When the Ca 2+ ions were uniformly distributed in the exterior or interior of the spherical vesicle, i.e., the spontaneous curvature of the entire vesicle was impacted by the asymmetric Ca 2+ ions, the shape transformation showed dramatically different trends between negative and positive values of c 0 . For negative spontaneous curvature c 0 < 0 (Ca 2+ outside the vesicle), the spherical shape remained the most stable one (data not shown). However, for positive spontaneous curvature c 0 > 0 (Ca 2+ inside the vesicle), we found branches of solutions such that the vesicle exhibited a multispherical shape in which several smaller vesicles are connected with narrow necks (Fig. ). There existed critical values of c crit 0 = 2 n/R 0 , where n is an integer, at which the neck is ininitesimally narrow (Fig. , indicated by the vertical lines) and the large vesicle would break into a num- ber of n smaller vesicles, so-called vesiculation. The physics behind the vesiculation phenomenon was investigated in reference. For a spherical vesicle of R 0 , when the spontaneous curvature was increased to c crit 0 , the vesicle would assume a shape of n connected spheres each with a radius of R 0 / n such that the total surface area of the n small vesicles equaled to the surface area of the original vesicle. The neck would assume a saddle shape whose mean curvature just matched c crit 0 . In this coniguration, the bending energy of the membrane was almost zero and, therefore was energetically more favorable.