23
strength (in terms of AU) such as tetrahydrothio~hene-1~6and TMA-S02 should behave so differently in this respect. We believe that further progress toward understanding solvent effects on the spectral parameters of CT complexes will depend upon two factors: (1) the acquisition of additional accurate data for both weak and strong complexes in the vapor phase; and (2) development of methods for the study of particularly weak complexes in solution, which may remove possible inadequacies in the interpretation of the solution data.20 The present study demonstrates the value of a combination of (20) P. J. Trotter and M. W. Hanna, J . Amer. Chem. Soc., 88, 3724 (1966).
spectral and nonspectral methods to obtain reliable spectral parameters in the vapor phase. For systems in which donor, acceptor and complex are not very volatile, additional techniques must be used. One new method, employing a mixture of polyiodides as a constant iodine activity source, has been successfully used to study the diethyl ether-iodine adduct; the technique is apparently applicable to systems in which either very weak or moderately strong complexes are present. Acknowledgment. This work was supported in part by National Science Foundation Grant No. GP-8029. (21) J. Childs, J. Grundnes, and S.D.Christian, in preparation.
Statistical Theory of Cooperative Binding to Proteins. The Hill Equation and the Binding Potential' Henry #A. Heck Contribution from the Department of Chemistry, University of California, Berkeley, California 94720. Received May 29, 1970 Abstract: The Hill equation and the binding potential are useful methods of describing the cooperative binding of ligands to proteins. Starting from the formal theory of solutions developed by McMillan and Mayer, statistical mechanical versions of both of these classical expressions are here derived. The fractional occupation of protein sites by ligand is expanded in powers of protein concentration, and is used to derive explicit expressions for the effects of intermolecular (protein-protein) interactions on both the Hill equation and the binding potential. The theory leads to a new definition of the apparent free energy of interaction between sites on a single molecule in terms of the free energy of a ligand-transfer reaction between two macromolecules, and provides insight into the significance of the slopes of Hill plots in terms of ligand-transfer processes. The results indicate that, in most cases, the Hill plot parameters may be expected to be influenced to only a minor or negligible extent by intermolecular forces.
T
he object of this paper is to discuss an empirical equation of A. V. Hill2 and the binding potential of Wyman3 from the point of view of McMillanMayer4 solution theory. The Hill equation, which has often been used to describe binding of ligands to prot e i n ~can , ~ be written as
P,/(l - P,)
=
Katn
where Pi is the fractional occupation of i sites by ligand i at ligand activity ai. The exponent n is, in general, a function of a , but is usually found to be essentially constant over a fairly wide range around the midpoint of the binding curve.5 As originally presented by Hill, this equation was a partially successful attempt to describe the cooperative binding of oxygen to hemoglobin. Its modern importance is derived from the demonstration that the Hill equation yields useful thermodynamic information (1) This work was supported by Grant No. 5-ROI-AM13164-02 from the National Institute of Arthritis and Metabolic Diseases, U. S . Public Health Service. (2) A. V . Hill, J . Physiol. (London), 40, ivP (1910). (3) J. Wyman, J . Mol. B i d , 11, 631 (1965). (4) W. G. McMillan and J. E. Mayer, J . Chem. Phys., 13,276 (1945). (5) J . Wyman, Aduan. Protein Chem., 19, 223 (1964).
about homotropic6 reactions, of whatever origin, in any ~ y s t e m . ~ If logarithms are taken of both sides of this equation, the resulting expression forms the basis for the wellknown Hill plot.5 Two important quantities can be extracted from such a plot: (1) the minimum value for the decrease in work per site required to saturate the macromolecule with ligand, which results from cooperative interactions among the sites; and (2) the slope of the Hill plot, n, at the midpoint of titration, which is a measure of the cooperativity of the binding reaction. The two quantities are closely related, and, in fact, one is a function of the othere5 The determination of the first of these two quantities from a Hill plot and its connection with the second are not based on the assumption of a particular cooperative model, but arise from general thermodynamic considerations. In the following sections, the methods of statistical mechanics are used to investigate more closely the significance of the two quantities which are derived from Hill plots. The approach taken is, in essence, an extension of the elegant theory of protein solutions pub(6) Homotropic is used for interactions between sites which bind the same type of ligand.
