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

Abstract

The relationships between the chemical properties of a system and the partition function algorithm as applied to the description of multiple equilibria in solution are explained. The partition functions Z m, Z A, and Z H are obtained from powers of the binary generating functions J j =(1+ k j γ j,i where i t j = p t j , q t j ,or r t j , represent the maximum number of sites in class j, for Y = M, A, or H, respectively. Each term of the generating function can be considered an element i j of a vector J j and each power of the cooperativity factor γ j,i i can be considered an element of a diagonal cooperativity matrix T j . The vectors J J are combined in tensor product matrices L l = { J 1} [ J 2]…[ J j]…, thus representing different receptor-ligand combinations. The partition functions are obtained by summing elements of the tensor matrices. The relationship of the partition functions with the total chemical amounts T M , T A , and T H has been found. The aim is to describe the total chemical amounts T m , T A , and T H as functions of the site affinity constants k j and cooperativity coefficients b j . The total amounts are calculated from the sum of elements of tensor matrices L l . Each set of indices { p j …, q j …, r j …} represents one element of a tensor matrix L l and defines each term of the summation. Each term corresponds to the concentration of a chemical microspecies. The distinction between microspecies M p j A q j H r j , with ligands bound on specific sites and macrospecies M PA QH R corresponding to a chemical stoichiometric composition is shown. The translation of the properties of chemical model schemes into the algorithms for the generation of partition functions is illustrated with reference to a series of examples of gradually increasing complexity. The equilibria examined concern: (1) a unique class of sites; (2) the protonation of a base with two classes of sites; (3) the simultaneous binding, of ligand A and proton H to a macromolecule or receptor M with four classes of sites; and (4) the binding to a macromolecule M of ligand A which is in turn a receptor for proton H. With reference to a specific example, it is shown how a computer program for least-squares refinement of variables k j and b j can be organized. The chemical model from the free components M, A, and H to the saturated macrospecies M PA QH R , with possible complex macrospecies M PA Q and AH R , is defined first. Subsequently, the binary functions compatible with the model, along with the initial values of the site affinity constants k j the number of sites in each class, and the cooperativity coefficients b j are entered. The chemical model controls the type of tensor product matrices L l which are generated and the limits of the lower-case letter indices p j , q j , and r j , which define the terms (microspecies) contributing to the total chemical amounts T M , T A , and T H.