Thermodynamic restrictions on the allosteric models through an analysis of the free energy of interaction between sites

I. theor. Biol. (1976) 63, 421-441

Thermodynamic Restrictions on the Allosteric Models Through an Analysis of the free Energy of Interaction Between Sites RICARDO P. GARAYS

Dcpartamento de Quimica Biologica, Facultad de Farmacia y Bioquimiro, Universidad de Buenos Aires, Argcntinet (Received 12 September 1975, and ipt revised form 15 January 1976) The free energy of interaction between protein sites for a protein in equilibrium with several ligands is analyzed thermodynamically. The free energy obtained contains homo- and heterotropic components which are not independent. A simple way of calculating the heterotropic free energy of interaction from the experimental data is given together with its limitations and thermodynamic significance. An expression is developed that relates the measurable protein affinity for one ligand with all the affinities that each kind of site can attain (“real” affinities). The approximation that all the real affinities for each ligand are the same for each protein conformation leads to the characteristic equations of the Monod, Wyman & Changeux (1965) model, without taking into account their molecular hypothesis. Furthermore, the approximation that almost all the empty sites have affinity equal to 0 leads to the characteristic equations of the Koshland (1966, 1970) model, again without taking into account their molecular hypothesis. Through the above thermodynamic analysis the approximations of the molecular models can be compared with the experimental data on hemoglobin. This analysis shows that the approximation made in the model of Monod et al. (1965) is too stringent to be adjustable to the experimental curves. On the contrary, a Koshland-like model permits the quantitative description of the alkaline Bohr effect in a simple form. 1. Table of Symbols .I, Y, z. P P XiYj

..

ligands. protein. protein with i sites occupied occupied by Y.

by .Y and .j sites

t Fellowship of the Consejo National de lnvestigaciones Cientificas y Tecnicas, Argentine. $ Present address: Service de Chimie Physique 2, FacultC de5 Sciences. ljniversitk SIN. de Bruxelles, CP 231 Bout. du Triomphe, 1050, Bruxelles. Belgium. 421

I< I’. (I .\ I
cortf’~~r-

I'.

ligand activities. fractional protein saturation with the ligands. Hill’s coefficients of each ligand. chemical potential of the ligands. interaction change of chemical potenlial with site occupation. interaction change of chemical potential with homotropic site occupation and/or heterotropic change in chemical potential. homo and heterotropic components of A~c,,~. Wcber’s espression of the free energy of heterotropic interaction. measurable protein affinity for .1;: thermodynamic affinities. “real” affinities. fraction of empty S,i-, sites. fraction of empty X sites or real allinily I’. )j

2. Introduction In a series of papers, Wyman described the thermodynamic equilibrium between proteins and ligands (Wyman, 1948, 1964, 1965, 1967. 1969). He studied the change in the chemical potential of a ligand in a protein site that results from the occupation, by the same ligand, of the other sites in the same protein molecule (“homotropic interaction”). He showed that the change in chemical potential (A,u~) was a function of either the ligancl activity (.u) or the fractional protein saturation with ligand (R), and thal using certain approximations, this change could bc calculated from the Flill plot of the saturation curve. In the present work this analysis is extended to more than one 1iganJ species, so that “heterotropic interactions” (i.e. interactions among sites fol different ligands) are also taken into account and can be calculated in a simpic form. The change in chemical potential due to the interaction is related IO a change in affinity of the site for the ligand. An expression can be developed relating the measurable protein affinity for the ligand to all the affinities that each kind of site can attain (“real” affinities). Now the change in the measurable affinity due to the occupation of a site can be expressed its a

l~l.STKlC-I’IONS

ON

‘THL

ALLOSTLRIC

4’3

MODLLS

change in the statistical distribution of “real” afl?nities among empty sites. Performing certain approximations in this expression, the characteristic equations of the molecular models may be developed in a way which does not take into account the molecular hypothesis on which these models are based. Finally, experimental evidence dealing with heterotropic interactions in hemoglobin is analyzed and the thermodynamic approximations of the models are tested. (A)

AN

LXPRESSION

PROTEIN

SITES

FOR

FOR A

THE

PROTElN

FREE IN

ENEKGY

OF

1iQUILlBRlUM

INTLRAC~ION WITH

BETWtl:N

SEVERAL

LIGANDS

In an equilibrium between several ligands (X, I’, Z, . .) and a protein (PI, having CJsites for X, s sites for I’, t sites for Z and so on, the chemical potential of these ligands when bound to the protein sites: (pX),,, (1~~)~. (I(Jp. . . . . will be equal to the chemical potential of these ligands in solution.+ For the ligand .Y this equilibrium can be expressedas: (/lx),, = po, + R7‘ In s.

(1)

Considering a particular A’ site of the protein molecule and excluding intermolecular interactions, we have that (c!,~\.),,in this site will depend on the occupation state of the other sites in the same protein molecule. This meansthat (F~)~ will be a function of the fractional saturation of the protein with the ligands (R, F, 2, . . .). Therefore:

? 111 :

SRI- -i?z

X.P,.

dZ+....

(2)

This equation in X includes the occupation of all the X sites, including the particular one under consideration. Subtracting in equation (2) the contribution of the occupation of this particular X site, we can obtain the change in chemical potential due to the interaction with the other sites (dp,,). This contribution can be obtained as follows: In each equilibrium state there will be a measurable protein affinity (K,) for the ligand .I’ defined by:

x-7 Kx = (I -I X)(s)

+ If the ligand as in hiochcmical term disappears

is electrically experiments in equation

charged, the ionic (2).

the force

electrochemical is maintained

(3) potential must almost constant,

be used; but the electrical

424

R.

