  Lateral diffusion of molecules on surfaces plays a very important role in
various biological processes, including lipid transport across the cell
membrane, synaptic transmission and other phenomena such as exo- and
endocytosis, signal transduction, chemotaxis and cell growth. In many cases,
the surfaces can possess spatial inhomogeneities and/or be rapidly changing
shape. Using a generalisation of the model for a thermally excited Helfrich
elastic membrane, we consider the problem of lateral diffusion on quasi-planar
surfaces, possessing both spatial and temporal fluctuations. Using results from
homogenisation theory, we show that, under the assumption of scale separation
between the characteristic length and time scales of the membrane fluctuations
and the characteristic scale of the diffusing particle, the lateral diffusion
process can be well approximated by a Brownian motion on the plane with
constant diffusion tensor $D$ which depends in a highly nonlinear way on the
detailed properties of the surface. The effective diffusion tensor will depend
on the relative scales of the spatial and temporal fluctuations and, for
different scaling regimes, we prove the existence of a macroscopic limit in
each case.
