Asymptotic Behavior of the Solution of the Space Dependent Variable Order Fractional Diffusion Equation: Ultraslow Anomalous Aggregation

Abstract

We find the asymptotic representation of the solution of the variable-order fractional diffusion equation, which remains unsolved since it was proposed in [Checkin et. al., J. Phys. A, 2005]. We identify a new advection term that causes ultra-slow spatial aggregation of subdiffusive particles due to dominance over the standard advection and diffusion terms, in the long-time limit. This uncovers the anomalous mechanism by which non-uniform distributions can occur. We perform Monte Carlo simulations of the underlying anomalous random walk and find good agreement with the asymptotic solution.Anomalous diffusion has attracted immense interest in the past due to many physical, chemical and biological processes characterized by the mean square displacement (MSD) involving the fractional exponent µ:Anomalous diffusion is observed also in many other areas, for instance, in finance and economics [7]. An influential paper by Metzler and Klafter [2] reviews anomalous diffusion in the scope of a constant exponent µ. However, anomalous transport in realistic inhomogeneous and complex environments [8], such as lipid granules [9], porous media [10] and entangled polymer liquids [11], requires a multi-fractional approach involving the space-dependent variable-order fractional exponent [12][13][14][15][16][17][18]. Important examples of anomalous transport involving multi-fractional exponents are lateral diffusion of proteins on crowded lipid membranes [19], intracellular subdiffusion of proteins [20], mRNA [21] and organelles [22] due in part to inhomogeneous crowding [23] and weak interactions between components in the cell [24]. Recent observations show that lysosomes, which are key organelles for cellular metabolism, predominantly move subdiffusively and maintain a non-uniform spatial distribution in the cell [24]. The majority of these organelles are concentrated in the perinuclear area. A fundamental unresolved question is how lysosomes are self-organized spatially to coordinate their roles [24]. In this Letter, we propose a new anomalous mechanism by which nonuniform distribution of subdiffusing organelles can occur.A generic model for anomalous diffusion in inhomogeneous media is the space-dependent variable-order fractional diffusion equation [12][13][14][15][16] ∂p(x, t) ∂twhere p(x, t) is the probability density function (PDF) of a particle at position x and time t. This function can be also interpreted as the mean number density of subdiffusive particles. In Eq. (1), D µ(x) = a 2 /2τis the fractional diffusion coefficient with the microscopic time scale τ 0 , length scale a, and space-dependent fractional exponent µ(x) ∈ (0, 1). The Riemann-Liouville deriva- (2) also involves spatial dependence. Equation (1) was first derived by Chechkin, Gorenflo and Sokolov [12], and since then, many attempts have been made to find a solution through composite regions with constant anomalous exponents and numerically [12,14,25]. However, Eq. (1) remains unsolved for the general case of a spacedependent anomalous exponent µ(x).In thi...

Variable-order fractional master equation and clustering of particles: non-uniform lysosome distribution

“…To verify that the master equation (2.1) does indeed correspond to the space-dependent variableorder fractional diffusion equation in [27], we need to introduce the densitŷ This is equivalent to the solution found in [27], shown as (1.3) in this paper. Monte Carlo simulations for the master equation (2.1) were performed and shown to have excellent correspondence with this asymptotic solution as shown in figure 3.…”

“…The simulations were made by generating random residence times, T, for particles in box i drawn from the PDF ψμifalse(τfalse)=−false(normal∂/normal∂τfalse)Eμifalse[−false(τ/τ0false)μifalse] (details can be found in [30]) after which the particle jumps right or left with equal probability. The fractional exponent μ i was increasing linearly as i increased (full details can be found in [27]). In the next section, we show that this asymptotic solution (2.15) and (1.2) corresponds to the distribution of lysosomes in living cells.…”

“…Recently, we found the asymptotic representation of the solution of the variable-order fractional diffusion equation analytically with an ultra-slow spatial clustering (aggregation) of subdiffusive particles [27]. This equation is normal∂pfalse(x,tfalse)normal∂t=normal∂2normal∂x2false[Dμfalse(xfalse)scriptDt1−μfalse(xfalse)pfalse(x,tfalse)false], where pfalse(x,tfalse) is the PDF of a particle at position x and time t.…”

“…The Riemann–Liouville derivative is defined as scriptDt1−μfalse(xfalse)pfalse(x,tfalse)=1Γfalse(μfalse(xfalse)false)∂normal∂t∫0tpfalse(x,t′false)false(t−t′false)1−μfalse(xfalse)dt′ For a monotonically increasing fractional exponent with domain x∈false[0,Lfalse], the asymptotic solution to (1.1) is [27] pfalse(x,tfalse)∼μ0′false(t/τ0false)normalΔμfalse(xfalse)Γfalse(1−normalΔμfalse(xfalse)false)[ln(tτ0)−ψ0(1−Δμ(x))], where normalΔμfalse(xfalse)=μfalse(xfalse)−μfalse(...…”

“…Recently, we found the asymptotic representation of the solution of the variable-order fractional diffusion equation analytically with an ultra-slow spatial clustering (aggregation) of subdiffusive particles [27]. This equation is…”