Turing patterns on rotating spiral growing domains.

We investigate the emergence of Turing patterns in a system growing as a rotating spiral in two dimensions, utilizing the photosensitivity of the chlorine dioxide-iodine-malonic acid (CDIMA) reaction to control the growth process. We observe the formation of single and multiple (double and triple) stationary spiral patterns as well as transitional patterns. From numerical simulations of the Lengyel-Epstein model with an additional term to account for the effects of illumination on the reaction, we analyze the relationship between the final morphologies and the radial and angular growth velocities, identify conditions conducive to the formation of transitional structures, examine the importance of the size of the initial nucleation site in determining the spiral's multiplicity, and evaluate the stability and robustness of these Turing patterns. Our results indicate how inclusion of rotational degrees of freedom in the growth process may lead to the formation of a diverse new class of patterns in chemical and biological systems.

Strong symmetry breaking rhythms created by folded nodes in a pair of symmetrically coupled, identical Koper oscillators.

The Koper model is a prototype system with two slow variables and one fast variable that possesses small-amplitude oscillations (SAOs), large-amplitude oscillations (LAOs), and mixed-mode oscillations (MMOs). In this article, we study a pair of identical Koper oscillators that are symmetrically coupled. Strong symmetry breaking rhythms are presented of the types SAO-LAO, SAO-MMO, LAO-MMO, and MMO-MMO, in which the oscillators simultaneously exhibit rhythms of different types. We identify the key folded nodes that serve as the primary mechanisms responsible for the strong nature of the symmetry breaking. The maximal canards of these folded nodes guide the orbits through the neighborhoods of these key points. For all of the strong symmetry breaking rhythms we present, the rhythms exhibited by the two oscillators are separated by maximal canards in the phase space of the oscillator.

Role of Fast and Slow Inhibitors in Oscillatory Rhythm Design.

In biological or abiotic systems, rhythms occur, owing to the coupling between positive and negative feedback loops in a reaction network. Using the Semenov-Whitesides oscillatory network for thioester hydrolysis as a prototype, we experimentally and theoretically analyzed the role of fast and slow inhibitors in oscillatory reaction networks. In the presence of positive feedback, a single fast inhibitor generates a time delay, resulting in two saddle-node bifurcations and bistability in a continuously stirred tank reactor. A slow inhibitor produces a node-focus bifurcation, resulting in damped oscillations. With both fast and slow inhibitors present, the node-focus bifurcation repeatedly modulates the saddle-node bifurcations, producing stable periodic oscillations. These fast and slow inhibitions result in a pair of time delays between steeply ascending and descending dynamics, which originate from the positive and negative feedbacks, respectively. This pattern can be identified in many chemical relaxation oscillators and oscillatory models, e.g., the bromate-sulfite pH oscillatory system, the Belousov-Zhabotinsky reaction, the trypsin oscillatory system, and the Boissonade-De Kepper model. This study provides a novel understanding of chemical and biochemical rhythms and suggests an approach to designing such behavior.

Transitions of Collective Motions Driven by Phase Resetting.

The front cover artwork is provided by Prof. Gao's group. The image shows the motion patterns transition of the active gel group under the step light intensity, which describes the mechanism of a new collective emergence structure. Read the full text of the Research Article at 10.1002/cphc.202300054.

Transitions of Collective Motions Driven by Phase Resetting.

Abrupt (i. e. step) environmental changes, such as natural disasters or the intervention of predators, can alter the internal dynamics of groups with active units, leading to the rapid destruction and/or restructuring of the group, with the emergence of new collective structures that endow the system with adaptability. Few studies, to date, have considered the influence of abrupt environmental changes on emergent behavior. Here, we use a model of active matter, the Belousov-Zhabotinsky (BZ) self-oscillating gel, to study the mechanism of formation and transition between modes of collective locomotion caused by changes of illumination intensity in arrays of interacting photosensitive active units. New forms of collective motion can be generated by step changes of illumination intensity. These transformations arise from the phase resetting and wave-signal regeneration induced by the abrupt parameter variation, while gradual change results in different evolution of collective motion. Our results not only suggest a novel mechanism for emergence, but also imply that new collective behaviors could be accessible via discontinuous parameter changes.

Symmetry-breaking rhythms in coupled, identical fast-slow oscillators.

