Some applications of statistical mechanics in enzymology 2. Statistical mechanical explanation on allosteric enzyme models

Z theor BioL (1990) 143, 455-464

Some Applications of Statistical Mechanics in Enzymology . Statistical Mechanical Explanation on Allosteric Enzyme Models ZHI-XIN

WANGt§

AND HIROSH!

KIHARA-~

t Laboratory of Molecular En=ymology, Institute of Biophysics, Academia Sinica, Beijing 100080, China and :~Jichi Medical School, School of Nursing, Minamikawachi, Tochigi 329-04, Japan (Received on 17 March 1989, Accepted in revisedJbrm on 16 November 1989) A unified explanation for several main models of allosteric enzyme are presented from the statistical mechanical point of view, The relationships and difference among these models as well as some problems concerned with distinction of different model of allosteric enzyme have been discussed, Introduction

Molecular basis of allosteric effect has aroused great interest a m o n g the enzymologists since the concept of allosterism was proposed by Monod el aL (1963). In order to explain the correlation of structure and function of allosteric enzyme, a great n u m b e r of molecular models for allosteric enzyme have been designed ( Dixon & Webb, 1979). In this paper, we will attempt, from the statistical mechanical point of view, to give a unified explanation for several main models of allosteric enzyme, and point out the relationships and differences among these models. Theory We now consider an otigomeric enzyme consisting of n subunits, each of which has one binding site for substrate. As already pointed out in preceding paper, at equilibrium, the probability that an enzyme molecule is in the quantum state tj(i)) with energy Ej,,, and combines exactly with i substrate molecules is; pL,,,, = Q el- i-,,,~,,i. ~r,

(11

then the corresponding enzyme concentration will be; [ E ],,ptj,, , _ [E],, el_ ~ +,~, l,~r

Q

(2)

where [E]o is the total concentration of the enzyme. If the rate constant for the formation of reaction product from enzyme-substrate complex in state [j(i)) is k,,,~ and the catalytic reaction of enzyme satisfies fast equilibrium condition, that is, the enzyme-substrate complexes are established much faster than the step leading to § Present address: Box 445, Baker Laboratory, Department of Chemistry, Cornell University, Ithaca, NY 14853-1301, U.S.A. 0022-5193/90/080455+ 10 $03.00/0

~) 1990 Academic Press Limited

456

z.-x.

WANG

AND

H.

KIHARA

the formation of reaction product, then the initial rate of enzyme catalytic reaction is; n

E E k~,,,[E],, el-%''+"l/kr v

=

~ B,[S]'

1.,,

,~,

,=, ,

E el-E""+i~'l/kr

i

(3)

E° A,[S]'

I=11 I i I l l )

t~O

where B, = [E].[~t,,,,~ k,,,, e I -E,,,,.,,-l/tr], A, = ~./,,,,, e l-E,,,,' ,,"t/tr (i = 1, 2 . . . . , n) and IS] is the substrate concentration. In some cases, the turnover number for all binding sites are identical, then, the number of substrate binding sites occupied, n, is proportional to the initial velocity, v, eqn (3) may therefore be written as; v = Vm~

Y-

no. of occupied binding sites total number of binding sites'

(4)

where Vm~ is the maximum reaction velocity. According to eqn (20) in Wang (1990), we then have; n

v ~M ~ = Y . . . . . Vm~ nM n

I 2-

iAi[ S] i

I

(5)

n

Ai[ S] i I=O

It can be seen from the derivation o f e q n (5) that by using the statistical mechanical method, we may obtain the general expression of reaction rate for allosteric enzyme without introducing any molecular models. This is impossible by using thermodynamic method, as already pointed out by Koshland, because we cannot write all possible molecular species a priori (Koshland, 1970). Now, we will derive the statistical mechanical expressions for several main allosteric models. (A) A D A I R

M O D E L (1925)

In the preceding derivation of Adair equation, we have assumed that in the absence of ligand only one state o f enzyme existed. In fact, this assumption is not necessary for derivation of Adair equation. Suppose that the enzyme molecule is composed of n identical subunits, and in the absence of substrate the enzymes are in the states Ij(0)). The energy decrement is Y.',=, e, (i = 1 , 2 , . . . , n) when an enzyme molecule combine with i substrates. The molecular partition function may therefore be written as; Q=~o i=

~ exp lJ(i))

{[

-Ej,,,+

--t

[

= i~,exp[-E,,),/kT]

e,+il.t .,=1

]// kT

{(,t ]"") {(~;,k e~+itz )/} i~(i kT . exp

(6,

APPLICATIONS

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MECHANISMS,

457

2

then,

[

i,~ e x p [ - E , , , , I k T ]

olnQ

a=kT--

} (7) {( ,~ i

exp

)// +'.)/<'}]

