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2022Apparent anomalous diffusion and non-Gaussian distributions in a simple mobile–immobile transport model with Poissonian switchingJ. R. Soc. Interface.1920220233http://doi.org/10.1098/rsif.2022.0233

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Apparent anomalous diffusion and non-Gaussian distributions in a simple mobile–immobile transport model with Poissonian switching

Timo J. Doerries

Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany

Contribution: Formal analysis, Investigation, Methodology, Validation, Visualization, Writing – original draft, Writing – review & editing

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Aleksei V. Chechkin

Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany

Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wrocław University of Science and Technology, Wyspianskiego 27, 50-370 Wrocław, Poland

Akhiezer Institute for Theoretical Physics, National Science Center ‘Kharkiv Institute of Physics and Technology’, 61108 Kharkiv, Ukraine

Contribution: Conceptualization, Formal analysis, Investigation, Supervision, Writing – review & editing

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Ralf Metzler

http://orcid.org/0000-0002-6013-7020

Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany

[email protected]

Contribution: Conceptualization, Funding acquisition, Project administration, Supervision, Validation, Writing – original draft, Writing – review & editing

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Timo J. Doerries

Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany

Contribution: Formal analysis, Investigation, Methodology, Validation, Visualization, Writing – original draft, Writing – review & editing

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Aleksei V. Chechkin

Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany

Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center, Wrocław University of Science and Technology, Wyspianskiego 27, 50-370 Wrocław, Poland

Akhiezer Institute for Theoretical Physics, National Science Center ‘Kharkiv Institute of Physics and Technology’, 61108 Kharkiv, Ukraine

Contribution: Conceptualization, Formal analysis, Investigation, Supervision, Writing – review & editing

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Ralf Metzler

http://orcid.org/0000-0002-6013-7020

Institute of Physics and Astronomy, University of Potsdam, 14476 Potsdam, Germany

[email protected]

Contribution: Conceptualization, Funding acquisition, Project administration, Supervision, Validation, Writing – original draft, Writing – review & editing

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Published:06 July 2022https://doi.org/10.1098/rsif.2022.0233

Abstract

We analyse mobile–immobile transport of particles that switch between the mobile and immobile phases with finite rates. Despite this seemingly simple assumption of Poissonian switching, we unveil a rich transport dynamics including significant transient anomalous diffusion and non-Gaussian displacement distributions. Our discussion is based on experimental parameters for tau proteins in neuronal cells, but the results obtained here are expected to be of relevance for a broad class of processes in complex systems. Specifically, we obtain that, when the mean binding time is significantly longer than the mean mobile time, transient anomalous diffusion is observed at short and intermediate time scales, with a strong dependence on the fraction of initially mobile and immobile particles. We unveil a Laplace distribution of particle displacements at relevant intermediate time scales. For any initial fraction of mobile particles, the respective mean squared displacement (MSD) displays a plateau. Moreover, we demonstrate a short-time cubic time dependence of the MSD for immobile tracers when initially all particles are immobile.

1. Introduction

Already in the 1960s, there was considerable interest in the transport of chemical tracers, especially pesticides, nitrates and heavy metals through water-carrying layers of soil [1]. A typical description for such contaminant transport was the diffusion–advection equation (sometimes called the convective–dispersive equation) [2]

\frac{\partial}{\partial t} C (x, t) = D \frac{\partial^{2}}{\partial x^{2}} C (x, t) - v \frac{\partial}{\partial x} C (x, t),

1.1

where C(x, t) is the contaminant concentration at distance x after time t, v is an advection velocity chosen as zero in the following and D is the diffusion constant (dispersion coefficient typically measured in units of cm² d⁻¹). Measurements revealed, however, that not all of the contaminant concentration was mobile at any given time, but that a fraction could be (transiently) trapped in stagnant volumes. Building on earlier models by Deans [3] and Coats & Smith [4], van Genuchten & Wierenga [5] analysed the exchange between mobile (C_m(x, t)) and immobile (C_im(x, t)) fractions. The mobile–immobile model under advection–diffusion conditions has been investigated intensively in the literature over the last 50 years [3,4,6–8]. Application of the mobile–immobile model includes the advective–diffusive transport of contaminants in the mobile domain of porous media [3,4,6–8], electrons in photoconductors [9], chemicals in chromatography [10] and adsorbing solutes in soil [11]. In many geophysical systems equations of the type (1.1) are modified to account for anomalous transport, in which molecular transport no longer follows the linear time dependence 〈Δx²(t)〉 = 〈x²(t)〉 − 〈x(t)〉² = 2Dt of Brownian motion, but follows laws of the type 〈Δx²(t)〉 = 2D_αt^α, for which α ≠ 1 [12]. Indeed, such transport anomalies were found on large field experiments, up to kilometre scales [13,14]. In such systems, the mobile–immobile transport model is replaced by models in which generalized transport terms are incorporated [15,16]. This type of model, in contrast to equation (1.1), is characterized by non-Gaussian distributions [12].

Motivated by concrete biological examples, we here study a seemingly simple version of the mobile–immobile transport model, in which particles switch between a freely diffusive phase and an immobile, stagnant phase. Even for the Poissonian switching dynamics considered here between the mobile and immobile phases and for biologically relevant parameters, we demonstrate the existence of a significant, transient anomalous–diffusive regime with a distinct non-Gaussian displacement distribution.

In fact, various components of biological cells, including tau proteins, synaptic vesicles in hippocampal neurons, glucocorticord receptors, calcium-sensing proteins and transcription factors at the junction of the endoplasmic reticulum and the plasma membrane, undergo diffusion with transient immobilization [17–24]. Another example is given by the diffusion and target search of DNA-binding proteins along DNA molecules. For instance, the core domain of the tumour suppressor p53, the damage detection complex Rad4-Rad23 and the architectural DNA-binding protein Fis repeatedly attach to and detach from the DNA during the target search [25–29]. We here focus on tau proteins, which transiently bind to microtubules in axons of neuronal cells and are immobilized in the bound state, as schematically depicted in figure 1. Tau proteins stabilize microtubules that give structure to cells [30]. Alzheimer’s disease is associated with tau proteins losing the ability to bind to microtubules [30,31]. This effectively destabilizes the microtubules and leads to neurodegeneration [30,31]. Owing to the extremely elongated shape of the axon, the motion of tau proteins can be effectively described in one dimension [17]. If the immobilization time follows an exponential distribution with mean τ_im and tracers immobilize with rate $τ_{m}^{- 1}$ , i.e. a Poissonian dynamics, as assumed in [17], the motion can be described by the mobile–immobile model

\begin{matrix} \frac{\partial}{\partial t} n_{m} (x, t) = - \frac{1}{τ_{m}} n_{m} (x, t) + \frac{1}{τ_{im}} n_{im} (x, t) + D \frac{\partial^{2}}{\partial x^{2}} n_{m} (x, t) \\ and & \frac{\partial}{\partial t} n_{im} (x, t) = - \frac{1}{τ_{im}} n_{im} (x, t) + \frac{1}{τ_{m}} n_{m} (x, t) . \end{matrix}}

1.2

Here, n_m(x, t) and n_im(x, t) denote the line densities of mobile and bound tau proteins, respectively, with physical dimension [1/length]. The diffusion coefficient of the mobile tracers is D. Since we are dealing with a system of non-interacting particles, we use a probabilistic formulation according to which the total concentration n_tot(x, t) = n_m(x, t) + n_im(x, t) is normalized to unity,

\int_{- \infty}^{\infty} n_{tot} (x, t) d x = 1

. The line densities n_m(x, t) and n_im(x, t) are then the respective fractions. Equations (1.2) were analysed in three dimensions for an equilibrium fraction of initially mobile tracers, finding Fickian yet non-Gaussian diffusion [32]. Accordingly, the mean squared displacement (MSD) of the total concentration n_tot grows linearly at all times, and under certain conditions a non-Gaussian distribution emerges [32].

