  We consider a coupled bulk-surface system of partial differential equations
with nonlinear coupling modelling receptor-ligand dynamics. The model arises as
a simplification of a mathematical model for the reaction between cell surface
resident receptors and ligands present in the extra-cellular medium. We prove
the existence and uniqueness of solutions. We also consider a number of
biologically relevant asymptotic limits of the model. We prove convergence to
limiting problems which take the form of free boundary problems posed on the
cell surface. We also report on numerical simulations illustrating convergence
to one of the limiting problems as well as the spatio-temporal distributions of
the receptors and ligands in a realistic geometry.