,~ e , + # l

kT

",+ilx")/kT}[S] ' '+~,l(:)exp{(,~le,+i#xii)/kT}[S] '' ,_~, /(7)exp {(,~,

-

{7)

which is identical with eqn (29) (Wang, 1990). For convenience, we will regard a group of quantum states which have identical property in binding of substrate as one state, unless otherwise specified. It is evident that the binding of substrate with enzyme is non-co-operative when e, = e,+,, positively co-operative when e, < e,, ,, and negatively co-operative when e, > e, +,. (B) S Y M M E T R Y

MODEL

OF

MONOD,

WYMAN

&

CHANGEUX

II~)651

We assume the following. (a) Enzyme molecule is an oligomer, consisting of n identical subunits. (b) In the absence of substrate, the enzyme molecules can exist in two quantum states, tR(0)) and IT(0)X in each o f w h i c h all subunits have identical conformation (R or T), and the two states of the enzyme are in equilibrium. The energies of other states are much higher than that of the states of IR(0)) and IT(0)), so that the enzyme concentrations in other states are negligible. (c) The energies of enzyme molecules in states IR(0)) and IT(0)) are zero and e, respectively. The energy change is - e , when the subunit in R form combines with a substrate and - e . when the subunit in T form combines with a substrate molecule. The binding of substrate to a subunit is independent of the ligated state of other subunits within the enzyme molecule. With these assumptions, the partition function of the enzyme molecule then can be written in the form of; n

Q=

~ ~ e [-E,l,l+ibl]/kT t_(l itltl) #l

= E [gR,,, e t t:u"'+'ul/~r+grii, el t'"'*'uli~r],

(8)

i =0

where g R ~ and gT~i~ are degeneracies corresponding to energy levels E r a , and E.r~,~, respectively. Since all subunits in the enzyme molecule are identical, then, the number of possible ways in which n binding sites are occupied by i substrate molecules is ('/), that is gR,,~ = gr~, = (','), therefore;

Q=i=oLkil

-,Ikr

= [ 1 + e ~' '+" ,/kr],, + e- ,/kr[ 1 + e "

e,,:+,~lk

:+~'t,,~r ] ,

(9)

458

Z.-X.

v

WANG

AND

H.

KIHARA

k T a In Q

-y

V,.,,~

n

a#

e,,,+,.,,,'kr[l+e,,,+,,.'~l],,-,+e .,~re,,:.,,> ~r[l+e,,:+., [l+¢,,,.,~r],,+e

~r],, ,

(10)

, ~ r [ l + e , , : + , . , tr],,

I n t r o d u c i n g e ' ` ' k r = e ' ~ " k r [ s ] into eqn (10), we o b t a i n ;

[ S ] ( I + [S]'~"

K.~\

v

w h e r e K sR= e -

I, i ~-ta"ilkT

K~-

, K sI= e

+L

1+

K.~/

[T,,S,][S] [T,,S,+,] IT,,]

L=

[R,,]

=e

(11)

I,,+ta"likT ' a n d L = e-,-IkT . In the M W C m o d e l ,

[R,,S,][S] e"',+"/kr[S] = [R,,S,+L] = e ' ' + ' ' l ' ' + ' ' / ~ r - e

K~

K~]

.

,,,+.",..~r

e '/~re"':''"~r[s]

It 2+~lPl'kT

e ' tre"""'-'+"'~r

e

,~T

H e n c e , the result o b t a i n e d from statistical m e c h a n i c a l a p p r o a c h is in a g r e e m e n t with that o b t a i n e d from t h e r m o d y n a m i c m e t h o d . I(')

SEQUENTIAL

MODEL

OF

KOSHLAND,

NEMETHY

&

FILMER

I1966)

We now consider the situation where the four identical subunits in an oligomeric enzyme are in the tetrahedral arrangement. The following assumptions are now made. (a) In the absence of substrate, all enzymes are in one quantum state, and all subunits in a given enzyme molecule have identical conformation A with energy e,. (b) When a subunit combines with a substrate molecule, it will undergo a conformational change A-~ B with energy change Ae = e~- e,. (c) The interaction energy between two subunits in conformation A is zero, and the interaction energies between two B subunits and between A and B subunits are eo~ and e.w~, respectively. With these assumptions, the energy of enzyme molecule in non-ligated state is 4e,, and the energy of enzyme molecule combined with i substrates is;

eats+

E, i l = ( 4 - i ) e , + i e ~ + -

1

(:)

e~.

