Generation of spatiotemporal calcium patterns by coupling a pH-oscillator to a complexation equilibrium

Unravelling diverse spatiotemporal orders in chlorine dioxide-iodine-malonic acid reaction-diffusion system through circularly polarized electric field and photo-illumination

Designing and predicting self-organized pattern formation in out-of-equilibrium chemical and biochemical reactions holds fundamental significance. External perturbations like light and electric fields exert a crucial influence on reaction-diffusion systems involving ionic species. While the separate impacts of light and electric fields have been extensively studied, comprehending their combined effects on spatiotemporal dynamics is paramount for designing versatile spatial orders. Here, we theoretically investigate the spatiotemporal dynamics of chlorine dioxide-iodine-malonic acid reaction-diffusion system under photo-illumination and circularly polarized electric field (CPEF). By applying CPEF at varying intensities and frequencies, we observe the predominant emergence of oscillating hexagonal spot-like patterns from homogeneous stable steady states. Furthermore, our study unveils a spectrum of intriguing spatiotemporal instabilities, encompassing stripe-like patterns, oscillating dumbbell-shaped patterns, spot-like instabilities with square-based symmetry, and irregular chaotic patterns. However, when we introduce periodic photo-illumination to the hexagonal spot-like instabilities induced by CPEF in homogeneous steady states, we observe periodic size fluctuations. Additionally, the stripe-like instabilities undergo alternating transitions between hexagonal spots and stripes. Notably, within the Turing region, the interplay between these two external influences leads to the emergence of distinct superlattice patterns characterized by hexagonal-and square-based symmetry. These patterns include parallel lines of spots, target-like formations, black-eye patterns, and other captivating structures. Remarkably, the simple perturbation of the system through the application of these two external fields offers a versatile tool for generating a wide range of pattern-forming instabilities, thereby opening up exciting possibilities for future experimental validation.

Recent advances in the temporal and spatiotemporal dynamics induced by bromate–sulfite-based pH-oscillators

The bromate–sulfite reaction-based pH-oscillators represent one of the most useful subgroup among the chemical oscillators. They provide strong H+-pulses which can generate temporal oscillations in other systems coupled to them and they show wide variety of spatiotemporal dynamics when they are carried out in different gel reactors. Some examples are discussed. When pH-dependent chemical and physical processes are linked to a bromate–sulfite-based oscillator, rhythmic changes can appear in the concentration of some cations and anions, in the distribution of the species in a pH-sensitive stepwise complex formation, in the oxidation number of the central cation in a chelate complex, in the volume or the desorption-adsorption ability of a piece of gel. These reactions are quite suitable for generating spatiotemporal patterns in open reactors. Many reaction–diffusion phenomena, moving and stationary patterns, have been recently observed experimentally using different reactor configurations, which allow exploring the effect of different initial and boundary conditions. Here, we summarize the most relevant aspects of these experimental and numerical studies on bromate–sulfite reaction-based reaction–diffusion systems.

