threshold number will be seen. As the intensity increases, a steadily in-
creasing proportion of flashes will exceed the threshold number. Finally,
intensities will be reached for which, despite the fluctuations, all flashes
will exceed the threshold number of photons.

The simple model and the fluctuation model are presented schemati-
cally in Fig. 17. A flash of a given intensity is presented repeatedly to
the subject, and the fraction of times he sees the flash is recorded.
Then, in succeeding experiments, using a series of intensities, similar
data are recorded. The data are then plotted to show the fraction of
times the flash is seen for each incident intensity.

SENSITIVITY OF THE EYE

43

100

1.5

2.0 2.5 1.5 2.0 2.5 1.5 2.0

Logarithm of average number of photons per flash

Fig. 19. This figure presents the experimental findings on three different sub-
jects. The lines are the theoretical curves given in the previous figure. As may
be appreciated by studying the theoretical curves, the choice of a curve with n
greater or smaller by unity would give an appreciably poorer fit to the experi-
mental points. (From Hecht, Shlaer, and Pirenne, J. Gen. Physiol. 25, 819,
1942; courtesy the authors and J. Gen. Physiol., Rockefeller Institute Press, New
York.)

The theoretical analysis utilizes the Poisson formula. For any average
number of incident photons, we can figure out the fraction of times
there will be a minimum of, say, 1 photon hitting the retina. This curve
is plotted in Fig. 18 with a 1 beside it. Similarly, we can figure out
the fraction of times there will be at least 2 photons hitting the retina
for any average intensity. And a similar calculation can be carried out
for any threshold number of photons. Figure 18 shows these curves for
various values of this threshold number. Next w 7 e try to fit the calculated
curves of Fig. 18 to the actual data obtained. Since the shape of the
curve depends strongly on the value of the threshold number, it is possible
to distinguish quite well between possible curves. In this way, by trying
successively 3, 4, 5, 6, 7, etc. photons for the threshold, it is possible to
show that the best number is about 6 photons. This means that at low
intensities there were few times that the flash was seen because there
w r ere few flashes which chanced to contain 6 photons, since the average
number of photons was less than 6. When an intensity w T as reached such
that an average of 6 photons reached the retinal absorbers, not all the
flashes were seen because some of the flashes contained 0, 1, 2, 3, 4, and
5 photons, none of which could be seen. When very high intensities were
used, all flashes had at least 6 photons, and so all were seen. In fact, as
shown in Fig. 19, the data for three different subjects gave 5, 6, and 7
photons as the number which best fitted the data. Note that this experi-
ment could have been done simply by fitting the curves to the data — it

44

PHYSICAL ASPECTS OF VISION

is not necessary to measure the absolute intensities, so long as we know
the ratio of the intensities.

It is of interest to point out that this experiment could have been done
in precisely this way 150 or more years ago. If someone had had the
inspiration to fit the results with the already known Poisson Distribution,
the quantum nature of light could have been discovered that long ago.

STRUCTURE OF THE RETINA

One of the problems which perturb biophysicists is the reason for the
size and shape of biological structures. In this category comes the
question of why we have a concave retina. In the early 1800's, one
Johannes Miiller was trying to figure out the answer to this question, and
in speculating on the problem he remarked on some interesting aspects.
Consider a flat, plane retina, as in Fig. 20(a). An object such as the
arrow to the left will send light into many of the rods shown schemati-
cally in the retina. As a result, only the presence or absence of an object
will be detected by such an eye. If the retina is made concave, as in
part (b) of the figure, some of the rods will still see many parts of the
arrow, so that little improvement in vision will result. But consider the
convex retina in part (c). Here, if the angular opening of each rod is
sufficiently small, only a very few rods see the various portions of the
arrow, so that what amounts to an erect image of the arrow is formed on
this retina.

(a)

(b)

(c)

Fig. 20. A schematic representation of the problem of detecting the shape of
an arrow as seen by retinas which are, respectively, Mat, concave, and convex
to the arrow.

Up until the time of Miiller no such retinal structures were known, and
it is to his credit that he took the trouble to go to Nature to discover
that this is essentially the insect eye. The rods are not really rods, but
are conelike structures called ommatidia, whose angular opening is about
one degree. It is therefore not surprising that the resolution of this insect
eye is about one degree. This information is gathered by making a grid
of black and white bars of such a thickness that at the eye of the insect

STRUCTURE OF THE RETINA 45

the individual bars subtend, in various experiments, 0.1 degree, 0.5 de-
gree, 1 degree, 2 degrees, etc. In response to a movement of the 2-degree
bars, the insect responds by moving away. When the 0.5-degree bars were
used, the insect made no response, thus showing that the fineness of the
bars was such that the images overlapped and looked grey to him; he
could perceive no motion in these circumstances.

The human eye can be studied in a similar way, and the resolution of
0.01 degree corresponds well with the angle subtended at the lens by
neighboring retinal elements. To give an idea of what these numbers
mean, a ringer at arm's length subtends about one degree. Thus an insect
can just make out the individual fingers of a man one yard away; a
human can make them out about 100 yards away.

It should be realized that the experiments described in this chapter
were done 20 years ago and that much work of a far more sophisticated
nature has been done in the intervening years. Also during that time,
enormous success has been achieved in studying the biochemistry of
vision. Some biophysicists feel that studies of vision will provide the
most direct route to an understanding of the nerve interplay which goes
by the name of neurophysiology. Accordingly, the neurophysiology of
vision is emerging as one of the most important and most exciting areas
of research today.

CHAPTER

4

Physical Aspects
of Hearing

INTRODUCTION

Physics has established that the thing we call sound travels through
matter in a manner similar to the way waves travel. Sinusoidal oscilla-
tion of the particles of matter causes one particle to collide with the next,
thereby transmitting its energy to the second particle. We therefore
speak of sound waves, and ascribe to sound the various properties of
waves, such as wavelength, amplitude, intensity, interference, diffrac-
tion, etc. The energy in the wave resides in the amplitude of the particle
motion and in the number of particles executing the motion. Thus the
expression for the intensity of the sound (energy passing through unit
area in unit time) has been shown to be proportional to the density of
the matter, the square of the sound frequency, and the square of the
amplitude.

These aspects of sound are usually taken up in an introductory course
in physics, and will therefore not be dealt with here. We will focus our
attention on the following questions. How does our brain get informa-
tion about the amplitude, frequency, and intensity of sound waves? To
make one of these questions more concrete, how does the frequency
information get to the brain? Is the frequency faithfully reproduced in
the auditory nerve? Or is some particular nerve fiber sensitive to a par-
ticular frequency, so that all that goes to the brain is the information that
particular nerve fibers have been fired, meaning that certain frequencies
have been received? Such questions clearly involve knowledge of proper-
ties of nerves and nerve fibers, so that we shall have to introduce this
study of sound with some aspects of nerve physiology.

The mechanism of transmission of an impulse down a nerve is believed
to be associated with an alteration of the electrical state of the nerve
membrane. The speeds of these impulses are of the order of 10 m/sec.
Once the impulse has passed each point, the recovery processes set in.
Upon the length of time required for these recovery processes depends the
frequency of impulses that the nerve can transmit. Another factor which
affects the frequency of transmission by a single nerve fiber is the time
taken to transmit the impulse across a synapse and the recovery time at

46

MODELS FOR SPECIFIC NERVES FOR SPECIFIC FREQUENCIES 47

the synapse. The events at the synapse by themselves show some of the
limitations involved. The law of diffusion is, approximately,

x 2 ~ Dt,

where x is the distance diffused by a substance in time t if D is the
diffusion coefficient. Putting in typical numbers for the acetylcholine
molecule (shown to be involved in many synaptic transmissions) t turns
out to be about one millisecond. Thus if a second impulse were to arrive
at the synapse during this time, the impulses would be blended together.
Therefore we can estimate an upper limit of 1000 cycles/sec for the
transmission frequency of nerves. Experimental measurements have
shown that this is indeed a good estimate, and that most nerve fibers
can transmit only up to a few hundred cycles per second. If this limit
holds for the auditory nerve, then it is clear that the nerve cannot trans-
mit input frequencies of up to 15,000 cycles/sec, which is roughly the
highest frequency that adults can hear. Accordingly, the early studies
of the hearing mechanism focused on the finding of individual nerve
fibers which responded characteristically to single frequencies. Several
of the models proposed will now be examined briefly.

MODELS FOR SPECIFIC NERVES FOR SPECIFIC FREQUENCIES

There are two kinds of models which have been envisaged. The first
proposes that certain nerves respond to certain frequencies, due to some
as yet unknown mechanism. The second proposes that the nerves which
respond to certain frequencies do so because they are connected to other
elements which themselves respond to the frequencies; the nerves them-
selves could transmit any frequency up to their maximum. In the latter
category are all the so-called resonance models, which propose, for
example, that there are resonating structural fibers in the ear which
respond only to characteristic frequencies. When these structural fibers
oscillate, they trigger the nerve fibers to which they are connected.
Proponents of the resonance theories have then had to demonstrate the
existence of such resonating structures in the ear.

To proceed further, we need a sketch of the relevant portions of the
ear. Figure 21(a) is a diagram of the snail-like cochlea. Part (b) is a
section of the cochlea, showing its division into three so-called canals
and the place of insertion of the nerve fiber into the basilar membrane.

The eardrum is connected by means of three auditory bones (malleus,
incus, and stapes) to the vestibular canal. The approximate area of the
stapes connection is also indicated in part (b) of the figure. Thus vibra-
tions of the eardrum are communicated to the bones of the inner ear.

48

PHYSICAL ASPECTS OF HEARING

which finally transmit the vibrations, and the associated pressures, to
the cochlea fluid filling the canals.

There are two kinds of resonance theories possible at this point: one
based on resonances in the fluid, the other on resonances in the mem-
branes. It can be shown that fluid resonances are physically excluded
for a variety of reasons. Consider the schematic representation of the
cochlea in Fig. 21(c). Here the cochlea has been shown as a straight
tube with its two parts separated by a membrane (we neglect the
cochlear duct here — it simply makes the dividing membrane more com-
plicated). The aperture (helicotrema) connecting the two parts is also
shown. The actual cochlea would differ from the schematic one chiefly
in being twisted around into the snail-like shape.

When the stapes moves in response to an impulse entering the ear, the
fluid is pushed. If the canals were filled with a gas or a compressible
liquid, waves could travel along the canal and bounce off the other end,

(a)

BM

(b)

Helicotrema

RW

Fig. 21. Sketch of the cochlea (a) and a cross section of the cochlea (b).
Part (b) shows the separation of the snail-like tube into Vestibular (V), Median
or Cochlear (C), and Tympanic (T) canals. The stapes insertion at the end
of the vestibular canal is indicated by the dotted line containing the .symbol S,
and the round window (RW) at the end of the tympanic canal is similarly indi-
cated. BM is the location of the basilar membrane. N is the auditory nerve as
it emerges from the cochlea. Part (c) shows a schematized version of the
cochlea unwound from its actual snail-like configuration. The basilar membrane
divides the two parts of the cochlea; the cochlear canal is omitted for simplicity.
The helicotrema allows passage of fluid from one part of the cochlea to the other.

MODELS FOR SPECIFIC NERVES FOR SPECIFIC FREQUENCIES 49

thus establishing standing waves, as in an ordinary musical instrument.
The positions of the maximum pressures in these waves would activate
pressure-sensitive nerve endings in the membrane. Thus the localization
of points of maximum pressure would activate different nerves, depend-
ing on the frequency of the incoming waves. Unfortunately, this simple
picture is entirely out of the question. It would require the existence of
a cochlea with dimensions similar to those of musical instruments, and
this is obviously not the case. In addition, the cochlear fluid is essentially
incompressible, so that a more refined analysis is required. (We shall
return to this aspect toward the end of this chapter.) Further, because of
the viscosity of the fluid, it is hard to imagine that the frequencies could
be so sharply separated from each other in what are called standing
waves that there could be good frequency (pitch) discrimination. There-
fore we confine our attention to membrane vibrations.

At first sight, standing waves in the membrane would seem to be ruled
out for the same reason as were waves in the fluid. There exists, however,
a phenomenon called, among other things, neural sharpening. We can
illustrate this with an experiment you yourself can perform. Touch the
tip of a finger with a reasonably sharp pencil point; it is best to do this
with your eyes closed. Then touch approximately the same place with
the tip of a ball-point pen whose point is retracted. The statement of
neural sharpening is that you cannot tell the difference. Schematically,
the statement is that your neural structure is such that you cannot dis-
criminate between a point (•) and an object (O) centered around the
point — provided the touch is not too strong. The stimulation of nerves
around the periphery of the circle is interpreted by the individual as a
stimulation of the center point itself.

A more readily appreciable experiment (though less directly illustra-
tive) that exhibits the neural sharpening phenomenon involves touching
the skin at the same time with two reasonably sharp pencils. If the
points touch the skin less than about 2 cm apart, only one stimulus is
felt, whereas at about 3-cm spacing one is aware of the two separate
pencil points.

The application to standing waves in the membrane comes in observ-
ing that even if a relatively broad area of the membrane is set into
vibration by the incoming wave, neural sharpening results in the sensa-
tion that only the central point of the vibrating area has been affected.
Thus only the particular frequency at that point will be communicated as
having been heard.

According to this kind of theory, the place which is stimulated in the
membrane is different for each frequency and is therefore the mechanical
agency of sorting out the frequencies. There are other varieties of the
place theory which will not be discussed.

50 PHYSICAL ASPECTS OF HEARING

An alternative resonance theory requires the existence of fibers which
are tuned (like the strings in a musical instrument) according to their
length and tension to produce a single note or frequency. Then, an in-
coming sound would cause the corresponding fibers to vibrate and the
vibrations would stimulate the nerves connected to them. But there
arises the purely experimental question of whether such fibers exist, and
if fibers can be found, there is the additional question of whether they
have sufficiently different lengths and tensions to respond to the range
of frequencies of human hearing (30 to 15,000 cycles/sec).

Studies of the membrane have shown that there are indeed structural
fibers in the membrane and that there are also hair cells in an organ
lying directly above the membrane, but that neither set of possible
resonators has a sufficient range of frequencies to be responsible for
human hearing.

Both kinds of theories discussed up to this point are included in the
general category of place theories, so called because the excitation of a
particular place in the cochlea is supposed to inform the brain of the
arrival of a particular frequency. It was a great disappointment that the
place theories didn't work, because there existed impressive experimental
evidence for their validity.

The kinds of evidence for the validity of place theories came from
direct examination of the membranes. The ears of individuals known to
be deaf to certain frequencies were examined after death, and lesions
were found in certain places along the membrane; the places found in a
study of many ears presented a consistent picture for the places of
various frequencies. It was also found that deafness to certain fre-
quencies could be induced in animals by subjecting an animal to such a
great intensity of a particular frequency that the ear was injured and
the animal subsequently no longer responded to the frequency. Examina-
tion of the resultant lesions in the membranes of such animals gave a
place representation of frequencies along the membrane which coincided
quite well with that derived from naturally occurring lesions. Thus there
would seem to be no question that there is at least a significant amount
of validity to the place theories, despite the inability of researchers to
find a suitable model.

The resonance theories discussed to this point stemmed from the
proposals of the German scientist Helmholtz, who began his studies in
the middle of the 19th century. You recall that we began our examina-
tion of place and resonance theories because of the finding that individual
nerve fibers cannot transmit impulses more often than at the rate of
1000 cycles/sec. Yet in recent years, it has become possible to measure
the impulse frequencies directly on the auditory nerve, and faithful

MODELS FOR SPECIFIC NERVES FOR SPECIFIC FREQUENCIES 51

replication of the incoming frequencies was found up to several thousand
cycles per second !

The explanation of the apparent discrepancy lies in the fact that
although individual nerve fibers cannot transmit above 1000 cycles/sec,
the groups of fibers making up the auditory nerve may cooperate to
transmit higher frequencies. In this so-called volley principle explana-
tion, nerve fiber #1 fires, say, every 0.001 second, neuron #2 fires every
0.001 second, but starting 0.00033 second later. Neuron #3 fires every
0.001 second, but starting after 0.00066 second. The combination of the
three neurons then fires three times every one-thousandth of a second,
thereby reproducing a frequency of 3000 cycles/sec. It is not known
where this "volley" action takes place, although the originator of the
theory, E. G. Wever, has proposed that it occurs in the neurons which
are connected to the inner ear membrane. According to Wever, hearing
takes place by a combination of frequency representation (using the
volley principle) and place representation (with neural sharpening).

The kind of place theory still has to be specified. We return to the
previously mentioned fact that the canals are filled with an incom-
pressible fluid. The properties of the fluid are such that waves travel
much faster through the fluid than through the membrane, which is
composed of reasonably stiff elements not very tightly coupled to each
other. When pressure is exerted by the stapes, the liquid wave reaches
the helicotrema and in part goes through to travel down to the round
window long before any appreciable part of the membrane can have be-
gun to move. Because the pressures above and below the membrane are
not precisely in phase, there is a pressure difference across the membrane
which displaces the membrane, producing a bulge. The place where this
bulge occurs depends on the frequency of the incoming sound; the
higher the frequency, the closer it is to the stapes. The positioning of the
bulges seems to depend on a surprising property of the waves which can
be predicted by the complicated mathematical theories now available.
The property is that almost all of the wave is confined to a thin region
of the fluid at the periphery, where the fluid is in contact with the canal
material and with the membrane. Thus the situation is somewhat similar
to that of surface waves coming in to a beach. Depending on the fre-
quency and amplitude of the waves, they will "break" at various dis-
tances from the shore.

Many of these aspects of the rapidly developing field of hearing re-
search are due to von Bekesy, who was awarded a 1961 Nobel Prize.

5

CHAPTER L-v Light Absorption Effects

INTRODUCTION

Quantum physics teaches us that we may talk about radiant energy as
traveling either as waves in space or as discrete packages called either
photons or quanta. When radiation impinges on matter, it has a proba-
bility of being absorbed which depends on (1) the energy in the photon
(or its equivalents in wave terms: the wavelength or the frequency of the
light), and (2) the nature of the matter. From the photon view, the
photon may be absorbed if and only if there are two energy states
(roughly two electron orbits) whose energy difference corresponds exactly
to the photon energy, as sketched in Fig. 22. And, of course, there must
be an electron in the lower of the two states, for there must be some
particle to absorb the energy. If the structure of the matter is such that
there is no energy difference equal to that of the impinging photon, the
photon may impart its energy to the electron, but the electron will
rapidly return to its original state and reradiate the energy. Thus, in
effect, this kind of matter doesn't absorb the energy at all; because of the
time taken for the absorption and re-emission, however, the reradiated
photon is delayed relative to its initial path, and we say that its phase
has been shifted.

