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 thermodn. of binding of a ligand A and (or) H to a macromol. M is treated by the partition function method. In complex systems, the representation of the equilibra by means of cumulative consts. βPQR used as coeffs. in partition functions Zm, ZA, and ZH is ill-suited to least-squares refinement procedures because the cumulative consts. are interrelated by common cooperativity functions Γj(i) and common site affinity consts. kj. There is therefore the need to express ZM, ZA, ZH as functions of site consts. kj and cooperativity coeffs. bj. This is done by developing an algebra of partition functions based on the following concepts: (1) factorability of partition functions; (2) binary generating function Jj = (1 + kj[Y])it for each class j of sites, represented by column {Jj} and row [Jj] vectors; (3) cooperativity between sites of one class described by functions Γj(i), represented by diagonal matrixes Γj; (4) probability of finding microscopies represented by elements of tensor product matrix Ll = {J1}[J2]; (5) statistical factors mij obtained from Newton polynomials, Jj; (6) power operators Oi, O(i-1)', and O(it-1)', transforming vectors Jj, and (7) operators Oi or O(i-7) indicating tensor products of i or (i - 1) 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 amts. TM, TA, and TH can be obtained as an appropriate sum of elements of matrixes Ll, each of which is represented in an index space by a combination p1, p2, ... q1, q2 ... r1, r2, ... of indexes ij. From these indexes the contribution of that element to partition function ZM, ZA, or ZH and to total amt. Tm, TA, or TH is calcd. in the affinity/cooperativity space as product of factors: [it!/i!(it - i)!]kji(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 macromol.-ligand-proton equil. are outlined.