Numerical Observation Of Nonaxisymmetric Starfish Vesicle

Abstract: Lipids amphiphilic molecules will aggregate into bilayer membranes in aqueous solution. These membranes form closed vesicles at low lipids concentration to reduce the large energy along the edges. The elastic curvature model for lipid vesicles was first proposed by P. B. Canham then by W. Helfrich (SC model) to explain the biconcave shape of red blood cells. According to the fluid mosaic model of cell membranes, the lipids bilayers play an important role in the shape of cell membranes. In the last two decades comprehensive studies have been made both on experiments and theories of vesicle shapes transitions. Study on the physical properties of lipid vesicles will provide us precious information of complex biological systems.According to the elastic curvature model, the equilibrium shape of a vesicle is determined by minimizing of the curvature energy of the vesicle under the constraints of constant area and enclosed volume. Previous studies on the vesicle shapes mainly focused on the case of axisymmetric shapes, since the corresponding Euler-Lagrange equation is an ordinary differential equation which can be integrated numerically by shooting method. Different solution branches may be found for the same pair of parameters, label the shape with the lowest energy for each pair of parameters, we can obtain the phase diagram of the model.Based on Helfrich spontaneous curvature (SC) model, other curvature models such as the the bilayer couple (BC) model and the area difference elasticity (ADE) model have been proposed to explain some shape transitions. All the three curvature models have the same solution catalogues, but their phase diagrams are quite different. Probing the complete phase diagram under various curvature models then comparing them with experimental data has always been a challenging task and an important direction in this field. Most of the researches on experiments and theories currently are restricted in axisymmetric shapes. From the current experimental results, the BC model and the ADE model are more reasonable then the SC model for pure lipid vesicles.So far more and more non-axisymmetric vesicles have been constructed in experiments, while the theoretical studies on these shapes are quite limited since the corresponding Euler-Lagrange equation is a nonlinear high order partial differential equation which still lacks numerical integrating methods. At present the method of direct minimization is often used to obtain the non-axisymmetric shapes and including the n-fold symmetric starfish shapes. However the stability of the starfish shapes and their stable regions are still not clear. In this thesis, we systematically study the stability region of n-fold symmetric starfish shapes under the SC model and the BC model. The main results can be summarized as follows:(1) The stability of discoid starfish vesicle shapes that were found in experiments are first studied under the SC model. They are also stable in the BC model. The stable regions of n-flod symmetric shapes are given for n=5,6,7, which are in good agreement with available experiments.(2) A new class of starfish shapes is found which is termed as n-armed starfish shapes, and the stable regions of reduced volume for n=5,6,7 are given.(3) We found the asymmetric transitions of 3-armed and 4-armed starfish shapes found in experiments can be obtained at low reduced volume. We have also found some new shapes with lower symmetry which are awaiting experimental proof.(4) For v=0.4, the transition from the 5-fold discoid starfish to the 5-armed starfish is calculated under the BC model, which is a discontinuous transition.