Substances (such as glass) in which the energy gap between the elec-
trons and the next higher energy state is so great that virtually none of
the incident energy of visible light is absorbed, make suitable insulators.
Electrical conductors, on the other hand, are composed of substances in
which, by definition, the electrons are free to move, which means that
there are so many permissible energy states that almost every incident
photon would be absorbed. In advanced physics texts it is shown why
the photons re-emitted by electrons in metals are virtually all in the
backward direction. The result is that even a thin sheet of metal (e.g.,
household aluminum foil) is quite opaque, because the incident light is
reflected backward.

When light is absorbed, the energy does not remain very long in the
absorbing substances. The fact that the colors of substances do not
change during illumination may be taken to mean that the absorbing
substances quickly revert to their original state, so that they arc 1 once
again ready to absorb the same wavelengths. There 1 are five main fates

52

INTRODUCTION

53

Electron is in nor-
mal or ground state
at time / = sec

Electron is in an
excited state after
absorbing photon
of energy E^ — Eq
after 1= 10~ 8 sec

(a)

Electron has lost
some excitation en-
ergy — is in lowest
allowed excited
state 10- n -10- 8
sec after reaching
Es

Emitted
photon

Electron is back
in ground state
after emitting flu-
orescence photon
of energy E l — E Q

10-

10-

sec

after reaching E x

Incident
photon

Emitted
photon

Electron is excited,
but not to an
energy level of the
material, Z«10~ 8
sec

Electron is back
in ground state
after <<10 -8 sec.
Emitted photon
has same energy
as absorbed photon

(b)

Fig. 22. (a) Light absorption and emission if photon energy equals an energy
difference in the material, (b) Light absorption and emission if photon energy
does not equal an energy difference in the material.

for the bulk of the light energy absorbed: (a) chemical reactions, (b)
fluorescence, (c) phosphorescence, (d) energy transfer, and (e) internal
conversion.

(a) Photochemistry — the study of the chemical reactions resulting
from the absorption of light — is among the more fascinating provinces of
current biochemical research. One of its most significant aspects is
photosynthesis. Since the latter is dealt with in standard introductory
biology courses, it will not be covered here. The more esoteric chemical
aspects are beyond the scope of these writings. Thus the only segment to
be covered in this monograph is that of action spectra, to be dealt with
in a later section.

54 LIGHT ABSORPTION EFFECTS

(b) Fluorescence is the simplest kind of result of light absorption:
the re-emission of most of the light. If the light is emitted very soon
after absorption (in no more than one microsecond), the re-emission is
termed fluorescence. Because the system cannot emit more energy than
it absorbed, the fluorescence is quite generally of a longer wavelength
(more toward the red end of the spectrum) than the incident absorbed
light. Because of rapid rearrangement of internal electron orbits [these
are the changes shown going from (b) to (c) in Fig. 22], fluorescence is
normally due to a transition from a particular excited electron orbit to a
particular lower orbit very near the normal electron orbit; thus the light
is usually monochromatic. If a substance absorbs in several regions of
the spectrum (e.g., chlorophyll absorbs strongly both in the blue and in
the red) the internal rearrangements referred to yield the result that the
fluorescence is as though only the longer wavelength is absorbed (chloro-
phyll fluoresces only in the red).

(c) Phosphorescence is the emission of light considerably later than
fluorescence emission — phosphorescences lasting a few seconds are not at
all uncommon. The light emitted is primarily the same as in fluorescence.
The reason for the delayed light emission in phosphorescence is that the
electron is trapped in an orbit (perhaps E 2 ) from which, according to
the rules of quantum physics, it cannot readily jump down to the normal
orbit. After a time, which may be as long as a few seconds, the electron
manages to get to the particular excited orbit from which it can jump
with the emission of the normal fluorescent light. Because the molecules
exhibiting appreciable phosphorescence are, de facto, in high energy states
for considerable periods, they tend to be very reactive chemically.

(d) Energy transfer refers to the transfer of the absorbed light energy
from the receiving molecule to another molecule. It will occur if the
molecules are sufficiently close to each other and if the energy of the
excited electron in the absorbing molecule chances to be matched to a
possible excited energy state of the other molecule.

(e) Internal conversion covers a number of experimental situations.
When electrons are excited by absorption of photons, they usually do
not reach the next highest orbit but an orbit somewhat higher, so that
they execute a vibration around the new orbit (the vibration, naturally,
must be one allowed by the rules of quantum physics). This extra vibra-
tional energy is dissipated by being communicated to the surrounding
medium or to vibrations of the lattice if the atom is part of a larger
structure. The electron thus reaches the lowest state of vibration of the
new orbit, and this is the particular excited electron orbit referred to
above in the discussion of fluorescence. The extra vibrational energy
may be the source of a number of other effects ranging from rearrange-
ments of the atoms composing a molecule to the dissociation of molecules.

LIGHT ABSORPTION 55

The general picture, then, is that photons absorbed by electrons are
usually re-emitted backward by free electrons, as in metals, and may be
re-emitted primarily in the incident direction if the photon energy does
not match any of the energy states available to the electrons.

These possibilities are sufficient to enable us to learn many things
through the use of light. The phenomena which we shall discuss include
(1) light absorption, (2) birefringence, (3) dichroism, and (4) action
spectra.

1 . Light absorption

If photons impinge on matter, each atom and molecule near which
the photons pass has a probability of absorbing the photon. If we call E
the probability per molecule of absorbing a photon of a particular
energy, then the total probability of a photon's being absorbed is pro-
portional to E and to the number of molecules along the photon path.
The number of molecules along the path of a photon is the product of
the number per centimeter and the path length in centimeters. Since the
number per centimeter is proportional to the concentration of molecules,
we can now write the average number of absorptions in a given solution of
these molecules as Enx, where n is the concentration in molecules per
cubic centimeter and x is the path length in centimeters.

A more usual form of this expression is obtained by multiplying and
dividing by Avogadro's number, N A , the number of molecules per mole:

This expression, whose magnitude has been changed not at all by our
operations, is now the product of the probability of absorption per mole,
the molar concentration, and the path length. This is written as E m cx.
This expression, as stated above, is the average number of absorptions
per photon in the particular experimental situation. We may find the
fraction of photons which is not absorbed by using the Poisson formula
for the fraction of zero cases where the average, a, is the expression just
above :

p (a) = e~ a = e- EmCX .

This fraction of photons not absorbed then represents the fraction of
light, measured in intensity units, which passes through the solution.
We write this as I x /I (the fraction of the incident intensity, I , which
gets through a thickness .r) :

* x _ P —E m cx
j e

Jo

56 LIGHT ABSORPTION EFFECTS

r*

DNA solution heated
to some temperature
lower than denaturation
temperature

DNA solution heated
above denaturation
temperature to produce
strand separation. This
is called melted DNA

Melted DNA after
sudden chilling

Fig. 23. A schematic representation of the partial and complete separation
of DNA strands by breaking some or all of the hydrogen bonds. If the com-
pletely separated strands are suddenly chilled, the strands form intra-strand
bonds before any inter-strand bonds can be formed. The opposite would have
been the case if the solution had been chilled slowly.

It is worth pointing out that machines which measure light absorption
— colorimeters, spectrophotometers, etc. — express the result in terms of
what is called the optical density, which is defined as

O.D. = log I /I x ,

since this equals E m cx and is therefore directly proportional to the con-
centration of molecules which absorb, which is the quantity usually being
sought experimentally.

This expression has been obtained for a single wavelength; the optical
density will vary with wavelength if E or E m varies with wavelength.
By measuring the O.D. at various wavelengths for a given setup (keeping
c and x constant), we can determine the dependence of E on wavelength.
The plot of E versus wavelength is the intrinsic absorption spectrum of
the molecule.

The absorption and scattering of light by a substance can depend very
strongly on its state of chemical combination and on the light-scattering
properties of the solvent in which it is suspended. We can see this from
an examination of two experimental situations which have proved of
interest and significance to biology.

(a) The hyperchromia effect. DNA is a long double-stranded molecule,
held together by hydrogen bonds along the length of the molecule. As a
solution of DNA is heated, the thermal agitation becomes great enough
to break more and more of these hydrogen bonds. At any temperature,
there will be equilibrium between the breakage of these bonds and their
re-formation, so that at any time when there are still some intact hy-

LIGHT ABSORPTION

57

0.15 -

■s-g

<u o
Ota

0.10 -

Temperature

Fig. 24. A sketch of the optical density (O.D.) of a DNA solution as the
temperature is slowly raised from room temperature to about 100°C. T m is the
melting temperature, defined as the temperature at which half the O.D. change
is found. The increase in O.D. is known as the hyperchromic effect.

drogen bonds, the strands will be held together. It is possible to raise the
temperature so high that all the bonds will be broken at once. If, at this
temperature, the solution is suddenly chilled (by placing the tube of
DNA solution in an ice bath), the chances are greater that the individual
strands will fold up and make internal hydrogen bonds than that they
will have time to find the complementary strands to re-form the DNA.
This is sketched in Fig. 23.

If the light absorption depends on the chemical state of the substance,
it would be possible for the solution to absorb different amounts of light
in each of the situations shown in the figure. In fact, when DNA is
"melted" there is a 40% increase in light absorption at the peak absorp-
tion wavelength (260 mfi) for nucleic acids. Figure 24 shows the light
absorption of a DNA solution at various temperatures. The DNA is said
to have a melting temperature T m at the temperature halfway between
the native and melted (or denatured) states.

It turns out that the melting temperature T m is quite characteristic of
the DNA of various organisms and that related organisms have melting
temperatures which are very close to each other, so that this light absorp-
tion measurement has phylogenetic significance. The increase of light
absorption is known as the hyperchromic effect, and is useful in a number
of experimental situations. Since RNA and single-stranded DNA have
very small hyperchromic effects, it is possible to deduce their presence or
absence by light absorption measurements.

(b) Matching of refractive indices. Light is bent in direction when it
goes from one medium to another of different optical properties (you
know this from the classical case of long objects appearing broken when
half in and half out of water). The light-bending property of a substance
is known technically as its refractive index. The contents of a cell will

58 LIGHT ABSORPTION EFFECTS

have some particular refractive index, and the cell will be visible if sus-
pended in a medium of another refractive index. This result may be used
in the opposite way. By varying the refractive index of the suspending
medium, perhaps by varying the sugar content of the medium, it is possi-
ble to find a situation of just the right refractive index so that the various
parts of the cell are no longer visible. Thereby, the refractive index of
these parts of the cell is determined. Since the refractive index is pro-
portional to the concentration of a substance, one can draw inferences
about the concentrations of various substances in different parts of cells.

2. Birefringence

As hinted at in the introduction to this section, light which is not
absorbed is re-emitted in all directions, producing what is called scatter-
ing of the light; we mentioned the two extreme cases of total reflection
and total transmission. Most cases are intermediate. If electron oscilla-
tions are equally possible in all directions, the incident unpolarized light
beam will emerge essentially as it entered the suspension — except for the
phase shift already mentioned. If, however, the particles or molecules in
suspension are not isotropic, i.e., the electrons can oscillate more readily
in one direction than in another, an incident unpolarized light beam will
be split, because waves oscillating in one direction will have their phases
shifted more than those oscillating in another direction. The net result is
that so-called anisotropic molecules produce two plane polarized emergent
beams. Since these emerge in somewhat different directions, the phe-
nomenon is called double refraction or birefringence.

Clearly, a measurement showing birefringence tells us at once that the
material being studied is anisotropic, and since the optical anisotropy is
due to anisotropy in the permissible movements of electrons, we can infer
something about the structure of the material.

The chief use of this phenomenon in biology presently is in the inverse
fashion. We shine plane polarized light on the substance of unknown
structure and see what happens to the plane of polarization. Further, if
we line up the molecules of a substance (as can be done in several ways),
then the refraction (bending of the light) measured along and at right
angles to the direction of alignment will tell us even more about the
electronic structure of the individual molecules.

Birefringent structure may arise in several ways, of which a few will
now be briefly presented.

(a) Intrinsic birefringence. The chemical nature and structure of
individual molecules may contribute birefringence if there is anisotropy in
the movability of the electrons of the molecules. Chemical bonds them-
selves may be highly anisotropic, the C — C bond being a good example.

BIREFRINGENCE 59

oftfla

Line-up Axis

Fig. 25. Disc-shaped and rod-shaped molecules lined up so that their bi-
refringence may be studied. (From G. E. Oster, "Birefringence and Dichroism,"
in Physical Techniques in Biological Research, Vol. I, p. 439, 1955; courtesy
Academic Press, Inc., New York.)

This bond, being a covalent bond due to shared electrons lying primarily
between the atoms involved, is difficult to distort at right angles to the
bond. On the other hand, the triple bond C = C has the atoms bound
about three times as strongly, so that the distortion of electron positions
which can be effected is mainly at right angles to the bond direction.

(b) Form (or organizational) birefringence. As a result of grouping
molecules in a regular array (or in a composite structure with an in-
herent regularity, e.g. nucleic acids) there may w T ell occur directions of
easier and harder electron movability in the structure as a w r hole.

(c) Flow birefringence. If a solution is caused to flow, the viscous
forces will tend to orient particles with their long axes in the direction of
the flow. Spherical particles may even be distorted by these forces.
Thus, during the flow, there will be organizational birefringence whose
magnitude will depend on the speed of the flow.

Consider the two separate cases of a collection of discs and a collection
of rods, as sketched in Fig. 25.

In the case of the discs, electrons can be moved more readily along the
planes of the discs than out of the plane; therefore polarized light inci-
dent at right angles to the lineup axis will be affected more than polarized
light incident along the axis. The case of the rods is precisely opposite,
since electrons can more readily be moved along the lineup axis. Thus,
the measurement of the birefringence for lined-up molecules tells us
whether the molecules have a rodlike or a disclike shape and how they
are oriented with respect to the lineup axis.

There are several important instances of the use of birefringence which
permitted significant deductions about structures in biology. The case of
muscle is the first one, in which the measurement of the birefringence
permitted the conclusion that there are both isotropic and anisotropic
regions in striated muscle — the bands called J and A are actually named

60 LIGHT ABSORPTION EFFECTS

for these properties. In the anisotropic region, the result of the bire-
fringence measurement led to the conclusion that the actomyosin fibers
must be oriented along the muscle fiber axis.

The second case is that of chloroplasts, whose birefringence showed
them to be composed of a stack of discs long before the electron micro-
scope made the structure visible.

The third case is that of DNA. By slowly drying a solution of DNA
stretched over a hole by capillary forces, a fiber can be produced. The
fiber birefringence shows that the individual elements of the DNA are
lined up in the disc configuration, even though it was known from vis-
cosity studies that the individual DNA molecules must be long and thin.
Thus it was possible to conclude that the elements composing the DNA
— the nucleotides — must be arranged as flat discs perpendicular to the
long axis of the molecule.

3. Dichroism

The section on birefringence has dealt with the transmission and
scattering of light by substances which do not absorb the light. By
examining the light scattered in two mutually perpendicular directions,
we were able to make inferences about the orientation of the subunits of
the substances.

In a very similar manner, we could examine the light transmitted by
substances which absorb appreciable amounts of the incident plane polar-
ized light. The results of the analysis are of the same type as for bire-
fringence. For example, DNA has been studied by shining plane polar-
ized light of wavelength 260 mfi, the wavelength of maximum absorption
by nucleic acids. There is twice as much absorption perpendicular to the
fiber axis as there is parallel to it. Thus one concludes that the absorbing
elements themselves lie chiefly perpendicular to the fiber axis, as already
deduced from birefringence studies.

If more light is absorbed at right angles to the fiber than parallel to
it, the fiber is said to exhibit negative dichroism; if the converse obtains
for the absorption, the fiber exhibits positive dichroism.

The energies involved in molecular vibrations and oscillations are such
that the wavelengths of light absorbed by these movements lie in the
infrared region of the spectrum. Polarized infrared sources now exist,
and many substances have been studied by this method. Such studies
have been of considerable use because of the characteristic absorption by
atomic groupings. For example, the stretching of the N — Ff bond involves
light of 3.0 fi wavelength, C — O stretching is at 6.0 fx, and the bending of
the N — H bond is at 6.5 fx. By measuring the dichroism at each of these
frequencies, the relative orientations of the groups may be inferred. For

ACTION SPECTRA 61

simple molecules, the information may be enough to permit a fairly good
preliminary model of the three-dimensional structure.

Finally, it should be noted that when light is absorbed by dichroic
structures, the re-emitted light (the so-called fluorescence) will be polar-
ized, provided only that the structures (or their constituent parts) do not
rotate appreciably during the time (about 10 -8 sec) required for the re-
emission of the light. The study of the fluorescent light then provides
additional information about the structures involved.

4. Action spectra

For light to produce an effect it must be absorbed, and for absorption
to take place there must be energy level differences equal to the energy
of the incident photon. Once the photon is absorbed, the pigment (the
substance absorbing the light) may itself produce the effect being ob-
served or it may pass the light energy on to another substance by one of
several mechanisms. For example, it may happen that a pair of energy
levels of the pigment chance to coincide with a pair of levels of an
adjacent molecule. If so, the energy may be directly transferred, with the
result that the electron of the pigment molecule ends up in its lowest
energy state, while an electron of the adjacent pigment is raised to the
excited energy state. If this adjacent molecule can produce the effect
being studied, the energy transfer will permit the utilization of light not
directly absorbed by the effective molecule, thereby expanding the effi-
ciency of the system.

As an example, if light is incident on chlorophyll, the wavelengths ab-
sorbed are found to be in the red and blue regions of the spectrum, so
that the chlorophyll color is that of the unabsorbed green light; the result
of the absorption is the production of photosynthetic activity. Carot-
enoids absorb in the blue-green, and it is found that such light is
effective in photosynthesis, thereby suggesting that the carotenoids and
chlorophyll are energetically linked. To prove the linkage, we would
have to show that the carotenoids themselves effect no photosynthesis.
This will be shown later.

To obtain this information, we measure the effectiveness of light of
various wavelengths in producing the effect (in our example of photo-
synthesis, the effect could be oxygen production, C0 2 fixation, etc.).
This is done by irradiating an organism at such an intensity that the
photosynthetic activity is not saturated, which means that we should
choose an intensity such that the amount of photosynthesis would double
if the intensity were doubled. Thereby we obtain the amount of photo-
synthesis per incident photon of the given wavelength. Then the entire
experiment is repeated with light of various wavelengths, so that finally

62

LIGHT ABSORPTION EFFECTS

1.75 -

C-22

1.50

<u

m j3

03 +j

1.25

£ o

o o

1.00

0; -G

SC ft

0J

0) c

0.75

>-s

-t-3 ^

0.50

oj o

« &

0.25

o o

400 450 500 550 600 650 700 750

Fig. 26. The effectiveness of various wavelengths of light in producing photo-
synthetic carbon dioxide fixation. Two clear peaks are in evidence. The fact
that the effectiveness does not go down to zero between the peaks indicates the
existence of at least one more peak in the wavelengths between the two.

we could plot, as in Fig. 26, the relative effectiveness of the various
wavelengths. Of course, to obtain these data we need to measure the
incident light intensity and the fraction of the light absorbed by the
system. This implies that we must use a solvent medium (photosynthetic
algae would be swimming in some specially designed medium) which is
entirely nonabsorbing in the spectral region studied, or that the fraction
of the light absorbed by the medium must be separately measured and
corrected for.

