I. theor. Biol. (1977) 67, 557-565

Thermodynamics of Allosteric Transitions HUEY W. HUANC Physics Department, Rice University, Houston, Texas 77001, U.S.A. (Received 14 September 1976) A new thermodynamic description of allosteric transitions in enzymes is introduced. Distinguished from previous approaches, the new method accounts for the effect of temperature on ligand binding aRnities as well as the effect of ligand concentration. When applied to hemoglobin, the method yields a generalized Hill equation. It reproduces the well known Hill equation when the oxygenation is neither too high nor too low, and it predicts two aspects of the temperature effect on oxygenation: (1) the shape of a Hill plot is invariant under temperature change, and (2) for a given degree of oxygenation, the oxygen activity is linearly dependent on temperature. The predictions are born out by experiment.

1. Introduction

A sigmoidal relationship is frequently shown between the input parameter and the output response of a biological system. At the molecular level many of such sigmoidal responses are results of allosteric transitions (or conformational equilibria) occurring in enzymes. Enzymes frequently contain two kinds of sites, those that activate or inactivate the enzyme, and those that exert the specific catalytic action. The influence of ligand binding at the control site on the activity of the catalytic site is said to be an allosteric interaction. Allosteric interactions in various kinds of enzymes appear to yield a similar input-output relationship, i.e. a sigmoidal one, and are presumably due to the same macromolecular mechanism, i.e., conformational transition in enzyme (Kvamme & Pihl, 1968). Much of our current knowledge about allosteric interactions are derived from extensive studies of hemoglobin. Theoretical interpretations mostly start with an assumed free energy spectrum for ligand bindings in a single enzyme. (For review see Wyman, 1967; Baldwin, 1975; Edelstein, 1975; Shulman, Hopfield & Ogawa, 1975.) Such an approach, although it accounts for the dependence of ligand binding on ligand concentration, does not explain the strong dependence of ligand binding on temperature. ss7

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In thermodynamics, different approaches based on different viewpoints toward the same phenomenon have proved to be fruitful. In this paper I shall present an alternative thermodynamic description of allosteric transitions. Instead of the discrete ligand-binding free energy spectrum of a single macromolecule, I study the binding free energy averaged over the ensemble of macromolecules, as a continuous function of ligand concentration. The ensemble-averaged free energy as a function of external variables is a thermodynamic property of a large system (the collection of macromolecules), not of a single macromolecule. Often in a complicated system certain regularities appear when a statistical average is taken of atomic properties, and such regularities are often independent of microscopic details. This is particularly true when the fluctuation correlation length is substantially longer than interatomic distance. Well known examples are critical phenomena, where systems of great varieties behave similarly when the fluctuation correlation length approaches infinity. In a typical enzyme the ligand binding sites are separated from one another by a distance of 20 A or longer, an order of magnitude longer than the interatomic distance, and allosteric interaction means fluctuation correlation exists among these sites (Huang, 1976). It is possible then that certain regularities independent of microscopic details may appear if we study the ensemble averaged properties of allosteric macromolecules. Indeed I found that a well known thermodynamic theory designed to deal with long-range correlations (or systems undergoing phase transitions) implies a generalized Hill equation for allosteric macromolecules. On the degree of ligand binding as a function of ligand concentration, the theory reproduces the well known Hill equation when the ligand saturation is neither too high nor too low. On the temperature effect the theory predicts that (1) the shape of a Hill plot is invariant under temperature change, and (2) for a given ligand saturation, the ligand activity in the solution is linearly dependent on temperature. All these predictions are in good agreement with hemoglobin data. 2. Cooperative Phenomena of Allosteric Transitions Hemoglobin will be used as an example to illustrate the idea. Consider a hemoglobin solution in equilibrium with gaseous oxygen. At a given (nonzero) partial pressure of oxygen, Po2, hemoglobins exist in five different forms as far as oxygenation is concerned : four deoxy hemes, three deoxy hemes and one oxy heme, . . . , and four oxy hemes. Taking the whole system as an ensemble of hemes, there is a well-defined average oxygen binding free energy per heme, AE. (As we shall see later it equals to kT {In PO2 - In [Y/(1 - Y)] + const), where Y is the percentage of hemes oxygenated.) The averaged free

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energy levels per heme are deplicted in Fig. 1. If there is no allosteric interaction, or in other words, each heme is independent of the others like in the case of myoglobin, then the binding free energy is a constant and the satur. . atron curve IS hyperbohc m PO,.

