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Reaction-diffusion equation modelling calcium waves with fast buffering in visco-elastic environment

Piechór K.

Archives of Mechanics

2012

Vol. 64, nr 5

477-477

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.

Diffusion of calcium in biological tissues and accompanying mechano-chemical effects

Peradzyński Z.

Archives of Mechanics

2010

Vol. 62, nr 6

423-440

In this paper we consider the coupling between chemical and mechanical effects accompanying the diffusion of calcium, either in biological tissues or in a single long cell. The tissue is treated either as a 3-D, or as a quasi-2-D thin layer, of visco-elastic medium, whereas the cell is represented as a thin long cylinder. In particular, the influence of viscosity on the properties of calcium travelling waves is studied. In principle, we explore here the simplest model of calcium diffusion which is based on an effective diffusion coefficient, thus neglecting the details of the role played by buffers. The mechano-chemical coupling in the model is realized by the presence of a traction tensor, in addition to the viscoelastic stress tensor in the mechanical equations, and the strain tensor in the source term of the calcium diffusion equation, as proposed in [1–4]. Our aim is to provide a simple and effective theory, which can be useful in studying various effects influencing propagation of calcium waves. Since in the absence of viscosity the whole mechano-chemical system for calcium and buffers is easily reduced to the “chemical one”, i.e. it consists only of reaction diffusion equations, therefore we decided to perform expansion with respect to the viscosity. Treating, thus, viscous forces as a perturbation, we reduce the problem in each case to a single reaction diffusion equation for the calcium concentration. In this way we avoid the question of the existence of travelling wave solutions as for the so obtained models, their existence follows simply from already known theorems [5–9].