The shape of the effectiveness spectrum or, as it is usually called, the
action spectrum, is then studied to see whether it resembles any known
pigment or collection of pigments. In the figure above, we see basically
three peaks. By comparing this spectrum with those in Fig. 27, we see
that two of the peaks are similar to those of chlorophyll, the third to that
of carotenoids. Thus we deduce that both pigments are involved.

To know that carotenoids are transferring their absorbed photon
energy to chlorophyll requires additional experimentation involving
fluorescence measurements. When a wavelength absorbed only by chloro-
phyll is used to illuminate the organisms, the typical red chlorophyll
fluorescence is found. When organisms are irradiated with wavelengths
absorbed only by carotenoids, no typical carotenoid fluorescence results;
only the chlorophyll fluorescence is observed. Thus we may deduce that
the energy transfer is only from carotenoids to chlorophyll. Since the
efficiency of production of photosynthesis by carotenoid absorption is
nearly as high as that of photons absorbed by chlorophyll, we conclude

ACTION SPECTRA

63

O.D.

400 440 480 520 560 600 640 680
Wavelength in m^

Fig. 27. The absorption spectra of chlorophyll and carotene. The chloro-
phyll peaks at about 430 and 665 m/x and the carotene peaks near 450 and
490 mfx are identified as being responsible for the action spectrum peaks in the
preceding figure.

that virtually all of the energy absorbed by carotenoids is transferred to
chlorophyll.

Action spectra then may be used to establish the absorption spectra
of compounds involved directly (or indirectly, by energy transfer) in
any given light-mediated process. Indeed, in many important investiga-
tions the action spectrum has given the first information about the
nature of the molecular species involved. The student should recall that
the comparison is between the action spectrum and the plot of the
extinction coefficient, E, against wavelength. Some additional examples
of action spectra utilization include the fitting of the action spectrum for
human vision to the absorption spectrum of rhodopsin, the action spec-
trum of phototaxis with the absorption spectrum of carotenoids, and the
action spectrum for flowering of certain plants with the absorption
spectrum for a molecule of what is now known as phytochrome.

Up to this point we have considered situations in which the absorbed
light promotes a given effect. There is a second type of action spectrum
in which the absorbed light inhibits or inactivates some process. A trivial
example is that of flowering inhibition by far-red light which reverses
the action of the phytochrome — this action spectrum has turned out to
be that of an excited state of the phytochrome molecule itself.

The more general case of inhibition results from the use of light of
such great energy as to damage the pigment. Ultraviolet light is the
usual agent of this kind of experimentation, since its photons are suffi-
ciently energetic to break chemical bonds, thereby altering the chemical
structure of the absorbing structures. Viruses, bacteria, and many kinds

64

LIGHT ABSORPTION EFFECTS

o

Nucleic acid

CO

CO
OS

Co

240 260 280 300
Wavelength in mn

(a)

240 260 280 300
Wavelength in m^u

(b)

Fig. 28. Part (a) shows the absorption spectra of equal concentrations of nu-
cleic acid and protein. Part (b) shows the action spectrum for inactivating the
capacity of Euglena cells to form photosynthetic progeny. Its peaks at 260 and
2S0 mfi show that both nucleic acids and proteins are involved in this experi-
ment, in which irradiated cells give rise to white, nonphotosynthetic colonies in
place of the normal green colonies from unirradiated cells. (After H. Lyman,
H. T. Epstein, and J. A. Scruff, Biochem. Biophys. Acta 50, 301, 1961.)

of single cells are inactivated by exposure to ultraviolet light to the
extent that they are actually killed.

In these instances, we measure the inactivation spectrum from about
220 m/A to about 350 m/u. In this region the nucleic acids and proteins
are the chief pigments, and one usually learns only whether one or both
are implicated in the inactivation. In the case of enzyme inactivation,
only a spectrum similar to protein absorption is obtained. In virus in-
activation usually a spectrum resembling nucleic acid absorption is
found, although there are instances of effectiveness peaks corresponding
to both nucleic acid and protein. The absorption spectra for nucleic
acids and proteins are shown in Fig. 28, along with an example of an
inactivation spectrum.

It must be carefully noted, however, that the process being studied
need not be viability. To emphasize this point, we present the following
list of inactivation spectra possible just for simple organisms such as
viruses; more complicated organisms would have an even longer list.
One can inactivate viruses with respect to (a) their ability to form viable
progeny, lb I their ability to kill cells on which they normally grow,
(c) their ability to adsorb to cells, (d) their ability to initiate the produc-
tion of various metabolic processes, (e) their ability to participate in
genetic recombination.

ACTION SPECTRA 65

Fig. 29. Light passing through a prism is broken up into the spectrum as
shown. A container C of finite size will necessarily intercept a whole range of
wavelengths.

For each of these functions an action spectrum can be obtained, and
there is no reason to expect the same spectrum for all effects; in fact,
different spectra are found. Therefore it is possible to learn some struc-
tural and functional lessons from action spectra of this kind, too.

The obtaining of an action spectrum is not as simple as it may seem,
due entirely to practical considerations. We have spoken as though it is
possible to irradiate with monochromatic light, i.e., light of a single
frequency. In practice, the source of light is some high-intensity lamp
whose light is used in one of two ways. As indicated in Fig. 29, the light
may be passed through a diffraction grating or through a prism to spread
the light out into a spectrum. If the container C being irradiated is
relatively large, or is placed relatively close to the spectrum-producing
device, it will intercept light of many wavelengths, thereby producing an
effect composed of all the wavelengths absorbed. If the container is
placed so far away from the source that it intercepts only a narrow
range of wavelengths, the total energy received becomes very low, be-
cause the intensity falls off inversely with the distance. Thus there is a
practical limit to the minimum range of wavelengths being absorbed.

A second device is a so-called interference filter which, as its name
suggests, uses an interference method to let through only a very small
range of wavelengths. This filter may be placed very close to an intense
light source to provide a high intensity. However, the band of wave-
lengths transmitted by such a filter is great enough to give trouble with
interpretation in terms of a single wavelength. Thus, in practice, action
spectra taken with a finite number of wavelengths are only roughly
accurate, so that only broad classes of compounds can be designated as
being involved in the action being studied. Happily, this is frequently
all the information that is desired.

CHAPTER

6

Physical Aspects
of Muscles

INTRODUCTION

The study of muscle action provides a good illustration of the different
points of view of the biophysicist and physicist. Consider a trapeze
performer hanging from a trapeze, supporting a (usually beautiful) lady
trapeze artist. The physicist's definition of work is the product of a
force on a body and the distance d that the body is displaced in the
direction of the force:

W = Fd.

From this point of view, the artist holding the lady is surely exerting
a force, but since there is no motion in the (upward) direction of the
force, the physicist claims that no work is being done. This is patent
nonsense, even to the physicist, as he discovers by trying to do this job
himself. Of course, the reason the physicist insists on talking this non-
sense is that he is talking about mechanical work being done against the
gravitational pull downward. The work we have in mind — the work that
results in sweat and fatigue of the performers — is biophysical and bio-
chemical work. It takes energy to keep the muscles in their tensed state,
and the biophysicist (and biochemist) is primarily concerned with this
aspect of the performance.

Today, the physics and chemistry of muscle action constitute one of the
more interesting research areas of biology. In describing the action of
muscles, we will first set forth the over-all aspects of the situation in
terms of forces exerted and the contraction and relaxation of the muscles.
Then we will inquire into the substructure of the muscles which make
possible the observed actions. Finally, we will indicate the molecular
aspects of the substructure, and will sketch the possible connection with
known biochemical facts of muscle action.

1 . The over-all action

Contraction of a muscle, or the attempt to contract it, results in a
force. Contraction is readily studied since the length of the muscle
before and after the contraction can actually be measured and (essen-

66

INTRODUCTION 67

tially by hooking the muscle to a very strong spring balance) the force
being exerted can be determined. The use of these data depends on the
questions being asked. One could ask for the total energy being ex-
pended, in which case a simple multiplication of the force and the distance
through which it moved (the contraction) gives the external work clone
by the muscle. This doesn't answer the entire question, because there are
quite obviously three energies to be reckoned with:

(a) the external work done,

(b) the work done in maintaining the readiness of the muscle, and

(c) the energy dissipated, as in heat, for there is always some in-
efficiency in any machine, even the human one.

There are still other questions. An important one derives from reason-
ing that the total external work done isn't always the important thing —
some situations might be better described in terms of the rate at which
work is being done, for that measures the extent of the immediate avail-
ability of the energy. In this case, as shown in an introductory physics
course, the rate can be computed from the product of the (assumed
constant) force and the velocity of contraction of the muscle. From this
rate of doing work the energy expended is computed by first multiplying
each rate by the duration of its persistence, and then adding the products,
in the same way that an electric company computes the electrical energy
we have used by adding the products obtained by multiplying the various
rates by the duration of use of each. The formula can be stated as

B = f - Pv,

where R is the rate of energy E expended in a time t if a pull (or force)
P is accomplished with a contraction velocity v. (These symbols are
chosen to conform to practice in muscle research laboratories.)

By studying a number of muscle preparations from different sources,
it has been found, empirically, by A. V. Hill (one of the pioneers of
biophysics and of muscle research) that a generalization can be made:

(P + a)v = constant.

Here a is a factor whose existence is readily rationalized. It is the internal
force expended to make the muscle contract, and therefore the product av
is the rate at which work is done to contract the muscle itself. The
magnitude of a can be determined from heat measurement, to be dis-
cussed later. This term is entirely similar to the corresponding correc-
tion to the ideal gas law.

PV = constant

68 PHYSICAL ASPECTS OF MUSCLES

Fig. 30. An experiment to show that with a given energy you can either
slowly exert a large force through a small distance or rapidly exert a small force
through a large distance. This is demonstrated by having your partner offer
either a large or small resistance to your push.

is corrected to

(P + a/V 2 )V = constant,

where the term with a is included to account for the work the gas mole-
cules do to pull apart from each other in the process of expanding to
exert a pressure P throughout the surface of the volume V of the con-
tainer.

There is a second correction to the perfect gas law which takes account
of the finite volume of the molecules by subtracting this volume, called
6, from the total volume, since the container cannot be made smaller
than the volume occupied by the molecules. In the case of muscle, there
is a similar correction for the contraction rate of the muscle itself. Thus
Hill's characteristic equation for muscle action,

(P + a)(V + b) = constant,

can, by suitable choice of the two arbitrary parameters, be made to fit an
extremely wide range of data on muscle contraction. The constant value
can be readily determined, since when maximum force is being exerted
the muscle will have its smallest contraction rate, which is b. Thus the
constant is simply P max b.

If the statement above seems implausible to you, you may familiarize
yourself with the point by a simple experiment. Have a friend hold his
hand facing you with palm vertical, as in Fig. 30. Now ask him to do
the following when you push on his hand. First, have him offer little

INTRODUCTION

69

o.-S

• - a

—

£ rt

£.5

50
Efficiency, %

Fig. 31. A representation of the power produced as a function of the effi-
ciency of using the force being applied. The efficiency of power production is
equal to 50% when the power delivered is at its maximum value. (From Odum
and Pinkerton, American Scientist, 43, 331, 1955; courtesy the authors and
American Scientist, Princeton, N. J.)

resistance. You will see that his hand is rapidly displaced, but that you
are exerting very little force. Second, have him offer his maximal re-
sistance. You will see that his hand is displaced at a minimal rate, but
that you are now exerting your own maximum force.

One interesting point seems worth an appreciable digression here. This
concerns the design of biological structure from the point of view of
energy expenditure. Are biological systems designed to give maximum
efficiency (least energy loss) or maximum power deliverable? Clearly,
we could conceive that some systems producing energy could be designed
for one purpose and some for the other, and perhaps some for other
purposes, too. What differences are involved in making this decision?

In an interesting paper on this topic, Odum and Pinkerton point
out that the work done depends on the forces and rates. When the
rates of displacement are very small, there is maximum force exerted.
The power delivered is the product of these two factors so that, since v
is very small, the power delivered is small. On the other hand, when the
force is very small, the value of v is large, but the product is again small.
It is then plausible, and they actually prove, that at intermediate forces
and displacement rates, the power delivered is greater than at the two
extreme values we have mentioned. This is sketched in Fig. 31. In the
cases they discuss, Odum and Pinkerton were able to deduce the exact
form of the curve. There is an optimal force to be expected for maximum
power delivered. At this value of the force, the efficiency proved to be
only 50% for ideal, loss-free systems; for real systems it is less than
50%. That is, by exerting either a very large force at a slow rate or a
very small force at a great rate, the physical efficiency is greatest, ap-

70 PHYSICAL ASPECTS OF MUSCLES

proaching 100%. But, if the energy-producing system is to produce the
most actual power, the system must be designed to work near 50%
efficiency. You may find further discussion of this point in relation to
biological systems in the paper whose reference is given at the end of this
section.

This digression allows us to conceive of a classification of muscles into
two types: those which exert large forces through small distances, and
others which exert small forces through large distances. Roughly speak-
ing, these types correspond to what is actually found in nature. Typical
striated muscles, such as facial, arm and leg, and heart muscles, contract
very little, but exert large forces. Smooth muscles, such as many involun-
tary muscles, e.g., bladder muscles, contract appreciably, and exert cor-
respondingly smaller forces over longer time periods.

Other than for measuring the external work done, the variables in
Hill's equation do not afford much scope for investigation of muscle
action problems. However, estimates of the energies involved in the
various processes may be obtained from heat measurements. The sensi-
tivity of the temperature measurements involved is fairly great but is
well within the scope of modern technology, which has no trouble measur-
ing temperature changes of millionths of a degree. When the temperature
changes in muscles are determined before, during, and after a muscle
contraction, it is found that there are three main stages of heat produc-
tion.

(a) Resting heat. This is the energy expended in just keeping the
muscles ready to perform. The metabolism within the muscle has the
function of keeping the metabolic machinery in readiness for action in
the extended position so that it can contract when the stimulus reaches it.

(b) Initial heat. This energy appears to be associated with the con-
traction process, being composed of two chief parts: the heat of develop-
ing the tension in the muscle, and the heat associated with the actual
contraction itself.

(c) Recovery heat. This, as the name implies, is associated with the
return of the muscle to its pre-contracted state, after which the resting
heat keeps the muscles in readiness. The recovery heat includes the
buildup of the molecules responsible for the quick availability of energy
which characterizes the muscle action. In this case, much of the quick-
energy activity is effected through the molecules of ATP and of creatine
phosphate, the former of which is a general energy storage battery for
the metabolic machinery of cells, the latter of which is an energy storer
quite specific for the needs of muscles.

Some of the points mentioned above may be directly observed. It was
first shown by Szent-Gyorgy that if an excised muscle is placed in
glycerol, many small molecules are dissolved out of the muscle, leaving

MUSCLE STRUCTURE 71

chiefly proteins called actin and myosin bound together in a complex
protein called actomyosin. Purified actin and myosin mixtures, which
can be made to form fibers, were used in later experiments. If ATP
molecules are added to the glycerinated muscle preparations or to the
synthetic fibers, the result is a readily measured (albeit slow) contrac-
tion. The ATP molecules are used up in this reaction, so that part of the
recovery heat must be associated with the resynthesis of ATP. Since in
living muscle the ATP is present during the resting state of the muscle,
it is clear that the mere presence of ATP molecules is not, by itself, the
trigger for the contraction.

It is of interest to point out that a similar role of contractile proteins
exists in an organism very different from those we ordinarily think about
as having muscle action. Bacterial viruses attach to the cells they infect
by means of appendages which have been called tails, though there is no
reason to think of these long slim appendages as tails any more than as
heads or necks. It has been shown that these viruses inject their nucleic
acids into the cells they attack, and that this injection is accompanied
by the contraction of the tails. Further, it has been shown that there are
about 100 molecules of ATP in the tail, and that the ATP is used up
during contraction in the same way as in the muscle contraction we
have been discussing.

2. Muscle structure

The light and electron microscopes have afforded us an excellent
picture of the substructure of muscle which should eventually lead to the
elucidation of the entire contractile mechanism. Unfortunately, up to this
point in time, the considerable technical problems in preparing materials
for electron microscopy have prevented the detailed substructure from
being determined.

The aspects of muscle visible to the eye resulted in classification of
muscles into striated and smooth types. Smooth muscles, by dissection
and examination in the light microscope, are found to be composed of
short (roughly 20 microns) cells of optically isotropic material. Striated
muscle, on the other hand, is found to be composed of bands of mate-
rials, the bands being distinguishable initially by being optically differ-
ent. There is a band of optically isotropic material, called the I-band,
and a band of optically anisotropic material, called the A-band. There
are two subdivisions in each band. The I-band is divided into two equal
parts by a dark-staining thin band called the Z-band (from the German
word Zwischen, implying a "between" band). The A-band is likewise
divided into equal parts by a slightly thicker clear band, called the H-
band (from the German word for clear — hell). These are shown dia-

72

PHYSICAL ASPECTS OF MUSCLES

Fig. 32. A schematic representation of a muscle showing the A, I, Z, and H
bands. The A-band is optically anisotropic. The I-band is optically isotropic.

grammatically in Fig. 32. More recent work has shown that the H-
band contains a thin central M-band, and that the Z-band is similarly
constructed. These refinements do not concern us here.

When a muscle is stimulated, there is a contraction, followed by a
relaxation and return to the original state; the entire process is called
a twitch. Observation with the light microscope reveals that during the
contractile phase of a twitch, the I-band and the H-band shorten by
about the same amount; the A-band width does not change. Since the
dark part of the A-band increases in size by the amount that the I-band
decreases in size, it is plausible that the material of the I-band is sliding
into the A-band, and that this is the mechanism of contraction. Of
course, the equality of change in the H-band and I-band could be mere
coincidence or have an entirely different interpretation. A once popular
theory ascribed the changes to alterations in configuration of the long
protein molecules making up the fibers; if the physico-chemical environ-
ment changed, the proteins would change in about the same amount.
This theory has foundered on the rock of measurement, for x-ray studies
have demonstrated that there is no such configuration change in the
proteins during muscle contraction. Indeed, this last result teaches us
that the muscle shortens while its constituent filaments do not; this is a
result which may be restated as showing that there are no contractile
proteins in muscle but that there are contractable configurations which
leave the filamentous proteins unaltered in length. It is further worth
noting that when muscles are stretched, the H- and I-bands both in-
crease by about the same amount, thereby supporting the notion that the
two changes are coupled, since the contractions and extensions occur
together.