Figure 1. Schematic of tau protein dynamics in axons of neuronal cells. Diffusing tau proteins bind to longitudinally aligned microtubules inside the axon with the rate $τ_{m}^{- 1}$ . Upon binding, they remain immobile for the average duration τ_im and unbind with the rate $τ_{im}^{- 1}$ . The green markers represent fluorescent proteins attached to the tau proteins. Because of the elongated shape of the axons, the tau protein dynamic can effectively be described in one dimension. In our model, we assume a homogeneous binding site density.

Such Fickian yet non-Gaussian diffusion has been shown to occur for the motion of colloidal beads on phospholipid bilayer tubes, molecules at surfaces and colloids in a dense matrix of micropillars, where the colloids can get trapped in pockets [33–35]. Fickian yet non-Gaussian diffusion with a finite correlation time beyond which the displacement probability density function (PDF) crosses over to a Gaussian with an effective diffusivity arises in diffusing–diffusivity models, in which the diffusivity of individual tracers varies stochastically over time [36–41]. Direct examples for such randomly evolving diffusion coefficients (mobilities) are indeed known from lipids in protein-crowded bilayer membranes [42], shape-shifting protein molecules [43] or (de)polymerizing oligomer chains [44,45]. In other systems, an intermittent plateau emerges in the MSD; for instance, for two-dimensional fluids confined in a random matrix of obstacles or a porous cavity, in which trapping in finite pockets plays a key role [46–48]. We also mention plateaus in the MSD of both two- and three-dimensional isotropic Lennard-Jones binary liquids [49]. In most of the systems mentioned here, the PDF crosses over from an exponential (Laplace) PDF to a Gaussian. In the following, we explicitly show how a Laplace distribution with fixed scale parameters arises at intermediate time scales in our mobile–immobile model, paired with transient anomalous diffusion.

In what follows, we consider three initial conditions: an equilibrium fraction of mobile tracers and a scenario in which initially all tracers are mobile or immobile. These experimentally feasible situations significantly change the diffusion at short and intermediate time scales, at which apparent anomalous diffusion arises with slow-down and plateau-like behaviour, or ballistic diffusion, respectively. Together with the transient non-Gaussian displacement PDF, this behaviour is remarkably rich, given the simplicity of the governing equation (1.2). We individually analyse the motion of the mobile and immobile population of tracers, made possible by the formulation of separate densities for mobile and immobile particles in this modelling approach. One physical incentive to do so is that the function of the tau proteins depends on their binding state [30]. Only bound tau proteins stabilize microtubules, or transcription factors modulate gene expression when bound to the DNA [21,30]. In some situations, only the mobile or immobile tracers can be measured. An example is given by combining total internal reflection fluorescence microscopy with fluorescently labelled single-stranded DNA, which binds to the microscope coverslip [50].

We present general results for the mobile and immobile concentrations and the MSD for arbitrary fractions of initially mobile tracers in §2. Sections 3–5 present concrete results and detailed discussions for different fractions of initial mobile particle concentrations; respectively, we start with the cases when all tracers are initially mobile and immobile and commence with an equilibrium fraction of mobile tracers. We conclude in §6.

2. Model and general solutions

We consider the mobile–immobile model equations (1.2) for the initial conditions n_m(x, 0) = f_mδ(x) and n_im(x, 0) = f_imδ(x), where f_m and f_im denote the fractions of initially mobile and immobile tracers, respectively, with the normalization f_m + f_im = 1. This formulation is suitable for typical single-particle tracking experiments used in biological and soft matter systems. They are also relevant for geophysical experiments, in which point-like injection of tracers is used. In this section, we keep the fractions f_m and f_im arbitrary and choose specific values in the following three sections.

In what follows, we use the concrete parameters D = 13.9 (μm)² s⁻¹, $τ_{m} = 0.16 s$ and $τ_{im} = 7.7 s$ from [17] in all figures and neglect the vanishingly small advection velocity v = 0.002 μm s⁻¹.¹ The values were obtained from experiments using the fluorescent decay after the photoactivation technique [17]. Let us briefly address the experimental origin of the time-scale separation between τ_m and τ_im. From single-particle tracking experiments of single-stranded DNA or tau proteins, immobilization times during the particle motion can be extracted [18,50]. The experiments for the tau proteins in [18] provided two-dimensional information and revealed relatively short residence times of the tau proteins on the microtubules, when compared with mobile times [18]. By contrast, the fluorescence decay after photoactivation (FDAP) experiment in one dimension along the axon direction, here denoted as the x variable, revealed long residence times and short mobile periods: τ_im ≈ 48τ_m [17]. This seeming contradiction can be resolved when examining more closely the two-dimensional trajectories in the electronic supplementary material of [18]. Namely, the microtubules inside the axon are aligned in parallel with the axon axis, as also shown in figure 1. While a single binding event is short, an unbound particle quickly rebinds to a parallel, nearby microtubule after a short distance covered by diffusion perpendicular to the axon axis. This perpendicular motion does not contribute to the one-dimensional motion in the x-direction and thus, while individual binding times are relatively short, effective binding times appear much longer in the projection to one dimension. Since we are only interested in the one-dimensional motion, we use the parameters of [17] and hence long immobilization times.

2.1. Mobile and immobile concentration profiles

We consider the Fourier–Laplace transform of the concentrations and solve for n_m(k, s), n_im(k, s) and n_tot(k, s) in expressions (A 1) and (A 2), in which the Fourier wavenumber k corresponds to the distance x in real space and the Laplace variable s is conjugated to time t; see appendix A for details. We denote functions in Fourier or Laplace space solely by replacing the explicit dependencies on the respective arguments. The relations in the Fourier–Laplace domain can be Fourier-inverted, and we obtain the expressions in the Laplace domain,

n_{m} (x, s) = (f_{m} + f_{im} \frac{1}{1 + s τ_{im}}) \frac{1}{\sqrt{4 ϕ (s) D}} e^{- \sqrt{ϕ (s) / D} | x |},

2.1

n_{im} (x, s) = (f_{m} + f_{im} \frac{1}{1 + s τ_{im}}) \frac{τ_{im} / τ_{m}}{1 + s τ_{im}} \frac{1}{\sqrt{4 ϕ (s) D}} e^{- \sqrt{ϕ (s) / D} | x |} + f_{im} \frac{τ_{im}}{1 + s τ_{im}} δ (x)

2.2

\begin{aligned} and n_{tot} (x, s) & = \frac{f_{m} + f_{im} \frac{1}{1 + s τ_{im}}}{s} ϕ (s) \\ \times \frac{1}{\sqrt{4 ϕ (s) D}} e^{- \sqrt{4 ϕ (s) / D} | x |} \\ + f_{im} \frac{τ_{im}}{1 + s τ_{im}} δ (x), \end{aligned}

2.3

as functions of x and s with

ϕ (s) = s [1 + τ_{im} τ_{m}^{- 1} / (1 + s τ_{im})]

. These expressions are valid for all s and hence for all times t. A numerical Laplace inversion then provides the densities for any specified time. Remarkably, it turns out that the density of mobile tracers, which were initially immobile, is proportional to the density of immobile tracers, which were initially mobile. This can be seen by setting f_m = 0 or f_im = 0 in (2.1) and (2.2), respectively. This proportionality holds for all s and hence at all times. We obtain the long-time Gaussian limit of the full concentration in B.3,

n_{tot} (x, t) \sim \frac{1}{\sqrt{4 π D_{eff} t}} \exp (- \frac{x^{2}}{4 D_{eff} t}), t ≫ τ_{m}, τ_{im},

2.4

with D_eff = D/(1 + τ_im/τ_m). Note that, for asymptotic equalities, we use the ∼ symbol. In fact, independent of the ratio f_m and f_im we asymptotically obtain a Gaussian distribution in which the diffusivity is reduced to the effective diffusivity D_eff. The mobile and immobile concentrations are asymptotically equivalent to (2.4) up to a scalar

f_{j}^{eq}

defined below [16].

