蜜糖直播

Leonid Berlyand, Department of Mathematics, Penn State University

PDE models of Active Matter

In this talk we attempt to demonstrate how mathematical analysis could be helpful in the study of active matter, with the focus on active gels and cell motility.聽聽聽聽聽

We first discuss mathematical challenges and developments of novel mathematical tools due to聽out-of-equilibrium state of active matter (e.g., active cytoskeleton gels, 聽bacterial suspensions, etc.).聽

Next we present three minimal PDE models of active gels: (i) phase-filed model (ii) mean curvature type free boundary model and (iii) Hele-Shaw type free boundary model.聽 These models are designed to capture key biophysical phenomena in cell motility such as persistent & turning聽 motion, symmetry breaking, and viscous fingering while having聽 minimal set of聽 parameters and variables.聽

Our goal is to provide theoretical understanding of cell polarity phenomenon via mathematical analysis of stability/instability 聽and bifurcation from steady states to traveling waves.聽 This is done by identification of key mathematical structures behind the models such as gradient coupling in Phase-Field model, Liouville type equation, Keller-Segel cross-diffusion, and nonlinearity due to the 聽free boundary. We employ mathematical techniques of (i)聽 sharp interface limit via asymptotic analysis,聽 (ii) construction of steady states and traveling waves via Crandall-Rabinowitz bifurcation theory and 聽(iii) topological methods such as Lerey-Schauder degree theory.聽

These are joint works with V. Rybalko (ILTPE, Kharkiv, Ukraine), J. Fuhrman (PSU & Mainz, Germany), 聽M. Potomkin (PSU, USA).