Continuous Time Random Walks with Reactions Forcing and Trapping

Abstract

One of the central results in Einstein's theory of Brownian motion is that the mean square displacement of a randomly moving Brownian particle scales linearly with time. Over the past few decades sophisticated experiments and data collection in numerous biological, physical and financial systems have revealed anomalous sub-diffusion in which the mean square displacement grows slower than linearly with time. A major theoretical challenge has been to derive the appropriate evolution equation for the probability density function of sub-diffusion taking into account further complications from force fields and reactions. Here we present a derivation of the generalised master equation for an ensemble of particles undergoing reactions whilst being subject to an external force field. From this general equation we show reductions to a range of well known special cases, including the fractional reaction diffusion equation and the fractional Fokker-Planck equation. Continuous Time Random Walks with Reactions and Forcingwas derived in [12][13][14]. This was extended to non-linear reactions in [15]. An alternative derivation with pure death processes was given earlier in [16].The CTRW model for sub-diffusion exhibits non-ergodic behaviour. The ensemble average mean square displacement scales as a sub-linear power law in time whereas time average mean square displacements scale approximately linearly with time, with different diffusion coefficients for different realisations [17,18]. An alternate model for sub-diffusion is fractional Brownian motion (fBm). This model has a Gaussian probability density function with a time dependent diffusion coefficient and it exhibits ergodic behaviour [19]. In this article we have confined our attention to the CTRW model, however tests to distinguish between different models for sub-diffusive behaviour are an active area of research [18,20].Among the experimental systems that exhibit diffusion, reactions and forcing, the motility of populations of cells is especially interesting because of the confounding aspect of chemotaxis, where particles experience a force in proportion to a chemical gradient. The governing equations to model sub-diffusion with linear reactions and chemotaxis have been derived in [21]. Sub-diffusion with non-linear reactions and chemotaxis was considered in [22].Fractional Fokker-Planck equations and fractional reaction-diffusion equations have become increasingly important because of the growing recognition that anomalous diffusion is ubiquitous [23]. Anomalous sub-diffusion is widespread in biological systems due to traps and crowding effects. In such systems the traps may be structural traps [24, 30] or binding spots [26]. Anomalous sub-diffusion has been measured in protein movements in cells [27], in movement of lipid granules in yeast cells [28] and movements of molecules in spiny neuronal dendrites [29, 30]. The consideration of the possible anomalous diffusion of ions in nerve cells prompted the development of a fractional cable equation to model the electrical...

Generalized Continuous Time Random Walks, Master Equations, and Fractional Fokker--Planck Equations

“…In general the effect of an external force in a CTRW model may be manifest through a change in the jump density [35,9,46,8,3], or a change in the waiting time density [15], or both. Here we suppose that the forcing is manifest through a bias in the nearest neighbor jumps.…”

“…We now seek the evolution equation for the conditional probability density for a walker starting from x 0 at time t = 0 to be at x i at time t. The GMEs can be obtained by differentiating ρ(x i , t|x 0 , 0) with respect to t. However, it is first necessary to take care of the singularities in the arrival fluxes at t = 0 [3]. We thus define…”

“…The motion of the particle is governed by waiting time, and step length, probability densities [40]. Generalized CTRWs have been used as a stochastic basis for deriving fractional diffusion equations [16,22,36], fractional Cattenao equations [12], fractional reaction diffusion equations [53,20,39,14,52,3,49], fractional cable equations [21,27,28], fractional Fokker-Planck equations (FFPEs) [35,46,8,3], and fractional chemotaxis equations [26,15]. The evolution equation for the probability density function (PDF) of the random walking particles in a CTRW can be written as generalized master equations (GMEs) [45,24].…”

Non-linear continuous time random walk models

“…Similar expressions for the switching terms have been obtained before in the context of proliferating glioma cells [37], reactions in spiny dendrites [38], and persistent random walks with death [39]. Exponential tempering factors have previously been found for subdiffusive reaction-transport processes [40,41]. For state 1 we can then use (13) and (19) to write…”

Continuous-time random walks on networks with vertex- and time-dependent forcing

“…One way of introducing the effect of a force field into a CTRW is by including a bias in the jump density [15][16][17]. A variety of force fields may be modeled with this approach.…”