P.

GARAY

If the X sites are identical lack of interaction between sites will imply that Kx is independent of X, Y, 2, . . . (if they are different, K,y diminishes with S but is still independent of iii, 2, . . .). Using equations (2) and (3), it can easily be seen that in the absence of interaction: KT

Wd, = X(1-X) -

dX.

(4)

Hence, d/c,, will be the difference between the total value [equation and the “no interaction value” [equation (4)] : dprx = x(l--x) _ !?.-

[(l/Q-l]

dX+R?‘“;;’

x, z,

(2)]

dY+ a Ill

x

dZ+... (5) az x,p,. where lzX is the tz of Hill at a given constant Y, 2, . . . . a In (W/(1-X)) ,lx = -- --aln ; --~(6) Y,z, 1 Equation (5) holds for identical sites; if the X sites are different, this equation gives us a minimum value of dpr,. In this equation we can see that the term which contains d8 is the homotropic contribution to dp,,, i.e. that due to the interaction between the X site in question and the other (q-- 1) X sites. The terms that contain d P, dZ, . . . are heterotropic contributions which result from the occupation of the Y, 2, . . . sites respectively. AprX, a magnitude that we want to obtain in an explicit form, will be the integral of equation (5) between the initial and final values of the independent variables. This integral can be taken along a path in which all the variables minus one are kept constant during each step of integration. Then Aplx is given by: (7) API, = &xx + AYIXU + btxz -t . . . . In this equation ApILIxX (the integral in dX), is the homotropic contribution to the interaction change in chemical potential. +RT-

(ApIxx)r . z. _, = RT ff[(lin,)-l]dX = RT r(l --nAu) d In X. (8) Xi X(1-X) xi This homotropic term coincides with that obtained for a single ligand by Wyman, when the fractional protein saturation with the heterotropic ligands is kept constant (Wyman, 1964). As Wyman pointed out, Allrxx can be estimated from a Hill plot, assuming the activity coefficients to be equal to one and taking into account the restriction of keeping Y, 2. constant.

RESTRICTIONS

ON

THE

ALLOSTERIC

425

MODFLS

APT APIXZ, . . ., are the heterotropic contributions of each heterotropic ligand to the interaction change in the chemical potential of X.

(4~xu)x.z.. = RT; “$

dY = RTyd x,z,.

In x.

xi

(9)

It is clear from this equation that this particular heterotropic term is simply the difference of In x between the initial and final values of f (at constant X, 2, . . .), multiplied by RT (see Fig. 1). Of course,: the other heterotropic terms (in Z, etc.) have the same form as equation (9).

FIG. 1. Calculation of the free energy of heterotropic interaction. In an equilibrium between a protein and two ligands (X and I’) the free energy of heterotropic interaction on the X sites (ApIxy) can be obtained from the protein saturation curve with X at two constant y. The AfiuIxYis equal to the difference in In x, at constant 8, between the two curves multiplied by RT.

In contrast with equation (S), which gives a minimum value of A/lrxx when the X sites are different, the equality in equation (9) holds even when the X, Y or Z sites are not identical among themselves. When the integration limits for all the independent variables are 0 and I, equation (7) gives us the total interacting change in the chemical potential of &A/I&, that takes place when the protein is saturated with all the ligands. By a similar method we can obtain AklTIy, A,uTIz, . . . . Hence, the total free energy of interaction per mol of protein (AC,,) will be: AGTI

=

~(A~LTIx)+s(AcLTI~)+~(AcITIz)+

.

. .

(10‘)

The individual components of the right-hand side of equation (7) have an interesting property in connection with the integration path used. Although

426

R.

I’.

GARAY

d/clx in equation (5) is an exact ditrerential and hence AI~,,~ in equation (,7) is independent of the integration path, this is not valid for Aprxx, Apr.ur, bxz, . . . . The difference arises because H,~ in equation (8) and the partial derivative 8 In x/a P),Xz,. in equation (9), depend on the other independent variables. The only restriction is therefore, that the .FM of all the contributions (homo- and heterotropics), be independent of the path 01 integration. The analysis of a particular case may illustrate this statement. Suppose, as in Fig. 2, that we saturate a protein with two ligands, .I. and Y.

FIG. 2. Dependence of the homo- and heterotropic components of the free energy of interaction on the integration path. ApTIx is independent of the integration path (A -: B or C i- D) but the same is not valid for their homo- and heterotropic components (A/I?, \ Y and AP~,.~,..)which depend on P and R respectively.

