Mathematical investigation of a spatially coupled model for cell polarization

细胞膜上和细胞内体积中的过程之间的精确相互作用对于许多功能至关重要。我们认为在这里出现的极化状态的对称性破坏的形式在某些蛋白质的分布，例如观察到的激活/失活周期的GTdR分子。GTdR循环的数学模型通常采用反应扩散系统的形式，并且一种提出的极化机制是存在图灵不稳定性。这需要扩散方面的显著差异，而对于活性和非活性GTdR的横向扩散而言，通常不是这种情况。另一方面，非活性GTdR的胞质扩散快得多。因此，膜和胞质过程的空间耦合可能导致一个现实的图灵机制。一个令人满意的数学证明这一假设是这个建议的动机之一。因此，我们考虑明确考虑耦合的膜和胞质过程的GTdR循环的反应扩散模型。这在以前的数学模型中经常被忽略，但对于证明允许图灵不稳定性的“更高有效扩散”假设是必不可少的。我们的目标是在一个全面的数学分析，包括线性化的稳定性分析的空间均匀的静止状态，调查系统的适定性，和一个理由的线性化的稳定性原则。空间耦合对数学处理提出了一些特殊的挑战。我们的分析将适用于一个大类的空间耦合反应扩散系统，因此是重要的GTdR循环的建模之外。

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.