Multispherical shapes of vesicles highlight the curvature elasticity of biomembranes

Giant lipid vesicles form unusual multispherical or "multi-balloon" shapes consisting of several spheres that are connected by membrane necks. Such multispherical shapes have been recently observed when the two sides of the membranes were exposed to different sugar solutions. This sugar asymmetry induced a spontaneous curvature, the sign of which could be reversed by swapping the interior with the exterior solution. Here, previous studies of multispherical shapes are reviewed and extended to develop a comprehensive theory for these shapes. Each multisphere consists of large and small spheres, characterized by two radii, the large-sphere radius, R_l, and the small-sphere radius, R_s. For positive spontaneous curvature, the multisphere can be built up from variable numbers N_l and N_s of large and small spheres. In addition, multispheres consisting of N_*=N_l+N_s equally sized spheres are also possible and provide examples for constant-mean-curvature surfaces. For negative spontaneous curvature, all multispheres consist of one large sphere that encloses a variable number N_s of small spheres. These general features of multispheres arise from two basic properties of curvature elasticity: the local shape equation for spherical membrane segments and the stability conditions for closed membrane necks. In addition, the (N_l+N_s)-multispheres can form several (N_l+N_s)-patterns that differ in the way, in which the spheres are mutually connected. These patterns may involve multispherical junctions consisting of individual spheres that are connected to more than two neighboring spheres. The geometry of the multispheres is governed by two polynomial equations which imply that (N_l+N_s)-multispheres can only be formed within a certain restricted range of vesicle volumes. Each (N_l+N_s)-pattern can be characterized by a certain stability regime that depends both on the stability of the closed necks and on the multispherical geometry. Interesting and challenging topics for future studies include the response of multispheres to locally applied external forces, membrane fusion between spheres to create multispherical shapes of higher-genus topology, and the enlarged morphological complexity of multispheres arising from lipid phase separation and intramembrane domains.