ORIGINAL RESEARCH published: 18 March 2019 doi: 10.3389/fphy.2019.00031 Diffusion Through a Network of Compartments Separated by Partially-Transmitting Boundaries Gorka Muñoz-Gil 1 , Miguel Angel Garcia-March 1*, Carlo Manzo 2*, Alessio Celi 1,3,4 and Maciej Lewenstein 1,5 1 ICFO - Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, Barcelona, Spain, 2 Facultat de Ciències i Tecnologia, Universitat de Vic - Universitat Central de Catalunya (UVic-UCC), Vic, Spain, 3 Center for Quantum Physics, University of Innsbruck, and Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Innsbruck, Austria, 4 Departament de Física, Universitat Autònoma de Barcelona, Bellaterra, Spain, 5 ICREA, Barcelona, Spain 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 L and the boundaries transmittance T. We identify two relevant spatio-temporal scales that provide alternative descriptions of the dynamics: (i) the microscale, in which the particle position is monitored at constant time intervals; and (ii) the mesoscale, in Edited by: which it is monitored only when the particle crosses a boundary between compartments. Carlos Mejía-Monasterio, Both descriptions provide—by construction—the same long time behavior. The analytical Polytechnic University of Madrid, Spain description obtained at the proposed mesoscale allows for a complete characterization Reviewed by: of the complex movement at the microscale, thus representing a fruitful approach for Diego R. Amancio, this kind of systems. We show that the presence of disorder in the transmittance is a University of São Paulo, Brazil necessary condition to induce anomalous diffusion, whereas the spatial heterogeneity Haroldo Valentin Ribeiro, Universidade Estadual de Maringá, reduces the degree of subdiffusion and, in some cases, can even compensate for the Brazil disorder induced by the stochastic transmittance. *Correspondence: Miguel Angel Garcia-March Keywords: random walk, anomalous diffusion, stochastic processes, complex systems, barriers miguel.garcia-march@icfo.eu Carlo Manzo carlo.manzo@uvic.cat 1. INTRODUCTION Specialty section: The characterization of the diffusive behavior in complex environments is crucial in many fields, This article was submitted to ranging from biology [1], via physics and chemistry, to geology [2]. Recently, it has been shown Interdisciplinary Physics, that a large number of systems display anomalous diffusion associated to spatial and/or energetic a section of the journal disorder of the environment. Often, the motion of particles in such systems has been shown to be Frontiers in Physics subdiffusive, i.e., x2 (t) ∼ t σ with anomalous exponent 0 < σ < 1. The characterization of this Received: 20 November 2018 movement provides important information on the disorder of the media and on the laws governing Accepted: 21 February 2019 the system [3]. The advances in this field have been mainly driven by developments in fluorescence Published: 18 March 2019 microscopy, which enable us to record movies of single particles diffusing in living matter, with a Citation: spatial precision of a few nanometers at the millisecond time scale [4]. Muñoz-Gil G, Garcia-March MA, The presence of barriers that prevent the particles to freely diffuse in the environments is Manzo C, Celi A and Lewenstein M a general mechanism used to explain subdiffusion [5]. Indeed, there exists a plethora of works (2019) Diffusion Through a Network of Compartments Separated by treating the effect of these barriers in various forms, from local maxima in potential landscapes [6] Partially-Transmitting Boundaries. to thin slices of poorly diffusive materials [7]. Recently, an analytical approach has been proposed Front. Phys. 7:31. for sufficiently regular geometries [8]. Recent experimental observations in cellular biology have doi: 10.3389/fphy.2019.00031 shown that the actin cytoskeleton acts as a compartmentalization scaffold for proteins diffusing Frontiers in Physics | www.frontiersin.org 1 March 2019 | Volume 7 | Article 31 Muñoz-Gil et al. Diffusion Through a Network of Compartments in the plasma membrane [9, 10], hence stressing the importance walk with rests, where flight times depend on the step size. of studying the motion in such environments. Moreover, In our system, the steps and rests are not alternate but have the evidence of the occurrence of ergodic and non-ergodic complementary probabilities at each event. We show how the processes in the diffusion of biomolecules [11] has triggered the existing theory for a Lévy walk with rests can be extended description of models in which geometric and energetic disorders to study such kind of walk. We determine the relationship coexist [12, 13]. between the stochasticity of the environment and the anomalous In this article, we study a general barrier model, where a diffusion of the particle by solving different configurations of our particle performs an unbiased random walk through a complex system, characterized by fixed or random compartment sizes and environment made by a mesh of compartments separated by boundary transmittances. barriers with random transmittance. A schematic of the system is shown in Figure 