In order to obtain the A/lTl.Y we can use two convenient integration A-t-B or Ci- D. Then:

paths:

APTLY = (A~,,,,)~:of(A~f,,,,)x:, = (A~t~,xy)ru:o+(A~~,~x)~:~. (‘1) In spite of the equality of Ap TIX by the two integration paths, (ApTIXXIY: (, is in general different from (A~T,XX)p:,, because ~1~is in general different for the two I’ [see equation (8)]. A similar analysis can be made for the heterotropic contribution ApTfXy at the two integration paths. Only when trX and the partial derivatives become independent of the integration path, do dpIXX, dp,,,, . . . become exact differentials. In [his particular case, using equation (9) and the linkage relation of Wyman: a”In x a In 11 (12) 4--=-ax y = s ----zY-~ c?Y I,y I we can easily see that:

RESTRICTIONS

ON

THE

ALLOSTERIC

427

MODELS

A more detailed analysis yields the general equality, of which equation (13) is a particular case. Using a different approach than ours, Weber has defined the total free energy of heterotropic interaction, as the difference in free energy in saturating the protein with one ligand, in the absence or in the presence of saturating concentrations of the other ligand (Weber, 1972). He called them AC,,, for the ligand X and AC,,, for Y. In our analysis (see Fig. 2), these are equal to: (14)

AC,,, = d(Alu~rxu)x: 1 +(APTIXX~: 1 -(&l.m)~:oI AC Y/X = 4W~r~xh: I +@PT,YY)X: I-(4+rrr)x:ol. It is easy to demonstrate (see, for instance, Weber, 1972), that:

i 15)

AC X/Y = AC,,,.

(16)

Now let [see equation (I I)]: A.Y = (A/I TIXX~: I -W+I~X)P:

o=

(brrxr)x:

I -(~~T~.uY~x:

o

(17)

and (IS) AY = (AI+IYY)X: I -(APTIYY)X: o = (4~rrr.x)~: I -(APT,Yx)Y: 01 using equations (ll), (14), (15), (IQ, (17) and (18), we can easily see that: (19) This equation is more general than equation (13). Ax and A)! represent a measure of the interaction difference in the integration path: if there is no difference (Ax = A? = 0), equation (19) becomes equation (13). ~@PT,XY)X:

I +A?(1

= $(APTIYx)~:

1 +AJI.

The kind of interaction The interaction between protein sites are usually divided into three kinds: (a) stabilizing or positive, if Ap,, c 0; (b) destabilizing or negative. if A[L,, > 0; and (c) nulI, if ApIx : 0. As 4+rns and bTIXY depend on F and ,Y respectively, the kind of the homo- and the heterotropic interactions CNIZchange with P or X. From equations (9) and (12) we can see thn!. if the kind of A!I~,~~ does not change with 2, the two reciprocal heterotropic interactions (A~~7.~sI and A/I~~).,\.) will also be of the Same kind. The change

of variables

In order to describe the equilibria between proteins and ligands (a! constant pressure, temperature and protein activity), we need as many independent variables as there are ligands. To obtain dp,,Y in equation (5) we chose the variables X, Y, 2, . . . , because the interaction between protein sites was defined as a change in chemical potential with site occupation (see above). However, these variables are usually difficult to measure and to

428

R.

P.

GARAY

control experimentally. This difficulty becomes greater when the protein concentration is small, as in enzymes (in which the study of the interactions between protein sites is of great importance). The problem can be solved for the heterotropic ligands by replacing P by In y, Z by In z and so on. When this is done Aprx, in equation (7), appears as a function of X, ~1.2. . . and ~~~~~~ AIJI~~, AP~x~, . . . T lose in general their meaning of homo- and heterotropic interactions as defined before. As we simply need to change variables in all the equations, the analysis with these new variables is essentially similar to the previous one. In order to differentiate between the two kinds of Apr, we shall call this new one as A&. Then, we can see that ApiXX is the interaction change in the chemical potential of X, at con.stutIt chemical potential of the heterotropic ligands. However, in these conditions the heterotropic variables Y, 2, . . . , may change, in which case this is not a “pure” homotropic interaction as defined before. On the contrary, as ,r’ is maintained constant, A&r, has the meaning of a real heterotropic interaction as defined before. Using the limit integration paths of Fig. 2, we have that the total interaction change in chemical potential A&x becomes equal to AP~,,~. For example : (20) W&H: o = G%,x)Y: 0. We have seen that this new formulation does not change x into X. This can be made, studying the interaction between protein sites as a change in the measurable protein affinity for one ligand (K,), with the activity of all the ligands (x, y, z, . . .). This formulation is equivalent to the one previously analyzed since ApIX, between an initial and final equilibrium states with affinities K,yi and Kxs respectively, is : A~~rx = -R T 111(K,//K,i). (21) Using the previous convention, an interaction will be positive if Kx,. is greater than Kxi, negative if it is less than Kxi and null if it is equal to K,yi. The interaction

as at1 qfinity

change

Reducing the system for simplicity to only two ligands, x’and Y, we have that in order to study the function K, = K,(x, v), we can decompose the protein P into each of its states PXiYj (which corresponds to the protein with i sites occupied by X and j sites occupied by Y). If Sii is an X site of PXiYj, the affinity of this protein state is: (22)

in which SijX is an occupied X site and Sij an empty site.

RESTRICTIONS

ON

THL

ALLOSTtRIC

MO1)l.l

S

429

Using equation (3) and taking into account that: (23)

in which “total” means that we take the double summation, of all the Sij or SijX, it can be easily seen that:

over i and j.

K,=i i w total Kxij =i i l,ijK,i,i i:l

j:O

i:l

In this equation Kx appears as a linear combination k,,ij, with the condition 2

j$

lSij

=

(74)

j:O

O~CI(J+ I) independent

I*

As the Kxij are independent of(x) and (y), the dependence of K, with these Iigands appears only in the coefficients l,, [whose meaning is the fraction of empty Sti _ , ,j sites]. This dependence can be easily obtained from equation (22).

