Phase diagrams and shape transformations of toroidal vesicles

Shapes of vesicles with toroidal topology are studied in the context of curvature models for the membrane. For two simplified curvature models, the spontaneous-curvature (SC) model and the bilayer-couple (BC) model, the structure of energy diagrams, sheets of stationary shapes and phase diagrams are obtained by solving shape equations for axisymmetric shapes. Three different sheets of axisymmetric shapes are investigated systematically: I) discoid tori; it) sickle-shaped tori and iii) toroidal stomatocytes. A stability analysis of axisymmetric shapes with respect to symmetry breaking conformal transformations reveals that large regions of the phase diagrams of toroidal vesicles are non-axisymmetric. Non-axisymmetric shapes are determined approxim~tely using conformal transformations. To compare the theory with experiments, a generalization of the SC and BC model, the area-difference-elasticity-model (ADlGmodel), which is a more realistic curvature model for lipid bilayers, is discussed. Shapes of toroidal vesicles which have been observed recently can be located in the phase diagram of the ADlGmodel. We predict the effect oftemperature changes on the observed shapes-The new class of shapes, the toroidal stomatocytes, have not yet been observed.