1. We show that even though the particle performs a Brownian motion within each compartment, the 2. MATERIALS AND METHODS stochasticity of the barrier’s transmittance induces anomalous diffusion for the overall movement. We also explore the effect The motion takes place on an environment characterized by a set of the stochasticity in the length of the compartments, showing of compartments with size {Li }N i=1 , with N ≫ 1 and Li ∈ [1, ∞). that it generally increase the anomalous exponent, up to restoring We treat the size of the compartments as a stochastic variable, normal diffusion. following the probability distribution function (PDF) g(L). The In order to study the behavior of the particle, we propose compartments form a meshwork with unbounded connectivity, a coarse-graining approach transforming the rather complex which we assume to be always sufficiently large such to make walk of the particle (mainly due to the interaction with the very unlikely that the particle returns to the same compartment boundaries) into two very well known theoretical models after leaving it. The boundary between the compartments is describing anomalous diffusion: continuous time random walks partially reflective, i.e., a particle reaching a boundary has a and Lévy walks. The former, introduced by Montroll and finite probability T of moving through the boundary to the next Weiss [14], consists in a random walk where the time between compartment and a complementary probability R = 1 − T successive steps is a stochastic value. Similarly, a Lévy Walk [15] of being reflected. The transmittance of each segment {Ti }N i=1 , consists in a random walk where not only the step time is T ∈ (0, 1] is a random variable drawn from the PDF q(T). a stochastic variable, but also its length. For a comprehensive For the sake of simplicity, we focus on the case where introduction to both models, we refer the readers to Klafter and the compartments consist in one-dimensional segments (see Sokolov [16] and Zaburdaev et al. [17]. Figure 1A). The extension of this theory to two- or three- In the most general description of our system, we show dimensional supports, like circles or spheres, is conceptually how the walk of the particle can be mapped into a Lévy straightforward but more elaborated and geometry-dependent, since it requires the determination of the stochastic time that the particle spends in each support. The particle performs an unbiased, discrete, random walk through the environment, temporarily confined between the boundaries until it is transmitted to the next compartment. The motion of particles in disordered media has been thoroughly studied in the past [18]. The usual approach is to explicitly solve the diffusion equation for the system under study. For instance, such direct approach has been recently applied to subdiffusive particles through the barrier separating two liquids [19]. However, when considering systems like the one presented above, where both the boundary transmittance and compartment length are stochastic variables, the direct approach is complicated and does not lead to exact analytical results. Therefore, we use an alternative method to solve the motion of the particle through such a system. First, we distinguish between a microscale description, in which the position of the particle is monitored at constant times << L2 /D with D being the diffusivity, and a mesoscale description, in which the position is FIGURE 1 | Schematic of the system. (A) Example of a one dimensional sampled at times subordinated to the exit from a compartment. compartmentalized environment, with compartment size L and boundary transmittance T. Higher boundaries represent lower transmittance. For We note here that, by definition, the asymptotic behavior of simplicity we plot the meshwork as formed by the segments of a line. (B) the motion of the particle coincides on both scales. Therefore, Motion of the particle in such environment. The dark line represents the studying the movement at the mesoscale provides a correct microscale description of the motion and the green one indicates its description of the movement at long times. mesoscale description, in this case a Lévy walk with steps given by their length and flight time (L, t). In the mesoscale description, the microscopic walk of the particle (represented by the black line in the same figure) is Frontiers in Physics | www.frontiersin.org 2 March 2019 | Volume 7 | Article 31 Muñoz-Gil et al. Diffusion Through a Network of Compartments reduced to a collection of lengths (Li ) and times (ti ) traveled to ϕr = 1/2, Equation (4) leads to the known result for the Lévy exit the compartments, as shown by the green line of Figure 1B. walk with rests [17]. As a matter of fact, the length traveled by the particle in each step However, when the previous condition is not fulfilled, solving corresponds to the size of the compartment itself. The flight time Equation (4) requires the calculation of ϕw (L, T). A case in which ti is the stochastic time the particle spent bouncing between the ϕw (L, T) is easily solvable is when the boundaries are completely boundaries before being