Open Membrane Vesicles Shape Of The Tubular Topology

Cell is the basic unit of life morphology and life activities. Cell membrane is the integrant component of the cell activities. People can observe the microstructure of the cell membrane since the electron microscope is applied to the biology. Lipid bilayer membranes are composed of the phospholipids amphiphilic molecules and the protein attached to the internal and external surface. As the liquid crystal found, people also recognized that the physiological temperature leads the bilayer membranes to the lamellar lyotropic liquid crystal phase. Moreover, the closed lipid vesicles shape has a high degree of variability with the environment of aqueous solution changing. In this paper, we study the shapes of the lipid bilayer vesicles. In the past 20 years, there is great progress on the vesicles shape research. Initially, people can use numerical calculations to study the vesicles shape only because there is not a better theory. The general shape equation given by Ou-Yang opens a new chapter to the problem with analytical solution.The experiment shows that periodic cylinders are common lipid bilayer membrane structures. Our work is based on the Helfrich spontaneous bending energy and the beyond Delaunay's surfaces. The beyond Delaunay's surfaces are found to be solutions of the Helfrich variation problem which is obtained firstly by Ou-Yang. The beyond Delaunay's surfaces not with constant mean curvature can describe two kinds of surfaces:the periodic unduloidlike shapes and the periodic nodoidlike shapes. In this paper, we have discussed the periodic unduloidlike shapes by numerical method. A vesicle of spherical topology leads toη=0,but a vesicle of toroidal has to leaveη≠0 in the shape equations. When the boundary conditions are added to the axisymmetric shape equations, we get the two-dimensional figures of the unduloidlike shapes with different spontaneous curvatures. And we can conclude that with the increase of the spontaneous curvature, both the parameter a and the period decrease.Secondly, the numerical and analytical study of the opening-up vesicles has become a challenging task due to the experimental observation. In this paper, we study the analytical solution of the opening-up shapes obtained from the Ou-Yang biconcave analytical solution. Recently, Tu and Ou-Yang derived the three boundary conditions for the spontaneous curvature model. By assuming axisymmetric deformations, we found only two of the three boundary conditions holding on the membrane edge are independent. In general, the Gaussian curvature modulus kg may affect the shape of the vesicle rim. The remanent two boundary conditions can be simplified to one equation with the Gaussian bending modulus kg=-2. Then we get the geometric equation for the rims. By analyzing the Ou-Yang analytic solution for closed circular biconcave vesicles and periodic noduloidlike vesicles, we get three kinds of shapes with tube topology, which are the convexlike tube, the toruslike tube, and the catenoidlike tube. The result shows the relationship between the opening of a hole and the line tension. We found that with the increase of the line tension,the holds became smaller and smaller and finally closed. And we get the conclusion that the shapes have higher total energy with large line tension through analyzing the relationship between the energy and the line tension.