Anomalous Transport of Heterogeneous Population and Time-Changed Pólya Process

Abstract

We propose a continuous-time unidirectional random walk model for a heterogeneous population of particles involving subdiffusive trapping effects. In this model, after escaping from the trap, each particle either jumps forward with a random probability or remains in the same place. The population heterogeneity is captured by modeling the jump probability as a beta-distributed random variable. The randomness in this transition parameter generates an effective jump probability with the ensemble self-reinforcement. We derive the limiting probability for the ensemble average of the particle position using an integral subordination formula. We show that the average particle position can be represented by a time-changed Pólya process involving an inverse stable subordinator.

1. Introduction

Recent studies demonstrated that anomalous transport in cellular and molecular biology is often heterogeneous in time and space [1]. As a result, a single anomalous exponent or generalised diffusion is insufficient to fully describe many anomalous transport phenomena. Various models have been developed to describe heterogeneous anomalous transport, including space-dependent variable order fractional subdiffusion [2,3,4,5,6] and space-dependent diffusivity [7,8]. A good example of heterogeneous anomalous transport is intracellular transport, which exhibits diverse non-Brownian behavior, including subdiffusion and Lévy walks [1,9]. Machine learning techniques have been used to analyze the heterogeneity of anomalous transport [10] (see also [11,12]). Another example of the impact of heterogeneity on transport in biological and soft matter systems is the linear growth of the mean-squared displacement, accompanied by a non-Gaussian distribution of increments [1,13,14]. A non-Gaussian distribution can be obtained by assuming random diffusion coefficients. The distribution of diffusivities can be obtained from the displacement distributions, as demonstrated for the diffusion of liposomes in nematic solutions of actin filaments [13].

The topic of random walks in heterogeneous, random environments has been covered extensively in the literature [15,16]. To describe random walks in heterogeneous disordered media, quenched and annealed models are widely used in physics and biology [1,15]. Anomalous diffusion may occur in quenched random media with a highly heterogeneous energy landscape with the random potential for which the waiting time for the jump or jump probability depend on the position of the random walker [15]. In annealed disorder, the random motion of particles in a heterogeneous environment can be described by time-dependent local diffusion coefficients [17] or a temporally fluctuating bias parameter [18].

Another approach to anomalous transport is based on population or ensemble heterogeneity [19]. In this approach, one can use a random parametrization of the stochastic force for the generalized Langevin Equation [20] or consider a heterogeneous ensemble of Brownian particles described by a distribution of diffusivities, rather than a single diffusion coefficient [21]. In cellular biology, populations of cells and organelles within cells are almost always heterogeneous [1]. Cells and organelles differ in size, shape, and behavior. In order to accurately quantify ’population-averaged assays’ [22] and properly identify the effects of subpopulations, stochastic models that account for static population heterogeneity have to be explored. As pointed out in [19], explaining experimental observables of intracellular transport dynamics requires both ensemble heterogeneity and time–space heterogeneity together.

In this paper, we deal with anomalous unidirectional transport of a heterogeneous population and derive an integral subordination formula for the averaged particle position probability. Specifically, we consider only ensemble heterogeneity where each member of the population has its own individual jump probability, and do not account for spatial and temporal random heterogeneity. It is well known [23,24] that in systems with anomalous transport, drift significantly influences the distribution and statistical properties of displacement, including ensemble and time variances. Recent studies have shown [25] that population heterogeneity, characterised by a distribution of jump probability (bias parameter), alters the fundamental nature of the fractional master equation. In particular, it has been shown that the variance can exhibit ballistic superdiffusion even as the fractional exponent approaches 1, in contrast to anomalous continuous-time random walk (CTRW) with a fixed bias parameter. Our research is motivated by the recent papers by Gavrilova et al. [26,27] who analysed the experimental trajectories of the unidirectional movement of dense core vesicles (DCVs) in the neurons of the C. elegans strains. DCVs are organelles in neurons responsible for storing neuropeptides, which neurons use to initiate and regulate complex behaviors and transport them along microtubules [28,29]. Gavrilova et al. [27] found that the dynein-driven motion of DCVs exhibits superdiffusive ballistic behavior with displacement variance $Var\left(X\left(t\right)\right)\sim t^2.$ It turns out that the distribution of DCV displacements fits a beta-binomial distribution, with the mean and variance following linear and quadratic growth patterns, respectively. They proposed a simple discrete space and time random walk model for the heterogeneous population to explain the observed superdiffusive ballistic retrograde transport behavior of DCV movement. In this paper, we propose a continuous in-time extension of the random walk model [27] by taking into account the subdiffusive trapping of DCVs that explain the frequent immobilization observed in experiments [26]. We introduce a continuous random walk model on a one-dimensional lattice, where after escaping from a trap, particles can either jump in one direction with a certain probability or remain at the same trap. For a heterogeneous population, each particle is characterized by its own jumping probability.