Heck
Statistical Theory of Cooperative Binding to Proteins
24
lished by T. L. and is thus quite general. The dependence of the Hill plot parameters on intermolecular (protein-protein) interactions is derived. In addition, a statistical mechanical version of the binding potential is presented which includes explicitly the influence of intermolecular interactions on this function. In the limit of infinitely dilute protein solutions, the equation for the microscopic binding potential is formally identical with the corresponding function for the macroscopic thermodynamic case. The Hill Equation in the McMillan-Mayer Theory We begin with some results of the McMillan-Mayer theory for polyatomic multicomponent solutions in order to define briefly the notation to be employed in later sections. We restrict ourselves to solutions containing one type of protein and one type of solvent ion or molecule whose binding t o the protein is considered explicitly. The absolute activity of this single solvent species is denoted by X = exp(p/kT). The activities of the remaining solvent species are signified by A. Let zs = exp(ps/kT) be the absolute activity of a protein molecule, with t sites, which contains s bound ligand molecules. We take the point of view that a given protein molecule to which s ligand molecules are bound has accessible to it the full set of energy states associated with all possible distributions of the s ligand molecules on the sites, that is, that the t ! / [ s ! ( t - s)!] protein subspecies are in “isomeric” equilibrium. It follows from the definition of chemical potential that, at equilibrium
(2) where z is the absolute activity of a protein molecule withs = 0. With these definitions, the equivalent of the grand canonical ensemble partition function for a multicomponent solute in an osmotic system of volume V can be written as 7,9,10 2,
exp(rV/kT)
=
XSZ
vent is implicit both in the single-molecule partition function, H8, and in the potential of average force on a fixed set of solute molecules m immersed in the outside solution which contains solvent species only, w ( ~ ) ({m), A, A, z = 0). The set of coordinates (translational and rotational) specifying the position, orientation, and conformation of each member of this set of solute molecules is denoted by { m}. We hereafter take ~(“’(0) t o mean d m ) ( { m ] ,A, A, z = 0) and employ w,(~)(O)to represent the spatial potential of average forceg on molecules of the set m, which includes averaging over all rotational configurations in the set, but which depends, in general, on the particular solute species comprising the set. In a fluid, w,(~)(O)approaches zero when the molecules of the set are widely ~ e p a r a t e d . ~ , The fractional occupation of sites is (5)
where p is the total density of solute molecules and Bz* and Ba* are, respectively, second and third virial coefficients for the solute. lo From eq 6 we obtain directly for the Hill equation
-- E:
1-Y
=
+ . .) C(t - s)H,Xs + CHsXS(pdB2*/bIn X + . . . ) CsH,Xs - CHsXS(pbBz*/bIn X s
s
s
In the limit p
= m)O
,
s
(7)
+0
_-1 - Y = -C(t - s)H8Xs 8
s
In this equation, r represents the osmotic pressure difference across a membrane separating a solution containing solute species and solvent species on one side from one containing solvent species on the other. Only solvent species are capable of permeating the membrane. HEis an effective partition function for a single solute molecule of species s in the solvent and includes interactions among the s ligand molecules bound to the protein. For a single molecule of species s, Hs is defined aslo
Equation 7 gives the first-order corrections to the Hill equation for finite protein concentrations. We shall discuss the importance of these and other corrections in a later section.
Hs = 4 ( w - Y S 0 ( x , A) (4) where y s o is the activity coefficient for this species at infinite dilution in the solvent. The influence of sol-
The Wm(0) are potentials of average force which vanish without integration over rotational coordinates when the solute molecules are widely separated in a fluid. The configuration integrals in eq 43 are related to those of the Hill theory according to
(11) I t should, perhaps, be stressed in order to avoid confusion that the present approach differs from that taken by T. L. Hill.’ The potential of average force employed in the present paper is related to potentials of average force in the Hill theory by w ( ~ ) ( o ) = Wm(O)
+ Z[W(~)((IJ,O) + . . . + W(l)((mJ,O)I
(a)
8
(7) T. L. Hill, J . Chem. Phys., 23, 623 (1955). (8) T. L. Hill, ibid., 23, 2270 (1955). (9) T. L. Hill, “Statistical Mechanics,” McGraw-Hill, New York, N. Y., 1956, pp 262-285. (10) T. L. Hill, “An Introduction to Statistical Thermodynamics,” Addison-Wesley, Reading, Mass., 1960, pp 353-362.
Journal of the American Chemical Society / 93:l / January 13, 1971
25
Apparent Interaction Energy It may be observed from eq 8 that, as X --+ 0 In [ P/(l - F)] = In ( H l / t H o ) In X (9) On the other hand, as X --+ In [ F/(l - P)] = In (tH1/H1-l) In X (10) The slope of the Hill plot at the two extremities is unity. These two limiting equations define the asymptotes of the Hill plot. The vertical distance between the asymptotes at any X is proportional to the minimum value for the decrease in work per site required to saturate the macromolecule with ligand,s and is termed the apparent interaction energy. When the final asymptote lies above the initial one, this energy is conventionally taken to be positive.5 From eq 9 and 10, this quantity is
+
+
let each protein molecule have TI sites of type 1, r 2 of ., r m of type m. Define effective partition type 2, functions for a site of type j to which zero or one ligand molecule is bound as h,(O) or hl(l). Then, if there is a total of s ligand molecules bound to a protein molecule, of which s1are bound to sites of type 1, sz are bound to sites of type 2, etc., we have that
..
where the zero of energy for intramolecular interactions is arbitrary. The sum is over all sets s = si,s2, . . . , sm satisfying the restrictions m
m
cs,=s
xr