In this equation, the product over all the affinities extends from 1 to the exponent of the concentration of the relevant ligand; the subindices of the summations extends from 0 to q or s; and the Krwo are the affinity constants of the Y sites (in this particular case for the state PX, Y,,). The dependence of the lxij on (x) and (y) describes the interaction in a thermodynamic sense, without molecular details (see later). From equations (24) and (25) it can be easily seen that K, tends to : K,, (I when (x) and (y) tend to 0; to K,,, when (x) tends to infinity and (y) to 0: to Kms when (x) tends to 0 and (y) to infinity; and to Kxqs when (x) and ( y) tend to infinity. Therefore, using equation (21) and remembering that the Kxij are independent among them, it can be seen that Ap,,, at P : 0 and P : 1, can be different; which implies, as previously anticipated, that the kind of the homotropic interaction can change with Y. A similar analysis for AnrIXY shows that the kind of this heterotropic interaction can also change with X. We have also: limit X < I, (26) .x,y+m the equality holding only when Kxqs # 0. Then, we can conclude that there is no thermodynamic restriction of the type investigated over the interaction.

430

R. P. c, A RA ‘y

Using equations (24) and (35) and rearranging equation (3), we can obtain the well-known function 8 = ,?(s,J~) with the K,ij. When this is done, we can obtain the following thermodynamic restriction:

However, nothing can be said about the sign of 8K,/Gx, a result previously anticipated and well known that implies that there is no thermodynamic restriction on the kind of the homotropic interactions. (B)

THERMODYNAMIC

IMPLICATIONS

AT

THE

MOLECULAR

LtVtI.

When a protein is in solution their sites can have different molecular conformations (or differences in other physical properties responsible of the binding). Using Weber’s nomenclature, let us call “real” affinity K,, the affinity of an X site in a given “real” conformation S, (Weber, 1972). Then : K Making

_ (S,W .ir - (s,j&f

(2X)

a similar analysis to the previous one, we can set that:

This equation gives us h:, as a sum over all the real affinities, each of them multiplied by a statistical coefficient, lXrr which is equal to the fraction of empty I’ sites, or which is the same, Kx appears as a statistical average of all the real affinities, in which the l,y, are probability factors (the probability of finding an unoccupied r site over all the unoccupied sites). The dependence of these statistical coefficients on (x) and (JJ), des~&es tlze infeructim at the molecular level. This implies that we must find the function I,, = lx,(x, .I’). Let us call P(la,,,, 2b,, . . ., rh,. . .)(lf,, . . .) a protein having 4 identical X sites in the first parenthesis and s identical Y sitesin the second one, with a number a of Xsites of real affinity I, of which RZare occupied, and so on. Using a similar method to that employed to obtain equation (25), we can seethat:

IXrE

- ~. ,;

”

rs ,c, &

~~,, *-l~~~s..-,I~

. .,gl ;;

(.v)

. .& ;;

,I!! L. -

-

+ I’-: -

‘KC] .-~

. . c(~~-~E’. (m+...+t+...)Kp (x)

.

Jq”

I

. .Gr.

..

‘L ,,,,

(30) .

(v) (r+...)Lla,. In this equation, C is a statistical factor, z is the total number of real affinities. L, (, is the equilibrium conslant of the empty protein species Yl . . .\_

RLSTRICTIONS

ON

‘I‘lIE

ALl.OS

I LRI(’

MODtLS

431

P (I a09 2b,, . , r/z,, . . )( If,, . . .) with an arbitrary entity and the A’,, arc the real affinity constants of the ligand Y. Using equations (3), (29) and (30) we can obtain the k:, = KX(s, .I’) and .P = X(-Y, JI) functions, with the molecular constants n-,,, K,, and L,,, . WC have seen before that we can describe these functions with 4(&v+ 1) independent constants Kxij and s independent constants KY,,,O. Then, it is not possible to obtain a number of molecular constants higher than q(.r+ I )-l-s from the experimental curve x = ,Y(s. ~3) and it could be very difficult to adjust this curve with a number of molecular constants less than this. Tlw tl~elnloc!~~trcrnlic approxitnatiotrs

that lead to the t~~olecular models

inspection of equations (29) and (30) shows that, as the number of sites per protein molecule or the number of real affinities increase, the resulting functions, K,Y = K,y(.x, y) and B = X(X, J,), become very complicated. of little practical usefulness. Things become simpler if certain approximation\ in equation (30) are made. With these approximations and using equations (3) and (?9), we can reach the characteristic equations of the molecular models. These equations now appear as a result rather of thermodynamic approximations than of the molecular hypothesis made by their authorc. Obviously, the validity of these approximations must bc experimentally tested. We can divide the approximations (and hence the models) into two kinds: (i) Ali the reai qjfinities for each ligartd arc the satw ,for each protein twlfornzatioi~. This implies that one protein conformation has only one site conformation for each ligand. In consequence. the only permitted specks arc P(r~l~)(<+.v,), wi:!i i and ,j going from 0 to f/ and .s rckpectivcly. With this ~tpproximation equation (30) is reduced to:

in which !.,.<,is an equilibrium constant [& = I’(!‘f/~,)(ci.r,)/P(ly,)( I.r,)]. We can also obtain the protein fraction in a given conformation

;is a function

of

ligaild

prg = 2 C L 1 j concentration.

