Reconciliation of classical and reacted-site probability approaches to allowance for ligand multivalence in binding studies

Abstract

The objective of this investigation is to engender greater confidence in the validity of binding equations derived for multivalent ligands on the basis of reacted-site probability theory. To that end, a demonstration of the theoretical interconnection between expressions derived by the classical stepwise equilibria and reacted-site probability approaches for univalent ligands is followed by the use of the traditional stepwise procedure to derive binding equations for bivalent and trivalent ligands. As well as demonstrating the unwieldy nature of the classical binding equation for multivalent ligand systems, that exercise has allowed numerical simulation to be used to illustrate the equivalence of binding curves generated by the two approaches. The advantages of employing a redefined binding function for multivalent ligands are also confirmed by subjecting the simulated results to a published analytical procedure that has long been overlooked.

Keywords: antigen−antibody affinity; binding equations; ligand multivalence; reacted-site probability theory.

Figure 1

Comparison of simulated binding curves calculated by means of expressions [eqns (15a,b) and (26a,b)] derived from reacted-site probability theory (solid symbols) and classical stepwise equilibria considerations (open symbols) respectively for the interaction between a bivalent ligand (B) and a bivalent acceptor (A). Diamonds refer to calculations with kC̄_A = 1; and triangles to calculations with kC̄_A = 10: ●, the rectangular hyperbolic dependence corresponding to the first term on the right-hand side of eqns (15a) and (26a).

Figure 2

Amalgamation of the two simulated sets of binding data from Figure 1 into a single set by the incorporation of a redefined binding function [eqn (36)] and analysis in terms of eqn (37), the counterpart of the Scatchard equation for a bivalent ligand.