Asymptotic stability of contraction-driven cell motion

We study the onset of motion of a living cell (e.g., a keratocyte) driven by myosin contraction with focus on a transition from unstable radial stationary states to stable asymmetric moving states. We introduce a twodimensional free-boundary model that generalizes a previous one-dimensional model [P. Recho, T. Putelat, boundary condition, and the Young-Laplace law with a regularizing term which precludes blowup or collapse by ensuring that membrane-cortex interaction is sufficiently strong. We find a family of asymmetric traveling solutions bifurcating from stationary solutions. Our main result is nonlinear asymptotic stability of traveling solutions that model observable steady cell motion. We derive an explicit asymptotic formula for the stabilitydetermining eigenvalue via asymptotic expansions in small speed. This formula greatly simplifies computation of this eigenvalue and shows that stability is determined by the change in total myosin mass when stationary solutions bifurcate to traveling solutions. Our spectral analysis reveals the physical mechanisms of stability.

Automated summary

This study focuses on the onset of motion of a living cell driven by myosin contraction. The results show that stable asymmetric moving states bifurcate from unstable radial stationary states, and these moving states have nonlinear asymptotic stability in modeling observable steady cell motion.