Fluctuating and deterministic intramolecular motion in proteins

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Fluctuating and deterministic intramolecular motion in proteins
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Christoph, Garrott William, 1946-
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Proteins -- Molecular rotation   ( lcsh )
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Thesis (Ph. D.)--University of Florida, 1981.
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Includes bibliographical references (leaves 163-166).
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by Garrott William Christoph.
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Typescript.
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Full Text










FLUCTUATING AND DETERMINISTIC
INTRAMOLECULAR MOTION IN PROTEINS












BY


GARROTT WILLIAM CHRISTOPH


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


1981































To my wife, Paulette














ACKNOWLEDGEMENTS


At this stage in one's life, it is appropriate to reflect upon and

acknowledge others who have contributed to one's achievements. It is with

the greatest possible pleasure that I do this now.

As with all their children, my parents invested a quarter century of

effort and care in me. In innumerable ways, understood by few other in my

experience, they provided us with roots to grow and wings to fly. My wife,

Paulette, is a sensible, clever woman of humane instincts. She has provided

me a solid foundation on which to rest at times when I needed it. I hope I

have done the same for her. Our son, Geoffrey, is a keen, mirthful child.

He is a particular delight to us. I dearly love them all.

There are some teachers to remember. In my primary years, I particu-

larly benefitted from the attention of Mrs. Noble and Mrs, Hays. I fondly

recall the memories of several secondary instructors: Marcelete Welker,

Marion T. Page, Dan Burton, A. Vestal, and James Majors. These teachers

variously provided their students with high standards, encouragement, dis-

cipline, inspiration, enlightenment, and independence.

In recent years I have been strongly influenced by several university

teachers and scholars. Dr. Harry McCloud taught me chemistry. Dr. Hitchens

and Dr. Kahn taught me something about American history and modern economics.

Dr. John M. Davis showed me that biology can be a science; more important,

that science is an adventure and a human enterprise. My education in physics

has been at the hands of some first rate people. Drs. Jim Dufty, Dwight








Adams, Chuck Hooper, Ralph Isler, and Tom Scott taught me courses with

clarity and precision. Dr. Benjamin Dunn did the same thing in enzyme bio-

chemistry.

A special word should be reserved for my friend and advisor, Professor

Raymond Pepinsky. He taught me how a physicist should think about biology,

He taught me how to have and criticize my own independent ideas, He showed

me how to recognize useful work in others. He taught me about music. He

helped me to work independently.

From time to time I have been befriended and helped by many others.

In one way or another I am indebted to them all. A partial list includes

Dr. Stanley Ballard, Dr, Dennis Schmidt, Dr. David Jankowsky, John R. Cavedo,

Dr. Mike Houst, D. Duffield, and Dr, Arthur Broyles.














TABLE OF CONTENTS


ACKNOWLEDGEMENTS

ABSTRACT

CHAPTERS


Page

iii

vii


DETERMINISTIC AND FLUCTUATING MOTION IN PROTEINS:
A BRIEF SURVEY

Introduction

Intramolecular Motion in Proteins

FLUCTUATING MOTION IN PROTEINS: A CRITICISM

Macromolecular Fluctuations in the Biological Milieu

Biochemical Fluctuations as Mechanism: A Review

Acceptable Roles of Thermodynamic Fluctuations in
Protein Mechanism

DETERMINISTIC AND FLUCTUATING MOTION IN PROTEINS:
A MODEL

Thermodynamic Fluctuations in Proteins

Protein Intramolecular Motion is Classical Physics

Can Thermodynamic Fluctuations Perform Chemical Work?

The Magnitude of Thermodynamic Fluctuations in
Proteins

The 6N Problem

The Enzyme as Maxwell's Demon

The Dynamics of Intramolecular Protein Motion: The
Langevin Equation

The Dynamics of Intramolecular Protein Motion: The
Fokker-Planck Equation

Conclusions


I










APPENDICES

A

B





C


D

E

F


G


THE LAW OF LARGE NUMBERS

A GENERALIZED ESTIMATION OF THE MAGNITUDE OF ENERGY AND
VOLUME FLUCTUATIONS IN PROTEIN

Legendre Transformations

Fluctuations in Proteins

THE UPPER LIMIT TO THE NUMBER OF DEGREES OF FREEDOM
CONSISTENT WITH CATALYTIC ACTIVITY IN A PROTEIN

SOLUTION TO LANGEVIN EQUATION FOR A BROWNIAN PROTEIN

THE FOKKER-PLANCK EQUATION

THE SOLUTION TO THE FOKKER-PLANCK EQUATION FOR A
BROWNIAN PROTEIN

SOLUTION OF THE FOKKER-PLANCK EQUATION FOR A BROWNIAN
PROTEIN COUPLED TO ELECTRIC FIELD FLUCTUATIONS IN AN
IONIC SOLUTION


BIBLIOGRAPHY

BIOGRAPHICAL SKETCH


Page


98


102

103

104


117

122

133


139



153

163

167














Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

FLUCTUATING AND DETERMINISTIC
INTRAMOLECULAR MOTION IN PROTEINS

By

Garrott William Christoph

December 1981

Chairman: Raymond Pepinsky
Major Department: Physics

There is debate within the biophysical community regarding the role

that structural fluctuations play in proteins and their surrounding aque-

ous environment. Certain investigators have argued that intramolecular

Brownian motion drives the mechanism of biomacromolecules; they contend

that the unique structures of protein, and perhaps other large molecules,

have evolved so that they advantageously extract energy from the thermal

heat bath in order to prepare the transition state and to impel the protein

mechanism forward. A theoretical model is constructed permitting an analysis

of the proposed mechanism. The model has three purposes. We derive some

thermodynamic consequences of the model, and the results imply the unaccept-

ability of the hypothesis that enzymes can be driven by fluctuating mech-

anisms. Essentially, these schemes fail to account for enzyme and protein

activity because they violate the second law of thermodynamics. Using the

model, we obtain certain kinetic consequences of both fluctuating and deter-

ministic mechanisms of protein intramolecular motion. In particular, a








general first order catalytic rate constant for rate-limiting conformational

changes is derived. These results suggest experimentally measurable quanti-

tative distinctions between deterministic and fluctuating intramolecular

motion. Finally, catalytic rate constants are found to have a reciprocal

solvent viscosity dependence when the configurational changes are a rate-

limiting step in the protein mechanism.

A brief review of intramolecular motion in proteins is presented; a

discussion of the limits that the second law imposes on protein mechanisms

is given; and a more critical discussion of specific studies of fluctuational

motion in protein is undertaken,


viii














CHAPTER I
DETERMINISTIC AND FLUCTUATING MOTION IN PROTEINS: A BRIEF SURVEY


Introduction

For nearly a century biochemists have realized that their discipline

entails more than an understanding of stoichiometry; they have realized

that questions of a macromolecular structure are inextricably linked to

the problem of macromolecular function, Recently this view has been

broadened. It has become clear that in addition to structural topography,

structural dynamics also plays a key role in macromolecular function.

Intramolecular motion is known to occur in biomolecules. Part of this

motion is chaotic Brownian motion in response to thermal agitation and

part is concerted or deterministic motion executed in response to internal

forces within the molecular complex itself. Both kinds of motion have been

objects of serious study in the last decade, Both kinds of motion will

variously account for certain experimental observations.

Two things seem certain about the problem of intramolecular motion in

biology. Ignorance of molecular dynamics is the major impediment to a

detailed understanding of macrobiopolymer mechanisms, The detailed eluci-

dation of this problem will lead to a "first principles" understanding

of the problem of protein mechanism. It is also clear that this problem

is very difficult. The effort required to understand the phenomenon

will be at least as great as the work invested in the first protein x-ray

structures,







This dissertation will address two elements of the problem. There is

a minor controversy among practitioners of molecular biology. A few people

argue that a protein mechanism uses thermodynamic fluctuations, rather than

the concerted control of energy and configuration, as the macromolecule exe-

cutes its function. This theory of fluctuating mechanism has been applied to

enzymes, transport, and contractile proteins. Sometimes these arguments only

exaggerate the possibilities of thermodynamic fluctuations in protein mechan-

isms. There is, however, a significant body of recent literature, accepted

as respectable, that is exaggerated in its assertion of fluctuating mechanisms.

The purpose of this dissertation is to address the question of fluctua-

ting mechanisms in proteins afresh, We will pursue two approaches. We pro-

vide a criticism of present theoretical models involving fluctuating mechan-

isms, and we will construct our own simple theoretical model of intramolecular

motion in proteins. With this model, we can theoretically distinguish between

mechanisms involving deterministic and fluctuating intramolecular motion in

protein. This is done by making quantitative estimates of the average relax-

ation and excitation time, as well as rate-limiting catalytic constants for

both kinds of motion. This model also suggests a quantitative experimental

relationship between intramolecular motion rate constants and solvent viscosity.


Intramolecular Motion in Proteins

Hemoglobin is a transport protein. It has been designated as an

honorary enzyme because its functional complexity mimics elements of pre-

sumed enzyme mechanism. It is the best understood protein (Perutz, 1978).

Its structure consists of four nearly identical polymers arranged tetra-

hedrally. Each subunit houses a heme structure which binds molecular

oxygen and communicates its liganded state by allosteric interactions with

the other monomers. Binding of four molecular oxygens to hemoglobin is








highly cooperative. The difference in oxygen binding affinity is several

orders of magnitude from the first to the last addition, This occurs

cooperatively with no direct interaction among the four oxygen binding

sites. The implication is that conformational change within the molecule

is responsible for this cross-molecular communication. When crystals of

deoxyhemoglobin are exposed to air, they shatter; this is in response to

conformational changes within the protein upon binding, Emerging from

x-ray crystallography is the view that oxyhemoglobin and deoxyhemoglobin

have divergent structures. Upon binding, one pair of the four subunits

slips and rotates some fifteen degrees with respect to the other; the

process is under the control of only a few amino acid side-chain contact

points between the segments. This motion is triggered by a shift of the

iron configuration into the heme plane upon oxygen binding; and it is coupled

to the motion of a peptide segment that connects the interaction region

between subunits and the iron moiety. In addition to the motion in response

to oxygen, ligands binding to other sites trigger conformational change.

The binding of hydrogen ions, carbon dioxide, and diphosphoglycerate at

separate surface binding sites influences hemoglobin affinity for oxygen,

There are other functionally important notions in the hemoglobin. The

structure of the protein pocket around the oxygen binding site on heme is

so tightly bound that oxygen cannot enter. This is necessary to protect

the iron atom from oxidation by water. The only way that molecular oxygen

can diffuse to the heme is through structural fluctuations of the protein

itself (Austin et al., 1975). The modern view of this, the most studied

protein, is clear only through an understanding of its intramolecular motion.

This motion has functional elements that are deterministic and fluctuating

in character.








The biophysical chemistry of hemoglobin is taken as the prototype for

the understanding of other protein mechanisms. However, the evidence for

mechanistically pertinent intramolecular motion is not limited to this

protein (Gurd and Rothgeb, 1979). Another system, not so well understood,

is the actin-myosin complex responsible for muscle contraction (Carlson and

Wilkie, 1974). This system undergoes a conformational change as part of

its mechanism. Interdigitating filaments of myosin and actin slide over

one another in response to the contraction of actin-myosin cross bridges

formed between the two. The energy for this work is provided by the hydro-

lysis of adenosine triphosphate (ATP), although the detailed mechanism is

not known. The intramolecular power stroke motion is actually the folding

of the myosin head while in contact with actin. It moves closer to the

helical rod structure below the head.

Evidence for global conformational change in phosphoglycerate kinase

has been obtained (Banks et al., 1979). The structure is a bi-globular

protein with the active site in the adjacent region. There are two x-ray

structures of this protein, one by itself, another bound to ATP. The con-

formation of the enzyme active site and the enzyme bound ATP are signifi-

cantly different from that of the free enzyme and ATP crystal structures.

Clearly, strain is induced into the active site as the two globular domains

rotate some fifteen degrees with respect to each another. If this rotation

does not occur, the ATP remains in contact with the solvent and water will

compete with the phosphate transfer.

Similar substrate induced conformational change has been observed in

other members of the kinase family. The enzyme hexokinase substantially

reorients its globular domains upon binding of its substrate, glucose

(Andersen et al., 1979). X-ray diffraction of alcohol dehydrogenase, a








dimeric enzyme, shows different structural forms when the enzyme is free

and when it is bound with substrate, coenzyme, or competitively bound

structural analogs. Conformational changes are observed around the cata-

lytically active zinc moiety. Upon coenzyme binding, lactate dehydrogenase

exhibits movement of one polypeptide chain. According to the x-ray map,

glycogon phosphorylase has a randomly oriented chain in the active site.

At the phosphorylation step, this flexible polypeptide settles into a well

defined tertiary structure (Fletterick et al., 1979; Weber et al., 1978).

This is reminiscent of the induced fit model (Koshland, 1958). The x-ray

structures of lysozyme and carboxypeptidase both indicate conformational

change upon substrate binding.

This is direct, recent, and incontrovertible evidence that proteins

undergo intramolecular motion in their mechanism. There is, however, much

other circumstantial evidence for this movement. The diffusion constant

of a macromolecule has contributions depending on intramolecular flexibility.

This contribution can be measured experimentally, and the relative motion of

lobes in hinged macromolecules has been detected this way (Cantor and Schim-

mel, 1980). Upon denaturation, proteins unfold. This can be detected by

a large increase in the UV absorbance of aromatic side chains (Cantor and

Schimmel, 1980). To a lesser degree, this same effect is observed in some

enzymes upon their binding to substrate or coenzyme. This indicates that

intramolecular motion is provoked upon binding. Spectroscopic reporter

groups inserted into the protein at positions far removed from the active

site can show spectral change during protein function. Optical rotatory

dispersion and circular dichroism of protein are partially due to the pre-

sence of asymmetric a-helix in its secondary structure. Changes in CD-ORD

spectra can occur upon enzyme substrate formation. High resolution NMR

and raman spectroscopy have also suggested intramolecular motion.








There is classical biochemical evidence of protein conformational

change. Liver alcohol dehydrogenase has fourteen thiol groups in its

interior, all of which are protected from reduction by iodoacetate. Upon

binding of the coenzyme, NADH, all of the -SH groups are available to

react with the reducing agent. Hydrolytic enzymes digest peptide bonds.

Any enzyme is subject to hydrolysis in the presence of enzymes like chy-

motrypsin. However, these enzymes are known to work faster on the target

enzyme as it consumes its own substrate. It seems that, during catalysis,

when the target enzyme has a more open structure, it is also more suscep-

tible to digestion by the proteolytic enzyme.

There is morphological evidence of molecular flexibility (Alexandrov,

1977). Studies in the comparative thermostability of plants and thermo-

resistant species of algae and protozoa collected from extreme environments

have been made. These suggest that there are evolutionary pressures on

their constituent protein's primary structure so that they remain moder-

ately flexible. When protein extracts from these species are studied in

vivo, their optimum temperature corresponds to the mean environmental

temperature of the protein. Further, the structures of high environmental

temperature proteins contain more prolines and other bulky stiff amino acids.

The implication is that a certain flexibility is required for proper protein

function. Suppressed motion due to low temperatures,and enhanced high tem-

perature fluctuations, both inhibit the proper mechanism of a protein.

Further, proteins evolve to match their flexibility to their environment.

Proteins are subject to structural fluctuations. Protein architecture

consists of a large hydrogen bonded network. Most of the hydrogen bonds in

the interior are protected from the water solvent. Hydrogen atoms, in hy-

drogen bond structures, will exchange with hydrogens from the solvent; of









course this does not happen if the hydrogen bond is shielded from the sol-

vent. If the protein is submersed in heavy water, one expects only the

surface moieties to undergo hydrogen-deuterium isotope exchange. This

effect can be monitored experimentally by changes in the IR vibrational

spectra and by NMR spectroscopy. The observation, however, is that heavy

water diffuses to the center of the protein; most hydrogen bonds throughout

the protein are subject to this kinetic exchange. This is only possible

if the close packed protein structure undergoes structural fluctuations

and creates micro channels for solvent diffusion. A water molecule is about
0
1.5A in diameter, so these channels must be at least this large. Of course

the exchange at the center is slower than exchange at the surface, and

studies of this exchange provide valuable insight to the fluctuating dy-

namical rate constants (Woodward and Hilton, 1979).

The fluorescence of aromatic amino acids is quenched by the presence

of dissolved oxygen or iodine in the solvent. Surface aromatic amino acids

of proteins do not fluoresce when the solution contains these species;

protected side chains in the center of the protein are presumably not af-

fected by this quenching process. However, these small molecules diffuse

into the fluctuating protein as it undergoes thermal agitation by the sol-
o 0
vent. Molecular oxygen has diameters of 3A and 1.5A. This is an indication

of fluctuation magnitude.

Careful evaluation of the Debye-Waller temperature factors in x-ray

crystallography indicates that root mean square fluctuations of atoms in
0
lysozyme and metmyoglobin are about .9A.

Nuclear magnetic resonance techniques promise to be the most useful

experimental probes of local dynamics in proteins. This method samples

the behavior and environment of particular nuclei having odd half integer








values of spin. Experiments can be designed that cover a very wide range

of time scales, but the method is relatively insensitive, and any individ-

ual experiment may not sample a sufficiently wide time domain to investigate

completely a particular dynamical event. The method is used to study

hydrogen-deuterium exchange kinetics, the limited rotational motion of

aromatic side chains, faster rotation of aliphatic side chains, and slower

backbone conformational mobility. One concludes from these studies that

proteins undergo a wide range of fluctuating intramolecular motion; this

motion has time constants from 10-12 to 1 sec.

