Biological membranes comprised of lipid membranes and proteins make up the boundary of the cell, as well as the boundaries of internal organelles. Lipid membranes and their interactions with proteins play an important role in many cellular processes, including endocytosis. Behavior of biological membranes is complex—they are elastic in bending, fluid in plane, and undergo several shape transitions. These shape changes include morphological transitions such as formation of invaginations, buds, and tubules from planar shapes in endocytosis, and topological transitions involving rearranging tubular networks in the endoplasmic reticulum. While these processes are well characterized by experiments in cell biology, the underlying mechanisms are poorly understood in a quantitative manner. One reason for this is the complex interplay between elastic bending and thermodynamically irreversible processes such as intra-membrane lipid flow, protein diffusion, and chemical reactions involving protein binding. Modeling these processes pose mathematical challenges as all these processes occur on arbitrarily curving lipid membranes.
In this talk, I will discuss recent advances in both the theoretical and numerical advances in modeling lipid membranes. To this end, I will discuss irreversible/non-equilibrium thermodynamics formalism for arbitrarily curved lipid membranes in the differential geometric setting to determine their dynamical equations of motion . Using this framework, we find relevant constitutive relations and use them to understand how bending, intra-membrane flows, diffusion of multiple transmembrane species, in-plane phase transitions and surface chemical reactions are coupled on deforming surfaces. Using recently developed numerical methodologies to study arbitrarily curved surfaces [2, 3], I will discuss some physical insights gained into the morphological transitions encoded in the biological process of endocytosis.
1. Sahu, A., Sauer, R. A., and Mandadapu K. K., Physical Review E. 96, 042409 (2017).
2. Sauer, R. A., Duong, T. X., Mandadapu, K. K., and Steigmann, D. J., Journal of Computational Physics, 330, 436-466 (2017).
3. Sahu, A., Omar, Y. A. D., Sauer, R. A., and Mandadapu, K. K., arXiv:1812.05086 (2018).