Symmetry-breaking in coupled, identical, fast-slow systems produces a rich, dramatic variety of dynamical behavior-such as amplitudes and frequencies differing by an order of magnitude or more and qualitatively different rhythms between oscillators, corresponding to different functional states. We present a novel method for analyzing these systems. It identifies the key geometric structures responsible for this new symmetry-breaking, and it shows that many different types of symmetry-breaking rhythms arise robustly. We find symmetry-breaking rhythms in which one oscillator exhibits small-amplitude oscillations, while the other exhibits phase-shifted small-amplitude oscillations, large-amplitude oscillations, mixed-mode oscillations, or even undergoes an explosion of limit cycle canards. Two prototypical fast-slow systems illustrate the method: the van der Pol equation that describes electrical circuits and the Lengyel-Epstein model of chemical oscillators.

Peptide-Modulated pH Rhythms.

Many drugs adjust and/or control the spatiotemporal dynamics of periodic processes such as heartbeat, neuronal signaling and metabolism, often by interacting with proteins or oligopeptides. Here we use a quasi-biocompatible, non-equilibrium pH oscillatory system as a biomimetic biological clock to study the effect of pH-responsive peptides on rhythm dynamics. The added peptides generate feedback that can lengthen or shorten the oscillatory period during which the peptides alternate between random coil and coiled-coil conformations. This modulation of a chemical clock supports the notion that short peptide reagents may have utility as drugs to regulate human body clocks.

Effect of obstructions on growing Turing patterns.

We study how Turing pattern formation on a growing domain is affected by discrete domain discontinuities. We use the Lengyel-Epstein reaction-diffusion model to numerically simulate Turing pattern formation on radially expanding circular domains containing a variety of obstruction geometries, including obstructions spanning the length of the domain, such as walls and slits, and local obstructions, such as small blocks. The pattern formation is significantly affected by the obstructions, leading to novel pattern morphologies. We show that obstructions can induce growth mode switching and disrupt local pattern formation and that these effects depend on the shape and placement of the objects as well as the domain growth rate. This work provides a customizable framework to perform numerical simulations on different types of obstructions and other heterogeneous domains, which may guide future numerical and experimental studies. These results may also provide new insights into biological pattern growth and formation, especially in non-idealized domains containing noise or discontinuities.

Capillarity-Induced Propagation Reversal of Chemical Waves in a Self-oscillating Gel.

In a self-oscillating gel, unidirectional chemical waves generated by the Belousov-Zhabotinsky reaction can drive locomotion, which results from the difference between the push and pull forces in the wavefront and waveback, respectively. In a narrow tube, such a gel is subject not only to the asymmetric force engendered by the propagation of the chemical waves but also to additional forces originating from the capillary effect in the polymer skeleton. The ends of a self-oscillating gel in a tube are squeezed unequally during unidirectional motion, causing new waves of higher frequency and ultimately giving rise to reversal of the direction of chemical wave propagation. This peculiar phenomenon of a self-oscillating gel in a narrow glass tube results in a nonmonotonic evolution of the gel locomotion velocity.

Autonomous reciprocating migration of an active material.

Periodic to-and-fro migration is a sophisticated mode of locomotion found in many forms of active matter in nature. Providing a general description of periodic migration is challenging, because many details of animal migration remain a mystery. We study periodic migration in a simpler system using a mechanistic model of a photosensitive, active material in which a stimulus-responsive polymer gel is propelled by chemical waves under the regulation of an illumination gradient sensed by the gel, which plays a role analogous to the environment in periodic animal migration. The reciprocating gel migration results from autonomous transitions between retrograde and direct wave locomotion modes arising from the gradient distribution of the illumination intensity. The local dynamics of the chemical waves modulates the asymmetry between push and pull forces to achieve repeated reorientation of the direction of locomotion. Materials that display similar intelligent, self-adaptive locomotion might be tailored for such functions as drug delivery or self-cleaning systems.

Programmed Locomotion of an Active Gel Driven by Spiral Waves.

Active media that host spiral waves can display complex modes of locomotion driven by the dynamics of those waves. We use a model of a photosensitive stimulus-responsive gel that supports the propagation of spiral chemical waves to study locomotive transition and programmed locomotion. The mode transition between circular and toroidal locomotion results from the onset of spiral tip meandering that arises via a secondary Hopf bifurcation as the level of illumination is increased. This dynamic instability of the system introduces a second circular locomotion with a small diameter caused by tip meandering. The original circular locomotion with large diameter is driven by the push-pull asymmetry of the wavefront and waveback of the simple spiral waves initiated at one corner of gel. By harnessing this mode transition of the gel locomotion via coded illumination, we design programmable pathways of nature-inspired angular locomotion of the gel.