One makes several general conclusions from this work taken as a whole.

There are two distinct modes of intramolecular motion in proteins: struc-

tural fluctuations and deterministic motion of parts of the macromolecule.

Nearly all of the protein is subject to continuous thermal agitation. In

the center of the protein, this motion is suppressed and the structure is

in some respects glass-like; nearer the surface, the fluctuations create a

semi-liquid structure. Bulky aromatic side groups move less and rotate more

slowly than smaller aliphatic side chains, but these larger side groups move

faster than elements of the protein backbone. Certain bulky side chains and

prolines probably play a stiffening role in enzyme structures. Concerted

cooperative motion of small and large regions of proteins occurs. This

certainly happens in catalysis, but this kind of motion is also randomly

driven in the closely packed center of the protein. Large regions must

move together in response to fluctuations because of excluded volume effects

of the tight structure. The time constants for the various movements range

from picoseconds to seconds. Some motion is continuous; some is episodic.

Allosteric communication through conformational changes occurs across the

entire protein and even through the quaternary structure of large protein








systems. Motion of charged and dipolar groups in protein disturbs the

large electrostatic stabilizing forces in proteins. The protein responds

to this by conformational change.

A serious proposal has been made concerning the role of these fluctua-

tions in protein mechanisms. Several investigators have made this argument

but the driving force behind it is from G. Careri (Careri et al., 1979).

Basically, his argument is that thermodynamic fluctuations are the secret

of the enzyme. Careri feels that the structure of proteins has evolved so

that they take advantage of fortuitous but continuous fluctuations from the

solvent in the completion of their biochemical function. He views energy

fluctuations from the solvent as being funneled or channeled into chemical

bonds at the active site of enzymes. Others have argued that thermodynamic

fluctuations are the driving force behind the mechanochemical conversion

process of muscle proteins and other proteins that perform muscle-like

tasks. Usually these writers argue that most, if not all, of the activation

energy barrier of protein mechanisms--transport, contraction, and catalysis--

is achieved by the clever filtering of omnipresent thermodynamic fluctuations

in the combined system of protein and solvent.

The main purposes of this dissertation are to discuss the possible

mechanism of protein operation and to construct a simple theoretical model

of rate-limiting intramolecular motion in proteins. In the next chapter

we will review some aspects of intramolecular fluctuational motion in

proteins. We will especially dwell on elements of fluctuations that bear

on the biophysics of protein mechanism. In the third chapter we will con-

struct a simple but general theoretical model, within which thermodynamic

fluctuating mechanisms must fit. We will derive certain thermodynamic

consequences of the proposal and compare them to the known properties of








proteins. We will investigate mechanisms driven by both hydrodynamic and

ionic charge fluctuations.

The essential conclusion of this study is that thermodynamic fluctua-

tions in protein chemistry are important, but only to the extent that fluc-

tuations are significant in any other chemistry (e.g., organic chemistry).

Small fluctuations, in a very few degrees of freedom at the enzyme substrate

transition state, permit the reaction to move in the forward or backward

direction. Fluctuations additionally provide a method for the protein to

dissipate surplus free energy to the solvent, However, the preparation

of the transition state is not a stochastic process. Elements of protein

reaction mechanisms, which are stochastic and rate-limiting, can achieve no

more than twenty percent of the activation energy barrier by fluctuations.

Other elements of protein mechanism that involve only structural fluctuations,

and not elastic strain or activation energy barriers, are not addressed in

this work. We make no criticisms of a theory of structural fluctuations that,

for example, permit the shape of an enzyme to fit around a substrate or other

ligand as long as the protein deformations do not involve large changes in

free energy. Enzymes work mostly by exploiting a unique protein structure

upon which deterministic concerted processes occur; this mechanism is mod-

ulated by a superimposed fluctuating motion, The fluctuations allow energy

dissipation to the heat bath, but that excess energy arises from chemical

potential energy and from the response to unbalanced forces within the

protein,














CHAPTER II
FLUCTUATING MOTION IN PROTEINS: A CRITICISM


Before we commence our modeling of intramolecular motion in proteins,

it seems appropriate to review the subject of thermodynamic fluctuations

in proteins. In this chapter we will discuss two kinds of research.

First, we will discuss some studies that provide an accurate understanding

of thermodynamic structural fluctuations and their role in protein biochem-

istry; then we will discuss some that do not. This chapter will not be an

exhaustive review of this field; rather, it will focus on several represen-

tative examples of the most important or frequently referenced work.

The idea that the shape of an enzyme has to fit precisely that of the

substrate is an old idea traced to Emil Fisher's lock-and-key hypothesis.

It seemed sensible to argue that the enzyme might more tightly bind to a

transition state of the substrate, thereby stabilizing the structure necessary

for catalysis. This led Koshland (1958) to suppose that an enzyme bound to a

substrate would undergo a conformational change, and induce the enzyme-

substrate active site into the necessary configuration allowing forward

catalysis. It has long been recognized that proteins probably undergo

fluctuations. However, no quantitative or experimental analysis of this

phenomenon were undertaken until recently. This is because no one really

knew how to approach the problem, and there seemed no particular reason to

do so.

Attention to the dynamic picture of proteins was set aside for a

decade by the arrival of x-ray diffraction maps of protein structures at







0 a
2-3A resolution. Many in the biochemical community, some not entirely

familiar with the implications of x-ray diffraction of solids, had many

reservations about the protein x-ray structures. Some concluded that

proteins were stiff, rigid structures. Even when presented with two

structures of the same protein, perhaps in its R or T state, their atten-

tion was focused on the rich structural detail of these maps, Only in the

last seven years or so have scholars begun to look closely at the dynamical

properties of protein and its structural significance. As we discussed in

Chapter I, recent work has found experimental evidence of concerted deter-

ministic motion in macromolecules, as well as fluctuational intramolecular

motion in proteins. The question of how a structurally complicated protein

exerts the right time-ordered forces in the active site environment is a

barely touched question for the coming generation of protein biophysical

chemists. It is a many-body problem of bizarre complexity. A conceptually

simpler hypothesis has been posed by some investigators, suggesting that

random fluctuations are somehow used by the periodic protein structure to

achieve catalysis in a way conceptually not available to the chemical

reactants in solution. This argument suggests that the exceedingly precise

structural architecture of proteins evolved for the purpose of exploiting

random fluctuations.


Macromolecular Fluctuations in the Biological Milieu

A great deal of sound, thoughtful effort has been invested, over the

years, in problems involving thermal fluctuations and their influence in

molecular biology. A great deal of this work has the physical sciences as

the point of embarkation. Consequently, some of this work, while physically

interesting and unquestionably right, is of unclear relevance to questions

of biological significance. In this section we will call this work to the








attention of the reader. Part of this work is several years old, and was

performed by some of the first modern physicists to take an interest in

biology. We will filter some of those elements that seem mechanistically

pertinent to modern biochemistry.

The earliest modern theoretical analysis of biological systems was

carried out by Kirkwood and his students. His collected papers are pub-

lished, and one entire volume is devoted to proteins (Oppenheim, 1965).

In this collection of Kirkwood papers are several addressing significant

questions of fluctuating phenomena in proteins. Chapter Nine of that book

is an analysis of the dielectric dispersion measurements of protein in an

aqueous solution. Previously, the decrement was assigned entirely to the

macrodipole moment of protein. This showed that fluctuations of the dipole

moment, in the region surrounding the protein, can significantly influence

this measurement. In another paper (Chapter Sixteen), Kirkwood's group

measured the mean square fluctuating charge of a protein and its immediate

environment by means of a light scattering experiment.

Chapter Thirteen is a paper of interest to modern enzymologists. It

presents a theory that partially explains the pH sensitivity of enzyme

reaction mechanisms. His argument is that the attraction between substrate

and enzyme can show a characteristic pH dependence, because the force between

them is mediated by fluctuations of charge density in the immediate environ-

ment of the two moieties. He estimates the effect of these fluctuations on

the Michaelis constant, Km, and the catalytic constant, kcat. This theory

predicts a maximum in kcat at the pK of the charged groups in the active

site. This effect is superimposed on the pH dependence in the charge state

of the catalytic group.

Spontaneous firing of neurons has been observed in sensory receptors.

These noise signals are perhaps triggered by fluctuations in the potential








on either side of the membrane (Fatt and Katz, 1950; Buller et al., 1953;

Buller, 1965, Naitoh, 1968; and Oosawa, 1973).

There have been several recent theoretical studies of fluctuations in

proteins. In a series of papers, N. Go has undertaken an idealized study

of intramolecular motion in protein (Go, 1974; Taketomi et al., 1975; Ueda

and Go, 1975; and Suezaki and Go, 1975). They present an idealized view of

a protein as a homopolymer of randomly assigned, equally weighted, pairwise,

long range interactions. Go does not try to model a particular protein,

and he does not allow for variation in the details of either the magnitude

or direction of the forces. He then permits transitions between the con-

structed ground state to fluctuation structures. The likelihood of struc-

tures that vary in energy is assigned by a priori Boltzmann probability

factors. In a determination of the probability distribution function for

the model, he determines the average number of broken bonds at various

temperatures as a function of time. The calculations are complex and re-

quire a Monte Carlo computational method. His systems are limited to only

fifty amino acid monomers. Go was motivated in this study by a desire to

predict protein tertiary structure from primary amino acid sequence data.

This is another formidable problem of biophysics. When he applied this

scheme to real proteins he found, as have all before him, too many local

minima and no clear folding path emerging from the study. He did see

structural fluctuations in his very small system. He saw a temperature

dependent nature-denature transition, and he can compute qualitatively

correct heat capacity functions.

His model does not easily extend so that it models a real protein.

It seems that his model is not structured to favor stochastic bond breaking

and reformation of other, spatially close, amino acids. If the calculations








allow any pair of monomers equal probability of reforming a bond, then

this suggests fast, wildly erratic conformational changes that are not

observed in real proteins. Go's calculations are reasonably sophisticated,

but apply only to overly idealized systems. This problem involves a fairly

complicated Monte Carlo computer experiment, and it yields some useful in-

formation. The main value of this work nay be that it provides a glimpse of

the true complexity of a real protein.

Work that is several orders of magnitude more sophisticated has recently

been undertaken by McCammon and Karplus. Some of their most important papers

are as follows: (McCammon et al., 1977; McCammon and Karplus, 1979; Northrup

and McCammon, 1980; McCammon and Karplus, 1980; art Karplus and McCammon,

1981). These works are magnificent tours-de-force. Essentially, they per-

form a computer experiment called a molecular dynamics calculation. McCammon

and Karplus assemble the atomic coordinates of a protein in a computational

box. These data are obtained from x-ray diffraction maps. Surrounding each

atom, they numerically insert a characteristic model potential of that par-

ticular atom. Hydrogens are excluded from the calculation, but hydrogen

bonds are modeled by extending the atom's energy and configuration parameters.

A set of coupled differential equations of motion are constructed, one for

each atom. The solutions to these equations are computed by the Gear algo-

rithm, and the computations proceed in finely graded time intervals of 10-15

sec. The temperature of the system is computed from an average of the

atomic kinetic energy. The temperature can be controlled by simultaneously

altering all the velocities by a constant multiplicative constant and pro-

ceeding for several steps until equilibrium is reached. About 10,000

iterations, each involving the solution of 1,500 coupled differential equa-

tions, are required to reach equilibrium at a preselected temperature for








a small protein. This corresponds to about 10 psec of real time. Because

of computational time limitations, the protein can only be studied for

these very short time periods, which are much shorter than catalytic time

constants.

The dynamic data acquired, however, are impressive, realistic, de-

tailed, and in agreement with equilibrium experiments in regions of overlap-

ping study where they can be checked. At computational thermal equilibrium,

the molecular dynamics computation yields rms fluctuations of atomic con-

figurations, time development of internal coordinates of motion, time and

spatial correlation coefficients, and cross-correlation coefficients.

There are several shortcomings to the procedure. The calculations

are extremely compl cated, expensive calculations. No outside group has

published work that replicates this effort. Because of the expense, the

calculations are terminated in a few picoseconds of real time. A great

deal of information about macromolecular dynamics is found in this time

domain; but most biochemically pertinent motion is much slower. Karplus

and McCammon have had some success in averaging over some of the motion,

effectively limiting the number of degrees of freedom, or limiting their

calculations to a small region. This extends the calculations to a few

hundred picoseconds. With this modification, they have been able to follow

small local relaxational processes; but it has not been possible for them

to follow conformational excitation, or an induced distortion process.

Most rate-limiting steps in enzymes take 103 sec to occur; and the mole-

cular dynamics scheme, as it is now practiced, simply cannot analyze this

time domain. Karplus has, therefore, focused his attention on processes

that can be studied. For example, he has modeled the short-lived transition

state of a small protein, pancreatic trypsin inhibitor.








Another shortcoming of this work is that it does not include the effect

of the aqueous solvent. Doing this requires that several layers of water

molecules be placed around the protein, and that exotic boundary conditions

be employed that allow the exchange of energy between the protein and the

solvent as a whole. As now constructed, the molecular dynamics calculation

allows only for exchange of energy among the degrees of freedom of the pro-

tein itself and a few captured water molecules. The method does not allow

for a high energy transition state to relax by dissipating energy to the

solvent. This problem is correctable, in principle, and it must be done in

order to look at long time protein mechanisms; but it costs at least a factor

of two in molecular structural complexity.

What these studies show are fluctuations. These studies conceptually

agree with the experimental results of Debye-Waller temperature factor

analysis. In pancreatic trypsin inhibitor, they find an average rms fluc-
o O
tuation displacement of .9A for all atoms, but fluctuations as large as 3A

for some atoms. The regions of large fluctuations are at terminal ends of

the primary sequence and at the B-turns. In a lOOps study of just the

active site of pancreatic trypsin inhibitor, McCammon and Karplus produce

potential energy profiles of the several possible barrier crossing trajec-

tories. This barrier crossing involves the ring flip of the tyr-35 side

chain. They find evidence of transient relaxation in the transition state

that is a response to concerted motion within the larger protein matrix.

This proposed structural relaxation is demonstrated to play a role in the

enzymatic mechanism of PTI. The potential energy barrier that the ring

flip process requires is too high to be overcome by thermal agitation in

only 100 picoseconds. The process is a response to concerted atomic dis-

placement within the entire protein.








Recently, a paper appeared in the chemical physics literature that

suggests a path whereby molecular dynamics studies can be practically ex-

tended to much longer time domains (Adelman, 1980). This paper discusses

a generalized Langevin equation approach to problems of many-body chemical

dynamics. Essentially, what molecular dynamics provides is a detailed

numerical view of the position and velocity of each atom. From these data,

one can compute all position and velocity, correlation and cross-correlation

functions. The Adelman formalism requires precisely this information to

construct response functions. The response functions are used to construct

a spectral density. With these one can calculate frequency moments. It

is necessary to evaluate the Einstein frequency describing the short-time

behavior and the adiabatic frequency that relates the low frequency or

static response to external forces. These constants and functions can be

used to construct an integro-differential equation of the damping function;

this function describes the viscous interaction among internal degrees of

freedom of the protein. Finally, one can construct functions that model

the effects of a fluctuating aqueous heat bath. This can be done because

the fluctuating forces and damping functions are coupled together by the

fluctuation-dissipation theorem. The net result of the Adelman analysis

is a set of coupled integro-differential equations in the displacement co-

ordinates of each degree of freedom. These equations are somewhat more

difficult to solve than the coupled equations of motion in molecular dy-

namics; but because of the averaging of microscopic details, one can take

larger time steps in the iterative procedure and still retain necessary

short-time information. Other simplifications can be made. Correlation

functions can be constructed for the center of mass motion of side chains

and peptide bonds from the molecular dynamics data. After averaging, this








reduces the number of degrees of freedom by a factor of 10 and the compu-

tation time by 103. This reduction in computer time makes the Adelman

procedure attractive for the study of long-time protein dynamics. It seems

sensible that a compound model can be constructed. This involves using

atom correlation functions at the active site and side chain, and construct-

ing peptide bond correlation functions in the remainder of the protein.

This proposal is not for an independent study of protein. The information

necessary to carry out this regimen must come from a Karplus-type molecular

dynamics calculation, which itself requires a structure derived from x-ray

analysis. However, the Adelman procedure provides a systematic scheme to

average the short-time information and permit long-time dynamic calculations

of protein mechanism and its interplay with fluctuating forces from the

solvent.

Recently, experimental work has become available that complements the

molecular dynamics calculation on proteins (Frauenfelder et al., 1979; and

Artymiuk et al., 1979). These papers present detailed analysis of the

Debye-Waller temperature factors in the crystal structures of metmyoglobin

and lysozyme. At sufficiently low temperature, irregularities in the

structure can be assigned to variations from unit cell to unit cell; but

a linear variation of with temperature suggests thermal fluctuations

of atomic positions. Solving crystal structures at several cryogenic

temperatures permits an extraction of this information from the data.

These two important papers substantially agree with the work of McCammon

and Karplus. They find fluctuations of varying magnitude throughout the
0
proteins. End groups and B-turns fluctuate about 3-4A, and rigid central
0
regions of the proteins move about .05A. Additionally, they find that

the c-helix structure does not remain rigid. The data suggest a possible








breathing or rippling motion of these structures. Larger conformational

fluctuations are found on the proximal side of the heme in metmyoglobin.

