1 Journal of Mokcular Liquids. 39, ( 1!388) BCi-206 EJaevisr Science Publishem B.V., Amsterdam - Printed in The Netherlands

CONFORMATIONAL CHANGE, FLUCTUATION AND DRIFT IN BIOLOGICAL

MACROMOLECULES: AN EMPIRICAL LANGEVIN APPROACH

ALAN COOPER

Chemistry Department. Glasgow University. Glasgow G12 8QQ. Scotland. UK.

(Received 14 December 1987)

SUMMARY

The flnlte size of biological macromolecules places them in an interesting class of rrresoscopic thermodynamic systems whose properties are influenced by in teractlons with surroundJng solvent. Not only the magnltudes but also the kinetics of thermodynamic and conformational fluctuations in such macromolecules can give rlee to novel propertles whJch are of bJologJcaJ slgniflcancc. The interplay between intra- and inter-molecular fluctuatlona in protein molecules is examJned hcrc in terms of an emplrical Langevln model for stochastic energy kinetks, in comparison with experimental observations. Particularly interesting is the possibility of thermal hystereele or memory effects which may play a role in bJologica1 control processes.

INTRODUCTION

During the course of evolution. biology has learnt to exploit to lts

advantage many aspects, frequently unexpected. of the physics tind chemistry

of molecular fluids. giving rise to a wide range of macromolecules capable of the

apecifjc tasks required to sustain life in an essentially aqueous environment.

These molecules may be catalysts (enzymes) , transporters. transducers. and so

forth. Without exception they arc macromolecules, i.e. with molecular weights

in excess of several thousand, and_, again without exceptlon, they perform their

assigned tasks with apecifichics and efficiencies that are the envy of the

modern chemist. Study of such systems presents a dlfflcuIt challenge to both

experimsntallst and theoretician alike. But the potential rewards are high.

both In the new physical insights that can emerge and. at a m&e pragmatic

level, Jn the adaptatJon of these principles ln technological and industrial

applications. I wish to discuss here, in a rather sPeculative fashion, some of the

general kinetic and thermodynamic consequences of the macromolecular nature of

biological molecules. I will concentrate on proteins. though many of the

conclualons are equally relevant to other macromolecules in solution. biological

or otherwise. Introductory reviews and background material may be found iu

refs. 1-4.

Proteins are, of course. polypepticles made up of long speciftc sequences of

amino acids, which fold up into untque conformations upon which their functional

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properties depend. Typical dimensions for globular proteins are in the region

of lo-100 Angstroms, and the structures.. unlike small molecules, have

reasonably well defined ‘insides’ and ‘outsides’ - But in contrast with familiar

macroscopic objet%, their surface area to volume ratios are large. Consequently,

such molecules fall into a class of mesoscopic systems, intermediate between

microscopic and macroscopic objects, in which surface interactions with

surrounding solvent molecules can be as important as the internal interactions

within the protein itself. One consequence of this is that proteh

conformations are not static. Many experimental techniques illustrate that,

despite the close-packed nature of the folded polypeptide revealed by X-ray

crystallography, the protein molecule is a quite fluid object in which perpetual

conformational fluctuation is the rule I It turns out that this is an inevit-able

consequence of the relatively small size of the system in thermodynamic terms

(refs. l-3) _ Despite bain g large molecuiea, by conventional standards, they

are still small (i.e. mesoocopic) thermodynamic systems and, aa such, arc

prone.to large thermodynamic fluctuations. Estimates using conventional

fluctuation theory show that the internal energy of a typical protein molecule

fluctuates by many tens of kcal/mole: energy fluctuations which are comparnblc

to or larger than the free energy of stabilization of the folded conformation

itself _ Similarly, volume fluctuations can occur which are large enough to allow

easy access of water, or other small molecules, Lnto the protein interior (refs.

l-3) - Consequently, the detailed conformations of proteins as described by

crystallographic or other structural techniques must be considcrcd only an

average picture upon which is superimposed chaotic motion induced by thermal

interaction (collisions) with the surrounding molecular fluid.

One might wonder whether such conformational fluctuations are an

embarrassment to a biological molecule. since so much of its specific function

seems to require unique stercochcmistry . But experience tells us that biology,

in the course of evolution, tends to take advantage of the inevitable and exploit

it to its own ends. This seems to be the case here. For example, many

proteins reversibly bind small molecules to perform chemical operations that must

be done in the absence of water. The binding and transport of oxygen by

haemoglobin , or the catalytic transfer of phosphate groups are typlcal.

Conformational flexibihty allows the protein to satisfy these rcquiremcnts by

allowing transient pathways for binding or release of substrates to an otherwise

inaccessible binding site. Other functions include the allosteric control of

enzyme function by binding at distant sites on the macromolecule. This too may

be. mediated by dynamic conformational processes (ref _ 5). Active transport by

molecules In membranes or energy transduction by protein in muscle contraction

alB0 ciearly involve dynz%ic .Bte& (refs.. 2.3) 1

197.

