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Title: Diffusion on rapidly-varying surfaces
Author: Duncan, Andrew
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 2013
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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. In this thesis we consider the problem of lateral diffusion on quasi-planar surfaces, which are fluctuating according to various models. Using homogenisation theory, we show that, under the reasonable assumption of well separated scales, the lateral diffusion process can be well-approximated by a Brownian motion on the plane with constant diffusion coefficient D. The diffusion coefficient D will depend in a complicated way on the different properties of the surface, such as the average excess surface area, and for biologically motivated models, the bending stress and surface tension. We consider three classes of surface fluctuation models. The first case we consider is a periodic fluctuation model, where the surface is time-independent possessing rapid, periodic fluctuations. Using classical homogenisation techniques we obtain an expression for D for a particle diffusing on such a surface and are able to study the various properties of D. Although D will not have a closed form expression in general, we identify a large class of two-dimensional surfaces for which the effective diffusion coefficient has an explicit form which depends only on the excess surface area. The second model we consider is a static, stationary random field model, where the surface is given by a rapidly fluctuating, random field with stationary, ergodic fluctuations. Under appropriate assumptions, we are also able to prove a homogenisation result for lateral diffusion on such a surface and prove results analogous to those for the first model. Generalising the thermally-excited Helfrich-elastic membrane model, the third case we consider is a fluctuating surface having both rapid spatial and temporal fluctuations. The effective diffusion coefficient will depend on the relative scales of the spatial and temporal fluctuations. For different scaling regimes, we prove the existence of a macroscopic limit in each case. In each of the cases, the theoretical results are supplemented with numerical experiments which highlight the theory as well as explore scenarios not covered by theory.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics