Last modified: 2014-03-31

#### Abstract

Following the suggestion of L. F. Jaffe [1], we propose a mathematical model of fast calcium induced calcium influx waves (so called CICI Waves). They can propagate at relatively high speeds (up to 1300 micrometers/s). According to [1], they propagate due to a mechano-chemical interaction, in which stretching a cell's membrane at one point opens nearby stretch-activated calcium channels. The stretching is resulting from the action of acto-myosin fibers of the cortex anchored in the membrane. In turn, the resulting influx of calcium implies stretching of a nearby region of the membrane. This phenomenon can be described by a single reaction diffusion equation for the calcium concentration coupled with the equation for the traction forces. We assume that the local stretching of the membrane is proportional to the traction forces. As follows from experiments, the calcium influx through the stretched activated channels is proportional to the speed of the membrane deformation. This leads to the dynamical boundary condition for the reaction diffusion equation of calcium concentration at the lateral boundary of the cell. That is to say, the boundary flux depends on the time derivative of the calcium concentration. Another important feature of the model is the expression of the local traction forces by an integral operator which takes into account fact that the fibers have the finite lengths.

Finally, making an approximation of thin cylindrical cell we perform the asymptotic analysis showing that, indeed, the proposed by L.F Jaffe mechanism of CICI waves propagation can work.

[1] L.F. Jaffe, "Stretch-activated calcium channels relay fast calcium waves propagated by calcium-induced calcium influx", Biol. Cell 99, 175-184 (2007)