Advances in the Theoretical and Computational Modeling of Lipid Bilayer Membranes

Abstract

This contribution presents an overview of theoretical and computational models for lipid bilayer membranes focusing on recent advances based on non-linear shell theory. Lipid bilayers are generally characterized by complex mechanical behavior that has both fluid-like and solid-like properties, leading to complex three-dimensional deformation states. This requires solving partial differential equations on evolving surfaces. To analyze those equations, a new computational model is proposed based on so-called isogeometric finite element methods. These methods are capable of representing surface topography, curvature and flow to high accuracy. The new model is suitable for analyzing and predicting the shapes changes observed during tethering, budding and endocytosis. The computational ingredients required for capturing the mechanics of lipid bilayers, like the bending kinematics, area-incompressibility, viscous surface flow and numerical stabilization techniques, are discussed. Several numerical examples are presented to illustrate the current simulation capabilities. The considered surface representation is also highly suitable for formulating surface interactions, like contact and adhesion, which are also discussed. This contribution also presents new work on the coupling of deformation to mass transport caused by diffusion or phase transitions and proposes computational formulations to handle these complex cases.