UC San Diego

To dynamically reshape the membrane, cells rely on a variety of intracellular mechanisms, ranging from forces exerted by the cytoskeleton to the spontaneous curvature induced by membrane–protein interactions. In this thesis, we present mathematical models in a continuum framework to understand the physics underlying membrane deformation by two different modes of curvature-generating mechanisms (1) protein-induced spontaneous curvature and (2) forces due to membrane-cytoskeleton interactions. In the first part of the thesis, we model the effects of curvature-generating proteins by extending the classical Helfrich-Canham bending energy and demonstrate how the local shape of the membrane can be used to infer the traction acting locally on the membrane. Particularly, we first propose a technique to extract effective line tension at the protein interface using the morphology and the composition of the membrane. We then analyze the beading morphology of membrane nanotubes due to heterogeneity in the membrane properties and protein distribution. We find that there exists a discontinuity in the energy that impedes two beads from fusing. Finally, we show the application of our continuum framework for studying curvature generation due to protein phase separation on membranes. In the second part of the thesis, we model the forces due to membrane-cytoskeleton interactions by adding an extra degree of freedom to the energy equation to account for heterogeneous forces representing the effects of actin polymerization and activity of molecular motors such as myosin on the plasma membrane. Using this framework, we show that a non-uniform force distribution coupled with membrane tension characterized the biconcave shape of Red Blood Cells (RBCs). We also explore the application of our mathematical framework to identify the possible force regimes that give rise to the classic shapes of dendritic spines which are bulbous protrusions along the dendrites of neurons and are sites of excitatory postsynaptic activity. We identify different mechanical pathways that are likely associated with different dendritic spine shapes, and find that some mechanisms may be energetically more favorable than others. We believe our models identify mechanisms of cell shape adaption by two modes of curvature generation, enabling future work to establish the contribution of cell membrane mechanics in many human diseases and designing better systems for drug and gene delivery.