2.2. Moments

In general, the fractions ${\bar{n}}_{m}$ and ${\bar{n}}_{im}$ of mobile and immobile tracers, initially fixed as f_m and f_im, change over time. To obtain the respective numbers, we integrate the tracer densities over space. This corresponds to setting k = 0 in the Fourier–Laplace transforms n_m(k, s) and n_im(k, s) of the densities. After Laplace inversion, we find

{\bar{n}}_{m} (t) = \frac{τ_{m}}{τ_{m} + τ_{im}} + \frac{f_{m} τ_{im} - f_{im} τ_{m}}{τ_{m} + τ_{im}} \exp (- [τ_{m}^{- 1} + τ_{im}^{- 1}] t)

2.5

and

{\bar{n}}_{im} (t) = \frac{τ_{im}}{τ_{m} + τ_{im}} - \frac{f_{m} τ_{im} - f_{im} τ_{m}}{τ_{m} + τ_{im}} \exp (- [τ_{m}^{- 1} + τ_{im}^{- 1}] t),

2.6

with

{\bar{n}}_{m} (t) + {\bar{n}}_{im} (t) = 1

. In the long-time limit t ≫ τ_m, τ_im, the fractions of mobile and immobile tracers reach the stationary values

f_{m}^{eq} = τ_{m} / (τ_{m} + τ_{im})

and

f_{im}^{eq} = τ_{im} / (τ_{m} + τ_{im})

, respectively. Our approach of splitting the total concentration into mobile and immobile fractions allows us to calculate the moments of the unbound, bound and total tau protein distributions individually,

{⟨ x^{2} (t) ⟩}_{j} = \frac{1}{{\bar{n}}_{j} (t)} \int_{- \infty}^{\infty} x^{2} n_{j} (x, t) d x,

2.7

where j stands for m, im and tot [16]. To shorten the notation, we use 〈x²(t)〉 = 〈x²(t)〉_tot in the remainder of this work. Using the Laplace inversion of

{\frac{\partial^{2}}{\partial k^{2}} n_{tot} (k, s) |}_{k = 0} = ⟨ x^{2} (s) ⟩,

2.8

we obtain the expression

\begin{aligned} ⟨ x^{2} (t) ⟩ & = 2 D_{eff} t + 2 D τ_{im} \frac{f_{m} τ_{im} / τ_{m} - f_{im}}{{(1 + τ_{im} / τ_{m})}^{2}} (1 - e^{- (τ_{m}^{- 1} + τ_{im}^{- 1}) t}) \end{aligned}

2.9

for the second moment. In the next section, we consider the initial conditions, when all tracers are initially mobile. This is chosen for didactic purposes, as this initial condition shows the plateau in the MSD and intermittent Laplace distribution most clearly. In §4, we consider immobile initial conditions and finally consider equilibrium initial conditions in §5, where the effects discussed in earlier sections are present at the same time.

3. All tracers initially mobile

We now consider the initial condition when all tracers are mobile, i.e. n_m(x, 0) = δ(x) and n_im(x, 0) = 0. This initial condition does not correspond to the experiment carried out by Igaev et al. [17]. However, this situation could be realized experimentally, e.g. by using the method of injection of fluorescently labelled tau proteins [53]. In what follows, we repeatedly use the time-scale separation τ_m ≪ τ_im observed for tau proteins and also relevant to other systems.

3.1. Concentration

We calculate the densities at short, intermediate and long times. In B.1, we obtain the Gaussian

n_{tot} (x, t) \sim \frac{1}{\sqrt{4 π D t}} \exp (- \frac{x^{2}}{4 D t})

3.1

in the short-time limit t ≪ τ_m, τ_im. The Gaussian (3.1) can be seen in figure 2 in the top left panel. In this figure n_m(x, t), n_tot(x, t) and a histogram are shown. The densities are obtained from Laplace inversions of the expressions in Laplace space (2.3), while the histogram is obtained from simulations, and colours denote the number of immobilization events N_im. Initially, all particles are mobile and diffuse freely, as denoted by the black colouring.

The concentration of freely diffusing particles that have not immobilized yet, i.e. have zero immobilization events N_im = 0, is given by the PDF of free Brownian motion multiplied by the probability of not having immobilized, i.e.

n_{m} (x, t | N_{im} = 0) = \frac{\exp (- t / τ_{m})}{\sqrt{4 π D t}} \exp (- \frac{x^{2}}{4 D t}) .

3.2

These mobile tracers immobilize with the position-dependent rate n_m(x, t|N_im = 0)/τ_m. Integrating from t′ = 0 to t′ = t, we obtain in the limit t ≪ τ_im (i.e. at short and intermediate times) that

\begin{aligned} n_{im} (x, t ≪ τ_{im}) & \sim \int_{0}^{t} \frac{\exp (- t^{'} / τ_{m}) / τ_{m}}{\sqrt{4 π D t^{'}}} \exp (- \frac{x^{2}}{4 D t^{'}}) d t^{'} \\ = \frac{\exp (- | x | / \sqrt{D τ_{m}})}{\sqrt{4 D τ_{m}}} \frac{1 - \erf (| x | / \sqrt{4 D t} - \sqrt{t / τ_{m}})}{2} \\ - \frac{\exp (| x | / \sqrt{D τ_{m}})}{\sqrt{4 D τ_{m}}} \frac{1 - \erf (| x | / \sqrt{4 D t} + \sqrt{t / τ_{m}})}{2} . \end{aligned}

3.3

Comparing (3.3) with the Laplace inversion of n_im(x, s) (2.3) in figure 8, we find very good agreement in the relevant range t ≪ τ_im.² For the total density, we obtain by adding n_m(x, t) (3.1) and n_im(x, t) (3.3)

\begin{aligned} n_{tot} (x, t) \sim \frac{\exp (- t / τ_{m})}{\sqrt{4 π D t}} \exp (- \frac{x^{2}}{4 D t}) + n_{im} (x, t ≪ τ_{im}), \\ t ≪ τ_{im} \end{aligned}

3.4

for the full tracer density. For t ≪ τ_m, we recover the Gaussian (3.1) from (3.4), while for τ_m ≪ t ≪ τ_im the distribution is distinctly non-Gaussian, as shown in figure 2. Up to around t = 0.6 s, the motion of the free tracers is dominated by the Gaussian n_m(x, t|N_im = 0) (see (3.2)), which spreads like free Brownian particles, shown as a white dotted line in figure 2. At around t = 1.6 s, most of the tracers with N_im = 0 immobilized and the majority of mobile tracers were immobile exactly once (N_im = 1) and transitioned back to the mobile zone, as shown by the red area. Because of the immobilization, these tracers have moved less than the free particles with N_im = 0 and a Laplace distribution emerges in the centre. For

x ≪ t \sqrt{D / τ_{m}}

and t ≫ τ_m, we can use the asymptotic lim _x→∞erf(x) = −lim_{x→ ∞}erf( − x) = 1 in n_im(x, t ≪ τ_im) (equation (3.3)), and obtain from n_tot(x, t) (3.4) the expression

n_{tot} (x, t) \sim \frac{1}{\sqrt{4 D τ_{m}}} \exp (- \frac{| x |}{\sqrt{D τ_{m}}}),

