  We prove an averaging principle which asserts convergence of diffusion
processes on domains separated by semi-permeable membranes, when diffusion
coefficients tend to infinity while the flux through the membranes remains
constant. In the limit, points in each domain are lumped into a single state of
a limit Markov chain. The limit chain's intensities are proportional to the
membranes' permeability and inversely proportional to the domains' sizes.
Analytically, the limit is an example of a singular perturbation in which
boundary and transmission conditions play a crucial role. This averaging
principle is strongly motivated by recent signaling pathways models of
mathematical biology, which are discussed towards the end of the paper.
