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Edited by: Carlos Mejía-Monasterio, Polytechnic University of Madrid, Spain

Reviewed by: Diego R. Amancio, University of São Paulo, Brazil; Haroldo Valentin Ribeiro, Universidade Estadual de Maringá, Brazil

This article was submitted to Interdisciplinary Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

We study the random walk of a particle in a compartmentalized environment, as realized in biological samples or solid state compounds. Each compartment is characterized by its length

The characterization of the diffusive behavior in complex environments is crucial in many fields, ranging from biology [^{2}(^{σ} with anomalous exponent 0 < σ < 1. The characterization of this movement provides important information on the disorder of the media and on the laws governing the system [

The presence of barriers that prevent the particles to freely diffuse in the environments is a general mechanism used to explain subdiffusion [

In this article, we study a general barrier model, where a particle performs an unbiased random walk through a complex environment made by a mesh of compartments separated by barriers with random transmittance. A schematic of the system is shown in

Schematic of the system.

In order to study the behavior of the particle, we propose a coarse-graining approach transforming the rather complex walk of the particle (mainly due to the interaction with the boundaries) into two very well known theoretical models describing anomalous diffusion: continuous time random walks and Lévy walks. The former, introduced by Montroll and Weiss [

In the most general description of our system, we show how the walk of the particle can be mapped into a Lévy walk with rests, where flight times depend on the step size. In our system, the steps and rests are not alternate but have complementary probabilities at each event. We show how the existing theory for a Lévy walk with rests can be extended to study such kind of walk. We determine the relationship between the stochasticity of the environment and the anomalous diffusion of the particle by solving different configurations of our system, characterized by fixed or random compartment sizes and boundary transmittances.

The motion takes place on an environment characterized by a set of compartments with size _{i} ∈ [1, ∞). We treat the size of the compartments as a stochastic variable, following the probability distribution function (PDF)

For the sake of simplicity, we focus on the case where the compartments consist in one-dimensional segments (see

The motion of particles in disordered media has been thoroughly studied in the past [^{2}/

In the mesoscale description, the microscopic walk of the particle (represented by the black line in the same figure) is reduced to a collection of lengths (_{i}) and times (_{i}) traveled to exit the compartments, as shown by the green line of _{i} is the stochastic time the particle spent bouncing between the boundaries before being transmitted to next compartment. In our case, this time is related to the transmittance

Once inside a compartment, the particle has two options: leaving through the same boundary through which it entered, or through the opposite one. Since our approach monitors the particle only when exiting a boundary, in the latter case, the particle has traveled a distance equal to the size of the compartment. However, in the former, the particle is not effectively moving, since it occupies the same position when entering and exiting the compartment. This translates into a rest with duration equal to the time taken to exit the compartment. Therefore, after entering each compartment, the particle has a probability of resting φ_{r}(_{w}(_{r}(

Through this coarse-graining approach, we convert the microscale walk into a Lévy walk with rests, with flight times depending on the jump length [_{r} + φ_{w} = 1, that can be used to calculate the PDFs of walk [ψ_{w}(_{r}(

and, in the spirit of Zaburdaev et al. [

where

Here, _{0}(_{k} to the Fourier transform of _{w} = φ_{r} = 1/2, Equation (4) leads to the known result for the Lévy walk with rests [

However, when the previous condition is not fulfilled, solving Equation (4) requires the calculation of φ_{w}(_{w}(

For _{w}(_{i}/2 to escape the _{w}(_{i}. For this reason, in the following we will refer to this approximation as the

From now on we will focus on the osmotic approach, which allows for a thorough theoretical description in the different configurations considered. In the osmotic approach, Equation (4) takes the much simpler form

where

To characterize the motion of the particle, we will use the mean squared displacement (MSD), defined as

As we will show later through numerical simulations of the microscopic walk, in spite of the simpler description, the osmotic approach displays the same long time behavior as the non-osmotic one.

