Membranes are crucial to the life movements as boundaries of living cells and living organelles. A membrane consists of lipids, proteins and carbohydrates etc. Lipids have hydrophilic heads and hydrophobic tails, thus they can form a bilayer in water environment spontaneously. The structure and property of a membrane is very complex and refined, but basically, a bio-membrane is an incompressible fluid of lipid bilayer which is defined on a two-dimensional evolving surface and embedded with some proteins. Researchers have shown their great interest on the shape transition of biomembrane. Most of the previous work are on the Helfrich's elastic model. By given different osmotic pressure and spontaneous curvature, Helfrich's model decides many different steady shapes. Someresearchers have also considered the dynamics of membrane. Waxman presented a integrated dynamcis, but we found some shortcomings in his model. We construct a directional dynamical model. Each material point on the membrane is endowed with a free "director", and an simple energies are generated for the directors. Part of the torques and stresses, which are induced by the directors and may result in bending resistance, are then decided by the principle of virtual work. This directional dynamical modelcan help to get some insight of the membrane. We simplify the model (a limit case, the directors are constricted to be the normals) and find an additional necessary stress for Waxman's model. Spontaneous curvature,which plays an important role in deciding the shape of red blood cell, are also considered. What's important, we also talk about the local spontaneous curvature. This local spontaneous curvature is important for many life processes. We also develop a suitable numerical method for these models. Some important life processes (endocytosis, exocytosis, cell division) are simulated. |
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Given different spontaneous curvature and osmotic pressure, our model can find different steady states.
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