Calculation of site affinity constants and cooperativity coefficients for binding of ligands and/or protons to macromolecules. I. Generation of partition functions and mass balance equations.

The thermodynamics of binding of a ligand A and/or proton H to a macromolecule M is treated by the partition function method. In complex systems, the representation of the equilibria by means of cumulative constants beta PQR used as coefficients in partition functions ZM, ZA, and ZH is ill-suited to least-squares refinement procedures because the cumulative constants are interrelated by common cooperativity functions gamma j(i) and common site affinity constants kappa j. There is therefore the need to express ZM, ZA, ZH as functions of site constants kappa j and cooperativity coefficients bj. This is done by developing an algebra of partition functions based on the following concepts: (i) factorability of partition functions; (ii) binary generating function Jj = (1 + kappa j[Y])i tau for each class j of sites, represented by column (Jj) and row (Jj) vectors; (iii) cooperativity between sites of one class described by functions gamma j(i), represented by diagonal matrices gamma j; (iv) probability of finding microspecies represented by elements of tensor product matrix Ll = (J1)[J2]; (v) statistical factors mij obtained from Newton polynomials, Jj; (vi) power operators Oi', O(i-l)', and O(i tau-l)', transforming vectors Jj; and (vii) operators Oi or O(i-l) indicating tensor products of i or (i-l) vectors Jj. Vectors Jj combined in tensors Ll give rise to both an affinity/cooperativity space and a parallel index space. The partition functions ZM, ZA, and ZH and the total amounts TM, TA, and TH can be obtained as an appropriate sum of elements of matrices Ll, each of which is represented in an index space by a combination p1, p2,...q1, q2,...r1, r2,... of indices ij. From these indices the contribution of that element to partition function ZM, ZA, or ZH and to total amount TM, TA, or TH is calculated in the affinity/cooperativity space as product of factors: [i tau !/i !(i tau-i)!]kappa ij(exp[bj (i-1)i])[X]i, i being any index p, q, r and X any component M, A, or H. Future applications of this algorithm to practical problems of macromolecule-ligand-proton equilibria are outlined.