Mori Yoichiro, Jilkine Alexandra, Edelstein-Keshet Leah
School of Mathematics, University of Minnesota, Minneapolis MN 55455, USA.
SIAM J Appl Math. 2011;71(4):1401-1427. doi: 10.1137/10079118X.
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.
我们描述并分析了一个用于两种相互转化化学物质的双稳反应扩散（RD）模型，该模型展现出一种波钉扎现象：其中一种物质的激活波在区域的一端引发，向区域内移动，减速，最终在区域内停止，形成一个静止前沿。在此过程中，第二种（“非活性”）物质被耗尽。这种行为出现在一个关于细胞因响应刺激而被Rho GTPases进行化学极化的模型中。最初空间均匀的浓度分布（代表静息细胞）发展为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特性：（1）有限区域内的质量守恒，（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差异。通过匹配渐近分析，我们解释了波钉扎的数学基础，并预测了波的速度和钉扎位置。对钉扎前沿解的分岔分析揭示了波钉扎状态如何依赖于诸如扩散速率和物质总质量等参数。我们描述了根据反应动力学细节钉扎解可能丧失的两种方式：鞍结分岔或叉形分岔。
