(398i) Gaussian-Curvature-Mediated Interactions of Elastic Inclusions in Fluid Membranes

The formation of solid domains in two-dimensional (2D) fluid films is integral to the structure and function of biological membranes. Lipid rafts, protein aggregates, and liquid-crystal phases all exhibit solid-like characteristics in fluid monolayers and bilayers. Inspired by the exotic geometries that manifest in real biological membranes, in this talk we present theoretical efforts to describe how membrane curvature couples to the elastic mechanics of solid inclusions. In the first part of the talk, we address the “rigid limit” of this problem by analyzing the force exerted on a rigid domain embedded in a tense, fluid film of variable Gaussian curvature. A connection is made to a controversial problem in the colloidal physics literature, where there remains a disagreement as to whether rigid, planar inclusions experience tension-driven forces in curved membranes. We argue that such forces exist by drawing an analogy to the dielectrophoretic force on a neutral, but polarizable, colloid in an electric field. Thus, we show unambiguously that the force on a planar, rigid inclusion scales in proportion to the Gaussian curvature of the host membrane. In doing so, we also uncover the source of disagreement in prior theoretical work. In the second part of the talk, we address the problem of a solid inclusion with finite stretching and bending elasticity in a tense membrane. We show that the nature of the force exerted on the inclusion depends upon the persistence length of the inclusion or, equivalently, its elastic modulus. When the inclusion is very stiff (large persistence length), we recover the “rigid limit” wherein the force scales in proportion to the Gaussian curvature. In this case, the small deformation of the inclusion is dominated by bending. On the other hand, when the inclusion is soft (small persistence length), then the force scales with the square of the Gaussian curvature and the deformation is primarily due to stretching. Thus, we find that soft inclusions are driven towards regions of minimum squared Gaussian curvature (e.g., developable surfaces). The same force explains the origin of elastic interactions between inclusions that are able to bend but not stretch. Several examples are considered to illustrate the static and dynamic consequences of elastic heterogeneities in curved membranes.