Vesicle deformation by an axial load: from elongated shapes to tethered vesicles

A sufficiently large force acting on a single point of the fluid membrane of a flaccid phospholipid vesicle is known to cause the formation of a narrow bilayer tube (tether). We analyze this phenomenon by means of general mathematical methods allowing us to determine the shapes of strongly deformed vesicles including their stability. Starting from a free vesicle with an axisymmetric, prolate equilibrium shape, we consider an axial load that pulls (or pushes) the poles of the vesicle apart. Arranging the resulting shapes of strained vesicles in dependence of the axial deformation and of the area difference of monolayers, phase diagrams of stable shapes are presented comprising prolate shapes with or without equatorial mirror symmetry. For realistic values of membrane parameters, we study the force-extension relation of strained vesicles, and we demonstrate in detail how the initially elongated shape of an axially stretched vesicle transforms into a shape involving a membrane tether. This tethering transition may be continuous or discontinuous. If the free vesicle is mirror symmetric, the mirror symmetry is broken as the tether forms. The stability analysis of tethered shapes reveals that, for the considered vesicles, the stable shape is always asymmetric (polar), i.e., it involves only a single tether on one side of the main vesicle body. Although a bilayer tube formed from a closed vesicle is not an ideal cylinder, we show that, for most practical purposes, it is safe to assume a cylindrical geometry of tethers. This analysis is supplemented by the documentation of a prototype experiment supporting our theoretical predictions. It shows that the currently accepted model for the description of lipid-bilayer elasticity (generalized bilayer couple model) properly accounts for the tethering phenomenon.