transmitted to next compartment. In our transmitting, i.e., q(T) = δ(T − 1). In that case, one finds case, this time is related to the transmittance T and length L of the compartment through the conditional probability φ(t|T, L). L ϕw (L, T = 1) = ϕw (L) = 1 − ∼ L−1 . (5) One can then write the joint probability for the particle to be L+1 in a compartment of length L and boundary transmittance T at time t as For T 6= 1, obtaining an analytical expression for ϕw (L, T) is a challenging task [21]. A trick commonly used to avoid this ψ(t, L, T) = φ(t|T, L)g(L)q(T). (1) difficulty consists in considering an annealed system [18], i.e., assuming that each time the particle exits a compartment, it Once inside a compartment, the particle has two options: leaving reappears at the center of the next one. In this case, the particle through the same boundary through which it entered, or through will always travel a distance Li /2 to escape the ith-segment, the opposite one. Since our approach monitors the particle independently on the exit side, hence eliminating the presence only when exiting a boundary, in the latter case, the particle of rests. In this case ϕw (L, T) = 1 ∀L, T and the motion of the has traveled a distance equal to the size of the compartment. particle is then a Lévy walk with flying times depending on the However, in the former, the particle is not effectively moving, jump length [20]. This is also analogous to the case in which, once since it occupies the same position when entering and exiting the particle enters a compartment, it cannot cross again the same the compartment. This translates into a rest with duration equal edge it entered from and thus will always travel a distance Li . For to the time taken to exit the compartment. Therefore, after this reason, in the following we will refer to this approximation entering each compartment, the particle has a probability of as the osmotic approach, in contrast with the general case that we resting ϕr (L, T) and the complementary probability of walking name non-osmotic. ϕw (L, T) = 1 − ϕr (L, T). From now on we will focus on the osmotic approach, which Through this coarse-graining approach, we convert the allows for a thorough theoretical description in the different microscale walk into a Lévy walk with rests, with flight times configurations considered. In the osmotic approach, Equation (4) depending on the jump length [20]. Previous works have takes the much simpler form extensively studied such kind of walks, both with alternating walks and rests [16] or with an equal probability of resting and (OA) 9(k, s) walking [17]. However, our system shows a substantial difference, P6 = , (6) 1 − ψ(k, s) since it displays different probabilities of resting or walking, ϕr + ϕw = 1, that can be used to calculate the PDFs of walk R1 where ψ(k, s) = 0 ψ(k, s, T)dT. [ψw (t)] and rest times [ψr (t)] as To characterize the motion of the particle, we will use Z ∞ Z 1 the mean squared displacement (MSD), defined as x2 (t) = ψw(r) (t) = dL ϕw(r) (L, T)ψ(t, L, T)dT, (2) −P′′ (k, s)|k=0 , which can be rewritten as Massignan et al. [22] 1 0 Z 1  −ψ ′′ (k, s)|k=0 −9 ′′ (k, s)|k=0  and, in the spirit of Zaburdaev et al. [17], to derive the density of x2 (s) = dT + . (7) 0 s[1 − ψw (s)] 1 − ψw (s) particles at position x and time t in the Fourier–Laplace space Z 1 As we will show later through numerical simulations of the P6 (k, s) = P6,T (k, s, T)dT, (3) microscopic walk, in spite of the simpler description, the osmotic 0 approach displays the same long time behavior as the non- osmotic one. where P6,T (k, s, T) = (4) 3. RESULTS  9r (s)P0 (k) + ϕw (x, T)9(x, s, T) k ψr (s)P0 (k) In the following, we will use the method described above to . 1 − {ϕr (x, T))ψ(x, s, T)}k ψr (s) solve the motion of the particle in different configurations of the system. We will first consider the case in which each boundary Here, P0 (x) Rcorresponds to the initial distribution of particles, has a different transmittance, drawn stochastically from the PDF ∞ ′ ′ 9(t) = t ψ(t )dt to the survival probability, i.e., the q(T), but all the compartments have equal length. We will then probability R∞ of not jumping until time t, 9(x, t, T) = briefly comment about the case in which the stochasticity is only ′ ′ t ψ(x, t , T)dt to the PDF of the displacement of the walker present in the compartment length. Last, we will consider the case during the last uncompleted step, and {f (x)}k to the Fourier where both the length and boundary transmittance are random transform of f (x). For constant step/rest probabilities, e.g., ϕw = variables. For each case, we will give the analytical solutions of Frontiers in Physics | www.frontiersin.org 3 March 2019 | Volume 7 | Article 31 Muñoz-Gil et al. Diffusion Through a Network of Compartments the mesoscopic walk and compare it to numerical simulations of We first analyze the case in which the compartments have all the the microscopic description. same size, i.e., the lengths {Li }N i=1 are distributed according to the The form of the conditional probability of the exit time PDF g(L) = δ(L − L̄), where L̄ ∈ [1, ∞). We refer to this system given a