Reaction-subdiffusion model of morphogen gradient formation.

We study gradient formation of subdiffusive morphogens. The morphogens are produced at a source point at a constant rate. From there they move subdiffusively and are also subject to degradation at a rate that may depend on location and on time. Our analysis is based on a reaction-subdiffusion equation obtained from a continuous time random-walk model with a long-tailed waiting time distribution that also incorporates an evanescence process. Spatially uniform degradation at a constant rate leads to an exponentially decreasing stationary concentration profile hardly distinguishable from that obtained with normal diffusion. On the other hand, with location-dependent degradation we find a rich gamut of profiles, some qualitatively quite different from those occurring with normal diffusion. We conclude that long-time morphogen concentration profiles are very sensitive to the spatial dependence of the reactivity and may also serve as a sensitive measure of the occurrence of anomalous diffusion.