Effect of ligand diffusion on occupancy fluctuations of cell-surface receptors

The role of diffusion in the kinetics of reversible ligand binding to receptors on a cell surface or to a macromolecule with multiple binding sites is considered. A formalism is developed that is based on a Markovian master equation for the distribution function of the number of occupied receptors containing rate constants that depend on the ligand diffusivity. The formalism is used to derive (1) a nonlinear rate equation for the mean number of occupied receptors and (2) an analytical expression for the relaxation time that characterizes the decay of equilibrium fluctuations of the occupancy of the receptors. The relaxation time is shown to depend on the ligand diffusivity and concentration, the number of receptors, the cell radius, and intrinsic association/dissociation rate constants. This result is then used to estimate the accuracy of the ligand concentration measurements by the cell, which, according to the Berg-Purcell model, is related to fluctuations in the receptor occupancy, averaged over a finite interval of time. Specifically, a simple expression (which is exact in the framework of our formalism) is derived for the variance in the measured ligand concentration in the limit of long averaging times.