This is the region that Perutz (1978) suggests is involved in a dynamic

control mechanism. The author describes the proteins as having a glassy

solid-like core surrounded by surface elements that are semi-liquid. The

lips of the active site in lysozyme have large average displacements, in-

dicating greater fluctuations.


Biochemical Fluctuations as Mechanism: A Review

All students who demand a special role for thermodynamic fluctuations

in macromolecular function refer to two old papers of Linus Pauling as

their inspiration (Pauling, 1946 and 1948). There seems no compelling

reason to conclude such a mechanism from his work. Pauling, in these two

unrefereed, invited papers, discusses structural fluctuations of enzymes but

makes no special claim for mechanistic fluctuations. In the second of these

papers, he says the following of generalized enzyme mechanisms:

This substrate molecule, or these molecules, would
then be strained by forces of attraction to the
enzyme which tend to deform it into the configura-
tion of the activated complex, for which the power
of attraction by the enzyme is greatest. The
activated complex would then, under the influence
of ordinary thermal agitation, either reassume the
configuration corresponding to the reactants, or
assume the configuration to the products.
(Pauling, 1948, p. 58)

Considering the date of the article and the state of knowledge of molecular

biology at the time, this is a statement of profound insight about protein

mechanism. Realizing this, one should allow for the fact that it places

excessive emphasis on only the elastomeric strain model of enzyme catalysis,

and that this statement about fluctuations in transition state chemistry

can be extravagantly interpreted. It seems that authors who argue for








fluctuation driven processes in biochemistry are inspired by the second

sentence, and are not paying attention to the first. Let us discuss

several of these.

There are several proposed mechanisms that address the dynamics of

muscular contraction. A microscopic model of Huxley and Simons has been

well regarded because with it one can derive the Hill equations for the

tension velocity relationship and the energy released from muscular tension

(Huxley, 1957 and 1973; and Huxley and Simons, 1971). One feature of this

model is that a segment of the model undergoes conformational change by

traveling up a potential barrier. The energy that drives this system is

derived from thermal fluctuations in the heat bath. This particle is ran-

domly driven out of a one-dimensional well to a higher energy state, about

6 kBT above the ground state. It is actually worse. The myosin cross

bridges seem to be elastically coupled to one another through microfilaments;

this probably has something to do with molecular cooperativity. This means

that a fluctuation that drives muscular contraction must be spatially cor-

related for a distance equal to the length of the thick filament, about

1 micron (104A).

A theoretical study by Kometani and Shimizu (1977) intelligently

addresses this question. These investigators distinguish between two modes

of intramolecular motion. They call one type stable motion, and the other

unstable motion. Stable systems are induced to motion only by fluctuating

forces; the macromolecule is in a local minimum of structural free energy,

and climbs out of that well in response to stochastic thermal agitation.

Unstable motion is involved when a system deterministically relaxes to a

ground state of lower energy. This motion is a response to nonstochastic

internal forces. Kometani and Shimizu theoretically distinguish between








these two models for the case of muscular contraction. They discard the

Huxley-Simons model because their theory predicts an irregular velocity

as the muscle shortens under light load. This irregular velocity is not

experimentally observed. The paper does not also mention that a stochastic

conformational change involving so much energy takes longer to occur than

is observed. They do, however, correctly conclude that muscular motion

must involve microscopic motion to a lower state of structural potential

energy. They identify the Huxley-Simons model as a Maxwell's demon. There

are some nontrivial compensating errors in their derivation, but the final

equation has the right limiting form. Then Kometani and Shimizu confuse

the issue.

These two types of energy conversion, the unstable
energy conversion and the thermal energy conversion,
are both possible from the principles of thermody-
namics and probably may be found in biological systems.
(Kometani and Shimizu, 1977, p. 426)

Later in the paper they also say this:

Thermal energy conversion may be found in the active
transport of substances across biomembranes, partly
because no (sic) mechanical energy is not always
necessary. (Kometani and Shimizu, 1977, p. 426)

This describes a classical Maxwell's demon. We will discuss such mechanisms

in the next chapter.

There is another theoretical paper that addresses this same problem

(Cooper, 1973; see also a review paper by Cooper, 1980). Cooper recognizes

the potential shortcomings of the Huxley scheme and provides a rationale

to circumvent them. His paper is one of those that explicitly uses thermo-

dynamics backwards, and creates a microscopic model. The model he creates

is a Maxwell's demon. He starts with the experimental observation that

ATP hydrolysis is not associated with the conformational change of myosin,

the power stroke of the cyclic process in muscle contraction. He thinks








that the ATP hydrolysis is just the trigger to release energy stored in

the extended myosin cross bridge. He thinks the energy is stored in the

myosin cross bridge by fluctuations from the solvent bath.

We will describe his model different from the way.. he does. His

model works as follows. The myosin cross bridges have a likely position

of lowest configurational energy, A. Through fluctuations, the myosin

heads occasionally attach to another site, B. This site involves elastic

strain energy stored in the myosin-actin complex. To achieve this bonding

configuration, the myosin heads must overcome an activation energy AEAA*.

Muscle works in only one direction, so these fluctuations can occur in

only one direction; site B is always above A. This result is accomplished

by Cooper with an undefined structural asymmetry in the myosin. It allows

fluctuations of the correct configuration to be selectively chosen and

stored within the myosin cross bridge. When one of the energetic cross

bridges is formed, it must bind to the site B; this binding energy is

AEA*B. The binding energy must be much greater than kBT; otherwise, other

thermal fluctuations would cause the myosin to disassociate from the high

energy binding site and relax. The product of Boltzmann's constant and

absolute temperature, k T, is the average thermal energy of one degree of

freedom in any equilibrium system. The cross bridge remains at site B

until its release is triggered by ATP. Of course the thermal energy that

drives the myosin to attach at point B must be larger than the binding

energy at site B; this is because of the induced strain that remains in

the cross bridge (see Figure 1). Upon demand by a neuron action potential,

ATP is hydrolized to form ADP. At least part of this released energy

raises the complex from state B to A*; subsequently, the cross bridge

relaxes and the thick filament slides over the thin filament in the muscular

power stroke. The process then repeats itself.

































7B
E "AB


IReaction Coordinate


This diagram describes the Cooper mechanism of muscle contrac-
tion. By means of a high energy fluctuation, the myosin head
moves from the ground state, A, to a neighboring actin site at
B. This means that AEAA* > kBT. The binding energy of the
bridge formation at B is large enough so that binding is not
subject to dislodgement by fluctuations; that is, AEA*B >> kBT.


0)
C:
LU)
0)


Figure 1.








This proposed mechanism is exactly the ratchet and pawl perpetual

motion machine discussed in volume one of the Feynman lectures (Feynman,

1963). The only difference is that the pawl is controlled by an intelli-

gent external force. Instead of the fluctuations doing useful work directly

on an external mass, the fluctuations are elastically stored and released

by a trigger whose energy is supplied by ATP. Cooper argues that this does

not violate the second law of thermodynamics, and he carefully points out

that the energy released by the ATP trigger is greater than the work per-

formed by the thermal engine. According to him, this nonequilibrium

element of the problem allows a Maxwell's demon to function. In the recent

review paper, he says the following of this model:

Paradoxically, at first sight, the energy releasing
step of the process--ATP hydrolysis--occurs when
the myosin is detached from the actin, with no mech-
anical connection existing between the filaments.
Clearly, no direct coupling of ATP hydrolysis to
force generation, by conformational change for in-
stance, is possible. There is no difficulty, however,
in describing the process in terms of nonequilibrium
thermodynamics in which thermal energy is stored in
the cross bridges during attachment is selectively
released by the ATP displacement reaction, and such
models can show many of the features of the contrac-
tion process, including the absolute magnitude of the
force generated. These models rely on the intrinsic
flexibility that the myosin molecules display, and
utilize the inevitable thermodynamic fluctuations
which result. We shall return to this general point
later, but in muscle the myosin-actin assembly acts
as a Maxwell demon, selecting appropriate conforma-
tional fluctuations in the system and converting the
thermal energy into work. No paradox is involved
because the system is not at thermodynamic equilibrium
and ATP is consumed in the process. (Cooper, 1980, p.484)

Later in the same review paper, he has the following to say about enzyme

mechanisms:

Such theories suggest that enzymes have evolved mech-
anisms for channeling thermodynamic fluctuations into








specific chemical bonds, thereby effectively lowering
the activation barriers for the catalyzed reaction.
(Cooper, 1980, p. 493)

Oosawa provides a solution to the problem of local electric field

fluctuations in ionic solutions (Oosawa, 1973). In a discussion at the

end, he superficially describes the influence these field fluctuations

have on a variety of microscopic biological systems. Most of this dis-

cussion is sound. However, he discusses the effects of these fluctuations

in macromolecules. He realizes that the relaxation of electric field

fluctuations around the protein is as fast as small but rapid conformation

changes in protein. He estimates the proLability of an electric field

induced conformational change in protein. This estimate is in error because

the electric force that was used is smaller than the restoring force of most

protein tertiary structures, and because the author neglects viscous damping

within the protein and between the protein and the solvent. Unfortunately,

this paper has been singled out by other workers in the field as providing

a model to analyze fluctuation induced enzyme mechanism.

1he most persistent proponent of the idea that thermodynamic fluctua-

tions are the "secret of the enzyme" is G. Careri (Careri, 1974; Careri and

Gratton, 1977; Careri et al., 1975; and Careri et al., 1979). Careri and

his group argue that spontaneous fluctuations in the protein superstructure

can be focused into its active site creating a structure suitable for catal-

ysis. At various times he has propounded both hydrodynamic and electric

field fluctuations as the driving force for this process. He is especially

fond of the idea that an electric-field-induced breaking of a hydrogen bond

at the surface of the protein can trigger a flow of pent-up strain in the

macromolecule to the active site. The strain is reestablished later when

the hydrogen bond reforms. No discussion is provided as to how a 5 kcal/mole








hydrogen bond holds back 10-15 kcal/mole of elastic strain. As we will

discuss in the next chapter, the model protein in this dissertation has

98 amino acids on the surface. Certainly half of them are hydrogen bonded

to the solvent; this adds a factor of 50 to the time necessary for the

protein to stochastically find the right hydrogen bond fluctuation. This

subtracts another 1 kcal/mole from the maximum 3.4 kcal/mole of energy that

the protein can extract from the solvent in 1 millisecond. Careri does not

provide a detailed model theory as Cooper does. It is difficult, therefore,

to find particular errors in the argument. A few passages from several of

his papers are quoted.

The secret of an enzyme lies in its ability to let relevant
conformational fluctuations occur with a well-defined time
correlation. (Careri, 1974, p. 20)

In other words we believe that correlated fluctuations are
the driving force which causes the enzyme substrate complex
to evolve along the chemical pathway. The space structure
of the molecule provides "the right" time correlation in
its conformation variables. (Careri, 1974, p. 21)

Here, instead, we propose that this fluctuating electric
field can act on some charged residues to produce those con-
formational motions and those changes in the chemical af-
finities that, properly coordinated in space and time, can
let the macromolecule work as a catalysist. (Careri, 1974,
p. 25)

Therefore the enzyme seems to operate as an ideal trans-
ducer of free energy from the bath to the active site.
(Careri and Gratton, 1977, p. 186)

Thus the internal array of hydrogen bonds is the transducing
network that propagates the conformational fluctuation ori-
ginated on the surface by a low free-energy interaction with
the solvent. (Careri et al., 1979, p. 78)

Finally, we will mention one more paper (Gavish, 1978). In it is

proposed the idea that a protein responds to temperature fluctuations

of the heat bath through its coefficient of thermal expansion. Gavish

further views enzymes as a molecular machine consisting of elastic








"pulleys, levers, and springs." He further views this system as being

driven entirely by fluctuating forces from the solvent.

Our model suggests that the specific biological function
of proteins are governed by transient states which are
developed during transitions between conformations .
The dynamic state of a protein, as it is presented here,
suggests to what extent, if at all, the second law of
thermodynamics is applicable in a micro-environment
like an enzyme. This problem arises since the specifi-
city of enzymes may involve the reaching of improbable
states of the substrate in the complex, and the storage
of energy in elastic strains Therefore the second
law of thermodynamics is not violated by the above tran-
sient strains since it is not applicable to such short
times. (Gavish, 1978, p. 46)

We close this section by reminding the reader that enzymes have typ-

ical time constants of 10-3 sec. Their microdynamics are governed by

molecular vibrations whose time constant is V/kBT = 1.6 x 1013 sec.

Enzymes do obey the second law of thermodynamics. Proteins are not Max-

well's demons.

Acceptable Roles of Thermodynamic
Fluctuations in Protein Mechanism

There are several roles that thermodynamic fluctuations play in enzyme

and protein mechanisms. Essentially, these are roles that do not entail

the localization of large quantities of free energy into a small region.

We will show in the next chapter that it takes longer than lO- seconds to

localize a few multiples of kBT (the average thermal energy) along one

degree of freedom in any general protein mechanism if that increase in

energy is achieved by fluctuations. Proteins, which typically take milli-

seconds to function, can thus extract no more than about 3 kcal/mole of

thermal energy for the purpose of completing its mechanism.

However, fluctuations that do not involve the extraction of large

amounts of energy from the heat bath can occur in macromolecules and in








particular are exploited by proteins as they function. For example,

structural fluctuations occur in some proteins that have ill-defined

tertiary structure. Such proteins are characterized as having high struc-

tural mobility. Under certain biochemical conditions, immunoglobin G has

a freely moving hinged structure. The two lobes of this structure change

their relative orientation by free rotational diffusion. As mentioned

earlier, lactate dehydrogenase has a free polypeptide end that is free to

wander in the solvent. Although not a protein, tRNA can have a flexible

ill-defined structure. Upon a change of chemical environment as the macro-

molecules encounter further chemical species (e.g., substrate), these dif-

fusive structures settle into well-defined tertiary shapes. These diffusive

motions have mechanistic implications especially as a method of binding,

but they do not involve the assembly of high energy intermediate states.

We describe these as fluctuations in structure, not as fluctuations in

energy. Such a description is consistent with the induced fit hypothesis

of Koshland (1958). In this scheme either deterministic or fluctuational

motion of relatively small (~ kBT) energy barriers can cause a protein to

achieve the tertiary shape associated with the bound or transition state.

Fluctuations are intimately connected with the reversible character

of enzymes and transport proteins. When the transition state of an enzyme

is prepared, small random fluctuations in structure and especially in

energy can cause the reaction to proceed in the forward direction to pro-

duce product or, in the backward direction, recreate substrate. The

detailed reaction mechanism of any particular protein molecule intimately

depends on the precise structural arrangement of the protein. Because

various particular structures are available in the ensemble of proteins

due to thermal fluctuations, an equilibrium distribution of forward and








reverse microreaction pathways is developed in a statistically large sample

of similar macromolecules.

Thermodynamic fluctuations have other roles. In the case of hemoglobin,

for example, structural fluctuations provide momentary channels that act as

diffusive pathways through which molecular oxygen can pass so that it can

bind at the iron site in heme. It seems that without this avenue of approach,

the oxygen binding pocket is so tightly bound against the introduction of

water that oxygen cannot enter the ligand binding site.

Proteins use macromolecular fluctuations in structure and energy to

relax topological states of high strain. They can relieve a post-reaction

state of chemical potential energy imbalance, or local variation in atomic

kinetic energy due to a chemical reaction. These fluctuations mediate

the redistribution of both exothermic and endothermic reactions. Fluctua-

tions provide a mechanism that allows a protein to relax to a configuration

- of minimum free energy without otherwise employing a photon emission process.

Of course this interaction with the solvent is deterministic in the same

sense that local regions of high energy density will inevitably disperse

that energy away from the center. The dispersal of energy is fluctuational

in the sense that it cannot happen unless there is continuous communication

and interaction among all the degrees of freedom of the protein and solvent.

This dispersal occurs because the magnitude of energy fluctuations in the

downhill direction are larger than those in the uphill direction.

The morphological evidence reviewed by Alexandrov implies that.

thermal fluctuations play a significant role in most protein mechanisms

(Alexandrov, 1977). He notes that there seems to be selective pressure

on protein genetics so that they maintain a certain degree of flexibility.

He reaches this conclusion by comparing similar proteins from species





31


found in environmentally extreme conditions. The control of protein

flexibility is maintained by genetic point substitutions and deletions

of bulky, stiff amino acid residues in the protein primary structure. No

particular protein mechanisms are invoked or discussed; however, his evi-

dence does seem to apply to very general classes of protein structures.














CHAPTER III
DETERMINISTIC AND FLUCTUATING MOTION IN PROTEINS: A MODEL


Equilibrium statistical thermodynamics is a theory that predicts

macroscopic behavior of physical quantities under specified conditions in

isolated systems. It provides a framework relating the average value and

the interrelationships between thermodynamic quantities such as internal

energy, temperature, pressure, volume, free energy, and entropy as the

system is perturbed and transforms from one equilibrium state to another.

These average values are constructed from appropriate microscopic models,

and are concocted as averages over a large ensemble of isolated systems

with identical states. Using probability distribution functions of appro-

priate internal coordinates, the average value of any function of thermo-

dynamic variables can be calculated. The particular macroscopic state of

a system normally reflects an average over a probability distribution

function describing a large number of microstates. Many of these states

may significantly deviate from the most probable microstate of the system.

Emerging from this study is a view of a system in one macroscopic equili-

brium state that is described by many dynamic and structurally divergent

microscopic states.

