Insights from chemical systems into Turing-type morphogenesis.

In 1952, Alan Turing proposed a theory showing how morphogenesis could occur from a simple two morphogen reaction-diffusion system [Turing, A. M. (1952) Phil. Trans. R. Soc. Lond. A 237, 37-72. (doi:10.1098/rstb.1952.0012)]. While the model is simple, it has found diverse applications in fields such as biology, ecology, behavioural science, mathematics and chemistry. Chemistry in particular has made significant contributions to the study of Turing-type morphogenesis, providing multiple reproducible experimental methods to both predict and study new behaviours and dynamics generated in reaction-diffusion systems. In this review, we highlight the historical role chemistry has played in the study of the Turing mechanism, summarize the numerous insights chemical systems have yielded into both the dynamics and the morphological behaviour of Turing patterns, and suggest future directions for chemical studies into Turing-type morphogenesis. This article is part of the theme issue 'Recent progress and open frontiers in Turing's theory of morphogenesis'.

On the possibility of spontaneous chemomechanical oscillations in adsorptive porous media.

We derive general conditions for the emergence of sustained chemomechanical oscillations from a non-oscillatory adsorption/desorption reaction in a gas/solid porous medium. The oscillations arise from the nonlinear response of the solid matrix to the loading of the adsorbed species. More particularly, we prove that, in order for oscillations to occur, adsorption of the gas must in general cause a swelling of the solid matrix. We also investigate the prototypical case of Langmuir kinetics both numerically and analytically.This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)'.

Speed of traveling fronts in a sigmoidal reaction-diffusion system.

We study a sigmoidal version of the FitzHugh-Nagumo reaction-diffusion system based on an analytic description using piecewise linear approximations of the reaction kinetics. We completely describe the dynamics of wave fronts and discuss the properties of the speed equation. The speed diagrams show front bifurcations between branches with one, three, or five fronts that differ significantly from the classical FitzHugh-Nagumo model. We examine how the number of fronts and their speed vary with the model parameters. We also investigate numerically the stability of the front solutions in a case when five fronts exist.

Oscillatory pulses in FitzHugh-Nagumo type systems with cross-diffusion.

We study FitzHugh-Nagumo type reaction-diffusion systems with linear cross-diffusion terms. Based on an analytical description using piecewise linear approximations of the reaction functions, we completely describe the occurrence and properties of wavy pulses, patterns of relevance in several biological contexts, in two prototypical systems. The pulse wave profiles arising in this treatment contain oscillatory tails similar to those in travelling fronts. We find a fundamental, intrinsic feature of pulse dynamics in cross-diffusive systems--the appearance of pulses in the bistable regime when two fixed points exist.

Wave propagation in a FitzHugh-Nagumo-type model with modified excitability.

We examine a generalized FitzHugh-Nagumo (FHN) type model with modified excitability derived from the diffusive Morris-Lecar equations for neuronal activity. We obtain exact analytic solutions in the form of traveling waves using a piecewise linear approximation for the activator and inhibitor reaction terms. We study the existence and stability of waves and find that the inhibitor species exhibits different types of wave forms (fronts and pulses), while the activator wave maintains the usual kink (front) shape. The nonequilibrium Ising-Bloch bifurcation for the wave speed that occurs in the FHN model, where the control parameter is the ratio of inhibitor to activator time scales, persists when the strength of the inhibitor nonlinearity is taken as the bifurcation parameter.

Arm splitting and backfiring of spiral waves in media displaying local mixed-mode oscillations.

The behavior of spiral waves is investigated in a model of reaction-diffusion media supporting local mixed-mode oscillations for a range of values of a control parameter. This local behavior is accompanied by the formation of nodes, at which the arms of the simple spiral waves begin to split. With further parameter changes, this nodal structure loses stability, becoming quite irregular, eventually evolving into turbulence, while the local dynamics increases in complexity. The breakup of the spiral waves arises from a backfiring instability of the nodes induced by the arm splitting. This process of spiral breakup in the presence of mixed-mode oscillations represents an alternative to previously described scenarios of instability of line defects and superspirals in media with period-doubling and quasiperiodic oscillations, respectively.

Pattern formation in a tunable medium: the Belousov-Zhabotinsky reaction in an aerosol OT microemulsion.

