Constriction force around closed membrane neck

To derive the constriction force generated by the spontaneous curvature around a closed membrane neck, we consider a convenient parametrization of a dumbbell shape, consisting of two hemispheres connected by two unduloid segments that form a narrow neck of neck radius R_ne^14,28. The dumbbell with a closed neck is obtained in the limit of zero R_ne. To reveal the curvature-induced constriction force f, we first consider an external constriction force f_ex compressing the neck. In such a situation, the bending energy E_be of the dumbbell has the form

up to first order in R_ne. The closed neck is stable if the term proportional to the neck radius R_ne increases with increasing R_ne which implies

In the absence of an external force, that is, for f_ex = 0, we then obtain the curvature-induced constriction force f = 8πκ(m − M_ne) as in Eq. (²). This constriction force is proportional to the curvature difference m − M_ne, which vanishes along the line L₁₊₁. Once we have crossed the line L₁₊₁ towards higher values of the shape parameter mR_ve, the curvature difference m − M_ne increases monotonically as we increase the spontaneous curvature m for fixed volume-to-area ratio v. Indeed, for constant v, the neck curvature M_ne is determined by the two-sphere geometry of the dumbbell-shaped vesicle which remains unchanged as we increase m for constant v. For the dumbbell shapes displayed in Figs. 2c, d and ​and3,3, the numerical values of the spontaneous curvature m, the curvature difference m − M_ne, and the constriction force f are displayed in Table 1.