One cannot use thermodynamics to predict unambiguously microscopic

detail of a system. If a macroscopic average is made over many different

microstates, one cannot unravel that averaging procedure and deduce a

particular microscopic detail without error bars sufficiently wide to allow

for other microscopic models yielding the same macroscopic result. The









reverse can be done. One can construct a microscopic model from direct

measurements or a theory of microscopic mechanism. A thermodynamic macro-

scopic average can be calculated in order to test the model. Two competing

models can be probed, studied, and manipulated to define thermodynamic

experiments that make diverging measureable macroscopic predictions. Sta-

tistical mechanics yields thermodynamics; the converse does not hold. Even

if one constructs one or several microscopic models from a macroscopic

measurement, one has no possible guarantee that all models have been ima-

gined. From thermodynamic measurement one can suggest or make plausible

some particular model; from several measurements one can narrow down the

possible choices, but before a thorough case can be made for any model a

variety of thermodynamic predictions and microscopic experiments must agree.

This fact is well known to scientists, and to biophysicists in particular.

It bears repeating at the beginning of a statistical treatment such as this

because thermodynamic analyses, which are usually easy to perform, have

been promiscuously misused for drawing unwarranted conclusions from time

to time.

Thermodynamic Fluctuations in Proteins

The discipline of thermodynamics intrinsically contains the notion of

fluctuations. Any system in equilibrium must display random fluctuations

about its most probable state. Deviations of a thermal variable are smaller

than the mean value of the variable. They are usually unobservably small.

Only in particularly favorable cases can these fluctuations even be detected.

The size of the fluctuations is proportional to the reciprocal square root of

the particle number density. Thus, a system described by a large number of

particles exhibits vanishingly small fluctuations. Conversely, experiments








that sample a very small volume of space, systems where many-body forces

extend over enormous distances, and systems with very few particles

exhibit fluctuations of relatively larger magnitude. There are phenomena

that are fundamentally a consequence of thermodynamic fluctuations. These

fluctuations areresponsible, for example, for the blue color of sky,

critical opalescence, electronic circuit noise, the oscillation of a pre-

cision galvanometer suspension, Brownian motion of a colloid suspension,

and the distribution of particles in a macromolecular sedimentation

ultracentrifuge experiment.

Thermodynamic fluctuations are usually small because of the law of

large numbers (see Appendix A). The implication is that the magnitude of

fluctuations in an extensive thermodynamic variable, one proportional to

the number of particles in the system, decreases with the size of the

sample. Typically one finds for the variable x that


<(Ax) <(x )2> 1
0N

There are special circumstances where the exponent of (1/N) differs from

one half; the law of large numbers only requires that fluctuations decrease

with large system size. This is a quite general consequence of any

probability distribution function that realistically reflects physical

properties at equilibrium. In particular, it holds for the Gaussian dis-

tribution function. This function, according to the central limit theorem,

is the probability distribution function for the limiting case in which

the number of independent random variables grows beyond bounds in most

discrete distribution functions. For example, the Poisson distribution

P(x) reduces to the normal or Gaussian distribution in the limit of small

intervals of x. For the most part, thermodynamic fluctuations obey a

Gaussian distribution function.








Another implication of the law of large numbers is that small systems

have relatively greater fluctuations. Large molecules, for example proteins

in dilute solution, can be thought of as independent small thermodynamic

systems subject to large fluctuations. The tertiary structure of a protein

must fluctuate about its average structural shape (Hopfinger, 1973). The

volume occupied by a single protein in solution changes so that its root

mean square volume fraction is of the order of N ,


<(AV) 2> ()
(2)


Similarly, the rms fractional internal energy can be expressed as


<(AE)2> _1 (3)


We will consider a single protein as a separate thermodynamic system.

This can be done in a dilute solution where the proteins act as independent

noninteracting units. This suggests that a beaker of dilute protein solution

at laboratory in vitro conditions constitutes a canonical ensemble of macro-

molecules in the classical thermodynamic sense. The individual independent

proteins are suspended in a constant temperature heat bath. The molecules

are free to exchange energy and expand into the bath. Because the covalent

bond energy that binds the atoms together is much greater than kBT, the

protein is not free to exchange particles with the heat bath. As long as

the solution is not too concentrated, the particles do not interact with

one another in any significant way. Finally, if the beaker is isolated, all

the requirements are fulfilled to permit description of the beaker as a canonical

ensemble of protein macromolecules at equilibrium. Even if the investigator








performs kinetic experiments, the protein is a system at one equilibrium

state transforming substrate reversibly into product at another equilibrium

state. Because of this, classical thermodynamics has been usefully applied

to such systems. The results of this analysis have been applied to pro-

teins in vivo, where the assumptions are not well satisfied but presumably the

protein behaves in a similar manner. The only difference in our thermodynamic

treatment and the usual treatment is the size of the system. A beaker of

water, a thermodynamic system of usual dimensions, contains on the order

of Avogadro's number of particles; an individual protein has very many fewer

particles--(50,000 dalton/protein) (13 dalton/atom) = 3,800 atoms. Fluc-

tuations in the solvent pressure are on the order of Ap ~ P N1 ~ 10-12 atm.
00
Fluctuations in protein volume are considerably larger: AV ~ V N .02V .

Similarly, AE ~ .02E Fluctuations in extensive variables of proteins are

a few percent of the variable's mean value. This is a consequence of

the law of large numbers.

In the spirit of a biophysical totalitarian principle, the question

has been raised by some as to whether these anomalously large thermodynamic

fluctuations are the basis of some mechanistically important biochemical

function. Scholars have asked the following kinds of questions in a

variety of ways. Are intramolecular motions associated with enzyme mech-

anisms really thermodynamic fluctuations? Does the unique structure of a

particular protein provide a kind of funnel to allow these fluctuations

in energy to perform chemical work? Do structural fluctuations of enzyme

substrate complexes provide transient intermediate states of high local

energy or substrate deformation that help explain the catalytic efficiency

of a protein? Can fluctuations act as a kind of triggering mechanism and

allow protein configurations of unstable equilibrium to relax to a state








of lower energy? Does the structure of a protein act as a kind of Max-

well's demon to pick out favorably those fluctuations associated with

useful catalysis? Do fluctuations provide the basis for dynamic binding

between substrate and enzyme so that an enzyme can distinguish between

stiff and pliable substrates of similar shape? Instead of providing energy

to a system, do fluctuations in the tertiary structure of a protein provide

a mechanism to allow a flexible protein to fit around a substrate, perhaps

even inducing strain in the substrate much like a hand fills out the shape

of an empty glove?

The purpose of this chapter is to address these questions. The

plausibility of the argument will be discussed. Reasonable postulates will

be investigated for thermodynamic consequences that might be observed.

Some reasonable alternative views will be discussed when that seems appro-

priate. One result of this study is the recognition that protein intramole-

cular motion needs to be segregated into three categories. These categories

are distinct because the forces that drive the motions are fundamentally

different. In particular, we will distinguish between two kinds of intra-

molecular motion: that driven by thermodynamic fluctuations and motion

driven by unbalanced forces within the protein.

It is self-evident that nonstructural proteins--enzymes, transport,

carrier, and contractile proteins--require a variety of structurally con-

sistent, dynamically ordered physical and chemical events to occur in order

for them to perform their assigned tasks. For example, a typical protein

exploits several of the following kinds of interactions. The enzyme reduces

the entropy of an uncatalyzed reaction by fixing the location and orienta-

tion of a substrate within an enzyme active site. This enables it to

increase the effective collision rate between the necessary moieties. In








order to facilitate nucleophilic or electrophilic catalysis, some enzymes

form transient covalent intermediate forms with all or part of the sub-

strate. An enzyme may hold a substrate in a transition state allowing a

proton to transfer from one moiety to another, thus executing general acid-

base catalysis. Electrostatic interactions may be controlled within the

enzyme as in the case of electrostriction of the protein. Ion pairs and

dipoles can act to decrease the energy gap between substrate in the ground

state and in its transition state. An enzyme provides a matrix where

dipole, dipole-dipole, or induced dipole interactions can be effectively

employed, as for example in side chain alignment or substrate binding.

Local hydrophobic and hydrophilic environments are created within the

enzyme. These can influence binding or change the relative energetic of

some process. Enzymes structurally stabilize and often distort cofactors

that may be necessary to complete a chemical reaction. An enzyme can

freeze out translational and rotational motions of a substrate that might

impede catalysis. By conformational change, a protein can induce a local

strain within the substrate, thereby amening it to ready chemical transfor-

mation. This strain can take the form of electronic distortion as well as

internuclear deformation. A macromolecular structure may evolve that

allows cooperative or allosteric effects to occur. This is a system in which

the occurrence of one step in the mechanism may trigger a subsequent step,

or where the occurrence of one step might allow a subsequent step to occur

more easily. Binding of inhibitors to some enzymes can trigger a change

in the enzyme, the effect of which is to alter the catalytic efficiency of

the protein. Mechanochemical effects are known to occur in contractile

proteins.









Most of the previous examples implicitly suggest that conformational

change within the protein or structural adaptability of the protein plays

a significant role in protein mechanisms. Generally these conformational

changes can occur in two ways. Conformational change can be a relaxation

from one conformation to another of lower free energy. This process might

be triggered by some event, say substrate binding. On the other hand,

thermodynamic fluctuations can occur whereby energy from the surrounding

solvent enters the enzyme and permits the protein momentarily to alter its

tertiary structure, perhaps allowing some subsequent event to occur.

We shall expand this catalog of conformational changes to three cate-

gories. It is necessary to sort out their differences and similarities,

and to quantitatively evaluate their contribution to enzyme reaction kinet-

ics. We have chosen labels for these motions--deterministic, undeterministic,

and fluctuational.

The deterministic mode of intramolecular motion in a macromolecule

occurs when the protein is initially in a structural state of high internal

energy, concomitant with local strain in regions of the molecule. By alter-

ing its structure, the protein can dissipate some of that energy to the

surrounding heat bath. One describes this motion as relaxation to a confi-

guration of lower free energy. We suggest that, for whatever reason

(fluctuation or some triggering effect), the enzyme structure initially

corresponds to a peak on the free energy versus reaction coordinate profile.

The tertiary structure subsequently relaxes to assume a state at a nearby

potential energy minimum on the same curve. Such a process is not a fluc-

tuation. However, in the length of time required for the enzyme to modify

its high energy structure, local fluctuations due to the fast exchange of

energy with the surrounding heat bath do occur. In fact these thermodynamic








fluctuations in energy provide the central mechanism by which the high

energy tertiary structure can dissipate that energy to the surrounding

heat bath. During relaxation, in this picture, fluctuations of energy

out of the macromolecule exceed those into the molecule. The time constant

for protein relaxation is intimately related to the decay constants of

these fluctuations. The rate constant of any macromolecular conformational

change is dependent upon the physics of the thermal fluctuations. However,

the fundamental point here is that the protein is initially in its high

energy state, and it relaxes deterministically to its low energy state via

superimposed fluctuating dynamics. The final shape of that protein is

determined by the detailed nature of the local forces within the molecule

and not by solvent or protein fluctuations.

The fluctuational mode of macromolecular motion is exactly the opposite.

In this case the protein is initially at a structure corresponding to a

local minimum in free energy. That is, by some process in its distant past,

the protein acquired its present equilibrium shape. Because the protein is

immersed in a solvent heat bath, there exists the necessary thermodynamic

consequence that energy from the bath can enter the macromolecule. This

energy permits the enzyme to achieve higher energy and a less favorable

configuration. Once the structure is altered to this high energy state, it

relaxes to the ground state iw-a deterministically. The process that creates

this high energy tertiary structure is absolutely not deterministic. It

depends solely on the thermodynamic fluctuating character of local energy

and particle density in the protein itself and its surrounding solvent. It

is important to realize that no violation of classical thermodynamics is

occurring here. One is not getting something (a more catalytically appro-

priate structure) for nothing (no expenditure of free energy). However,








it is necessary to be careful in invoking such a chaotic structural change

as part of an enzyme mechanism. This is for two reasons. First, it is

just as likely that any conformational change can occur--those that favor

a catalytic configuration as well as those that oppose such a configuration.

In a typical enzyme there are very few (on the order of one) configurations

that optimally favor catalysis. However, for a large system of 3N degrees

of freedom, there exists at least 6N qualitatively distinct configurations

in which an enzyme might find itself as a consequence of thermodynamic

fluctuations. This problem will be discussed presently.

The second reason to be careful about fluctuations as a catalytic

mechanism is that these fluctuations are really not that large. By invoking

the equipartition theorem they are on the order of NkBT = N(.5) kcal/mole,

where N is the number of degrees of freedom. The N can be large for a pro-

tein, but that energy is spread out roughly uniformly over a protein. The

excess energy due to thermal fluctuations in the active site region that

is available to transform the substrate into the transition state is on the

order of 10 kBT =5 kcal/mole, or about the energy released in forming one

hydrogen bond. This is rather small compared to a typical activation energy

of 10-20 kcal/mole (Guetfreund,1972).

Nonetheless, nothing in the preceding paragraph precludes the possi-

bility that such fluctuations may be mechanistically pertinent to catalysis.

For example the protein may be divided into two, or at most a very few,

stiff regions. These regions are rigid enough so that intra-region ther-

modynamic fluctuations would be over-damped, but in the spaces between

these stiff parts the bonds are weak enough so intramolecular thermally

induced motion can occur. The structure of hexokinase suggests such a

possibility, but we also have in mind more subtle features universal to








proteins. For example, the c-helix structure is internally stiff and

is found to remain rigid under a variety of experimental conditions. The

protein domains catalogued by Richardson (1979) suggest themselves as

likely candidates for one or a very few degrees of freedom. The point is,

that for a system of a very few degrees of freedom, it is quite conceivable

that configurationally advantageous fluctuations might be pertinent to the

question of how enzymes provide catalytic power. The quantitative dynamics

of such fluctuations need to be studied.

The third mode of intramolecular motion, the undeterministic mode, is

similar in character to the fluctuational mode, but with an important differ-

ence. In this case two or more distinct configurations can occur with the

same free energy; further, these states are intraconvertible by means of a

pathway that does not involve passing over any activation barrier or perhaps

a very small activation barrier, AEA << kBT. In the language of a free

energy versus reaction coordinate profile, one would say that the free

energy minimum, in which the protein finds itself, is very broad on the reac-

tion coordinate axis and very shallow on the free energy axis. Under such

circumstances, only minimal investment of free energy extracted from the

solvent heat bath through a fluctuational mechanism is required to transform

the enzyme from one structure to another. Such structural transformations

can quite easily occur through such a mechanism. Of course the same caveat

that applied to fluctuational motions must also apply in this mode. That is,

in all likelihood, stiff regions of the protein must constrain the number of

configurations to a very few so that particularly advantageous configurations

do not take too long to occur.

In this dissertation we will not dwell at length on undeterministic

motion. Such motion is essentially diffusive. The dynamics of this motion







in a two-lobed enzyme exhibits unrestricted Brownian motion between reflec-

tive boundaries. It is elementary to estimate the time constant for this

process; it can be obtained from the simple diffusion constant, t = /6D.

Disordered proteins are not observed in biology, but if they were, Flory con-

figurational statistics could be applied to them (Morawetz, 1975). The most

reasonable undeterministic mechanism I can think of is the floppy-enzyme,

induced-fit model for substrate binding. In this case, the enzyme starts as

an undeterministic system, but the protein quickly envelops substrate. This

binding means that in the presence of substrate, the enzyme no longer flops

around but arranges its structure to that of a lower energy state. The ori-

ginal motion was undeterministic; however, the biochemically interesting

binding is deterministic.


Protein Intramolecular Motion is Classical Physics

A small but important point needs to be made. Any kind of conforma-

tional change in a protein is a consequence of classical physics. These

motions are not quantum mechanical phenomena, thus a full quantum mechanical

many-body treatment is not required for the study of intramolecular motion

in proteins. This will now be shown.

A convenient yardstick used to determine the degree to which quantum

mechanical phenomena is important in a physical problem is the Heisenberg

uncertainty principle. For our purposes it is convenient to state it as

AEAt h, where AE is the change in energy of some mechanical process and

At is the time necessary for the system to undergo this transformation. The

product of these two numbers must always be greater than Planck's constant, h.

If the product is very much greater, then the process involves the transfer

of enough energy over a long enough period of time to be classical. We

will write this equation as At > t/AE. A small conformation alteration of








a protein might be associated with as small a perturbation in energy as

5 kcal/mole (3 x 10-20 Joules). This is the amount of energy associated

with rupturing one hydrogen bond or a few Van der Waal interactions. To

completely denature or unfold a protein requires about 20-40 kcal/mole

(2 x 10-19 J). The Heisenberg uncertainty principle requires that such a

process take at least At > t/2rAE ~ 6.63 x 10"3 J.S/(2r x 3 x 10-2.J) ~

10-15 sec to occur. This is a time two orders of magnitude shorter than

one atomic vibrational time (/kBT ~ 10-13 sec). The time constants asso-

ciated with configurational changes of the smallest molecular groups in

proteins is no faster than 10-12 sec. The time constant for catalytically

relevant conformational changes is usually 10-3-10-4 sec, and the time

necessary to denature a protein can be as long as 103 sec. These are all

times several orders of magnitude longer than those associated with quan-

tized effects. Thus, the mechanics of intramolecular motion in proteins

is classical.