The basic principles of thermodynamic and conformational fluctuations in

proteins are now well established and have abundant support from experiment and

molecular dynamics simulations. The nature, magnitudes and, to some extent.

the functional consequences of these fluctuations are reasonably understood..

What we lack at present is a clear description of ths rates of these fluctuations

and the role of these rates in controlling macromolecular function. Experimentally

the rates of conformational fluctuation span a vast range: from less than a

picosecond for small group vibrations or rotations, to several seconds or longer--

for global conformational changes. No single model can adequately cover this

range, but I will present here some preliminary observations using an empirical

Langevin approach which illustrates some of the possible consequeneee of this

large dynamic range and which may heIp ln understanding some of the Ltereating

properties of protein molecules.

FLUC%UATION KINETICS

A macromolecule has many possible states consisting of different

conformationalarrangements of the polypeptide and its side chains, and of

dffferent states of motion of the individual atoms or groups. In the course of

time, any one macromolecule will evolve from state to state as a consequence of ’

thermal motion and interactions with the molecules of the surrounding solvent.

On average, of course. the system will tend to cluster around some mean

conformational state, with mean thermodynamic properties that are observed by

macroscopic measurement. But fluctuations are observed. and these might be

broadly classified into two types: (f) adiabatic or internal transitions involving

redlstrlbution,of energy within the protein molecule, and (ii) non-adiabatic or

external changes involving energy exchange with the surrounding bath due to

solvent molecule collisions, frktlonal damping. and so forth.

Adiabatic fluctuations have been extensively studied by molecular dynamics

simulations over time scales of the order of 1OOpeec (ref. 4)_ Such studies show. .

details both of the relatively fast local motions of molecular groups within the

protein and the somewhat slower evolution of global changes between iso-energetic

large scale conformations. But, despite the wealth of atomic detail that might be

apparent in such computer simulations, they give only a very reatrlcted view of

the @osslbilities. The limited time scale of the simulations implice-that, without

special techniques. the slower transformations relevant to chemical or bioIogica1

function are not covered. Even more important is the lack. so-far, of any

reasonable simu_lation of non-adiabatic fluctuations due to interactions with

solvent molecules, which results ln the exclusion of important activation and

dissipation processes associated with function. We have earlier shown (ref. 3)

how the non-adiabatic fiuetuations in a pro&h molecule r&y de.treatkd In a

general phenomenological fashion using a Langcvin equation approach. essentially

as a sort of Brownian motion in energy space. This lacks the fine structural

detail apparent in molecular dynamics computations, but does allow general

features-to be identifi&d.

If we consider a single protein molecule in solution as a mesoscoplc

thermodynamic system in equilibrium with an infinite thermal bath. then the

time-dependent gnternalenergy of the molecule may bc described, in the linear

approximation limit, by a stochastic differential equation formslly identical to the

Langevin equation:-

dE a= - X(E - <E>) + R(t) (1)

where X is a phcnomenological linear relaxation coefficient representing energy

dissipation to the bath, <E> is the th ermodynamic mean energy, and R(t) is a

random impulse term representing interactions (collisions) with the bath molcculos.

Following standard procedures (ref. 6). the fluctuation and dissipation parameters

are notindependentbut can be related by way of the equtiibrium properties of

the distribution function which. in this case, givesr-

<dE2> = CR2>d2X = kT2cv (2)

where bE=E- (E>; <R'>is the mean square impulse; -ris an arbitrarfly small

perfodoftimeover which each impulse mightbethoughttoact: and the second

equality arises from-the classical exact value of the mean square energy

fluctuation in term& of the heat capacity, cv, of the molecule.

Solutio& of th& Langevin equation gives. at equilibrium, a Gaussian energy

probability profile for the system which is essentially identical to the precise

result obtained L-y tit&r means from experimental data for protein molecules under

normal con&tions (ref. 3). But we also know from this previous work that

protein energy distributions are not always simply Gaussian, particularly under

conditions where the protein might be undergoing gross changes in menn

conformation, unfolding and so forth. To understand this in +x-ma of the

stochastic r&e equation we must examine in more detail some of the underlying

presumptions;

Equation (1) is based on the assumption that the energy dissipation and

fluctuation parameters, X and R, are single-valued quantities for each protein

molecule. But this is potentially anoversimplification since different types of

conformation&l motion within the macromolecule &ill couple differently with the

surrounding medium and will, therefore, show different relaxation properties.

For tinstance. certain dynamic states of the protein might censist~of short-rtige.

.

/. .-. ..,,‘_ .I_. , 1 . ..

