Application of Fractional Calculus to Reaction-Subdiffusion Processes and Morphogen Gradient Formation

TL;DR

This paper applies fractional calculus to model reaction-subdiffusion processes, specifically analyzing morphogen gradient formation in developmental biology, revealing how spatial reactivity influences concentration profiles.

Contribution

It introduces a fractional calculus-based approach to reaction-subdiffusion equations and explores their application to morphogen gradient formation with both uniform and non-uniform reactivity.

Findings

Exponential stationary profiles in uniform reactivity case.

Profiles are highly sensitive to spatial reactivity variations.

Rich phenomenology observed in non-uniform reactivity scenarios.

Abstract

It is a well known fact that subdiffusion equations in terms of fractional derivatives can be obtained from Continuous Time Random Walk (CTRW) models with long-tailed waiting time distributions. Over the last years various authors have shown that extensions of such CTRW models incorporating reactive processes to the mesoscopic transport equations may lead to non-intuitive reaction-subdiffusion equations. In particular, one such equation has been recently derived for a subdiffusive random walker subject to a linear (first-order) death process. We take this equation as a starting point to study the developmental biology key problem of morphogen gradient formation, both for the uniform case where the morphogen degradation rate coefficient (reactivity) is constant and for the non-uniform case (position-dependent reactivity). In the uniform case we obtain exponentially decreasing stationary concentration profiles and we study their robustness with respect to perturbations in the incoming morphogen flux. In the non-uniform case we find a rich phenomenology at the level of the stationary profiles. We conclude that the analytic form of the long-time morphogen concentration profiles is very sensitive to the spatial dependence of the reactivity and the specific value of the anomalous diffusion coefficient.