2. Random Walk Model for Anomalous Unidirectional Transport

We consider a continuous time random walk on the lattice for which a particle experiences trapping with heavy-tailed waiting times with infinite mean [30]. When the particle escapes from its current position x, it either jumps with probability q to the right at the point $x+a$ or stays in the same position x (zero jump) with the probability $1-q$. In what follows we consider the heterogeneous population of particles with random $q.$ Anomalous unidirectional movement of a particle on a lattice can be described by the fractional master equation [25,30],

$${\displaystyle \frac{\partial p}{\partial t}}=-q{\tau }_0^{-\mu }{\mathcal{D}}_t^{1-\mu }(p(x,t|q)-p(x-a,t|q)).$$

(1)

Here $p(x,t|q)$ is the conditional probability of finding the particle at position $x=ka$ with $k=0,1,2,\dots$ at time t for given q, ${\tau }_0$ is the time parameter, $0<\mu <1$, and ${\mathcal{D}}_t^{1-\mu }p$ is the Riemann–Liouville derivative

$${\mathcal{D}}_t^{1-\mu }p(x,t)={\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\displaystyle \frac{\partial }{\partial t}}{\int }_0^t{\displaystyle \frac{p(x,t^{\prime })}{{(t-t^{\prime })}^{1-\mu }}}d{t}^{\prime }.$$

(2)

To account for the population’s heterogeneity, we assume that q is a random variable with a beta distribution defined on the interval $[0,1]$:

$$f\left(q\right)={\displaystyle \frac{q^{\alpha -1}{(1-q)}^{\beta -1}}{B(\alpha ,\beta )}},$$

(3)

where $B(\alpha ,\beta )$ is the beta function. This distribution takes different geometrical forms depending on the values of parameters $\alpha$ and $\beta$ [31]. If, for example, $\alpha >\beta$, the beta distribution is skewed to the right. It corresponds to the case of a high chance of successful particle movement to the right. If $\alpha <\beta$, the beta distribution describes more failures in movement. Small values of $\alpha$ and $\beta$ correspond to a very broad distribution, explaining the high heterogeneity of the population in terms of their motility. Increasing values of these parameters leads to a narrow distribution with the mean value, ${\displaystyle \frac{\alpha }{\alpha +\beta }}$. This corresponds to a reduced heterogeneity of the population. In what follows, we consider the limit $\beta \to \infty$ with a finite value of the parameter $\alpha$ which corresponds to the very small probability of jumping to the right. Recently, a beta distribution (3) has been used for the analysis of heterogeneous nanoparticle internalization by cells, where cellular uptake has been represented as a compound Poisson process with a random probability of success [32].

It is convenient to introduce the non-dimensional time

$$\tau ={\displaystyle \frac{t}{{\tau }_0}}$$

(4)

and rewrite the master equation (1) in terms of the Caputo fractional derivative [33]

$${\displaystyle \frac{{\partial }^{\mu }p}{\partial {\tau }^{\mu }}}=-q(p(x,\tau |q)-p(x-a,\tau |q)).$$

(5)

Using subordination [30,31], we can write the solution to the master equation (5) as follows

$$p(x,\tau |q)=\sum _{n=0}^{\infty }P(x,n|q)Q_{\mu }(n,\tau ),$$

(6)

where $P(x,n|q)=\mathrm{Pr}\{X_n=x\}$ is the probability of the position of the underlying discrete time random walk $X_n$. It can be described by the difference equation $X_{n+1}=X_n+{\xi }_{n+1},$ where the random jump ${\xi }_n=a$ with probability q and ${\xi }_n=0$ with the probability $1-q.$ The probability $P(x,n|q)$ obeys the master equation

$$P(x,n+1|q)=qP(x-a,n\left|q\right)+(1-q)P(x,n|q).$$

The solution to this equation is well-known binomial distribution [31]

$$P(x,n|q)=\left({\displaystyle \genfrac{}{}{0pt}{}{n}{\frac{x}{a}}}\right)q^{{\displaystyle \frac{x}{a}}}{(1-q)}^{(n-{\displaystyle \frac{x}{a}})}.$$

(7)

The particle reaches the point x at time n if it makes $x/a$ positive jumps, and $n-x/a$ is the number of immobilization events (zero jumps).

