  We derive the equations that describe adsorption of diffusing particles onto
a surface followed by additional surface kinetic steps before being transported
across the interface. Multistage surface kinetics occurs during membrane
protein insertion, cell signaling, and the infection of cells by virus
particles. For example, after nonspecific binding, additional kinetic steps,
such as binding of receptors and coreceptors, must occur before virus particle
fusion can occur. We couple the diffusion of particles in the bulk phase with
the surface kinetics and derive an effective, integro-differential boundary
condition that contains a memory kernel describing the delay induced by the
surface reactions. This boundary condition takes the form of a singular
perturbation problem in the limit where particle-surface interactions are
short-ranged. Moreover, depending on the surface kinetics, the delay kernel
induces a nonmonotonic, transient replenishment of the bulk particle
concentration near the interface. Our approach generalizes that of Ward and
Tordai, and Diamant and Andelman, to include surface kinetics, giving rise to
qualitatively new behaviors.
