Two-Fluid Variable Length Model for Cell Crawling

Cells crawl along substrates by exploiting the biochemically driven dynamics of their internal structure: actin filaments are constantly polymerized from solution and the resulting actin network is contracted by molecular motors. We describe these processes with a two-fluid poroviscous one-dimensional model. Importantly, when spreading on substrates, a cell's length in general is not constant. The model, which reflects this by accounting for different densities of the two phases forming the cell (actin network and cytoplasm), has hence to be formulated for a movable domain of varying length. We solve the model by a numerical scheme based on the deformation as the time-integrated velocity. Our results yield an onset of motion upon perturbing a stationary state, and moving cells have different lengths and velocities depending on the parameters, in agreement with experimental evidence.