Journal of Nonlinear Science

, Volume 25, Issue 2, pp 389–449

A Multiscale Analysis of Diffusions on Rapidly Varying Surfaces

  • A. B. Duncan
  • C. M. Elliott
  • G. A. Pavliotis
  • A. M. Stuart
Article

Abstract

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 generalization 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 homogenization theory, we show that, under the assumption of scale separation between the characteristic length and timescales 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 \(D\) that depends on 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.

Keywords

Homogenization Laplace–Beltrami Lateral diffusion Multiscale analysis Helfrich elastic membrane  Effective diffusion tensor 

Mathematics Subject Classification

35Q92 60H30 35B27 

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • A. B. Duncan
    • 1
  • C. M. Elliott
    • 2
  • G. A. Pavliotis
    • 1
  • A. M. Stuart
    • 2
  1. 1.Department of MathematicsImperial College LondonLondonUK
  2. 2.Mathematics InstituteWarwick UniversityCoventryUK

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