Fractional Chemotaxis Diffusion Equations

TL;DR

This paper develops fractional differential equations and Monte Carlo methods to model chemotaxis with anomalous subdiffusion, capturing transport hindered by obstacles in biological systems.

Contribution

It introduces new mesoscopic and macroscopic models for chemotaxis with anomalous subdiffusion, including formulations with fractional derivatives and a simulation method.

Findings

Models effectively describe hindered biological transport.

Simulation results match numerical solutions.

Framework can replace Keller-Segel equations in complex environments.

Abstract

We introduce mesoscopic and macroscopic model equations of chemotaxis with anomalous subdiffusion for modelling chemically directed transport of biological organisms in changing chemical environments with diffusion hindered by traps or macro-molecular crowding. The mesoscopic models are formulated using Continuous Time Random Walk master equations and the macroscopic models are formulated with fractional order differential equations. Different models are proposed depending on the timing of the chemotactic forcing. Generalizations of the models to include linear reaction dynamics are also derived. Finally a Monte Carlo method for simulating anomalous subdiffusion with chemotaxis is introduced and simulation results are compared with numerical solutions of the model equations. The model equations developed here could be used to replace Keller-Segel type equations in biological systems with transport hindered by traps, macro-molecular crowding or other obstacles.