Rate constant for diffusion-influenced ligand binding to receptors of arbitrary shape on a cell surface

The theory of ligand binding to receptors on a cell surface suggested by Berg and Purcell and generalized by Zwanzig and Szabo uses the assumption that receptors are circular absorbing disks on an otherwise reflecting sphere. One of the key ingredients of this theory is a solution for the rate constant for ligand binding to a single circular receptor on a reflecting plane. We give an exact solution for the rate constant for binding to a single elliptic receptor and an approximate solution for binding to a single receptor of more general shape. The latter was tested by Brownian dynamics simulations. We found that the approximate formula predicted the rate constant with better than 10% accuracy for all studied receptor shapes. Using our solutions one can find the rate constant for ligand binding to a cell covered by N noncircular receptors by means of the Zwanzig-Szabo formula.