Turing patterns on radially growing domains: experiments and simulations.

We study Turing pattern formation in a system undergoing radial growth in two dimensions. The photosensitive chlorine dioxide-iodine-malonic acid reaction is illuminated to inhibit patterning, with a growing non-illuminated circular domain in which the pattern develops. We examine the relationship between the linear radial growth rate and the resulting pattern morphology. Faster growth causes the pattern to form parallel to the growing boundary as concentric rings, while slower growth leads to pattern formation perpendicular to the growing boundary. We observe three distinct growth modes for the Turing patterns, which also depend on the radial growth rate. The experimental results are qualitatively reproduced in numerical simulations using the Lengyel-Epstein model with an additional term to account for the photosensitivity of the reaction. These results may provide new insight into how patterns form in growing biological systems.

Modulation of Turing Patterns in the CDIMA Reaction by Ultraviolet and Visible Light.

We have carried out the first systematic study of the effects of ultraviolet light, both alone and in combination with visible white light, on Turing patterns in the chlorine dioxide-iodine-malonic acid (CDIMA) reaction. The ultraviolet light used has a sharp peak at 368 nm and can perturb the system selectively. It primarily decomposes chlorine dioxide in a zeroth-order reaction, and when it is used to illuminate Turing patterns, shrunken spots are formed with an imperfect hexagonal arrangement. The ultraviolet light competes directly with the visible white light via the photoreaction with dissolved chlorine dioxide, which prevents the total suppression of patterns at intermediate intensities of white light. These results suggest that specific wavelengths of light in the ultraviolet spectrum selectively modify the chemistry behind the pattern formation and can be utilized to generate novel self-organized structures under forcing conditions. We propose a modified Lengyel-Epstein model to incorporate the effect of ultraviolet illumination and obtain good qualitative agreement between simulations and experiments. These results support the idea that chlorine dioxide photoreaction is a key step in modulating CDIMA patterns under ultraviolet illumination.

Effect of Reaction Parameters on the Wavelength of Pulse Waves in the Belousov-Zhabotinsky Reaction-Diffusion System.

The wavelength of Belousov-Zhabotinsky (BZ) traveling waves is the key factor that limits the scale of BZ self-oscillating gel motors. To achieve control of the wavelength, it is necessary to evaluate the wavelength dependence on species concentrations and temperature. In this work, the effect of reaction parameters on the wavelength of BZ pulse waves was studied. The most effective way to reduce the wavelength of pulse waves is to increase the concentration of organic species and/or the temperature. Decreasing the concentration of bromate, hydrogen ion, or metal catalyst also reduces the wavelength of pulse waves. This work provides a convenient and direct method to produce sub-millimeter BZ waves, which could be applied to designing BZ wave-driven small-scale gel motors as well as providing insight into other emergent behaviors of self-oscillating gels.

The smallest chimera: Periodicity and chaos in a pair of coupled chemical oscillators.

Symmetrically coupled identical oscillators were once believed to support only totally synchronous or totally asynchronous states. More recently, chimera states, in which a subset of oscillators behaves coherently while the other subset exhibits disorder, have been found in large arrays of oscillators, coupled either locally or globally. We demonstrate for the first time the existence of a chimera state with only two diffusively coupled identical oscillators, one behaving nearly periodically (coherently) and the other chaotically (incoherently). We attribute this behavior to a "master-slave" interaction, which arises via a symmetry-breaking canard explosion.

Phase-frequency model of strongly pulse-coupled Belousov-Zhabotinsky oscillators.

We demonstrate that the dynamical behavior of strongly pulse-coupled Belousov-Zhabotinsky oscillators can be reproduced and predicted using a model that treats both the phase and the instantaneous frequency of the oscillators. Model parameters are extracted from the experimental data obtained using a single pulse-perturbed oscillator and are used to simulate the temporal dynamics of a system of two coupled oscillators. Our model exhibits the out-of-phase and anti-phase synchronization and the 1:N and N:M temporal patterns as well as the oscillator suppression that are observed in experiments when the inhibitory coupling is asymmetric. This approach may be adapted to other systems, such as coupled neurons, where the oscillatory dynamics is affected by strong pulses.

