Temporal dependence of ligand dissociation and rebinding at planar surfaces

In a previous work, conditions for which a ligand reversibly bound to a planar surface dissociates and then rebinds to the surface were theoretically examined (Lagerholm, B. C.; Thompson, N. L. Biophys. J. 1998, 74, 1215). The coupled differential equations that describe the reversible interaction of ligands in three-dimensional solution with sites on a planar surface were solved to find analytical solutions for the probabilities of finding a ligand on the surface or in solution, given initial placement on the surface. An expression was also found for the probability that a ligand rebinds to the surface at a given position and time after its release. In this work, the formalism is extended to calculate analytical, closed form expressions for the average number and rate of rebinding events that have occurred, on the average, at a given lime after placing a ligand on the surface. These functions depend only on the intrinsic dissociation rate and a "rebinding parameter," which depends on a group of constants including the intrinsic association and dissociation rates, the density of surface binding sites, and the diffusion coefficient in solution. The results are interpreted in terms of typical conditions for biologically relevant ligands and their receptors.