Numerical Analysis of Hybrid Models for Particles in Biological Membranes

Abstract

Biological cells interact with their environment through molecular processes in the cell membrane. By this means, the interplay between the lipid membrane and specialized membrane proteins determines cell morphology and controls a variety of specific cellular functions. For modelling these interactions on relevant time and length scales, so-called hybrid models are widely-used and established. Thereby the proteins are modelled as dicrete particles, while the lipid membrane is seen as a two-dimensional elastic continuum, whose energy is given by a Canham-Helfrich-like functional. Particularly focusing on the accessibility for efficient numerical methods and numerical analysis, this thesis investigates the somehow most detailed hybrid model. Thereby, the particles are described as discrete rigid objects with finite size, which, according to their particular properties and geometry, induce local constraints along the membrane-particle-contour. First of all, the variational formulation of the membrane-particle hybrid model problem is provided, along with a suitable solution space, particularly incorporating the particle-induced constraints and their degrees of freedom. Furthermore, existence and uniqueness of the corresponding weak solution is proven, followed by the main part of this work - the derivation of a suitable discretization and the provision of an appropriate a priori error estimate. For discretization, the non-conforming Morley finite element is chosen. Due to its simple structure and its few degrees of freedom, it is particularly attractive. However, as Morley's element does not even provide C⁰-continuity, estimating the discretization error is not straightforward, even not for simple model problems with homogenous boundary data on convex, polygonal domains. For the membrane-particle problem, which constitutes the basis of this work, the curved boundaries, the non-homogeneous boundary conditions - particularly depending on the particles' degrees of freedom - and the lacking convexity of the domain are additionally aggravating factors.