  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.
