Mathematical studies of a mechanobiochemical model for 3D cell migration

This work presents the development, analysis and numerical simulations of a model for cell deformation and movement, which couples biochemical reactions and biomechanical forces. The way that cells move is key to the creation and development of most organisms on earth. Consequently a deeper understanding of cell motility is likely to have significant applications to medicine. We propose a mechanobiochemical model which considers the actin filament network as a viscoelastic and contractile gel. The mechanical properties are modelled with a force balancing equation for the displacement. The pressure and contractile forces are influenced by actin and myosin and we model these with a system of reaction-diffusion equations. The model consists of highly non-linear partial differential equations. To analyse the model, we carry out linear stability analysis to determine key bifurcation parameters and find analytical solutions close to bifurcation points. We then approximate the equations and produce numerical solutions in multi-dimensions, using an evolving finite element method. The solutions predicted from linear stability theory are replicated in the early stages of cell movement. Subsequently, both simple and complex deformations, such as expansions, protrusions, contractions and translations of the cell are observed. This theoretical and computational framework allows the study of more complex and experimentally driven reaction kinetics involving, actin, myosin and other molecular species that play an important role in cell movement and deformation.