Can Thermodynamic Fluctuations Perform Chemical Work?

Another point that seems to be not well understood needs to be dis-

cussed. Some workers in this field explicitly state (and many others imply)

that thermodynamic fluctuations in energy can perform chemical, even mech-

anical work, or that these fluctuations can act as some kind of energy

source. It is argued that the structure of an enzyme is such that if the

right fluctuation comes along, that energy can be channeled into a chemical

bond. This is not correct.

In the first place, a thermodynamic system cannot perform work by ex-

tracting heat from only one heat reservoir at only one temperature. This

is the Kelvin-Planck statement of the second law of thermodynamics. Devia-

tions from this law are the basis for so-called perpetual motion machines of







the second kind. Of course no one suggests that an enzyme is such a

machine.

In the second place, this effect is not observed. Consider the fol-

lowing. Thermodynamic fluctuations in energy of a protein can be large

compared to one hydrogen bond (3-7 kcal/mole). The breaking of one hydrogen

bond or a few Van der Waal contacts is nearly the minimum structural fluc-

tuation that can occur in a protein and still trigger a conformation change

in the molecule. This amount of energy is only 5 to 10 times kBT. Suppose

that 5 kcal/mole of energy is extracted from the solvent and somehow inserted

into a chemical bond. The loss of heat from the water bath would cool the

solvent. Let us estimate the rate of cooling the solvent undergoes if every

catalytic event involves such a process. Consider a representative biochemi-

cal in vitro experiment involving enzyme in a aqueous buffer at a protein

concentration of [Et] = 10 micromolar. Arrange a large substrate concentra-

tion [s] ~ 10 KM, where KM is the Michaelis constant. At this substrate con-

centration, the enzyme will work at maximum velocity, vmax (Lehninger, 1975).

Vmax = k3 [Et]. (4)
We will choose a representative value of the catalytic rate constant, k3 =

104 sec-1. This means that one enzyme will transform 104 substrate molecules

into product molecules per second, or that the turnover time for one catalytic

event is 10-4 sec. Substrate molecules are consumed at the following rate.

vmax = k3 [Et] ~ (104S-1)(10-5 moles/liter) = 10-1 moles/sec.

If each catalytic event consumes energy of one hydrogen bond from the heat

bath (5 kcal/mole), this means that energy is consumed from the heat bath

at a rate of

dH (5 kcal/mole)(10-1 mole/sec) ~ .5 kcal/sec 500 cal/sec. (5)
dt








Because of the loss of this thermal energy from the water, the bath will

cool. We can estimate the rate of cooling knowing the heat capacity of

water. If

dH = mcdT
then

dH dT dT 1 dH
S- mc or dt c dt (6)

The specific heat of water is 1 cal/gmk and the mass of 1 liter of water

is 103 gm. We then obtain the following:

dT -1 dH -1
dT -l d 500 cal/sec
(103 gm)(l cal/gmk)


dT
dT -.50k/sec .(7)


We conclude that a moderately fast enzyme mixed with a moderately high

concentration of substrate would lower the temperature of its beaker at a

rate fast enough to have been observed by investigators a century ago. Such

an effect could certainly provide the basis of commercial refrigeration

without the necessity of politically less reliable energy sources. It is

not possible.

It is clear that if thermodynamic fluctuations are to play any role

whatsoever in biology, then it must be a fully reversible one. Fluctuations

can only provide a protein with structurally or energetically transient

states. These states may enable the enzyme to undertake a function it could

not otherwise perform, but the energy invested in the formation of this state

must be returned to the solvent bath and cannot otherwise be consumed. This

analysis suggests a constraint on any proposed fluctuating mechanism.








The Magnitude of Thermodynamic Fluctuations in Proteins

The study of thermodynamic fluctuations in a system does not require

the introduction of any new physical ideas beyond that of equilibrium ther-

modynamics. It is necessary to construct a probability distribution function,

w(xi), of the fluctuating extensive variable. This expresses the likelihood

that a particular variable, xi, can take on some particular value. From

this function one can evaluate the average value of xi and the average value

of any function of xi. For example, the mean square deviations of xi from

its mean, ,is <(xi )2>. It is necessary to square the deviations

from the mean so that the averaging is done over positive numbers. The root

mean square of xi, or <(x. )2> provides a quantitative estimate of

the range in magnitude of the variable x.. The ratio <(xi )2>/

provides a fractional dimensionless estimate of the fluctuation magnitude

in the variable xi. One visualizes this as the fractional scatter in the

data xi.

Consider once again a beaker filled with a protein solution. We will identify

the protein as an independent subsystem. This protein subsystem is free to

exchange energy with the surrounding heat bath of solvent. It is also free

to alter its volume by extending momentarily into the buffer. These fluc-

tuations in energy and size occur freely and spontaneously; they are balanced

by corresponding fluctuations of opposite sign in the immediate region of

the solvent surrounding the protein. These microstates occur even in the

state of strict thermodynamic equilibrium.

Appendix B provides a mathematical derivation of the magnitude of

energy and volume fluctuations in protein. The calculation holds the protein

particle number fixed. The proteins are treated as systems in equilibrium

with the surrounding heat bath. This is a completely general derivation and








does not rely on any microscopic protein properties whatsoever. It rests

on these three pillars: the assumption of classical thermodynamics; the

treatment of the protein as a member of a canonical ensemble; and the

experimental measurements of heat capacity, isothermal compressibility,

and the coefficient of thermal expansion. The analysis applies to proteins

because a flexible dynamic macromolecule conforms to these three criteria.

All the mechanistic biophysics and biochemistry of proteins is ensconced

in the three experimental measurements. Any system whose microscopic details

suggest the same experimental numbers would behave in precisely the same way.

The numerical fluctuations presented below are so large because of the small

size of the protein. This is in agreement with the converse of the law of

large numbers.

In this calculation, as in the entire body of this dissertation, it

should be realized that the numerical calculations are estimates only. Argu-

ments are made with respect to an average globular water soluble protein of
0
a molecular weight ~ 50,000 daltons with a radius of ~ 20A.

Numerical results are not meant to be exact but are to reflect only the

appropriate order of magnitude. Improvements on these calculations can be

made and these were discussed in Chapter Two. It has not been the purpose

of this study to look at a particular protein system; rather, it is the pur-

pose to look at the general role of fluctuations in proteins. When a specific

protein is discussed, it will be made clear in context. These are model

calculations on model systems yielding only order of magnitude estimates.

The following results are obtained in Appendix B:

<(E )2> = kBT2Cp 2kBPvT28 + kBP2vTK (8)

<(V )2> = kBvTK (9)

<(E )(V )> = kBvT2B kBPvTK (10)








< > denotes an average over a canonical ensemble

E Internal energy of the protein subsystem

= 4r3 = 3.3 x 10-26m/molecule, volume of a representative
globular protein with
radius of 20A

kB = 1.38 x 1038 J/K/particle, Boltzmann's constant

T = 300K, temperature

P = 1 atm = lONt/m2 the average pressure on the protein

Cp ~ .4 cal/gm.k 1700 J/kg.k the constant-pressure specific
heat of protein

S~ 240 x 106 K-1 the representative coefficient of volume
expansion for proteins and carbon-based
polymers

K l10-2 m2/Nt the representative isothermal compressibility
for proteins and carbon-based polymers (see
text)

One also needs

M = 50,000 gm/mole the molecular weight of a representative
globular protein, and

N = 6.02 x 1023 particles/mole, Avogadro's number.


One notices in the literature a rather careless disregard for the

distinction between a protein C and C This is acceptable because, in

general, Cv = C Tvy2/K. For protein the correction between the two heat

capacities (TvB2/K) is, at most, a variation in the third significant figure.

Therefore, C ~ Cv.

To make numerical evaluation of these fluctuations we will take care to

modify the dimensions to those commonly used in biochemistry.

1 kcal/mole ~ 6.9 x 10-21 J/particle ~ .0434 ev/particle

~ 500 Kelvin/particle ~ 1.68 kgT at T = 3000K

The numerical evaluations are as follows:








<(E )2> {(3624 .0406 + 2.82 x 10-42) kcal2/mo12}

~ 60.2 kcal/mole (11)

<(V )2> ~ {1.37 x 10-542 m6/part2} ~ 1.17 x 10-271 m3/part

~ 704 x 10 cm'/mole (12)

<(E -)(V )> ~ (-852 + 1.18 x 10-52) kcal/mole .cm3/mole

-852 kcal/mole cm3/mole. (13)

Thus a protein experiences fluctuations in energy from time to time,

on the order of 60 kcal/mole. This is about one to two times the energy

associated with one C-C covalent bond (Atkins, 1978). Energy fluctuations

within a protein are more than enough to form or break a covalent chemical

bond if the protein structure could take advantage of this fact.

The fluctuation in volume of 704 x 101 cm3/mole requires discussion.

First, let us discuss how large are the fluctuations in diameter consistent

with this fluctuation in volume. (It is not the cube root of <(V )2>).

Consider the volume of a sphere, V = (4/3)wr3, and its differential,

6V = 47r26r. Solving for 6r, we obtain the following:

6r = -. (14)

This corresponds to fluctuations in diameter of 26r = .46 x 101 A, or
0 0
.046A < 26r < 4.6A. Fluctuations in the linear dimension of a protein are

no more than a few atomic diameters or the size of one amino acid side

chain. They can be vanishingly small.

Secondly, consider the range of values of 6V and Sr. The spread in

values is related to the uncertainty of protein compressibility. The upper

bound of K, 10-6 m2/nt, corresponds to a very soft protein or one with a

very loose structure. This value was obtained by considering the force








necessary to bend or stretch a macromolecular assembly consisting of two

lobes that connect at a hinge. Its reciprocal is a measure of the relatively

small force needed to distort the lobed structure about its hinge. In par-

ticular, one might think of the two-headed myosin molecule or immunoglobin-G.

In this case one expects the fluctuations in linear dimensions to be at the
0
upper limit of 26r or a few angstroms, conceivably up to 10A. The lower

bound of K, 10-10m2/nt, is a representative compressibility for a stiff pro-

tein of well-defined tertiary structure. Such a protein might have a mech-

anism associated with conformational changes but probably not as large as

those of myosin. Lysozyme, myoglobin, and protolytic enzymes are represen-

tative proteins of this type. The fluctuations in linear dimension for such
0
proteins are quite small, about .05A.

It is reasonable to argue that proteins are either designed to have

segmental flexibility in which the force necessary to bend them is small,

or proteins are designed to have a well-defined tertiary structure and are

much harder to distort. Their compressibilities are either 10'6 or 10-1o

*m2/nt and it is not particularly useful to consider only the mean compress-

ibility of 10-8 m2/nt. On the other hand, hexokinase provides a possible

counter example. Its two lobed structure and mechanism suggests a large

conformational change about a particularly wide hinge. It can reasonably

be expected to have this median compressibility. However, these mechanical

measurements have not been made. It is easy to imagine other examples of

intermediate compressibility. For example, a globular protein with two or

a few stiff regions or segments bound to one another at the boundaries

through weaker chemical bonds. Suitable candidates for these subregions

can be found in the Richardson (1979) taxonomy of protein folding patterns.

Segments of stiff proteins (e.g., the terminus) can be expected to have








considerable flexibility. Proteins with quaternary structure seem likely

to have this intermediate compressibility.

The value of <(E )(V )>,or the correlation,deserves parti-

cular attention. If the correlation between deviations of energy and

volume is zero, then a positive fluctuation in energy is just as likely to

be correlated with a positive or a negative fluctuation in volume, that is,

the two dynamical variables have completely uncorrelated values. Enzymes

do not have structures carefully constructed to exploit the fluctuating

opportunities provided by their small size if the energy and volume fluc-

tuations are uncorrelated. If the normalized correlation, <(E ) *

(V )>/{<(E )2><(V )2>}1 equals 1, then one would say that

the two phenomena are different manifestations of the same physics. In

the case of protein, <(E )(V )> = -850 kcal*cm3/mole2. To see

what this means, we construct the normalized correlation

r <(E )(V )
<(E )2> <(V )2>

-852 kcal cm3/mol 2 -.02 x 10+ (15)
(60.2 kcal/mole)(704 x 101 cm3/mole)

or -.2 < r > -.002. Consider the implication of this fact. First, because

r is negative, an increase in energy in the protein is partially correlated

with a decrease in volume of the protein. This is physically reasonable

because an increase in internal energy is likely to occur as the protein

elastically shrinks as a consequence of an increased local pressure in the

solvent around the molecule. The most favorable case of r =-.2 corresponds

to the inflexible stiff protein. In such a protein, only twenty percent
of the fluctuations in energy are ever correlated with fluctuations in
volumes. Eighty percent of the fluctuations are completely uncorrelated








and cannot contribute to any enzyme mechanism involving precise structural

alignments; they only contribute to noise and dissipation in the system.

It does not follow that twenty percent of the fluctuations do correlate

to enzyme mechanism, only that an increase in energy corresponds to a

decrease in volume. That change in volume can occur with equal likelihood

in one part of the protein as another. On the other hand, if the number of

degrees of freedom were drastically reduced, or if there were features of

the structure that in some way optimized the many thermodynamic fluctuations,

then the possibility of their role in enzymatic mechanism is not excluded.

The opposite case of r = -.002, corresponding to loose or easily distorted

protein structures, suggests an even more severe limitation with regard to

these proposals. Proteins with larger structural fluctuations have smaller

correlations. Thus the outlook provided by the normalized correlation of

fluctuations in energy and structure to possible enzyme mechanism is, at

best, pessimistic.

The 6N Problem

For over two decades thermodynamic fluctuations have been regularly

invoked as a possible mechanism in enzyme biophysics by at least a few

investigators. At times fluctuation mechanisms have been quite fashionable

in some circles. It is time to address a question not previously discussed

by these scholars. Fluctuations are not deterministic intramolecular motions,

they are radnom and their magnitude disperses away from their center of

occurrence. In principle, they exist in all degrees of freedom of a protein.

In particular, if the fluctuation in energy is on the order of 60 kcal/mole

(see Equation 11), all that energy is not confined to one region, say the

active site. That energy is distributed roughly equally over all degrees of

freedom of the protein. At any particular time, all of the energy may be








localized; but over the time of many fluctuations, that energy will be

distributed uniformly over the entire structure. The carbon, nitrogen,

and oxygen atoms that build up the protein in the active site, and the

remainder of the protein, are similar in size and energy so we can estimate

the amount of energy available to the active site,


kcal Volume of active site
Eact 60 mole Volume of protein


S60kcal (4/3)7 (8A)3 3. kcal
mole (4/3)7r (20A)3 mole


On the average, the amount of energy available to the active site due to

fluctuations is considerably less than the representative 10-15 kcal/mole

activation energy of an enzyme substrate complex (Gutfreund, 1972). The

60 kcal/mole is thus misleadingly optimistic for this model. Exceptions

are possible only if some element of the structure is constructed so that

energy can be "focused" into one small region of the enzyme. Otherwise

there is simply not sufficient thermal energy available materially to

augment the enzyme substrate complex in its effort to escalate the activa-

tion energy barrier. We will return to this possibility at a later time.

For now, let us assume that all degrees of freedom in an enzyme are avail-

able for excitation by the heat bath. Let us discuss the consequences of

this case.

Consider the number of degrees of freedom in a molecule. To be strictly

precise, each atom is associated with three degrees of freedom. Each atom

is allowed to vibrate in any combination of the three orthogonal directions

of the coordinate system centered on the atom. These are called transla-

tional degrees of freedom. If the molecule has side groups, they may have








additional rotational degrees of freedom and the molecule as a whole has

some rotational and translational degrees of freedom that are not internal

degrees of freedom. A polyatomic crosslinked molecule has, therefore,

3N 6 + nrot internal degrees of freedom, where N is the number of atoms

and nrot are internal rotational degrees of freedom. A protein has so many

atoms that, for practical purposes, it is sufficient to say it has 3N

degrees of freedom.

This very detailed view is not too useful. Certainly it is not helpful

to distinguish between two proteins whose structural differences rest only

on whether one particular atom, far removed from the active site, is at

some vibrational amplitude extreme compared to another. (It may be very

important at the catalytically active center of the enzyme-substrate com-

plex. We will not account for this because to do so only makes the 6N

constraint more severe). Clearly, in order to make mechanistically perti-

nent distinctions, some kind of averaging over degrees of freedom is neces-

sary. A unique feature of protein chemistry allows us to do this.

Proteins are polymerized amino acids linked to one another by peptide

bonds. The peptide bond is a stiff planar structure due to quantum mech-

anical resonance stabilization. The protein forms its secondary and

tertiary structure by rotations about the a-carbon, not through distortions

of the peptide bond. Side chains of some twenty-one different types are

also connected to the a-carbon. Only differentiation in the structure of

these side chains distinguishes one amino acid from another. These side

chains usually bind to other moieties only through weak interactions; hydro-

phobic, electrostatic, hydrogen bonds, and Van der Waal bonds. They do

not form strong covalent bonds, but there are a few exceptions. Often the

side chain is also a stiff planar structure. Typically, some side chains









are free to rotate and move about with considerable configurational freedom

while fluctuations within the stiffer side chain are of lesser magnitude.

We will conservatively assign each amino acid two degrees of freedom,

one associated with the peptide bond and one associated with the side chain.

