Equilibrium budding and vesiculation in the curvature model of fluid lipid vesicles

According to a model introduced by Helfrich [Z. Naturforsch. 28c, 693 (1973)], the shape of a closed lipid vesicle is determined by minimization of the total bending energy at fixed surface area and enclosed volume. We show that, in the appropriate regime, this model predicts both budding (the eruption of a satellite connected to the parent volume via a neck) and vesiculation (the special case when the neck radius goes to zero). Vesiculation occurs when the minimum is located at a boundary in the space of configurations. Successive vesiculations produce multiplets, in which the minimum-energy configuration consists of several bodies coexisting through infinitesimal necks. We study the sequence of shapes and shape transitions followed by a spherical vesicle of radius ${\mathit{R}}_{\mathit{V}}$, large on the scale ${\mathit{R}}_0$ set by the spontaneous curvature, as its area A increases at constant volume V=4π${\mathit{R}}_{\mathit{V}}^3$/3. Such a vesicle periodically sheds excess area into a set of smaller spheres with radii comparable to ${\mathit{R}}_0$. We map out this (shape) phase diagram at large volume. In this region the phase diagram is dominated by multiplets and reflects the details of the shedding process. The overall effect of successive vesiculations is to reduce the energy from a quantity of order ${\mathit{R}}_{\mathit{V}}^2$ down to zero or near zero when the area reaches 3V/${\mathit{R}}_0$; however, the decrease is not uniform and the energy E(A,V) is not convex.