Shape transformations of vesicles: Phase diagram for spontaneous- curvature and bilayer-coupling models

Vesicle shapes of low energy are studied for two variants of a continuum model for the bending energy of the bilayer: (i) the spontaneous-curvature model and (ii) the bilayer-coupling model, in which an additional constraint for the area difference of the two monolayers is imposed. We systematically investigate four branches of axisymmetric shapes: (i) the prolate-dumbbell shapes; (ii) the pear-shaped vesicles, which are intimately related to budding; (iii) the oblate-discocyte shapes; and (iv) the stomatocytes. These branches end up at limit shapes where either the membrane self-intersects or two (or more) shapes are connected by an infinitesimally narrow neck. The latter limit shape requires a certain condition between the curvatures of the adjacent shape and the spontaneous curvature. For both models, the phase diagram is determined, which is given by the shape of lowest bending energy for a given volume-to-area ratio and a given spontaneous curvature or area difference, respectively. The transitions between different shapes are continuous for the bilayer-coupling model, while most of the transitions are discontinuous in the spontaneous-curvature model. We introduce trajectories into these phase diagrams that correspond to a change in temperature and osmotic conditions. For the bilayer-coupling model, we find extreme sensitivity to an asymmetry in the monolayer expansivity. Both models lead to different predictions for typical trajectories, such as budding trajectories or oblate-stomatocyte transitions. Our study thus should provide the basis for an experimental test of both variants of the curvature model.