Hierarchy of PDE Models of Cell Motility

We consider mathematical PDE models of motility of eukaryotic cells on a substrate and discuss them in a broader context of active materials. Our goal is to capture mathematically the key biological phenomena such as steady motion with no external stimuli and spontaneous breaking of symmetry.

We first describe the hierarchy of PDE models of cell motility and then focus on two specific models: the phase-field model and the free boundary problem model.

The 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 mathematical properties of this system are (i) the presence of gradients in the coupling terms and (ii) the mass (volume) preservation constraints. These properties lead to mathematical challenges that are specific to active (out of equilibrium) systems, e.g., the fact that variational principles do not apply. Therefore, standard techniques based on maximum principle and Gamma-convergence cannot be used, and one has to develop alternative asymptotic techniques.

The free boundary problem model consists of an elliptic equation describing the flow of the cytoskeleton gel coupled with a convection-diffusion PDE for the density of myosin motors. This PDE system is of Keller-Segel type but in a free boundary setting with nonlocal condition that involves boundary curvature. Analysis of this system allows for a reduction to a Liouville type equation which arises in various applications ranging from geometry to chemotaxis. This equation contains an additional term that presents an additional challenge in analysis.

In the analysis of the above models our focus is on establishing the traveling wave solutions that are the signature of the cell motility. We also study breaking of symmetry by proving existence of non-radial steady states. Bifurcation of traveling waves from steady states is established via the Schauder's fixed point theorem for the phase field model and the Leray-Schauder degree theory for the free boundary problem model.

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