Homogenization of Lateral Diffusion on a Random Surface

Abstract

Abstract. We study the problem of lateral diffusion on a static, quasi-planar surface generated by a stationary, ergodic random field possessing rapid small-scale spatial fluctuations. The aim is to study the effective behaviour of a particle undergoing Brownian motion on the surface viewed a projection on the underlying plane. By formulating the problem as a diffusion in a random medium, we are able to use known results from the theory of stochastic homogenization of SDEs to show that, in the limit of small scale fluctuations, the diffusion process behaves quantitatively like a Brownian motion with constant diffusion tensor D. While D will not have a closed-form expression in general, we are able to derive variational bounds for the effective diffusion tensor, and using a duality transformation argument, obtain a closed form expression for D in the special case where D is isotropic. We also describe a numerical scheme for approximating the effective diffusion tensor numerically and illustrate this scheme with two examples.Key words. stochastic homogenisation, random media, Laplace-Beltrami, diffusion, multiscale-analysis. AMS subject classifications. 35Q92, 60H30, 35B271. Introduction. Lateral diffusion of particles along interfaces is a frequently occurring phenomenon in cellular biology. In the case of lipid bilayer membranes, the lipid molecules and integral membrane proteins which constitute the cell membrane themselves undergo lateral diffusion as a result of thermal agitation [4]. The mobility of membrane proteins has far-reaching implications for many cellular processes, in particular protein transport, signalling and morphology [2, 6, 9], thus there has been considerable interest in measuring how the mobility of these proteins is affected by the membrane and its surrounding environment. The dynamics of protein diffusion within a fluid membrane was first considered by Saffman and Delbrück [43] who proposed a continuum hydrodynamic model for a laterally diffusing particle in a flat, homogeneous fluid membrane. The model predicted a relationship for the diffusion tensor of a particle in the membrane in terms of the particle radius, the thickness and viscosity of the membrane and the viscosity of the bulk medium. Continuum models for studying the influence of shape fluctuations of the membrane on the macroscopic protein diffusion rate were subsequently considered in [18,19,32,41] in which the proteins undergo Brownian motion laterally along an infinitesimally thin two dimensional surface embedded in R 3 . The equilibrium fluctuations of the surface are characterised by the Canhan-Helfrich Hamiltonian [10,20]. In [32], an expression for the effective diffusion tensor was derived by considering the joint Markov process for the coupled particle and surface, applying an adiabatic elimination to average out the rapid temporal surface fluctuations. Another approach, based on a path-integral formulation was considered in [41]. More recently,in [13], a multiscale approach to the problem of lateral diffusion on rapi...

Noise-induced transitions in rugged energy landscapes

“…Brownian motion in disordered media (or rugged energy landscapes) is a problem of great scientific and technological interest, and applications are found in a wide range of different areas, such as e.g. collective transport of particles in random media [13,12,31,17,33,28], molecular motors [22,29], and protein reaction dynamics and folding [35], to name but a few. In the latter example in particular, proteins are dynamic macromolecules that exhibit many scales of molecular motion which is governed by a hopping mechanism through the local minima of the free-energy surface, the so-called conformational substrates or microstates.…”