FIG. 1. The average free energy levels of a hemoglobin oxygen binding to a subunit.

subunit. AC is the free energy of

What makes allosteric macromolecules interesting is that their AE varies as a fun&on of Po2. As PO2 increases, de varies from a positive value to a negative value. Now if a thermodynamic system has two macroscopic states with their energy levels separated by a barrier as in Fig. 1, and the energy difference Ae varies across zero as a function of a thermodynamic variable, it is a (first-order) phase transition (Callen, 1960). Indeed, binding oxygen to a hemoglobin causes tertiary structural change as well as quaternary structural change in protein (Shulman et al., 1969; Ogawa & Shulman, 1971, 1972). Without going into the microscopic details as to how the protein structure changes at each step of oxygenation, it is safe to say that on average the macroscopic conformation changes continuously from one form to another as saturation proceeds from zero to 100%. We therefore propose that a phenomenological theory for structural phase transitions is applicable to describe the binding free energy of hemoglobin oxygenation (Huang, 1976). The phenomenology of phase transitions is described by Landau’s theory (Landau & Lifshitz, 1969) which is a thermodynamic theory independent of microscopic details and is applicable to all phase transitions. The central idea of Landau’s theory is that a phase transition is characterized by an order parameter which takes on the value zero at temperatures above the transition point and a greater than zero value below the transition point. The thermodynamics are contained in a free energy (Landau’s free energy) expanded as a power series of the order parameter. The order parameter for a given phase transition is not necessarily uniquely defined. Sometimes many different thermodynamic quantities can be used as order parameters for a transition. If one considers an allosteric transition as a structural phase transition, the natural choice for its order parameter is the degree of completion of the conformation change, q, from the fully deoxy conformation to the fully oxy conformation. Again it should be emphasized that q is a quantity averaged over the entire ensemble of hemoglobins. Landau’s free energy, P, per

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oxygen-heme

system, can be written as (Landau & Lifshitz, 1969) F = -gOCL~+(A/2)r12-(8/3)113+(C/4)114+ ..., (1) where the first term expresses the coupling between the chemical potential of 02, p, and the order parameter (go being an unknown coupling constant), A has the form as assumed by Landau in order to describe a phase transition (A,, To constants), and both B and C vary slowly in the transition region compared with A, so they are regarded as constants to the lowest order approximation [the factors 4, 3, & in equation (1) are introduced for convenience]. The first term is analogous to a magnetic coupling-p is thermodynamically equivalent to a magnetic field and 1 is equivalent to a magnetic moment. The free energy equation (l), including the specific form for A, is also a typical result of the so-called molecular field approximation of most statistical models (Stanley, 1971; see discussion and references in Huang, 1975). The stable (and metastable) states of an oxygen-heme system are determined by the minima of Landau’s free energy. As we shall see, this free energy by design has two minima (like Fig. 1): one represents the deoxy state and the other the oxy state. A phase transition takes place when the two minima have the same energy level. Landau’s theory, being of thermodynamic nature, does not predict the phase transition points but describes the nature of free energy around the phase transition points and correlates the transition points on the p-T plane. We choose p = 0 at a phase transition point. (This always can be done since chemical potential is defined only relative to an arbitrary reference point.) For p = 0, Landau’s free energy F is depicted in Fig. 2. For temperatures close to To, or more precisely 0 < (T- To) < B2/4AoC (see Appendix), F has two minima on the positive q axis corresponding to stable (or metastable) states: one at q = 0, and the other at q = ‘lo > 0. At T = T, z To +2B2/9AoC, two minima have the same energy level (see Appendix). As Fig, 2 shows, at T > T, the level at q = 0 is lower than the level at q. and uice uersa at T < T,. Since experimentally Y is a decreasing function of T,

FIG. 2. The Landau free energy F vs. the order parameter q at p = 0 for temperature above, equal and below To.

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the level at q = 0, which has a lower free energy at high temperatures must correspond to the deoxy state, while the level at q. is the oxy state. The binding free energy As is the energy level of the oxy state minus the energy level of the deoxy state. We see that AE vanishes at the point p = 0, T = T,; (0, T,) is a phase transition point in the p-T plane. In general the first order phase transition points fall on a line (e.g. Huang, 1975). The other phase transition points near (0, T,) can be obtained by expanding AE to first order in p and (T- T,) (see Appendix) AE N -go(2B/3C)~+(2AoB2/9C2)(TT,). (2) The equation AE = 0 then defines the phase transition line (Fig. 3).

