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
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Category: Seminar
Description:
We study two types of models describing the motility of eukaryotic cells on substrates. The first, a phasefield model, consists of the AllenCahn 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 nonlinear free boundary problem. It consists of an elliptic equation describing the flow of cytoskeleton gel coupled with a convectiondiffusion PDE for the density of myosin motors. The key properties of this problem are (i) presence of the cross diffusion as in the classical KellerSegel 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 nonradial steady states. Existence of both traveling waves and nonradial steady states is established via LeraySchauder degree theory applied to a Liouvilletype 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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