Reaction-diffusion equation modelling calcium waves with fast buffering in visco-elastic environment (Archives of Mechanics - Test, 2012)
The model we consider treats a cell or a group of cells as a viscoelastic medium whose stress tensor has a term - the traction- representing the stresses generated in the medium by the actomyosin molecules. We consider three kinds of domains (“shapes” of cells): the thin circular cylinder mimicking a long cell, the thin slab being a cari-cature of a tissue, and the unbounded space. We assume that the viscous effects are much weaker than the elastic ones and consider two extreme cases: either the body force is negligible or it is strong. This leads to three pairs, one pair for each domain, of approximations for the dilatation. We interpolate between the approximated ex-pressions forming one pair and as the result we obtain a single calcium conservation equation and a system of buffer equations. Using the rapid buffering approximation we reduce the problem to a single reaction-diffusion equation. We study the travelling wave solutions to these equations. We show that not only the high affinity buffers but also the mechanical effects alone can prevent the formation and propagation of the waves if the supply of calcium is not sufficiently substantial.