Pulse-coupled Belousov-Zhabotinsky oscillators with frequency modulation.

Inhibitory perturbations to the ferroin-catalyzed Belousov-Zhabotinsky (BZ) chemical oscillator operated in a continuously fed stirred tank reactor cause long term changes to the limit cycle: the lengths of the cycles subsequent to the perturbation are longer than that of the unperturbed cycle, and the unperturbed limit cycle is recovered only after several cycles. The frequency of the BZ reaction strongly depends on the acid concentration of the medium. By adding strong acid or base to the perturbing solutions, the magnitude and the direction of the frequency changes concomitant to excitatory or inhibitory perturbations can be controlled independently of the coupling strength. The dynamics of two BZ oscillators coupled through perturbations carrying a coupling agent (activator or inhibitor) and a frequency modulator (strong acid or base) was explored using a numerical model of the system. Here, we report new complex temporal patterns: higher order, partially synchronized modes that develop when inhibitory coupling is combined with positive frequency modulation (FM), and complex bursting patterns when excitatory coupling is combined with negative FM. The role of time delay between the peak and perturbation (the analog of synaptic delays in networks of neurons) has also been studied. The complex patterns found under inhibitory coupling and positive FM vanish when the delay is significant, whereas a sufficiently long time delay is required for the complex temporal dynamics to occur when coupling is excitatory and FM is negative.

Photo-Controlled Waves and Active Locomotion.

Waves of chemical concentration, created by the interaction between reaction and diffusion, occur in a number of chemical systems far from equilibrium. In appropriately chosen polymer gels, these waves generate mechanical forces, which can result in locomotion. When a component of the system is photosensitive, light can be used to modulate and control these waves. In this Concept article, we examine various forms of photo-control of such systems, focusing particularly on the Belousov-Zhabotinsky oscillating chemical reaction. The phenomena we consider include image storage and image processing, feedback-control and feedback-induced clustering of waves, and phototropic and photophobic locomotion. Several of these phenomena have analogues in or potential applications to biological systems.

Testing Turing's theory of morphogenesis in chemical cells.

Alan Turing, in "The Chemical Basis of Morphogenesis" [Turing AM (1952) Philos Trans R Soc Lond 237(641):37-72], described how, in circular arrays of identical biological cells, diffusion can interact with chemical reactions to generate up to six periodic spatiotemporal chemical structures. Turing proposed that one of these structures, a stationary pattern with a chemically determined wavelength, is responsible for differentiation. We quantitatively test Turing's ideas in a cellular chemical system consisting of an emulsion of aqueous droplets containing the Belousov-Zhabotinsky oscillatory chemical reactants, dispersed in oil, and demonstrate that reaction-diffusion processes lead to chemical differentiation, which drives physical morphogenesis in chemical cells. We observe five of the six structures predicted by Turing. In 2D hexagonal arrays, a seventh structure emerges, incompatible with Turing's original model, which we explain by modifying the theory to include heterogeneity.

pH-regulated chemical oscillators.

The hydrogen ion is arguably the most ubiquitous and important species in chemistry. It also plays a key role in nearly every biological process. In this Account, we discuss systems whose behavior is governed by oscillations in the concentration of hydrogen ion. The first chemical oscillators driven by changes in pH were developed a quarter century ago. Since then, about two dozen new pH oscillators, systems in which the periodic variation in pH is not just an indicator but an essential prerequisite of the oscillatory behavior, have been discovered. Mechanistic understanding of their behavior has grown, and new ideas for their practical application have been proposed and, in some cases, tested. Here we present a catalog of the known pH oscillators, divide them into mechanistically based categories based on whether they involve a single oxidant and reductant or an oxidant and a pair of reductants, and describe general mechanisms for these two major classes of systems. We also describe in detail the chemistry of one example from each class, hydrogen peroxide-sulfide and ferricyanide-iodate-sulfite. Finally, we consider actual and potential applications. These include using pH oscillators to induce oscillation in species that would otherwise be nonoscillatory, creating novel spatial patterns, generating periodic transitions between vesicle and micelle states, stimulating switching between folded and random coil states of DNA, building molecular motors, and designing pulsating drug delivery systems. We point out the importance for future applications of finding a batch pH oscillator, one that oscillates in a closed system for an extended period of time, and comment on the progress that has been made toward that goal.