Thin-film theories for two-phase reactive flow models of active cell motion

The aim of this paper is to develop a broadly-applicable and self-consistent thin-film biphasic modelling framework for the full crawling cycle of a single animal cell. A hierarchy of thin-film two-phase 'reactive flow' models is derived; between them these cover a wide range of biologically relevant parameter regimes. The mathematical properties and biological implications of the resulting systems of high-order nonlinear degenerate parabolic-elliptic evolution equations are investigated. Linear-stability arguments suggest the formation of highly localized regions of high or low network density associated with small irregular oscillations or 'ruffling' of the plasma membrane. Local analyses at the contact line identify the classes of admissible contact-line conditions, through which we study for the first time the effect on the cell-scale motion of the 'mesoscopic' contact-line physics, which consists of the chemical and mechanical mechanisms for protrusive and retractive force generation near the outer cell periphery. One of the formulations is used to develop a minimal model for cell body translocation over a thin pseudopod, which predicts that myosin-driven contraction is not essential for rapid translocation. An analytic prediction for the translocation speed is given in terms of the network viscosity and slip coefficient (a parameter measuring the adhesion strength), of the membrane tension and of the thicknesses of the pseudopod and actin cortex; this is in good agreement with the translocation speed of osteoblasts on biomaterial substrates commonly used for orthopaedic implants. Limitations of the modelling approach and directions for future work are outlined.