(12)

2

T h u s the p a r t i t i o n f u n c t i o n o f e n z y m e m o l e c u l e can be written as

Q:

Y. ( 4 1 e x p { [ - ( 4 - i ) e i - i e ~ _ - ( 4 - i ) i e a o - O ' S i ( i - l ) e , ~ , l + i t . L ] / k T i ~ o\ I /

= Ei o i Ks~ 4 - , ,L, K A,,4-,, ,,, ,,_~[S],, ~ Kml w h e r e Ks = e -''/gT,

L

=

e'-":+""~/kT, KA~

=

e - " ,~/kr a n d K~u = e "uu'kT

(13)

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MECHANISMS.

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459

The rate o f e n z y m e catalytic reaction is; v

V,,~,,,

kTOlnQ

4

a/.t

L K ~ •K K ~•+ 4 L K

[S]olnQ

4

a[S]

.~, , [ S ] + 3 L : K . ~2 K a~. K , , [ S ] .~s K a~. [ S ]

2+3L'K~K~a,

+ 6 L ._K s2K . , u.~K m d S ]

K m, , [ S ]

, + L .~K mt,, [ S ] .~

2 + 4 L ~K s K .~. , , K , ~, [ S ]

+ L 4 K m~,, [ S ] .~.

(14) In the K N F model, define; [AB][A]

[BB][A] 2

Kan-[AA][B~]'

L

[B]

Ks - [A][S]"

K,.,-[AA][B]2.

(15)

Introducing the expressions o f various species into eqn (15L we then have. K.t/~ =

KI~;Ii =

e " AI e' ', ,: , , , . , ,

e-2"

~t

At e' 2 , . , 2 .

L

e ~ ~_,,,~r AI

Ks

e " AJ[S]

- e ,.,, AI

-e

'"" t r

(16)

Again. it can be seen from the expressions o f K a . . Km~ and L~ K s that the results obtained from these two different m e t h o d s are in agreement. The energy difference between the e n z y m e species associated with i substrates and those with i + 1 substrates is E , , , . ~, - E , , , , = e. - e~ + 3 e a . + if em~ - 2 e a . L When e.~ > 2ean. the energy increment E , , , , - E,,, ~, created by the c o m b i n a t i o n of enzyme molecule with the ith substrate is smaller than with the ( i + l)th substrate, E , , , , ~ , E,,,,. which s h o w e d that the binding o f substrate is negatively co-operative, otherwise. when 2 e a . > em~. positively co-operative.

(I)1 H I L L

MOI)EL

(19t0~

Now. let us consider a case o f extreme positive co-operativity, for which either all sites on an e n z y m e molecule are e m p t y or all sites are occupied, with no intermediate species. This extreme model o f positive co-operativity was first introduced by Hill in 1910. S u p p o s e that the energies o f e n z y m e molecule u n c o m b i n e d with substrate and c o m b i n e d with n substrate are zero and ne, respectively, the molecular partition function then is, Q=l+e

..... ,,,A f,

(17)

460

Z.-X.

WANG

AND

H. K I H A R A

and,

v

kTa In Q

=y--.

V.,~

n

8/z

e "('+~''/kr 1 + e "'+~'~/kr

e""+~'"'/kr[s]"

K[S]"

l+e"(~+~'°'/kr[s]"

1+ K[S]""

(E) B E R N H A R D M O D E L ( M A C Q U A R R I E & B E R N H A R D , MALHOTRA & BERNHARD,

(18)

1971; S E Y D O U X ,

1974)

In recent years it has been realized that m a n y oligomeric proteins and enzymes may be frozen into conformations which are c o m p o s e d of asymmetric dimers, that is, pair o f polypeptides o f identical covalent structure which have different tertiary structures. Schematic representation of an asymmetric tetramer, using the convention of squares and circles which are called A conformation and B conformation

dimer

tetramer

FIG. 1. The simple asymmetric dimer and an asymmetric tetramer.

TABLE 1

The expressions of A~ in ligand binding equations for different models General equation

•..V glci~e-E, ,+ ~°/k'T

MWC model KNF model (Tetrahedral) Adair model Bernhard model ( n t = n: = n / 2 )

T

1

l

(i=0, 1,2)

APPLICATIONS

OF STATISTICAL

MECHANISMS.