Turing structures, standing waves, oscillatory clusters, and accelerating waves have been found in the spatially extended Belousov-Zhabotinsky system dispersed in water droplets of a reverse AOT microemulsion. The variety of patterns is determined by the tunable microstructure of the medium, i.e., by the concentration and size of water droplets. We propose a simple model to describe this system.

Spatial periodic forcing of Turing structures.

Spontaneously evolving Turing structures in the chlorine dioxide-iodine-malonic acid reaction-diffusion system typically exhibit many defects that break the symmetry of the pattern. Periodic spatial forcing interacts with the Turing structures and modifies the pattern symmetry and wavelength. We investigate the role of the amplitude and wavelength of spatial periodic forcing on the hexagonal pattern of Turing structures. Experimental results and numerical simulations reveal that forcing at wavelengths slightly larger than the natural wavelength of the pattern is most effective in removing defects and producing ordered symmetric hexagonal patterns.

Inwardly rotating spiral waves in a reaction-diffusion system.

Almost 30 years have passed since the discovery of concentric (target) and spiral waves in the spatially extended Belousov-Zhabotinsky (BZ) reaction. Since then, rotating spirals and target waves have been observed in a variety of physical, chemical, and biological reaction-diffusion systems. All of these waves propagate out from the spiral center or pacemaker. We report observations of inwardly rotating spirals found in the BZ system dispersed in water droplets of a water-in-oil microemulsion. These "antispirals" were also generated in computer simulations.

Turing pattern formation induced by spatially correlated noise.

The effect of spatially correlated noise on Turing structures is analyzed both experimentally and numerically. Using the photosensitive character of the chlorine dioxide-iodine-malonic acid reaction-diffusion system, spatial randomness is introduced in the system. In the presence of noise, Turing patterns appear and are stable at levels of average illumination that would be more than sufficient to suppress pattern formation in the case of homogeneous illumination.

Dynamics of kinks in one- and two-dimensional hyperbolic models with quasidiscrete nonlinearities.

We study the evolution of fronts in the Klein-Gordon equation when the nonlinear term is inhomogeneous. Extending previous works on homogeneous nonlinear terms, we describe the derivation of an equation governing the front motion, which is strongly nonlinear, and, for the two-dimensional case, generalizes the damped Born-Infeld equation. We study the motion of one- and two-dimensional fronts finding a much richer dynamics than in the homogeneous system case, leading, in most cases, to the stabilization of one phase inside the other. For a one-dimensional front, the function describing the inhomogeneity of the nonlinear term acts as a "potential function" for the motion of the front, i.e., a front initially placed between two of its local maxima asymptotically approaches the intervening minimum. Two-dimensional fronts, with radial symmetry and without dissipation can either shrink to a point in finite time, grow unboundedly, or their radius can oscillate, depending on the initial conditions. When dissipation effects are present, the oscillations either decay spirally or not depending on the value of the damping dissipation parameter. For fronts with a more general shape, we present numerical simulations showing the same behavior.

Resonant suppression of Turing patterns by periodic illumination.

We study the resonant behavior of Turing pattern suppression in a model of the chlorine dioxide-iodine-malonic acid reaction with periodic illumination. The results of simulations based on integration of partial differential equations display resonance at the frequency of autonomous oscillations in the corresponding well stirred system. The resonance in Turing pattern suppression is sharper at lower complexing agent concentration and is affected by the waveform of the periodic driving force. Square wave (on-off) periodic forcing is more effective in suppressing Turing patterns than sinusoidal forcing. We compare the dynamics of periodically forced Turing patterns with the dynamics of periodically forced nonhomogeneous states in a system of two identical coupled cells. Bifurcation analysis based on numerical continuation of the latter system gives good predictions for the boundaries of the major resonance regions of the periodically forced patterns.

Oscillatory clusters in the periodically illuminated, spatially extended Belousov-Zhabotinsky reaction.

Cluster-cluster transitions in the periodically illuminated photosensitive Belousov-Zhabotinsky (BZ) reaction-diffusion system exhibit the same scenario as in the autonomous BZ system with negative global feedback: two-phase clusters <--> three-phase clusters <--> irregular clusters <--> localized clusters. Transitions induced by changing the dark ( TD) or light ( TL) phases of the periodic external square wave illumination are dependent not only on the frequency of illumination at constant TD/TL, but also on the ratio TD/TL at constant frequency (when TD+TL = const).

