Receptor clustering on a cell surface. I. theory of receptor cross-linking by ligands bearing two chemically identical functional groups

Abstract

We present a general mathematical analysis of the problem of bivalent ligands, containing two identical reactive groups, interacting with cell surface receptors. The initial binding of such ligands to a cell is followed by further reactions on the surface, leading to the formation of ligand-receptor clusters. If the receptors are monovalent, at most two receptors can be cross-linked by ligand. However, if the receptors are bivalent (or multivalent), large cell surface aggregates can form, the clusters being distributed in size. Our mathematical analysis of ligand binding and receptor cross-linking is based upon the equivalent-site hypothesis and encompasses both equilibrium and kinetic aspects of the problem. In principle, the mathematical description of cluster formation kinetics requires the solution of an infinite system of coupled nonlinear differential equations. We show that in the absence of ligand-receptor rings, the problem of obtaining the entire aggregate size distribution as a function of time can be reduced to the solution of two nonlinear differential equations. Approximate solutions to these equations are developed via singular perturbation methods. When ring formation is included in the model, additional equations need to be solved. We show that when the rate constant for ring closure varies inversely with chain size, the infinite system of differential equations needed to describe the growth of linear aggregates can be reduced to a system of three integral and integrodifferential equations. To determine the concentration of rings, an additional equation must be solved for each ring size of interest. The mathematical development is general and may be applicable to the analysis of a number of biological problems, including histamine release from basophils and mast cells, B-cell triggering, and the action of hormones, such as insulin and epidermal growth factor.