This deduction about material from the I-band sliding into the A-band
cannot be the whole story. For if the I-band were to disappear entirely,
by sliding into the A-band, the shortening would have a maximum value
of about 40%, since the I-band is roughly 40% of the total length of the
A- plus I-bands. Yet measurements of intact muscles reveal a shortening

MUSCLE STRUCTURE 73

^— A-

1 H ' ' H ' ' H

Fig. 33. A diagram of the apparent arrangement of actin and myosin mole-
cules in a striated muscle. The thin lines represent the actin molecules; the
thick lines the myosin molecules. When the actin molecules come together
end-to-end, the muscle is presumed to shorten maximally, with the disappear-
ance of the H-band. The maximum shortening on this model would be 40%.

of up to 80%. Further, this shortening of 80% may also be observed in
smooth, unstriated muscles which presumably are very differently con-
structed. Furthermore, the synthetic actomyosin filaments, when in-
cubated with ATP, have been observed to contract more than 40%, and
in this case there is no possibility of a "sliding" structure.

It is therefore of some weird interest that electron microscopy of
muscles has revealed an internal arrangement which dovetails neatly
with the sliding structure model. The structure is represented diagram-
matically as in Fig. 33. What we see are thick myosin filaments and thin
actin filaments. These long protein molecules are placed so that the dark
parts of the A-band contain both kinds of molecules, the clear H-band
contains only myosin molecules, and the I-band contains only actin
molecules. As the sliding structure model would have it, the actin mole-
cules of the I-band slide past the myosin molecules until, as sketched in
the next figure, they have gone as far as they can. Thus, the I-band has
shortened by as much as the H-band. The evidence for these diagrams
derives from studies of cross sections in the I-, A-, and H-regions. As
shown in Fig. 34, there are only thick filamentous elements in the H-
band. In the dark part of the A-band, there is the same number of thick
elements, and twice as many thin elements. In the I-band there are only
this doubled number of thin elements. Some clever chemistry has demon-
strated that the thick elements are myosin, and that the thin elements
are mainly actin.

As may be deduced from the discussion thus far, the structural basis
of muscle contraction is simply not known, despite a wealth of detailed
information about the nature and arrangement of the materials making
up the muscle. Either some vital information is lacking or else there is
some misinformation in w r hat we think w r e already have established.

74

PHYSICAL ASPECTS OF MUSCLES

Cross section
through I-band

Cross section

through end part

of A-band

Cross section
through H-band

Fio. 34. A diagrammatic representation of the results of electron microscope
studies of sections through various parts of a striated muscle. From these
sections, the model given in Fig. 33 was deduced.

Some of those working on muscle already feel that there is likely to be
something incorrect about the x-ray results, which showed no change
of configuration of the muscle proteins during a twitch. Clearly, much
remains to be done in this field. It is of particular interest to biophysi-
cists because, since the structural and chemical aspects have been reason-
ably established, it is likely that the essential missing features of muscles
will turn out to be encoded in some esoteric aspects of the physics of
molecules and their aggregates. These parts of biophysics belong to a
more advanced presentation than we can give here.

CHAPTER

7

Methods for Determining
Molecular Size and Shape

INTRODUCTION

1 . A survey

One of the important aspects of biological studies is the determination
of the size, shape, and molecular weight of molecules associated with
various biological functions. Until the advent of the electron microscope,
these properties could be determined directly only if the molecules were
larger than, say, 0.4 micron, so that they could be studied in the light
microscope. As a result, indirect methods of determining particle size
and shape were developed. Even now that the electron microscope has
made it possible to obtain visual images of particles as small as 1 to 5 m/x,
there are still many instances in which the less direct methods are more
practical than electron microscopy. And there are still other instances in
which the indirect methods are the only ones which can give the desired
information. Furthermore, some of these indirect methods operate on a
fineness of structure still inaccessible to the electron microscope. For all
these reasons, therefore, it is advisable and worth while to present a sur-
vey of such biophysical methods. After presenting these surveys, several
of the methods will be presented in more detail.

(a) Kinetic methods

Diffusion. If a particle moves through a liquid, its movement will de-
pend on its size, shape, and mass. Therefore, by following the movement,
deductions may be made about these properties. If there is no net force
acting on the particle, it moves only by thermal agitation in a random
fashion. The bigger it is, the less mobile it will be. Therefore the average
displacement of such particles will be a measure of the size and shape.
If the shape chances to be spherical, it can be shown that the actual
diameter can be deduced. If the shape is other than spherical, the obser-
vation of the displacement must be coupled with some other measurement
to enable the investigator to deduce the actual size and shape.

For a spherical particle, the relationship that is used is ^* - «^^

75 \uj\ L IBRARY ]3>

\;zz \ I - 1

76 METHODS FOR DETERMINING MOLECULAR SIZE AND SHAPE

where x 2 is the average of the square of the displacement of the particles
from their starting point, k is the molecular gas constant, t is the time in
seconds at which the displacement is x, T is the absolute temperature, r)
is the viscosity of the medium (to be measured or looked up in hand-
books) , and r is the radius of the particle.

Sedimentation. If a gravitational or centrifugal force is applied
to a particle in solution, the particle will accelerate until its velocity is
such that the accelerating force is just balanced by the frictional force.
A measurement of this terminal velocity then gives a relationship be-
tween the mass, size, and shape of the particle and the frictional force.
If the shape is spherical, it is possible to obtain a direct measure of the
diameter and the mass of the particle. If the shape is not spherical, one
obtains only a relation between the mass and the diameter, and informa-
tion must be added from other methods to permit the knowledge of the
mass m and diameter separately. The relationship for spherical particles
is

mv m{\ — dL/dp)
F Girrir

where the symbols are the same as in the relation for the diffusion
method, F is the magnitude of the applied force, d L is the density of the
liquid solution, d v is the density of the particle, and v is the velocity of
the particle in cm/sec.

In both the diffusion and sedimentation methods, it is necessary to
measure the position of the particle at various times. Since the particles
of interest are normally invisible even in the light microscope, it is
necessary to use some clever method for determining these properties.
This is normally done for diffusion by carefully putting two solutions
together at a sharp, flat interface. One solution contains the particles of
interest, the other is identical except that these particles are missing.
Under these conditions, as the particles move they advance like the front
line of a parade of soldiers, as indicated in Fig. 35. By optical methods
which will not be discussed here, this boundary can be made visible and
its progress measured, thereby giving an average displacement for the
particles.

Electrophoresis. If an electric force is applied to a particle in solution,
the particle will accelerate until it reaches a constant velocity, as in sedi-
mentation. A measurement of this terminal velocity then gives a rela-
tionship between the size and shape of the particle, the electric driving
force, and the resisting force. Since the electric force is proportional to
the electric charge on the particle, and since that charge is unknown, this
relationship docs not permit deductions concerning the size and shape of

KINETIC METHODS 77

1 1 Filler tube

Buffer only

■ 1 1 i 1 1 ill Mil

• -•— : —

Buffer plus
particles

JL

^ e. e.

(a)

(b)

JL

(c)

Fig. 35. The setting up of a diffusion experiment. In (a) the lower tube has
been filled with a solution containing the particles whose movement is being
studied. The tube has flanged ground-glass ends. A similar tube is inverted
while empty onto the lower tube in a position such that the flange of one tube
covers the chamber portion of the other tube. The top tube is then filled with
the solution minus the particles. As shown in (b), the experiment is started by
sliding the top tube until the chambers are coincident. Particle diffusion can
then begin, and the particles can be found farther and farther up the tube as
time progresses. By placing a light-sensitive film behind the joined tubes, and
by shining light absorbed by the particles only from the front side, an absorption
image of the situation can be obtained at any time.

the particle. Nevertheless, the electrically driven movement, the so-
called electrophoretic mobility, is characteristic for each particle in a
given solution, and can therefore be used to ascertain the presence or
absence of the particles. Further, one can find the number of electro-
phoretically different constituents of unknown solutions, and such infor-
mation frequently permits important deductions to be drawn. Still
further, electrophoretic separations may be used to prepare pure solu-
tions of these constituents. An important example concerns the study of
blood. When blood is subjected to an electric force, a number of con-
stituents are found in an entirely reproducible fashion. The constituents
are made visible by the boundary detection method described for diffu-
sion and sedimentation. The fastest moving substance has been isolated,
and turns out to be serum albumin. The next three fastest substances all
turn out to be globular proteins. In the order of speed, these have been
called alpha, beta, and gamma. The ignorance of particle size and shape
has not prevented investigators from finding that the entire antibody
content of blood is in the gamma globulin fraction. The electrophoretic
method is used to isolate this fraction for clinical use and for clinical and
laboratory research.

78 METHODS FOR DETERMINING MOLECULAR SIZE AND SHAPE

Recent developments of the electrophoretic technique include placing
the solution on long strips of special filter papers in a moist atmosphere,
and applying the electric force across the ends of the paper strip. The
separation of the constituents is then achieved simply by using a scissors,
and the constituents may then be eluted from the paper in suitable
buffers.

(b) Static methods

If the particles are stationary, then the physical agent for extracting
information has to move into and out of (or at least away from) these
particles. Thus, we are led to examine the irradiation of particles by
visible light, ultraviolet and x-radiation, and various charged elementary
particles.

Visible light scattering. If a particle is very tiny, light waves scattered
off its front and rear will be so nearly in the same phase that they
scarcely interfere with each other in the sense of waves reinforcing or
cancelling each other. If, as indicated in Fig. 36, the particle is compara-
ble in size to the wavelength of light being used, then the wave scattered

Eye or
photographic film

Fig. 36. A light beam whose wavelength is comparable to the dimensions of
the scattering particle is reflected in various directions by the particle. The
bottom wave travels about a half wavelength more than the middle wave be-
fore being reflected, so that it is possible to find a direction, as indicated, in
which the eye or a light-sensitive film will record that the reflected beams arrive
out of phase, so that there is a cancellation of beams. Other directions could be
found in which the beams arrive so that thev reinforce each other.

STATIC METHODS 79

from the front and rear can produce appreciable, and measurable, inter-
ference effects whose magnitudes depend on the angle A from which the
scattered light is viewed. By measuring the interference for all angles of
viewing, it is possible to make significant deductions about the size and
shape of the particles even though they may be too small to be visible,
even in the light microscope.

A question that has been studied by light scattering is whether heating
of DNA separates the two Watson-Crick strands from each other. In
one set of measurements it was found that the volume of the scattering
molecules was unaltered by heating to temperatures previously believed
to be high enough to separate the strands completely. If the separation
had occurred, the volume of the individual scattering molecules would
have been halved.

Light absorption. Chemical bonds, such as C — H, C — O, C=0, etc.,
have specific absorption regions in the infrared. Series of atoms with
alternating double and single bonds, such as — C=C — C=, absorb at
wavelengths which increase progressively with the length of the series.
The shorter series absorb in the ultraviolet. Among the materials which
absorb in that region of the spectrum, the biologically important ones
are proteins and nucleic acids, which contain molecules with short series
of these alternating bonds.

Light of the infrared and ultraviolet regions mentioned above, when
absorbed by a solution of molecules, will be absorbed in proportion to
the concentration of the molecules. Thus instruments designed to measure
this light absorption, called colorimeters or spectrophotometers, can give
information as to concentrations, but not as to size and shape.

X-ray absorption. In contradistinction to light absorption, the ab-
sorption of x-rays can be used for studies of size and shape. When an
x-ray photon passes through a molecule, the properties are such that
there is little likelihood that it will actually interact with any molecules.
Therefore the chance of interacting is strictly proportional to the number
of opportunities for interaction, and therefore to the thickness of the
molecule. In addition, the greater the cross-sectional area of the mole-
cule, the greater the probability of the photon's passing through it.
Therefore the total probability P of interaction to produce a biological
effect is proportional to both factors, i.e., to the thickness t and the cross
section A of the molecule:

P a t A = V,

where V = tA = the volume of the molecule.

Thus, by measurement of the rate at which biological entities are af-
fected by x-rays, it has been possible to make an estimate of the volume
of the molecules involved.

80 METHODS FOR DETERMINING MOLECULAR SIZE AND SHAPE

2. Diffusion

It is well known that thermal motion effects a homogeneous distribu-
tion of molecules in solution if enough time elapses for the equilibrium
state to be reached. The Brownian motion is the visible result of thermal
motion, being due to numerous impacts by molecules on a visible par-
ticle; the actual motion observed is the resultant of the collisions occur-
ring practically simultaneously. It should be plausible that a molecule's
motion will be inversely related to its size. Since thermal equilibrium
means equal average molecular energies, the bigger molecules will have
smaller velocities. Thus, if we could follow the individual molecules well
enough to measure their average velocities, we could compute their mass
from the average thermal energy, using the formula

average thermal energy = \ mass X (velocity) 2
or

#therm = 2 ^^ ■

In practice, however, this approach is unworkable, because the mole-
cules of interest arc not large enough to be seen in the light microscope.
Indeed, if they could be seen, their masses could be estimated from their
volumes and some reasonable estimate or measurement of their densities.
Thus, when we wish to obtain information about masses of invisible
molecules, we have to resort to various stratagems. In the section on
centrifugation, it is shown how the information can be obtained by using
high speed rotors. Here we indicate how similar information can be ob-
tained from measurement of the thermal velocity.

Since we cannot make visible the motion of a single molecule, we use
the device of measuring the motion of a large number of molecules. In
Fig. 37 we picture a tube on its side set up with the molecules of interest
in only one of the two compartments. When the compartments are slid
so that molecular interchange can occur, there is an average motion of
the molecules of interest, and it is this advancing boundary that gives us
a handle on the measurement. In the figure we see schematically what
happens when the two compartments are slid together.

In the left-hand column of sketches, we represent the concentration of
solute molecules. In the right-hand column, we present the plot of con-
centration at various positions in the connected compartments. After a
while, the sharp concentration difference becomes blurred, as molecules
move towards the right. In the second column of sketches, we further in-
dicate the positions where the concentrations are '-\'\ of the original con-
centration and also Vi of the original concentration. As can be seen, as
time goes on these positions separate more and more from each other. By
measuring the actual concentration curve by optical absorption means

DIFFUSION

81

i i a i i i l

i'','i."^V- -

* ,. »** . o « i>

° ' _? — dr-*.-*-:^-2_==

-» o f m . <9 = H 1

T—

" n o°

c *

m —Q-

TT -

o

-

a — e_

3

O i

^^

•- —

#

- a -a—

a *

9

,—

* «

•. a

o O

c j «

m

° B *

" o °

-C_

o °

o~ ° - a

O - O

J£

1

a

100

o

03

75

—

50

<v

S3

25

O

U

Initial arrangement

(a)

6§

^

100

a

.2

75

03

f-<

50

a

<D

'25

o

O

U

A short time later

(b)

65

c

100

o

-tJ

75

a!

t*

50

<t>

25

o

U

A still later time

(c)

65

a

100

o

+^

75

crt

e

50

03

(5

25

O

(J

After an infinite time

(d)

Position in joined tubes

x 3/4 x l/4

x 3/4

•r 1/4

Fig. 37. The initial arrangement has all the particles in one half of the setup.
The concentration, plotted on the right, is 100% up to the division point, then
falls abruptly to zero. After a short time there is a net flow of particles to the
right, so that the concentration now drops more gradually to zero. After more
time for diffusion, the particles have reached almost to the right-hand end of
the tube, and the concentration falls almost linearly to zero. After an infinite
time the particles are evenly distributed through the tube which is twice as
large as the starting tube, so the concentration throughout is half the starting
concentration. The positions in the tube at which the concentration is 75% or
25% of the starting value are indicated. The distance between these two points
is a direct measure of the time elapsed since diffusion commenced.

82 METHODS FOR DETERMINING MOLECULAR SIZE AND SHAPE

which we shall not discuss here, we obtain curves like those in the right-
hand column. The smaller the molecules involved, the faster the curves
fall to the even distribution situation in the bottom sketch. The move-
ment of the molecules can be computed from the actual distribution of
molecular velocities, which is known from physicochemical theory.

The rate of spreading of the boundary can be related to the so-called
diffusion coefficient, D, by the approximate formula

x 2 = 4Dt,

where x is the distance between the % and ^4 concentration points at
time t seconds after starting.

This formula is a very handy one for biophysicists. For example, if
we introduce a few enzyme molecules into a solution and these are taken
up by a 10-micron cell, can these enzyme molecules possibly reach the
substrates on which they act, simply by diffusing? In the table below we
give typical values of D for spherical molecules of various sizes.

Molecular weight Diffusion coefficient, D

Glycine

75

95 X 10 -7 cm 2 /sec

Cytochrome C

1.3 X 10 4

10 X 10~ 7

Hemoglobin

6.8 X 10 4

6.2 X 10 -7

Urease

4.8 X 10 5

3.5 X 10 -7

Southern bean

mosaic virus

6,000,000

1.34 X 10~ 7

Bacteriophage

T2

200,000,000

3 X 10 -8

A typical enzyme might have a molecular weight of 100,000, and we
see that its diffusion coefficient D is therefore about 5 X 10~ 7 cm 2 /sec.
Putting this in the formula, we obtain the value of t, the time taken to
diffuse just once across the cell:

, x 2 (1(T 3 ) 2 , „ r

Thus, this enzyme molecule will diffuse in a waterlike medium so fast
that it could travel back and forth across this large cell two times per
second, and therefore could well be able to catalyze reactions anywhere
in the cell where the substrate happened to be.

To return to the problem of estimating molecular weights, it is neces-
sary to have a connection between the diffusion coefficient and the mass
of the molecule. Advanced physical theory gives us the result that

hTrjr

CENTRIFUGATION 83

where k is the gas constant per molecule, T is the absolute temperature,
7] is the viscosity of the solution, and r is the radius of the molecule.

The temperature and viscosity of the solution are obtained by stand-
ard measuring instruments, so that the radius of the molecule can be
found if D is known. This formula holds only for spherical molecules.
For nonspherical molecules, as indicated in the section on centrifugation,
we combine data on sedimentation and diffusion to obtain the desired
information.

For spherical molecules, we replace r in the formula by its equivalent
in terms of molecular weight, using the relationship

M = d p V = %irr 3 d p .

