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 two-dimensional free-boundary PDE model that generalizes a previous one-dimensional model by combining a Keller-Segel model, Hele-Shaw kinematic boundary condition, and the Young-Laplace law with a novel nonlocal regularizing term. This nonlocal term precludes blowup or collapse by ensuring that membrane-cortex interaction is sufficiently strong. We found a family of asymmetric traveling solutions bifurcating from stationary solutions. Our main result is the nonlinear asymptotic stability of traveling wave solutions that model observable steady cell motion. We derived and rigorously justified an explicit asymptotic formula for the stability determining eigenvalue via asymptotic expansions in a small speed of cell. This formula greatly simplifies the 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. It also leads to interesting mathematics due to non-selfadjointness of the linearized problem which is a signature of active matter out-of-equilibrium systems. If time permits, we will discuss work in progress on fingering instability in multicellular tissue spreading. This is joint work with V. Rybalko and C. Safsten published in Transactions of AMS (2022), and Phys. Rev.B, (2022).

Bio: Leonid Berland received his Ph. D. in 1985 from Kharkiv University (Ukraine). He joined the Pennsylvania State University (PSU) in 1991 and he is currently a Professor of Mathematics and a member of the Materials Research Institute and Huck Life Sciences Institutes at PSU. He is a founding co-director of PSU Centers for Interdisciplinary Mathematics and for Mathematics of Living and Mimetic Matter. He works at the interface of mathematics and other disciplines such as physics, materials sciences, life sciences, and most recently computer science. His research has largely been concerned with homogenization theory, Ginzburg–Landau theory, PDE models of active matter such as phase field & free boundary models, and most recently stability, convergence, and multiscale in deep learning algorithms. He co-authored three books and more than 100 refereed publications.