Reaction?diffusion models of development with state-dependent chemical diffusion coefficients

C ROUSSEL
2004 Progress in Biophysics and Molecular Biology  
Reaction-diffusion models are widely used to model developmental processes. The great majority of current models invoke constant diffusion coefficients. However, the diffusion of metabolites or signals through tissues is frequently such that this assumption may reasonably be questioned. We consider several different physical mechanisms leading to effective diffusion coefficients in biological tissues which vary with the local conditions, including models in which juxtacrine signaling results in
more » ... the diffusion of a signal in the absence of material transport. We develop a mathematical formalism for transforming local transport laws into diffusive terms. This procedure is appropriate when the typical length scale over which the concentrations change significantly is much greater than the dimensions of a cell. We review previous developmental models which considered the possibility of state-dependent diffusion coefficients. We also provide a few new motivating examples. r molecules of a certain type then, in the absence of other effects, this random motion will result in a net flow of molecules from the region of high concentration to the region of lower concentration. This is a simple statistical effect which does not depend on the detailed mechanism by which molecules transit from one region to the other. The kinetics of diffusion, on the other hand, do depend on the transport mechanism. When the number of molecules in a system is sufficiently large, the fact that molecules are countable entities becomes less and less significant. The concentration (the number of molecules per unit volume) then becomes a continuous and, generally, differentiable variable of space and time. In this limit, one can describe reaction-diffusion systems using differential equations. This is the regime which we will consider in this paper. In a simple unstructured medium (e.g. an ordinary solvent), ideally behaving solutes (i.e. ones for which solute-solute interactions are negligible) obey Fick's two laws of diffusion (Crank, 1956) . However, transport in biological systems, even when it is purely diffusive in nature, frequently violates the assumptions which lead to Fick's laws. First, the medium is cellular in nature so that over distance scales which are relevant for development, the medium can hardly be described as unstructured. In fact, the cellular interior is such that even on the scale of a single cell, classical Fickian diffusion is at best a first approximation (Luby-Phelps, 1993; Agutter et al., 1995) . Furthermore, solute-solute interactions are often of considerable importance, particularly since many key metabolites are ionic species. In some cases, these added complexities require elaborate mathematical descriptions of the diffusion process. In many others, the chief effect is to make the diffusion coefficient concentrationdependent or, in general, dependent on some other aspect of the state of the system. In this paper, we examine several questions relating to the formulation of reaction-diffusion models, with a particular emphasis on issues of relevance in the modeling of development: In what cases can we describe diffusion of biochemical species through biological tissues using Fick's first law with state-dependent diffusion coefficients? What analytic forms for the diffusion coefficients are we likely to encounter in practice? What are the consequences of including state-dependent diffusion coefficients in a model? Before proceeding further, it is useful to review a few of the key concepts used in the mathematical description of diffusion. To start, imagine a small surface of area dA oriented perpendicular to one of the coordinate axes, say the x-axis. The flux of material in the x direction, J x , is defined as the number of molecules (generally counted in moles) which pass through the surface per unit area per unit time. In other words, the number of molecules passing through the surface in time dt is just J x dA dt. The net flux ought to depend on the number of molecules in small regions to either side of the surface: If there are more molecules on the left then, averaged over a small period of time dt, we would expect a left-to-right flux which grows in size as the difference in concentration to either side of our surface increases. Moving our little surface from one point in space to another, we may find that this local difference changes. Thus, the flux depends on the position in space, i.e. J x =J x (x,y,z). Of course, we could just as easily talk about the flux through surfaces perpendicular to the y or z axes. Accordingly, flux is a vectorial quantity. The idea that the flux should depend in some way on the concentration difference across our imaginary surface(s) immediately leads us to the question of the precise mathematical form of this dependence. The simplest possibility is Fick's first law, namely that the flux is proportional to the local concentration difference or, more accurately, to the local derivative of the concentration (c) with respect to the spatial variables: J x =ÀD @c/@x in one dimension, or J=ÀDrc in three
doi:10.1016/j.pbiomolbio.2004.03.001 pmid:15261527 fatcat:fz45sgaokjcgbaym7t6rp6xhxm