This treatment seems physically extreme, but this averaging makes biochem-

ical sense. When a biochemist discusses a mechanistically pertinent confor-

mational change, he thinks of motion at least as large as the reorientation

of one side chain and perhaps as large as a partially unfolded backbone, or two

regions within a protein slipping over one another. Ascribing two degrees

of freedom to an amino acid provides a minimum basis to discuss biochemi-

cally relevant configurational changes and more complicated cooperative

motion as well. The extreme simplification introduced by this model, and

the disregard of side chain rotational degrees of freedom, clearly provides

a lower limit to an enumeration of the degrees of freedom in an enzyme. Any

improvement of this model certainly yields a greater number of internal

coordinates and makes the argument that follows more extreme.

Consider only one degree of freedom within a protein. How many internal

phase states are associated with that molecular group? The answer is that

one group can occupy a denumerably infinite number of continuous positions

and velocities between vibrating extremes. Again we will simplify this

tremendously by ignoring the different momentum states and focus our attention

only on the configurational states. Let the origin for a degree of freedom

be centered on the center of mass position of that molecular group. We

will say that one position is structurally distinct from another by dis-

criminating between opposite sides of the origin in each of the three

orthogonal directions. Thus the molecular group associated with one degree

of freedom will move up, down, right, left, front, back, or in linear








combinations of these six possibilities. This view is probably extreme

and certainly not the entire story at the active site. Critical catalytic

side chains have more than one degree of freedom, and it can make a dif-

ference to enzymatic mechanism if some special side chain is in the right

position but moving in the wrong direction. Many side chains are freely

flexible and undergo internal free rotation. So again, six qualitatively distinct

configurations per degree of freedom underestimates the number of phase

states available to a molecular group in a protein, but we will let it

suffice as a lower bound assessment.

Knowing that one degree of freedom is associated with six configura-

tions, we can estimate the total number of structurally distinct confi-

gurations in which a protein might find itself due to thermodynamic fluc-

tuations. If the first degree of freedom has six distinct configurations,

the second degree has six configurations, the third a similar number, etc.;

then the total number of configurations can be found by multiplying the

number of configurations of each degree of freedom. This yields 6N quali-

tatively distinct configurations available to the protein. If the protein

has a molecular weight of 50,000 daltons and the average molecular weight

of its constituent amino acids is 137 daltons, this suggests that the model

protein contains 365 amino acid monomers. If each amino acid has two bio-

chemically relevant degrees of freedom, then the number of qualitatively

distinct configurations that a protein may possess is 62(365) = 6730 10568

What this means is that even though the fluctuations in energy are large,

about 60 kcal/mole (or .08 kcal/mole/degree of freedom), there are 10568

different ways that the protein structure might arrange itself in order to

accommodate this energy. The equipartition theorem says that any of these

10568 structures are equally probable.








Now consider the number of configurations associated with catalysis.

Enzymologists study an enzyme with their attention centered at the active

site. They view the surrounding protein matrix as structural housing and

environmental shielding for the catalytic chemistry. These scientists

carefully discuss the precise arrangement of this or that side chain in the

active site, and the various time-ordered electronic and configurational

changes that must occur for a particular mechanism to take place. This is

done because the intimidating complexity of any enzyme requires it. How-

ever, enzymologists will also be the first to argue that it is more compli-

cated. It is well known that small deviations in primary and tertiary con-

figuration, in regions far removed from the active site, affect the activity

of the enzyme. Most of these effects are at least inhibitory and often

debilitating to the functioning of an enzyme. Mechanistic enhancement due

to complicated allosteric effects are exceptions that very simply prove the

assertion--namely, that the most subtle changes in regions of a protein far

removed from the active site drastically alter the mechanism of the enzyme.

It is reasonable to argue that there is only one protein configuration

that optimizes the action of an enzyme. Any significant deviation from this

optimum structure impedes the operation of the enzyme. However, for the

purposes of this discussion we will significantly abate the constraint. It

does appear strictly true in the active site, about 1-15% of the protein

volume. Perhaps this condition is relaxed somewhat for the rest of the

protein, but the number of configurations associated with optimum enzyme

function is very small compared to the number of possible states. Appendix C

provides a reasonable upper limit to the number of distinct configurations

associated with efficient catalysis. The estimate is realistically no more

than 10171 configurations.








Let us ask: What is the probability that one of the twenty percent

of energy correlated structural fluctuations corresponds to catalytically

contiguous structures? From the previous discussions, it is clear that the

probability is between 1/10568 ~ 10-568 and 10171/10568 10-397. Another

way of saying this is that a protein, on the average, must pass through

about 10400 equally likely structures that are not catalytically effective

before it reaches one that is. If the reader remembers that small systems

have large fluctuations and fewer degrees of freedom, he might wonder what

this model predicts for a smaller protein. If the protein is 20,000 daltons,
0
and also has an 8A active site, the probability reduces to 10-159. This

is a consequence of the equipartition theorem applied to the enumerated

degrees of freedom.

There is another way the enzyme fluctuation mechanism hypothesis can

be posed, although I have never found it stated quite this way. There are

motions, and there are lots of ways for the energy to distribute; but

the movement is fast and before too long (say in a time kcat) an enzyme

might reach a catalytically favorable configuration, thus inducing the

necessary substrate strain and lower the activation energy of the reaction,


E
substrate product.
.-<


This suggests that an enzyme goes through a quasi-periodic Poincare cycle

when submerged in a heat bath. This idea is preposterous for large systems;

but it might be acceptable, in this case, because proteins are such small

systems compared to macroscopic thermodynamic ones. Certainly, the Poincare

cycle time for a diatomic molecule is only a few molecular vibrations (about

10'- sec), and the idea seems superficially reasonable for proteins.








Consider a quantitative estimate of a Poincare cycle time for proteins.

The Poincare cycle time for coupled quasi-harmonic oscillators have not

been successfully evaluated (Chandrasekhar, 1943; Wang and Uhlenbec, 1945).

We can, however, place a lower limit to this time by using the appropriate

configuration probabilities previously calculated. The time associated

with a configurational change in one degree of freedom, a side chain rear-

rangement or a backbone vibration, is about 10'911 sec. Assume the average

protein must pass through about 10400 structures to reach one that is cat-

alytically optimized. The enzyme must wait 10400*10-10 sec = 10390 sec for

one appropriate fluctuational configuration. If the protein is the very

small 20,000 dalton protein discussed above, the Poincare cycle time reduces

to 10150 sec. The age of the universe is 1017 sec. Thermodynamic fluctua-

tions in energy or a quasi-periodic Poincare process, spread through all

possible degrees of freedom in a protein, do not contribute in a special

structural way to enzymatic mechanisms.

We can sensibly ask the following question. If an enzyme has a turn-

over time of about 10-3 sec, and if the time constant to rearrange one

degree of freedom is 10-10 sec, to how many degrees of freedom does this

limit a fluctuating or a Poincare process? The number of steps that a sys-

tem must go through is l0-1/1010 = 107 configurations. The number 107 is

nearly equal to 69 structurally distinct configurations. Thus a system like

a protein, limited to working in 10~3 sec in steps of 10-10 sec, can only

have nine degrees of freedom. A fluctuating protein mechanism involving

all of its degrees of freedom is excluded. This kind of mechanism is only

allowed if the structure of the system is arranged so that there are only

a very few degrees of freedom in the system. Fluctuation of all atomic

elements, or a few side chains, provide a suitable mechanism if they occur








only in a very small part of the protein: for example, at the substrate in

the transition state.

There may be an alternative view of this problem that maintains the

mechanistic role of fluctuations. We have seen this proposed in several ways,

some of which are at least curscrily reasonable. The unique structure of an

enzyme might "focus" thermal energy from other regions of the molecule to the

active site, and these could provide exactly the right strain upon demand in

the substrate to ease the chemical transformation over a large activation

barrier. The protein structure acts as a kind of Maxwell's demon, filtering

from the many fluctuations those that might be catalytically useful. What

both of these statements mean is that large regions of the protein are so

stiff that they do not materially share in the available thermodynamic fluc-

tuations; they have no more than one or a few degrees of freedom. The fluc-

tuations in energy and configuration are shared in a system of drastically

fewer degrees of freedom than the 6N suggested by their primary structure.

In the simplest case, two rigid lobes are connected by a weaker spring and

submerged in a heat bath. The active site in the center is alternately ex-

panded and compressed by thermal fluctuations. Such a system has only one

degree of freedom. More complicated schemes can be constructed with only

a few degrees of freedom, but we will study only the simplest case of a

simple harmonic oscillator in a heat bath.

This view is not without biochemical validity. When x-ray structures

of an enzyme are available in both the relaxed and tense state, it is clear

that the secondary structure of the protein remains the same; and large

parts of the tertiary structure remain constant as these regions are shifted

with respect to one another. Only a brief glance at the enzyme hexokinase

and its mechanism suggests that a simple harmonic oscillator submerged in a

heat bath provides a quite reasonable initial model on which to start a








theoretical study. It is a favorite of those who study thermodynamic

fluctuations in enzymes.


The Enzyme as Maxwell's Demon

Adopting the notion of enzyme as Maxwell's demon is to accept the

possibility of a perpetual motion machine. Perhaps more needs to be said

about this, because we have read several scholarly references to such an

idea in papers published in the last decade.

The idea of a Maxwell's demon was proposed by Maxwell in his 1871

classic book, Theory of Heat. The idea has consumed the interest of phy-

sicists ever since. It is not a trivial idea, but alas, it also is not

correct. The definitive modern analysis of the idea is found in Brillouin

(1962). The argument can be made several ways; consider a most provocative

one. Assume a piston has a small trap door operated by an extremely clever

and observant demon. Whenever a molecule is about to hit the dorr on one

side, while at the same time the other side of the door is to be devoid of

collisions, the demon opens the door and allows the molecule to pass from

the first chamber to the second. The pressure builds in the second chamber

and does resultant work on the piston. The piston can be connected to a

machine, and this work can be extracted from a single temperature heat bath.

This is a violation of the second law of thermodynamics. The argument can

be constructed so the demon segregates molecules by speed, between two fixed

volume chambers, and spontaneously creates a temperature difference.

As early as 1871, Lord Kelvin thought the second law of thermodynamics

did not apply to biology; and thus began a rich tradition of applying the Max-

well's demon argument to living systems. From its inception, the demon has

been necessarily imbued with attributes of intelligence, free will, memory, and

microscopic dimensions. Arguments have been made that only the law of large








numbers, and the necessity of micro-machinery, prevented a macroscopic

Maxwell's demon from being built; but that certainly is was conceptually

possible.

Smoluchowski (1908) presented the first detailed study of the pheno-

menon. He noticed that an automatic system (for example, a trap door on

a spring just weak enough to let fast particles through) would itself be

subject to thermal Brownian motion. The interaction with hot particles

would warm the door. The subsequent motion of the door is of large ampli-

tude and is irregular enough to confuse purposeful openings with thermal

fluctuations. Finally, the system reaches equilibrium between the chambers

by particle exchange in the forward, as well as the backward, direction.

A beautiful and elementary discussion of this is in Feynman (1963).

Smoluchowski concluded that an automatic Maxwell's demon was impossible,

and that only microscopic fluctuations can occur between the two chambers.

He explicitly avoids the possibility that Maxwell's demon might be an

intelligent being.

The crucial relationship between entropy and information was made by

Szilard (1929). He realized that the operation of obtaining information

about a system necessarily meant that the entropy of that system decreased;

and further, he found that the price of that information through any exper-

imental procedure introduces an even greater residual entropy to the system.

Hence, observing a system increases its entropy. Szilard also provided a

way to quantify informational entropy.

Finally, after several papers, Brillouin (1962) wrote a monograph that

sorted out most of the remaining problems and laid the demon to a well-

deserved rest. Brillouin realized that before a demon can work, it must

have immediate detailed information about the state of all the particles








in the chamber. To acquire information, he must observe the system. The

easiest way to do this is to use light as a probe of the system. The demon

must be careful because the box is at some temperature T, and the cavity

is bathed in blackbody radiation. This radiation has properties proportional

to temperature, not particle distribution or even particle number. The

demon must use radiation of significantly higher energy than that pro-

vided by the blackbody effect. The demon's flashlight will pump energy and

entropy into the system. This influx into the system will enable the demon

to extract the necessary information about the system through light scat-

tering; but the loss of entropy into the system is always greater than the

reduction of entropy provided by the information.

Light is the easiest practical way to obtain information about fluc-

tuations in a system, but of course an enzyme does not use light to monitor

the heat bath. The enzyme must exploit information gathered through some-

thing like Van der Waal forces with the solvent, and between interacting

groups. Unfortunately, these weak forces are short range and decrease with

distance at least as fast as r-2. By the time the protein can sample these

forces, the same forces would act on the "trap door." To open the "trap

door" requires additional work from some other source. This work irre-

versibly adds energy and entropy to the system.

We will briefly consider a nonenzymatic system that superficially

seems to be a Maxwell's demon. Like an enzyme, it does not use scattered

light to obtain information about surrounding particles. It relies on

automatic operation of a micromechanism employing short range forces to

sort out fluctuations in the surrounding heat bath. We will briefly dis-

cuss thermal excitations in a solid state rectifier. The details of this

are found in Brillouin (1962). First, consider a passive resistance R








connected to an extremely sensitive galvanometer of inductance L. The

meter needle in the circuit spontaneously fluctuates in response to a

microcurrent induced by a random time-dependent electromotive force, V(t),

thermally created across the resistor, R. The average value of the EMF

is zero; thus,

di
L F + Ri = V(t). (16)

The circuit has only one degree of freedom--the current--so by the equipar-

tition theorem the average value of the energy associated with the current

is

= kBT = L = R

where
kBT 2kBT
= A. (17)
L R

Physically, this is explained by mechanical thermal excitations of atoms or

molecules in the resistor losing their energy through collisions with

electrons in the lattice. The subsequent motion of the electrons constitute

the fluctuating current that is measured by the ammeter. In a detailed

study of this process, one needs to do a Fourier analysis of the stochastic

differential equations and employ the Wiener-Khintchine theorem (McQuarrie,

1973). One can obtain the average square EMF in a frequency range Av, where

v ranges from zero to Vmax by the relation

= 4RkBT Av (18)


This is the so-called Nyquist relation. Using the total impedance and

Ohm's law, one can also obtain the average square current in the same

frequency range through the equation








4RkBT Av
<2> B = 0, (19)
R2 + X2

and X is the AC reactance of the circuit.

Up to now, we have discussed the thermal fluctuations in a resistor;

now let us consider the Maxwell's demon part. A solid state rectifier is

a nonlinear device that provides a large resistance to current going in

one direction, but only a small resistance to a current going in the oppo-

site direction. It seems reasonable that such a device, sampling all

fluctuations in current produced by the resistor connected in series with

the rectifier, selects out those current fluctuations in the favorable low

resistance direction and produces a net current in the circuit, f 0.

Several such devices could be connected in series and be used to charge a

battery or perform work. It uses only a single temperature resistance

heat bath.

Clearly something is wrong, as this violates the second law. What

happens is that the average current remains zero and the voltage generated

across the rectifier is always balanced by an identical voltage in the

resistor, so that the loop voltage is zero. No power, P = iV, is created

by the rectifier; however, an average voltage, f 0, is created across

the rectifier. The value of this average voltage is proportional to the

nonlinear character of the rectifier. If the voltage current characteristic

equation of the solid state device is a simple quadratic, then

Vrect ~ Vi + bi2 (20)


Then the average voltage generated across a rectifier in series, with a

thermally equilibrated resistor, is proportional to b; and








V = V + V

= 0


= Vo = b (21)


= 4(R + r) kBT Av

For the average square current, one obtains the following:

= 0

4(R + r) kBT Av
B (22)
(R + r)2 + X2 (R + r)2 + X2

This means that the DC component of the voltage fluctuations just

equals the rectified voltage due to thermal fluctuations in the resistance

around the circuit. This is in agreement with Kirchoff's law. There is

no rectified current and no net potential develops around the circuit.

Experiments always show that any voltage fluctuation in the rectifier is

always compensated by equal fluctuations of voltage in the resistance ele-

ments of the circuit.

The conclusion is that a Maxwell's demon is impossible and, as such,

cannot be invoked as a biophysical mechanism. We have read papers that

accept most of this premise but hold, for example, that myosin can exploit

ATP as an external energy source in a nonequilibrium way to make Maxwell's

demon an appropriate biophysical mechanism for doing mephanochemical work

in muscle. This is incorrect because to use ATP as an energy source to

sort out useful fluctuations is conceptually no different from using a

light beam as an information retrieving energy source. In addition, the

use of chemical potential energy has the additional problem of very weak

short range electromagnetic interactions (discussed previously). The








essential feature of both processes is that one must introduce irreversible

elements for a Maxwell's demon to function. The introduction of energy and

entropy into the system is always greater than the net work accomplished by

the demon. Of course, once this is done it is no longer a Maxwell's demon,

but a conventional thermodynamic device. This must be true by the second

law of thermodynamics.


The Dynamics of Intramolecular Protein Motion:
The Langevin Equation

The next step in this discussion is to begin an investigation of the

dynamic details of intramolecular motion in proteins. Generally, we will

follow two paths. In this section we will discuss the Langevin equation

for protein fluctuations. In the next section we will pursue a more

general discussion involving the Fokker-Planck equation. These two treat-

ments will provide two points of view to some of the same questions and

suggest equivalent answers to them. The reasons for doing this are more

important than as an exercise to derive the same answer from two somewhat

different approaches. Using the Fokker-Planck formalism we can obtain

kinetic rate constants and use a formalism that can be numerically extended

to simple systems with complex interaction potential functions.

