Keywords: 
• 开口膜泡 /
• 双层耦合模型 /
• 黏附

Abstract: 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 71011913, 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 ?a, and the latter corresponds to relatively large value of ?a. 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?a 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 ?a 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 (?a<1 with an outward hole near?a=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 that there is a gap near ?a = 1 between the two solution branches. For ?a 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 ?a is exactly equal to 1 for ?p = 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 ?a 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 ?a, 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 ?a parametric space, the solutions are folded for a solution branch, which means that there exist several shapes corresponding to the same ?a value in the folding domain. This phenomenon has never occurred in previous study of the closed vesicles under the BC model. The influences of?a on the shape and energy of the free vesicles and adhesive vesicles are also studied.