Calculation of site affinity constants and cooperativity coefficients for binding of ligands and/or protons to macromolecules. II. Relationships between chemical model and partition function algorithm

The relationships between the chem. properties of a system and the partition function algorithm as applied to the description of multiple equil. in soln. are explained. The partition functions ZM, ZA, and ZH are obtained from powers of the binary generating functions Jj = (1 + Kjγj,i[Y]i,j, where itj = ptj, qtj, or rtj represent the max. no. of sites in class j, for Y = M, A, or H, resp. Each term of the generating function can be considered an element {ij} of a vector Jj and each power of the cooperativity factor γj,ii can be considered an element of a diagonal cooperativity matrix Γj. The vectors Jj are combined in tensor product matrixes Ll = {J1} [J2] ... [Jj] ..., thus representing different receptor-ligand combinations. The partition functions are obtained by summing elements of the tensor matrixes. The relationship of the partition functions with the total chem. amts. TM, TA, and TH has been found. The aim is to describe the total chem. amts. TM, TA, and TH as functions of the site affinity consts. kj and cooperativity coeffs. bj. The total amts. are calcd. from the sum of elements of tensor matrixes Ll. Each set of indexes {pj ..., qj ..., rj ...} represents one element of a tensor matrix Ll and defines each term of the summation. Each term corresponds to the concn. of a chem. microspecies. The distinction between microspecies MpjAqjHrj with ligands bound on specific sites and macrospecies MpAQHR corresponding to a chem. stoichiometric compn. is shown. The translation of the properties of chem. model schemes into the algorithms for the generation of partition functions is illustrated with ref. to a series of examples of gradually increasing complexity. The equil. examd. concern: (1) a unique class of sites; (2) the protonation of a base with 2 classes of sites; (3) the simultaneous binding of ligand A and proton H to a macromol. or receptor M with 4 classes of sites; and (4) the binding to a macromol. M of ligand A which is in turn a receptor for proton H. With ref. to a specific example, it is shown how a computer program for least-squares refinement of variables kj and bj can be organized. The chem. model from the free components M, A, and H to the satd. macrospecies MPAQHR, with possible complex macrospecies MPAQ and AHR, is defined first. Subsequently, the binary functions compatible with the model, along with the initial values of the site affinity const. kj, the no. of sites in each class, and the cooperativity coeffs. bj, are entered. The chem. model controls the type of tensor product matrixes Ll which are generated and the limits of the lower-case letter indexes pj, qj and rj which define the terms (microspecies) contributing to the total chem. amts. TM, TA, TH.