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Phase field models for small deformations of biomembranes arising as Helfrich energy equilibria
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Hatcher, L. (2020) Phase field models for small deformations of biomembranes arising as Helfrich energy equilibria. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b3519303~S15
Abstract
In this thesis we develop and analyse a model for phase separation on biological membranes. Motivated by observations of raft formation we couple the local membrane composition to the local membrane curvature. Using a perturbation method we describe the geometry of the membrane as a graph over an undeformed Helfrich minimising surface, which we here take to be a sphere. The resulting energy is a small deformation functional that is coupled to the interface.
We first explore the energy for a diffuse interface. We prove that suitable minimisers exist for this energy and explore the critical points using a gradient flow with conserved Allen-Cahn dynamics. We prove the existence of unique solutions for this gradient ow. Additionally, we perform numerical simulations and validate that the model predicts the formation of stable raft-like structures. We then explore the parameter dependence of these subdomains.
Subsequently, we relate the diffuse interface approach to a sharp interface approach. We calculate the equilibrium equations of the diffuse and sharp interface energies and use these to define what we call the reduced diffuse and sharp interface energies. The minimisers of these reduced energies coincide with those of the original energies. However, the reduced energies depend only on a single parameter. We demonstrate that the reduced diffuse interface energy
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics Q Science > QP Physiology |
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Library of Congress Subject Headings (LCSH): | Membranes (Biology), Perturbation (Mathematics), Homotopy theory | ||||
Official Date: | September 2020 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Elliott, Charles ; Stinner, Björn | ||||
Sponsors: | Engineering and Physical Sciences Research Council | ||||
Format of File: | |||||
Extent: | ix, 160 leaves: illustrations, plates, charts. | ||||
Language: | eng |
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