compartment of size L and transmittance T is common as the spatially ordered case. In the osmotic approach, the walk to all the cases. A reasonable assumption based on the consists on a collection of steps of size L̄ with flight times drawn Brownian motion is that, independently on the expression of from the PDF this conditional probability, it should give an average time Z 1 for exiting a compartment hti which depends on the length ψ(t) = φ(t|L̄, T)q(T)dT ∝ t −1−α/2 . (12) as L2 . We can further assume that the dependence on T is 0 such that hti ∝ (L/T)2 . We checked that this behavior is As all the steps have equal length, the walk reduces to a consistent with the numerical results for a collection of T and continuous time random walk with waiting time PDF given by L, finding that the average exit time follows an exponential Meroz et al. [12]. Thus, in the spatially ordered case the MSD is behavior, ∝ exp −tL2 /T 2 , for large L and small T. For all given by Charalambous et al. [26] cases numerically considered, even when the distribution did not match an exponential behavior, we found a quadratic dependence (SO) t→∞ on L/T. Therefore, we assume the simplest distribution which x2 (t) −−−→ t α/2 , (13) produces the expected behavior of the average exit time, which is showing that the particle undergoes subdiffusive motion for 2 φ(t|T, L) ∼ δ(t − (L/T) ). (8) 0 < α < 1. In Figure 2A we show the numerical results corresponding to MSD calculated for a single value of α = 0.2 This form of the conditional time also has the advantage of and different values of L̄ by using the microscale description simplifying the analytical expressions and, as we discuss below, for the spatially ordered case. The plot shows that the motion allows us to correctly model the microscopic motion in all the is initially Brownian and become subdiffusive at longer times. cases considered. The analytical calculation of this conditional The time at which the onset of subdiffusion occurs increases as L̄ probability falls beyond the scope of this work. We note that grows, corresponding to the time needed to reach the boundary, previous works have focused in the investigation on the exit of a compartment. The asymptotic value of the MSD for any time in similar structures [23, 24], but do not provide a L̄ is given by Muñoz Gil et al. [13]. This is a first indication useful expression for our particular system nor a practical way that anomalous diffusion can only be obtained by considering to derive it. stochastic boundary transmittance with a heavy-tail PDF. In the We will now consider the case in which the boundaries have spatially ordered case, the distribution of transmittances of the all the same transmittance, i.e., q(T) = δ(T − T̄), with T̄ ∈ (0, 1]. media can be directly inferred from the asymptotic behavior of We will consider that each compartment has a different length, the MSD of the particle. retrieved from the PDF We will now consider the case where both compartment length and boundary transmittance are stochastic variables. As g(L) = βL−1−β . (9) stated before, this situation can be modeled at the mesoscale as a Lévy walk with flight times depending on the step size. Our first step is to calculate the distribution of flight times, which We consider that the transmittances are distributed according is done by convolving Equation (1) over all possible values of L to Equation (11) and the compartment lengths as described by Z ∞ Equation (9). Following the method used to derive Equation (12), ψL (t) = φ(t|L, T̄)g(L)dL ∝ t −1−β/2 . (10) we can calculate the PDF of flight times by convolving the 1 conditional probability φ(t|T, L) with Equations (11) and (9), to find Using this result and Equations (1) and (7), we find that x2 (t) ∼ ( t ∀ T, i.e., the particle performs normal diffusion. Therefore, −1−γ α if β > α, ψf (t) ∝ t , with γ = (14) the stochasticity of the compartment length does not imply any β if β < α. effect on the MSD and, indeed, similar results are obtained when using regular compartment size. We would like to emphasize that By using the previous result and Equation (1) we can determine this result holds for any finite T different from zero. In fact, as the MSD through its Laplace transform as in Equation (7). In the shown in Lapeyre [25], for T = 0 subdiffusion occurs. In the time domain we find presence of transmitting boundary, there is no mechanism that (SD) t→∞ 1 confines the particle for pathologically long times, so particles x2 (t) −−−→ t 2 (2−β+γ ) . (15) diffuse normally in the asymptotic limit. A very different result arises when considering disordered The values of the MSD exponent σ = 12 (2 − β + γ ) obtained for boundary transmittances {Ti }N different values of α and β are shown in Figure 2B. In Figure 2C i=1 distributed according to a power law PDF we further show the values of the MSD exponent calculated from numerical simulations for the microscale description of the walk  1−α 1 (dashed lines) and the theoretical value given by Equation (15). q(T) = α . (11) The numerical calculation and the theoretical prediction show T Frontiers in Physics | www.frontiersin.org 4 March 2019 | Volume 7 | Article 31 Muñoz-Gil et al. Diffusion Through a Network of Compartments 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 FIGURE 2 | (A) MSD of a particle moving in an system of compartments of increases the value of the anomalous exponent toward one. We equal length and boundary transmittances distributed following (11), with α = 0.2. All curves are calculated for the microscale and tend to the predicted thus fully characterize the mean-square displacement exponent subdiffusive motion given by (13). A larger L leads to a larger time for the onset as a function of the parameters controlling the heavy-tailed of subdiffusion to occur. The dashed bottom line corresponds to the distributions of both the lengths and barrier heights. mesoscale and coincides with the theoretical prediction. (B) Value of the The model presented in this article might be a useful exponent of the MSD in a system with stochastic compartment sizes and framework to interpret diffusion in a variety of systems boundary transmittance, given by (15). (C) Comparison between the predicted results of the previous case and numerical simulations of the microscopic walk composed of compartment of varying size and barriers. A striking (dashed lines). 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 [27]. The presence of compartments has been shown to affect the a good agreement. It can be noticed that, when α > β (and diffusion of transmembrane proteins in the plasma membrane, thus γ = β), the particle movement is normally diffusive (see e.g., as a consequence of a self-similar actin network acting as Equation (15) and Figures 2B,C). Therefore, the stochasticity semipermeable barrier [9]. in the length of the segments is capable of compensating for An interesting outlook of our model could consist in the the disorder that would be induced by the stochasticity in possibility of its further generalization, as to include previously transmittance, that would generate a subdiffusive motion with proposed models for diffusion in complex environment. For anomalous exponent σ = α/2 in the case the segments lengths example, our approach shares important features with the were regular. In addition, for β > α, the motion is subdiffusive, previously proposed comb model [28]. In fact, the comb model but with a higher anomalous exponent as compared to the case can be considered as a continuous-time random walk with in which the lengths were regular. Therefore, in this case the two stochastic waiting time, the latter derived from first-passage disorders compete, producing a weaker subdiffusion. time. This system can be analyzed through our coarse-grained approach upon conversion of the waiting time distribution into 4. DISCUSSION a stochastic transmittance. The realization of the comb model including convective terms [29], could be further implemented in In this article, we introduce a coarse-graining method that we use our approach, e.g., through the use of asymmetric transmittance. 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 AUTHOR CONTRIBUTIONS a description of the motion in such cases, we propose a procedure that allows one to transform the microscopic walk into well- All authors contributed conception and design of the study. known theoretical models, such as Lévy Walks or continuous GM-G, AC, and MG-M developed the theory. GM-G performed time random walks. The coarse-grained transformation maps the simulations. GM-G, MG-M, and CM wrote the paper. the original walk performed at the microscale into a simplified ML supervised research. All authors contributed to manuscript movement at a larger scale (which we term mesoscale) that revision, read and approved the submitted version. Frontiers in Physics | www.frontiersin.org 5 March 2019 | Volume 7 | Article 31 Muñoz-Gil et al. Diffusion Through a Network of Compartments FUNDING Catalunya (AGAUR Grant No. 2017SGR940). AC acknowledges financial support from the ERC Synergy Grant UQUAM This work has been funded by the Spanish Ministry MINECO and the SFB FoQuS (FWF Project No. F4016-N23). GM (National Plan15 Grant: FISICATEAMO No. FIS2016-79508-P, acknowledges financial support from Fundació Social La Caixa. SEVERO OCHOA No. SEV-2015-0522, FPI), European Social AC acknowledges support from the UAB Talent Research Fund, Fundació Cellex, Generalitat de Catalunya (AGAUR program and from the Spanish Ministry of Economy and Grant No. 2017 SGR 1341 and CERCA/Program), ERC Competitiveness under Contract No. FIS2017-86530-P. AdG OSYRIS, EU FETPRO QUIC, and the National Science Centre, Poland-Symfonia Grant No. 2016/20/W/ST4/00314. ACKNOWLEDGMENTS CM acknowledges funding from the Spanish Ministry of Economy and Competitiveness and the European Social Fund We acknowledge Oriol Rubies for the initial numerical through the Ramón y Cajal program 2015 (RYC-2015-17896) exploration of the problem and John Lapeyre and Vasily and the BFU2017-85693-R, and from the Generalitat de Zaburdaev for inspiring and useful discussions. REFERENCES 18. Bouchaud JP, Georges A. Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys Rep. (1990) 195:127–293. 1. 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