The mass probability $Q_{\mu }(n,\tau )=\mathrm{Prob}\{N_{\mu }\left(\tau \right)=n\}$ for the fractional Poisson process $N_{\mu }\left(\tau \right)$ is given by [34,35,36]

$$Q_{\mu }(n,\tau )={\tau }^{n\mu }\sum _{k=0}^{\infty }{\displaystyle \frac{(k+n)!}{n!k!}}{\displaystyle \frac{{\left(-\tau \right)}^{k\mu }}{\Gamma \left(\mu \right(k+n)+1)}}.$$

(8)

The function $Q_{\mu }(n,\tau )$ gives the probability that in the time interval $[0,\tau ]$, the particle has n jumps, including zero jumps.

Let us define the average probability

$$\overline{p}(x,\tau )={\int }_0^{1}p(x,\tau |q)f\left(q\right)dq.$$

(9)

From (6) we get

$$\overline{p}(x,\tau )=\sum _{n=0}^{\infty }\overline{P}(x,n)Q_{\mu }(n,\tau ),$$

(10)

where $\overline{P}(x,n)={\int }_0^{1}P(x,n|q)f\left(q\right)dq$. It is well-known [37] that a mixture of binomial distribution with the parameter q having a beta distribution is a beta-binomial distribution:

$$\overline{P}(x,n)=\left({\displaystyle \genfrac{}{}{0pt}{}{n}{\frac{x}{a}}}\right){\displaystyle \frac{B\left(\frac{x}{a}+\alpha ,n-\frac{x}{a}+\beta \right)}{B(\alpha ,\beta )}}.$$

(11)

2.1. Underlying Discrete in Time Random Walk with Ensemble Self-Reinforcement

To obtain the master equation for $\overline{P}(x,n)$ we need to average the master equation for $P(x,n|q)$ by using beta distribution $f\left(q\right).$ One can find [25]

$$\overline{P}(x,n+1)=u_n(x-a)\overline{P}(x-a,n)+(1-u_n\left(x\right))\overline{P}(x,n),$$

(12)

where the transition probability $u_n\left(x\right)$ is

$$u_n\left(x\right)={\displaystyle \frac{{\int }_0^{1}qP(x,n|q)f\left(q\right)dq}{{\int }_0^{1}P(x,n|q)f\left(q\right)dq}}.$$

(13)

Using a binomial distribution for $P(x,n|q)$ we obtain

$$u_n\left(x\right)={\displaystyle \frac{\alpha +\frac{x}{a}}{\alpha +\beta +n}}.$$

(14)

The solution to the master equation (12) is the beta-binomial distribution (11) [37] (Pólya–Eggenberger distribution [38]).

It follows from (14) that the randomness of the parameter q generates effective transition probability, $u_n\left(x\right)$, which describes the ensemble self-reinforcement phenomenon [25]. One can think of the underlying random walk ${\overline{X}}_n$ for which the probability of the step in the positive direction (see (14)) increases as more steps in those directions are made in the past, which is known as self-reinforcement. It was demonstrated in [25] that there exists a link between the random walk model described by (12) with (14) and random walks with transition probabilities dependent on the entire history of its past (strong memory). Feller discussed this phenomenon [31,39], where population inhomogeneity can create an apparent self-reinforcing effect (contagion) by emphasizing that a good fit of the Pólya–Eggenberger distribution to data does not necessarily imply an underlying contagion process.

2.2. Comparison of the CTRW Model with Fixed Jump Probability and the Heterogeneous Model

The aim of this subsection is to compare the classical CTRW model with a fixed jump probability q to a heterogeneous model in which q is a random variable with a beta distribution. We analyse the variance in both models and discuss the fundamental differences between them. The main reason for this difference is that the variance of the underlying discrete-time random walk $X_n$ for a fixed q is a linear function of the number of steps $n,$ whereas in heterogeneous model, the variance is quadratic [27]. It follows from (7) that

$$Var\left(X_n\right)=q(1-q)a^{2}n,$$

(15)

while for (11) one can obtain

$$Var\left({\overline{X}}_n\right)={\displaystyle \frac{\alpha \beta a^2}{(\alpha +\beta )(1+\alpha +\beta )}}n+{\displaystyle \frac{\alpha \beta a^2}{{(\alpha +\beta )}^2(1+\alpha +\beta )}}{n}^2.$$

