The Scientific World Journal

Volume 2014 (2014), Article ID 150826, 8 pages

http://dx.doi.org/10.1155/2014/150826

## Diagrammatic Analysis of Nonhomogeneous Diffusion

Sección Biofísica, Facultad de Ciencias, Universidad de la República, Iguá Esq. Mataojo, 11400 Montevideo, Uruguay

Received 5 November 2014; Accepted 13 December 2014; Published 31 December 2014

Academic Editor: Boris Martinac

Copyright © 2014 Julio A. Hernández. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

By virtue of its complexity, realistic approaches to describe diffusion in cellular media require the employment of computational methods. Among others, this type of studies has shown that the apparent diffusion coefficient of a macromolecular solute through a cytoplasmic-like medium exhibits a power-law dependence with the excluded volume. Power laws are ubiquitous findings in diverse systems, such as metabolic processes, population dynamics, and communication networks, and have been the object of many interpretative formal approaches. This study introduces a diagrammatic algorithm, inspired in previous ones employed to analyze multicyclic chemical systems, to derive expressions for nonhomogeneous diffusion coefficients and to study the effects of volume exclusion. A most noteworthy result of this work is that midsize diagrams of nonhomogeneous diffusion are already able to exhibit an approximate power-law dependence of the diffusion coefficient with the excluded volume. The employment of the diagrammatic method for the analysis of simple situations may thus prove useful to interpret some properties of larger network systems.

#### 1. Introduction

The cellular compartment is a highly crowded medium of great structural heterogeneity [1–4]. Due to this complexity, the realistic approaches to represent diffusion in cellular media usually employ computational simulations [5–10]. Among other properties, this type of studies has shown that the apparent diffusion coefficient of a macromolecular solute through a cytoplasmic-like medium exhibits a power-law dependence with the excluded volume [8], in agreement with theoretical predictions from the study of mechanical models of polymers in solution [11–13] and consistent with experimental evidence [14–17]. Power laws are ubiquitous findings in many different types of processes, ranging from metabolism to communication networks, and have been the subject of many interpretative formal approaches (e.g., [18–20]). For a thorough revision and a historical perspective of this topic, the reader should see the articles in Newman et al. [21], and more recent surveys can be found in Clauset et al. [22] and Pinto et al. [23].

The general objective of this study is to contribute to the formal analysis of diffusion of solutes in cellular media. The specific purposes are to introduce a diagrammatic algorithm to derive explicit expressions of nonhomogeneous diffusion coefficients and to employ this method to study the dependence of the diffusion coefficient with the excluded volume. Since this work is not intended to contribute with complex realistic examples of nonhomogeneous diffusion but to introduce a formalism to interpret some basic aspects of this type of processes, the models analyzed here are relatively simple. Nevertheless, they already embody some properties characteristic of systems with a high degree of complexity, such as the aforementioned power-law dependence of the apparent diffusion coefficient with the excluded volume.

#### 2. Diagrammatic Method for the Derivation of the Diffusion Coefficient of Solute Transport in Nonhomogeneous Media

The diagrammatic method was originally developed to analyze steady-state kinetics in chemical systems of intermediate complexity [24, 25] and was further employed to interpret diverse biochemical and biophysical processes, for instance, water and solute transport through biological membranes [26, 27]. As shown here, the method can be extended to obtain diffusion coefficients of steady-state diffusion in nonhomogeneous media. For this purpose, the nonhomogeneous medium is conceived as a network of transitions between selected positions or nodes, each one characterized by a specific concentration of the diffusing species. Discrete network approaches to represent nonhomogeneous processes of transport have been employed, for example, to understand the basic aspects of percolation [28]. The multicompartment representation adopted in this study permits expressing the transition of the solute between nodes via kinetic expressions. This type of strategies has been utilized, for instance, to understand the role of diffusion in brain processes [29] and to describe sarcomeric calcium movement [30].

The flux of a permeating species through a membrane has been classically analyzed assuming the existence of a series of potential energy barriers. In this one-dimensional case, the kinetic formalism permits obtaining explicit expressions for the net flux in terms of the kinetic constants of jumping between neighbor positions in a rather straightforward fashion [31]. Similarly, the flux of a solute through a two- or three-dimensional nonhomogeneous medium can be conceived as mediated by transitions between positions separated by potential energy barriers. As mentioned, in these situations the derivation of explicit expressions for the solute fluxes may benefit from the employment of a simplifying algorithm, such as the diagram method proposed in this study. Instead of deriving general expressions, the procedure to obtain a kinetic expression for the nonhomogeneous diffusion coefficient is illustrated here employing the diagram shown in Figure 1(a). The basic assumption is that the diffusion of a solute between positions “” and “” only occurs via one or more specific paths that connect intermediate positions or nodes. As in chemical kinetics, the transitions connecting two neighbor positions are governed by rate constants. This work assumes that, for each transition, the rate constants in the two directions are equal. This assumption guarantees the accomplishment of the detailed balance condition in all of the cases. In steady state, the entrance flux into node equals the exit flux at node . The concentrations of the transported substance in these nodes ( and ) and in the intermediate nodes ( and ) are determined by the externally imposed steady-state flux () and by the rate constants (). The rate constants (’s) have dimensions LT^{−1}.