细胞极化反应扩散模型中波钉扎的渐近与分支分析
Asymptotic and bifurcation analysis of wave-pinning in a reaction-diffusion model for cell polarization
摘要：
我们描述和分析了一个反应扩散（RD）模型的两个相互转换的化学物种，表现出的波钉扎现象：一波的激活的物种之一是在域的一端发起，移动到域，减速，并最终停止在域内，形成一个固定的前。第二种（“非活性”）物质在这一过程中被耗尽。这种行为出现在一个模型中的细胞的化学极化的Rho GTP酶响应刺激。最初的空间均匀的浓度分布（代表静息细胞）发展成不对称的静止前沿分布（典型的极化细胞）。这里的波钉扎基于三个性质：（1）有限域中的质量守恒，（2）允许多个稳定稳态的非线性反应动力学，以及（3）两种物质扩散的足够大的差异。利用匹配渐进分析，解释了波钉扎的数学基础，并预测了波的速度和钉扎位置。钉扎前解决方案的分叉的分析揭示了如何波钉扎制度取决于参数，如扩散率和总质量的物种。我们描述了两种方式，其中钉扎的解决方案可以失去取决于反应动力学的细节：一个鞍节点或干草叉分叉。
英文摘要：
We describe and analyze a bistable reaction-diffusion (RD) model for two interconverting chemical species that exhibits a phenomenon of wave-pinning: a wave of activation of one of the species is initiated at one end of the domain, moves into the domain, decelerates, and eventually stops inside the domain, forming a stationary front. The second ("inactive") species is depleted in this process. This behavior arises in a model for chemical polarization of a cell by Rho GTPases in response to stimulation. The initially spatially homogeneous concentration profile (representative of a resting cell) develops into an asymmetric stationary front profile (typical of a polarized cell). Wave-pinning here is based on three properties: (1) mass conservation in a finite domain, (2) nonlinear reaction kinetics allowing for multiple stable steady states, and (3) a sufficiently large difference in diffusion of the two species. Using matched asymptotic analysis, we explain the mathematical basis of wave-pinning, and predict the speed and pinned position of the wave. An analysis of the bifurcation of the pinned front solution reveals how the wave-pinning regime depends on parameters such as rates of diffusion and total mass of the species. We describe two ways in which the pinned solution can be lost depending on the details of the reaction kinetics: a saddle-node or a pitchfork bifurcation.
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