:

.199

high frequency motions of individual molecular groups (rotation, vibration,

l i b r a t i o n , e t c . ) w h o s e d i s s i p a t i o n m i g h t r e s e m b l e r e l a x a t i o n - v i a t h e r m a l . •

d i f f u s i o n a c c o r d i n g t o t h e p h e n o m e n o l o g i c a l l i n e a r F o u r i e r l a w o f t h e r m a l

c o n d u c t i o n . ~k i n t h i s c a s e w o u l d r e p r e s e n t s o m e k i n d o f e f f e c t i v e t h e r m a l " " '

C o n d u c t i v i t y p a r a m e t e r . O t h e r d y n a m i c s t a t e s wi l l , a l t e r n a t i v e l y , i n v o l v e l a r g e

s c a l e , l o w - f r e q u e n c y c o o p e r a t i v e m o t i o n s o f e x t e n d e d r e g i o n s o f t h e

m a c r o m o l e c u l e o r r e l a t i v e m o t i o n s o f d i f f e r e n t s t r u c t u r a l d o m a i n s ( e . g . t y p e s o f

m o t i o n t h a t h a v e b e e n d e s c r i b e d f i g u r a t i v e l y a s ' h i n g e - b e n d i n g ' o r ' b r e a t h i n g '

o f t h e m o l e c u l e ) . D i s s i p a t i o n in s u c h c a s e s w o u l d b e m o r e a k i n t o t h e e x p o n e n t i a l

d e c a y o f d a m p e d m e c h a n i c a l s y s t e m s , a n d ~ w o u l d b e s o m e e f f e c t i v e f r i c t i o n a l

o r d a m p i n g c o e f f i c i e n t o f t h e m e d i u m . M o r e o v e r 0 t h e r e l a x a t i o n a n d i m p u l s e

p a r a m e t e r s wil l a l s o v a r y w i t h t h e c o n f o r m a t i o n a l s t a t e o f t h e p r o t e i n . A m o r e

o p e n , ' u n f o l d e d ' s t a t e o f t h e p o l y p e p t i d e will c l e a r l y e x p e r i e n c e m o r e c o n t a c t w i t h

t h e s u r r o u n d i n g s o l v e n t a n d wi l l , t h e r e f o r e , b e s u b j e c t t o e f f e c t i v e f r i c t i o n a l ,

d a m p i n g o r t h e r m a l e x c h a n g e p a r a m e t e r s w h i c h a r e d i f f e r e n t f r o m m o r e c o m p a c t

s t a t e s o f t h e p r o t e i n . C h a n g e s i n a v e r a g e e n e r g y a n d r e l a x a t i o n p r o p e r t i e s

m i g h t a l s o c o m e a b o u t a s a r e s u l t o f f u n c t i o n a l p r o c e s s e s , s u c h a s l i g a n d b i n d i n g

o r p r o t e i n a g g r e g a t i o n .

We m u s t , t h e r e f o r e , e x t e n d t h e s t o c h a s t i c L a n g e v i n d e s c r i p t i o n t o

i n c o r p o r a t e t h i s p o s s i b i l i t y t h a t t h e r e e x i s t a r a n g e o f c l a s s e s o r t y p e s , i , o f

c o n f o r m a t i o n a l s t a t e s w i t h d i f f e r e n t r e l a x a t i o n ( h i ) a n d r a n d o m i m p u l s e ( R i )

parameters. Thus:

d E = ~ ~i (E,t) { - ~ i ( E - ~Ei)) + Ri ( t ) ) (3)

i

where ~i(E,t) is a stochastic function giving the probability that the system is

in a particular conformational class at any one time, and (Z i) is the mean energy

of that class (i.e. the mean energy that would be observed if aH other classes

were inaccessible.

Figuratively this describcs the dynamic conformatlonal motion of the protein

as a sort of weighted random walk across a multi-dimensional energy surface.

This surface might be mapped into different domains representing different

classes of motion or relaxation parameters. Within one domain the motion will be

described by a single Langevi~ equation with parameters appropriate to that

class. But whenever the system encounters a domain boundary it will transfer,

iso-energetically, to the next class and will continue to evolve according to the

new Langevin parameters until another classboundary Is encountered. Clearly,

i n r e a l i t y , t h e b o u n d a r i e s b e t w e e n d i f f e r e n t c l a s s e s wil l b e i l l - d e f i n e d , b u t t h i s

simple model will allow Us to proceed. Under these circunlstances ~b i = 1 when .

the system is in class i, @j = 0 for j ~ i; and the time or ensemble average of

-~..,

• ", ~ . . . . . . . .

• . . . "

200

qb r e p r e s e n t s t h e f r a c t i o n a l o c c u p a n c y Of e a c h c l a s s a t d y n a m i c e q u i l i b r i u m :

NO s t r a i g h t f o r w a r d a n a l y t i c a l s o l u t , o n o f E q u . ( 3 ) i s a v a i l a b l e , s i n c e w e k n o w

l i t t l e a b o u t t h e t i m e d e p e n d e n c e o [ ~b i n r e a l p r o t e i n s . B u t t h e r e a r e t w o e x t r e m e

c a s e s in w h i c h i n t u i t i v e s o l u t i o n s a r e p o s s i b l e .

