Theoretical Study On The Shape Of The Adhered Open Bubble

The adhesion of vesicles is very important for the vesicles to adapt to the environment.It is also a fundament biological process.In the previous study,more efforts are focused on the adhesion of closed vesicles.Dong Ni et al.made some primary study on the adhesion of opening-up vesicles in the SC model,and they gave some solutions similar to the cup vesicles.In this paper we will study adhesion of open vesicles in the BC model in detail.In order to study the phase transformation under the influence of adhesion coefficient and the line tension stress,the first step is to obtain as many branches of solutions as possible.Since the adhesion opening-up vesicles have to satisfy more boundary conditions,it is quite difficult to solve the shape equations.Based on the previous study of T.Umeda et al.and Kong et al.for free opening-up vesicles,and the relationship between the free vesicles and the adhesion vesicles,we find enough solutions for adhesion opening-up vesicles.Based on these solutions we study the phase transformations between adhesion vesicles and free vesicles in the BC model.The main results we get are as follows:1.The relationship ?c = 2kcHf2 between the critical adhesion coefficient ?c for vesicles with vanishing adhesion radius and the mean curvature Hf at the symmetric axis for free vesicles is obtained by studying the boundary conditions of the free opening-up vesicle and the adhesive opening-up vesicle with infinitesimal adhesion radius.The solutions of adhesive vesicle and the free vesicle can be obtained from each other by this method.2.Two physical ensembles are studied.Ensemble-1 has fixed adhesion coefficient while the adhesion radius is determined by solving the shape equation.Ensemble-2 has fixed adhesion radius while the adhesion coefficient is determined by solving the shape equation.The shape equation and the solutions are exactly the same for these two ensembles.But the solutions for ensemble-2 have better connectedness,and this fact is helpful for us to find solutions with particular parameters.3.In general two branches of solutions are found for infinitesimal adhesion radius,one branch with ?a<1,and the other branch with ?a>1.Around ?a=1,the branch with ?a<1 has convex holes,while the branch with ?a>1 has concave holes.In the previous study,T.Umeda et al.claimed that the convex shape can change to closed spheres continuously.Strictly speaking,we find that the two branches cannot change into closed spheres,because closed shapes and shapes with one hole have different topological structure.What's more,the two branches of opening-up solutions also cannot change from one to the other by continuous parametric transformation.The branch with ?a>1 can change into opening-up dumbbell shapes continuously through the inward self-intersection shapes.This also explains why T.Umeda et al.cannot find the opening-up dumbbell shapes from the convex shapes with ?a<1.4.The long ellipsoid vesicle' solution is found when the reduced line tension is large.It is easy to obtain the solution of the dumbbell-shaped and long ellipsoid vesicles if the adhesion radius is smaller or the adhesion coefficient is appropriate.When the adhesion radius or the adhesion coefficient is large,the vesicle is easy to self.5.There are no concave vesicles solutions in the minimum curvature model with no constraint on the reduced area difference if the Gaussian curvature is zero.6.We discussed the effects of adhesion coefficient on the shape and energy about cups,dumbbells and 3budding adhesive vesicles in the examples of ?a = 0.9??a= 1.2 and ?a = 1.5.The results showed that the phase transition between the adhesive vesicles and the free vesicles is nearly continuous when the adhesive radius is nearly zero.