The cellular integrity of animal cells is maintained by a specialised structure, the cytoskeleton. Apart from its scaffolding and cell shaping properties, it acts as a dynamic structure which resists, generates and transmits cellular forces. The tension dependent assembly of the actin and myosin, forming stress fibres, transmits the forces through focal adhesions to the integrin-extra cellular matrix(ECM) interface on the cell membrane. In addition to sensing the internal forces, focal adhesions also transmit external force that act on the ECM to the cytoskeleton. The inositol triphosphate (IP3) molecules produced at the focal adhesion junction, diffuse through the cytoplasm, attach themselves to the calcium gates on the endoplasmic reticulum and release calcium ions. These free calcium ions in the cytoplasm are responsible for the formation of the actin-myosin bridge and hence the force in the cytoskeleton. It has been verified experimentally that the behaviour of the cell is significantly influenced by the mechanical forces. Mathematical models simulating these experiments help to predict the cellular behaviour.

Coupling stress fibre growth to the focal adhesions results in a bio-chemo-mechanical problem. Solving the coupled system of equations in a staggered manner restricts the time step that can be used. In this contribution, a monolithic approach is developed to relax this restriction and is found to be superior to the staggered approach for a wide range of time steps. The stress fibre is assumed to obey the Hill type growth, where the stress depends on the strain rate, even in non-muscle cells. A computationally stable smooth non-linear Hill model for the stress fibre growth is coupled with the focal adhesion development and is compared with a piecewise continuous, linear approximation of the Hill model. Furthermore, the effect of the diffusion of unbound integrins on the growth of focal adhesions is analysed. The difference between the systems with and without feedback mechanism is distinguished and compared with experimental results.