Elasticity and Morphology of Flexible Solid Domains in Lipid Vesicles*

In biological systems, lipid rafts play important roles in controlling structure and function in membranes.  While analogous phenomena are well-studied in the context of fluid-fluid domain separation in multi-component system, far less is understood about the interplay between elasticity and shape in multi-phase vesicles, where one of those phases is a 2D solid.  Motivated to understand the rich arrange of emergent shapes and morphologies in fluid-solid composite vesicles, we study the elasticity of the simplest case, a single solid domain embedded in a fluid vesicle with a spherical topology.  As nanometrically thin materials, both solid and fluid regions are highly-flexible to bending deformations. Yet, the resistance to in-plane shear imbues solid domains (i.e. elastic sheets) with a geometrically-nonlinear resistance to changes of Gaussian curvature, which frustrates the preference of bending energy for globally spherical shapes.  We first describe the analytical results for axisymmetric shapes under the assumption that the solid domain remains strictly planar, for which we show that elastic energy of the composite vesicle to be a non-monotonic function of internal volume, with an optimal degree of inflation dependent on the size of the solid domain.  Next, we describe the more complex shapes that emerge when the vesicle relaxes from axisymmetric shape based on Surface Evolver computations.  We find that, even in the limit of infinitely large solid stretching to bending costs (otherwise the limit of large vesicle radius), there exists a rich pattern of nearly-isometric deformations of the solid domain that lead to pronounced relaxation of the elastic cost of vesicle inflation.  We explore what controls the geometric pattern of nearly-isometric “folding” of the solid domain, as well as the transition to non-isometric shape equilibria when the ratio of solid stretching to bending modulus (or radius) decreases.  Motivated by observations of highly symmetry-broken solid domains in experiments, we describe preliminary results for the elastic energy of non-axisymmetric solid domain shapes on vesicles, motivated by recent observations of solidification process in multi-component vesicles exhibiting non-convex, flower-like solid domains.