3.5

in the intermediate-time regime τ_m ≪ t ≪ τ_im. Combining the conditions t ≪ τ_m and

x ≪ t \sqrt{D / τ_{im}}

leads to

x ≪ τ_{im} \sqrt{D / τ_{m}} = 71 μ m

, which is large compared with the standard deviation

\sqrt{2 D τ_{m}} = 2.1 μ m

of the Laplace distribution (3.5). This means that the distribution follows such a Laplace shape for a large range of positions. The total concentration, in turn, therefore follows a Laplace distribution with fixed parameters. This is a pronounced deviation from a Gaussian distribution. This result can also be obtained from calculations in Laplace space, as shown in B.2. By contrast, for times significantly longer than τ_im, many immobilizations take place, as shown by the bright yellow area in figure 2, where the distribution follows the Gaussian (2.4) with the effective diffusivity D_eff = D/(1 + τ_im/τ_m).

3.2. Mean squared displacement

From the general expression for the MSD (2.9) for immobile initial conditions, we obtain the expression

⟨ x^{2} (t) ⟩ = \frac{2 D}{1 + τ_{im} / τ_{m}} [t + \frac{τ_{im}^{2} / τ_{m}}{1 + τ_{im} / τ_{m}} (1 - e^{- (τ_{m}^{- 1} + τ_{im}^{- 1}) t})] .

3.6

At intermediate times, the MSD (expression (3.6)) exhibits a plateau-like behaviour with the constant MSD

⟨ x^{2} (t) ⟩ \sim 2 D τ_{m}, τ_{m} ≪ t ≪ τ_{im},

3.7

corresponding to free Brownian particles that moved for the duration τ_m. This requires the condition τ_m ≪ τ_im, which is satisfied in the tau protein case [17], with τ_m = 0.16 s and τ_im = 7.7 s. Such plateaus are often found when tracers diffuse in porous media or for dynamics in crowded membranes or environments with obstacles, in which the tracer can be transiently confined [38,46,48,54,55]. The MSD (3.6) is shown in figure 3a as the black solid line.

Figure 3. Second moments for different initial conditions on a log–log scale. In (a), all tracers are initially mobile, as in §3. After a linear growth, the second moment 〈x²(t)〉 of all tracers (equation (3.6)) shows a plateau for τ_m ≪ t ≪ τ_im. The second moment of the mobile particles (equation (3.10)) in (a) has a peak immediately before the total particle moment and the mobile particle moment reach a plateau value. Immobile tracers spread ∼Dt at short times, and the second moment (3.11) has a plateau at intermediate times. In (b), all tracers are initially immobile, as in §4. The second moment of all tracers (equation (4.3)) grows ∼Dt²/τ_im at short times, owing to the decaying number of particles located at x = 0. The immobile tracers spread ∼Dt³/(3τ_mτ_im) at short times, while the full expression is given in equation (4.3). The mobile tracers in (b) spread exactly like the immobile tracers in (a), where all tracers are initially mobile. (c) The equilibrium case, §5, in which the second moment grows like 2Dt/(1 + τ_im/τ_m) (equation (5.3)) for all times. The mobile and immobile moments exactly match the moments of the total distribution with mobile and immobile initial conditions, respectively.

When calculating the moments of the mobile and immobile tracers (2.7), the time-dependent normalizations of the tracer densities (2.6),

{\bar{n}}_{m} (t) = \frac{τ_{m}}{τ_{m} + τ_{im}} [\frac{1 + τ_{im}}{τ_{m} e^{- (τ_{m}^{- 1} + τ_{im}^{- 1}) t}}]

3.8

and

{\bar{n}}_{im} (t) = \frac{τ_{im}}{τ_{m} + τ_{im}} [1 - e^{- (τ_{m}^{- 1} + τ_{im}^{- 1}) t)}],

3.9

need to be taken into account, yielding the moments of the mobile and immobile densities (2.7) [16]

\begin{aligned} {⟨ x^{2} (t) ⟩}_{m} & = \frac{2 D}{(1 + τ_{im} / τ_{m}) (1 + τ_{im} / τ_{m} e^{- (τ_{m}^{- 1} + τ_{im}^{- 1}) t})} [t (1 + \frac{τ_{im}^{2}}{τ_{m}^{2}} e^{- (τ_{m}^{- 1} + τ_{im}^{- 1}) t}) \\ + \frac{2 τ_{im}^{2} / τ_{m}}{1 + τ_{im} / τ_{m}} (1 - e^{- (τ_{m}^{- 1} + τ_{im}^{- 1}) t})] \end{aligned}

3.10

and

\begin{aligned} {⟨ x^{2} (t) ⟩}_{im} & = \frac{2 D}{1 - e^{- (τ_{m}^{- 1} + τ_{im}^{- 1}) t}} [\frac{t}{1 + τ_{im} / τ_{m}} (1 - \frac{τ_{im}}{τ_{m}} e^{- (τ_{m}^{- 1} + τ_{im}^{- 1}) t}) \\ + \frac{τ_{im}^{2} / τ_{m} - τ_{im}}{{(1 + τ_{im} / τ_{m})}^{2}} (1 - e^{- (τ_{m}^{- 1} + τ_{im}^{- 1}) t})] . \end{aligned}

3.11

As shown in figure 3, the mobile second moment exhibits a peak at around t = 0.6 s, followed by a plateau. This peak arises as the density of mobile tracers initially consists of mobile tracers that have never immobilized. Once t ≫ τ_m, the mobile density mainly consists of tracers that were immobile (at least) once and mobilized, as discussed above. Since the latter had less time to move, they have spread less and the MSD temporarily decreases.

The immobile MSD (3.11) has the short-time behaviour 〈x²(t)〉_im ∼ Dt for t ≪ τ_m, τ_im. The factor $\frac{1}{2}$ when compared with the mobile tracers arises because immobile tracers effectively average over the history of the mobile tracers. Namely, for t′ ≪ τ_m, τ_im, mobile particles immobilize with the constant rate p(t′) = 1/τ_m. A particle that immobilized at time t′ before moved for the duration t′ and thus contributes 2Dt′ to the second moment for t > t′; see figure 4a for a schematic drawing. When averaging over different mobile periods t′ and normalizing with the fraction of immobile tracers $\int_{0}^{t} p (t^{'}) d t^{'}$ , we obtain

\begin{aligned} {⟨ x^{2} (t) ⟩}_{im} \sim 2 D \frac{\int_{0}^{t} t^{'} p (t^{'}) d t^{'}}{\int_{0}^{t} p (t^{'}) d t^{'}} = \frac{2 D \int_{0}^{t} t^{'} / τ_{m} d t^{'}}{t / τ_{m}} = D t, \\ for t ≪ τ_{m}, τ_{im} . \end{aligned}

3.12

As mentioned above, the long-time limits of the MSDs of all densities remain equal to 2D_efft, regardless of the fractions f_m and f_im.

Figure 4. Schematic showing the short-time behaviour of tracers for mobile (a) and immobile initial conditions (b) at three snapshots of time. In both panels, the tracers change the mobilization state at times t′₁ and t′₂, respectively. For mobile initial conditions in (a), the number of immobile tracers grows ∼t/τ_m at short times. Namely, the later a tracer immobilizes, the longer it was previously mobile. In (b), the number of mobile tracers grows ∼t/τ_im. Namely, the earlier a tracer mobilizes in (b), the longer it is mobile.

