Boundary behaviour of open vesicles in axisymmetric case

A continuous transformation from a closed vesicle to an open vesicle requires that the area of open hole enlarges from zero. Since the shape equation and boundary conditions of lipid open vesicles with free edges have been obtained, we want to know whether this process can be achieved with valid parameters. By studying the boundary conditions in the axisymmetric case, the analytic expression of the boundary edges is obtained generally. It reveals that the radius and line tension of boundary edges are confined strongly by bending moduli. In some cases, there is the minimal nonzero boundary radius and the line tension needs to surmount the maxim following the increase of boundary radius. Without the spontaneous curvature, the line tension will trend to infinite when the boundary radius shrinks to zero. The continuous opening up process requires that the spontaneous curvature is nonzero and the ratio between the bending moduli of Gauss curvature and mean curvature satisfies [Formula: see text], which is smaller than the value [Formula: see text] from experiments and simulations. This result indicates that the opening up process is discontinuous.