We study a simple reaction scheme in a two-dimensional lattice of particles/molecules with a refractory state. We are interested in the dynamics of the propagating front as a function of physicochemical properties of the host medium. Anisotropy of the medium greatly affects the smoothness of the wave front. Similarly, if we allow the particles/molecules to diffuse slowly to neighboring sites, then the front wave is more likely to be discontinuous. Both situations affect the ability of the whole system to relax to the original state which is a required feature in biological cells. Attempts to map this simple reaction scheme to reactions involved in the intracellular pathways suggest that in some cases the signal transduction might take both connotation of a random walk and a propagating wave, depending on the local density of the medium. In particular, a sufficient condition for the appearance of waves in high density regions of the media, is the existence of at least one reaction in the chain of reactions characterizing the pathway.
F. Castiglione, M. Bernaschi, S. Succi R. Heinrich, and M.W. Kirschner. Intracellular signal propagation in a two-dimensional auto-catalytic reaction model. Phys. Rev. E 66, 031905 (2002)