Reaction-diffusion Patterns Research Articles

Pattern formation in the ferrocyanide-iodate-sulfite reaction: The control of space scale separation

We revisit the conditions for the development of reaction-diffusion patterns in the ferrocyanide-iodate-sulfite bistable and oscillatory reaction. This hydrogen ion autoactivated reaction is the only example known to produce sustained stationary lamellar patterns and a wealth of other spatio-temporal phenomena including self-replication and localized oscillatory domain of spots, due to repulsive front interactions and to a parity-breaking front bifurcation (nonequilibrium Ising-Bloch bifurcation). We show experimentally that the space scale separation necessary for the observation of stationary patterns is mediated by the presence of low mobility weak acid functional groups. The presence of such groups was overlooked in the original observations made with hydrolyzable polyacrylamide gels. This missing information made the original observations difficult to reproduce and frustrated further experimental exploitation of the fantastic potentialities of this system. Using one-side-fed spatial reactors filled with agarose gel, we can reproduce all the previous pattern observations, in particular the stationary labyrinthine patterns, by introducing, above a critical concentration, well controlled amounts of polyacrylate chains in the gel network. We use two different geometries of spatial reactors (annular and disk shapes) to provide complementary information on the actual three-dimensional character of spatial patterns. We also reinvestigate the role of other feed parameters and show that the system exhibits both a domain of spatial bistability and of large-amplitude pH oscillations associated in a typical cross-shape diagram. The experimental method presented here can be adapted to produce patterns in the large number of oscillatory and bistable reactions, since the iodate-sulfite-ferrocynide reaction is a prototype of these systems.

Patterns of the Ferrocyanide−Iodate−Sulfite Reaction Revisited: The Role of Immobilized Carboxylic Functions

We experimentally demonstrate that the standing lamella reaction-diffusion patterns initially observed 17 years ago in a gel-filled open spatial reactor operated with the ferrocyanide-iodate-sulfite bistable reaction requires an upper critical concentration of low-mobility species with weak acid functional groups, a parameter that was overlooked at the time and had made observations difficult to reproduce. The present approach enables the control of the space scale separation between activatory and inhibitory processes. It makes the wealth of exotic pattern dynamics observed earlier easier to reproduce and understand. This contribution should considerably revive the interest in this reaction and boost the search for the control of reaction-diffusion patterns in other bistable systems.

Chemical origin of the sustained-like pattern formation observed in the bromate — Dual substrate — Dual catalyst oscillatory batch system

The BrO3− — BrAc — Ru(bpy)32+ subsystem is shown to represent the core oscillator that serves as source of the long lasting temporal and spatial periodic behaviors observed in the BrO3− — H2PO2− — acetone — Mn2+ — Ru(bpy)32+ — acid “double substrate-double catalyst” oscillatory batch system. The BrAc — the substrate of the core oscillator — is formed and accumulated in the reactions taking place in the six-component system. BrAc was produced in a separate experiment with bromide, acetone, acid and excess bromate and the mixture was used for bringing about patterns in the thin solution layer after adding the Ru(bpy)32+ catalyst. The two-dimensional reaction-diffusion patterns that appear in the subsystem and its parent system are very similar in wave speed, wavelength, color and in the duration of the pattern evolution, therefore a common chemical origin is supposed to exist in their formation. The role that the BrAc may play in the mechanism of the BrO3− — reductant — acetone — catalyst type oscillators (∼ 30 variants) is also pointed out.

Spatiotemporal Dynamics of the Landolt Reaction in an Open Spatial Reactor with Conical Geometry

In a previous study, the iodate-sulfite proton autoactivated reaction (Landolt reaction) was shown to exhibit spatial bistability and spatiotemporal oscillations when operated in an open spatial reactor with fixed "thickness", i.e., feed boundary to core distance. Here, we show that the spatial reactors with conical geometry enable one to rapidly probe the sensitivity of the above phenomena over a large range of the "thickness" parameter. This often-neglected parameter in chemical pattern studies plays an important role on the selection and stability of states. We reveal that the quenching capacity of slow diffusing polyacrylate ions on the spatiotemporal oscillations depends on this "thickness". The presented results should be useful for further research on reaction diffusion patterns and chemomechanical structures.

