Statistical Mechanics of the Stability of Multivalent Ligand−Receptor Complexes

A multivalent ligand can bind to a multivalent receptor to form a multivalent complex that is orders of magnitude more stable than its “monovalent” counterpart formed by the binding of monovalent ligands, each of which is equivalent to the corresponding ligating unit of the multivalent ligand. This “enhancement effect” is frequently rationalized qualitatively in terms of the concept of “effective concentration” borrowed from polymer physics. The purpose of this article is to present the details of a fundamental theory of the enhancement effect that was only summarized recently. Classical statistical mechanics is used to compute closed analytic expressions for the binding constants of the various ligand−receptor complexes associated with the prototypal system, divalent ligand binding to a divalent receptor. A simple analytic expression for a defined “enhancement factor” reveals insight into the mechanism of enhancement. The enhancement factor is proportional to a generalized effective concentration of one ligating unit with respect to the other, which depends not only on the separation between the binding sites of the receptor but also on the thermodynamic state of the system. The theory is related to typical experiments and is used to analyze data on the activation of ion channels (tetravalent receptors) in cell membranes by polymer-linked divalent ligands.