(16)

Now we compare the variances for the continuous case. Let $X_q\left(\tau \right)$ be the position of the particle at time $\tau$ for CTRW with fixed q. Then

$$\mathbb{E}\left(X_q\left(\tau \right)\right)=qa\mathbb{E}\left(N_{\mu }\left(\tau \right)\right),\mathbb{E}\left(X_q^2\left(\tau \right)\right)=q(1-q)a^2\mathbb{E}\left(N_{\mu }\left(\tau \right)\right)+q^2{a}^2\mathbb{E}\left(N_{\mu }^2\left(\tau \right)\right).$$

(17)

Using the well-known first and second moments of the fractional Poisson process [34], we compute the variance:

$$Var\left(X_q\left(\tau \right)\right)=q{a}^2{\displaystyle \frac{{\tau }^{\mu }}{\Gamma (\mu +1)}}+q^2{a}^{2}A\left(\mu \right){\tau }^{2\mu },$$

(18)

where

$$A\left(\mu \right)={\displaystyle \frac{2}{\Gamma (2\mu +1)}}-{\displaystyle \frac{1}{{\Gamma }^2(\mu +1)}}.$$

(19)

When the anomalous exponent $\mu \to 1$, the second superdiffusive term in (18) disappears because $A\left(1\right)=0$.

Let us now show that, in contrast to the CTRW model with fixed q, the variance in the heterogeneous model becomes ballistic and superdiffusive as $\mu \to 1.$ By using the subordination formula (10) and the variance (16), we obtain

$$Var\left(\overline{X}\left(\tau \right)\right)={\displaystyle \frac{\alpha a^2}{(\alpha +\beta )}}{\displaystyle \frac{{\tau }^{\mu }}{\Gamma (\mu +1)}}+{\displaystyle \frac{2\alpha \beta a^2}{{(\alpha +\beta )}^2(1+\alpha +\beta )}}{\displaystyle \frac{{\tau }^{2\mu }}{\Gamma (2\mu +1)}}+{\displaystyle \frac{{\alpha }^2{a}^2}{{(\alpha +\beta )}^2}}A\left(\mu \right){\tau }^{2\mu }.$$

(20)

It follows from (20) that for $\mu =1$ the last term becomes zero while the second term describes the ballistic superdiffusion. Therefore, when the jump probability q is random, we cannot replace q with its average value, as the variance behaves differently in the two models.

3. Integral Subordination Formula and Time-Changed Pólya Process

The purpose of this section is to derive the subordination integral for the average mass probability function of the particle positions for a heterogeneous population. We make long-time scaling

$$\tau \to {\displaystyle \frac{\tau }{ϵ}},$$

(21)

where $ϵ$ is a small positive parameter. Using subordination formula (10) we represent rescaled mass probability

$${\overline{p}}^{ϵ}(x,\tau )=\overline{p}\left(x,{\displaystyle \frac{\tau }{ϵ}}\right)$$

(22)

as follows

$${\overline{p}}^{ϵ}(x,\tau )=\sum _{n=0}^{\infty }\overline{P}(x,n)Q_{\mu }^{ϵ}(n,\tau ).$$

(23)

Here $\overline{P}(x,n)$ is the beta-binomial distribution defined in (11) and $Q_{\mu }^{ϵ}(n,t)=Q_{\mu }(n,t/ϵ).$

Our main purpose is to find the limiting probability

$$g(x,\tau )=\underset{ϵ\to 0}{lim}{\overline{p}}^{ϵ}(x,\tau )$$

(24)

and represent it in the integral subordination form by using the operation time u as [40,41]

$$g(x,\tau )={\int }_0^{\infty }{\overline{P}}_u(x,u)Q_u(u,\tau )du.$$

(25)

In what follows, we show that the limiting mass probability $g(x,\tau )$ describes a time-changed Pólya process $X\left(E\right(\tau \left)\right)$, where $X\left(u\right)$ is the Pólya process [31] and $E\left(\tau \right)$ is the inverse stable subordinator [41]. The probability density function $Q_u(u,\tau )$ is the density of the inverse stable subordinator $E\left(\tau \right)$ and $\overline{P_u}(x,u)$ is the probability mass function for the Pólya process:

$${\overline{P}}_u(x,u)={\displaystyle \frac{\Gamma (\alpha +\frac{x}{a})}{\Gamma (1+\frac{x}{a})\Gamma \left(\alpha \right)}}{\left({\displaystyle \frac{u}{1+u}}\right)}^{{\displaystyle \frac{x}{a}}}{\left({\displaystyle \frac{1}{1+u}}\right)}^{\alpha }$$

(26)

for $x=0,a,2a,\dots$. The Pólya process can be derived as the continuous limit of the Pólya urn model (see Feller’s book [31] for details).