Thus the diffusion coefficient varies inversely as the cube root of the
molecular weight. This means that diffusion measurements are quite
insensitive to variations in molecular weight, so that other methods
should be used for more than approximate values of the molecular weight.
That this conclusion is borne out experimentally may be seen by looking
at the table given above. There we see that the diffusion coefficient
changes from 95 X 10~ 7 to 0.3 X 10~ 7 , while the molecular weight
changes from 75 to 2 X 10 8 . That is, a 300-fold change in diffusion co-
efficient corresponds to a 3 million-fold change in molecular weight.

3. Centrifugation

There is a radially directed force on a particle which moves along a
curved path. If the particle is within a container filled with a liquid
which is less dense than the particle, the particle will move radially out-
ward; if the liquid is more dense than the particle, the particle will float
inward. As indicated in Fig. 38, the important physical parameters are

r — the radius of circulation of the particle p,

a) — the frequency of rotation of the centrifuge tube, in radians per

second,
d p — the density of the particle, and
d L — the density of the liquid.

A centrifuge is a device for spinning tubes containing particles dis-
solved in a liquid. Centrifugation is used currently in five different ways
in biology.

(a) By centrifuging a suspension of particles for a sufficiently long
time in a liquid that is less dense than the particles, the particles are
packed into a pellet at the bottom of the tube. They are then collected by
decanting the supernatant fluid. We say that the particles have been
pelleted or sedimented.

84

METHODS FOR DETERMINING MOLECULAR SIZE AND SHAPE

Axis of rotation

Fig. 38. Tubes spinning about an axis of rotation with an angular velocity a)
and with the particle of interest at a distance r from the axis. The particle is at
P and is suspended in a liquid L.

Inner edge

>

Boundary

I

Aijfw*:**

Axis of rotation

Particles in solution
moving to the right

Particles which have

reached the outer end

of the tube and have

formed a pellet

Fig. 39. A representation of the situation some time after a tube containing
particles denser than the suspending liquid has been started spinning in a cen-
trifuge. The particles which were at the inner edge at time zero move together
to the right and thus form a boundary whose movement can be photographed
for determination of the velocity of the movement.

(b) The rate at which a particle will move when being spun at a
given frequency w at a given radius r will depend on its density, the
density and viscosity of the liquid, and on the size of the particle, as wall
be demonstrated below.

The individual particle cannot be studied, but the particles at the inner
edge of the tube form a boundary which moves radially outward as a
result of the centrif ligation, as indicated in Fig. 39. This boundary can
be made visible by optical methods, so that the rate of displacement of
the particles can be deduced by photographing the transparent tube at
various times after starting the centrifuge. Since the density and vis-
cosity of the liquid can be measured, it is possible to compute the ratio of
the mass of the particle (its molecular weight) to its size. If the particle
is spherical, the molecular weight can be deduced directly. If it is not
spherical, a somewhat more complicated procedure is required which
gives us the size of the particle by measuring the diffusion of particles
through a liquid. Then, by a combination of these measurements, the
molecular weight can be deduced. The mathematics of this method will

CENTRIFUGATION 85

1.6

\J_____

Fig. 40. The situation in a CsCl density gradient in which the solution den-
sity varies from 1.6 at the inner edge of the tube to 1.8 at the outer edge of the
tube. The point of density 1.7 is indicated, and particles of that density will be
found in a band at that point. If particles were to the left of the band, they
would be more dense than the solution and thus would be moved to the right
by the centrifuging force. If to the right of the band, they would be less dense
than the solution and thus would be floated inward to the left. Thus, after
many hours of spinning the mixture of particles and CsCl solution, the situation
sketched will be obtained.

be presented below. This is called the method of the sedimentation
velocity.

(c) If the centrifuge is run for a sufficiently long time (days), all the
particles will tend to be at the bottom of the tube. But, due to the
Brownian (heat) motion, some of the particles will actually go backward
toward the axis of rotation. When these two processes (centrifugal sedi-
mentation and backward diffusion) have come to equilibrium, there will
be a gradient of particle concentration from essentially zero at the inner
edge of the tube to a maximum at the outer edge. This gradient can be
made visible by the optical methods referred to above. The bigger the
particles, the more they will tend to be concentrated near the outer edge
of the tube. From a measurement of the actual concentration gradient,
then, the molecular weight of the particles may be directly deduced. This
is called the method of sedimentation equilibrium.

(d) If a low molecular weight salt were in solution, the sedimentation
equilibrium method would result in a very gradual gradient. Indeed, it
should be obvious that a salt could be chosen w r ith so low a molecular
weight that it is hardly sedimented at all. Table salt (NaCl), for exam-
ple, would hardly be expected to be recovered from a solution by this
method. But it will in fact give a measurable gradient. In practice, for
example, using a certain concentration of CsCl instead of NaCl, we can
set up a salt gradient so that at the inner edge the density is 1.6 gm/ml
and rises essentially uniformly to about 1.8 gm/ml at the outer edge.
Other gradients may be obtained with other CsCl concentrations.

Now, if particles of an intermediate density, say 1.7 gm/ml, are placed
in t; e solution before centrifuging, they will all end up in a band at their
own characteristic density, as indicated in Fig. 40. The reasoning is that
when the salt equilibrium has been established, those particles finding

86 METHODS FOR DETERMINING MOLECULAR SIZE AND SHAPE

Proteins Cell debris and membranes

I

Fig. 41. A representation of the results of a centrifugation of a mixture of
crushed cells layered onto the inner edge of a sucrose gradient which increases
to the right. The fastest moving substances are cell membranes and internal
membranes, which are represented as just having reached the outer edge of the
tube. The next most rapidly moving parts are the nuclei, followed by the pro-
teins, with the lipid material still floating at the starting place.

themselves to the left of the place where the density is 1.7 gm/ml will be
forced centrifugally outward until they reach that place. Particles find-
ing themselves to the right of that place will be less dense than the
solution, and will therefore be floated inward until they reach the same
place. Again, the density gradient and the particle band can be photo-
graphed by suitable optical methods, so that the particle density can be
measured.

There is a further result from this density gradient sedimentation
method. Because of the thermal (Brownian) motion of the particles, they
will not be precisely in a thin band but will be in a band of some finite
thickness. Indeed, the smaller these particles, the faster and further they
will go at any given temperature. Thus it should be plausible that the
width of the band is related inversely to the size of the particles: the
bigger the particles, the smaller the band. This proposition can be
demonstrated mathematically to be true, and therefore the molecular
weight of the particles can also be obtained from the same photograph.
This method then gives both molecular weight and density from a single
photograph !

(e) Density gradient sedimentation can be used in another very fruit-
ful way. The density gradient is first established (more usually with
sugar than with salt, for technical reasons) and afterwards the solution
to be studied, containing a mixture of particles, is placed on top of (at the
inner edge of) the centrifuge tube. When the machine is then run for a
while, the various particles in the solution will have reached different
points in their travel towards the other end of the tube. If the centrifuge
is stopped, the different kinds of particles will be found to have been
separated into bands, as sketched in Fig. 41. In the sketch, we have
assumed that some cells have been broken and that the method has
separated the solution into the components indicated. By cutting the tube
carefully, the various components can be separated from each other.

THE USE OF CENTRIFUGATION IN BIOLOGY 87

The purpose of the sugar (or salt) is to stabilize the contents so that
minor shaking or convection will not remix the separated components.
The viscosity of the sugar solution contributes greatly to the stabiliza-
tion. Ordinary buffer solutions are not stable enough. This density
gradient separation method gives no absolute quantitative data, but does
give quantitative separation of components.

AN EXAMPLE OF THE USE OF CENTRIFUGATION IN BIOLOGY

One of the fundamental problems of genetics is whether the nuclear
material, especially the nucleic acid, stays in one piece during replication.
In principle three different fates are possible for the nucleic acid. First,
it could always remain intact. This would imply that the new nucleic
acid which goes to one of the daughter cells after mitosis is built by
copying the parental nucleic acid without disrupting it at all. Since such
a mechanism entirely conserves the parental molecules, it is called the
conservative method of replication.

Second, the Watson-Crick model of DNA affords the possibility that
the nucleic acid comes apart into just two pieces, which are separately
replicated by pairing with the proper components. Thus each daughter
cell would receive exactly half of the parental nucleic acid, in one single
piece. If this model is correct, then in a second generation, resulting in
four daughter cells, two would contain one each of the parental half-
strands; the other two cells would contain none of the parental nucleic
acid atoms. This scheme conserves half of the parental molecules of
nucleic acid and is therefore called semiconservative.

Third, the nucleic acid might come apart into a number of pieces which
are incorporated at random into various maturing DNA strands by some
unknown mechanism. In this scheme the parental nucleic acid molecules
are dispersed among the daughter cells and the method of replication is
therefore called dispersive.

A combination of isotope utilization and density gradient centrifuga-
tion has permitted an unequivocal decision about the mechanism for
bacteria. The bacteria in question were grown in a nutrient medium con-
taining much heavy nitrogen (N 15 ) instead of normal nitrogen (N 14 )
and heavy carbon (C 13 ) rather than normal carbon (C 12 ). Thus the
nucleic acids of such cells would be more dense than the nucleic acids of
cells grown in a normal medium.

In this experiment, cells were grown for many generations in the dense
medium and then transferred to a normal medium. Samples of cells
were taken immediately upon transfer to the normal medium and at
various times thereafter; the DNA of the sample cells was extracted and
run in a CsCl density gradient. The results are shown in Fig. 42.

88 METHODS FOR DETERMINING MOLECULAR SIZE AND SHAPE

Density increases

Generation number

1.9

Band is at position
of unlabeled normally
dense DNA

and 1.9
mixed

Band is at position
of isotopically labeled
dense DNA

^ f

Band is at a density
intermediate between
that of isotopically
labeled dense DNA and
that of unlabeled
normally dense DNA

Fig. 42. Schematic drawings of photographs of DNA extracted from iso-
topically labeled dense organisms at various times after organisms were placed
in unlabeled normally dense medium. The DNA was placed in CsCl and banded
as explained in Fig. 40. Since the DNA absorbs highly specifically and strongly
at 260 mfx, the band positions can be photographed by absorption of this light,
as explained in Fig. 35.

In this sketch, each section represents a photograph of the centrifuge
tube taken so as to show materials which absorb ultraviolet light, since
nucleic acids absorb that part of the spectrum much more strongly than
other biological materials. The single band at generation zero is clue to
the dense nucleic acid. After one generation, there are no molecules of
the parental density, demonstrating unequivocally that a completely
conservative mechanism of replication does not exist. The density of the
nucleic acid molecules found after one generation is actually midway
between that of the dense parental molecules and that of molecules of
cells grown in a normal medium, as can be seen by looking at genera-
tions 1.9 or 3.0, in which most of the molecules are from cells which were
formed after the transfer to a normal growth medium. It should be noted
that at all times there are molecules of this intermediate density. Thus
we see evidence that molecules which have obtained half their nucleic
acid from the original parents and half from new normal density mate-

THE USE OF CENTRIFUGATION IN BIOLOGY 89

Exp. no.

2
1
1
1
2
2
2

U .. .

M* and 1.9

— J ^ — ^ mixed

(b)

and 4.1
mixed

Fig. 43. The actual figure on which Fig. 42 is based. (From Meselson and
Stahl, Proc. Natl. Acad. Sci., U.S., 44, 671, 1948; courtesy the authors and
National Academy of Sciences.)

rials always persist. After the parental molecules have separated into
two pieces, they do not separate further; there are no molecules of any
density intermediate between these "half-dense" molecules and normally
light molecules. Accordingly, w T e conclude that a semiconservative mech-
anism of replication is operating, and the Watson-Crick model has yet
another piece of supporting data. For your study, the actual data pub-
lished by the researchers is presented in Fig. 43.

90 METHODS FOR DETERMINING MOLECULAR SIZE AND SHAPE

It should be remembered that this, evidence concerns only bacterial
DNA. It is possible that bacteria which have no nucleus of the con-
ventional type (there is no nuclear membrane and no evidence for
mitosis) are a special case. Students should be cautious in extrapolating
this result to all organisms.

THE MATHEMATICS OF THE SEDIMENTATION VELOCITY METHOD

Quantitative data are most frequently obtained by the use of the sedi-
mentation velocity method. The essential features of the calculation may
be derived from an analysis of the forces involved.

If a centrifugal force acts on a particle in solution, the particle will
move if its density differs from that of the solvent. The particle will
speed up but, in accelerating, the frictional resistance will be increased
because (for low speeds) the resisting force is proportional to the particle
speed. Thus the particle will accelerate until the radially outward cen-
trifugal force is just balanced by the radially inward frictional force.
From that point on the particle will move with constant speed outward,
since there is no net force acting on it. This force equilibrium point may
be written as

F c = F r ,

where F c is the net centrifugal force and F r is the resistance force. As
already mentioned, F r is proportional to the velocity v.

Fr = fv,

where / is the proportionality constant which is known as the friction
coefficient.

The centrifugal force is given by F c = mv 2 /r = mo) 2 r, where m is the
effective mass of a particle spinning at a rate of u> radians per second at
a distance r from the axis of rotation. The effective mass is the difference
between the mass of the particle and the mass of an equal volume of
solvent liquid (if the particle had the same density as the solvent liquid,
there would be no resultant motion) :

4-1)

m = m p - triL = m p 11 - I = m

where d is the density, and the ratio of, the masses is replaced by the
equivalent ratio of the densities. The force equilibrium condition may
now be written as

m p (l - d L /d p )u) 2 r = fv

or, alternatively, as

THE SEDIMENTATION VELOCITY METHOD 91

m P {\ ■ - d L /d p )

co 2 r /

The left-hand side of this expression is defined as the sedimentation
coefficient s. It is in terras of s that many biological materials are de-
scribed, since it is equivalent to stating the mass or molecular weight of
the particle. Since s is a ratio between the velocity attained by a particle
and the acceleration a) 2 r given it by the particular machine used, it will
be the same for all centrifugation machines. Also, the right-hand term
must be independent of the particular machine used, and it can be seen
that it depends only on the properties of the particle and the solvent
liquid.

To proceed further, we must, in effect, measure /. The usual procedure
is to utilize a relation first derived by Einstein:

, kT

where T is the absolute temperature, k is the molecular gas constant, and
D is the so-called diffusion coefficient, which can be determined from the
distance particles diffuse (by Brownian motion) in a measured time
interval. Thus, by using Einstein's expression for /, we obtain an expres-
sion for s which depends on measurable quantities:

m p (l — d L /d p ) D Nm p (l — d L /d p )D
S ~ ' ~kf~ NkT

where we have multiplied numerator and denominator by N, Avogadro's
number. Here Nm p is the molecular weight M of the particle and Nk is
the more usual form of the (molar) gas constant R, as used in the gas
law, pV — RT. Thus the final expression for the sedimentation coefficient
is

M(l - d L /d p )D
S ~ RT

In practice, s, T, and d L are obtained in one experiment and D in a
separate experiment. The particle density d p may be obtained by cen-
trifuging the particles in liquids of various densities to determine the
density at which the particles do not move ; this liquid density must then
equal the particle density. Thus, finally, we can obtain the molecular
weight, which is the object of all this work and analysis.

For spherical particles, we have indicated that the diffusion coefficient,
D, is inversely proportional to the cube root of the molecular weight.

92 METHODS FOR DETERMINING MOLECULAR SIZE AND SHAPE

Thus the sedimentation coefficient containing the product MD is propor-
tional to the two-thirds power of the molecular weight. Accordingly, the
sedimentation velocity method is much more sensitive to changes in
molecular weight than is the diffusion method. Some typical sedimenta-
tion coefficients are given in the table below.

, , , , . , , Sedimentation

Molecular weight — .

coefficient

Botulinus toxin

950,000

17

Southern bean mosaic virus

6,300,000

115

Bacteriophage T2

200,000,000

900

The unit in which the sedimentation coefficients is given is the svedberg,
which is 10~ 13 /sec. This is the standard unit found in the literature.

4. X-ray diffraction

When x-rays impinge upon matter, most of the rays pass through
unaltered and undeflected, if the matter is not very thick. Some of the
rays, however, are scattered, and the pattern of scattering depends on
the pattern of the atoms and molecules making up the piece of matter.
Given the structure of the piece of matter, the scattering of the radiation
can be computed, but given the pattern of the scattered radiation, it is
much more difficult, and in some cases impossible, to deduce the structure
of the scattering material. We shall first discuss the use of x-ray dif-
fraction methods for determining the structures of crystalline materials.
The usual experimental procedure yields a photograph of a pattern of
lines or spots of varying spacing and intensity. The spacings will be
shown to be related to the distances between the molecules (or atoms)
in the crystal. Analysis of the intensities of the lines or spots can be
made to yield the detailed structure of the (polyatomic) molecules them-
selves. The mathematical techniques needed to achieve these results are
far beyond the scope of these writings, but an attempt will be made to
suggest the essential factors involved in the methods.

We start with a standard diagram (Fig. 44) showing a beam of mono-
chromatic x-rays hitting and bouncing off a crystal which is represented
as an array of atoms, indicated by circles. The wavy lines are the paths
of two of the rays making up the x-ray beam. The parallel lines are to
be thought of as planes perpendicular to the paper and passing through
the atoms as shown. Thus we are really talking about a three-dimen-
sional crystal of which we see a plane section. The angle is that be-
tween the direction of the incident beam and the plane of the atoms.

The two rays are shown bouncing off atoms in such a way that the
path of wave II becomes the same as that of I. This second ray travels

X-RAY DIFFRACTION

93

To detector

Figure 44

Figure 45

farther than the first ray by the amount indicated by the heavy line. A
detector (such as a photographic film) into which these rays go will yield
one of three possible results:

(a) If the extra distance is just equal to a whole wave, then the two
rays arrive at the detector in the same phase of their motion (that
is, crests are with crests, and troughs with troughs). They add up
to give an intensity greater than that from a single ray.

(b) If the extra distance is just equal to half a wave, then the two
rays arrive at the detector exactly out of phase (the crest of ray
I would be superposed on the trough from ray II) and the rays
cancel, giving zero intensity.

(c) If the extra distance is neither of these two alternatives there will
also be a cancellation, because in actuality the detector receives
many rays, not just two, and except for the two cases presented
the rays will be at all phases of the motion with respect to each
other and there will be an over-all cancellation.

For any spacing between planes, there will be precisely one angle 6
which causes the rays to be exactly one wave apart, so that they reinforce
each other; at any other angle general cancellation will occur.

Bragg was the first to point out that arrays of atoms in a crystal could
be thought of as forming planes in the way shown. This is clearly an
approximation, but the exact treatment of the problem by advanced
methods yields precisely the same criterion for reinforcement as Bragg
obtained from his simple model.

