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'.

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.

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 a model of the photosensitive belousov-zhabotinsky reaction system with global feedback

Oscillatory cluster patterns are studied numerically in a reaction-diffusion model of the photosensitive Belousov-Zhabotinsky reaction supplemented with a global negative feedback. In one- and two-dimensional systems, families of cluster patterns arise for intermediate values of the feedback strength. These patterns consist of spatial domains of phase-shifted oscillations. The phase of the oscillations is nearly constant for all points within a domain. Two-phase clusters display antiphase oscillations; three-phase clusters contain three sets of domains with a phase shift equal to one-third of the period of the local oscillation. Border (nodal) lines between domains for two-phase clusters become stationary after a transient period, while borders drift in the case of three-phase clusters. We study the evolving border movement of the clusters, which, in most cases, leads to phase balance, i.e., equal areas of the different phase domains. Border curling of three-phase clusters results in formation of spiral clusters-a combination of a fast oscillating cluster with a slow spiraling movement of the domain border. At higher feedback coefficient, irregular cluster patterns arise, consisting of domains that change their shape and position in an irregular manner. Localized irregular and regular clusters arise for parameters close to the boundary between the oscillatory region and the reduced steady state region of the phase space.

Oscillatory cluster patterns in a homogeneous chemical system with global feedback

Oscillatory clusters are sets of domains in which nearly all elements in a given domain oscillate with the same amplitude and phase. They play an important role in understanding coupled neuron systems. In the simplest case, a system consists of two clusters that oscillate in antiphase and can each occupy multiple fixed spatial domains. Examples of cluster behaviour in extended chemical systems are rare, but have been shown to resemble standing waves, except that they lack a characteristic wavelength. Here we report the observation of so-called 'localized clusters'--periodic antiphase oscillations in one part of the medium, while the remainder appears uniform--in the Belousov-Zhabotinsky reaction-diffusion system with photochemical global feedback. We also observe standing clusters with fixed spatial domains that oscillate periodically in time and occupy the entire medium, and irregular clusters with no periodicity in either space or time, with standing clusters transforming into irregular clusters and then into localized clusters as the strength of the global negative feedback is gradually increased. By incorporating the effects of global feedback into a model of the reaction, we are able to simulate successfully the experimental data.

Noise-based switches and amplifiers for gene expression.

The regulation of cellular function is often controlled at the level of gene transcription. Such genetic regulation usually consists of interacting networks, whereby gene products from a single network can act to control their own expression or the production of protein in another network. Engineered control of cellular function through the design and manipulation of such networks lies within the constraints of current technology. Here we develop a model describing the regulation of gene expression and elucidate the effects of noise on the formulation. We consider a single network derived from bacteriophage lambda and construct a two-parameter deterministic model describing the temporal evolution of the concentration of lambda repressor protein. Bistability in the steady-state protein concentration arises naturally, and we show how the bistable regime is enhanced with the addition of the first operator site in the promotor region. We then show how additive and multiplicative external noise can be used to regulate expression. In the additive case, we demonstrate the utility of such control through the construction of a protein switch, whereby protein production is turned "on" and "off" by using short noise pulses. In the multiplicative case, we show that small deviations in the transcription rate can lead to large fluctuations in the production of protein, and we describe how these fluctuations can be used to amplify protein production significantly. These results suggest that an external noise source could be used as a switch and/or amplifier for gene expression. Such a development could have important implications for gene therapy.

A theory for controlling cell cycle dynamics using a reversibly binding inhibitor.

We demonstrate, by using mathematical modeling of cell division cycle (CDC) dynamics, a potential mechanism for precisely controlling the frequency of cell division and regulating the size of a dividing cell. Control of the cell cycle is achieved by artificially expressing a protein that reversibly binds and inactivates any one of the CDC proteins. In the simplest case, such as the checkpoint-free situation encountered in early amphibian embryos, the frequency of CDC oscillations can be increased or decreased by regulating the rate of synthesis, the binding rate, or the equilibrium constant of the binding protein. In a more complex model of cell division, where size-control checkpoints are included, we show that the same reversible binding reaction can alter the mean cell mass in a continuously dividing cell. Because this control scheme is general and requires only the expression of a single protein, it provides a practical means for tuning the characteristics of the cell cycle in vivo.