P(l'c/ii(Ysj)lp,,,:,I

I

Using equation (3), (29) and (31) we can obtain the function X = B(.u, ~9) which governs protein saturation when this approximation is used. This function was first developed by Monad, Wyman 6r Changeux (1965) for a

432

K. I’.

tiAHAY

model of this kind, with only two protein conformations and hence five molecular constants (the affinities Ku,, K,,, Kvr, KY2 and the equilibrium constant between the two protein conformations, I,) (Monod et a/., 1965). In obtaining it, they start from six molecular postulates; we only need one thermodynamic approximation. Obviously these six molecular postulates have a great biological significance, an aspect which is not taken into account in the thermodynamic analysis. We name “MWC” the general model and “common MWC” the particular one with only two protein conformations. We can also obtain the different AP~,~ (homo- and heterotropics) predicted by this model, using equations (21), (29) and (31). For the total A/r,,, wc can use equation (21) and the limiting values of KS of Table I. TABLE

Limiting

1

values of K,t

Therm0 dynamic

j’ These limiting values can be different if there are some k’,, and;or k’,-, equal to 0 (see text). $ Actually this model has some extra assumptions on the values of the L,, (EC text).

We studied above the thermodynamic restrictions for ligand binding to proteins. Now we are going to see that the approximation of the MWC model impose additional restrictions. (a) The model excludes the possibility of tlegative homotropic irlteractioru. This well-known restriction implies that not only 8X/& > 0 [see equation (27)] but also that 8&/8x 3 0. This restriction arises from the particular dependence of I,, on K,, [see equation (31)], which, as (x) increases, raises the fraction of empty sites of higher affinity (a null homotropic interaction implies q : 1 or only one K,,). The common MWC model imposes other restrictions on the heterotropic interactions :

RESTRICTIONS

ON THE

ALLOSTERIC

433

MODELS

(b) The kind of the heterotropic interaction cannot change with 8. This is so because changes in the unique equilibrium constant due to the heterotropic ligand are always in one direction. However, this restriction has some exceptions: the negative and positive heterotropic interactions can be transformed to a null one when B tends to 0 or 1. Thus, there are four exceptions which arise in certain particular conditions (see Table 2). The particular characteristics of these exceptional cases are given in Fig. 3. (c) The only heterotropie interaction that can be maintained constunf, OS a function of x, is the null interaction. These additional restrictions predict that certain curves cannot be found experimentally (see Fig. 3) and allow us to confront this model with the experimental data (see later). The MWC model does not impose additional restrictions on the limiting value of x of equation (26) (it can be less than I, in the particular cast in which each P,, has one real affinity, K,Y, or KYg, equal to 0). Equation (j2) allows us to explore the dependence of the protein conformations on (s) and (>I). We can see that when (s) tends to infinity, we obtain a mixture of the conformations of higher K,,; when (y) tends to infinity, we obtain a mixture of the conformations of higher KYq: and when TABLE

2

The ,four exceptions of the common M WC model for the predicted invariance with respect to the kind of the interaction with x Kind of interaction

Transforms to a null interaction when R tends to :

Required conditions on the model’s constants

0

L,c ;> 1 L,cQ-’ 66: 1 LgP-1

Negative 1

..’ I

L,c .- I L,P ‘: I L, I

Positive

In this Table, L, and L, represent the values of L{[l + K,&y)]/[l + KY,(r)])“, at the initial and final values of y, y, and y, respectively, with ( JJ,) > (J:) (see Fig. 1). The required conditions for positive and negative interactions in each limit, are equivalent changing L, by L, and vice versa. The real affinities for the ligand X are given in order to make c -1 &,IK,,) < 1.

434

R. I’.

GARA’I

x FIG. 3. Imposed restrictions on the ligand binding to proteins by the common MWC model. The common MWC model excludes certain heterotropic interactions which arc thermodynamically allowed (dotted curves). The full curves are heterotropic interactions permitted by this model: (a) a negative (or positive) heterotropic interaction which ir transformed into a null one as R tends to 1 (in the example of the figure the model constants are: c = 0.01; Li = lo3 and L, = lo”) (the curves for these particular interactions cannot exhibit relative maxima or minima); (b) a positive (or negative) heterotropic interaction which is transformed into a null one as x tends to 0 (in the example of the figure, c = 0.01; L, = lo7 and L, = 105); (c) a null interaction constant as a function of 2 (which implies L, = L, or c -= I); and (d) a negative (or positive) interaction in the entire range of x (in the example of the figure, c -- 0.01 ; L, - IO and L, :; 10-j. The temperature is assumed to be 25°C. For more explanations see Table 3 and tcut. both (x) and (~3) tend to infinity, we obtain LX mixture of the conformations of higher n/,, and/or ky, and 1101ir.termediate (new) conformations, a4

supposed by Weber (Weber, 1973).-i(33) !J

If some real affinities are Lqunl to 0, in certain casesequation (33) must be modified. For example, in a common MWC model with K,, = K,, =L0, we have three possibilities: (1) if y = s (which is in fact one of the molecular postulates of Monod et al., 1965), the limiting value of P2 is equal to K%,L/(K”,, +K$,L); (2) if q < s, it is equal to 0; and (3) if 4’ ;. ,T, it is t Weber comes to this conclusion from the analysis of AC,,,. for a particular system. This AGXjl- can be easily obtained from equation (14) and Table I.