To obtain the integral subordination Formula (25), we need to represent (23) as a Riemann sum in the limit $ϵ\to 0$. Taking the Laplace transform of (23), we get

$${\widehat{\overline{p}}}^{ϵ}(x,s)=\sum _{n=0}^{\infty }\overline{P}(x,n){\widehat{Q}}_{\mu }^{ϵ}(n,s).$$

(27)

Here

$${\widehat{Q}}_{\mu }^{ϵ}(n,s)={\widehat{\psi }}_{\mu }^n\left(ϵs\right){\displaystyle \frac{1-{\widehat{\psi }}_{\mu }\left(ϵs\right)}{s}},$$

(28)

where ${\widehat{\psi }}_{\mu }\left(s\right)$ is the Laplace transform of the waiting time probability density function ${\psi }_{\mu }\left(\tau \right)$ [30]. This function corresponds to the survival probability ${\Psi }_{\mu }\left(\tau \right)$ given by the Mittag–Leffler function $E_{\mu }$: ${\Psi }_{\mu }\left(\tau \right)=E_{\mu }\left[-{\tau }^{\mu }\right].$ In the limit $ϵ\to 0$, one can write

$${\widehat{\psi }}_{\mu }^n\left(ϵs\right)={\displaystyle \frac{1}{{(1+{\left(sϵ\right)}^{\mu })}^n}}\sim e^{-n{\left(sϵ\right)}^{\mu }}.$$

(29)

Therefore

$${\widehat{Q}}_{\mu }^{ϵ}(n,s)\sim {ϵ}^{\mu }{s}^{\mu -1}{e}^{-n{\left(sϵ\right)}^{\mu }}.$$

(30)

This formula allows us to obtain the Laplace transform of the density $Q_u(u,\tau )$ in the integral subordination formula (see (25)). We introduce the dimensionless operational time

$$u=n{ϵ}^{\mu }$$

(31)

related to the index n in the subordination formula (6). Then

$${\widehat{Q}}_u(u,s)=\underset{ϵ\to 0}{lim}{\displaystyle \frac{{\widehat{Q}}_{\mu }^{ϵ}(n,s)}{{ϵ}^{\mu }}}=s^{\mu -1}{e}^{-u{s}^{\mu }}.$$

(32)

The inverse Laplace transform of ${\widehat{Q}}_u(u,s)$ is the the density of the inverse stable subordinator $Q_u(u,\tau ).$ It obeys the fractional drift equation [41,42]

$${\displaystyle \frac{{\partial }^{\mu }{Q}_u(u,\tau )}{\partial {\tau }^{\mu }}}=-{\displaystyle \frac{\partial Q_u(u,\tau )}{\partial u}},Q_u(u,0)=\delta \left(u\right),$$

(33)

where ${\displaystyle \frac{{\partial }^{\mu }{Q}_u}{\partial {\tau }^{\mu }}}$ is the Caputo fractional derivative [33].

To obtain ${\overline{P}}_u(x,u)$ in (26), we need to find the continuous-time limit of the beta-binomial distribution (11). Let us rewrite it in another form:

$$\overline{P}(x,n)=\left({\displaystyle \genfrac{}{}{0pt}{}{n}{\frac{x}{a}}}\right){\displaystyle \frac{\Gamma (\frac{x}{a}+\alpha )\Gamma (n-\frac{x}{a}+\beta )\Gamma (\alpha +\beta )}{\Gamma (n+\alpha +\beta )\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}}.$$

(34)

Taking the limits $n\to \infty$, $\beta \to \infty$ and using the formula $\Gamma (\alpha +\beta )\approx \Gamma \left(\beta \right){\beta }^{\alpha }$ as $\beta \to \infty$, we get

$$\overline{P}(x,n)\sim {\displaystyle \frac{\Gamma (\alpha +\frac{x}{a})}{\Gamma (1+\frac{x}{a})\Gamma \left(\alpha \right)}}{\left({\displaystyle \frac{n}{n+\beta }}\right)}^{{\displaystyle \frac{x}{a}}}{\left({\displaystyle \frac{\beta }{n+\beta }}\right)}^{\alpha }.$$

(35)

To be consistent with the limit (32), we set

$$\beta ={\displaystyle \frac{1}{{ϵ}^{\mu }}}.$$

(36)

By using (36) and the non-dimensional operational time u defined in (31), we obtain from (35) the limiting distribution (26).