F i r s t l y , i f %b V a r i e s r a p i d l y c o m p a r e d t o t h e r e l a x a t i o n p a r a m e t e r s ( i . e . t h e

i n t e , : n a l a d i a b a t i c t r a n s f o r m a t i o n s a r e f a s t c o m p a r e d t o e n e r g y e x c h a n g e w i t h t h e

b a t h ) , E q u . ( 3 ) r e v e r t s t o a s i m p l e L a n g e v i n e q u a t i o n ( E q u . 1) i n w h i c h A. R

a n d <E> a r e w e i g h t e d m e a n s o f t h e v a l u e s f o r i n d i v i d u a l c l a s s e s . T h e e q u i l i b r i u m

d i s t r i b u t i o n I n s u c h c a s e s w o u l d b e a s i n g l e G a u s s i a n , w i t h a p p r o p r i a t e m e a n

e n e r g y a n d s t a n d a r d d e v i a t i o n .

A t t h e o p p o s i t e e x t r e m e , t h e d i f [ e r e n t c l a s s e s m i g h t b e s o f a r a p a r t t h a t

i n t e r c o n v e r s i o n i s r a r e o n t h e r e l a x a t t o n a l t i m e s c a l e , s u c h l h a t t h e r m a l

e q u i l i b r a t i o n i n a n y o n e c l a s s o c c u r s w e l l b e f o r e t r a n s i t i o n t o a n o t h e r c l a s s . A s

a r e s u l t , t h e e n r g y p r o b a b i l i t y d i s t r i b u t i o n c o n s i s t s o f a s u m o f G a u s s i a n s w i t h

m e a n s a n d w i d t h s r e p r e s e n t a t i v e o f e a c h i n d i v i d u a l r e l a x a t i o n a l c l a s s o f t h e

p r o t e i n .

I n g e n o r a l , h o w e v e r , t h e s i t u a t i o n i s m o r e c o m p l e x . E n s e m b l e a v e r a g i n g o f

E q u . ( 3 ) i n t h e s t e a d y - s t a t e l i m i t s h o w s t h a t , a t e q u i l i b r i u m :

[ ~i <~i E> = X xi <#i><Ei> i i

This leads to the important conclusion that the average internal energy of the

system and, therefore, its thermodynamic properties can depend on the relative

rates of relaxation of the different classes. For example, even in the trivial and

p h y s i c a l l y u n r e a l i s t i c c a s e t h a t a d i a b a t i c c l a s s t r a n s i t i o n s a r e u n c o r r e l a t e d w i t h

energy : -

<E> = ~- Ai <#i><Ei> / [ Ai <~bi> i i

so that the mean energy depends not or, ly on the fractional occupancies of each

class, <~bi> . but also on the relative values of A i. We shall see later how this

sort of effect is manifest in reM protein systems.

NUMERICAL SIMULATION: A Z-CLASS MODEL

In the absence of a general analytical solution to Equ. 3, numerical

slmulation, using a simple two-class model, can be used to illustrate some of the

general conclusions, Here we assume that the protein molecule may exist in Just

two conformational classes, A and B, with potentially different mean energies and .. . : .

"'- relaxation parameters. Numerical integrations of the appropriate Langevln

• equation' following the evolution of the system in energy Space, wore performed

: 2 0 1 =

h e r e o n a n A p p l e I I m i c r o c o m p u t e r u s i n g a G a u s s i a n r a n d o m i m p u l s q v a r i a b l e . . . .

( R ) . I n m o s t c a s e s t h e m a g n i t u d e s o f R a n d ~ w e r e s c a l e d , u s i n g E q u . Z, t o :~

g i v e t h e e q u i l i b r i u m d i s t r i b u t i o n p a r a m e t e r s r e l e v a n t t o a t y p i c a l p r o t e i n m o l e c u l e

o f 25000 m o l e c u l a r w e i g h t . S i m u l a t i o n s t y p i c a l l y r u n f o r 10000 s t e p s , o r m o r e , :

w i t h t h e p r o b a b i l i t y o f r a n d o m A - - > B o r B - - > A t r a n s i t i o n s d u r i n g a n y

a r b i t r a r y t i m e s t e p c o n t r o l l e d b y a s c a l e d r a n d o m n u m b e r g e n e r a t o r . T h e s e

a d i a b a t i c t r a n s i t i o n s c a n b e s e l e c t e d t o o c c u r e i t h e r o v e r t h e e n t i r e t m e r g y r a n g e ,

o r o n l y w i t h i n d i s c r e t e e n e r g y w i n d o w s t o s i m u l a t e p o s s i b l e a c t i x - a t e d t r a n s i t i o n

p r o c e s s e s . A t t h i s s t a g e t h e s i m u l a t i o n s a r e i n t e n d e d a s a n e m p i r i c a l g u i d e

r a t h e r t h a n a n a t t e m p t t o d o f u l l q u a n t i t a t i v e a n a l y s i s o f n o n - a d i a b a t i c p r o t e i n

d y n a m i c s , w h i c h i s b e y o n d c u r r e n t c o m p u t a t i o n a l s c o p e .