RESTRICTIONS

ON

THF

ALLOSTCRIC

MODl-1.S

435

equal IO I A\ in this example K,y, :.= 0, the second pos~,ibility lcads to &I limiting value of g equal to 0 [see equation (26i]. (ii) Almost nil the enzptJ* sites Irtrvc 0 r!fit?itj,. ‘This implies that, in equnlion (‘29), each ligand has one real affinity equal to 0 (A:,, = ICY, = 0). which is the affinity of almost all the empty sites ( Ix, and I y, are nearly I i. This implics that the major existing protein species are of the type: P( IN,,, 2b,, . rh,,, . . .)( If,, . . . . cjk,, . .) and that the major existing Ypecies, lvith empty sites of real nflinily different frown 0, arc: /‘(In,, 2b,, . ., rh,-,, . . .)( I.f,, . . , .4X,<,. _ . ). Th>n, wilh this approximation equation (29) takes the form of a summation starting from r : ?. in which the I Yr have the form of equation (30). but wi:h I’ =: II- I (which makes one of the summations vanish). In the particular case in which each jigantI ha? only one real afini!! diKerent from 0, equation (29) takes the simple form: in which equal to :

K, = ’ 92 Kx,. I y2 @ I and therefore 1,, z I. In thi\

(33)

particular

case 1y2 is

in which Lij = /‘[I((!--i),, 2i,][ I(.s-j)o, Q,,]/P(!q,,, 20,,)( IX,, 20,). As before, using equations (3), (34) and (35), we can obtain the X = X(.y. J’) L‘unction predicted by this approximation. Star?ing from molecular postulate3 and making certain assumptions on the Lii, Koshland, Nemethy & Filmel ( I966) obtained it for several particular cases (Koshla.nd et crl.. I966: Koshland, 1970). Taking into account the pariicular condition (/ = .s. Koshland assumes that if Y is an activator L,, = Lii (the matrix of the fdfi is symmetric) and if Y is an inhibitor, a11 the L,, for which i+; > (! at-c equal to 0. In obtaining the different A,u,, predicted by this approximafion. we mu*t use equations (21), (34) and (35) and Table 1. As opposed to the previous model. this approximation does not give additional restrictions and has more ilexibiliry in the number and propertica or the molecular constants. (C)

ANALYSIS

OF

SOME

DA’r‘A

I:KOM

TIllI

I ITI.RATl:RC

The objective of this work is to provide a useful tool for estimating the free energy of ligand-ligand interactions in protei:is and for studying the thermodynamic consistency of the molecular models. I I!. ‘,Q

436

R.

P.

CARAY

As an example of application, we can explore ligand-ligand interaction\ in hemoglobin (Hb), which are the most widely studied protein-ligand equilibria. Homotropic

internctions

in Hb

Wyman extensively studied the homotropic interaction between O2 sites in Hb (see, for example, Wyman, 1964). He calculated the free energy ot homotropic interaction at constant pH and not at constant fractional saturation of Hb with H+ (I?). Then, Wyman’s free energy of homotropic interaction becomes our ApiXx (A&,, in this case). As it has been demonstrated by their authors the common MWC and Koshland models accurately describe this homotropic interaction by a close fit to the experimental saturation curve of Hb with O2 f0 = 0(02)] (Monod et trl., 196.5; Koshland et nl., 1966). The agreement between theory and experimental findings is not of a general nature, because there are certain oligomeric proteins which have negative homotropic interactions, which are excluded by the M WC model (Koshland, 1970). Care must be taken here because, in the MWC model, this exclusion arises only for binding interactions and not fat reactivity interactions (Goldbeter, 1974). Heterotropic i!lterurtions in Hb Wyman explored the heterotropic interactions in Hb, using his “linkage relations” [equation (12)] ( see, for instance, Wyman, 1948, 1964). The general treatment of the free energy of interactions, developed here, allows us to explore these interactions in more detail and to test the approximations of the molecular models. The estimation of the heterotropic free energy of interaction has two advantages over the estimation of the homotropic one: (a) it is easier to obtain AWL,,,, than ApTIXx, because this latter requires the rather precise measurements of the x = X(.X) curve at x nearly 0 or 1: (b) even when each ligand has no identical sites, A/llsv is not a limiting value as it is the case for AprX.; which is very important, because 0, sites in Hb seem not to be identical (Gibson, 1973). We will study the heterotropic interactions of 0, with H+ and with organic phosphates. (a) Heterotropic interaction between O2 and H’. As is well known, at a pH above 6, H+ diminishes the affinity of Hb for O2 and vice versa (alkaline Bohr effect). The experimental data give us the function 0 = O(0,) at different constants pH’s (see, for instance, Allen, Guthe & Wyman, 1950). Then, we can calculate the interaction change in the 0, chemical potential with pH (APL;& as a function of 0, by the procedure given in Fig. I (replacing I? by pH). As can be seen in Fig. 4, the Apiorr obtained is independent of 0. This

RESl-RICTIONS

ON

THE

ALLOSTERIC

MODELS

437

FIG. 4. Free energy of heterotropic interaction in hemoglobin as a function of it\ fractional saturation with 0, (0). The heterotropic change in the O2 chemical potential in Hb due to H+ (AP’~~~) is calculated from the experimental data of Antonini et al. (1962) for a pH interval from 6.5 to 9.2, at 2O”C, in 0.2 M phosphate. The heterotropic changes in the O2 chemical potential due to DPG (Ap’ ,& and to IHP (AJL~~~)are calculated from the experimental data of Tyuma et al. (1973) for a variation of (DPG) from 0 to 2 mM and of ([HP) from 0 to 1.7 mM, at 25°C in 0.05 or 0.01 M Tris buffer (pH 7.40). By comparing these curves with Fig. 3 and Table 2, it can be seen that the common MWC model does not satisfactorily explain these heterotropic interactions (see text). The Ajl’r,,tt cnrve can be exactly fitted using the constants of Table 3 and 4 and equations (3). (21 1. (24) and (25).

