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ISMCM: "Phase Field and Free Boundary Models of Cell Motility"

Leonid Berlyand Department of Mathematics Pennsylvania State University

Monday, November 20, 2017
  12:10–1:10 p.m.


Location: Surge Building 268
  Parking Information

Category: Seminar

Description:

We study two types of models describing the motility of  eukaryotic cells on substrates. The first, a phase-field model, consists of  the Allen-Cahn equation for the scalar phase field function coupled  with a vectorial parabolic equation for the orientation of the actin filament network. The  key properties of this system are (i) presence of gradients in the coupling termsand (ii) mass (volume) preservation constraints. We pass to the sharp interface limit to derive the equation of the motion of the cell boundary, which is mean curvature motion modified by a novel nonlinear term. We establish the existence of two distinct regimes of the physical parameters and prove existence of traveling waves in the supercritical regime. The traveling waves describe persistent motion of the cell without external cues or stimuli which is a signature of cell motility.


The second model  is a non-linear free boundary problem. It consists of an elliptic equation describing the flow of cytoskeleton gel coupled with  a convection-diffusion  PDE  for the  density of myosin motors. The  key properties of this problem are (i) presence of the cross diffusion as in the classical  Keller-Segel  problem in chemotaxis and (ii) nonlinear nonlocal free boundary condition that involves curvature of the boundary. We establish the bifurcation of the traveling waves from  a family of  radially symmetric steady states. We also study breaking of symmetry by proving existence of non-radial steady states. Existence of both traveling waves and non-radial steady states is established via Leray-Schauder degree theory applied to a Liouville-type equation (which is obtained via a reduction of the original system) in a  free boundary setting. 


These results were obtained in collaboration  with J. Fuhrmann, M. Potomkin, and V. Rybalko


 



Additional Information: ISMCM

Open to: General Public
Admission: Free
Sponsor: Mathematics

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