With the aid of (30), (35), and (36), we write the subordination formula (27) as a Riemann sum

$${\widehat{\overline{p}}}^{ϵ}(x,s)\sim \sum _{n=0}^{\infty }{\displaystyle \frac{\Gamma (\alpha +\frac{x}{a})}{\Gamma (1+\frac{x}{a})\Gamma \left(\alpha \right)}}{\left({\displaystyle \frac{n{ϵ}^{\mu }}{1+n{ϵ}^{\mu }}}\right)}^{{\displaystyle \frac{x}{a}}}{\left({\displaystyle \frac{1}{1+n{ϵ}^{\mu }}}\right)}^{\alpha }{s}^{\mu -1}{e}^{-n{\left(sϵ\right)}^{\mu }}{ϵ}^{\mu }$$

(37)

as $ϵ\to 0.$ By using operational time $u=n{ϵ}^{\mu }$, the distribution ${\overline{P}}_u(x,u)$ (26), and taking the limit $ϵ\to 0$, we obtain

$$\widehat{g}(x,s)={\int }_0^{\infty }{\overline{P}}_u(x,u){\widehat{Q}}_u(u,s)du.$$

(38)

The inverse Laplace transform of (38) gives the subordination integral (25).

As an example, let us obtain the subordination integral (25) for the continuous-time random walk (CTRW) described by the fractional master equation (5) with fixed probability $q.$ To be consistent with (32), we set

$$q={ϵ}^{\mu }.$$

(39)

In the limit $ϵ\to 0$ and $n\to \infty$ such that $n{ϵ}^{\mu }$ is finite, the binomial distribution (7) can be written as

$$P(x,n)\sim {\displaystyle \frac{{\left(n{ϵ}^{\mu }\right)}^{\frac{x}{a}}{e}^{-n{ϵ}^{\mu }}}{\left(\frac{x}{a}\right)!}}.$$

(40)

The Laplace transform of rescaled probability $p^{ϵ}(x,\tau )=p\left(x,{\displaystyle \frac{\tau }{ϵ}}\right)$ can be written as a Riemann sum

$${\widehat{p}}^{ϵ}(x,s)\sim \sum _{n=0}^{\infty }{\displaystyle \frac{{\left(n{ϵ}^{\mu }\right)}^{\frac{x}{a}}{e}^{-n{ϵ}^{\mu }}}{\left(\frac{x}{a}\right)!}}{s}^{\mu -1}{e}^{-n{\left(sϵ\right)}^{\mu }}{ϵ}^{\mu }$$

(41)

as $ϵ\to 0.$ Finally, we obtain the limiting probability $g(x,\tau )={lim}_{ϵ\to 0}{p}^{ϵ}(x,\tau )$ as

$$g(x,\tau )={\int }_0^{\infty }{\displaystyle \frac{u^{\frac{x}{a}}{e}^{-u}}{\left(\frac{x}{a}\right)!}}{Q}_u(u,\tau )du$$

(42)

for $x=0,a,2a,...$ One can recognise this function as the mass probability for a time-changed Poisson process $N\left(E\right(\tau \left)\right).$ This probability obeys the fractional equation

$${\displaystyle \frac{{\partial }^{\mu }g}{\partial {\tau }^{\mu }}}=-g(x,\tau )+g(x-a,\tau ).$$

(43)

Thus, we obtain the well-known result [43] that the fractional Poisson process can be derived from the Poisson process by applying a time change with the inverse stable subordinator $E\left(\tau \right).$ In this case, as the anomalous exponent $\mu$ tends to one, the variance becomes a linear function of time, in contrast to the behavior described by (25).

3.1. Underlying Pólya Process

It is instructive to find the relationship between the discrete-time master equation (12) and its continuous-time counterpart. It turned out that the position of particle $X\left(u\right)$ at time u is the well-known Pólya process [31]. Let us divide the time interval $[0,u]$ into n sub-intervals $h=u/n$ and consider the limit $n\to \infty$ such that

$$\beta ={\displaystyle \frac{n}{u}}.$$

(44)

It follows from (14) that the conditional probability of jumping from point x to $x+a$ in infinitesimal time h is

$$\mathrm{Pr}(X(u+h)=x+a|X\left(u\right)=x)={\displaystyle \frac{\alpha +\frac{x}{a}}{1+u}}h+o\left(h\right),$$

