Breathing and moving vesicles in a geometric mechanochemical model

Abstract: We consider a geometric mechanochemical model of vesicles which couples the Helfrich flow for the shape of a lipid bilayer vesicle membrane $X$ with a reaction-diffusion equations for a single morphogen'' $φ$ on $X$. The Helfrich flow is the $L^2$ gradient flow of the elastic bending energy $E(X)=\int_X (H-c_0)^2 dS$ of $X$, typically supplemented by area or volume constraints, or both. The morphogen $φ$ adsorbs/desorbs at places of high/low mean curvature $H$, i.e., the kinetics of $φ$ depend on $H$, and conversely $φ$ modifies the spontaneous curvature $c_0$ on $X$. The flow is no longer gradient, and hence allows for more complicated dynamics, including time periodic orbits, e.g.,breathing and moving'' vesicle shapes. We show how to compute bifurcation diagrams for such solution branches via numerical continuation and bifurcation methods. We mostly focus on ``planar'' vesicles (1D closed curves) but also give an outlook on 3D vesicles (2D closed membranes).