4. All tracers initially immobile

We now discuss the case when all tracers are immobile at t = 0, n_im(x, 0) = δ(x) and n_m(x, 0) = 0.

4.1. Concentration

In B.1, we obtain the short-time behaviour

\begin{aligned} n_{tot} (x, t) & \sim \frac{2 t / τ_{im}}{\sqrt{4 π D t}} e^{- (x^{2} / 4 D t)} - \frac{| x | (1 - \erf (\frac{| x |}{\sqrt{4 D t}}))}{2 D τ_{im}} \\ + (1 - \frac{t}{τ_{im}}) δ (x), for t ≪ τ_{m}, τ_{im} \end{aligned}

4.1

by applying approximations for large s in Laplace space. Expression (4.1) is shown in the left panel of figure 5 as the black dashed line. In particular, note the distinctively non-Gaussian shape of the distribution in contrast to the case of mobile initial conditions. The Gaussian in equation (4.1) has the normalization ∼2t/τ_im, while the second term has the normalization ∼−t/τ_im, and thus the whole expression (4.1) is normalized to unity. In B.2, we obtain the total density at intermediate times τ_m ≪ t ≪ τ_im

n_{tot} (x, t) \sim \frac{t / τ_{im}}{\sqrt{4 D τ_{m}}} \exp (- \frac{| x |}{\sqrt{D τ_{m}}}) + (1 - \frac{t}{τ_{im}}) δ (x),

4.2

as shown in figure 5 in the top row (except for the leftmost panel) as a black–white striped line. Compared with the mobile initial condition, the coefficient of the Laplace distribution has the linear growth t/τ_im. Most tracers remain immobile at the origin at t = 1.6 s. In B.4, we find expression (B 11) for n_tot(x, t), which is valid for t ≪ τ_im and contains equations (4.1) and (4.2) as limits. In figure 5, the lower panels show the transition from the Laplace distribution to the Gaussian (2.4).

4.2. Mean squared displacement

From the general expression for the MSD (2.9), we obtain the expression

⟨ x^{2} (t) ⟩ = \frac{2 D}{1 + τ_{im} / τ_{m}} [t - \frac{τ_{im}}{1 + τ_{im} / τ_{m}} (1 - e^{- (τ_{m}^{- 1} + τ_{im}^{- 1}) t})] .

4.3

The MSD (4.3) has the ballistic short-time behaviour

⟨ x^{2} (t) ⟩ \sim \frac{D t^{2}}{τ_{im}} + O (t^{3}), t ≪ τ_{m}, τ_{im} .

4.4

The Landau symbol O(·) represents higher order terms. The ballistic behaviour at short times t ≪ τ_im arises because the fraction exp(−t/τ_im) ∼ 1 − t/τ_im of tracers are immobile at x = 0 and hence do not contribute to the second moment. For t′ ≪ τ_m ≪ τ_im, immobile particles mobilize with the constant rate p(t′) = 1/τ_im. A particle that mobilized at time t′ moved for the duration t − t′ and thus contributes 2D(t − t′) to the second moment for t > t′; see figure 4a for a schematic drawing. When integrating over different mobilization times t′, we find

\begin{aligned} ⟨ x^{2} (t) ⟩ & \sim 2 D \int_{0}^{t} (t - t^{'}) p (t^{'}) d t^{'} = 2 D \int_{0}^{t} \frac{t - t^{'}}{τ_{im}} d t^{'} \\ = D \frac{t^{2}}{τ_{im}}, t ≪ τ_{m} ≪ τ_{im} . \end{aligned}

4.5

We obtain the number of free and bound tracers from the general expression (2.6),

{\bar{n}}_{m} (t) = \frac{τ_{m}}{τ_{m} + τ_{im}} [1 - e^{- (τ_{m}^{- 1} + τ_{im}^{- 1}) t)}]

4.6

and

{\bar{n}}_{im} (t) = \frac{τ_{im}}{τ_{m} + τ_{im}} [\frac{1 + τ_{m}}{τ_{im} e^{- (τ_{m}^{- 1} + τ_{im}^{- 1}) t}}] .

4.7

This produces the normalization of the immobile moment, and we find

\begin{aligned} {⟨ x^{2} (t) ⟩}_{im} = \frac{2 D t}{1 + τ_{m} / τ_{im}} \frac{1 + e^{- (τ_{m}^{- 1} + τ_{im}^{- 1}) t}}{τ_{im} / τ_{m} + e^{- (τ_{m}^{- 1} + τ_{im}^{- 1}) t}} \\ - \frac{4 D τ_{im}^{2} / τ_{m}}{{(1 + τ_{im} / τ_{m})}^{2}} \frac{1 - e^{- (τ_{m}^{- 1} + τ_{im}^{- 1}) t}}{τ_{im} / τ_{m} + e^{- (τ_{m}^{- 1} + τ_{im}^{- 1}) t}} . \end{aligned}

4.8

This MSD has the short-time behaviour 〈x²(t)〉_im ∼ Dt³/(3τ_imτ_m) for t ≪ τ_m, τ_im. The cubic scaling emerges as the only immobile tracers, which are not located at the origin, have previously mobilized and then immobilized again. The mobile concentration grows ∼t/τ_im at short times t ≪ τ_im. Integrating over the time t′ spent in the mobile phase yields the cubic scaling

{⟨ x^{2} (t) ⟩}_{im} \sim 2 D \int_{0}^{t} \frac{1}{τ_{m}} n_{m} (t - t^{'}) t^{'} d t^{'} = \frac{2 D}{τ_{m} τ_{im}} \int_{0}^{t} (t - t^{'}) t^{'} d t^{'} = D \frac{t^{3}}{3 τ_{m} τ_{im}},

4.9

where, in the first step, we took the limit t ≪ τ_m. Since the mobile concentration with immobile initial conditions is proportional to the immobile concentration with mobile initial conditions, 〈x²(t)〉_m is equal to 〈x²(t)〉_im in (3.11) with mobile initial conditions. This can be seen in figure 3a,b. As for the mobile initial condition considered in §3, the MSDs of all densities grow ∼2D_efft asymptotically.

5. Equilibrium initial fractions of initial mobile tracers

In this section, we use the equilibrium values $n_{m} (x, 0) =$ $f_{m}^{eq} δ (x)$ and $n_{im} (x, 0) = f_{im}^{eq} δ (x)$ as initial conditions.

5.1. Concentration profiles

From the general expressions (2.1) and (2.3) for the densities n_m(x, s) and n_tot(x, s), we find that the mobile concentration of the equilibrium case discussed here is proportional to the total concentration for the mobile initial condition in §3 at all times. To understand why this is true, we note that both concentrations at all times contain mobile tracers that were initially mobile. Moreover, from equations (2.1) and (2.3), we see that the mobile concentration of the equilibrium case contains initially immobile tracers, while the total concentration contains immobile tracers that were initially mobile. In equations (2.1) and (2.3), the respective terms that appear in addition to the initially mobile fractions that are still mobile are proportional to each other at all times, as described in §2.1. An analogous relation holds between the immobile concentration with equilibrium initial conditions and the total concentration with immobile initial conditions, as can be seen in equations (2.1) and (2.3).

