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Abstract

Thin elastic surfaces containing molecules infuencing the mechanical prop- erties of the surface itself are wide spreaded structures of different scales in biological systems. Prominent examples are bilayer membranes and cell tis- sues. In this paper we present a continuous dynamical model of deforming lateral inhomogeneous surfaces, using the example of biological membranes. In agreement with experimental observations the membrane consists of dif- ferent molecule species undergoing lateral phase separation and influencing the mechanical properties of the membrane. The presented model is based on the minimization of a free energy leading to a coupled nonlinear PDE system of fourth order, related to the Willmore flow and the Cahn-Hilliard equation. First simulations show the development of budding structures from stochas- tic initial conditions as a result of the gradient flow, which is comparable to experimentally observed structures. In our model mechanical properties are described via macroscopic mechanical moduli. However, the qualitative and quantitative relationships of mechanical moduli and the local composi- tion of the membrane are unkown. Since the exact relationship significantly influences the emerging structures, this study motivates the development of techniques allowing for upscaling from the molecular scale.