A r t i f i c i a l t h o u g h i t i s , t h i s 2 - c l a s s m o d e l c o u l d b e r e p r e s e n t a t i v e o f t h e i

e x t r e m e c o n f o r m a t i o n a l c l a s s e s r e c o g n i z e d f o r p o l y p e p t i d e s , n a m e l y t h e f o l d e d

( n a t i v e ) f o r m a n d t h e u n f o l d e d ( d e n a t u r e d ) f o r m . B e c a u s e o f t h e d i f f e r e n c e s i n

e x p o s u r e t o t h e s o l v e n t i n t h e s e e x t r e m e s , t h e r e w i l l s u r e l y b e d i f f e r e n c e s i n

t h e i r r e l a x a t i o n a l p r o p e r t i e s , a n d t h e r e a r e c e r t a i n l y d i f f e r e n c e s i n t h e i r m e a n

e n e r g i e s .

F i g u r e 1 s k e t c h e s s o m e o f t h e e f f e c t s t h a t c a n b e o b s e r v e d i n s i m u l a t i o n s o f

t h i s s y s t e m f o r t w o p r o t e i n c l a s s e s s e p a r a t e d , a r b i t r a r i l y , b y a n e n e r g y o f a b o u t :

200 k c a l / m o l e t o g e t h e r w i t h o t h e r p a r a m e t e r s t y p i c a l f o r p r o t e i n s . F i g . 1 ( a a n d

c ) i l l u s t r a t e s a s i t u a t i o n w h e r e b o t h c l a s s e s a r e e q u a l l y p o p u l a t e d , b u t i n w h i c h

i n t e r c o n v e r s i o n b e t w e e n t h e c l a s s e s i s e i t h e r v e r y f a s t o r s l o w c o m p a r , e d t o t h e

n o n - a d i a b a t i c r e l a x a t i o n s . T h e s e c o n f i r m o u r p r e v i o u s e x p e c t a t i o n s t h a t w e

o b t a i n e i t h e r a p a i r o f G a u s s t a n d i s t r i b u t i o n s s i t u a t e d a b o u t t h e a p p r o p r i a t e m e a n s

( s l o w e x c h a n g e ) , o r a s i n g l e G a u s s i a n m i d - w a y b e t w e e n t h e t w o ( f a s t e x c h a n g e

l i m i t ) . I n t h e i n t e r m e d i a t e r e g i m e ( F i g . l b ) , w h e r e r e l a x a t i o n a n d t r a n s i t i o n

r a t e s a r e c o m p a r a b l e , m u c h b r o a d e r d i n t r i b u t i o n s a r e o b t a i n e d a n d t h e i n d i v i d u a l

c l a s s d i s t r i b u t i o n s a r e d i s t i n c t l y n o n - G a u s s i a n a n d m a r k e d l y s k e w e d t o w a r d s o n e

a n o t h e r . . I n t h e s e l a t t e r e x a m p l e s t h e r e i s t h e i n t e r e s t i n g c o n s e q u e n c e t h a t t h e

p r o t e i n w o u l d s p e n d m u c h o f i t s t i m e i n r e g i o n s o f c o n f o r m a t i o n a l e n e r g y s p a c e

t h a t a r e s o m e w h a t r e m o v e d f r o m t h e m e a n s o f e i t h e r o f t h e t w o c o n f o r m a t i o n a l

c l a s s e s .

T w o e x t r e m e v i e w s o f w h a t m i g h t o c c u r d u r i n g a s h i f t i n o v e r a l l

c o n f o r m a t i o n a l e q u i l i b r i u m , aB m i g h t b e s e e n d u r i n g t h e r m a ~ o r c h e m i c a l

d e n a t u r a t i o n t r a n s i t i o n s f o r e x a m p l e , a r e s h o w n i n F i g . 2 . I n t h e o n e c a s e

( s l o w a d i a b a t i c l n t e r c o n v e r s i o n , F i g . Z a ) w e s e e s i m p l y a c h a n g e i n r e l a t i v e

p o p u l a t i o n o f t h e t w o , a l b e i t b r o a d , c l a s s d i s t r i b u t i o n s . T h i s c o n t r a s t s w i t h t h e

g e n e r a l d r i f t o f t h e d i s t r i b u t i o n t h a t wouid b e s e e n a t t h e 0 p p 0 s i t e e x t r e m e .