We consider the short-time approximation t ≪ τ_m, τ_im for which initially immobile tracers have not yet mobilized and initially mobile tracers have not yet been trapped. Therefore, we can neglect the terms with the rates $τ_{m}^{- 1}$ and $τ_{im}^{- 1}$ in (1.2) and solve n_m(x, t) and n_im(x, t) separately, yielding

\begin{aligned} n_{tot} (x, t) \sim \frac{f_{m}^{eq}}{\sqrt{4 π D t}} \exp (- \frac{x^{2}}{4 D t}) + f_{im}^{eq} δ (x), \\ t ≪ τ_{m}, τ_{im}, \end{aligned}

5.1

with a Gaussian distribution describing free diffusion in addition to a Dirac-δ distribution of initially immobile tracers that have not yet moved. This behaviour can be seen in the top left panel of figure 6. The same result as (5.1) can be obtained by combining the short-time expressions for the mobile (3.1) and immobile (4.1) initial conditions for t ≪ τ_m, τ_im, as done in equation (B 5). At short times, the total density (5.1) behaves like the case of mobile initial conditions with an additional delta peak. At intermediate times τ_m ≪ t ≪ τ_im, we obtain

n_{tot} (x, t) \sim \frac{t / τ_{im}}{\sqrt{4 D τ_{m}}} \exp (- \frac{| x |}{\sqrt{D τ_{m}}}) + (1 - \frac{t}{τ_{im}}) δ (x)

5.2

by combining the mobile (3.5) and immobile expression (4.2), respectively.³ In fact equation (5.2) is the same as expression (4.2) for the case of immobile initial conditions, in the intermediate-time regime. This result is shown in figure 6, where this approximation is compared with the full concentration from t = 0.5 to t = 2. This result is the one-dimensional equivalent to the findings in [32]. The lower right panels of figure 6 show the Gaussian long-time limit (2.4) as a black–white striped line.

Figure 6. Concentration profiles for equilibrium initial conditions. At t = 0, all tracers are at x = 0 and the equilibrium fraction τ_im/(τ_m + τ_im) is immobile. For a description of the legend, see figure 2. The top left panel shows the short-time behaviour consisting of a Gaussian and a δ-distribution (equation (5.1)). At t = 1, almost all initially mobile tracers immobilized at least once and the total concentration follows the Laplace distribution (5.2), as shown by the black–white striped line for t = 0.5–2 s. At longer times, after several immobilizations the concentration profiles cross over to a Gaussian, as witnessed by equation (2.4), shown as a black–white striped line for t = 50 s and t = 200 s.

5.2. Mean squared displacement

The number of mobile and immobile tracers remains constant for equilibrium initial conditions. At all times, the second moment of all tracers (2.9) thus simplifies to

⟨ x^{2} (t) ⟩ = \frac{2 D}{1 + τ_{im} / τ_{m}} t .

5.3

The second moment is similar to that of a free Brownian particle, with the effective diffusion coefficient D_eff = D/(1 + τ_im/τ_m), as shown in figure 3. This is a known result from models for Fickian yet non-Gaussian diffusion [32]. The mobile and immobile moments, 〈x²(t)〉_m and 〈x²(t)〉_im, are equivalent to the moments of the full density with mobile (3.6) and immobile (4.3) initial conditions, as can be seen in figure 3. This relation holds because the respective densities are proportional, as discussed above. The mobile and immobile moments show clear anomalous diffusion for t ≪ τ_im, with a quite long crossover dynamics, as depicted in figure 3c. The mobile moment has a plateau in the intermediate regime τ_m ≪ t ≪ τ_im and the immobile moment behaves ballistically at short times t ≪ τ_m.

In the long-time limit, all mobile and immobile second moments grow like the moments of the total concentration, i.e. 〈x²(t)〉_m ∼ 〈x²(t)〉_im ∼ 2D_efft for t ≫ τ_im, τ_m.

6. Discussion and conclusion

We considered a quite simple mobile–immobile model according to which tracer particles switch between a mobile diffusing state and an immobilized state. On average, the tracers remain mobile for the duration τ_m and immobile for τ_im. We considered the particular case, motivated by experiments on tau proteins binding to and unbinding from microtubules in axons of dendritic cells [17], when the two time scales are separated, τ_m ≪ τ_im. We analysed three different initial conditions with varying fractions of mobile to immobile tracers at the origin, which can, in principle, all be realized in experiments. The initial condition of mobile tracers can be realized by injecting fluorescently labelled proteins [53]. Initially, immobile tracers could in principle be obtained in single-particle tracking experiments, by focusing on the tracks of immobile tracers. Equilibrium fractions of mobile tracers naturally occur when the tau proteins were in proximity to the microtubules for t ≫ τ_m, τ_im before the start of the data acquisition.

First, we studied the case when all tracers are initially mobile, as described in the experiment in [53]. Second, we assumed all tracers to be initially immobile. Third, we considered an equilibrium fraction, corresponding to the experiment in [17]. For non-equilibrium fractions of initially mobile tracers, we find anomalous diffusion at short and intermediate time scales, at which initially mobile tracers display a plateau in the MSD at intermediate times and initially immobile tracers spread ballistically at short times. At t ≪ τ_m and an initial equilibrium fraction, the tracer density consists of a Gaussian and a delta peak. Initially, mobile tracers follow a Gaussian distribution at short times. When all tracers are initially immobile, the short-time distribution consists of a delta peak and a non-Gaussian distribution. At intermediate times τ_m ≪ t ≪ τ_im, the distribution is made up of a Laplace distribution and a delta distribution of initially immobile tracers that have not moved yet. The coefficients of the two distributions depend on the specific initial conditions. We additionally obtain expressions for the densities that are valid for the whole range t ≪ τ_im. We stress that the distribution is non-Gaussian at intermediate times, regardless of the initial conditions. By contrast, the distribution asymptotically at long times matches a Gaussian for all initial conditions. The densities of mobile and immobile tracers with equilibrium initial conditions match the total tracer densities of mobile and immobile initial conditions, respectively, at all times. Moreover, the immobile tracer density from mobile initial conditions is proportional to the mobile tracer density from immobile initial conditions at all times. As a special case for equilibrium initial conditions, our model corresponds to the one-dimensional version of the model used in [32] to describe Fickian yet non-Gaussian diffusion. We find the same linear MSD for all times and obtain a closed expression for the Laplace distribution at intermediate time scales.