Spatial bistability: A source of complex dynamics. From spatiotemporal reaction-diffusion patterns to chemomechanical structures

We show experimentally and theoretically that reaction systems characterized by a slow induction period followed by a fast evolution to equilibrium can readily generate "spatial bistability" when operated in thin gel reactors diffusively fed from one side. This phenomenon which corresponds to the coexistence of two different stable steady states, not breaking the symmetry of the boundary conditions, can be at the origin of diverse reaction-diffusion instabilities. Using different chemical reactions, we show how stationary pulses, labyrinthine patterns or spatiotemporal oscillations can be generated. Beyond simple reaction-diffusion instabilities, we also demonstrate that the cross coupling of spatial bistability with the size responsiveness of a chemosensitive gel can give rise to autonomous spatiotemporal shape patterns, referred to as chemomechanical structures.

Gene Expression Time Delays and Turing Pattern Formation Systems

The incorporation of time delays can greatly affect the behaviour of partial differential equations and dynamical systems. In addition, there is evidence that time delays in gene expression due to transcription and translation play an important role in the dynamics of cellular systems. In this paper, we investigate the effects of incorporating gene expression time delays into a one-dimensional putative reaction diffusion pattern formation mechanism on both stationary domains and domains with spatially uniform exponential growth. While oscillatory behaviour is rare, we find that the time taken to initiate and stabilise patterns increases dramatically as the time delay is increased. In addition, we observe that on rapidly growing domains the time delay can induce a failure of the Turing instability which cannot be predicted by a naive linear analysis of the underlying equations about the homogeneous steady state. The dramatic lag in the induction of patterning, or even its complete absence on occasions, highlights the importance of considering explicit gene expression time delays in models for cellular reaction diffusion patterning.

Spiral Flow Wave in a Reaction–Diffusion–Convection System

The excitable Belousov–Zhabotinsky (BZ) reaction coupled with diffusion can exhibit a large variety of spatial patterns. In this letter, we report on superimposed spiral structures, providing evidence of a hierarchical self-organized order that connects two complex phenomena involving the coupling of a reaction–diffusion pattern with convection. A macroscopic propagating spiral flow wave (wavelength about 50 mm) is induced spontaneously by the preexcited reaction–diffusion structure of chemical spiral waves (wavelength about 1 mm). The pattern dynamics links two different hierarchical levels of structure formation in a nonlinear system.

Reaction−Diffusion Patterns of the Chlorite−Tetrathionate System in a Conical Geometry

We experimentally study the spatiotemporal reaction-diffusion patterns exhibited by the chlorite-tetrathionate reaction in a conical piece of gel. Since the nonequilibrium states of this reaction-diffusion system depend on the thickness of the gel, one can observe simultaneously such phenomena as spatial bistability, excitability waves, and a monostable state coexisting at different height over the cone. This new geometry allows for the exploration of gel size effects on pattern selection down to smaller dimensions than in previous experiments.

SOME SPATIO-TEMPORAL MODELS IN IMMUNOLOGY

Brief descriptions of spatio-temporal structures that arose in the author's explorations of theoretical immunology are provided. Topics range from singular perturbation theory for a biological assay and a model for skin rashes (reaction–diffusion patterns), through immunotherapy for allergy and selection of appropriate antibody types (compartment models), to a somewhat abstract "shape space".

DIFFUSIVE INSTABILITIES AND CHEMICAL REACTIONS

Diffusive instabilities provide the engine for an ever increasing number of dissipative structures. In this class autocatalytic chemical systems are prone to generate temporal and spatial self-organization phenomena. The development of open spatial reactors and the subsequent discovery in 1989 of the stationary reaction–diffusion patterns predicted by Turing [1952] have triggered a large amount of research. This review aims at a comparison between theoretical predictions and experimental results obtained with various type of reactors in use. The differences arising from the use of reactions exhibiting either bistability of homogeneous steady states or a single one in a CSTR are emphasized.

