Scale-invariance in reaction-diffusion models of spatial pattern formation.

Abstract

We propose a reaction-diffusion model of spatial pattern formation whose solutions can exhibit scaleinvariance over any desired range for suitable choices of parameters in the model. The model does not invoke preset polarity or any other adboc distinction between cells and provides a solution to the French flag probleit without sources at the boundary. Furthermore, patterns other than the polar pattern that usually arises first in a growing one-dimensional syst6m described by Turing's model can be obtained. Evidence is given that suggests that the model may apply in the slug stage of Dictyostelium discoideum. (5).The assumption that the kinetic and diffusion coefficients in a reaction-diffusion model are constant is certainly only an approximation, and our purpose here is to show that any desired degree of scale-invariance can be achieved via a simple, yet plausible, mechanism by which the overall size of the system is reflected in these coefficients. We treat in detail a case in which it is assumed that the diffusion coefficients of the morphogens depend on the concentration of a diffusible regulatory species that is produced at a constant rate by all cells, and we show that the desired invariance can be obtained by adjusting the production rate and the rate at which the regulatory species leaks into the sourroundings.There are several ways in which such concentration-dependence could arise. For instance, the permeability of the cell membrane (either junctional or ordinary) could be increased by the regulatory species; in the continuum description we adopt here, this would be reflected in an increased diffusion coefficient. Alternatively, the regulatory species could facilitate transport by increasing adsorption of the morphogens on cellular structures, thereby enhancing surface diffusion, or through the formation of specialized transport structures. Recent experimental evidence shows that the postulated effect on membrane permeability actually occurs in some developing systems. For example, the molting hormone ecdysone stimulates a period of increased epidermal communication, as measured by the degree of electrical coupling between cells, in the prepupa stage of the beetle Tenebrio (7), and there is evidence that gap junctions in some amphibian oocytes are hormonally controlled (8). In these examples the source of the regulatory species is, extracellular, but the same effect can be achieved when the source is intracellular. SCALE-INVARIANCE IN TURING'S MODELLet S denote the region of space occupied by the developing system, whose boundary is assumed to be impermeable to the morphogens but not to the regulatory species, and suppose that transport within S is solely by diffusion. Suppose that there are N morphogens whose concentration is C = (C1, . . ., CN)T, and let CN+ 1 denote the concentration of the regulatory species.We first consider the case in which the regulatory species affects only the morphogen diffusivities and, for simplicity, we assume that the effect is linear in CN+ 1. Then the govern...

Reaction and Diffusion on Growing Domains: Scenarios for Robust Pattern Formation

“…This semi-scale invariance is particularly significant in that it allows regulation, a property of many biological systems, whereby a specific number of pattern elements is laid down despite significant variation in the domain size and without the need to finely tune parameter values. Various authors have considered the problem of scale invariance in reaction-diffusion systems (Othmer and Pate, 1980;Hunding and Sørensen, 1988) and have concluded that some form of feedback from the domain size into the kinetic parameters is necessary, requiring close parameter tolerances. The semi-scale invariance in the present system arises as a natural consequence of the sequence formation mechanism.…”

Pattern formation in reaction-diffusion models with spatially inhomogeneous diffusion coefficients

“…There are several ways in which this can be done, and here we focus on one possibility. The composite model that we consider is one in which the spatial variation in diffusion coefficients is controlled by a regulatory chemical c. In our model we assume, as before, that the diffusion coefficient of u is constant, but we now assume that the diffusion coefficient of v is modulated by c. This could, for example, reflect an increase in gap junction permeability for v due to the presence of c (Othmer & Pate, 1980). We consider a mechanism for the production of c, in which c is secreted at one end of the domain, diffuses and is degraded throughout the domain, but does not flow through the other boundary.…”

“…In this paper we consider how an underlying spatial prepattern in diffusion coefficients modifies the pattern forming properties of a reaction-diffusion model. In developmental biology, spatial variation in morphogen diffusivity may be controlled by the concentration of a regulatory chemical which, for example, affects morphogen transport by binding to the morphogen or by modulating gap junction permeability (Othmer & Pate, 1980, Hunding & Serenson, 1988Brummer et al, 1991). Several authors have investigated numerically the effects of spatially varying reaction terms in reaction-diffusion systems (Gierer & Meinhardt, 1972;Hunding et al, 1990, Lacalli, 1990) and the effects of spatially varying diffusion (Hunding, 1987(Hunding, , 1989.…”

Scale invariance of morphogen-mediated patterning by flux optimization

“…However, appropriate changes must also be made to the input flux , lest the profile will increase or decrease in amplitude depending on whether diffusion or decay rates are modified. We have previously analyzed this system in detail and refer the interested reader to Othmer et al, 1980, Umulis et al 2008 and Umulis 2009 [6], [7], [8]. Herein we focus on an alternative and remarkably simple mechanism that confers approximate scale invariance in Drosophila embryos.…”