The model developed here is, of course, much more general. We provided the framework for any ratio of the characteristic time scales τ_m and τ_im, such that the model will be useful for scenarios ranging from geophysical experiments with Poissonian (im)mobilization statistics to molecular systems such as protein (un)binding to DNA in nanochannel set-ups. To the best of our knowledge, the transient Laplace distribution of tau proteins has not been observed yet. We now discuss possible experiments that could reveal the anomalous diffusion regime and the Laplace distribution, which depend on the time scales τ_m and τ_im. For the present analysis, we used the parameters $τ_{m} = 0.16 s$ and $τ_{im} = 7.7 s$ for tau proteins, which were obtained from an FDAP experiment [17]. FDAP experiments do not directly allow the observation of single-particle displacement densities and the moments thereof. However, a single-molecule tracking (SMT) study of tau proteins [18] with two-dimensional trajectories of $2.2 s$ length was conducted, where we expect the transient Laplace distribution to be visible in the marginal distribution, given that the sample size is large enough. From SMT experiments, the moments can be obtained, although, in [18], the moments of the distribution were not evaluated. Another example for a system with τ_im > τ_m is given by synaptic vesicles [19]. In [19], fluorescence correlation spectroscopy reveals $τ_{im} = (4.2 \pm 0.4) s$ and $τ_{m} = (2.0 \pm 0.4) s$ . In addition, glucocorticoid receptors show long immobilization events with $τ_{im} = 13 ms$ compared with $τ_{m} = 2 ms$ , as revealed by fluorescent recovery after a photobleaching experiment [20]. The Laplace distribution cannot be observed in this experiment, owing to the missing information on single tracers. SMT experiments of the transcription factor p53 [22] show a switched separation of time scales with $τ_{im} = 1.80 s$ and $τ_{m} = 8.4 s$ . Here, SMT allows us to measure the exponential binding time distribution corresponding to a single binding rate, as in our model. The second moment is measured for up to 0.6 s, where longer trajectories would allow for a comprehensive comparison with the moments calculated in this work. For τ_m > τ_im, the Laplace distribution does not arise. We now look at another SMT experiment in more detail. In [29], the architectural DNA-binding protein Fis was observed to have a linear MSD and a non-Gaussian displacement distribution, as depicted in figure 7. The authors of [29] fitted two Gaussians to the distribution and deduced the presence of two sliding modes of Fis on the DNA. Since the motion during the slow sliding mode is within experimental accuracy, it is plausable to assume that the non-Gaussian distribution emerges as a result of immobilization. In figure 7, we show fits with a Laplace distribution and a Gaussian distribution in a logarithmic presentation. The Laplace distribution matches the general shape with few exceptions around $- 0.4 μ m$ , while the Gaussian distribution does not capture the peak in the centre. We note that the Laplace distribution requires a single fitting parameter, compared with the two Gaussians with advection used in [29] requiring five parameters. The apparent Laplace distribution and linear MSD translate to equilibrium initial conditions in our model.

Figure 7. Displacement distribution of the Fis DNA-binding protein. The histogram depicts the data measured by Kamagata *et al*. [29]. The solid and dashed lines depict fits with a Laplace and a Gaussian distribution, respectively. The Laplace fit exp( − |x|/a)/2a yields $a = 0.27 μ m$ . The Gaussian fit $\exp (- x^{2} / (4 D t)) / \sqrt{4 π D t}$ with $t = 0.28 s$ yields $D = 0.21 μ m^{2} s^{- 1}$ , which is within the margin of error of the value $(0.19 \pm 0.02) μ m^{2} s^{- 1}$ of the fast diffusion coefficient reported in [29]. The data were extracted from the PDF file of [29].

Figure 8. Comparison of the Laplace inversion of n_im(x, s) (expression (2.3)) and the analytic expression for n_im(x, t) (equation (3.3)) that holds for t ≪ τ_im. Both overlap almost perfectly for t < τ_im = 7.7 s.

We note that (non-)exponential (im)mobilization has been studied using a Langevin equation with switching diffusivities [56,57] and the continuous time random walk framework, where the waiting time probability distribution function consists of a combination of two exponentials with different time scales [58]. It will be a topic of future research to study the effect of a drift velocity on the pre-asymptotic behaviour for different initial conditions, as well as what happens when non-exponential (im)mobilization is considered in a mobile–immobile model in connection with chemical reactions.

Data accessibility

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Authors' contributions

T.J.D.: formal analysis, investigation, methodology, validation, visualization, writing—original draft, writing—review and editing; A.V.C.: conceptualization, formal analysis, investigation, supervision, writing—review and editing; R.M.: conceptualization, funding acquisition, project administration, supervision, validation, writing—original draft, writing—review and editing. All authors gave final approval for publication and agreed to be held accountable for the work performed herein.

Conflict of interest declaration

We declare we have no competing interests.

Funding

We acknowledge funding from the German Science Foundation (DFG, grant no. ME 1535/12-1). A.V.C. acknowledges the support of the Polish National Agency for Academic Exchange (NAWA).

Footnotes

Endnotes

1 The slow directed motion only plays a role when very long times are considered [17,51,52].

2 Equations (3.2) and (3.3) can also be obtained by taking the limit τ_im → ∞ in (1.2) and solving the equations directly.

3 An approximation for the whole range of t ≪ τ_im can be obtained for any fractions of initially mobile tracers f_m by combining equations (3.4) and (B 11) from the mobile and immobile initial conditions, respectively. This yields equation (B 12) and is shown in figure 10.

Appendix A. General equations

Starting with equation (1.2), we apply the Fourier–Laplace transform $f (k, s) = \int_{- \infty}^{\infty} \int_{0}^{\infty} e^{- s t + i k x} f (x, t) d t d x$ to the rate to obtain

\begin{matrix} n_{m} (k, s) = \frac{f_{m} + f_{im} (1 / (1 + s τ_{im}))}{ϕ (s) + k^{2} D} \\ and & n_{im} (k, s) = \frac{τ_{im}}{1 + s τ_{im}} (f_{im} + τ_{m}^{- 1} \frac{f_{m} + f_{im} (1 / (1 + s τ_{im}))}{ϕ (s) + k^{2} D}) \end{matrix}}

A 1

as well as

n_{tot} (k, s) = n_{m} (k, s) + n_{im} (k, s) = \frac{f_{m} + f_{im} (1 / (1 + s τ_{im}))}{s} \frac{ϕ (s)}{ϕ (s) + k^{2} D} + f_{im} \frac{τ_{im}}{1 + s τ_{im}}

A 2

with

ϕ (s) = s [1 + τ_{im} τ_{m}^{- 1} / (1 + s τ_{im})]

Appendix B. Asymptotics calculated in Laplace space

We go from a short-time limit to a long-time limit.

B.1. Short-time limit

For t ≪ τ_m, τ_im, we obtain sτ_im ≫ 1 and sτ_im ≫ 1. This yields ϕ(s) ∼ s in this limit. With (A 1) for f_m = 1 and f_im = 0, we obtain the expression

n_{m} (k, s) \sim n_{tot} (k, s) \sim \frac{1}{s + k^{2} D},

B 1

which produces the Gaussian (3.1). We now consider f_im = 1 and f_m = 0 and obtain the expression

n_{tot} (k, s) \sim \frac{1}{s τ_{im}} \frac{1}{s + k^{2} D} + (\frac{1}{s} - \frac{1}{s^{2} τ_{im}})

B 2

from (A 2) in the limit sτ_m ≫ 1 and sτ_im ≫ 1. Fourier–Laplace inversion yields the expression

\begin{aligned} n_{tot} (x, t) \sim \frac{1}{τ_{im}} \int_{0}^{t} \frac{1}{\sqrt{4 π D t^{'}}} e^{- (x^{2} / 4 D t^{'})} d t^{'} + (1 - t / τ_{im}) δ (x) \\ for t ≪ τ_{m} ≪ τ_{im} . \end{aligned}

Solving the integral in (B 3) gives the expression

\begin{aligned} n_{tot} (x, t) \sim \frac{2 t / τ_{im}}{\sqrt{4 π D t}} e^{- (x^{2} / 4 D t)} - \frac{| x | (1 - \erf (| x | / \sqrt{4 D t}))}{2 D τ_{im}} \\ + (1 - \frac{t}{τ_{im}}) δ (x), for t ≪ τ_{m} ≪ τ_{im}, \end{aligned}

B 4

where normalization is conserved. By combining expression (B 4) for immobile initial conditions and (3.1) for mobile initial conditions, we obtain the expression

\begin{aligned} n_{tot} (x, t) \sim (\frac{f_{m} + 2 f_{im} t}{τ_{im}}) \frac{1}{\sqrt{4 π D t}} e^{- (x^{2} / 4 D t)} \\ - f_{im} \frac{| x | (1 - \erf (\frac{| x |}{\sqrt{4 D t}}))}{2 D τ_{im}} + f_{im} (1 - \frac{t}{τ_{im}}) δ (x), \end{aligned}

B 5

for t ≪ τ_m ≪ τ_im for arbitrary fractions of initially mobile tracers.

