## Monday, May 25, 2009

### Cell invasion (5) : Cooperation via Chemotaxis

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It is known that cells can communicate with each other via signalling molecules that are diffusing in the extracellular matrix (e.g. cytokines). Would such chemotaxis be compatible with the apparent backward drift component observed in the motion of invading cells ?

To include chemotaxis into the cell invasion model, we assume that the effective drift velocity is proportional to the local gradient of the "cytokine" concentration. This leads to the following set of coupled equations:

Drift-diffusion of cells:
\frac{\partial}{\partial t}c(z,t) = - \frac{\partial}{\partial z}\left[v(z,t)\;c(z,t)\right] + D_c \frac{\partial^2}{\partial z^2}c(z,t),
Gradient controlled drift velocity:
v(z,t) = s\; \frac{\partial}{\partial z} g(z,t)
Production, diffusion and decay of the guiding substance:
\frac{\partial}{\partial t}g(z,t) = p\!\cdot\!c(z,t) + D_g \frac{\partial^2}{\partial z^2}g(z,t) - d\!\cdot\!g(z,t)
Here, the two coupled fields are the cell concentration profile c(z,t) and the concentration profile of the guiding substance (the "cytokine") g(z,t). The diffusion constants for cells and guiding substance molecules are denoted by D_c and D_g, respectively. In addition, we introduce a sensitivity parameter s, a production parameter p and a decay parameter d. It is assumed that the guiding substance is locally produced by each cell at a constant rate and that it decays within a characteristic time 1/d.