Designing Stationary Reaction-Diffusion Patterns in pH Self-Activated Systems

Since Alan Turing's 1952 pioneering work, reaction-diffusion (RD) processes are regarded as prototype mechanisms for pattern formation in living systems. Though suspected in many aspects of morphogenetic development, pure RD patterns have not yet been demonstrated in living organisms. The first observations of an autonomous development of stationary chemical patterns were made in the early 1990s. In this Account, we discuss the recent developments for producing stationary pH RD patterns in open spatial reactors. The theoretical analysis of the early experiments anticipated the possibility of finding Turing patterns in a wide range of oscillatory reactions if one could control the kinetic and diffusional rate of some key species. However, no experimentally effective method to produce stationary Turing patterns was attained before 2009, and the number of systems stagnated at two until then. The two precursor reaction systems benefited from unplanned favorable chemical properties of the RD media. Theoretical studies point out that appropriate diffusion rate differences are necessary to produce stationary patterns since a competition between an effective short distance self-activation and a long distance inhibitory process is required. This differential diffusion would naturally lead to differential exchange rates between the RD system and its feed environment, an aspect somewhat overlooked in theoretical and in primal experimental approaches. Our pattern design method takes this aspect into account. A slower diffusion of a self-activated species (here, protons), produced in the RD part of the spatial reactor, generates the accumulation of this species compared to the other species. This accumulation has to be at least partly compensated by an independent scavenging reaction. The above requirement naturally brought us to focus on two-substrate pH oscillatory reactions. Stationary RD patterns are now well documented in six pH driven reaction systems. Furthermore, the coupling with a pH dependent metal ion complexing agent led to stationary patterns in calcium ion concentration. Our effective semiempirical design method does not require a detailed knowledge of the reaction kinetics; thus it is applicable to a broad spectrum of reactions and even to synthetic biological systems. It is based on simple dynamic arguments and on general topological characteristics of a nonequilibrium phase diagram. We first illustrate our method with numerical simulations, based on a realistic but idealized general model of the two-substrate pH-oscillator reaction family, and provide a refined view of the topology of the resulting phase diagrams. Then, we exemplify its effectiveness by observations made in distinct pH self-activated systems. Analogies and differences between experiments and the model calculations are pointed out. Besides standard hexagonal arrays of spots and parallel stripes, hitherto undocumented dynamic phenomena, such as randomly blinking areas and complex dynamic and stationary filamentous structures, were observed. The main challenge, to find low-mobility complexing agents that would selectively and reversibly bind a species controlling the self-activatory kinetic path of the reaction, was readily overcome in multiple ways by anions of weak acids: not only by polymeric substances but in some cases by a pH color indicator or even smaller molecules, depending on their proton binding affinity.

From Master-Slave to Peer-to-Peer Coupling in Chemical Reaction Networks

Design strategy through linking a driving pH oscillator (master system) to a pH sensitive complexation, precipitation, or protonation equilibrium (slave system) has been widely used to create and control concentration oscillations of chemical entities (e.g., monovalent cations, DNA, nanoparticles) not participating in the pH oscillatory system. No systematic investigation has been carried out on how the components of these equilibria affect the characteristics of the driving pH oscillators, and this feedback effect has been often neglected in previous studies. Here we show that pH sensitive species (hydrogen carbonate, EDTA) through a pH-dependent equilibrium could significantly affect the characteristics (time period and amplitude) of the driving pH oscillators. By varying the concentration of those species we are able to control the strength of the chemical feedback from slave system to master system thus introducing a transition from master-slave coupling to peer-to-peer coupling in linked chemical systems. To illustrate this transition and coupling strategies we investigate two coupled chemical systems, namely, the bromate-sulfite pH oscillator and carbonate-carbon dioxide equilibrium and the hydrogen peroxide-thiosulfate-copper(II) and EDTA complexation equilibrium. As a sign of the peer-to-peer coupling the characteristics of the driving oscillatory systems can be tuned by controlling the feedback strength, and the oscillations can be canceled above a critical value of this parameter.

Kinetic and Diffusion-Driven Instabilities in the Bromate-Sulfite-Ferrocyanide System

The spatiotemporal dynamics of the bromate-sulfite-ferrocyanide (BSF) reaction-diffusion system in a open one-side-fed reactor (OSFR) is investigated by numerical simulations. The results of the simulations are compared with experiments performed in an annular shape OSFR. Both kinetic and diffusion-driven instabilities are identified in the model. There are two hydrogen ion consuming pathways in the mechanism: the partial oxidation of sulfite to dithionate and the oxidation of ferrocyanide by bromate ions. Their dynamical effects are similar, as they support the same negative feedback loop via sulfite ion. However, the time scale of the oxidation of ferrocyanide by bromate ions can be conveniently controlled by the input feed concentrations, thus it provides a more flexible way to find spatiotemporal oscillations. Long-range activation due to the relative fast diffusion of hydrogen ions compared to the other reactants can also result in oscillations in this mechanism. We show that the spatial extent of the reaction-diffusion medium along the direction of the diffusive feed (the thickness) acts as a general control parameter of the dynamics. Oscillations, either originated in kinetic or in diffusive instabilities, can only develop in a narrow range of the thickness. This property explains the experimentally often observed spatial localization of the oscillations. A reciprocal relationship is found between two main control parameters of the dynamics, which are the thickness and the hydrogen ion input feed concentration.

