Uniqueness and traveling waves in a cell motility model

Abstract

<p style='text-indent:20px;'>We study a non-linear and non-local evolution equation for curves obtained as the sharp interface limit of a phase-field model for crawling motion of eukaryotic cells on a substrate. We establish uniqueness of solutions to the sharp interface limit equation in the subcritical parameter regime. The proof relies on a Grönwall estimate for a specially chosen weighted <inline-formula><tex-math id="M1">\begin{document}$L^2$\end{document}</tex-math></inline-formula> norm. <p style='text-indent:20px;'>As persistent motion of crawling cells is of central interest to biologists, we next study the existence of traveling wave solutions. We prove that traveling wave solutions exist in the supercritical parameter regime provided the non-linearity of the sharp interface limit equation possesses certain asymmetry (related, e.g., to myosin contractility). <p style='text-indent:20px;'>Finally, we numerically investigate traveling wave solutions and simulate their dynamics. Due to non-uniqueness of solutions of the sharp interface limit equation we numerically solve a related, singularly perturbed PDE system which is uniquely solvable. Our simulations predict instability of traveling wave solutions and capture both bipedal wandering cell motion as well as rotating cell motion; these behaviors qualitatively agree with recent experimental and theoretical findings.