  We consider a coupled bulk/surface model for advection and diffusion of
interacting chemical species in biological cells. Specifically, we consider a
signalling protein that can exist in both a cytosolic and a membrane-bound
state, along with a variable that gives a coarse-grained description of the
cytoskeleton. The main focus of our work is on the well-posedness of the model,
whereby the coupling at the boundary is the main source of analytical
difficulty. A priori $L^p$-estimates, together with classical Schauder theory,
deliver global existence of classical solutions for small data on bounded,
Lipschitz domains. For two physically reasonable regularised versions of the
boundary coupling, we are able to prove global existence of solutions for
arbitrary data. In addition, we prove the existence of a family of steady-state
solutions of the main model which are parametrised by the total mass of the
membrane-bound signal molecule.