The Langevin equation for simple harmonic motion is physically unam-

biguous and easy to solve. It provides a foundation for a qualitative

understanding of dynamical properties in proteins. For the first time, we

can theoretically estimate the time constant for a protein conformational

relaxation. This section will imply another microscopic constraint on

fluctuating enzyme mechanisms. The Langevin equation can be generalized

to more complicated and realistic forces of interaction, but these equations

require numerical solutions. This generalization does provide a formalism








that can be used for a full-blown classical many-body treatment of protein

mechanism. The Fokker-Planck equation is conceptually more obscure to

understand and technically more difficult to solve than the Langevin equa-

tion, but the investment of effort is worthwhile for several reasons. The

central one is that we can easily distinguish between the dynamics of

fluctuational changes and deterministic motion. By an estimation of first

passage time, we can compute the average time for a protein to relax or for

a protein to undergo thermal fluctuations. With this information, we can

estimate chemical rate constants for conformational motion or for an enzyme

mechanism in which conformational change is the rate-limiting step. We can

make a prediction about the behavior of enzymes whose mechanism undergoes a

considerable configurational modification in solutions of varying viscosity.

We can unambiguously show that configurational intramolecule fluctuational

motion is not a possible complete mechanism in a system that must achieve

an activation energy typical of enzyme catalysis. Finally the Fokker-Planck

equation, which is a diffusion equation with drift, has been solved for a

particle in several different fields of force. In fact the equation can be

cast into a Schrodinger like equation, so that an immense literature of

mathematical physics is available to solve it.

If one wishes to solve a problem in the classical dynamics of a system,

he must set up and then solve Newton's second law as it applies to the sys-

tem. When this equation of motion has at least one time-dependent random

and fluctuating force, it is called a Langevin equation. Because we need

to solve the problem of a protein subject to fluctuating bombardment by the

solvent, we need to solve this equation of motion.

We have argued in this chapter that if conformational fluctuations are

important in an enzyme mechanism, they cannot involve all the possible struc-

tural degrees of freedom in protein. For fluctuations to be the central








driving feature of enzyme mechanism, the protein must be constrained so that

thermal excitations from the solvent will distort the molecule in only a

very few ways--less than nine. For the sake of argument, we will consider

the simplest case. Let our model protein have only one degree of freedom.

The easiest way to imagine this is to let the enzyme be divided into two

equal parts. These two parts have stiff internal structures that are imper-

vious to fluctuations from the solvent. The two lobes are connected to one

another by a region of much weaker bonds, so that any fluctuation in the

buffer is likely to manifest itself as strain (Ax/x) in these bonds. One

degree of freedom does not mean one chemical bond. This weakly bound region

could encompass many bonds over a large area; it is only necessary that the

two lobes remain relatively rigid, and the binding site be small compared

to the dimensions of the lobes. We imagine, for example, that the active

site is in this weakly bound region and the substrate is anchored to both

lobes. Fluctuations in the system could induce strain in the substrate and

facilitate catalysis. Systems with more than one, but only a few, degrees

of freedom are more complicated to model but conceptually they are no dif-

ferent from our one-dimensional case. So we will view the protein as a

small elastic system with masses connected at the ends. The system tries

to restore its equilibrium separation in the face of fluctuations from the

solvent. Any reasonable equilibrium position in a conformational potential

energy curve is a continuous concave potential well that can be approximated

by a quadratic harmonic potential, E ~ (1/2)kx2. As long as deviations from

the minimum energy position are not too great, this potential works very

well. The harmonic approximation is qualitatively correct and quantitatively

tolerable to within an order of magnitude.

The question--what is the influence a fluctuating Brownian force has

on a simple harmonic oscillator--is a classic problem in physics. It is








solved in many places; for example, see Chandrasekhar (1943), and Wang

and Uhlenbec (1945). We have never seen a treatment of this problem applied

to a protein model of any kind, so one is provided to the reader in Appendix

D. This calculation accomplishes three things. First, it devises a formal

solution to the harmonic Langevin equation (see Equation 32). Notice that

the solution is an exponentially damped equation of motion, with a long-

time residual mean square displacement from equilibrium. Second, the appen-

dix suggests a quantitative estimate of the long time rms displacement and

decay constants using the mechanical values for our model protein on page 49.

Finally, we can obtain a feel for the numerical range of decay times sug-

gested by the variation in protein bulk modulus.

For a protein made entirely of rigid bonds, even at the point of con-

figurational strain, the relaxational response of the macromolecule to

thermal fluctuations from the solvent is a damped harmonic process (see Equa-

tions 29 and 31 in Appendix D).


kBT
= Ae-t/2m eit+ + k (23)


The decay constant for this process is of the order of 1011 sec-'. The

relaxation of this motion occurs in about the same time necessary to excite

the conformational change itself as estimated by Equation (1) in Appendix D.

In the time the fluctuation has occurred in a stiff protein, the distortion

in structure has relaxed back to its long time equilibrium configuration.

Conformational fluctuations in soft regions of a protein behave differ-

ently (see Equation 34, Appendix D). The response of a protein to a solvent

fluctuation is viscous damping of its structural motion as it relaxes back

to its equilibrium position,








~ Ae'2kt/c + (24)
k

Because of the variation in bulk modulus discussed earlier, the decay con-

stant of this process is considerably larger--about 108 sec-1. Thus the time

necessary for the protein to reestablish its equilibrium structure is on the

order of 10 ns. This time is not as large as an average enzyme reciprocal
-1
catalytic rate constant, kcal, but it is much longer than the time of a quantum

of vibrational energy. Further, it begins to suggest a theoretical answer

to the old question of why an enzyme takes so long to perform its particular

chemistry compared to that of the uncatalyzed chemical relaxation.

Another feature that evolves from Appendix D is the appearance of an

equilibrium root-mean-square displacement. The enzyme is continually sub-

ject to thermal fluctuations from the heat bath. Before the enzyme relaxes

to its ground state configuration, another fluctuation comes along. On the

average, the enzyme is always subject to some tension. This deviation varies

for soft to stiff proteins as follows:

O 0
8.3A > 7kx2> > .083A. (25)


This microscopic estimation of the rms displacements agrees with the macro-

scopic thermodynamic calculation made earlier in this chapter.

These numbers represent dynamic extremes in the protein. We certainly

expect intermediate cases to be observed in enzyme structures. For example

McCammon et al. (1977), in their molecular dynamics study of bovine pancre-

atic trypsin inhibitor, find fluctuations of atomic displacement in tightly
0 0
bound regions less than .01A and as large as 2A in more flexible regions.

Similarly beautiful studies of Debye-Waller temperature factors in the met-

myoglobin structure (Frauenfelder et al., 1979) and the lysozyme structure








(Artymiuk et al., 1979), report rms displacements in atomic coordinates of
0 0
.3-.5A and main chain and side chain displacements of 3-4A. The larger

fluctuations correspond to regions at the elbow of a B-turn with long arms,

or at terminal ends of a protein. These parts of a protein are expected to

be held less rigidly, and might reasonably be expected to move about with
0
displacement between the Langevin equation prediction of 10 A.


The Dynamics of Intramolecular Protein Motion:
The Fokker-Planck Equation

The Langevin equation is a stochastic differential equation of motion.

Using it one can compute the average value and the time dependence of a

stochastic variable--for example, position or velocity--under the influence

of an external force. The Langevin equation does not explicitly tell us

about the distribution of variables, or how that distribution evolves in

time. To obtain this information, we need a probability distribution func-

tion for the stochastic variable. This can be obtained by solving the

Fokker-Planck equation. This equation describes the manner in which an

arbitrary distribution of Brownian particles approaches the state of thermo-

dynamic equilibrium. It can be derived in a number of ways and is usually

discussed in graduate texts of statistical mechanics. In addition to the

Langevin equation, one can start from the master equation--the Chapman-

Kolmogarov equation--or the Liouville equation; and with appropriate

approximations, derive the Fokker-Planck equation.

As a theoretical tool of physics, this equation is mostly used to

study Brownian motion phenomenology; but it is also used in the study

of kinetic theory and the problem of approach to equilibrium. Mathematical

biologists in the last decade have used it to study a variety of diffusive

effects in birth and death processes, epidemiology, population dynamics,

population genetics, neuron excitation, chemical kinetics, and excitation








diffusion in photosynthetic centers. Appendix E is neither a derivation,

nor a thorough study of the multidimensional phase space Fokker-Planck

equation; but it does provide the reader w-th an outline of the one-dimensional

configuration space equation and all the relevant mathematical properties

and results we will use in this study. The Fokker-Planck equation is a

sophisticated diffusion equation. It is often called the diffusion equation

with drift. The equation describes the diffusive motion of an ensemble of

particles while under the influence of an additional force. For example,

the Fokker-Planck equation might describe the motion of a sea of electrons

in a conductor while being pushed along by an externally applied electric

field.

Because of the 6N problem, we have seen that fluctuations in all pos-

sible degrees of freedom in a protein involve so many configurations that

it takes too long for the enzyme to reach a catalytically appropriate

structure. We have seen that if an enzyme takes 10-3 seconds to work, then

there must be less than 10 degrees of freedom in a protein driven by a

fluctuating process. In fact, this statement is optimistic; a protein must

have fewer degrees of freedom than ten if it is to extract much strain

energy from the reservoir. In our enumeration of structurally distinct

configurations, we did not address the question of the magnitude of an

elastic distortion. For example, two masses connected by a spring stretched

an amount 6 is not structurally distinct from the same system distorted 26.

These two cases were not counted as structurally distinct in Appendix C.

Energy fluctuations of kBT in one degree of freedom occur all the time in

any system; fluctuations of 10 kBT are rare. The argument for fluctuation

induced catalysis involves the enzyme waiting a long time, kcat, until the

large, structurally acceptable fluctuation in energy occurs at the active








site. A protein of 10 degrees of freedom needs 10-' seconds to sample all

10' structurally distinct configurations only once; most of these are only

kBT = .59 kcal/mole. Such a protein must wait much longer to reach one

particular configuration of an activation energy 10 kcal/mole.

The purpose of this section is to address several questions. We will

ask how much thermal energy can a protein extract from the surrounding heat

bath through thermodynamic fluctuations. We will ask how long a protein

requires to execute intramolecular motion that is a consequence either of

fluctuating or deterministic motions. We will use that information to

construct analytic rate constants when intramolecular motion is the rate-

limiting step in the protein mechanism. We will ask what influence solvent

viscosity has on intramolecular motion in proteins.

We will study these problems in the simplest possible way, then embellish

the results with some complications at the end. We will model a fluctuating

protein that has only one structural degree of freedom. Myosin, hexokinase,

or allosteric proteins, whose quaternary structure consists of two subunits,

are systems I have in mind for this study. We imagine a protein constructed

of two lobes connected together at the active site or allosteric control

site. The lobes of the protein are internally stiff structures whose ter-

tiary configuration corresponds to a deep, narrow potential well. Their

secondary structure is made of well-packed a-helices, B-sheet, a generous

portion of cystine crosslinks, directionally well-satisfied hydrogen bonds,

and salt bridges. The structure of these lobes is sufficiently stable so

that very large solvent fluctuations induce only translational or rotational

diffusion of these lobes, while the internal degrees of freedom are essen-

tially frozen to any significant structural distortion. At the junction of

the two fragments are weaker bonds. These bonds are easy to distort and








the two rigid regions can twist, bend, and separate about this region.

Many biomolecules are known to have this "hinge" structure.

We will perform a model calculation of a two-lobed protein with one

degree of freedom. We will assume the two lobes have an equilibrium con-

figuration separated by x Due to thermal fluctuations, the lobe separa-

tion coordinate, x, can change, The separation coordinate can be an angle,

a pure displacement, or a displacement measure of one lobe slipping over

another, breaking and reforming weak Van der Waal bonds. We immerse many

similar systems in a solvent heat bath to construct a canonical ensemble of

proteins. We calculate the conditional probability distribution function

that describes the time evolution of their state, given that they all start

in the ground state x In particular we ask how long it takes this popu-

lation of one-dimensional systems, on the average, to reach a configuration

larger than xo by an amount 6, given that they all start in the state x .

The rigid lobes separate from one another under the influence of fluctuations,

but they must do this in the face of a tendency for the protein to reassert

its own original equilibrium separation of x This process is not simple

diffusion, but diffusion coupled to a restoring force. Eventually a large

fluctuation, or many similarly directed small fluctuations, will influence

a protein to distort its internal coordinate by an amount 6. When this

happens, the system has developed the requisite conformational change that

enables a subsequent chemical process to occur. After the chemistry has

occurred, the system relaxes and starts over.

The thrust of this dissertation is an effort to distinguish between

intramolecular conformational changes that are a consequence of fluctuations,

and those that are a relaxational response of the protein to its own internal

forces. The forces occur as the protein responds to the changing structures








between free enzyme, enzyme-substrate complex, and the transition state.

We will, therefore, make another model calculation. In this case we will

consider a protein that has reached an energetically unfavorable state with

the lobes separated by x The protein may have reached this distorted

state by a fluctuation, or it may have been placed in a state of unstable

equilibrium by a previous process and awaits the appropriate structural

trigger that allows it to relax to the ground state. If there is no

impediment, the protein simply relaxes to the ground state. Otherwise, a

trigger--which might be a fluctuation, the binding of substrate, a conforma-

tional change of a catalytic side chain, or the completion of a chemical

transformation at the transition state--allows the process to proceed. Once

this process is initiated, the two lobes repel or attract one another

reducing their potential energy until they reach a lower equilibrium state

of free energy. During this process, fluctuations still occur as the lobes

are buffeted about by the solvent bath. However, these fluctuations are not

the driving force of separation. Fluctuations carry the configurational

strain arising from an imbalance of forces from within the protein and

releases energy to the solvent, but the fluctuations that tend to drive the

lobes back toward xo must work against the tendency of the protein to relax.

Only very large, or a series of unusual directed fluctuations, will drive

the two lobes to their original separation. Whenever this happens, the

system starts the process over and the molecule relaxes again. Forward

fluctuations from the solvent enhance the fluctuations of the lobes to the

new ground state.

Essentially this is a two-state model. Initially the system is at

state x The system evolves to the state x0 + 6, where a chemical trans-

formation occurs. In a fluctuating mechanism, the protein tries to








deterministically reassert its equilibrium state, x in the face of

fluctuations that will drive it up the potential barrier to state x0 + 6.

In the deterministic mechanism, the protein is in the unstable state, x ,

and falls to the ground state, x0 + 6, while it is meanwhile subject to

fluctuating forces from the heat bath.

The biochemically relevant information that one would like is the

first order rate constant for conformational change. This number is essen-

tially the reciprocal of the average time required of the protein ensemble

to undergo conformational change. We want to compute the mean first passage

time for our two-lobed protein to extend from xo to x0 + 6; that is, to

stretch by an amount 6. We perform this calculation for one system driven

by fluctuations and another driven by intramolecular relaxational forces.

It turns out that finding the mean first passage time of a process governed

by a Fokker-Planck equation does not require a formal solution to the equa-

tion, it requires only the knowledge of the functional coefficients to the

derivatives in the equation. From these functions, we can mathematically

construct the mean first passage time. From the reciprocal of the time,

we can estimate the first order rate constant for fluctuational and relax-

ational conformational changes.

These calculations are presented in Appendix F. In this calculation,

we derive the time necessary for the enzyme to relax to a lower energy

state due to a configurational change. This is Equation (24) in Appendix F:


M%(610) = t e- + 1 (26)


We also calculate the time necessary for the average enzyme to distort

through a fluctuating mechanism. It is Equation (22) in Appendix F:









M (610) = to e 1 (27)


In these formulas, two sets of constants have been gathered for convenience:

E is a dimensionless parameter, it is a ratio of the elastic strain energy

in the protein to thermal energy kBT; to is the time constant of the process.

Thus,
2K6
k T (28)


to = (29)
"o K

Equations (26) and (27) can be simplified somewhat. A typical enzyme cata-

lytic mechanism must supply at least 10 kcal/mole of activation energy to

the substrate in order to effect its transformation to product. If all or

at least a significant part of that energy is induced through a strain in

the active site, corresponding to either a relaxation or a fluctuation

induced conformational change in the enzyme, then we can estimate e to be


S- (2)(10 kcal/mole) ~ 37
.595 kcal/mole

This number is much larger than 1, so we can simplify the equations for

mean first passage time as follows:

M (610) = t (1 ) t for E >> 1 (30)
and

Mt(610) = t ( e) for e >> 1. (31)

Let us estimate to, using the numerical values for a standard protein.

These can be found in this chapter and Appendix D:








~ 6nma = 67r(103 Ntms)(15.9 x 10-1 m) 2.99 x 10-11 Nt-s/m,


6 ~ 5 x 10-1o m,


K ~ K 10-1 Nt for a soft protein (perhaps as small as 10-12 Nt,
see Appendix D),


t ~- C 3 x 10-11 Nt-s/m)(5 x 10-~ m) ~ 1.5 x 10-9 sec to 1.5x 10-sec.
S K 10-11 to 10-12 Nt


The reader should note that this relaxation time numerically agrees with the

reciprocal of the decay constant for the Langevin equation treatment (see

Equation 30 in Appendix D).

The graph in Appendix F shows the fundamental difference between Equa-

tions (26) and (27). It takes essentially no time to separate the protein

lobes if no energy barrier is involved, and so both equations vanish at

c = 0. Of course we expect it takes a long time for fluctuating motion to

impart a separation, compared to relaxational motion. For very large fluc-

tuational energy barriers, we expect to wait a very long time for the appro-

priate catalytic configuration to occur.