FIG. 3. The phase diagram near (0, Tc) on the p-T plane. The line is the phase boundary separating the oxy state from the deoxy state.

If the energy levels described above were belonging to an infinite system, the transition would take place discontinuously, i.e. the system would occupy the deoxy state on one side of the transition line in the p-T plane and occupy the oxy state on the other side of the line. Since hemoglobin is a finite system, its thermodynamic behavior will not be discontinuous. Each heme can occupy either state at any given fi and T, with the probability determined by the grand canonical distribution law. Thus the ratio of the probability occupying the oxy state to the probability occupying the deoxy state has the following form

Y/U- Y>= exp[8(v - WI or

in [Y/(1 - Y)] = &-A&)

(3) In order to express equation (3) in terms of PO2 and T, we write the chemical potential of gaseous oxygen in the form (Landau & Lifshitz, 1969, p. 127) P = km C~‘O,/P~(T)I and (T- TJ, one obtains

(4)

To first order in In [Po,/Po(Tc)]

P = kT, In C~‘O~/P~K)I - So2(T- TJ (5) where So, is the specific entropy of gaseous oxygen at PO, = P,(T,) and T.B. 31

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T = T,. P,,(T,) is the partial pressure of oxygen which makes p equal to zero at T,. Combining equations (2), (3), and (5), one obtains

InCY/U- VI = Cl+ C&oJVC)lIni?o,/CK)l - WJ-‘(so, + &XT--K),

(6)

where S,, = (2g,B/3C)Soz+2A,,B2/9C2 is the entropy change associated with the transition from the deoxy state to the oxy state at p = 0, T = T,. Since P,(T,) and T, cannot be determined from saturation data, we rewrite equation (6) in the form

ln CY/U- Y)l N

np In PO2 - n,T+ K,

(7) where n,, ~tr and K are constants to first order in In [P,,/P,(T,)] and CT- T,). Equation (7) is a generalized Hill equation including the temperature dependence. The original Hill equation, consisting of the first term on the right-hand side of equation (7) is well known to describe the experimental data of mammalian hemoglobin from 10% to 90% saturation with a Hill constant nP about 3.0 (Roughton & Lyster, 1965; Tyuma, Imai & Shimizu, 1973). The regions above 90% and below 10% saturation correspond to very high and very low oxygen pressures respectively. Very high and very low oxygen pressures in turn correspond to large chemical potentials, positive and negative respectively. Landau’s expansion is not applicable in these regions. Although there is no quantitative theory of phase transitions applicable to large chemical potentials, let us call on our experience in one well studied phase transition, i.e. para to ferro-magnetism, to see what to expect. A large chemical potential here corresponds to a large magnetic field in magnetism. In a magnetic hysteresis a large field forces the energy levels of the metastable and stable states to merge into one. Thus we expect As in equation (3) to approach zero as the absolute value of p increases to a large value (the high and low PO2 limits). That of course agrees with the experimental observation that the slope of the asymptotes in the Hill plot (In [Y/(1 - Y)] vs. In Po2) approaches unity (n,, + 1). The temperature term in equation (7) predicts two:aspects of the temperature effect on hemoglobin oxygenation. (1) The shape of the Hill plot is invariant under temperature change. Two saturation curves at different temperatures can be brought to coincide with each other by changing the scale of oxygen pressure in one of them:

In [Y/(1 - Y)l = npIn CPO~/W’~)I= npIn CPo2/W2)I,

(8)

where G(T) is a function of temperature only. This prediction is in very good agreement with experiment (Roughton, Otis & Lyster, 1955). (2) For a given Y, In PO2 is linear in T. This is also found to be true in reality. Figure 4

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shows In PO2 vs. T at Y = 50% in a standard dilute hemoglobin solution (Benesch, Benesch & Yu, 1969). Figure 5 shows the same curve for hemoglobins in serum (Dittmer, 1961). The values of nP, &, and K for Y between 10% and 90% are also shown in the figure captions.

FIG. 4. Log PO2 vs. Tat 50%saturation(Y = 0.5).Data (dots)weremeasured by Benesch et al. (1969)underthe condition:hemoglobinconcentration0.4%. his-TRJS buffer 0.05M, pH 7.3, total chlorideO-1M. The straightline is the predictionof the theoryequation(7). Togetherwith the data of Tyuma et al. (1973),one getsnPN 3.0, nTN 0.19 deg-I, K N 52(PO2 in mmHg,Tin kelvin) for equation(7).

