Solving The Shape Equation Of Multi - Component Vesicles Based On Bidirectional Targeting Method

The real biological membranes are made up of various lipid molecules and cholesterols, so the different sections of the membranes which consist of different ingredients have different function. This is the so-called lipid raft model. People think that formation of lipid raft due to the phase separation different lipid compositions. On experiments, breakthrough for phase separation of spherical vesicles is made. Scientists find the giant vesicles which composed of saturated phospholipids molecules, unsaturated phospholipids molecules and cholesterols will generate phase separation, because of interaction of different lipid molecules. Consequently, two or more concomitant domains are formed. On the theoretical side, F. Julicher et al. studied shape of two-domain vesicles on the condition of rotational symmetry through numerical calculations. We have to emphasize that they only think about the effect of the line tension coefficient and osmotic pressure under the same two-domain mean curvature modulus, while the affection of different two-domain curvature modulus which are important for experiment has not been considered. On the premise of spherical vesicles phase separation, M. Yanagisawa et. al find in experiment that vesicles will form rotational symmetry three-domain vesicles which are oblate and prolate, if they change the osmotic pressure of the two sides of vesicles. Furthermore, they find the pattern of phase separations of the two kinds of shapes are different. In an approximate calculation, they use the parametric surface of ellipsoid to represent the shapes obtained in experiments. The transition point they got does not coincide with the experimental results.In this paper, we will introduce the influence of curvature modulus on the shape of two-domain vesicles. Besides, we compare the calculated result of Euler-Lagrange equation of shape with experiment. The conclusion that we get are as follows:1.We study the Euler-Lagrange shape equation and their boundary conditions of two-domain vesicles which have different curvature modulus with rotational symmetry by variational method. Then the numerical solutions of shape equation of two-domain vesicles under definite boundary conditions are obtained through shooting mthods to the fitting point, which is consistent with experiment. The results show that our numerical method is reasonable for study the equilibrium shape of two-domain vesicles, which set a foundation for further study of experiment-related problems for two-domain vesicles with different curvature modulus.2. We study the influences of the ratio of mean curvature modulus of two-domain vesicles εK on the shapes and get the solutions of Euler-Lagrange shape equations under area fractionX(α) of different a domain and different line tension coefficient λ. We could find that the shapes of vesicles are the results of competition of the curvature energy of the two parts and the line tension energy at the boundary of the two parts. In addition, on one hand, if the value of A is small and εK is relative large, the β domain play the main role in formation of shapes. On the other hand, in the condition of big value of A, two-domain vesicles will have three branches of solutions that correspond to different shapes of vesicles for one εK. Thus the transition of shapes of vesicles will be discontinuous with the increasing of .3. We discuss the influence of X(α) on shapes of two-domain vesicles. Because of the value of mean curvature modulus of P domain is larger than that of a domain, for small value of x(α), the shapes of vesicles of β domain trend to be a ball. While for large value of X(α), the shapes of vesicles of β domain trend to be flat. This tendency becomes more obvious when X(α)tends to 1. For large value of εK, The shapes of β domain play the important role in control the shapes of vesicles with the change of X(α).4. We observe the influence of the value of λ on two-domain vesicles and find that two-domain vesicles also have three bifurcation solutions, each one of which corresponds to different shapes of vesicles. What’s more, the transition of shapes of vesicles will be discontinuous. We could find the value of regional boundary radius of vesicles will become smaller with increasing λ for two branches of solutions. However, the value of regional boundary radius of vesicles will become larger with increasing λ for the rest branch of solution.5.We also get the numerical solutions of Euler-Lagrange shape equations of three-domain vesicles through changing one of the initial value of boundary that belong to shooting methods of two-domain vesicles and calculate the free energies of two regional modes under reduced volume constraint and no reduced volume constraint. The results what we obtain not only show that the critical points of the two modes on the condition of reduced volume constraint close to the experimental results that M. Yanagisawa et al. observed in the laboratory. While their approximate calculation of parametrizing the vesicle shape by an ellipsoid surface differs with the experimental results.