Mathematics of Active Gels: Bifurcations & Stabilit

Mathematics of Active Gels: Bifurcations & Stabilit

-
Leonid V Berlyand, Penn State University

Please note that this seminar will take place online via Zoom. You can connect to this seminar via the following link:

https://princeton.zoom.us/j/9148065146

                           

In this talk we present analytical studies of active gels and cell motility.  We start from a brief   overview of out-of-equilibrium active matter systems (e.g., active gels, bacterial suspensions, etc.) and  mathematical  challenges  in their  modeling and analysis.

Next we present three minimal models of active gels: a phase-field model and two free boundary models. While having a minimal set of parameters and variables, these models capture key biophysical phenomena such as persistent & turning motion, and spontaneous symmetry breaking. The main focus of the talk is stability analysis of our recently developed 2D Hele-Shaw type free boundary (FBP) model. Bifurcation analysis of this model reveals the transition from sub- to supercritical bifurcations of the rest states to traveling waves and bistability in subcritical bifurcation.  We show that this transition occurs due to the gradient coupling, which is a signature mathematical feature of   active matter. Our analysis is based on the center manifold consideration for the underlying non-linear dynamical system. The challenges in the stability analysis are due active terms in the FBP model that manifests  as the  non self-adjoint form of the corresponding linearized operator.

In summary, we identify the key mathematical structures behind the models of active matter such as gradient coupling, reduction of the vectorial models to a scalar Liouville type equations, Keller-Segel cross-diffusion in free boundary settings, and center manifold analysis of  free boundary models.

If time permits,  we will discuss recent work in progress on free boundary models of tissue growth  (joint with  J. Casademunt, et al) and numerics (with A. Safsten).

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