# American Institute of Mathematical Sciences

June  2019, 24(6): 2811-2835. doi: 10.3934/dcdsb.2018315

## Uniqueness and traveling waves in a cell motility model

 1 Department of Mathematics and Statistics, The College of New Jersey, 2000 Pennington Road, Ewing, NJ 08628, USA 2 Institute of Natural Sciences and Department of Mathematics, Key Laboratory of Scientific and Engineering Computing; Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, China

* Corresponding author

Received  March 2017 Revised  May 2018 Published  January 2019

Fund Project: The work of MSM was completed at Pennsylvania State University, supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program. He also received partial support from NSF grants DMS-1106666 and DMS- 1405769. The work of PZ was partially supported by the National Natural Science Foundation of China grant 11471214 and the One Thousand Plan of China for young scientists

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 $L^2$ norm.

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).

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.

Citation: Matthew S. Mizuhara, Peng Zhang. Uniqueness and traveling waves in a cell motility model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2811-2835. doi: 10.3934/dcdsb.2018315
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##### References:
Sketch of functions $w_B$, $w_F$ and $w_{new}$ from the proof of Theorem 3.2
(Left) Plot of $I_2(V, \lambda)$ with $\tilde{\Phi}_\beta(V)$ and $\beta = 100$ (Right) Traveling wave profile for $\tilde{\Phi}_\beta(V)$, $\beta = 100$, $V \approx 2.15$ (in positive $y$ direction), $\lambda \approx 9.75$
(Left) Plot of $I_2(V, \lambda)$ with $\Phi_\beta(V)$ derived from (4)-(6) with $W$ as in (37); $\beta = 100$. (Right) Traveling wave profile with $V\approx 1.7$, $\lambda \approx 0$
(Left) Sketch of rotating cell; intervals in red (dotted) represent unstable velocities (Right) Graph of the isoperimetric inequality $Q$ over time for various $\varepsilon$
(Left) Plot of $V-\Phi_\beta(V)$ tracked for a point on a curve evolving by system (38)-(39) shows approximate hysteresis jumps. (Right) Trajectory of center of curve in (38)-(39) when $\varepsilon = .01$; after short transience period, convergence to "zig-zag" motion
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