Mathematical investigation of a spatially coupled model for cell polarization

细胞膜上的过程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是必不可少的。我们在这里考虑了某些蛋白质在分布上以对称性破坏的形式出现的极化状态，例如在GTPase分子的激活/去激活周期中观察到的。GTPase循环的数学模型通常采用反应-扩散系统的形式，一种提出的极化机制是图灵不稳定性的存在。这需要在扩散上有很大的不同，这通常不是活性和非活性GTP酶的横向扩散的情况。另一方面，非活性GTP酶的胞内扩散要快得多。因此，膜和胞质过程的空间耦合可能导致一种现实的图灵机制。对这一假设的令人满意的数学证明是这一提议的动机之一。因此，我们考虑GTPase循环的反应-扩散模型，该模型明确地解释了膜和胞质过程的耦合。这在以前的数学模型中常常被忽视，但对于证明允许图灵不稳定的“更高有效扩散”假设是必要的。我们的目标是一个全面的数学分析，包括空间齐次定态的线性化稳定性分析，系统适定性的研究，以及线性化稳定性原理的证明。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大类空间耦合的反应-扩散系统，因此其重要性超出了对GTPase循环的建模。

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.