Continuum-based Modeling and Analysis of Lipid Membranes induced by Cellular Function

  • Author / Creator
    Belay, Tsegay
  • Lipid membranes represent a critically important interface in biological cells and cellular organelles and mediate all interactions between cells and their surrounding environment. Although quite fragile and negligibly thin, they can be homogeneous down to molecular dimension. Consequently, their mechanical properties can be described by idealizing their structure as a thin-walled continuum approximated by a two-dimensional surface. In this context, theoretical approaches based on continuum mechanics are becoming powerful tools to examine lipid bilayer membrane models to explain various aspects of the mechanical deformability of the membrane. However, the corresponding analysis most often involves heavy numerical treatments due to the highly nonlinear nature of the resulting systems of equations. For example, some analytical description of lipid membranes assembled into non-axisymmetric shapes such as rectangular and elliptical shapes remains largely absent from the literature. In addition, most bilayer membrane studies have been conducted using the classical elastic model of lipid membranes which cannot account for simultaneous changes in membrane shape and membrane tension arising from certain biological phenomena such as protein absorption or surface diffusion of proteins on the membrane surface. To address these issues, in the present work we employ the theory of continuum mechanics to develop a comprehensive model for predicting the deformation behavior of both uniform and non-uniform lipid bilayer membranes. For the uniform lipid membrane, our emphasis is to develop an analytical description for the membrane morphology when different membrane shapes are subjected to various types of boundary forces or membrane lipid-protein interactions. In this regard, we supply a complete analytical solution predicting the deformation profile of rectangular lipid membranes resulting from boundary forces acting on the perimeter of the membrane. We also give a complete semi-analytic analysis for the deformation profiles of lipid membranes induced by their interactions with solid elliptical cylinder substrates (e.g. proteins). In both problems, the theoretical framework for the mechanics of lipid membranes is described in terms of the classical Helfrich model. A linearized version of the shape equation describing the membrane morphology is obtained via a limit of superposed incremental deformations for the respective problems. Thus, complete analytical solutions are obtained by reducing the corresponding problem to a single partial differential equation and formulating the resulting shape equations with suitable coordinate systems to accommodate the shapes of the membrane. Each of the analytical results successfully predict smooth morphological transitions over the respective domain of interest. Membrane proteins play a vital role in various cellular activities (such as endocytosis, vesiculation and tubulation) yet the study of the contribution of membrane proteins presents a major challenge with one of the main difficulties being the lack of a full understanding of the mechanics of membrane-protein interaction. Therefore, a portion of this work is devoted to the study of the mechanics of vesicle formation on a non-uniform flat bilayer membrane where the vesicle formation process is assumed to be induced by surface diffusion of transmembrane proteins and acting line tension energy on the membrane. Much attention is also given to the discussion of the role of thickness deformation (distension) in the vesicle formation of the bilayer membrane. Since the classical elastic model of lipid membranes cannot account for simultaneous changes in membrane shape and membrane tension due to surface diffusion proteins, we propose a modified Helfrich-type model for non-homogeneous membranes. The proposed model is based on the free energy functional accounting for the bending energy of the membrane including the spontaneous curvature, thickness distension and the acting line tension energy on the boundary of the protein concentrated domain and the surrounding bulk lipid. In the analysis, the protein concentration level is coupled to the deformation of the membrane through the spontaneous curvature term appearing in the resulting shape equation. Our emphasis in this research is the rigorous mathematical treatment of this model, in particular to find the numerical solution of the membrane shape equation with associated boundary conditions. Accordingly, we supply numerical solutions by reducing the corresponding problem to a coupled two-point boundary value problem by the use of collocation method. These results successfully predict the vesicle formation phenomenon on a flat lipid membrane surface with a smooth transition of membrane thickness variation inside the boundary layer where the protein-free membrane and the protein-coated domain is observed.

  • Subjects / Keywords
  • Graduation date
    Fall 2016
  • Type of Item
  • Degree
    Doctor of Philosophy
  • DOI
  • License
    This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for non-commercial purposes. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.