Pattern Formation in the Bromate-Sulfite-Ferrocyanide Reaction

Mixed Landolt-type pH oscillators are versatile systems that allow the experimental study of a wide range of nonlinear phenomena including multistability, oscillations, and spatiotemporal patterns. We report on the dynamics of the bromate-sulfite-ferrocyanide reaction operated in a open one-side-fed reactor, where spatial bistability, spatiotemporal oscillations, front and Turing-type patterns have been observed. The role of different experimental parameters, like the input flow concentrations of the hydrogen and the ferrocyanide ions, the temperature and the thickness of the gel medium (which affects the rate of the diffusive feed) have been investigated. We point out that all these parameters can be efficiently used to control the spatiotemporal dynamics. We show that the increase of ionic strength stabilizes the uniform states at the expense of the patterned one. Some general aspects of the spatiotemporal dynamics of mixed Landolt type systems, which are based on the oxidation of sulfite ions by strong oxidants, are emphasized.

Chemical gardens without silica: the formation of pure metal hydroxide tubes

Contrary to common belief, hollow precipitation tubes form in the absence of silicate if sodium hydroxide solution is injected into solutions of various metal ions. In many cases, the growth speed has a power law dependence on the flow rate. For vanadyl, we observe damped oscillations in the tube height.

Contribution to an effective design method for stationary reaction-diffusion patterns

The British mathematician Alan Turing predicted, in his seminal 1952 publication, that stationary reaction-diffusion patterns could spontaneously develop in reacting chemical or biochemical solutions. The first two clear experimental demonstrations of such a phenomenon were not made before the early 1990s when the design of new chemical oscillatory reactions and appropriate open spatial chemical reactors had been invented. Yet, the number of pattern producing reactions had not grown until 2009 when we developed an operational design method, which takes into account the feeding conditions and other specificities of real open spatial reactors. Since then, on the basis of this method, five additional reactions were shown to produce stationary reaction-diffusion patterns. To gain a clearer view on where our methodical approach on the patterning capacity of a reaction stands, numerical studies in conditions that mimic true open spatial reactors were made. In these numerical experiments, we explored the patterning capacity of Rabai's model for pH driven Landolt type reactions as a function of experimentally attainable parameters that control the main time and length scales. Because of the straightforward reversible binding of protons to carboxylate carrying polymer chains, this class of reaction is at the base of the chemistry leading to most of the stationary reaction-diffusion patterns presently observed. We compare our model predictions with experimental observations and comment on agreements and differences.

Solvent-starved conditions in confinement cause chemical oscillations excited by passage of a cathodic delamination front

In situand operando Raman spectroscopy shows oscillations in pH during delamination of a polymer coating.

Spatiotemporal behavior induced by differential diffusion in Landolt systems

The spatiotemporal dynamics of Landolt type reactions, for instance reactions between sulfite ions and strong oxidants, performed in a medium fed by the diffusion from a boundary is investigated numerically by using a simple general reaction scheme. In these conditions, the model can generate various spatiotemporal phenomena, e.g., spatial bistability, simple and complex oscillations, birhythmicity, and chaos, even though in a spatially homogeneous open reactor only steady-state bistability can develop. The model consists of two reactions, a protonation equilibrium of the reductant and an oxidation step that is autoactivated by hydrogen ions. The rich dynamics is the result of two factors: (i) long-range activation, through the rapidly diffusing hydrogen ions, (ii) the presence of two parallel autocatalytic pathways. Long range activation provides the necessary negative feedback, whereas the dual nature of the autocatalysis leads to the appearance of two different oscillatory modes. The presence of a second-order term in the rate law of the autocatalytic reaction is linked to a large amplitude oscillations that affect the system as a whole, whereas the third-order one gives rise to localized front oscillations. The complex phenomena, like mixed mode and chaotic oscillations are observed at the meeting of these different oscillatory modes.