Mathematical investigation of a spatially coupled model for cell polarization

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A precise interplay between processes on the cell membrane and in the inner cell volume is essential for many functions. We consider here the emergence of polarized states in form of a symmetry breaking in the distribution of certain proteins, as for example observed in the activation/deactivation cycle of GTPase molecules. Mathematical models of the GTPase cycle often take the form of a reaction-diffusion system and one proposed mechanism for polarization is the presence of a Turing instability. This requires a substantial difference in diffusion, which in general is not the case for the lateral diffusion of active and inactive GTPase. Cytosolic diffusion of inactive GTPase on the other hand is much faster. Therefore the spatial coupling of membrane and cytosolic processes may lead to a realistic Turing mechanism. A satisfactory mathematical justification of this hypothesis is one motivation of this proposal. We therefore consider reaction-diffusion models for the GTPase cycle that explicitly account for the coupling of membrane and cytosolic processes. This has often been neglected in previous mathematical models but is essential to justify the `higher effective diffusion' hypothesis allowing for a Turing instability. We aim at a comprehensive mathematical analysis, including a linearized stability analysis of spatially homogeneous stationary states, an investigation of the well-posedness of the system, and a justification of a linearized stability principle. The spatial coupling presents some particular challenges for the mathematical treatment. Our analysis will apply to a large class of spatially coupled reaction-diffusion system and is therefore of importance beyond the modeling of the GTPase cycle.