implies that, in this range of pH and without other interactions, d/l;,,o and d&rr are exact differentials; which is equivalent to saying that n, is independent of pH and i3In (O,)/JH is independent of 8. The independence of 11~ with pH is a well-established experimental fact for a wide range of pH (Allen et al., 1950; Antonini et al., 1962; Bonaventura & Riggs, 1968: Bunn & Guidotti, 1972). As we have seen before, this independence 1:; a necessary and sufficient condition for reducing equation (19) to equation (13) and for obtaining the disappearance of the homotropic components of the Weber’s expression of the free energy of heterotropic interaction [see equations (14) and (15)]. As there are four 0, sites per Hb molecule and presumably four H“‘ site\ responsible of the alkaline Bohr efl’ect (Wyman. 1939, 1964), we need 20

438

It . 1’. c; A K A Y

“thermodynamic” afinities h’oij and four Kllio in order to Iit the cspcrimcnial 0 = 0(01, H+) curves. The independence of kzOwith respect io pH imposes severe restrictions on the value of the thermodynamic affinities, but this is not sufficient I;)I obtaining them uniquely. The calculaled constants of Table5 3 and 4 arc one particular set of constants thal quantitatively accounts for the cxperimental data studied as they exactly fit the A,&,,, = ,f(Ti) curve of Fig. 4. These constants were obtained using equations (3), (24) and (25) for tkc analysis of the experimental data of the references of Table 3 and the hometropic experimental data of Imai (Imai, 1973). In Table 3 the four /i,,i,, arc assumedlo be identical (K,,) and an average K,, of all the experiments was used. The A&,,, =.f(@ curve of Fig. 4 was fitted using equations (31, (?I), (24) and (35), with the numerical values of Tables 3 and 4. However, by comparing Figs 3 and 4. ~vc can set that :his curve is “forbidden” h!f the common MWC model. The intrinsic difliculty of all the MWC modck, including complicated ones with many protein conformations, is that they do not permit to obtain a H -‘- site linked o/z/y to the Oz sile of :hc same Hb subunity, which, as Wyman pointed out, is a necessarycondition for explaining the Bohr effect (Wyman, 1965). On the contrary, ali the thermodynamic constants of Tables 3 and 4 can be obtained from a simple model which contains the thermodynamic approximation that leads to the Koshland et al. (1966) model (see above). In this simple model it is assumed that almost all the empty O2 sites have zero, affinity and that it is not possible lo obtain the occupation of the H+ and 0, sites, at the same time, in the same subunity. This means that each 13’. silt is only linked to tile O? site

& (mM-‘)

Source of data for calculation of A’,i

Antonini ct

al., 1962-V

Bonaventura & Riggs, 1968 Roughton 8.z Lyster, 1965: Allen et c/l., 1950:;: Nagel ct ul., 39675 The small difference in the calculated K,, can be explained by the different experimental conditions. -t This data was also e:nployed for obtnlning h/d’,r,,i of Fig. 4. $I The curves at ditferent pH were ohtained wit11 different solvent,;. $ This csperimcntnl data was obtained at 10 C‘.

RLSTRICTIONS

ON

‘I’HL:

ALLOSTLRIC

439

MODELS

TAULE 4 Calculated O2 u$?iuities, K,ij, jbr tllc alkahe 0.248 0.359 2,545 8,787

i:

0.186 0.139 I .I!71 0

0.124 0.13. 0 0

The /Co,, of this Table are expressed I, . . . . 4and.j:0, . . . . 4.

Bohr ej’&*t 0 0 0 0

0.062 0 0 0

in mm Hg

I and arc given

in a matrix

form,

with

of the same subunity (see above). In order to show that this model quantitalively explains the experimental data we can see that, in comparing equations (24) and (25) with equations (34) and (35), the following relation is obtained : K”;j

KO~Lii

=

L(i-

(30) I),'

in which Lo2 is the real afEnity for 0, of equations taking into account that: L;, =

(34) and (35). Then.

Lij Li"

=

a;,

(37)

Li

LiO

we can easily obtain the aij of this model (which, as can be seen in equation (371, are probability factors). In matrix form they are: 1 I I I I

1 3/4 I/” l/4 0

1 Ii2 l/6 0 0

If we now give the following ,f, = 0*148,

1 l/4 0 0 0

1 0 0 0 0

i~.j : 0. . . . , 4.

values to,/; : h;,LLi/Li-

f2 = 0.359,

,r; = 2.545,

(3X)

1: /:, = 8.787

(39) wc can obtain all the Koii of Table 4 by replacing the values of (38) and (39) in equations (36) and (37).-T Then, the use of equations (3). (?I), (34), (25). (34) and (35) with the numerical values of (38), (39) and an average K,, of Table 3, permit us to account for the experimental data studied from a molecular point of view. It is interesting to see that, by means of structural analysis, Perutz found that each subunity has an histidine whose combination with H+ stabilizes 1 In fact. the calculated However, this indelermination to the previous one.

&,,

with i -: .j :a 5 have disappears in equation

an indeterlnitlcltit~n of the tvpe OjO. (25) and the obtained resultSis equal

440

K.

