Reseach On Vesicle Shapes Of Higher Topological Genus

Amphiphilic molecules are composed of a hydrophilic head, which favors in contact with water, and two hydrophobic hydrocarbon chains. When introduced into an aqueous environment, in order to avoid the exposure of hydrophobic tails of the lipid molecules with water, these amphiphilic molecules aggregate spontaneously into two mono-molecular layers held together by weak non-covalent forces due to the hydrophobic effect. The bilayers will form closed vesicles at low lipid concentration. Experimentally a vesicle observable over an hour can be considered as a system in a well defined constrained equilibrium. One striking point is the configuration of a vesicle is not determined by its surface tension but rather by its bending elasticity. Curvature models based on this assumption allow the theoretical study of a rich variety of different vesicle shapes.Vesicles observed by experiments can also be classified by their topologies. The topology of a vesicle is characterized by its topological genus g which counts the number of "handles" that have to be attached to or "holes" that have to be penetrated through a sphere to obtain a surface of given topology. Vesicles with zero ( spherical topology), one (toroidal topology), two, and even more "handles" or " holes" have been observed experimentally.According to the SC (spontaneous-curvature) model proposed by Helfrich, scientists have obtained some analytic solutions, such as spherical shapes, normal red blood cells' shape, Clifford torus, periodic cylindrical surfaces, Delaunay surfaces and beyond Delaunay surfaces. In the general case, this problem is studied by numerical methods, and many interesting numerical solutions, most of which are axisymmetric shapes have been found. They were placed on the so-called phase diagrams.As for Vesicle shapes of higher topological genus g=2, only the Willmore problem and Willmore surfaces were discussed explicitly, and some rough speculations were made on the phase diagram beyond Willmore region. So far there still lacks systematic study on this region. Only some shapes similar to the vesicles found in experiments have been obtained by Surface Evolver. And the stability of various shapes and the transitions between various branches have not been studied so far.This paper focuses on the research of the shapes of Topological Genus 2 beyond Willmore region in different curvature models. First, we choose two initial configurations with different symmetries in Surface Evolver. We made our calculation under SC model which has a high convergence rate. At last, we construct our phase diagram under BC model in order to make comparison with the one former researcher have done. Through adjusting some parameters that contributes to the bending elasticity and stability analysis, we get some new results as follows:(1) a wilder rang of mean curvature integral for the phase diagram.(2) several new shapes in SC model with C2v, D3h and C3v symmetry, respectively.(3)several semicontinuous phase transitions in SC model by tracing the Hessian matrix and the corresponding phase diagram in BC model.(4)calculations were made in BC model for some discontinuous phase transitions in SC model and regions of overlapping were found in BC model.(5)a relatively complete phase diagram in BC model with which we show the above results.