Pattern Formation in Reaction–Diffusion Models with Nonuniform Domain Growth

Recent examples of biological pattern formation where a pattern changes qualitatively as the underlying domain grows have given rise to renewed interest in the reaction–diffusion (Turing) model for pattern formation. Several authors have now reported studies showing that with the addition of domain growth the Turing model can generate sequences of patterns consistent with experimental observations. These studies demonstrate the tendency for the symmetrical splitting or insertion of concentration peaks in response to domain growth. This process has also been suggested as a mechanism for reliable pattern selection. However, thus far authors have only considered the restricted case where growth is uniform throughout the domain. In this paper we generalize our recent results for reaction–diffusion pattern formation on growing domains to consider the effects of spatially nonuniform growth. The purpose is twofold: firstly to demonstrate that the addition of weak spatial heterogeneity does not significantly alter pattern selection from the uniform case, but secondly that sufficiently strong nonuniformity, for example where only a restricted part of the domain is growing, can give rise to sequences of patterns not seen for the uniform case, giving a further mechanism for controlling pattern selection. A framework for modelling is presented in which domain expansion and boundary (apical) growth are unified in a consistent manner. The results have implications for all reaction–diffusion type models subject to underlying domain growth.

Mode-doubling and tripling in reaction-diffusion patterns on growing domains: a piecewise linear model.

Reaction-diffusion equations are ubiquitous as models of biological pattern formation. In a recent paper we have shown that incorporation of domain growth in a reaction-diffusion model generates a sequence of quasi-steady patterns and can provide a mechanism for increased reliability of pattern selection. In this paper we analyse the model to examine the transitions between patterns in the sequence. Introducing a piecewise linear approximation we find closed form approximate solutions for steady-state patterns by exploiting a small parameter, the ratio of diffusivities, in a singular perturbation expansion. We consider the existence of these steady-state solutions as a parameter related to the domain length is varied and predict the point at which the solution ceases to exist, which we identify with the onset of transition between patterns for the sequence generated on the growing domain. Applying these results to the model in one spatial dimension we are able to predict the mechanism and timing of transitions between quasi-steady patterns in the sequence. We also highlight a novel sequence behaviour, mode-tripling, which is a consequence of a symmetry in the reaction term of the reaction-diffusion system.

Reaction-diffusion patterns in confined chemical systems

We review on the effects of the feed mode on pattern selection observed in chemical systems operated in open spatial reactors. In two-side-fed reactors, strong parameter ramps naturally confine patterns in a stratum. The reactor thickness acts both as a genuine bifurcation parameter and on the pattern dimensionality. Depending on that thickness, standard 2D hexagon and stripe Turing patterns or more complex 3D planforms are observed. In thin one-side-fed reactors, patterning process must escape the imposed fixed boundary conditions either by devices introducing mixed boundary conditions or by an intrinsic phenomenon dubbed “spatial bistability.” We show that in most cases, for a comprehensive understanding of experimental observations, the full 3D aspects have to be taken into account.

Catalysis on mesoscopic composite surfaces: Influence of palladium boundaries on pattern formation during CO oxidation on Pt(1 1 0)

The effect of well-defined palladium boundaries on the formation of reaction-diffusion patterns during the oxidation of CO on Pt(1 1 0) has been investigated. The Pd boundaries were created by microlithography technique. Reaction-diffusion patterns formed on this surface were observed by photoemission electron microscopy (PEEM). On a multilayer Pd-covered Pt(1 1 0) surface, depending on the partial pressures, Pd areas may act as a reservoir of CO for adjacent Pt regions. Therefore, patterns forming on the bare Pt areas exhibit a pronounced interaction with the artificial Pt Pd boundaries.

Reaction-diffusion patterns on a disk or a square in a model with long-range interaction

A condensed model that captures the main features of high- or low-pressure catalytic oscillators is used to simulate spatiotemporal patterns in a catalytic disk or square. This model includes a single autocatalytic variable (activator), a slowly changing and localized inhibitor, and a very fast and highly diffusive variable that provides the long-range interaction. The extremely rich plethora of patterns is classified according to their symmetries, capitalizing on the inversion symmetry of the model. The simpler case of the bistable system (with no inhibitor) exhibits a very high sensitivity to initial conditions that leads to large multiplicity of stationary patterns. The effect of the parameter that defines the system stability (oscillatory, excitable, or bistable) is investigated, in the three variable model, either by using the same initial conditions for all simulations or, in an “experimental mode,” by stepping up or down the parameter. Patterns on a disk may be classified as circular, like stationary or oscillatory or moving (inwards or outwards) target pattern, rotating patterns, like stationary or oscillatory or moving spiral waves, and other patterns. Successive bifurcations, with changing parameter, reduce the system to states with lower symmetries and to asymmetric or even chaotic motions. Motions on a square are similar to those on a disk and include target-like pattern, propagating pulses, and oscillating or breathing stationary pulses.

