Theory for ligand rebinding at cell membrane surfaces

Conditions for which a ligand reversibly bound to a cell surface dissociates and then rebinds to the surface have been theoretically examined. The coupled differential equations that describe reaction at the interface between sites on a plane and three-dimensional solution have been described previously (Thompson, N. L., T. P. Burghardt, and D. Axelrod. 1981. Biophys. J. 33:435-454). Here, we use this theoretical formalism to provide an analytical solution for the spatial and temporal dependence of the probabilities of finding a molecule on the surface or in the solution, given initial placement on the surface at the origin. This general analytical solution is used to derive a simple expression for the probability that a molecule rebinds to the surface at a given position and time after release at the origin and time zero. The probability expressions provide fundamental equations that form a basis for subsequent modeling of ligand-receptor interactions in specific geometries.