单开口膜泡形状转变的研究

Shape transformations of opening-up vesicles with one hole

基于双层耦合模型,先通过求解无黏附能情况下满足给定边界条件的欧拉-拉格朗日方程组,找到了约化面积差?a稍大于1的内凹开口形状解,并发现以往Umeda和Suezaki(2005 Phys.Rev.E 71 011913)给出的杯形解是对应?a<1的另一支解,该支解在?a趋于1时开口是外凸的.进而在无黏附能和有黏附能的情况下对开口膜泡的两支解进行了深入研究,发现在?a=1附近这两支解之间有一个间隙,在该间隙内不存在开口解.随着黏附半径的增大,该间隙的位置较缓慢地向右移动.在?a=1附近,在无黏附能时的闭合形状只有球形一个解,而在有黏附能的情况下,闭合形状在1附近的一个区间内都有解.在无黏附及有黏附情况下的计算结果都表明这两支开口解及闭合形状属于不同的分支,它们之间不能连续演化.在间隙右侧的这一支解随着?a的增大可以通过自交形状连续地演化到开口哑铃形.在有黏附的情况下,在?a参数空间,同一支解会发生折叠,即出现同一?a值对应多个解(形状)的情况,这在以往双层耦合模型的计算中没有出现过.讨论了?a对无黏附和有黏附开口膜泡的形状和能量的影响.

So far two kinds of solutions to the problem of opening-up vesicles with one hole have been found. One is cup-like shape found by Umeda and Suezaki （2005 Phys. Rev. E 71 011913）, the other is dumbbell shape with one hole, found by our group. As seen in the context of the bilayer coupling （BC） model, the former corresponds to relatively small reduced area difference Aa, and the latter corresponds to relatively large value of Aa. The relationship between these two kinds of shapes is not clear. Viewing from the angle of the cup-like shape, whether one can obtain the dumbbell shape by increasing Aa is not known. In this paper, we try to clarify this problem by solving the shape equations for free vesicles and adhesive vesicles based on the BC model. Firstly, we solve the set of Euler-Lagrange shape equations that satisfy certain boundary conditions for free vesicles. A branch of solution with an inward hole is found with the reduced area difference Aa slightly greater than 1. It is verified that the solution named cuplike vesicles, which was found by Umeda and Suezaki, belongs to another solution branch （Aa 〈 1） with an outward hole near Aa ----- 1. According to this result, we make a detailed study of these two solution branches for free vesicles and vesicles with adhesion energy. We find tha~ there is a gap near Aa = 1 between the two solution branches. For Aa in this gap, there is no opening-up solution. For adhesive vesicles, the gap will move towards the right side slowly with increasing adhesive radius. In order to check whether the two solution branches can evolve into closed shapes, we also make a calculation for closed vesicles. For free closed vesicles, we find that there is only the sphere solution when Aa is exactly equal to 1 for Ap _-- 0 （in order to comply with the opening-up vesicle, no volume constraint is imposed on it）, while for adhesive vesicles there exist closed solutions in a region of Aa without volume constraint. Both studies for free vesicles and adhesive vesicles show that these two kinds of opening-up vesicles belong to different solution branches. They cannot evolve from one to the other with continuous parameter changing. And strictly speaking, they cannot evolve into the closed vesicles. With increasing Aa, the opening-up branch on the right side of the gap can evolve into an opening-up dumbbell shape with one hole via the self-intersection intermediate shapes. Another interesting result is that for adhesive opening-up vesicles, in the Aa parametric space, the solutions are folded for a solution branch, which means that there exist several shapes corresponding to the same Aa value in the folding domain. This phenomenon has never occurred in previous study of the closed vesicles under the BC model. The influences of Aa on the shape and energy of the free vesicles and adhesive vesicles are also studied.