A numerical approach to studying cell dynamics

The focus of this thesis is to propose and implement a highly efficient numerical method to study cell dynamics. Three key phases are covered: mathematical modelling, linear stability analytical theory and numerical simulations using the moving grid finite element method. This aim is to study cell deformation and cell movement by considering both the mechanical and biochemical properties of the cortical network of actin filaments and its concentration. These deformations are assumed to be a result of the cortical actin dynamics through its interaction with a protein known as myosin II in the cell cytoskeleton. The mathematical model that we consider is a continuum model that couples the mechanics of the network of actin filaments with its bio-chemical dynamics. Numerical treatment of the model is carried out using the moving grid finite element method. By assuming slow deformations of the cell boundary, we verify the numerical simulation results using linear stability theory close to bifurcation points. Far from bifurcation points, we show that the model is able to describe the deformation of cells as a function of the contractile tonicity of the complex formed by the association of actin filaments with the myosin II motor proteins. Our results show complex cell deformations and cell movements such as cell expansion, contraction, translation and protrusions in accordance with experimental observations. The migratory behaviour of cells plays a crucial role in many biological events such as immune response, wound healing, development of tissues, embryogenesis, inflammation and the formation of tumours.