On a poroviscoelastic model for cell crawling

In this paper a minimal, one–dimensional, two–phase, viscoelastic, reac- tive, flow model for a crawling cell is presented. Two-phase models are used with a variety of constitutive assumptions in the literature to model cell motility. We use an upper–convected Maxwell model and demonstrate that even the simplest of two– phase, viscoelastic models displays features relevant to cell motility. We also show care must be exercised in choosing parameters for such models as a poor choice can lead to an ill–posed problem. A stability analysis reveals that the initially station- ary, spatially uniform strip of cytoplasm starts to crawl in response to a perturbation which breaks the symmetry of the network volume fraction or network stress. We also demonstrate numerically that there is a steady travelling–wave solution in which the crawling velocity has a bell–shaped dependence on adhesion strength, in agreement with biological observation.