Optimal Shapes of Thin Solid Domains in Fluid Vesicles: Inflated vs. Deflated

Lipid rafts in biological membranes play a central role in controlling membrane structure and function. While the physics of fluid–fluid domain separation in multicomponent membranes is well studied, much less is understood about the interplay between elasticity and geometry in vesicles containing solid domains—i.e., 2D elastic phases embedded within an otherwise fluid membrane. Unlike fluid membranes, solid domains resist in-plane shear and exhibit nonlinear elastic responses to changes in Gaussian curvature, with this resistance governed by the elastic thickness of the solid. Due to the topological constraint of fixed total Gaussian curvature on a closed vesicle, this generates morphological frustration in the system, competing with the bending energy's tendency to favor spherical shapes. To investigate the rich morphological behaviors arising from this frustration, we study a model system: a circular solid domain embedded in a fluid vesicle with spherical topology. Using a theoretical approach, we explore a wide parameter space—varying the size of the domain, the degree of vesicle inflation, and the elastic thickness of the solid. When the elastic thickness is large, solid stretching energy is negligible, and vesicle shapes are governed primarily by bending energy, resembling those of homogeneous vesicles with nonzero Gaussian curvature on the solid. In contrast, for small elastic thickness, stretching becomes energetically costly and must be minimized. The solid domain therefore tends to adopt isometric configurations with zero Gaussian curvature. At low inflation, such isometries can be realized through cylindrical bending with minimal deformation from homogeneous vesicle shapes. However, in highly inflated vesicles, isometric deformations either incur high bending energy or become geometrically impossible as the surface approaches a spherical shape. In this regime—where morphological frustration is most pronounced and experimentally relevant—we observe a series of transitions in the solid domain, including folding, crumpling, and wrinkling.