Steady-state reaction rate of diffusion-controlled reactions in sheets

In many biological situations, a species arriving from a remote source diffuses in a domain confined between two parallel surfaces until it finds a binding partner. Since such a geometric shape falls in between two- and three-dimensional settings, the behavior of the macroscopic reaction rate and its dependence on geometric parameters are not yet understood. Modeling the geometric setup by a capped cylinder with a concentric disk-like reactive region on one of the lateral surfaces, we provide an exact semi-analytical solution of the steady-state diffusion equation and compute the diffusive flux onto the reactive region. We explore the dependence of the macroscopic reaction rate on the geometric parameters and derive asymptotic results in several limits. Using the self-consistent approximation, we also obtain a simple fully explicit formula for the reaction rate that exhibits a transition from two-dimensional to three-dimensional behavior as the separation distance between lateral surfaces increases. Biological implications of these results are discussed.