Diffusion across proton collecting surfaces

A model for a proton collecting apparatus on a protein surface includes a ring-shaped collecting domain on which reversible receptors are scattered. These transport a proton by surface diffusion, until it reaches a central orifice where it is absorbed. Mathematically, this scenario is represented approximately by two-dimensional diffusion on the ring, with steady-state rate coefficients for adsorption/desorption of protons from the bulk (which depend on the bulk diffusion coefficient), and a boundary condition mimicking an irreversible reaction on the orifice perimeter. The ensuing differential equation is of the modified Bessel type, and can therefore be solved analytically in terms of modified Bessel functions. The most general solution involves a reflecting boundary condition on the outer perimeter of the ring, and a radiation one on its inner perimeter. This solution admits numerous special cases, such as when the ring becomes infinite, the inner boundary absorbing, or some parameter small or large. These various limits are discussed, as well as possible implications to experiment.