Electromechanical and curvature-driven molecular flows for lipid membranes

Abstract

Lipid membranes play a crucial role in sustaining life, appearing ubiquitously in biology. Gaining a quantitative understanding of the flows of lipid membranes is critical to understanding how living systems operate. Additionally, the mechanical properties of lipid membranes make them ideal material for nanotechnology, further motivating a need for accurate computational models. This thesis is organized in three projects that model important features of lipid membranes. First, we define the mechanical energy of vesicle lipid membranes and propose a fast numerical algorithm for minimizing this energy. The mechanical energy is well known, and existing computational techniques for minimizing this energy include solving the Euler-Lagrange equations for axisymmetric shapes or approximating the minimization problem by minimizing over a subspace of membrane configurations. We choose the latter approach, making no restrictive symmetry assumptions. Specifically, we use surface harmonic functions to parameterize the membrane surface, drastically reducing the degrees of freedom compared to similar existing approaches. Numerical equilibrium shapes are presented, including conformations exhibited by red blood cells. The numerical results are verified against analytical values of axisymmetric shapes. Second, we develop the electrostatic potential energy for lipid bilayer membranes in the context of lipid-protein interactions. We extend the electrostatic potential energy of a protein-solvent system to include charged lipids in a protein-membrane-solvent system. Here, we model the bilayer membrane as a continuum with general continuous distributions of lipids charges on membrane surfaces. Key geometrical properties of the membrane surfaces under a smooth velocity field allow us to apply the Hadamard-Zolésio structure theorem of shape calculus, and we compute the electrostatic force on membrane surfaces as the shape derivative of the electrostatic energy functional. Third and finally, we develop the mathematical theory and the computational tools for curvature-driven flow of proteins within lipid membranes. Recently, much attention has been devoted to understanding curvature generating and curvature sensing properties of proteins in vesicle membranes. That is, certain proteins prefer regions of specific curvature and naturally flow to these regions. We develop the mathematical theory for curvature-driven diffusion along these membranes, which involves a variable diffusion coefficient. Finite element and finite difference methods have been used to solve diffusion equations on surfaces, but these methods require costly spatial resolution and adaptive mesh refinements for dynamic membrane surfaces. Instead, we use a phase field model with Fourier spectral methods so that no explicit tracking of the surface is required. Furthermore, the spectral accuracy allows for uniform mesh with no refinement near the boundary. The numerical solution of the diffusion equation and the numerical solution of the membrane shape equation is performed in a consistent framework to allow for the coupling of membrane shape with the curvature-driven surface diffusion. Results which capture the curvature preference are presented.