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461

respectively, are shown in Fig. 1. Suppose that in the absence of substrate, the enzymes are in the ground-state in which the energy is zero, and the energy changes are - e l and - e 2 when the A and B subunits combine with a substrate molecule, respectively. If the subunits in the enzyme molecule are independent from each other, then, according to the eqn (32) in Wang (1990), the rate of enzyme-catalyzed reaction is

o -YV~

K,[S] K2[S] F 2(1 + K,[S]) 2(1 + K_.[S]) (K, + K . ) [ S ] + 2K, K._[S]2 - 2{ 1 + (K, + K2)[S] + K, K2[S]-~}"

(19)

where Kj = e ~*,+~'''}/kr, K2 = e ~*2+~'"vkT. Discussion We have given the statistical mechanical expressions for several main models of co-operativity above. Other models can also be treated with similar methods. It can be seen from the previous section that all models for allosteric enzyme are the simplifications of general case in varying degrees. The differences among them are only in the expressions o f A~ (see Table 1). Since many arbitrary assumptions are made for the various models, it is almost impossible to apply a specific model to various enzymes which have different structures and functions. Moreover, it is of little significance to discuss generally which model is better rather than aiming at a specific enzyme. It is of considerable interest to note that the several models listed in Table 1 are just corresponding to several specific cases which can give rise to co-operative phenomena. In the MWC model, there is no interaction related to the binding of ligand between the subunits within the enzyme molecule; the aliosteric effect is due to the fact that the enzyme molecule may exist in two convertible conformations with different properties for ligand binding. In the Bernhard model, there is also no interaction between subunits within the enzyme molecule; the co-operativity stems from the existence of two kinds of nonconvertible conformations with different properties for ligand binding among the subunits. In the K N F model, all subunits in various conformations have identical property for binding of ligand, and there is no interactions between the binding sites on the enzyme molecule; the interactions between subunits result in allosterism. In the Adair model, the aliosteric phenomena is caused by interactions between binding sites on an enzyme. In summary, cooperative effect may result from two different causes. Although both o f them may give rise to co-operativity, their molecular bases are completely different. For the MWC model, it is worth noting that there is indeed interaction between the subunits within the enzyme molecule, which is independent of the binding of ligand. To make this argument more concrete, let us consider the following example (Fig. 2). It is, in fact, the Eigen's dimer model (1967). Suppose that e~ and ea express the energies of A and B subunits, and eA~, eaa and eAR, represent the

462

Z.-X+ W A N G

AND

H, K I H A R A

Q

i

J 1

If

r [

FIe;. 2. The general dimer model for the binding of ligand, S, Circles and squares represent different conformations of the protomer,

interaction energies between two A subunits, between two B subunits, and between A and B subunits respectively; the energy changes are - e t and - e 2 when A and B subunits combine with a substrate molecule, respectively. Let eAA be equal to zero, the molecular partition function of enzyme then is,

+ 2 e ~'-'+'"+'~"'/kr[l +e~"+"'/kr][1 +e~'-~+"~/~r],

(20)

when ~'an~oo, this equation simplifies to the MWC model, and when e t - - > - ~ , e2--,oo, eB-->oo and e 2 - e t ~ = f i n i t e value the K N F model is given. Assuming that there is no interaction between the subunits in the enzyme, that is e/ta = e , a t j = ~nu = 0 , eqn (20) will then take the form of; Q = [ i (=e, ~, (.2i )

__+,_.,,~re , , +,~,kT+e__~,. /kr e+,,. +,~/kr)]

+ 2e-' " ÷ ' . ' / ~ r [ 1 + e' '+,+~' ,/t r][ 1 + e' ~-'+~',/kr].

(21)

Although we can obtain an equation which is identical in form with eqn (9) by deleting the part outside the square bracket, there is no reason for us to do so. In brief, only by considering the interaction between subunits can we get a reasonable explanation for MWC model, and it is the interaction between the st bunits that result in concerted conformational transition of the subunits. Even if all models of allosteric enzyme mentioned previously are simplifications of general case to some extent, these models have played an important role in understanding the molecular mechanism of co-operativity. If there are some clear evidences to show that a specific enzyme obeys one of the specific models of co-operativity rather than the other, it is still of some significance for understanding

APPLICATIONS

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MECHANISMS.

