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 Zm, ZA, and ZH are obtained from powers of the binary generating functions Jj =(1+kjγj,iwhere it =pt,qt,or rt, 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 elementij of a vector Jj and each power of the cooperativity factor γj,iican be considered an element of a diagonal cooperativity matrix Tj. The vectors JJ are combined in tensor product matrices Ll= {J1} [J2]...[Jj]..., 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 TM, TA, and TH has been found. The aim is to describe the total chemical amounts Tm, TA, and TH as functions of the site affinity constants kj and cooperativity coefficients bj. The total amounts are calculated from the sum of elements of tensor matrices Ll. Each set of indices {pj..., qj..., rj...} represents one element of a tensor matrixLl and defines each term of the summation. Each term corresponds to the concentration of a chemical microspecies. The distinction between microspecies Mp Aq Hr, with ligands bound on specific sites and macrospecies MPAQHR 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 kj and bj can be organized. The chemical model from the free components M, A, and H to the saturated 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 constants kj the number of sites in each class, and the cooperativity coefficients bj are entered. The chemical model controls the type of tensor product matrices Ll which are generated and the limits of the lower-case letter indices pj, qj, and rj, which define the terms (microspecies) contributing to the total chemical amounts TM, TA, and TH. © 1990.
