Breathing and moving vesicles in a geometric mechanochemical model

RESUMO

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'' $\phi$ 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 $\phi$ adsorbs/desorbs at places of high/low mean curvature $H$, i.e., the kinetics of $\phi$ depend on $H$, and conversely $\phi$ 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).