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463

the correlation of enzyme structure and function. We now discuss the method for distinguishing different models of co-operativity according to the experimental binding curve. The more sophisticated a p p r o a c h is to use the equations predicted by the various models to fit to the experimental binding curve obtained and compare which models can give the best fitting. Mathematically, if one builds a complex enough model with sufficient parameters, it will become easier to fit almost any experimental set of results, as has often been pointed out. However, it will be clear from inspection of eqn (5) that, in general, the n u m b e r of parameters in the equation is just the n u m b e r of binding sites on an enzyme molecule. So a rational function of substrate concentration with n independent parameters should be adequate to fit any binding curve of allosteric enzyme with n binding sites. Any attempt at increasing the number of parameters by building a more complex model to obtain better fitting curve is meaningless (for example, using three states MWC model with five independent parameters to fit the binding curve of a four sites protein). In the case of a four-subunits enzyme, there are four and three parameters in Adair and MWC model respectively, but only two truly independent parameters in K N F model. It can therefore be predicted that better fitting curves will be obtained by using Adair or MWC model, and perhaps poor result will be obtained if the K N F model is used to fit the experimental binding curve. The binding of oxygen to haemoglobin has offered a good example in this regard (Imai, 1982). Consequently, although the curve fitting sometimes is a useful method for testing whether a specific model can provide an explanation for the co-operative behavior of an enzyme, it is limited by the fact that more than one model may provide equally satisfactory fitting to a particular set of experimental data and therefore a choice cannot be easily made. In fact, from experimental data, we can only obtain the values of A, by the curve fitting method. For any allosteric model with n independent parameters, /3,,/32, t33 . . . . . /3,, we always have; A; = f , ( f l , , t32,/33 . . . . . ft,). Solving the set of equations, we have/3; = qb;(Ai, A:, A 3 , . . . , A, ) ( i = 1 , 2 , . . ' . , n ). Hence, any model with n independent parameters will provide equally satisfactory explanation for the same set experimental results. As listed in Table 1, the differences a m o n g the various models are only in the expressions of A,s. in general, A, is the sum of several terms, but to a specific model, A, is only c o m p o s e d of one or two terms. According to this point, in principle, the Adair model and MWC model as well as Adair model and Bernhard model can be distinguished by studying the binding of ligand with enzyme at different temperatures. In practice, however, as shown in Fig. 3, since the allowed range of temperature change is too narrow, it may be difficult to distinguish between a straight line and a curve. In addition, it can also be seen from this example that the apparent straightness of plots of In A, against 1/T, in general, should not be taken as evidence of the single exponential law. Distinguishing between these possible mechanisms usually requires experiments other than equilibrium binding, in some cases, binding experiments in the presence of competing ligands are useful ( Henis & Levitzki, 1979).

464

Z.-X.

WANG

AND

H.

KIHARA

13

iz! /

/

/

O, /

Y Ii

o

c ...I

.¢

p, Y x¢ 10

,ox5 x5 ~5

I

1

1

I

t

3-O

3"2

5-4

3"6

5"8

4-O

IOS/T Ft(;. 3. C o m p a r i s o n between the plots of In A t vs. I / T of Adair model and Bernhard model. (- - -L Adair dimer model in which K t = e t',÷~'''~'t~' = 1 , 5 x 10 4 M - I , K : = e ~':+""~/~r = 0 - 8 3 3 4 × 10~ M i an d A =2K~ = 3 x 104 M °~ at 25°C. (©), Bernhard dimer model in which K~ =e~",+~'"~/tr=0.5x 10"~M -z, K 2 - - e ~''+'''~/~1- -,,-'~ 5 X I 0 4 M i and A ~ = K I + K , = 3 x l O a M -I at 25°C. REFERENCES

bioL Chem. 63, 529-545. D r x o N , M. & WH~I~, E. C. 11970). En--ymex. 3rd edn. pp. 399-467. London: Longman. Etcil~N, M. (1967). Nobel Syrup. 5, 333. Ht-NIS, Y. I. & LI..VlTZKI, A. (1979). Eur. J. Biochem. 102, 449. H t L t , A. V. (1910). J. PhysioL 40, iv-vii. IMA1, K, (1982). AIIoxteric Effects in Haemoglobin. Cambridge: C a m b r i d g e University Press. KOSHLANI), D. E., J r . (1970). The En'.ymes ( Boyer, P., ed.) 3rd edn. Vol. 1. p. 342. New York: Academic Press. KOSHLANI), D. E., .It., Nt-.MI~Ttt',, G. & FILMUR, D. (1966). Biochemistry 5, 365-385. MA('QUARrlt:, R. A. & BFRNHArl), S. A. (1971). J. molec. Biol. 55, 181-192. MONOD, J., CHAN(;EUX, J.-P. & JA(OI~, F. (1963). J. molec. BioL 6, 306-329. MONO1), J., WYMAN, J. & CHANC;EUX, J.-P. 11965). J. raolec BioL 12, 88-118. SEYI)OUX, F., MALHOTRA, O. P. & BLRNHARI), S. A. (1974). Crit. Rec. Biochem. 2, 227-257. WANG, Z.-X. (1990). J. theor. Biol. 143, 445-453. A D A I R , G. S. (1925). Z