Models, Numerics and Simulations of Deforming Biological Surfaces

Abstract

Thin, elastic surfaces are a fundamental building block in each biological system. Their main representative on the small scale are biomembranes; an important example on the larger scale are cell tissues. In both cases, the surfaces define a mechanical and chemical border, indispensable for the genesis and maintenance of each organism. An essential property of the surfaces is a lateral inhomogeneous composition of the surfaces themselves: without these inhomogeneities, the complexity of shapes, mechanochemical properties and dynamics would not be possible. In this thesis, we develop continuous mechanobiological models of membranes and tissues. Since these surfaces are experimentally often difficult to access, our approaches allow to investigate their behavior theoretically. The developed mathematical models are coupled nonlinear systems of partial differential equations (PDE) of fourth order. To enable simulations of these models, we significantly extend numerical algorithms for surface deformation based on the finite-element method (FEM). Extensive systematic simulations of the different models - in close comparison to recent experimental and theoretical studies on different scales - lead to new findings in membrane as well as tissue research. The key findings are the prediction and characterization of new mechanisms of communication between the two monolayers of a biomembrane, the investigation of the elusive role of the Gaussian rigidity in different fundamental membrane processes (like budding and lateral sorting), and moreover, the postulation and investigation of a new model for pattern formation in biological tissues, leading to experimental evidences for a new key mechanism for symmetry break in Hydra polyps.