In silico experiments bear the potential for further understanding of biological transport processes by allowing a systematic modification of any spatial property and providing immediate simulation results. Cell polarization and spatial reorganization of membrane proteins are fundamental for cell division, chemotaxis and morphogenesis. We chose the yeast Saccharomyces cerevisiae as an exemplary model system which entails the shuttling of small Rho GTPases such as Cdc42 and Rho, between an active membrane-bound form and an inactive cytosolic form. We used partial differential equations to describe the membrane-cytosol shuttling of proteins. In this study, a consistent extension of a class of 1D reaction-diffusion systems into higher space dimensions is suggested. The membrane is modeled as a thin layer to allow for lateral diffusion and the cytosol is modeled as an enclosed volume. Two well-known polarization mechanisms were considered. One shows the classical Turing-instability patterns, the other exhibits wave-pinning dynamics. For both models, we investigated how cell shape and diffusion barriers like septin structures or bud scars influence the formation of signaling molecule clusters and subsequent polarization. An extensive set of in silico experiments with different modeling hypotheses illustrated the dependence of cell polarization models on local membrane curvature, cell size and inhomogeneities on the membrane and in the cytosol. In particular, the results of our computer simulations suggested that for both mechanisms, local diffusion barriers on the membrane facilitate Rho GTPase aggregation, while diffusion barriers in the cytosol and cell protrusions limit spontaneous molecule aggregations of active Rho GTPase locally.

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The unfitted discontinuous Galerkin (UDG) method allows for conservative dG discretizations of partial differential equations (PDEs) based on cut cell meshes. It is hence particularly suitable for solving continuity equations on complex-shaped bulk domains. In this paper based on and extending the PhD thesis of the second author, we show how the method can be transferred to PDEs on curved surfaces. Motivated by a class of biological model problems comprising continuity equations on a static bulk domain and its surface, we propose a new UDG scheme for bulk-surface models. The method combines ideas of extending surface PDEs to higher-dimensional bulk domains with concepts of trace finite element methods. A particular focus is given to the necessary steps to retain discrete analogues to conservation laws of the discretized PDEs. A high degree of geometric flexibility is achieved by using a level set representation of the geometry. We present theoretical results to prove stability of the method and to investigate its conservation properties. Convergence is shown in an energy norm and numerical results show optimal convergence order in bulk/surface H 1 {H^{1}} - and L 2 {L^{2}} -norms.

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