(45)

so the transition rate T depends on space and time variables as

$$T(x,u)={\displaystyle \frac{\alpha +\frac{x}{a}}{1+u}}.$$

(46)

The master equation (12) in continuous limit takes the form

$${\displaystyle \frac{\partial {\overline{P}}_u(x,u)}{\partial u}}=-{\displaystyle \frac{\alpha +\frac{x}{a}}{1+u}}{\overline{P}}_u(x,u)+{\displaystyle \frac{\alpha +\frac{x-a}{a}}{1+u}}{\overline{P}}_u(x-a,u).$$

(47)

The solution of this equation can be written in the form of a negative binomial distribution (26).

3.2. First Moment and Variance

The first and second moments of the position of the particle described by the Pólya process $X\left(u\right)$ can be written as follows [31]

$$\mathbb{E}\left(X\left(u\right)\right)=\alpha au,\mathbb{E}\left(X^2\left(u\right)\right)=\alpha a^{2}u+\alpha (\alpha +1)a^2{u}^2.$$

(48)

The variance $Var\left(X\left(u\right)\right)=\alpha a^{2}u(1+u).$

It follows from the subordination integral (25) that a time-changed Pólya process $X\left(E\right(\tau \left)\right)$ with the inverse stable subordinator $E\left(\tau \right)$ has the following first and second moments:

$$\mathbb{E}\left(X\left(E\left(\tau \right)\right)\right)=\alpha a\mathbb{E}\left(E\left(\tau \right)\right),\mathbb{E}\left(X^2\left(E\left(\tau \right)\right)\right)=\alpha a^2\mathbb{E}\left(E\left(\tau \right)\right)+\alpha (\alpha +1)a^2\mathbb{E}\left(E^2\left(\tau \right)\right),$$

(49)

where

$$\mathbb{E}\left(E\left(\tau \right)\right)={\displaystyle \frac{{\tau }^{\mu }}{\Gamma (\mu +1)}},\mathbb{E}\left(E^2\left(\tau \right)\right)={\displaystyle \frac{2{\tau }^{2\mu }}{\Gamma (2\mu +1)}}.$$

(50)

For a particular case $\mu =1$,

$$Var\left(X\left(E\left(\tau \right)\right)\right)=\alpha a^2\tau +\alpha a^2{\tau }^2$$

(51)

that corresponds to ballistic superdiffusive movement.

3.3. Governing Equation for the Limiting Probability

Although we have an explicit formula for the limiting probability $g(x,\tau )$ in terms of the subordination integral (25), we cannot obtain a closed equation for $g(x,\tau ).$ The main reason is that the transition rate (46) for the Pólya process depends on both space and time variables. To show the problem of obtaining the closed-form equation for g we now multiply both sides of Equation (33) by ${\overline{P}}_u(x,u)$ and integrate from 0 to ∞ with respect to u. One can obtain

$${\displaystyle \frac{{\partial }^{\mu }g(x,\tau )}{\partial {\tau }^{\mu }}}=-{\int }_0^{\infty }{\overline{P}}_u(x,u){\displaystyle \frac{\partial Q_u(u,t)}{\partial u}}du.$$

(52)

Integration by parts together with (47) gives

$${\displaystyle \frac{{\partial }^{\mu }g(x,\tau )}{\partial {\tau }^{\mu }}}={\int }_0^{\infty }\left[-{\displaystyle \frac{\alpha +\frac{x}{a}}{1+u}}{\overline{P}}_u(x,u)+{\displaystyle \frac{\alpha +\frac{x-a}{a}}{1+u}}{\overline{P}}_u(x-a,u)\right]Q_u(u,\tau )du.$$

It is clear that we cannot obtain a closed equation for $g(x,\tau ).$ It can be achieved only for the case $\mu =1$ that corresponds to the standard Poisson process $N_1(\tau$) with $\mu =1$ and $\tau =\lambda t.$ The drift Equation (33) takes the form

$${\displaystyle \frac{\partial Q_u(u,\tau )}{\partial \tau }}=-{\displaystyle \frac{\partial Q_u(u,\tau )}{\partial u}}.$$

(53)

It has a solution $Q_u(u,\tau )=\delta (u-\tau ).$ Then $g(x,\tau )$ obeys the master equation (47) and has a form of negative binomial distribution (26). It is interesting that for the case $\mu =1$ the probability of a particle remaining at the position $x=0$ follows a power-law distribution

$$g(0,\tau )={\left({\displaystyle \frac{1}{1+\tau }}\right)}^{\alpha }.$$

(54)

