Herbert, P. J. (2020) Point particle interactions on surface biomembranes: second order splitting and surface finite elements. PhD thesis, University of Warwick.

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Abstract

We study the well-posedness and approximation of mathematical models for small deformations of biological membranes where the deformations are due to point constraints. The differentiability of of the membrane energy with respect to the movement of the point constraints is studied. We begin by reviewing mathematical theory related to the shape of biomembranes and embedded proteins. We show that modifications of established theory hold and introduce notation which allows us to easily discuss the movement of many proteins embedded into the surface.

We then discuss the well-posedness of an abstract second order splitting method with linear constraints, which we will apply to the energy minimising biomembrane with embedded proteins. We also consider a penalty method to weakly enforce the constraints. It is shown that the solution of this penalty method converges strongly to the solution of the constrained problem. We consider the abstract numerical analysis of these problems. Numerical experiments are given, demonstrating the convergence theory presented.

After this, we consider the differentiability of the energy of the optimal membrane with point constraints with respect to a tangential movement of the points. We demonstrate that the energy is differentiable and give a convenient characterisation of the derivative which is efficient to evaluate. This numerically accessible derivative is employed in some numerical experiments.

We conclude by discussing some directions to extend the theory presented, or ideas which are highly related to the studied theory. In particular, we discuss the extension to consider small deformations of a near-tube membrane.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics Q Science > QC Physics Q Science > QH Natural history
Library of Congress Subject Headings (LCSH):	Membranes (Biology) -- Mathematical models, Surfaces, Deformation of, Finite element method, Constraints (Physics)
Official Date:	September 2020
Dates:	Date Event September 2020 UNSPECIFIED
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Elliott, Charles M. ; Stinner, Bjorn
Sponsors:	Engineering and Physical Sciences Research Council
Format of File:	pdf
Extent:	ix, 106 leaves : illustrations
Language:	eng

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