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

Spatial programming of self-organizing chemical systems using sustained physicochemical gradients from reaction, diffusion and hydrodynamics

Living organisms employ chemical self-organization to build structures, and inspire new strategies to design synthetic systems that spontaneously take a particular form, via a combination of integrated chemical reactions, assembly pathways and physicochemical processes. However, spatial programmability that is required to direct such self-organization is a challenge to control. Thermodynamic equilibrium typically brings about a homogeneous solution, or equilibrium structures such as supramolecular complexes and crystals. This perspective addresses out-of-equilibrium gradients that can be driven by coupling chemical reaction, diffusion and hydrodynamics, and provide spatial differentiation in the self-organization of molecular, ionic or colloidal building blocks in solution. These physicochemical gradients are required to (1) direct the organization from the starting conditions (e.g. a homogeneous solution), and (2) sustain the organization, to prevent it from decaying towards thermodynamic equilibrium. We highlight four different concepts that can be used as a design principle to establish such self-organization, using chemical reactions as a driving force to sustain the gradient and, ultimately, program the characteristics of the gradient: (1) reaction-diffusion coupling; (2) reaction-convection; (3) the Marangoni effect and (4) diffusiophoresis. Furthermore, we outline their potential as attractive pathways to translate chemical reactions and molecular/colloidal assembly into organization of patterns in solution, (dynamic) self-assembled architectures and collectively moving swarms at the micro-, meso- and macroscale, exemplified by recent demonstrations in the literature.

Provenance of life: Chemical autonomous agents surviving through associative learning

We present a benchmark study of autonomous, chemical agents exhibiting associative learning of an environmental feature. Associative learning systems have been widely studied in cognitive science and artificial intelligence but are most commonly implemented in highly complex or carefully engineered systems, such as animal brains, artificial neural networks, DNA computing systems, and gene regulatory networks, among others. The ability to encode environmental information and use it to make simple predictions is a benchmark of biological resilience and underpins a plethora of adaptive responses in the living hierarchy, spanning prey animal species anticipating the arrival of predators to epigenetic systems in microorganisms learning environmental correlations. Given the ubiquitous and essential presence of learning behaviors in the biosphere, we aimed to explore whether simple, nonliving dissipative structures could also exhibit associative learning. Inspired by previous modeling of associative learning in chemical networks, we simulated simple systems composed of long- and short-term memory chemical species that could encode the presence or absence of temporal correlations between two external species. The ability to learn this association was implemented in Gray-Scott reaction-diffusion spots, emergent chemical patterns that exhibit self-replication and homeostasis. With the novel ability of associative learning, we demonstrate that simple chemical patterns can exhibit a broad repertoire of lifelike behavior, paving the way for in vitro studies of autonomous chemical learning systems, with potential relevance to artificial life, origins of life, and systems chemistry. The experimental realization of these learning behaviors in protocell or coacervate systems could advance a new research direction in astrobiology, since our system significantly reduces the lower bound on the required complexity for autonomous chemical learning.

Synthetic spatial patterning in bacteria: advances based on novel diffusible signals

Engineering multicellular patterning may help in the understanding of some fundamental laws of pattern formation and thus may contribute to the field of developmental biology. Furthermore, advanced spatial control over gene expression may revolutionize fields such as medicine, through organoid or tissue engineering. To date, foundational advances in spatial synthetic biology have often been made in prokaryotes, using artificial gene circuits. In this review, engineered patterns are classified into four levels of increasing complexity, ranging from spatial systems with no diffusible signals to systems with complex multi-diffusor interactions. This classification highlights how the field was held back by a lack of diffusible components. Consequently, we provide a summary of both previously characterized and some new potential candidate small-molecule signals that can regulate gene expression in Escherichia coli. These diffusive signals will help synthetic biologists to successfully engineer increasingly intricate, robust and tuneable spatial structures.

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.

A bottom-up approach to construct or deconstruct a fluid instability

Fluid instabilities have been the subject of study for a long time. Despite all the extensive knowledge, they still constitute a serious challenge for many industrial applications. Here, we experimentally consider an interface between two fluids with different viscosities and analyze their relative displacement. We designed the contents of each fluid in such a way that a chemical reaction takes place at the interface and use this reaction to suppress or induce a fingering instability at will. This process describes a road map to control viscous fingering instabilities in more complex systems via interfacial chemical reactions.