We can write down the first order rate constant for a system that under-

goes a relatively large energy conformational change. This rate constant

is the reciprocal of the mean first passage time and can be expressed as

k (l + 1 for >> 1, (32)
Sto E t
0 0
or
-E:
k -e for E >> 1. (33)
t


Several interesting conclusions can be drawn from this calculation.

The relaxation results (Equations 26, 30, and 32) are not very interesting

because they are not at all surprising. In particular, they agree with








other estimates in this paper and a host of other experimental and theoret-

ical observations, as we have already discussed. They reflect the deter-

ministic character of the process. As K6 gets larger, the influence of

fluctuations abates and the relaxation time reaches a limiting time to

We will for the most part focus our attention on the up ramp potential

(Equations 27, 31, and 33). These equations describe the kinetics and

dynamic properties of a true one-dimensional fluctuating mechanism. In

what follows, we will continue to discuss the standard protein as we have

done throughout this paper. Enzymes have catalytic rate constants of

kcat ~ 1033 s-1, but typically most have rate constants of 103 s-1.

This means that the turnover times of such enzymes are on the order of

10-3+1 sec/reaction.

We offer, as a suitable constant to any proposed fluctuation mechanism,

the experimental observation that enzymes take about one millisecond to work.

Consider the value of the up ramp, mean first passage time for an energy

parameter, e = 37 (10 kcal/mole). This is obtained from Equation (33) to be


toe (10-8 sec)(e") 3
t t0E 37 ~ 3 x 106 sec 37 days.


No enzyme takes 37 days to perform one catalytic turnover. We conclude

that enzyme-substrate complexes do not surmount a typical activation energy

barrier through a fluctuational process. In the language of this chapter's

introduction, one would say that this microscopic model of fluctuating

mechanisms provides a macroscopic thermodynamic consequence that is not

observed.

It turns out that there are investigators who are more circumspect in

their claims on this phenomena. The most cautious arguments suggest that








an enzyme is a very complicated system; and its mechanism involves many

intermediate steps. One of the many steps in some enzyme facilitated reac-

tions may involve or be triggered by a fluctuation in the tertiary structure

of the enzyme-substrate complex. We are in a position to estimate the

largest possible energy barrier that an enzyme may overcome by means of

thermodynamic fluctuations. The two requirements placed on the system under

study are the Fokker-Planck analysis of the fluctuating rate constant and

the experimental observation of a 10-311 second turnover time. We can com-

pute the energy parameter E that is associated with an enzyme taking only

10-31 seconds to function. This is done by using Equation (31) to obtain

toe
t-

E
10-3"' sec = (10-8 sec) e,

and

e = 1051

where

16.7 > e > 11.7.

But e = 2EA/kBT, where EA is the activation energy barrier of a fluctuation.

Therefore, we obtain


3.4 kcal/mole > EA > 4.9 kcal/mole. (34)


If the enzyme is to act in a time of 10-' seconds, the largest amount

of heat energy the enzyme can momentarily extract from the heat bath, for

the purpose of inducing an energetically unfavorable (but catalytically

useful) structure, is less than 5 kcal/mole. This is approximately the

energy of one hydrogen bond. Extracting more energy than this requires the








newly formed enzyme-substrate complex to persist longer than the observed

enzyme-substrate lifetime. This maximum of 4 kcal/mole can only be extracted

by the simplest imaginable system: one with only one internal coordinate.

A protein can achieve less than half of its activation energy by fluctuations.

The rest of its energy must be obtained very quickly, through other chemical

means. It must be obtained with only one degree of freedom.

Let us consider what happens when we remove the constraint that the

enzyme has only one degree of freedom. Now we must wait longer than t e-le-"

for a structural fluctuation of appropriate configuration and sufficient

energy to augment catalysis. The mean first passage time is just long

enough for only one of those 6N structurally distinct configurations to

achieve the right energy. For an enzyme of several degrees of freedom, we

clearly must wait 6N times longer for the catalytically appropriate struc-

ture to occur by fluctuations. Thus,

t = N e-_ (35)


We will construct a table of calculations similar to Equation (34)

and estimate the maximum energy associated with an experimentally observed

rate-limited fluctuation mechanism in a protein of N degrees of freedom.

These calculations are predicated upon an enzyme reciprocal catalytical

rate constant of k-c ~ t 10-3 sec. These calculations also use to = 10-9
cat o
sec, rather than 10-8 sec; this is the lower limit estimation of Equation

(29). Using the smaller time constant, one calculates activation energy

barriers that are larger by about one quarter of a kcal/mole. These esti-

mates of activation energy are relatively insensitive to large changes in

relaxation time constants. For purposes of tabulation, the case yielding

the most generous quantitative estimate of EA was chosen (see Table 1).





84



Table 1. This table suggests the maximum activation energy barrier that
can be achieved by a fluctuating mechanism in a protein of N
degrees of freedom. These calculations are subject to the
experimental constraint that typical enzymes have catalytic
rate constants of 103 sec-1.


-1 e = t6-N/t


e EA(kcal/mole)


1.67 x 105


2.7 x 104


4.6 x 103


7.7 x 102


1.2 x 102


6 4.66 x 104


7 2.79 x 105


8 1.67 x 106


21.4


3.58 *


.598 *


the others in this
6 the others in this


216


1276


7776


14.7


12.75


10.82


8.83


6.7


4.59


2.55


.87


4.37


3.79


3.2


2.62


1.9


1.4


.758


.258


*These calculations use Equation (27) modified by
column use Equation (31).








We conclude from Table 1 that if the enzyme has as many as 8 degrees

of freedom, it takes the protein 10-3 seconds to sample all fluctuations

as large as kBT in energy. In no case does an enzyme achieve catalysis

through a fluctuating mechanism by increasing its energy more than about

4 kcal/mole. If it does this, the protein must arrange for the number of

structural degrees of freedom in the system to be very small indeed.

There is another feature of this study that merits notice. All the

rate constants associated with conformational changes are proportional to

reciprocal solvent viscosity. Thus,

k c t K K 1 (36)
cat o ~7 667ar n"

This rate constant is analogous to kcat if conformational changes in

the enzyme are the rate-limiting step in the protein mechanism. This

equation says that a reciprocal solvent viscosity dependence of kcat is a

signature to enzyme mechanisms that involve significant conformational

changes. It is generally understood in the biochemical community that

conformational changes in a macromolecule are slowed in solutions of higher

viscosity, butwearenot aware of any quantitative or experimental study of

this; and weknow of no prediction of the appropriate functional dependence

between the two ideas.

The reader should not be misled by the fact that general second order

rate constants can be proportional to n-1. This is because the rate con-

stant is occasionally limited by the rate of diffusion of the two species

toward one another, and the diffusion constant has a viscosity dependence

(Atkins, 1978). Thus the binding constant of substrate to enzyme in the

Michaelis-Menton scheme has a trivial viscosity dependence. Once the sub-

strate is bound, no more bulk diffusion occurs and one expects the kcat

(the rate that E+p is formed from [E-S]) to be viscosity dependent. If








the chemistry of product formation does not involve global conformational

change in the enzyme, then there is no viscosity dependence between kcat.

On the other hand, if segments of the protein have to sweep through a vis-

cous solvent, then one expects this to influence the rate of the transfor-

mation. The reciprocal viscosity dependence of kcat does not distinguish

between a fluctuating and a deterministic mechanism.

There is one more fluctuating enzyme mechanism that we can discuss.

Recently, Careri (1974) has proposed that structural fluctuations in

enzymes might be driven by a different kind of fluctuating force; namely,

one due to a fluctuating electric field. All biological fluids are buf-

fered salt solutions. These solutions are overall charge neutral, but the

dissolved components carry positive and negative charge. This charge

arranges itself in solution so that each ion is surrounded by a counter

distribution of opposite charge and polarized solvent. Associated with each

of the floating charges in the solution is an electric field that emanates

into the solution. Any macroscopic estimation of this field vanishes be-

cause of global charge neutrality. On the microscopic level, this field

does not vanish; and it is not even very small. Also, on the microscopic

level, the induced polarization and the counter ion distribution has the

effect of reducing the electric field; but, it does not eliminate it.

Biochemically relevant ionic solutions have ionic concentrations of about

1 millimolar to 1 molar. If one calculates the average distance between
0
the ions, one finds that from 10-100A separates ions in such an ionic solu-

tion. This is a distance comparable to the size of proteins, and we conclude

that macromolecules are individually influenced by the isolated charges. It

is more complicated. The ions drift about under the influence of solvent

bath fluctuations; consequently, there is a microscopic fluctuating electric








field in the solvent. These field variations extend over distances much

larger than a protein, and they last for a period of time about equal to

the structural decay time of a protein.

Generally, proteins contain some charged amino acids that carry charge.

Even if a protein does not have a net charge, it usually does have several

oppositely charged ionic side chains in its primary structure. When charged

moieties are buried within the protein, opposite charges attract one another,

and they usually are effectively neutralized through their close proximity

with one another. This structure is called a salt bridge. Nonetheless, a

protein is often found to have non-neutralized charged side chains in its

structure. In particular, charged groups--especially histidine side chains--

are often found at the active site in a hydrophobic pocket. The charge on

these side chains is shielded from interaction with the solvent. The ionic

form of histidine is often involved in catalysis because its pK is near 6.

The hydrophobic pocket controls its local pK and protects the positive charge

of the amino acid. If at least two (or several) charges are found in a

protein and they are found on adjacent degrees of freedom, one can sensibly

ask how these charges respond to a fluctuating electric field in an ionic

solvent.

Does the response of the charged residues transfer distortion to the

protein that can be used by the enzyme-substrate complex in a catalytic

mechanism? We address this question quantitatively in Appendix G. This

appendix repeats again the same calculation in Appendix F. The difference

is that the microscopic ionic solution electric field generates the force

on the protein. Comparing Equation (12-b) in Appendix G with Equation (9)

in Appendix D suggests the first quantitatively important difference between

the two mechanisms. The fluctuating electric force on a protein is an order








of magnitude smaller than the fluctuating hydrodynamic force. We expect

the influence of the electric fluctuations to be comparably less.

It turns out that the time involved for proteins to make a biochemi-

cally useful conformational change is extremely long (see Equation 22 in

Appendix G). Protein conformational changes driven by ionic fluctuations

requires so long a time to occur that one concludes such mechanisms are

not pertinent to intramolecular motion in proteins. The reason for this

is more subtle than just the fact that electric fluctuating forces in ionic

solutions are smaller. We will digress slightly to explain why this time

is so long.

Consider fluctuations in a galvanometer spring suspension. These

fluctuations are responsible for the noise, and limit any physical measure-

ment involving this instrument. The analysis of this problem is identical

to the Langevin treatment of a harmonic system that we used to model the

protein in Appendix D. The fluctuations in the mirror position are driven

by hydrodynamic fluctuations in the air. The Brownian motion of the gal-

vanometer is opposed by the spring and damped by the viscous drag of the

air. The system reaches an equilibrium state of continuous fluctuations

in which the theoretical root-mean-square deviations in amplitude reach an

experimentally confirmed number. One might naively suspect that if the

same experiment was performed in a vacuum, or in a low molecular weight

gas atmosphere, then the mirror fluctuation magnitude would diminish. This

is not true. The mean square deviations of the spring is given by Equa-

tion (33) in Appendix D as

kBT
=-7- (37)


The magnitude of deviations depends on the spring constant and the








temperature of the media. It does not depend upon pressure, particle

density, particle size, or intramolecular interactions. Even at the great-

est attainable vacuum many molecules bombard the mirror. At low pressure

the molecules collide with the mirror less frequently; but the viscous drag

force on the mirror is also less. The mirror amplitude fluctuations are

therefore larger. When the pressure is large, more collisions occur but

the influence of each collision is less. The net effect is that at all

pressures the mean square amplitude of thermal fluctuations is the same,

but at low pressure the power spectrum is shifted to much lower frequencies.

Therefore the time between fluctuations is, on the average, longer.

In our original problem we compared hydrodynamic fluctuations and

electric field fluctuations. A one millimolar solution of ions has charged
0
species about 100A apart. In an aqueous buffer, the water molecules are

relatively close together. Ionic fluctuation centers are dilute, and

hydrodynamic fluctuation centers are concentrated. The effective ionic

pressure is less than the effective hydrodynamic pressure. Crudely speaking,

the reduction in pressure is proportional to the reduction in density; so

n/V ions P 10- moles/L P (ls-).
Pi w n/V water w 55.5 moles/L w

The ionic pressure in this experiment is much less than the hydrodynamic

pressure on the protein. We expect, by analogy to the galvanometer, that

the mean square deviations are about the same size in both hydrodynamic and

ionic fluctuations. One also expects that the time between fluctuations

for ionic fluctuations is much longer. This is part of the explanation.

Unfortunately, it is slightly more complicated. Ionic electric field

fluctuations are the driving force, but hydrodynamic viscous drag forces

are the damping mechanism. This retarding force is constant, and it does








not depend on ion concentration. What this means is that the fluctuations

are not only infrequent, but when they occur the motion is quickly damped

out. Consequently, it takes a very long time for electric field fluctua-

tions in an ionic solution to affect the structure of a macromolecule in

any biochemically useful way. Ionic fluctuations are not significant for

the purpose of inducing conformational change in molecules.

The last paragraph of Appendix G presents the results of a calculation

in which one recasts the problem. Here we ask how long it takes for ionic

fluctuations to induce cleavage in a hydrogen bond structure. The result

is still too long to be biochemically useful. This does not mean that

fluctuations do not break hydrogen bonds in water. Hydrodynamic fluctuations

continuously break hydrogen bonds, but ionic electric field fluctuations do

not.

It is useful to view this entire question in one more way. Investi-

gators who argue for thermodynamic fluctuation mechanisms in biochemistry

have perhaps misled themselves by looking too closely at Eyring's absolute

reaction theory (Atkins, 1978). According to this theory, the absolute

rate of a microchemical process is determined by fluctuations in an ensemble

of similar systems. The rate is given by

kBT AE
k exp(k A). (38)


These workers plug-in a typical activation energy barrier of 10 kcal/mole

and obtain

S(1.38 x 10-23 J/K)(300K) exp (-10 kcal/mole)
6.6 x 10- J.Sp 95 kcal/mole


k ~ 3.15 x 105.


(39)








This is a high but respectable catalytic rate constant for an enzyme.

Eyring deduced and applied Equation (38) to a very simple chemical process.

The surprising fact is that it works, if used carefully, in most chemical

processes. This is because the assumptions on which it is based are so

general. Any chemical process that must achieve an activation energy to

occur will proceed, if the high energy tail of the reactants probability

distribution function has enough particles with enough energy. These

functions do not usually depend on the details of the molecules. The

particles usually do not deviate significantly from the Maxwell-Boltzmann

distribution. The probability that a molecule has energy AEA is then given

by the Boltzmann factor, exp (-AEA/kBT). Therefore, the Eyring theory

rests on fluctuations and the form of molecular probability distribution

functions. Perhaps it is only natural to think of enzymes the same way.

The weakness of the Eyring theory, when indiscriminantly applied to

enzymes, is that the rate of an enzyme catalyzed process critically depends

on the shape, energy, and configuration of the substrate and the enzyme.

Further, enzyme chemistry is not a microchemical process. A substrate

works with one enzyme not because of stochastic influences, but because the

protein evolved to perform that particular task. Usually slight deviations

in the structure of a substrate molecule drastically reduce the enzyme reac-

tion rate. It is difficult to understand enzyme specificity if the system

is driven by fluctuations. One would think fluctuations of slightly dif-

ferent configurations are just as likely to perform their task on a substrate

of slightly different shape.

The global Eyring model of enzyme reactions has another problem. The

pre-exponential factor, kBT/h, is the frequency of one quantum of vibrational

energy. This is the smallest frequency that nonelectronic atomic vibrations








can have in a molecule. The number t/kBT is roughly one period of one

atomic vibration; it equals 10-13 seconds. This time, 10-13 seconds, is a

suitable time for discussions of micromolecular processes, such as biomole-

cular collision reactions. It is not suitable for macromolecular biochemi-

cal reactions. Certainly, at the critical moment of catalysis, in the

transition state, some parts of the active site dynamics occur in picoseconds

and the vibration of certain atoms is important. For the rest of the 10-3

seconds it takes an enzyme to work, it is just not relevant whether some far

flung hydrogen--nowhere near the active site--is in one quantized vibratory

state or another. During most of a catalytic mechanism, a biochemist worries

about side chain orientation and changes in secondary structure. The motion

of side chains and the rearrangement of backbone tertiary structure are

critical biochemical degrees of freedom used by the enzyme for catalysis.

The time for this process is what should be in an Eyring type rate equation,

not an atomic vibratory time. The time for a side chain fluctuation is

10l--10-9 seconds.

Perusing a table of amino acid structures will reveal to one that amino acids

have side chains made of about ten atoms. Peptide bonds have six atoms.

Using the 6N argument, we can estimate the number of structurally distinct

configurations in one of these degrees of freedom as follows:

6102 ~ 1072 for amino acid side chains, and

66 ~ 104.6 for peptide bonds.


Because of thermal fluctuations, each of these configurations is developed

from time to time. Of those 105 configurations, only six are significant

for the purposes of forming structurally distinct biochemical configurations.

Finally, we ask how long it takes a side chain to transform between two of