  We present a position Langevin equation for overdamped particle motion on
rough two-dimensional surfaces. A Brownian Dynamics algorithm is suggested to
evolve this equation numerically, allowing for the prediction of effective
(projected) diffusion coefficients over corrugated surfaces. In the case of
static surface roughness, we find that a simple area-scaling prediction for the
projected diffusion coefficient leads to seemingly quantitative agreement with
numerical results. To study the effect of dynamic surface evolution on the
diffusive process, we consider particle diffusion over a thermally fluctuating
elastic membrane. Surface fluctuation has the effect of increasing the
effective diffusivity toward a limiting annealed-surface value discussed
previously. We argue that protein motion over cell surfaces spans a variety of
physical regimes, making it impossible to identify a single approximation
scheme appropriate to all measurements of interest.