Statistical approach for parameter identification by Turing patterns

Prevailing theories in biological pattern formation, such as in morphogenesis or multicellular structures development, have been based on purely chemical processes, with the Turing models as the prime example. Recent studies have challenged the approach, by underlining the role of mechanical forces. A quantitative discrimination of competing theories is difficult, however, due to the elusive character of the processes: different mechanisms may result in similar patterns, while patterns obtained with a fixed model and fixed parameter values, but with small random perturbations of initial values, will significantly differ in shape, while being of the "same" type. In this sense each model parameter value corresponds to a family of patterns, rather than a fixed solution. For this situation we create a likelihood that allows a statistically sound way to distinguish the model parameters that correspond to given patterns. The method allows us to identify model parameters of reaction-diffusion systems by using Turing patterns only, i.e., the steady-state solutions of the respective equations without the use of transient data or initial values. The method is tested with three classical models of pattern formation: the FitzHugh-Nagumo model, Gierer-Meinhardt system and Brusselator reaction-diffusion system. We quantify the accuracy achieved by different amounts of training data by Bayesian sampling methods. We demonstrate how a large enough ensemble of patterns leads to detection of very small but systematic structural changes, practically impossible to distinguish with the naked eye.

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.

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.

Mechanism of the Ferrocyanide-Iodate-Sulfite Oscillatory Chemical Reaction

Existing models of the ferrocyanide-iodate-sulfite (FIS) reaction seek to replicate the oscillatory pH behavior that occurs in open systems. These models exhibit significant differences in the amplitudes and waveforms of the concentration oscillations of such intermediates as I(-), I3(-), and Fe(CN)6(3-) under identical conditions and do not include several experimentally found intermediates. Here we report measurements of sulfite concentrations during an oscillatory cycle. Knowing the correct concentration of sulfite over the course of a period is important because sulfite is the main component that determines the buffer capacity, the pH extrema, and the amount of oxidizer (iodate) required for the transition to low pH. On the basis of this new result and recent experimental findings on the rate laws and intermediates of component processes taken from the literature, we propose a mass action kinetics model that attempts to faithfully represent the chemistry of the FIS reaction. This new comprehensive mechanism reproduces the pH oscillations and the periodic behavior in [Fe(CN)6(3-)], [I3(-)], [I(-)], and [SO3(2-)]T with characteristics similar to those seen in experiments in both CSTR and semibatch arrangements. The parameter ranges at which stationary and oscillatory behavior is exhibited also show good agreement with those of the experiments.

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.

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.

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

Sustained spatiotemporal pH and calcium patterns are produced in a non-equilibrium inorganic reaction-diffusion system by coupling two modules, the bromate-sulfite-ferrocyanide pH-oscillator and the pH-sensitive complexation of Ca(2+) by ethylenediaminetetraacetate. The development of chemical waves is mainly determined by the oscillatory module, however, the formation of the localised stationary patterns results in the synergistic interaction between the modules.

Interaction of multiarmed spirals in bistable media

We study the interaction of both dense and sparse multiarmed spirals in bistable media modeled by equations of the FitzHugh-Nagumo type. A dense one-armed spiral is characterized by its fixed tip. For dense multiarmed spirals, when the initial distance between tips is less than a critical value, the arms collide, connect, and disconnect continuously as the spirals rotate. The continuous reconstruction between the front and the back drives the tips to corotate along a rough circle and to meander zigzaggedly. The rotation frequency of tip, the frequency of zigzagged displacement, the frequency of spiral, the oscillation frequency of media, and the number of arms satisfy certain relations as long as the control parameters of the model are fixed. When the initial distance between tips is larger than the critical value, the behaviors of individual arms within either dense or sparse multiarmed spirals are identical to that of corresponding one-armed spirals.

