It is a well known fact that subdiffusion equations in terms of fractionalderivatives can be obtained from Continuous Time Random Walk (CTRW) models withlong-tailed waiting time distributions. Over the last years various authorshave shown that extensions of such CTRW models incorporating reactive processesto the mesoscopic transport equations may lead to non-intuitivereaction-subdiffusion equations. In particular, one such equation has beenrecently derived for a subdiffusive random walker subject to a linear(first-order) death process. We take this equation as a starting point to studythe developmental biology key problem of morphogen gradient formation, both forthe uniform case where the morphogen degradation rate coefficient (reactivity)is constant and for the non-uniform case (position-dependent reactivity). Inthe uniform case we obtain exponentially decreasing stationary concentrationprofiles and we study their robustness with respect to perturbations in theincoming morphogen flux. In the non-uniform case we find a rich phenomenologyat the level of the stationary profiles. We conclude that the analytic form ofthe long-time morphogen concentration profiles is very sensitive to the spatialdependence of the reactivity and the specific value of the anomalous diffusioncoefficient.
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