The relation for the condition of reinforcement is

X = 2d sin 0,

where X is the wavelength of the x-rays used and d is the spacing between
planes. As will be indicated below, X and are usually measured, which
permits the computation of d.

94

METHODS FOR DETERMINING MOLECULAR SIZE AND SHAPE

There are in fact many possible sets of parallel planes that can be
formed by arrays of atoms, as indicated in Fig. 45. Each of these sets of
planes will produce reinforcement of incident x-ray beams scattered off
them, provided only that the planes are at the proper angle to the inci-
dent beam. It should be readily appreciated that some planes have many
more atoms than others, and since the over-all intensity obtained depends
on the density of atoms in the planes, the more "natural" planes, in
practice, yield the most intense reinforcements.

In general, the orientation of a single crystal will not be such as to
produce reinforcement, but there are several means of obtaining the
proper orientation. We shall first discuss the so-called powder method.
As the name of the method implies, a crystalline material is first smashed
into small pieces, and the resultant crystal powder is placed in the path
of the x-ray beam. Since there are now very many small crystals, in
essentially all possible orientations with respect to the beam, many will
chance to be in the precise orientation needed to give ray reinforcement.
These correctly oriented crystals will not all be parallel to each other,
since a properly oriented crystal can rotate about the beam and still
maintain its proper angle. As shown in Fig. 46(a), the result will be a
reinforcement sufficient to produce circles of blackening of a photographic
film.

If a strip of film (indicated by the dashed lines in the sketch) is used
instead of a very large piece of film, the positions of the reinforcements
will be given by the (somewhat curving) lines at various distances from
the intense blackening produced by the undeflected x-ray beam. Such a
strip is shown in part (b) of the sketch; it is the experimental result
frequently presented in research papers.

(b)

Figure 46

X-RAY DIFFRACTION 95

To detector

Figure 47

Consider the three measurable parameters indicated in the sketch:
L, the distance from the crystal to the film ;
s h the distance from the undeflected beam to the individual lines ;
X, the wavelength of the x-rays.
Since in practice the angles are always small,

sin 20 « 2 sin
and

Si

L

X = 2d sin « 2d - S f « ^ •

Thus, as indicated earlier in this section, the spacing d can readily be
computed from the measured quantities. For the simple monatomic
crystal we are discussing, it is only moderately difficult to relate the
calculated spacings to the distances between atoms, but it is beyond the
scope of this presentation.

It remains to indicate how the intensities can yield the structures of
molecules placed at the crystal positions. To this end we return to Fig.
44 and, as in Fig. 47, insert atoms midway between the planes. It is true
that there will now be some new powder lines, but we shall neglect them
for the present and consider only what happens to the already existing
powder lines. In the case presented in the sketch, the extra distance
traveled by ray III is just one-half that for ray II. Since the latter dis-
tance is exactly one whole wave more than for ray I in the case of ray
reinforcement, ray III travels exactly one half wave more than ray I.
Thus it cancels ray I because its troughs are superposed on the crests of
ray I, and vice versa. In this instance, then, the line will be missing.
Now, if the extra atoms are not placed so that their contribution is pre-
cisely one half wave different, then there will not be a total cancellation
of ray I. In general, then, the insertion of the extra atoms will diminish

96 METHODS FOR DETERMINING MOLECULAR SIZE AND SHAPE

some of the line intensities. It should not be difficult to construct the
argument leading to the conclusion that other lines could actually be
enhanced.

Now imagine that these extra atoms are moved so that they touch the
original crystal atoms; the result will still be increases and decreases in
the intensities of some of the lines. When two atoms are close together
at a crystal position, the situation is entirely equivalent to having a
diatomic molecule at that position. Accordingly, we have deduced that
the effect of making a crystal out of diatomic molecules is to produce a
pattern of lines that is the same as that of a single atom crystal, but
with a different set of intensities. Continuing in this vein, we argue that
if we place polyatomic molecules at the crystal positions, once again the
basic pattern of lines is obtained, but the intensities will differ in a very
complicated fashion. From the lines and their intensities, it should be
possible to make deductions about the structures of the polyatomic
molecules.

To say this more directly, if we make crystals of enzymes (or any
other polyatomic molecules), we obtain patterns of lines and intensities
which apparently can be used to deduce the three-dimensional structure
of the molecules. We say apparently because in practice the difficulty
increases enormously as the number of atoms in the molecules increases.
For biologically important molecules, the unraveling of the structures is
presently a matter of years, even utilizing the fastest of modern com-
puters for the mathematical work and using some important new devices
developed for such studies.

It is not always possible, even theoretically, to deduce the structures of
molecules from the intensities. The theoretical difficulty stems from the
so-called phase problem, which we shall now attempt to picture for you.

Consider part (a) of Fig. 48. We have indicated that the scattering
from the two black atoms is such that crests are leaving them at the
particular instant at which a trough is leaving the uncolored atom, which
we will use as the reference atom. In part (b) the black atoms have been
moved so that they are emitting troughs, but you will notice that at their
old positions, indicated by the dotted circles, crests are present. Therefore
the blackening on the film will be the same in both instances. Thus we
have indicated that a given blackening can be produced by two (and of
course there are many other possibilities) different configurations of the
atoms within the molecule. If we knew the relative phases of the scatter-
ing (whether crests or troughs arc being emitted when the reference atom
is emitting a trough) we could deduce the structure directly. But since
we don't know the phases, we can't know the positions of the atoms.

In parts (c) and (d) of the figure we show a way of getting around
this difficulty. The molecule will be part of many different planes and,

X-RAY DIFFRACTION

97

(a)

(b)

(c)

Figure 48

depending on where they are, the contributions of the two atoms differ
widely. Thus, by combining the results of several different reinforce-
ment situations, it is possible to deduce the correct structure of molecules
containing a few atoms. However, if the number of atoms in the molecule
is very large, as it is for almost all biologically important molecules
whose structure is unknown, it is not feasible to deduce the structure
from the observed intensities alone.

We now turn briefly to a method other than the powder method for
obtaining properly oriented crystals. If a single crystal is placed in the
path of the x-ray beam, it will not in general be at the proper angle to
give reflections. But if the crystal is rotated slowly, it will at some in-
stant be in the proper position to give reinforcement for a particular set
of planes, and the x-rays will form a spot on the photographic film.
When rotated to a somewhat different position, still another set of planes
will be at the proper angle for reinforcement, and a spot will be formed
at a different place on the film. Thus, by rotating the crystal, a set of
spots of various intensities is obtained. The analysis is similar to that
for the powder method, in that the distances between spots are related to
the crystal plane spacings and the intensities of the spots are related to
the structures of the (polyatomic) molecules at the crystal positions.
The materials of biological interest usually do not lend themselves to the
powder method, and other methods, especially rotating crystal methods,
are usually employed.

As pointed out in the first paragraph of this section, it must be
realized that, given the structure, the x-ray diffraction pattern is com-

98 METHODS FOR DETERMINING MOLECULAR SIZE AND SHAPE

Fig. 49. Drawing of a model of the myoglobin molecule. The model shows
the shape of the chain seen at 6 A resolution. The dark disc-shaped region is
the haem group.

putable in a straightforward way. Thus if the structure of the molecule
being studied is either known or suspected, the pattern can be computed
and compared with the experimental pattern. A skilled worker can use
the comparison to make reasonable guesses about what modifications
may be needed to make the theoretical pattern agree better with the
observed pattern. Thus, by a series of trial and error modifications, it is
frequently possible to obtain a structure which fits the data very well.
The phase problem still exists, but known bond lengths and angles and
auxiliary information from infrared spectroscopy and other chemical
means can frequently decide uniquely among alternatives.

Through various clever stratagems and much hard work, the structures
of several complex molecules of biological interest have been obtained
in recent years. One of these is sketched in Fig. 49. The number of
molecules whose structures have been determined is too small for any
generalizations about three-dimensional structures to have emerged. But
despite the complexities and the time needed for such x-ray diffraction
studies of molecular architecture, there is rapidly increasing interest in
these methods because at present the information can be obtained in no
other way.

8

CHAPTER >< Isotope Methods

INTRODUCTION

There are two kinds of isotopes: radioactive and nonradioactive. Both
kinds have important roles in biological studies. They may both be
used as tracers in a given substance during its uptake and metabolism
by any organism, and also in test-tube syntheses. Radioactive isotopes
have two further uses. First, they may be used as indicators of localiza-
tion within cells or within organisms without having to take the organism
apart. Second, their disintegration may be used to effect an alteration
in the functioning or the structure of biological systems. We will give
examples of all three uses later in this chapter.

Some isotopes have a special use because they occur chiefly in one or
a few types of biologically important compounds. The following table
gives some examples, along with the lifetimes of the radioactive isotopes.

Isotope Primary occurrence in Lifetime (half-life)

Carbon- 14
Phosphorus-32
Tritium-3
Iodine-131
Sulfur-35
Nitrogen- 15
Oxygen- 18
Deuterium-2

everything

nucleic acids

everything

thyroid compounds

proteins

most compounds

most compounds

everything

5700 years
14.3 days
12.5 years
8.1 days
87.1 days
stable
stable
stable

Even though a number of these isotopes are seen to occur in many
biological compounds, they may nevertheless be used in a specific way
by synthesizing special compounds containing them. For instance, if
thymidine is synthesized with tritium, it can be used to study nucleic
acids, because thymidine is known to be incorporated almost entirely
by deoxyribonucleic acid. The same isotope could be used to study RNA
by using uridine which had been synthesized in the presence of tritium.

The basic relation for the survival of undecayed radioactive atoms
may be obtained by methods readily available to us. If there has been

99

100 ISOTOPE METHODS

an average of d disintegrations, then the zero class of the Poisson dis-
tribution lets us write the fraction of atoms in which there has been no
decay as

N_ -d
No" 6 ■

Since the number of disintegrations increases steadily with time, it may
be written as

d = \t.

Thus the final expression for the survivors of radioactive decay is

_N_ -u
No" 6 '

In a time equal to 1/X, the surviving fraction is e _1 , or 37%. For purely
historical reasons, 1/X is not used directly as the index of the average
lifetime of these atoms, but rather the index is taken as the time to
reach 50% survival. From the Poisson formula, this value may be shown
to be related to X by the expression

0.693
T= -A-'

where r is the time for half of the atoms to disintegrate; it is r, the
half-life, which is listed in the table above.

The units in which radioactivity are measured are, basically, the
number of disintegrations per second, chosen because it is proportional
to the number of radioactive atoms present. Physicists chose for the
unit of activity that of one gram of radium: 3.7 X 10 10 disintegrations
per second; this unit is called a curie. For biological purposes, this unit
is between 1000 and 1,000,000 times too high to be practical, and you
will find reference to millicuries and microcuries.

In any given experiment, the total number of disintegrations per
second need not be the most important aspect, however, for one can have
the same number with different total amounts of the chemical species.
For example, the fraction of all carbon atoms which is radioactive could
vary from 100% downward (if pure isotope were being used). The frac-
tion of atoms which is radioactive is ordinarily expressed in terms of the
specific activity of the substance, i.e., disintegrations per second per
gram, although in biological studies it is more usual to express it in terms
of disintegrations per minute per microgram of substance.

There are difficulties involved in the detection of disintegrations, and
the number registered by the counting apparatus is generally consider-

TRACER EXPERIMENTS 101

ably less than the total number that actually occur. As a result, activity
has to be corrected for the efficiency of the counter (by using a standard
source of radioactivity), or else the results are given as counts per min-
ute per microgram, which is entirely valid so long as only relative radio-
activity levels are necessary for the conclusions of the study.

1 . Tracer experiments

(a) One of the most significant metabolic studies using the tracer
aspects of isotopes has been the determination of the pathway of uptake
of C0 2 by plants. In these experiments, C 14 2 was bubbled into chambers
containing photosynthesizing algae, and the algal metabolism was
stopped after various times by taking aliquots and placing them in
metabolic poisons. The algae were then fractionated by standard bio-
chemical techniques, and the amount of radioactive carbon in the vari-
ous compounds was determined. In this fashion it was found that the
carbon first appears in the 3-carbon molecule of phosphoglyceraldehyde.
The subsequent metabolic journey could be deduced by noting which
compounds contained the tracer in the next aliquot, etc. Finally, a sub-
stantial proportion of the metabolic pathway was inferred. The pathway,
of course, then had to be checked by establishing the existence of the
requisite enzymes and of energy available in the proper amounts and in
the proper forms (i.e., in the proper energy-rich compounds) to effect the
chemical conversions indicated by the tracer study.

(b) An equally significant genetic study was carried out using sulfur
and phosphorus labeling of bacteriophages. At the time of this study
(1952) it was not known with compelling certainty that the nucleic
acids carried the genetic information for the entire organism. Hershey
and Chase grew two batches of phages, one in S 35 , the other in P 32 . They
infected unlabeled bacteria with the S 35 phages, and at various times
after infection, aliquots were taken to be placed in a blender for study.
Operation of the blender was shown to strip phages from the bacterial
surface. In the experiment, about 95% of the sulfur-labeled protein
could be stripped at all times during the latent period. If the blenderiz-
ing was done just after adsorption of the phages (within 2 or 3 minutes)
the number of cells yielding progeny bacteriophages was very small. By
about 3 minutes after adsorption was initiated, the stripping had no
effect on the infective process; all cells yielded progeny phages. This
experiment showed that the protein is almost irrelevant (to within about
5% of the total protein), since normal phage synthesis could ensue with-
out the presence of about 95% of the protein.

When this experiment was repeated with the P 32 phages, the amount
of strippable P 32 decreased steadily from 100% when the phages were

102 ISOTOPE METHODS

first added to practically 0% after about 3 minutes of adsorption; the
phosphorus after that time was always found firmly attached to, or
within, the bacteria. Here again, after 3 minutes there was 100% in-
fection of the cells. But during the first 3 minutes, the percentage of
cells yielding progeny phages was strictly equal to the percentage of the
P 32 which could no longer be stripped. Thus, since the P 32 is contained
entirely in the DNA of the phages, it was deduced that the nucleic acid
was all injected into the cells within the first 3 minutes, and after that
time the S 35 -labeled protein was all that remained on the bacterial
surface. Thus the injected nucleic acid was able to replicate itself and
the normal protein coats of the phage. The genetic information and the
control of metabolic functioning were thus shown both to reside entirely
in the DNA (except for the possible 5% of the protein which was not
strippable).

(c) The lifetime of human red blood cells has been determined through
the use of the nonradioactive isotope N 15 . The amino acid glycine was
synthesized with the isotope, and fed to human beings. This amino acid
is incorporated into the protein hemoglobin, so that within a few days,
there was a significant amount of N 15 in the red blood cells of the sub-
jects. The amount of N 15 in the red blood cells began to show a signifi-
cant decrease only after about 100 days, and reached half its peak value
after about 140 days. More refined calculations, taking into account
the time required to build the N 15 -containing glycine into the hemoglobin
molecule, yield a result of 127 days for the average life span of human
red blood cells. The fact that the content of N 15 drops, as expected from
the complete loss of label, leads to the conclusion that very little, if any,
of the hemoglobin molecule is re-utilized for the synthesis of new
erythrocytes.

2. Localization experiments

In these experiments, the information sought is generally of a strictly
qualitative nature, although some quantitative information is not in-
frequently obtained in addition. The most obvious experiments have to
do with cytological studies of localization within cells, for if the cell
is large enough, the information can be obtained by radioautography.

Radioautography involves feeding the isotope to the organism and, at
suitable later times, stopping the metabolism by appropriate agents. The
intact cells are used, if possible, or else the cells are sectioned by the
now-standard techniques for preparation of specimens for light and
electron microscopy. The material is then placed in contact with a photo-
graphic emulsion sensitive to the radiation emitted by the isotope in use.
The material sits in contact with the film for a period found to be

STUDIES UTILIZING DISINTEGRATION 103

sufficient for the emitted rays to have darkened the film adequately.
The film is then developed, and the material is studied with the micro-
scope. If developed grains occur over a particular part of the cell, it
follows that the isotope is located in that part. It is necessary to focus
alternately on the material and on the film to make the identification.

This technique was used by Taylor in a study of the fate of the
chromosome materials during cell division. Taylor fed onion root tips
with tritium-labeled thymidine for a while, and then stopped further
detectable labeling by adding a large excess of unlabeled thymidine. The
cells were allowed to go through controlled numbers of divisions, and
after each division, the roots were sectioned and radioautographs ob-
tained. It was found that grains appeared over both chromatids at
the beginning of the experiment. After all rounds of cell division,
chromosomes usually contained either one labeled and one unlabeled chro-
matid, or else completely unlabeled chromatids. There was some chromo-
some breakage, and in such instances the part of one chromatid which
was unlabeled was matched precisely with an equal length of labeled
sister chromatid. This experiment shows directly the doubled nature of
the chromosome and that the chromatids are retained intact during
mitosis except for some occasional breakage and reunion.

By similar methods it has been demonstrated that strontium is de-
posited almost entirely in bone, while other atoms, such as phosphorus,
are much more generally deposited. The very high localization of iodine
in the thyroid has been amply confirmed by these techniques. Localiza-
tions of this kind are extremely useful clinically. For example, the
administration of radioactive strontium could be of use in treatment of
bone diseases, and treatment of thyroid disorders by giving radioactive
iodine is so standard that thyroidectomy is now very infrequent.

Radioautography is limited for several reasons, the most important of
which is the limitation on the resolution due to the finite size of the
grains in the photographic emulsion and to the fact that the radiations
are emitted in all directions, so that to each source of radiation there is
a much larger zone of blackening of the film.

3. Studies utilizing the disintegration itself

We have already mentioned the use of localization of various atoms in
the clinical treatment of disease. There has been important use of dis-
integration in fundamental biological studies. One of the most significant
series of studies has been on the effect of P 32 decay on the course of
virus infection. It was first shown that P 32 -labeled phages, kept in a
refrigerator, exhibited a progressive decrease of viable phages. By plac-
ing unlabeled phages in a solution of P 32 which would give them about

104 ISOTOPE METHODS

the same amount of irradiation, it was shown that this decrease was
due to the disintegration and not to the bombardment of the other
phages by the emitted beta particles. This kind of experiment is
called a reconstruction experiment, and is frequently used to test hy-
potheses. In the picturesque jargon appropriate to the experiment, it
was shown that very little "murder" occurred in the reconstruction
experiment, so that, by inference, the deaths of the P 32 -labeled phages
had to be due to "suicide" coming from the disintegration of P 32 which
had been incorporated into the phages.

