Analytical And Numerical Studies on Minimal Models of Crawling Cell Motion

The motility of eukaryotic cells is ubiquitous in biological systems and is central to various processes such as wound healing and the immune response. Understanding the biophysical mechanisms driving such crawling cell motion has long attracted biologists, biophysicists, and applied mathematicians alike. Experimental results exhibit a wide range of modes of motility including persistently moving cells, wobbling (bipedal) motion, and rotating cells. Although the biological pathways driving cell motility are complicated, various mathematical models have had great success in both replicating experimental results as well as predicting new phenomena.;In this dissertation we study two minimal models of cell motion which are derived from a phase-field model of keratocyte motion. The first is derived via the so-called sharp interface limit of the phase-field model. In this limit one recovers a non-linear and non-local geometric evolution law for the motion of a planar curve representing the boundary of the cell membrane. In a particular physical parameter regime we prove well-posedness by establishing existence/uniqueness of solutions. We next demonstrate necessary conditions for the existence of traveling wave solutions which correspond to persistently moving cells. We additionally investigate the sharp interface limit equation numerically: we introduce novel algorithms which resolve the difficulties of non-linearity, non-locality, and non-uniqueness of solutions of the sharp interface limit equation. Simulations reveal wobbling motions as well as rotating cells corresponding to experimental results.;The second minimal model is derived by reducing the full phase-field system to a differential algebraic equation (DAE) system. In this simplified system we investigate the effect of patterned substrates on the direction of cell motion. In particular we are interested in understanding the motility of cells on substrates with alternating adhesive and non-adhesive stripes. We validate the DAE system by showing qualitative agreement with full phase-field simulation results wherein either parallel and perpendicular motion to stripes are observed depending on physical parameters. Additionally we predict the effect of changing biophysical parameters and substrate geometry on the direction of cell motility; these results have applications to directed cell motion and sorting.