细胞运动自由边界模型中行波的出现及其稳定性
Emergence of traveling waves and their stability in a free boundary model of cell motility
摘要：
本文介绍了真核细胞在基质上运动的二维Hele-Shaw型自由边界模型。该模型的关键成分是细胞骨架凝胶（活性凝胶）过阻尼运动的Darcy定律与肌球蛋白密度的平流-扩散方程耦合，导致椭圆-抛物Keller-Segel系统。该系统补充了Hele-Shaw型边界条件：压力和速度连续性的Young-Laplace方程。我们首先表明，径向对称的固定的解决方案变得不稳定和分歧行波解的总肌球蛋白质量的临界值。接下来，我们执行这些行波解的线性稳定性分析，并确定分叉的类型（亚或超临界）。我们的研究揭示了该模型中不稳定/稳定转换的数学基础。具体而言，我们表明，这些过渡发生通过广义本征向量的线性化运营商。
英文摘要：
We introduce a two-dimensional Hele-Shaw type free boundary model for motility of eukaryotic cells on substrates. The key ingredients of this model are the Darcy law for overdamped motion of the cytoskeleton gel (active gel) coupled with advection-diffusion equation for myosin density leading to elliptic-parabolic Keller-Segel system. This system is supplemented with Hele-Shaw type boundary conditions: Young-Laplace equation for pressure and continuity of velocities. We first show that radially symmetric stationary solutions become unstable and bifurcate to traveling wave solutions at a critical value of the total myosin mass. Next we perform linear stability analysis of these traveling wave solutions and identify the type of bifurcation (sub- or supercritical). Our study sheds light on the mathematics underlying instability/stability transitions in this model. Specifically, we show that these transitions occur via generalized eigenvectors of the linearized operator.
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