These P 32 phages were used to infect unlabeled cells in an unlabeled
medium, and at various times many aliquots were taken. The aliquots
were frozen rapidly, using liquid nitrogen as coolant, and stored at
liquid nitrogen temperature (about — 200°C). Each day an aliquot of
each infection time was warmed, and the number of cells yielding phage
progeny was determined. In this way, for instance, it was found that
after one minute of infection, the ability to yield phage progeny de-
creased at the same rate as the viability of the free phage particle. That
is, the injected nucleic acid had apparently not been altered at all, for
it suicided at the same rate as it would have suicided outside the cell.
After about one-third of the latent period, the cell was completely in-
sensitive to the suicide, in that there was no decrease at all in the number
of cells yielding viable phage progeny. The obvious interpretation is
that by this time in the latent period the nucleic acid has replicated
itself out of nonradioactive materials, as is necessarily the case since
there are only nonradioactive materials present.

A complementary experiment was made using nonradioactive phages
to infect radioactive cells in a radioactive medium. In this case, the
cells were always stable to disintegrations, since the infecting phage
contained no radioactive atoms. The stability of the phage-yielding is
additional evidence that "murder" makes very little contribution to the
total effect, else the nonradioactive phage DNA would have been killed.

It is of great interest that the designers of this experiment decided
to do a complete experiment, however foolish it may have seemed before
the results were known. They infected radioactive cells with radioactive
phages in a radioactive medium. Since all nucleic acids are now radio-
active, it is evident that there can never be any stabilization of the
cell's ability to yield phages. Nevertheless, when the experiment was
made, a stabilization very similar to that in the first of the experiments
was found: when about one-third of the latent period had passed, there
was no decrease in the number of cells yielding viable phage progeny!
This experiment remains to be satisfactorily interpreted.

The suicide method has been used also in studies of bacterial growth
and function. In addition, suicide has been found to occur when atoms

STUDIES UTILIZING DISINTEGRATION 105

other than phosphorus were used as the source of the disintegration. The
mechanism of suicide is only incompletely known. Using P 32 , when
single-stranded RNA and DNA viruses are labeled, it is understandable
that death will ensue because the nucleic acids are held together by a
phosphorus backbone which is broken by the recoil of the S 32 to which
the P 32 is transformed by the disintegration. For double-stranded DNA,
the efficiency of the suicide is much less (about 0.1 efficient) and is
probably due to the fraction of times there occurs a scission of the
opposite strand originating from the recoiling S 32 atom.

9

CHAPTER Vj Radiobiology

INTRODUCTION

Radiation is used in biology in several ways. First, light of various
visibLe, wavelengths is used as a source of energy for biological functions.
Second, as detailed in the section on light absorption effects, light may
be used to analyze such diverse subjects as the pigments responsible for
the reception of biologically important light and for analysis of the con-
centration and shape of molecules. The two kinds of light which have
not yet been discussed here are the ionizing and the infrared radiations.
The latter, since they have thus far given few if any biologically impor-
tant results, will not be discussed. The former include short wavelength
ultraviolet radiation (which has been briefly treated) and the radiation
composed of the physical particles: electrons, neutrons, protons, alphas,
and photons in the x-ray and gamma-ray energies. These particles have
the distinguishing property of being so energetic that they are easily
able to ionize one or more of the atoms in the organisms, thereby creating
a disturbance great enough to break one or more chemical bonds. A
drastic effect might then be expected to ensue, in terms of the alteration
of the biological integrity of the irradiated material. Indeed, most
studies with these ionizing particles have concerned themselves with the
killing of the target organisms. We will see, however, that the particles
can do many other things.

The fundamental mechanism of action of these ionizing radiations
rests on the ejection of an electron, leaving an ionized atom or molecule
behind. Since the outermost electron is most weakly bound to its atom
or molecule, it is this electron which is usually ejected. And, since it is
this same outer electron which is involved in chemical bonds, the bond
is readily broken by the ionization.

The ejection of the electrons takes place by two mechanisms, depend-
ing on the type of particle involved. Electrically charged particles
(electrons, protons, and alphas), when flying past an atom or molecule,
exert an electric force on the electrons, and thereby effect their ejection.
The uncharged particles (x-rays, gamma-rays, and neutrons) then must
have a somewhat less direct mode of action. For those who have studied
physics, it is only necessary to say that the photoelectric effect and the

106

INTRODUCTION 107

Compton effect play the major role for the photons. For those who
have not studied physics, it may suffice to say that the photons, acting
like billiard balls, strike the electrons and knock them out of their orbits.
Even if an inner electron is ejected, the result will normally be an
ionization and breaking of bonds because the electrons outside the orbit
of the ejected electron will now cascade inward, and finally the outer-
most (bonding) electron will fall into an inner orbit and the bond will
be broken. Neutrons act primarily also in a billiard-ball fashion, hitting
protons (hydrogen nuclei) which have an almost identical mass, and
knocking them out of their places. These protons now serve as agents
of charged-particle irradiation of other atoms and produce the effects
observed.

In all these instances, the ejected electrons (and the occasionally
occurring photons) will also produce effects, termed secondary effects,
which must be taken into account in any complete reckoning of the
results of irradiation.

Up to this point, we have treated only direct action of radiation on
the biological materials of interest. Radiation experiments are usually
done on organisms suspended in water. Also, even single cells are pri-
marily water in makeup, and complicated organisms contain internal
fluids within vessels. Therefore, if radiations can produce effects on
water, there is another source of biological effects if the products of
water irradiation can react with the biological materials.

There has been extensive study of the effects of radiations on water.
Water has been split into two kinds of products. The ions H + and OH~
are the well-known kind. The kind less known to students in their intro-
ductory studies of science is known as the free radical. An atom of
hydrogen is not normally found alone in nature, because it is too reactive
a substance to endure as such very long. Indeed, the stable elementary
form of hydrogen is the molecule with two atoms of hydrogen. The
hydrogen atom or radical is not normally free, so that when it is found
uncombined, it is called a free radical. This particular radical, H, differs
from H + in that it has an electron in orbit about it, and is therefore
electrically neutral. A more general definition of free radicals is based
on the electrically neutral form of a normally electrically charged atom
or group of atoms. Thus any neutralized ion would be termed as free
radical. And, if this is not its normal state, it is very reactive chemically.

Water can be split into its component free radicals: H and OH. The
latter is a powerful oxidizing agent, pulling electrons strongly to itself
in order to make itself into the stable OH - ion. Thereby it will break
chemical bonds and produce consequent biological effects. Experiments
have shown that these free radicals are indeed produced by ionizing
radiation, so that indirect effects of radiation may be expected, due to

108 RADIOBIOLOGY

the migration of these radicals to the biological materials. If the purpose
of the irradiation is to produce a biological effect, it makes no difference
that there are both direct and indirect actions. If, however, we wish to
study some property of the biological substance itself, it is useful to
restrict the situation to one of direct actions only.

This can be done in several ways, the most important of which is to
add biologically inert materials to the solution of organisms in such
concentration that the diffusing radicals and ions are much more likely
to encounter the added materials than the organisms in question. For
example, if we add glycerol to a solution of viruses, a not atypical
situation would find 0.01 M glycerol and 10 13 viruses per liter. The
glycerol would then be present in a concentration of 10 21 molecules per
liter, or 10 8 more concentrated than the viruses. So, neglecting the size
difference between virus and glycerol, any diffusing radical or ion will
encounter (and give up its energy to) 10 8 glycerol molecules before a
single virus will be affected, on the average. Thus the direct hits on the
viruses will far outnumber the indirect hits, and so we may neglect
these indirect effects.

In practice, broth, gelatin, and many small sulfur-containing molecules
have been found to be highly efficient protective agents. The protective
action of these agents is still incompletely understood, and forms an
important part of current research in radiobiology.

THE TARGET THEORY

In this section, we assume that only direct effects of radiations occur.
If ionizing radiation impinges on a solution of organisms, the probability
of producing an effect will depend, first, on the probability that the in-
cident particle goes through the organism. Clearly, the bigger the or-
ganism, the greater the probability that the line of flight of the incident
particle will pass through it. Indeed, the probability is proportional to
the projected cross section of the organism. Second, given that the line
of flight is through the organism, there will be an effect only if an ioniza-
tion is produced within it. If the incident particle produces an average of
one ionization every micron of its path, the probability of producing
an ionization within the organism will depend strongly on its thickness.
If, for instance, the organism is 0.1 micron thick, then only one organism
in ten will be ionized. If, on the other hand, the organism is 10 microns
thick, then each organism will be ionized 10 times, on the average. Thus
we are led to an important parameter of radiations: the number of
ionizations per unit of path length. Currently, this aspect of ionizing
particles is described in terms of the LET (linear energy transfer),
which is the amount of energy transferred per unit of path length. This

THE TARGET THEORY

109

X-rays •

Alpha rays-

Low LET

High LET

Fig. 50. A diagrammatic presentation of the difference between radiations.
The densely ionizing alpha rays produce a high linear energy transfer (LET).
The sparsely ionizing x-rays produce a low linear energy transfer. Thus every
alpha ray produces many ionizations within a particle, while a particle would
have to be very large for x-rays to produce at least one ionization in each particle.
More often, the x-ray produces no more than a single ionization (if that) in a
particle of a size found in typical biological experiments.

LET concept differs from the number of ionizations per unit path length
by the factor of the energy transferred per ionization.

For low LET (sparsely ionizing radiations, such as very high-energy
charged particles, x-rays, and gamma-rays) the probability of an ioniza-
tion's occurring within the organism is directly proportional to its thick-
ness, since the thicker the organism the more likely it is that an
ionization will occur before the incident particle has passed entirely
through the organism. (See Fig. 50.) Thus we can formulate the prob-
ability of producing an effect by irradiation as

P = probability that the line of flight goes through the organism
X the probability that an ionization will occur within the
organism.

The first probability, as we have already argued, is proportional to
the cross-sectional area A of the organism, and the second probability is
proportional to its thickness t. The total probability is proportional to
their product, At, which equals the volume of the organism. Thus low
LET particles produce an effect which is proportional to the volume of
the organism.

If we utilize radiation at the other extreme of high LET (densely
ionizing particles, such as low-energy charged particles), then, as in
Fig. 50, every incident particle whose line of flight goes through the
organism will produce an ionization and therefore the biological effect
being observed. This amounts to setting the second probability factor
equal to unity. Then the total probability of producing an effect is pro-
portional to the area A of the organism.

Consequently, if we do two separate irradiations with low LET
particles and with high LET particles, w 7 e obtain results whose ratio is

110 RADIOBIOLOGY

Fig. 51. Each shaded block represents a virus particle. The other areas are
volumes equal to the virus particle volume but within which there are no virus
particles.

proportional to the thickness of the organism. In this way we can obtain
estimates of the size and shape of the organism.

If we knew enough about the physics of the incident particles, we
could also deduce the actual volume of the organism. We now look at
what is involved in this deduction. Consider a solution containing some
viruses and imagine that the solution is broken up into subvolumes each
equal to the volume of the virus particle. As indicated in Fig. 51, most
of these hypothetical volumes will contain no viruses, for the virus con-
centration will never be more than, say, 10 12 per ml, which is approx-
imately 10 -8 molar. Suppose next that we shine enough sparsely ionizing
(low LET) radiation so that we have created an average of one ioniza-
tion in each of these small compartments. This situation is entirely
similar to the one we have previously dealt with in our discussion of
biometry, when we were expounding the Poisson distribution. When
there is an average of one ionization per compartment, there will be
e~ l = 0.37 of the compartments which have no ionizations; the others all
have one or more ionizations per compartment.

If the viruses are randomly placed in the solution, as of course they
actually are, then 37% of the viruses, too, will escape ionizations. Thus
the dose which leaves 37% of the viruses in a viable condition will be
that dose which gives an average of one ionization per compartment the
size of the virus. The way in which the 37% dose is actually found is
by giving a series of doses, measuring the surviving fraction of viruses
at each dose, and then interpolating to the 37% survival values, as in
Fig. 52.

If we know the physics of the radiation, we can actually calculate the
radiosensitive volume of the virus. Physicists have measured the ionizing

THE TARGET THEORY

111

20 40 60 80

X-ray dose in units of 1000 roentgens

Fig. 52. The x-ray inactivation of bacteriophage alfa. The dose for produc-
ing 37% survival is found by the dashed lines to be about 23,000 roentgens.

properties of all the radiations we have mentioned earlier, and the num-
ber of ionizations produced per cc of substance is a number which can be
found by looking in the proper books on radiation biology. It depends
on the nature of the substance, for it depends on how many electrons are
contained per unit volume. For typical biological material (wet tissue)
there will be something like 10% hydrogen, 12% carbon, 4% nitrogen,
73% oxygen, 0.1% sodium, 0.2% phosphorus, 0.2% sulfur, 0.35% potas-
sium, 0.1% chlorine, 0.04% magnesium, and 0.01% calcium. Knowing
this, it is possible to compute the number of ionizations produced per cc
by each kind of ionizing particle. The unit of dose was chosen by the
physicists, who called the amount of radiation that will produce 1.8 X 10 12
ionization per cc in air one roentgen. Note that the unit is expressed as
ionizations per cc. For these sparsely ionizing radiations, such a unit is
reasonable, since we have indicated that the probability of producing an
ionization is proportional to the volume traversed. For substances such
as water or normally occurring organisms, the density of the substances
is about 1000 times greater than that of air; therefore this density
factor must be taken into account in computing the number of roentgens
delivered to the substance.

The upshot of the physicists' calculations is that the number of ioniza-
tions per cc of substance can be computed from the nature of the radia-
tion and of the substance. Thus, suppose that the dose given to produce
37% survival of our viruses corresponds to 3 X 10 18 ionizations per cc.
If these are distributed so as to average one ionization per volume
equal to the virus volume, the virus volume must be 1/(3 X 10 18 ) cc.
Thus, from the biological measurement of the 37% survival dose (D 37 in

112 RADIOBIOLOGY

roentgens) and a knowledge of the ionizations produced by the radia-
tions utilized, we can deduce the volume of the virus.

There are important corrections to be made before the number of
ionizations can be used as in the illustrative calculation above. The
most important is the correction for the clustering of ionizations. It
turns out that the (secondary) ejected electrons produce enough ioniza-
tions near their own origin that the ionizations are clustered in threes,
on the average. This effect depends, however, on the size of the or-
ganism. For large organisms, we would simply divide the total number
of ionizations by three to find the effective number. For very tiny
organisms, it is possible that the clustering is on a scale larger than the
virus diameter, so that we would actually count all the ionizations.
Since this and other corrections require specialized knowledge of the
physics of radiations, we will not deal with them further.

It is extremely important, however, to point out that we have tacitly
assumed that a single ionization anywhere in the virus (or whatever
biological organism is being used) will produce the effect being measured.
This is simply not generally true, any more than it would be true that a
single bullet passing through any part of your body would kill you.
There are parts of any organism which may be damaged without affect-
ing some of the properties of the organism. For viruses, for example, it
turns out that the volume deduced from irradiation experiments is defi-
nitely less than the volume of the virus; it turns out to be equal to the
volume of the nucleic acid of the virus. This is to be expected, since
radioactive tracer experiments have shown that the nucleic acids of
viruses can, by themselves, produce new viruses, complete with protein
coats. Accordingly, we should refine our thinking and our terminology
by talking about the radiosensitive volume — the volume within which
an ionization will produce the effect being studied.

This definition includes a second refinement in our thinking. It is not
at all necessary to study only viability. There have been studies of the
radiosensitive volume associated with many other properties, such as the
ability of the virus to kill a cell, the ability to adsorb to a cell, the
ability of viruses to agglutinate red blood cells, the ability of cells to
make enzymatic adaptations, etc.

We have discussed how sparsely ionizing radiations are used. If we
use densely ionizing radiations, we need a physical device to count the
number of incident particles per cm 2 . Then, as before, we give a series
of doses of radiation, interpolate to the 37% survival point, and find the
number of particles per cm 2 which give this survival. In a way entirely
similar to that used for volume calculations, we here calculate the area
of the target organism or, more accurately, the radiosensitive area. As
we mentioned above, use of the two kinds of radiation in parallel

MULTITARGET ANALYSIS 113

experiments gives us the actual volume, area, and thickness of the target.
From a knowledge of the structure of the organism, it may be possible
to use this knowledge of shape to identify the part of the organism in-
volved in the effect being studied.

Still another way in which these radiations have been employed in-
volves the use of electrons so low in energy that they can penetrate only
small thicknesses of materials. By increasing the energy of these elec-
trons, an energy can be found at which the electrons begin to produce a
particular effect. From the known penetration thickness of such elec-
trons, we then deduce how far under the surface of the organism lies the
portion associated with the effect being studied.

MULTITARGET ANALYSIS

For those who can follow a little algebra, it is possible to show how
the number of targets per organism can be discovered. Such a situation
could obtain, for example, in polynucleate cells. If it is necessary to in-
activate all the nuclei to kill the organism, several targets must be hit
in each cell. Below are given the few mathematical steps needed to make
this analysis for a cell with n nuclei :

Let p be the probability that a nucleus survives a given dose D of

radiation.

(1 — p) is the probability that it does not survive this dose D.

(1 — p) n is the probability that all n nuclei do not survive this dose

D.

1 — (1 — p) n is then the probability that at least one of the n nuclei

survives the dose D.

This last expression then represents the fraction of cells which survive
the dose D because each of such cells has at least one intact nucleus. We
write this as

£-1 -(!-„)",

where N is the number of cells surviving a dose D if there were A T cells
at zero dose.

If there were only one nucleus (n = 1), this would reduce to p, as
expected. When very large doses have been given, the survival prob-
ability p becomes very small. In this case, we can expand the term in
parentheses by the binomial theorem to give

(1 - p ) n = l - np + n(n - l)p 2 /2

If p is very small, then the right-hand side of this expression is ad-

114

RADIOBIOLOGY

equately approximated by the first two terms: 1 — np. Thus the sur-
viving fraction of cells becomes

N_
N

= 1 - (1 ■ - np)

lip.

That is, the surviving fraction is n times what it would have been
for a mononucleated cell. This is entirely understandable, because n
times as much radiation is required to hit n times as many targets.

Since p is the survival probability corresponding to a dose D, if we
express D in terms of the average number of ionizations produced in
the target, we can at once write the expression for p, for p is the fraction
of cells having no hits, if the aver-
age is D hits. This is just the zero
class of the Poisson distribution
again, so

-D

p = e

Thus for mononucleate cells the
surviving fraction is

N_
No

= p = e

-D

or

i N
l ° g N- =

D.

For multinucleate cells, we had

N_
No

np

ne

-D

so that

log

N_

Nn

log n - D.