( f a s t i n t e r c o n v e r s t o n , F i g . E b ) . E x p e r i m e n t a l o b a e r v a t i o n s s h o w t h a t t h e t h e r m a l

u n f o l d i n g t r a n s i t i o n s o f t h e m a j o r i t y o f s m a l l g l o b u l a r p r o t e i n s , a t l e a s t , : :

: . . : :

: :i ! : : i i i:: . • • . i • : i ii il i . ::i .) . / : i : . : " . , - . ~ . . : , . - . : . . ~ : i ~ , ~ " " , ~ " ! - ? . . ~ ." : . . : : . . . . ~ ; " . : , : 2 : : ~ : ' ~ , " " : : : ~ - ' . ~ , ' = '

202

C

1 -_- 0 200

Relative Energy (kcal/mole)

Fig-l. Typical energy distribution functions obtained from numerical 8imulation of the Langevin behaviour (Equ.3) of a 25000 mol.wt. protein at 25C assuming two discrete conformational classes separated by an average energy of 200 kcaIfmole (1 cal = 4.1845). (a) Slow adiabatic, internal transitions. (b) Comparable internal and external, non-adiabntic exchange rates. (c) Rapid internal transitions.

Fjg.2. Extreme views of the variation in protein cncrgy distr5bution functions ’ during conformational transitions. (a) Slow adiabatic transition, fast thermal

equilibration. (b) Fast adiabatic transition. slow external equilibration. .Dlstributions are shown for various-cquiIibrium populations during a shift from lowem to higher energy conformational classes, as indicated by the arrows.

208

correspond more closely to the results of Fig. 2a (refer. 317). This lndicat&:

that the major conformational rearrangements required in folding or unfolding 1

the protein are relatively rare and infrequent comparedtoth&rmal equ~ibr~~~

with the surroundings. as is perhaps not unexpected in this case.

It is worth noting here that conformational transitions can be studied by a

wide variety of experzmental techniques that do not always give the same result.

The 'two-statet hypothesis or approximation is frequently used to analyse the

data, though it is not always clear how these two states or classes are to be

characterized. Spectroscopic techniques. in particular, may sample different

regions of the protein on widely different time scales. We have seen how the

relative time scalea for different physical processes may affect our perception of

the number of 'states' present (i.e. ningle or double Gaussian distributions. for

instance). It 3s feasible, therefore, that different techniques, sampling on

differing time scales. may lead to significantly different pictures of protein

transitions. One particularly striking -ample of such a case in a real protein

system is discussed in ref. 8.

C~~FORMATI~N~ DRIFT

We have seen above that for major conformational transitionsin proteins the

adiabatic, intunal rearrangements are usually quite rare compared to non-

adiabatic thermal equilibrations. Eut there are other processes involving

proteins where that is not necessarily the cask. In some circumstances the

conformational class of a protein might be changed ~~tantaneously by extra-

molecular events such as, for example, the binding of a ligand or substrate

molecule, or by an encounter and association with a second protein mole&ule to

form an aggregate. The equilibrium conformational energy of the reaufting

protein-ligand complex or protein-protein dimer will likely be different from that

of the free protein, so this constitutes a change of class in our model. Imagi+ie

the following scenario, first outlined in a classic paper by Xu and Weber (ref. 9):

a protein molecule, fluctuating about its mean conformation , encounters and binds,

to another molecule. This complax is initially relatively weak because ef

unfavourabla molecular contacta, but the system may now begin to drift towards

a new (lower} mean energy appropriate for the complex. Molecular associati&ns

are. however, reversible and, when spontaneous dissociation occurs, the proteh

molecule reverts to ita original class and drifts back towards the original mean,

For long-lived complexes this is of'little consequence since the complex has time.

to thermally equilibrate before dissociation occurs , and the thermodynamic and

kinetic behaviour can be describad &n conventional 2-stateterms. But&zppoee

the lifetime of the complex is relatively short compared to conformational

relaxation and energy exchange.with the surroundings. In this case tipontaneous-

2 0 4 ~ : :

d i s s o c i a t i o n m a y t a k e p l a c e b e f o r e t h e r m a l e q u i l i b r a t i o n a n d w h i l s t t h e s y s t e m i s

s t i l l u n d e r g o i n g ~ c o n f o r m a t t o n a l d r i f t * t o w a r d s t h e n e w m e a n e n e r g y s t a t e .

