Diffusion Through a Network of Compartments Separated by Partially-Transmitting Boundaries

Gorka Muñoz-Gil,Maciej Lewenstein,Carlo Manzo,Miguel Angel Garcia-March,Alessio Celi

doi:10.3389/fphy.2019.00031

Abstract

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 which it is monitored only when the particle crosses a boundary between compartments. Both descriptions provide --by construction-- the same long time behavior. The analytical description obtained at the proposed mesoscale allows for a complete characterization of the complex movement at the microscale, thus representing a fruitful approach for this kind of systems. We show that the presence of disorder in the transmittances is a necessary condition to induce anomalous diffusion, whereas the spatial heterogeneity reduces the degree of subdiffusion and, in some cases, can even compensate for the disorder induced by the stochastic transmittance.

Highlights

The characterization of the diffusive behavior in complex environments is crucial in many fields, ranging from biology [1], via physics and chemistry, to geology [2]
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
In order to study the behavior of the particle, we propose a coarse-graining approach transforming the rather complex walk of the particle into two very well known theoretical models describing anomalous diffusion: continuous time random walks and Lévy walks

Summary

Introduction

The characterization of the diffusive behavior in complex environments is crucial in many fields, ranging from biology [1], via physics and chemistry, to geology [2]. It has been shown that a large number of systems display anomalous diffusion associated to spatial and/or energetic disorder of the environment. The characterization of this movement provides important information on the disorder of the media and on the laws governing the system [3] The advances in this field have been mainly driven by developments in fluorescence microscopy, which enable us to record movies of single particles diffusing in living matter, with a spatial precision of a few nanometers at the millisecond time scale [4]. The presence of barriers that prevent the particles to freely diffuse in the environments is a general mechanism used to explain subdiffusion [5]. Recent experimental observations in cellular biology have shown that the actin cytoskeleton acts as a compartmentalization scaffold for proteins diffusing

Methods

Results

Conclusion