A new chemical system for studying pattern formation: bromate-hypophosphite-acetone-dual catalyst.

A modified version of the short-lived BrO3(-)-H2PO2(-)-Mn(II)-N2 oscillator, the BrO3(-)-H2PO2(-)-acetone-dual catalyst system, where the catalyst pair can be Mn(II)-Ru(bpy)3SO4, or Mn(II)-ferroin, or Mn(II)-diphenylamine, shows long-lasting batch oscillations in the potential of a Pt electrode and in colour, accompanying periodic transitions between the oxidised and reduced forms of the catalysts. Experimental conditions for the oscillations are established. The origin of the batch oscillations and the role of the catalyst pair in the oscillatory behaviour are discussed. The new system is ideally suited to the study of waves and patterns in reaction-diffusion systems, since in addition to the longevity of its spatial behaviour in batch, it produces no gaseous or solid products and exhibits significant photosensitivity.

Calcium waves in a model with a random spatially discrete distribution of Ca2+ release sites.

We study the propagation of intracellular calcium waves in a model that features Ca2+ release from discrete sites in the endoplasmic reticulum membrane and random spatial distribution of these sites. The results of our simulations qualitatively reproduce the experimentally observed behavior of the waves. When the level of the channel activator inositol trisphosphate is low, the wave undergoes fragmentation and eventually vanishes at a finite distance from the region of initiation, a phenomenon we refer to as an abortive wave. With increasing activator concentration, the mean distance of propagation increases. Above a critical level of activator, the wave becomes stable. We show that the heterogeneous distribution of Ca2+ channels is the cause of this phenomenon.

The consequences of imperfect mixing in autocatalytic chemical and biological systems.

When chemical reactions whose rate increases with the concentration of a product species are carried out in imperfectly mixed systems, a variety of complex behaviours can occur. These phenomena, which have relevance for biological processes as well, include chaotic and stochastic behaviour and selection of one final state over an equally probable alternative.

Glycolytic pH oscillations in a flow reactor.

A new type of flow reactor (UCSTR) has been developed that uses anisotropic ultrafiltration membranes in a continuous flow stirred tank reactor (CSTR) to facilitate the study of nonlinear enzyme catalyzed reactions. The design allows the study of enzymes with subunit molecular weights > or = 9000 dalton and protein concentrations up to at least 2 mg/ml under flow conditions with a residence time of 3 min or more, in a reactor of volume 1.67 ml. The UCSTR allows continuous potentiometric or spectrophotometric measurement without design change. Calibration of reactor performance was carried out by reproducing pH oscillations in the ferrocyanide-hydrogen peroxide reaction. Experimental verification of oscillatory glycolysis in the UCSTR was carried out with extract of rat skeletal muscle. Input feeds were fructose-6-phosphate and ATP with low concentrations of phosphate as buffer. Oscillations in pH, sustained for over eight hours, were observed. A six-step mechanism, including product activation and substrate inhibition, seven concentration variables, and four enzymes sufficed simulate the pH oscillations observed in the UCSTR.

Transient turing structures in a gradient-free closed system.

Transient, symmetry-breaking, spatial patterns were obtained in a closed, gradient-free, aqueous medium containing chlorine dioxide, iodine, malonic acid, and starch at 4 degrees to 5 degrees C. The conditions under which these Turing-type structures appear can be accurately predicted from a simple mathematical model of the system. The patterns, which consist of spots, stripes, or both spots and stripes, require about 25 minutes to form and remain stationary for 10 to 30 minutes.

A chemical approach to designing Turing patterns in reaction-diffusion systems.

A systematic approach is suggested to design chemical systems capable of displaying stationary, symmetry-breaking reaction diffusion patterns (Turing structures). The technique utilizes the fact that reversible complexation of an activator species to form an unreactive, immobile complex reduces the effective diffusion constant of the activator, thereby facilitating the development of Turing patterns. The chlorine dioxide/iodine/malonic acid reaction is examined as an example, and it is suggested that a similar phenomenon may occur in some biological pattern formation processes.