F u r t h e r m o r e , a s t h e f r e e p r o t e i n S y s t e m r e l a x e s b a c k t o i t s o r i g i n a l s t a t e , b u t

b e f o r e r e a c h i n g i t , i t m a y w e l l e ~ x c o u n t e r a n o t h e r l i g a n d a n d t h e p r o c e s s r e p e a t s

i t s e l f . T h u s w e m a y s e e , i n q u a l i t a t i v e t e r m s , t h a t i n s u c h c i r c u m s t a n c e s , i n

: t h e s t e a d y s t a t e , n e i t h e r f r e e n o r c o m p l e x e d p r o t e i n w i l l a t t a i n t h e i r t r u e m e a n

e q u i l i b r i u m s t a t e s , b u t w i l l s p e n d m o s t o f t h e i r t i m e a t s o m e i n t e r m e d i a t e e n e r g i e s .

M o r e o v e r , b e c a u s e r a t e s o f m o l e c u l a r e n c o u n t e r a n d a s s o c i a t i o n w i l l d e p e n d o n

c o n c e n t r a t i o n s , t h e o b s e r v e d m e a n e n e r g i e s o f t h e f r e e p r o t e i n a n d i t s c o m p l e x e d

s p e c i e s w i l l t h e m s e l v e s d e p e n d o n c o n c e n t r a t i o n . T h i s l e a d s t o t h e I n t e r e s t i n g

c o n c l u s i o n t h a t t h e r m o d y n a m i c p a r a m e t e r s f o r t h e m o l e c u l a r a s s o c i a t i o n p r o c e s s ,

s u c h a s t h e e q u i l i b r i u m c o n s t a n t , w i l l d e p e n d o n c o n c e n t r a t i o n i n a n o n - c l a s s i c a l

f a s h i o n a n d w i l l b e c o n t r o l l e d b y t h e r e l a t i v e k i n e t i c s o f i n t e r - a n d I n t r a -

m o l e c u l a r r e l a x a t i o n p r o c e s s e s . E f f e c t s s u c h a s t h e s e h a v e , I n d e e d , - b e e n

o b s e r v e d e x p e r i m e n t a l l y I n s e v e r a l p r o t e i n s y s t e m s ( r e f s . 9 - 1 1 ) . [ T h i s

i n t e r p r e t a t i o n , i n c i d e n t a l l y , h a s b e e n s u g g e s t e d ( e r r o n e o u s l y ) t o b e i n v i o l a t i o n

o f t h e l a w s o f c l a s s i c a l t h e r m o d y n a m i c s ( r e f s . 1 2 , 1 3 ) . S u c h c r i t i c i s m i s b a s e d

o n t h e m i s g u i d e d n e g l e c t o f t h e m u l t i t u d e o f a c c e s s i b l e c o n f o r m a t t o n a l s t a t e s o f

p o l y p e p t i d e s w h i c h , t h e r e f o r e , d i f f e r f r o m s i m p l e r a s s o c i a t i o n s o f a t o m s o r s m a l l

molecules. ]

T h e s i t u a t i o n o u t l i n e d a b o v e c a n b e r e a d i l y s i m u l a t e d b y o u r e m p i r i c a l

L a n g e v i n p r o c e d u r e , a n d s o m e e x a m p l e s c o n f i r m i n g t h e q u a l i t a t i v e e x p e c t a t i o n s

a r e s k e t c h e d i n F i g . 3 . I n t h e s e c a l c u l a t i o n s , a d i a b a t i c c l a s s t r a n s i t i o n s a r e

a s s u m e d t o b e p o s s i b l e a t a l l e n e r g i e s a n d a r e c o n t r o l l e d b y r a n d o m . c o n c e n t r a t i o n

d e p e n d e n t e n c o u n t e r s w i t h o t h e r m o l e c u l e s . A t l o w c o n c e n t r a t i o n s m o l e c u l a r

a s s o c i a t i o n s a r e r a r e a n d t h e c o m p l e x i s t o o s h o r t - l i v e d t o a l l o w m u c h r e l a x a t i o n

t o t h e m o r e f a v o u r a b l e c o n f o r m a t i o n e n e r g y . B u t a t h i g h c o n c e n t r a t i o n s t h i s

s i t u a t i o n i s r e v e r s e d : s p o n t a n e o u s d i s s o c i a t i o n , b e i n g a f i r s t o r d e r p r o c e s s ,

c o n t i n u e s a t t h e s a m e r a t e , b u t t h e f r e e p r o t e i n I s n o w t o o s h o r t - l i v e d f o r

c o m p l e t e r e l a x a t i o n . C o n s e q u e n t l y , t h e m e a n i n t e r n a l e n e r g i e s o f f r e e o r b o u n d

p r o t e i n , w h i c h c o n t r i b u t e t o t h e t h e r m o d y n a m i c a s s o c i a t i o n c o n s t a n t , fo : - e x a m p l e ,

d e p e n d o n c o n c e n t r a t i o n a n d d e g r e e o f a s s o c i a t i o n i n a m a n n e r u n e x p e c t e d f r o m

c o n v e n t i o n a l v i e w s o f m o l e c u l a r a s s o c i a t i o n e q u i l i b r i a , b u t c o m p a r a b l e t o

" e x p e r i m e n t a l o b s e r v a t i o n s o n s o m e p r o t e i n s ( r e f s . 9 - ] 1 ) .

