The composition-dependence of the extent of reaction between a multivalent acceptor and a bivalent ligand: Implications of self-interaction of the ligand in precipitin effects

Theory is presented relating to the reversible interaction of an f-valent acceptor, A, with a bivalent ligand, B, which leads to the formation of a series of complexes comprising networks of alternating A and B molecules. An explicit expression is derived for the overall extent of reaction in terms of the total molar concentrations of reactants ( m A and m B ) , the valency of the acceptor and the site-binding constant, k, governing the equilibria. It is shown by differentiation of this expression holding m A (or m B ) fixed that relations are available for the independent evaluation of f and k from a combination of precipitin and radioimmunoassay experiments. Moreover, it is established that dilution with solvent ( m A / m B fixed) cannot lead to the appearance of a precipitate with this type of crosslinking system. The latter observation forms the background for the development of theory pertaining to the joint operation of ligand dimerization, 2 B⇌ B 2, and crosslinking of the multivalent acceptor with bivalent B 2. The theoretical examination of this system is developed in terms of site-probability functions and involves the delineation of unique solutions for the extent of crosslinking reaction aided by the definition of the extent of binding in defined limits. It is shown with the use of numerical examples that the system involving self-associating ligand may result in the appearance of a precipitate on dilution with solvent and the conditions for the operation of this phenomenon are elucidated. It is noted that other types of ligand self-interaction may lead to similar effects in crosslinking systems, and the general principles emerging from this study are discussed in terms of systems in which antibody ligands are known to be involved in association reactions or are suspected to be so involved on the basis of precipitation effects observed on dilution with solvent.