B.2. Density at intermediate time scales

We now investigate the intermediate time τ_m ≪ t ≪ τ_im, corresponding to sτ_m ≪ 1 and sτ_im ≫ 1. In this regime, we have $ϕ (s) \sim τ_{m}^{- 1}$ , yielding the expression

n_{tot} (x, s) \sim \frac{f_{m} + f_{im} \frac{1}{s τ_{im}}}{s} \frac{1}{\sqrt{4 D τ_{m}}} e^{- (| x | / \sqrt{D τ_{m}})} + f_{im} (\frac{1}{s} - \frac{1}{s^{2} τ_{im}}) δ (x)

B 6

from (2.3) for the total concentration. The inverse Laplace transform of (B 6) yields the expression

n_{tot} (x, s) \sim (\frac{f_{m} + f_{im} t}{τ_{im}}) \frac{1}{\sqrt{4 D τ_{m}}} e^{- (| x | / \sqrt{D τ_{m}})} + f_{im} (\frac{1 - t}{τ_{im}}) δ (x)

B 7

for τ_m ≪ t ≪ τ_im.

B.3. Density in the long-time limit

We obtain the long-time limit t ≫ τ_m, τ_im from n_tot(k, s) (A 2) using s ≪ 1/τ_im, 1/τ_m and ϕ(s) ∼ s(1 + τ_im/τ_im). This yields the expression

n_{tot} (x, t) \sim \frac{1}{\sqrt{4 π D_{eff} t}} e^{- (x^{2} / 4 D_{eff} t)}, for t ≫ τ_{m}, τ_{im},

B 8

with D_eff = D/(1 + τ_im/τ_m).

B.4. Density at short to intermediate time scales

Here, we analyse the regime t ≪ τ_im. The case f_m = 1 and f_im = 0 is considered in §3. We consider the case f_im = 1 and f_m = 0 here. From n(x, s) (2.3), we obtain, with sτ_im ≫ 1 and ϕ(s) ∼ s + 1/τ_m,

\begin{aligned} n_{tot} (x, s) & \sim \frac{s τ_{m} + 1}{s^{2} τ_{im} τ_{m}} \frac{1}{\sqrt{4 D (s + 1 / τ_{m})}} e^{- \sqrt{(s + 1 / τ_{m}) / D} | x |} \\ + (\frac{1}{s} - \frac{1}{s^{2} τ_{im}}), for s τ_{im} ≫ 1. \end{aligned}

B 9

In the time domain in the limit t ≪ τ_im, this corresponds to the expression

\begin{aligned} n_{tot} (x, t) & \sim \int_{0}^{t} \frac{t + τ_{m} - t^{'}}{τ_{im} τ_{m}} \exp (\frac{- t^{'}}{τ_{m}}) \frac{\exp (- \frac{x^{2}}{4 D t^{'}})}{\sqrt{4 D t^{'}}} d t^{'} + (1 - t / τ_{im}) δ (x) \\ = \frac{e^{- (x^{2} / 4 D t)}}{\sqrt{4 π D t}} e^{- t / τ_{m}} \frac{t}{τ_{im}} + (1 - t / τ_{im}) δ (x) \\ + \frac{\exp (- | x | / \sqrt{D τ_{m}})}{\sqrt{4 D τ_{m}}} (t / τ_{im} - | x | \sqrt{\frac{τ_{m}}{4 D τ_{im}^{2}}} + τ_{m} / 2 τ_{im}) \frac{1 - \erf (| x | / \sqrt{4 D t} - \sqrt{t / τ_{m}})}{2} \end{aligned}

B 10

- \frac{\exp (| x | / \sqrt{D τ_{m}})}{\sqrt{4 D τ_{m}}} (t / τ_{im} + | x | \sqrt{\frac{τ_{m}}{4 D τ_{im}^{2}}} + τ_{m} / 2 τ_{im}) \frac{1 - \erf (| x | / \sqrt{4 D t} + \sqrt{t / τ_{m}})}{2} .

B 11

Normalization is preserved, as can be seen by integrating (B 10) over x. The first summand in (B 10) then resolves to t/τ_im. In the limit t ≪ τ_m, τ_im, we recover the short-time behaviour for n_tot(x, t) (B 4), as shown in figure 9. For τ_m ≪ t ≪ τ_im and

| x | ≪ \sqrt{4 D τ_{im}^{2} / τ_{m}}

, we recover the Laplacian intermediate regime in (B 7) with f_im = 1 and f_m = 0. In figure 9, we show a verification of (B 11). For arbitrary fractions of initially mobile tracers, we combine equation (B 11) for immobile initial conditions with equation (3.4) for mobile initial conditions, as follows:

\begin{aligned} n_{tot} (x, t) & \sim \frac{e^{- (x^{2} / 4 D t)}}{\sqrt{4 π D t}} e^{- t / τ_{m}} (f_{m} + f_{im} \frac{t}{τ_{im}}) + f_{im} (1 - t / τ_{im}) δ (x) \\ + [f_{m} + f_{im} (t / τ_{im} - | x | \sqrt{\frac{τ_{m}}{4 D τ_{im}^{2}}} + τ_{m} / 2 τ_{im})] \\ \times \frac{\exp (- | x | / \sqrt{D τ_{m}})}{\sqrt{4 D τ_{m}}} \frac{1 - \erf (| x | / \sqrt{4 D t} - \sqrt{t / τ_{m}})}{2} \\ - [f_{m} + f_{im} (t / τ_{im} + | x | \sqrt{\frac{τ_{m}}{4 D τ_{im}^{2}}} + τ_{m} / 2 τ_{im})] \\ \times \frac{\exp (| x | / \sqrt{D τ_{m}})}{\sqrt{4 D τ_{m}}} \frac{1 - \erf (| x | / \sqrt{4 D t} + \sqrt{t / τ_{m}})}{2} . \end{aligned}

B 12

In figure 10, expression (B 12) is compared with the Laplace inversion of the exact expression of n_tot(x, s) (2.3).

Figure 9. All tracers initially immobile. Comparison of the exact Laplace inversion of (2.3), the short-time approximation (B 4), the intermediate-time approximation (B 7) and the short- to intermediate-time approximation (B 11).

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Journal of The Royal Society Interface cover image

July 2022

Volume 19Issue 192

Article Information

DOI:https://doi.org/10.1098/rsif.2022.0233
PubMed:35857918
Published by:Royal Society
Online ISSN:1742-5662

History:

Manuscript received22/03/2022
Manuscript accepted15/06/2022
Published online06/07/2022

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Apparent anomalous diffusion and non-Gaussian distributions in a simple mobile–immobile transport model with Poissonian switching

Abstract

1. Introduction

2. Model and general solutions

2.1. Mobile and immobile concentration profiles

2.2. Moments

3. All tracers initially mobile

3.1. Concentration

3.2. Mean squared displacement

4. All tracers initially immobile

4.1. Concentration

4.2. Mean squared displacement

5. Equilibrium initial fractions of initial mobile tracers

5.1. Concentration profiles

5.2. Mean squared displacement

6. Discussion and conclusion

Data accessibility

Authors' contributions

Conflict of interest declaration

Funding

Footnotes

Endnotes

Appendix A. General equations

Appendix B. Asymptotics calculated in Laplace space

B.1. Short-time limit

B.2. Density at intermediate time scales

B.3. Density in the long-time limit

B.4. Density at short to intermediate time scales

This Issue