I t s e e m s i i k e l y t h a t k i n e t i c a l l y - d e t e r m t n e d c o n f o r m a t t o n a l d r i f t s a n d

• h y s t e r e s i s e f f e c t s o f t h i s k i n d ; w h i c h a r e h a r d t o e n v i s a g e e x c e p t t n m a c r o -

m o l e c u l a r S y s t e m s . h a v e e v o l v e d t o b e o f u s e i n b i o l o g i c a l c o n t r o l s y s t e m s u n d e r

s o m e c i r c u m s t a n c e s . T h e y a r e n o t n e c e s s a r i l 3 ; a u n i v e r s a l p r o p e r t y o f

m a c r o m o l e c u l e s , s i n c e i t c l e a r l y r e q u i r e s s o m e q u i t e p r e c i s e e n g i n e e r i n g o f

, - i i n t e r n a l a n d e x t e r n a l m o l e c u l a r r e l a x a t i o n p r o p e r t i e s a n d i s o n l y o b s e r v e d i n . - . -

.i. ::. i _ _' ' . : " : " . . . . .

: : . . . .

~'~:i~:i~_~~..~~, !: : ' :::~::: i-::i~i:i~ : i. i . ~i::: : ' i : i : ' : : : ::~ :: : ' : : : : : : i , ~ : : ~ i , : : : : : : - : : :: i

C - O-8

- o-2

relatively few protein3, at least so far.

Fig.3. Conformational enekgy drift in a simulated protein-ligand equiIlbrium in which the mean Hfetime of tha complex is comparable to the rate of thermal conformational relaxation _ Steady-state energy distributions are shown separately for free protein (P) and complex (C) , respectively, and the numbers indicate the fraction of complex formed ln each case, (Note that the energy separation of the two classes has been exaggerated here for clarity . )

But a protein molecule capable of

functionally connected conformational. drift has distinct advantages in some

sltuationa because it is endowed, to some extent. with a .memory of its recent

functional history. An enzyme, for example. which 1s only rarely used under

normal circumstances n.ay.spcnd much of lta time in a relatively inactive state and

.would respond sluggishly to sudden fluctuations ln substrate or metabolite

concentration. But, lf elevated substrate concentrations persist. the enzyme.

could drift to a more active conformation to rcdrcss the balance. And it might

remain in this more active state for a short while In case of further bursts of

substrate. Such elegant dlffcrcntial and integral control at the molecular level

may contribute much to tha stability of the complex. non-cquilit ,-ium systems of

living organisms.

ACKNOWLEDGEMENT

I am grateful to D-T-F. Drydcn for useful discussions nt various &age& of this

work.

REFERENCES

1

2

3

4

5

6 7 8

9

A. Cooner. Thermodynamic fluctuations in protein molecules, Proc. Natl, Acad. Sci.- US, 73 (l-976) 2740-2741- A. Cooper, Conformational fluctuation and change in biological macromolecules, Science Progress, Oxford, 66 ( 1980) 473-497. A. Cooper, Protein fluctuations and the thermodynamic .uncertainty principle, Frog. Biophys. Molec. Biol.. 44 (1984) 181-214. J.A. McCammon and S.C. Harvey, Dynamics of Protdna and Nucleic Acids, Cambridge University Press, Cambridge, 1987. A. Cooper and D.T.F. Dryden. Alloatery without conformatlonal change, Eur. Biophys. J., 11 ( 1984) 103-109. D.A. McQuarrie, Statistical Mechanics, Harper & Row, New York, 1973. P.L. Privalov, Stability of Proteins, Adv. Protein Chem., 33 (19’19) 167-241. A. Cooper, Spurious conformational transitions in proteins. Proc. Natl. Acad. Sci. US, 78 (1981) 3551-3553. G-J. Xu and G. Weber. Dynamics and time-averaged chemical potentials bf protains: importance in oligoaier association, Proc . NdU. Acad : Sci. US, 79 ( 1982) 5268-5271_

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L. King and G. Weber, Conformational drift of lactate dehydrogcnasa, Biophys. J., 49 ( 1986) 72-73. G. Weber. Phenomonologlcal description of the association of protein subunitsi subjected to conformational drift. Effects of dilution and of hydrostatic pressure, Biochemistry, 25 ( 1986) 3626-3631. 0-G. Berg, Time-averaged chemical potential of balance principle, Proc. Natl. Acad. Sci. US, 80 ( 1983) G. Weber, Stability of ollgomeric proteins and equilibria, Proc. Natl_ Acad. Sci. US. 80 ( 1983)’ 5304-5395.