Both of these expressions have a
slope of —I). As shown in Fig. 53,
where log N/N is plotted against
D for several values of n, the
asymptotic slopes are parallel and
are n times higher than the value
for the mononucleate cell. Thus, if

2 4 6 8 10
Dose in units of 37% survival dose

Fk;. 53. Theoretical survival curves

where there are n equally radiosensitive
nuclei per cell. The curves are plots
of the expression in the text: 1 —
(1 — e~ D ) n , where // is the number of
nuclei and D is the radiation dose in
units of the 37% survival dose. The
dashed-line extrapolations of the
straight portions of the curves give
the number of nuclei as the intercepts
on the zero dose axis.

MULTITARGET ANALYSIS

115

10 20 30

X-ray dose (kr)

40

Fig. 54. Bacterial survival after various x-ray doses. The age of the culture in
hours is listed with the survival curve for bacteria of that age. (From Stapleton,
J . Bacteriology, 20, 357, 1955; courtesy the author and Williams and Wilkins Co.,
Baltimore, Md.)

we obtain a multitarget curve as a result of an experiment, we simply
draw a line of the same slope so as to pass through the 100% survival
point, and we can then calculate the value of n. In addition, from the
mononucleate survival curve, we can find the 37% survival dose, and
then obtain the radiosensitive volume as before.

The value of n can also be found by extrapolating the straight-line
portion of the survival curve back to the zero dose line; the intercept is
the nuclear number itself. That this is true can be seen from the last
equation: log (N/N ) = log n — D.

If we set D equal to zero, we obtain log (N/N ) = log n. This way of
finding n is illustrated in Fig. 53 by extrapolating the n = 2 and n = 10
curves back to the zero dose line.

The student should be warned that there are serious possible com-
plications which can arise in a radiobiological experiment. For example,
there are cases of irradiation of polyploid yeasts in which the target

116 RADIOBIOLOGY

1.0

10"

Lag

Logarithmic

I l_

Maximum
-stationary -

I

2 4 6 8 10

Age of culture (hours)

12

Id 1

Fig. 55. A replotting of some of the data of Fig. 55, showing the cycling of
radio sensitivity. The cell concentration is shown as a dashed line. (From Staple-
ton, J. Bacteriology, 20, 357, 1955; courtesy the author and Williams and Wilkins
Co., Baltimore, Md.)

numbers simply do not follow the ploidy. Another example is the varia-
tion in inactivation curves of bacteria in various stages of their growth
cycle. The experimental results are given in Fig. 54, taken from Staple-
ton's work. This information can be summarized as in Fig. 55, also taken
from Stapleton, which shows clearly the cyclic variation in radio-
sensitivity. This variation is not altogether out of the purview of target
theory, for rapidly dividing cells could well have a lesser number of
nuclei than lagging or stationary phase cells in which nuclear division
may have occurred without corresponding cell division. Furthermore,
the metabolic activity of the nucleic acids of the nuclear apparatus
could also result in different sensitivities at different times in the growth
cycle. However, there still remain the problei is of a change in target
number from unity to six, which seems outside any reasonable increase
in nuclear number.

Opponents of the use of target theory maintain that such failures
show the terrible weakness of the theory, and some even go so far as to
say that target theory should therefore really not be used at all. Our
point of view is that since there are many instances in which target
theory gives well-documented success, there is nothing basically wrong
with the theory, but that where the theory fails it must be due to the

BIOLOGICAL EFFECTS OF RADIATION 117

inapplicability of one or more of its assumptions. From an analysis of
the failures, one should then learn even more about the nature of the
biological entities and about radiobiology.

The target theory may be applied, in part, to studies utilizing ultra-
violet radiation, for these are also energetic enough to ionize atoms. The
restriction stems from the fact that although the probability of an ultra-
violet photon's being absorbed by matter is proportional to the electron
density (and thus to the volume of the matter), there is no theoretical
connection between the radiosensitive volume and the volume of any
substantial portion of the organism involved. The photon's absorbing
element is really the electron taking part in a bond, and its volume is not
greatly dissimilar to that of the bond itself. Therefore, although a vol-
ume can be computed, it doesn't teach us anything. Furthermore, if we
break our substance into pieces, the absorption probabilities of the pieces
do not total to the probability of the united substance. In sum, then,
one cannot compute a volume from ultraviolet inactivation data. How-
ever, since the action appears to be due to a single ionization process,
the kind of target analysis we have presented may still be applied to
deduce the number of targets. Also, from the asymptotic slopes of the
inactivation curves, we can say whether the targets are the same or not.

BIOLOGICAL EFFECTS OF RADIATION

Up to this point, we have been analyzing the physical action of
radiations on biological systems. There remains the subject of the bio-
logical effects. We shall discuss the following examples of biological
effects: mutation, ultraviolet-light inactivation and reactivation.

1 . Mutation

Since the initial demonstration by Muller that x-rays can induce
mutations, an extensive body of literature has developed on this topic.
Aside from the straightforward use of radiations to induce mutations
for the sake of obtaining mutant organisms, the kinds of mutations in-
duced permit deductions as to the nature of mutation itself. X-ray
mutagenesis has been found to include chromosome breaks, transloca-
tions, and other cytological effects which offer strong evidence that the
x-ray simply breaks the chromosome. Ultraviolet light mutagenesis has
recently been given an enormous stimulus by the finding that physiolog-
ical doses effect the formation of dimers of one of the four bases making
up nucleic acids. In such a situation, as the nucleic acid is copied during
its replication, the copying mechanism is very likely to err when it
reaches the dimer, thereby producing a mutation. Such a mechanism is
restricted to the locations having two identical bases in series along the
nucleic acid chain. It is not known whether x-rays can also act this

118 RADIOBIOLOGY

way, but since a respectable fraction of the x-ray energy is dissipated
in amounts similar to those of ultraviolet light photons, it is not im-
probable that it sometimes does.

Since nucleic acids have been shown to include double-stranded struc-
tures as the major component of the genetic apparatus, it is conceivable
that one strand could be affected without at all altering the other strand.
Then, when copying of both strands ensues, if the copying mechanism
reaches an ionized base it could make an error by inserting one of the
other three bases, thereby producing a mutation.

2. Ultraviolet-light inactivation and reactivation

The formation of standard chemical bonds across the two strands of
the DNA would surely provide adequate impedance to DNA replication,
thereby inducing death of the organism at the time of its attempt to
replicate the DNA. This picture is supported by the finding that UV-
inactivated DNA does not separate into two strands on prolonged heat-
ing, indicating a "sewing together" of the strands. In addition, dimeriza-
tions within a strand, as discussed above, can lead to mutation, and some
of these may be lethal mutations in that they affect an indispensable
function of the DNA.

Two types of reactivation of UV inactivation have been discovered.
The first type is called photoreactivalion. If UV-inactivated organisms
are subsequently exposed to visible light, a substantial fraction of the
organisms is found to be viable again. In microbial systems, this has
been shown to be due to the light supplying the energy needed for a
photoreactivating enzyme to work. This enzyme has been greatly puri-
fied, and it is found to work in the test tube. If DNA is treated with
UV, as mentioned above, it no longer separates on being heated, but if
it is then treated with the photoreactivating enzyme (in the light), its
separability is restored.

The second type of reactivation is potentially more instructive, al-
though its promise has not yet been realized. Bacteriophages inactivated
by UV have been shown to be able to cooperate to produce live progeny.
Specifically, cells infected with two or more UV-inactivated phages are
found to yield viable phage progeny, even though each phage, by itself,
could produce no progeny. Presumably, the phage DNA which is in-
jected into the cells can undergo a reaction which allows the DNA copy-
ing mechanism to produce one viable copy by copying the good parts of
the two UV-damaged phages. Such an interaction should provide us
with great insight into the functioning and replication of DNA, but thus
far there has been no really successful theory of reactivation in multiple
infection or, as it is normally called, multiplicity reactivation.

References

GENERAL REFERENCES

Biophysical Science, by Eugene Ackerman. Prentice Hall, Inc., Englewood
Cliffs, N. J., 1962. This is the best general textbook to appear to date. There
are some limitations which are undoubtedly due to the fact that a single author
has attempted to cover the entire broad scope of biophysics.

Molecular Biophysics, by Richard Setlow and Ernest Pollard. Addison-
Wesley Publishing Co., Inc., Reading, Mass., 1962. Some parts of this text are
very clever and illustrate the point of view of biophysicists very much better
than does Ackerman's text. However, some important concepts are not defined,
and the text is less up to date and less inclusive than Ackerman's.

Medical Physics, edited by 0. Glasser. Year Book Publishers, Inc., Chicago.
This collection of short articles is intended for those whose background is not
greatly different from that of beginning science students. It includes many
aspects of science which are within its intended scope of medical physics as
distinguished from tiophysies.

SELECTED REFERENCES

Chapter 1

Errors of Observation and Their Treatment, by J. Topping. Reinhold
Publishing Corporation, New York, 1958. An excellent short treatment of many
aspects of statistical analysis. Students should own this little book, which
will serve them well throughout their science careers.

Facts from Figures, by M. J. Moroney. Penguin Books Inc., Baltimore,
1958. A wordy (but interestingly written) exposition of the ideas of elementary
statistics.

Chapter 2

So far as I know the only book that treats the material discussed in this chapter
on the desired level is the text by Setlow and Pollard. However, the mathe-
matical and physical discussions of the topics is beyond the background of most
beginning students. A much more suitable treatment will be contained in a book
being readied for publication by Addison- Wesley: Biophysical Principles of
Structure and Function, by Fred M. Snell, Sidney Shulman, Richard P. Spencer,
and Carl Moos.

Chapter 3

There are good discussions of this topic in both Ackerman, and Setlow and
Pollard.

Vision and the Eye, by M. H. Pirenne. Chapman and Hall, London, 1948.
An excellent book for an introduction to a substantial fraction of the work in
this area. It is now a little out of date, but is recommended both for its content
and for the interesting presentation of an interesting subject.

119

120 REFERENCES

Chapier 4

There is a good presentation of this topic in Ackerman.

Theory of Hearing, by Ernest G. Wever. John Wiley and Sons, Inc.,
New York, 1949. This interestingly written book covers the historical aspects
of the subject and then presents the details of current research. Although
somewhat out of date, its only real defect is the inadequate presentation of the
work of Von Bekesy. The level is fine for beginning students.

"The Ear," by G. von Bekesy. Scientific American, August, 1957.

Chapter 5

Both the Setlow and Pollard, and the Ackerman texts have adequate presenta-
tions of these topics. The one in Setlow and Pollard is better written.

Chapter 6

Ackerman and Setlow and Pollard have sections on this subject. Ackerman's
is longer, more detailed, and better written.

Essentials of Biological and Medical Physics, by Ralph W. Stacy, David
T.Williams, Ralph E. Worden, and Rex O.Morris. McGraw-Hill Book Company,
Inc., New York, 1955. This is a general textbook. I have not listed it among
the general references because it contains little more than an outline of most
of the topics I have chosen to present in this monograph. The excellent chapter
on muscle is almost entirely understandable by beginning students.

Chapter 7

There are good treatments of most of these topics in Ackerman, Setlow and
Pollard, and Glasser.

Chapter 8

Both Ackerman, and Setlow and Pollard contain reasonably inclusive, well-
written expositions of this subject.

Isotopic Tracers in Biology, by Martin Kamen. Academic Press Inc.,
New York, 1957. This book, written by one of the pioneers in the field, contains
a wealth of information on all aspects of the subject. It is especially good for
beginning students, who will gain an appreciation of the scope of isotope methods.

Chapter 9

There are good treatments of this topic in both Ackerman and in Setlow and
Pollard. The presentation in the text by Setlow and Pollard is excellent for stu-
dents seeking an inclusive treatment, since it contains an exposition of the
methods used and results obtained in the laboratory directed for a decade by
Prof. Pollard.

Actions of Radiations on Living Cells, 2nd edition, by Douglas Lea. Cam-
bridge University Press, New York, 1955. A classic in the field which almost
created the field when it appeared in 1947. A revision due to the late Douglas
Lea's associates altered very little of the original, as would be expected for a
classic. Many parts are understandable by beginning students, even though the
mathematics and physics may be beyond their grasp.

Index

Absorption spectrum, 56, 63, 64

Acceleration, 26, 90

Acetylcholine, 47

Actin, 71, 73

Action spectrum, 53, 55, 61-65

Actomyosin, 60, 71

Albumin, 77

Arithmetic mean, 1-3

Atoms, 28-33, 35, 38

ATP, 38, 70, 71

Avogadro's number, 55

Bacteriophage, 82, 101-104
Bands (muscle), 59, 71-74
Basilar membrane, 47-51
Biological variability, 41
Birefringence, 55, 58-60

flow, 59

form, 59

intrinsic, 58
Blood, 77, 102
Bond energies (table), 33
Bonds, chemical, 25, 27-36, 38, 58, 59,

covalent, 30, 33, 35, 59

electric, 30

hydrogen, 32, 34-36

ionic, 29, 30, 33

partial ionic, 30
Bonds and energy exchange, 35
Bragg's law, 93
Brownian motion, 36, 80, 85, 86, 91

Carotenoids, 61-63
Centrifugation, 76, 83-92
Cesium chloride, 85, 87, 88
Chi-squared, 12, 16-19
Chlorophyll, 54, 61-63
Chloroplasts, 60
Chromatids, 103
Chromosomes, 103, 117
CO 2 uptake, 101
Cochlea, 47-50
Colorimeters, 79
Compton effect, 107
Conductors, 52
Cones, 39, 44
Correlation, 22-24
Covalent bond, 30, 31, 33, 36
Creatine phosphate, 70
Crystals, 92-97

Curie, 100

Cytochrome, 82

Deafness, 50

Density gradient centrifugation, 85-89
Deviations, 2, 3, 5, 6, 12, 13, 16, 18, 20,
23

least-squared, 3, 20, 31

standard, 4, 7, 10-13
Dichroism, 55, 60, 61
Diffusion, 47, 75-77, 80

coefficient, 10, 82, 83, 91
Dipole, 32, 34, 35
Distributions, 5-10
DNA, 56, 57, 60, 79, 87-90

bacterial, 90

melted, 57

Watson-Crick model, 87
Double refraction (see Birefringence)

Efficiency, 69, 70
7g Electron orbits, 36, 52-54
Electrophoresis, 76-78

filter paper, 78
Electrophoretic mobility, 77
Energy, 25-38, 61-63, 67-70

in biology, 63, 69

electrical, 28, 29, 33

exchange, 30, 31, 36

states, 52

transfer, 53, 54, 61-63

van' der Waals, 32
Errors, 2, 3, 5

compounding, 10, 11

standard, 7, 13
Eye, 39-45

Fluorescence, 53, 54, 61, 62
Force, 25-29, 34, 35, 68, 69

centrifugal, 90

electrical, 27-29, 32, 34, 35, 76, 77, 106

frictional, 76, 90

gravitational, 26

magnetic, 26

viscous, 27, 59

van der Waals, 27, 32
Free radicals, 107, 108
Frequency curves, 4-10
Frequency discrimination, 46-51

121

122

INDEX

Frequency of light waves, 52
Friction coefficient, 90

Gamma globulin, 77
Gamma rays, 90

Half-life, 99, 100
Hearing, 46-51
Helieotrema, 48, 51
Hyperchromic effect, 56, 57

Inactivation spectrum, 64
Infrared light, 106
Insulators, 52
Interference, 46
Internal conversion, 53, 54
Ion clusters, 112
Ionizing radiation, 106-118
Ions, 28-30, 32-35, 107
Isotopes, 87, 99-105

Least-squares line, 20-24
Least-squares principle, 3, 20
Light absorption, 52-65, 79, 106
Light scattering, 78, 79
Linear energy transfer, 108-110

Measurements^ 1-24

independent, 13, 18, 19
Membrane vibrations, 49-51
Metals, 30, 52

Molecular weight, 10, 75, 82-85, 91, 92
Multiplicity reactivation, 118
Multitarget analysis, 113-116
Muscle, 66-74

bands, 59, 60, 71-73

heats, 70

Hill's equation, 68

smooth, 70, 71

striated, 70, 71
Mutations, 14-16, 117, 118
Myosin, 71, 73

Nerves, 41, 46-51

transmission speeds of, 46, 47
Neural sharpening, 40
Nucleic acids, 56, 57, 60, 64, 79, 87-90,
102-105, 112

Ommatidia, 1 1

( Jptical density, 56, 57

Phase, 52, 58
Phase problem, 96, 97
Phosphorescence, 53, 51
Phosphorus-32 decay, 103 105

Photochemistry, 53

Photons, 39, 40, 52, 54, 55, 61-63

Photoreactivation, 118

Photosynthetic activity, 61, 62, 64

Pigment, 61-63

Poisson distribution, 9, 10, 17, 41, 43, 44

zero class, 9, 10, 55, 100, 110, 114
Polarized light, 58-60
Probability, 40, 55
Proteins, 64, 79, 101, 102

Quanta, 52

Quantum physics, 25, 27, 29-31

Radioactivity, 99-105
Radioautography, 102
Radiobiology, 106-118
Radiosensitive volume, 109-112
Red blood cells, 102
Refraction, 57, 58
Replication, 87-89
Resolution, 44, 45
Resonance, 31, 32
Retina, 39, 40, 43-45
Rods, 39, 40, 44
Roentgen, 111

Sedimentation, 10, 76, 83-86, 90-92

coefficient, 91, 92

equilibrium, 85

velocity, 85, 90-92
Solutions, 33-35
Sound, 46
Standing waves, 49
Specific activity, 100
Spectrophotometers, 79
Stapes, 51
Student's t-test, 13
Suicide, 104, 105
Svedberg, 92
Synapse, 46, 47

Thermal equilibrium, 80
Tracers, 101-103

Ultraviolet light, 63, 64, 79, 106, 117, 118

Variance, 4, 7, 12-16
Viruses, 71, 82, 103 105, 112
Vision, 39 45
Volley principle, 51

Waves, 46 52

X-ray diffraction, 72, 92-98
X-rays, 79, 100, 100, 111, 115-118

THE AUTHOR

Hern
Univer
Mic>'
Prof
and
pr-

°iophysics at Brandeis

the University of

.mi, : ty of Pittsburgh.

iter^st are virus genetics

.s c ember of the Bio-

'~ 4 ion for the Ad-

of ' '"•iversity

• is intended t
ology. Its purpo:
.ics into the elementc.
contention that the trend in biolbv >wc.>_

chemical biology, and today's students will be bette.
if they are taught to think in physio-chemical terms
...^ with their first course in biology.

Among the topics covered are statistics (from the point of
view of the problems faced in evaluating measurements);
physical forces and chemical bonds; physics of vision, hear-
ing, and muscles; and biophysical methods.

AODISON-WESLEY PUBLISHING COMPANY

READING, MASSACHUSETTS • PALO ALTO • LONDON • DON MILLS, ONTARIO

TRINTED IN U.S.A.