In the following, we will use the method described above to solve the motion of the particle in different configurations of the system. We will first consider the case in which each boundary has a different transmittance, drawn stochastically from the PDF

The form of the conditional probability of the exit time given a compartment of size ^{2}. We can further assume that the dependence on ^{2}. We checked that this behavior is consistent with the numerical results for a collection of ^{2}/^{2}, for large

This form of the conditional time also has the advantage of simplifying the analytical expressions and, as we discuss below, allows us to correctly model the microscopic motion in all the cases considered. The analytical calculation of this conditional probability falls beyond the scope of this work. We note that previous works have focused in the investigation on the exit time in similar structures [

We will now consider the case in which the boundaries have all the same transmittance, i.e.,

Our first step is to calculate the distribution of flight times, which is done by convolving Equation (1) over all possible values of

Using this result and Equations (1) and (7), we find that 〈 ^{2}(

A very different result arises when considering disordered boundary transmittances

We first analyze the case in which the compartments have all the same size, i.e., the lengths

As all the steps have equal length, the walk reduces to a continuous time random walk with waiting time PDF given by Meroz et al. [

showing that the particle undergoes subdiffusive motion for 0 < α < 1. In

We will now consider the case where both compartment length and boundary transmittance are stochastic variables. As stated before, this situation can be modeled at the mesoscale as a Lévy walk with flight times depending on the step size. We consider that the transmittances are distributed according to Equation (11) and the compartment lengths as described by Equation (9). Following the method used to derive Equation (12), we can calculate the PDF of flight times by convolving the conditional probability ϕ(

By using the previous result and Equation (1) we can determine the MSD through its Laplace transform as in Equation (7). In the time domain we find

The values of the MSD exponent

In this article, we introduce a coarse-graining method that we use to study diffusion through complex environments. This method is useful to study systems in which the microscopic behavior of the particles is too involved to be described analytically. To obtain a description of the motion in such cases, we propose a procedure that allows one to transform the microscopic walk into well-known theoretical models, such as Lévy Walks or continuous time random walks. The coarse-grained transformation maps the original walk performed at the microscale into a simplified movement at a larger scale (which we term mesoscale) that captures the relevant properties of the environment. This allows for a complete analytical characterization of the diffusion in terms its observables, such as the mean square displacement.

To illustrate the use of the proposed method, we consider the diffusion in an environment consisting of compartments with random sizes and/or transmittances. To resolve the diffusion of the system at the microscale, one needs to consider the complex interaction of the particle with the boundary of each compartment. For some simple systems, e.g., when all the compartments have the same size, it is possible to get an analytical solution of the microscale motion. In this cases, we show that a heavy-tailed distribution of boundary transmittance is a necessary requirement to induce subdiffusion. However, for more intricate spatially-disordered environments, it is often difficult to obtain an analytical solution at the microscale. This is the scenario where our method allows to get insights on the motion while neglecting microscopic details. As an example, we demonstrate that when the compartments length is a stochastic variable, geometric disorder alone cannot generate subdiffusion. However, it can affect the one generated by the heterogeneity in the boundary transmittance. Namely, increasing the geometric disorder reduces the degree of subdiffusion, as it increases the value of the anomalous exponent toward one. We thus fully characterize the mean-square displacement exponent as a function of the parameters controlling the heavy-tailed distributions of both the lengths and barrier heights.

The model presented in this article might be a useful framework to interpret diffusion in a variety of systems composed of compartment of varying size and barriers. A striking example of such kind of system is provided by eukaryotic cells, highly compartmentalized at different spatial scales to provide optimal conditions to perform specific functions [

An interesting outlook of our model could consist in the possibility of its further generalization, as to include previously proposed models for diffusion in complex environment. For example, our approach shares important features with the previously proposed comb model [

All authors contributed conception and design of the study. GM-G, AC, and MG-M developed the theory. GM-G performed the simulations. GM-G, MG-M, and CM wrote the paper. ML supervised research. All authors contributed to manuscript revision, read and approved the submitted version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We acknowledge Oriol Rubies for the initial numerical exploration of the problem and John Lapeyre and Vasily Zaburdaev for inspiring and useful discussions.