In the long-time limit $\tau \to \infty$, $g(x,\tau )$ takes the form of power-law distribution

$$g(x,\tau )\sim {\displaystyle \frac{\Gamma (\alpha +\frac{x}{a})}{\Gamma (1+\frac{x}{a})\Gamma \left(\alpha \right)}}{\tau }^{-\alpha }.$$

(55)

3.4. The Probability of First Arrival to the Given Point

Gavrilova et al. [27] analyzed the time distribution for dense core vesicles to reach a certain threshold for the first time. They found that a beta-negative binomial distribution fits the empirical data well.

Let us find the continuous-time probability density function $\overline{f}(\tau ,z)$ for the first arrival to the point z when $\mu =1.$ It can be written as

$$\overline{f}(\tau ,z)=g(z-1,\tau )T(z-1,\tau ),$$

(56)

where the transition rate $T(z,\tau )$ is defined in (46). By using the negative binomial distribution (26) for $g(z,\tau )$, we obtain

$$\overline{f}(\tau ,z)={\displaystyle \frac{\Gamma (\alpha +\frac{z}{a})}{\Gamma \left(\frac{z}{a}\right)\Gamma \left(\alpha \right)}}{\left({\displaystyle \frac{\tau }{1+\tau }}\right)}^{{\displaystyle \frac{z}{a}}-1}{\left({\displaystyle \frac{1}{1+\tau }}\right)}^{\alpha +1}.$$

(57)

For large $\tau$

$$\overline{f}(\tau ,z)\sim {\displaystyle \frac{\Gamma (\alpha +\frac{z}{a})}{\Gamma \left(\frac{z}{a}\right)\Gamma \left(\alpha \right)}}{\displaystyle \frac{1}{{\tau }^{1+\alpha }}}.$$

(58)

As follows from above formula, the mean of the first arrival time does not exist for $\alpha \le 1.$ The reason for the infinite mean first arrival time is as follows. It is well-known that when the beta distribution parameters satisfy $\alpha \le 1$ and $\beta >>1$, the probability density is concentrated near zero. This implies that, in the context of particle motion, most particles are effectively “trapped”. As a result, the average time it takes for the particles in a heterogeneous population to reach a certain point z diverges, since a significant fraction of particles do not move.

4. Summary

Heterogeneity is always present in any cell population, and the ensemble behavior of the population can differ significantly from that of individual cells [22]. Inside cells, organelles move as heterogeneous populations, in such a way that their motion varies significantly in speed, rest durations, etc. [1]. One of the key factors influencing organelle motility is size. Larger organelles move more slowly due to drag forces in the crowded environment of the cells. However, size is not the only parameter that contributes to the heterogeneity of movement. The other factors are local viscosity, energy (ATP) and microtubules availability, etc. The question is how these important factors can be implemented into the random walk models. An indirect method of taking into account these factors is to tune parameters $\alpha$ and $\beta$ of the beta distribution. For example, the influence of the size of organelles can be implemented as follows. The parameter $\alpha$ that represents the level of success in organelle movement should be greater for smaller organelles.

Motivated by the recent experimental analysis of the movement of dense core vesicles in C. elegans neurons by Gavrilova et al. [26,27], we introduce a continuous-time unidirectional random walk model for a heterogeneous population of particles with subdiffusive trapping effects. To model the heterogeneity in population motility, we assume a random jump probability governed by a beta distribution. This randomness in the jump parameter leads to an effective self-reinforcement, leading to superdiffusive motion at the ensemble level. After time rescaling, we derived the limiting probability for the ensemble average of the particle position, which is described by a time-changed Pólya process involving the inverse stable subordinator. We should also mention that the random walk with a temporally fluctuating bias parameter [18] is fundamentally different from our model for a heterogeneous population, as it does not, to our knowledge, lead to ensemble self-reinforcement.

It has been found [27] that a simple discrete random walk model accurately captures the empirical ballistic superdiffusive behavior of the displacement variance over the time span of $1.5$ to $5.5$ s. Additional experiments with longer observation durations of DCB movement are needed to validate our model with anomalous rest states. Further developments might include estimating the parameters $\alpha$ and $\beta$ of the beta distribution from the analysis of DSB trajectories, similar to how diffusivity distributions were obtained in [13]. As a further application, it would be interesting to employ our random walk models for the analysis of cell population movement, given that cellular heterogeneity is a widespread phenomenon [22].