The objective of this project is to investigate the fundamental mathematics and biophysics that underlie the equilibrium shape and dynamics of a lipid bilayer interacting with surfactants and proteins. The lipid bilayer is the main structural component of the elastic biological membranes that consist of different lipid species, and contain many kinds of proteins, and many other molecules such as cholesterols and surfactants. In addition, the biomembranes are asymmetric double leaflets coupled to a cytoskeleton that is crucial for cellular adaptation of different shapes and motility. It is now known that lipid composition may set limits to the possible shapes and deformations, while the actual shape of the cellular membrane depends on both the force (from the cytoskeleton, for example) and membrane-deforming proteins. Much progress has been made to uncover and decipher the mechanisms by which proteins can sense and/or induce membrane curvature. However, recent experiments have shown that interaction between lipid membranes and amphipathic molecules, such as surfactant and amphipathic peptides, also leads to changes in total membrane area, curvature and rigidity. No mathematical modeling of these effects have yet been obtained to help elucidate the asymmetry in the lipid bilayer due to surfactants and proteins, or how cells may utilize these mechanisms to perform diverse biological functions. This project aims to establish mathematical understanding of the roles of surfactants and proteins in causing the asymmetries in lipid double leaflets and their subsequent shape and dynamics. The particular focuses are on (1) surfactant-induced membrane area regulation and shape change, and (2) effects of transport, assembly, and organization of trans-membrane proteins on membrane shape and dynamics. In addition, the experiments (by Stone) will validate and refine the modeling approach (by Young), which will in turn facilitate the numerical simulation (by Veerapaneni) of the elastic lipid bilayer interacting with surfactant and protein. Results will help uncover the complex organization and interplay between proteins and lipid domains in cellular membranes.
The theme of this project is the development and analysis of mathematical models of the interaction between lipid membranes and surfactant or protein over large spatial (microns) and long time (seconds to minutes) scales, which is difficult to achieve using state-of-the-art coarse-grained molecular dynamics simulations. Quantitative comparison between experiments, mathematical modeling, and numerical simulations will be conducted. Broader impacts of the research include the development of new mathematical models, numerical methods and complementary analysis and experiments that will be of benefit to scientists and engineers studying rafts dynamics as well as emerging applications in drug delivery, cellular adhesion, motility and mechanosensing. The concepts and methods described here go beyond the context of moving boundary dynamics mediated by an active species, and extend to other problems featuring reaction-diffusion of chemicals. The approach has the potential to be adapted to more complicated biological membranes that are similar from a mathematical point of view. The students involved in this project will receive valuable interdisciplinary training in a wide range of mathematics, biophysics and computer science.