Chemical morphogenesis: recent experimental advances in reaction-diffusion system design and control

In his seminal 1952 paper, Alan Turing predicted that diffusion could spontaneously drive an initially uniform solution of reacting chemicals to develop stable spatially periodic concentration patterns. It took nearly 40 years before the first two unquestionable experimental demonstrations of such reaction-diffusion patterns could be made in isothermal single phase reaction systems. The number of these examples stagnated for nearly 20 years. We recently proposed a design method that made their number increase to six in less than 3 years. In this report, we formally justify our original semi-empirical method and support the approach with numerical simulations based on a simple but realistic kinetic model. To retain a number of basic properties of real spatial reactors but keep calculations to a minimal complexity, we introduce a new way to collapse the confined spatial direction of these reactors. Contrary to similar reduced descriptions, we take into account the effect of the geometric size in the confinement direction and the influence of the differences in the diffusion coefficient on exchange rates of species with their feed environment. We experimentally support the method by the observation of stationary patterns in red-ox reactions not based on oxihalogen chemistry. Emphasis is also brought on how one of these new systems can process different initial conditions and memorize them in the form of localized patterns of different geometries.

Multiple types of spatio-temporal oscillations induced by differential diffusion in the Landolt reaction

The acid autoactivated iodate-sulfite redox reaction (Landolt reaction) exhibits bistability but no oscillatory dynamics when operated in a continuous stirred tank reactor (CSTR). However, it has been previously found experimentally that this reaction can exhibit both spatial bistability and oscillations when carried out in a one side diffusely fed spatial reactor. The precise origin of the oscillatory instability remained mainly elusive. We unambiguously show, in numerical simulations based of a kinetic model recently proposed by Csekõet al., J. Phys. Chem., 2008, 112, 5954), that the observed oscillations are due to the faster diffusion of the proton relative to the other feed species (long range activation instability). Furthermore, our calculations account for the previous experimental observation of two different oscillatory modes. The first one is associated to localized front oscillations, as already reported in another reaction. The other one is a periodic switch between the two states of the spatial bistability and affects the system as a whole. This oscillatory mode was undocumented in the previous studies of long range activation instabilities. More complex dynamical behaviors that mix these two types of oscillations are also reported.

Spatiotemporal dynamics of mixed Landolt systems in open gel reactors: effect of diffusive feed

In this report we present an experimental study on the spatiotemporal dynamics of the iodate-sulfite-ferrocyanide and the iodate-sulfite-thiourea systems. Both systems are capable of producing nontrivial reaction-diffusion patterns when they are operated in a one-side-fed open spatial reactor. An important parameter of these types of reactors is the time scale of the diffusive feed, which is determined by the "thickness" of the gel and diffusion coefficients of the chemicals. A conical shape gel is used to study the effect of the thickness gradient on the dynamics. We show that spatiotemporal oscillations stop below a critical thickness. It is demonstrated that the period of the oscillations is determined by the time scale of the inhibitory kinetics and the time scale of the diffusive feed together. In the case of the iodate-sulfite-thiourea system we observed the appearance of a stationary iodine front in the presence of the oscillating pH front. An experimentally supported kinetic explanation is given to account this phenomena.

Temperature oscillations, complex oscillations, and elimination of extraordinary temperature sensitivity in the iodate-sulfite-thiosulfate flow system

Temperature oscillations and complex pH oscillations in the IO(3)(-)-SO(3)(2-)-S(2)O(3)(2-) system were observed in a continuously flow stirred tank reactor. During one period of oscillation, the temperature increases rapidly while the pH shows an extremely sharp change. High-amplitude pH oscillations undergo 1(1) complex oscillations (L(S), oscillations with L large peaks and S small peaks per period) to another kind of higher-amplitude regular oscillations upon increasing the concentration of sulfite step by step. Importantly, the longstanding experimental phenomena, the extraordinary temperature sensitivity of oscillatory behavior reported 20 years ago by Rabai and Beck, can be eliminated by premixing of sulfite and sulfuric acid before entering into the reactor, avoiding local acidification, which brings out fluctuation and temperature sensitivity. The temperature oscillations can be understood by taking into account the interaction between thermal effect of various reactions and heat transfer. Experimental observations, both temperature oscillations and 1(1)-type pH oscillations, are reproduced with a four-step Horvath model by addition of an energy-balance equation. This new detailed dynamical behavior would have potential applications in designing complex chemical waves and pH responsive gels with rhythmical motion.

Introduction to focus issue: design and control of self-organization in distributed active systems

Spatiotemporal self-organization is found in a wide range of distributed dynamical systems. The coupling of the active elements in these systems may be local or global or within a network, and the interactions may be diffusive or nondiffusive in nature. The articles in this focus issue describe biological and chemical systems designed to exhibit spatiotemporal dynamics and the control of such dynamics through feedback methods.