FIG. 5. LogPO2 vs. Tat 50%saturationfor hemoglobins in serumpH 7.4.Thedatawere taken from Dittmer (1961).n, =2-4, nT~~0.11 deg-I, K N 26 (PO2 in mmHg, T in kelvin) for equation(7). The straightline is the predictionof the theory equation(7).

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In conclusion we found that the allosteric transition, at least in the case of hemoglobin oxygenation, closely resembles a phase transition phenomenon, and that our new thermodynamic approach agrees with experiment. The author wishesto thank Professor H. J. Morowitz for his valuable comments. This work was supported in part by the Office of Naval Research Contract NOOO14-76-C-0273 and by PHS Research Grant No. GM21721-02 from National Institute of General Medical Sciences.

REFERENCES BALDWIN, BENFXH,

CALLEN, D-R,

J. M. (1975).Progr. Biophys. molec. Biol. 29, 235. R. E., BENESCH, R. & Yu, C. I. (1969). Biochemistry 8,2567. H. B. (1960). i’?termodynamics, Chap. 9. New York: John Wiley. D. S. (ed.) (1961). Biological Handbook-Blood and Other Body

Flui&, p. 164.

Washington, D.C. : Federation of American Societiesfor Experimental Biology. EDE~STEIN, S. J. (1975). A. Rev. Biochem. 44,209. HUANG, H. W. (1975). HUANG, H. W. (1976). KVAMME,. E. & &-IL,

Phys. Rev. B 12,216. Collect. Phenom. A. (eds)(1968). Regulation of Enzyme Activity and Allosteric Znter-

actions. New York: Academic Press. L. D. & Lnsnrrz, E. M. (1969). Statistical Physics, Chap. 14. Reading, Mass.:

LANDAU,

Addison-Wesley.

OGAWA, S. Jc SHIJLMAN, R. G. (1971). Biochem. biophys. Res. Comm. 42,9. R. G. (1972). f. molec. Biol. 70, 315. OGAWA, S. & SHULMAN, ROUGHTON, F. J. W. & LYSTBR,R. L. J. (1965). Hvalradets Skrifter 48, 185. ROUGHTON, F. J. W., OTIS, A. B. & LYSTER, R. L. J. (1955). Proc. R. Sot. B 144,29. SHULMAN, R. G., HOPFDXD, J. J. & OOAWA, S. (1975). Q. Rev. Biophys. 8, 325. SHIJLMAN, R. G., OGAWA, S., WUTHRICH, K., YAMAME, T., PEISACI%, JR. & BLIJMBERG, W. E. (1969). Science, N.Y. 165, 251. STANLEY, H. E. (1971). Introduction to Phase Transitions and Cooperative Phenomena.

Oxford: Oxford University Press.

TYUMA, I., IMAI, K. & SHIMIZU, K. (1973). Biochemistry 12,419l. WYIVIAN, J. (1967). J. Am. them. Sot. 89,2202.

APPENDIX (i) The condition that, at p = 0, F bus two minima on q 2 0 aF/@ = Arl-&r’+ C$ = 0 must have three real roots. That requires B2 > 4AC or B2/4AoC > (T-To). (ii) The phase transition point T, at ~1= 0 F = q2[(A/2)-

(B/3)1 +(C/4)q2]

= 0

has an obvious double root at rl = 0 which corresponds to the minimum at q = 0. The other minimum has a non-zero energy level unless the other two

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roots coincide. That occurs when (B/3)2 -4(A/2)(C/4)

= 0 or

T = T, +2B2/9A,C.

(iii) The free energy of binding AE to first order in p and (T - TC) To first order in p and (T- T,) the energy level of the deoxy state is zero. To calculate the energy level of the oxy state, we first calculate its order parameter q0 at ,u = 0. q,, is the largest root of aF/aq = 0, i.e. tjo = (B/2C) + (+C)(B’-4AC)*

(AZ>

Substituting it into F one obtains the energy level of the oxy state to first order in (T- T,) F. 2 (2AoB2/9C2)(T-

Tc).

(A3)

To first order in p and (T- T,), F, E -go(2B/3C)p+(2AoB2/9C2)(T-

which is also AE to the same order.

T,),

(A4)