Reaction-diffusion patterns on a disk or a square in a model with long-range interaction

Reaction-diffusion patterns on a disk or a square in a model with long-range interaction

Order and localization in reaction-diffusion pattern

The present work is concerned with two aspects of pattern formation: pattern localization and degree of pattern order. In reaction-diffusion models, there are three major effects. These stem from the reaction terms, the diffusion terms and the presence or absence of precursor gradients. Global analysis of reaction terms at late stages of pattern formation is at present unavailable. Therefore, we study the effect of the diffusion terms and of precursor gradients with numerical solution in two models: the Brusselator and the Gierer-Meinhardt. Differences in response to changes in the diffusion terms and the precursor gradients are related to contrasts between the nonlinear kinetics of the two models. These models both have Hill coefficient 2; effects of higher cooperativities have recently been discussed by Hunding and Engelhardt [1].

Phase differences in reaction-diffusion-advection systems and applications to morphogenesis

The authors study the effect of advection on reaction-diffusion patterns. It is shown that the addition of advection to a two-variable reaction–diffusion system with periodic boundary conditions results in the appearance of a phase difference between the patterns of the two variables which depends on the difference between the advection coefficients. The spatial patterns move like a travelling wave with a fixed velocity which depends on the sum of the advection coefficients. By a suitable choice of advection coefficients, the solution can be made stationary in time. In the presence of advection a continuous change in the diffusion coefficients can result in two Turing-type bifurcations as the diffusion ratio is varied, and such a bifurcation can occur even when the inhibitor species does not diffuse. It is also shown that the initial mode of bifurcation for a given domain size depends on both the advection and diffusion coefficients. These phenomena are demonstrated in the numerical solution of a particular reaction–diffusion system, and finally a possible application of the results to pattern formation in Drosophila larvae is discussed.

EXPERIMENTAL STUDY OF STATIONARY TURING PATTERNS AND THEIR INTERACTION WITH TRAVELING WAVES IN A CHEMICAL SYSTEM

We give a brief review of recent observations of Turing patterns in an isothermal single-phase chemical system. The basic principles of open spatial reactors used in the experiments are described. Different types of one-, two- and three-dimensional symmetry breaking reaction-diffusion patterns are discussed in relation with the geometric dimensions of the reactors and with the localization of the patterned regions in the concentration ramps. We also present a set of new spatiotemporal structures resulting from the interaction of the Turing (spatial) and Hopf (temporal) instabilities. Among other things, these interactions lead to antisymmetric wave sources in quasi-one-dimensional systems and to spatiotemporal intermittency in quasi-two-dimensional systems. We also report on a “cell splitting” growth mechanism of stationary patterns after a supercritical change in parameter value beyond the onset of the Turing instability.

Evolution of reaction-diffusion patterns in infinite and bounded domains

Abstract We introduce a semi-analytical method to study the evolution of spatial structures in reaction-diffusion systems. It consists in writing an integral equation for the relevant densities, from the propagator of the linear part of the evolution operator. In order to test the method, we perform an exhaustive study of a one-dimensional reaction-diffusion model associated to an electrical device - the ballast resistor. We consider the evolution of step and bubble-shaped initial density profiles in free space as well as in a semi-infinite domain with Dirichlet and Neumann boundary conditions. The piecewise-linear form of the reaction term, which preserves the basic ingredients of more complex nonlinear models, makes it possible to obtain exact wave-front solutions in free space and stationary solutions in the bounded domain. Short and long-time behaviour can also be analytically studied, whereas the evolution at intermediate times is analyzed by numerical techniques. We paid particular attention to the features introduced in the evolution by boundary conditions.