  This work presents the development, analysis and numerical simulations of a
biophysical model for 3D cell deformation and movement, which couples
biochemical reactions and biomechanical forces. We propose a mechanobiochemical
model which considers the actin filament network as a viscoelastic and
contractile gel. The mechanical properties are modelled by a force balancing
equation for the displacements, the pressure and concentration forces are
driven by actin and myosin dynamics, and these are in turn modelled by a system
of reaction-diffusion equations on a moving cell domain. The biophysical model
consists of highly non-linear partial differential equations whose analytical
solutions are intractable. To obtain approximate solutions to the model system,
we employ the moving grid finite element method. The numerical results are
supported by linear stability theoretical results close to bifurcation points
during the early stages of cell migration. Numerical simulations exhibited show
both simple and complex cell deformations in 3-dimensions that include cell
expansion, cell protrusion and cell contraction. The computational framework
presented here sets a strong foundation that allows to study more complex and
experimentally driven reaction-kinetics involving actin, myosin and other
molecular species that play an important role in cell movement and deformation.