P. GARAY

the subunity conformation of deoxyHb which has a very small affinity for O2 (zero in our model), which is in accord with the two principal suppositions of our model (Perutz, 1970). In addition, it can be seen from Table 3 that the average pK of the H+ site (7.63) lies, as Wyman supposed a long time ago, in the range of pK of the histidine (Wyman, 1939, 1948, 1964). (b) Heterotropic interactiom between 0, and organic phosphates. Many organic phosphates, as 2-3,diphosphoglycerate (DPG) and inositol hexaphosphate (IHP), interact negatively with 0, in Hb. Tyuma, lmai & Shimizu (1973) studied the effect on the 0 : &O,) curves, of constant concentrations of DPG and IHP (Tyuma et al., 1973). As before, using a similar procedure to that given in Fig. 1, we can obtain from these data the interaction changes in the 0, chemical potential with DPG (A&,) and with IHP (A&J. We can see in Fig. 4 that, while A,nio, is always a negative interaction, Apic,n becomes a null interaction as 0 tends to I. As in these cases, n, and the partial derivatives depend on the integration paths, d/l;,,, d&D and &h are not exact differentials. The disagreement between the MWC model and the experimental data on the independence of ilo with pH, was explained by the authors saying that H+ is not an “allosteric ligand” (Monod et al., 1965). Then, it is interesting to test this model with the experimental data on the heterotropic interaction between O2 and organic phosphates because, at least DPG can be considered an allosteric ligand. This is so for the following reasons: (1) as it is different from H+, it is a stereospecific ligand (Perutz, 1970) ; (2) it has a specific affinity for deoxyHb (Benesch, Benesch & Yu, 1968); and (3) its interaction with 0, in Hb plays a physiological role (Benesch & Benesch, 1969).? As we can see in Table 2 and Fig. 3, the calculated A& :f(@ curve corresponds to the general case in which Lc 9 I, but the A&r, :f(O) curve corresponds to the particular condition in which Lc 4 1. Then, these two interactions cannot be explained with a unique model. Moreover, the Ap;,-,n : f(O) curve has a relative maximum which, in this particular case, is “forbidden” by the common MWC model (see Fig. 3). In conclusion we can say that the common MWC model makes certain approximations which, while adequately describing the homotropic interactions in Hb, are too stringent to explain the heterotropic interactions studied in this system. On the contrary, the thermodynamic approximation t In fact, one of the molecular hypothesis of Monod er at. (1965) is that all the ligands have the same number of sites per protein molecule. However, as we have seen before. the form of the equations does not change if the number of protein sites is different, as is the case for O2 and organic phosphates in Hb. Moreover, the MWC model has been employed in many circumstances in which the number of protein sites are different.

RESTRICTIONS

ON

THE

ALLOSTERIC

hfODE1.S

441

that leads to the characteristic equations of the Koshland et 01. (1966) model is a good one to quantitatively account for the thermodynamic analysis of the alkaline Bohr effect. However, this is not a conclusive test for this modeI, as the constants of Tables 3 and 4 are not unique and the Koshland model is not a unique description of these constants. The author is greatly indebted to Dr P. J. Garrahan, Departamento de Quimica Biologica, F.F. y B., Universidad de Buenos Aires, for reading this manuscript and for helpful discussion. The author is also indebted to Dr J. W. Turner, Service de Chimie Physique 2, Universite Libre de Bruxelles, who helped with the English. REFERENCES ALLEN, D., GUTHE, K. & WYMAN, J. (1950). J. biol. Chcm. 187, 393. ANTONINI, E., WYMAN, J., BRUNORI, M., FRONTI~ELLI, C., Buccr. E. & ROSS-FANFI I I. A. (1962). J. biol. Chem. 237, 2773. BENESCH, R., BENESCH, R. E. & Yu, C. (1968). Proc. nurn. Acad. Sci. U.S.A. 59, 5X. BENESCH, R. & BENESCM, R. E. (1969). Nature, Lond. 221, 618. BONAVENTURA, J. & RIGGS, A. (1968). J. biol. Chem. 243,980. BUNN, F. & GUIDOTTI, G. (1972). J. biol. Chem. 247,2345. GIBSON, Q. (1973). Proc. natn. Acad. Sci. C/&4. 70, 1. GOLDBETER, A. (1974).J. molec. Biol. 90, 185. IMAI, K. (1973).Biochemistry 12,798. KOSHLAND, D., NEMETHY, G. & FILMER, D. (1966). Biochemisfry 5, 365. KOSHLAND, D. (1970).T/ieEn:ymes (Boyer, P. D., ed.).Vol. 1,p. 341.NewYork: Academic

Press.

NAGEL, R. MONOD, J., PERUTZ, M. ROUGHTON, TYUMA, I., WEBER, G. WYMAN. J. WYMAN, J. WYMAN, J. WYMAN, J. WYMAN, J. WYMAN, J.

(1967). Biochemistry 6, 2395. WYMAN, J. & CHANGEUX, J. P. (1965). J. molec. Biol. F. (1970). Nature, Lond. 228, 734. F. & LYSTER, R. (1965). Hvalraders Sk. 48, 191. IMAK, K. & SHIMIZU, K. (1973). Biochemistry 12, 1491. (1972). Biochemistry 11, 864. (1939). J. biol. Chem. 127, 581. (1948). Adv. Protein Chem. 4, 407. (1964). Adv. Protein Chem. 19, 223. (1965). J. molec. Biol. 11, 631. (1967). J. Am. Chem. Sot. 89, ??01. (1969). J. molec. Biol. 39, 523.

12. 88.