Emergence of collective behavior in groups of excitable catalyst-loaded particles: spatiotemporal dynamical quorum sensing

Collective chemomechanical oscillations in active hydrogels

We report on the collective response of an assembly of chemomechanical Belousov-Zhabotinsky (BZ) hydrogel beads. We first demonstrate that a single isolated spherical BZ hydrogel bead with a radius below a critical value does not oscillate, whereas an assembly of the same BZ hydrogel beads presents chemical oscillation. A BZ chemical model with an additional flux of chemicals out of the BZ hydrogel captures the experimentally observed transition from oxidized nonoscillating to oscillating BZ hydrogels and shows this transition is due to a flux of inhibitors out of the BZ hydrogel. The model also captures the role of neighboring BZ hydrogel beads in decreasing the critical size for an assembly of BZ hydrogel beads to oscillate. We finally leverage the quorum sensing behavior of the collective to trigger their chemomechanical oscillation and discuss how this collective effect can be used to enhance the oscillatory strain of these active BZ hydrogels. These findings could help guide the eventual fabrication of a swarm of autonomous, communicating, and motile hydrogels.

Collective Behavior of Urease pH Clocks in Nano- and Microvesicles Controlled by Fast Ammonia Transport

The transmission of chemical signals via an extracellular solution plays a vital role in collective behavior in cellular biological systems and may be exploited in applications of lipid vesicles such as drug delivery. Here, we investigated chemical communication in synthetic micro- and nanovesicles containing urease in a solution of urea and acid. We combined experiments with simulations to demonstrate that the fast transport of ammonia to the external solution governs the pH-time profile and synchronizes the timing of the pH clock reaction in a heterogeneous population of vesicles. This study shows how the rate of production and emission of a small basic product controls pH changes in active vesicles with a distribution of sizes and enzyme amounts, which may be useful in bioreactor or healthcare applications.

Seeding Crystallization in Time

We introduce the concept of seeding of crystallization in time by studying the dynamics of an ensemble of coupled continuous time crystals. We demonstrate that a single subsystem in a broken-symmetry phase acting as a nucleation center may induce time-translation symmetry breaking across the entire ensemble. Seeding is observed for both coherent and dissipative coupling, as well as for a broad range of parameter regimes. In the spirit of mutual synchronization, we investigate the parameter regime where all subsystems are in the broken-symmetry phase. We observe that more broadly detuned time crystals require weaker coupling strength to be synchronized. This is in contrast to basic knowledge from classical as well as quantum synchronization theory. We show that this surprising observation is a direct consequence of the seeding effect.

Chemical micro-oscillators based on the Belousov–Zhabotinsky reaction

The results of studies on the development of micro-oscillators (MOs) based on the Belousov –Zhabotinsky (BZ) oscillatory chemical reaction are integrated and systematized. The mechanisms of the BZ reaction and the methods of immobilization of the catalyst of the BZ reaction in micro-volumes are briefly discussed. Methods for creating BZ MOs based on water microdroplets in the oil phase and organic and inorganic polymer microspheres are considered. Methods of control and management of the dynamics of BZ MO networks are described, including methods of MO synchronization. The prospects for the design of neural networks of MOs with intelligent-like behaviour are outlined. Such networks present a new area of nonlinear chemistry, including, in particular, the creation of a chemical ‘computer’.

The bibliography includes 250 references.

Reaction-diffusion hydrogels from urease enzyme particles for patterned coatings

The reaction and diffusion of small molecules is used to initiate the formation of protective polymeric layers, or biofilms, that attach cells to surfaces. Here, inspired by biofilm formation, we present a general method for the growth of hydrogels from urease enzyme-particles by combining production of ammonia with a pH-regulated polymerization reaction in solution. We show through experiments and simulations how the propagating basic front and thiol-acrylate polymerization were continuously maintained by the localized urease reaction in the presence of urea, resulting in hydrogel layers around the enzyme particles at surfaces, interfaces or in motion. The hydrogels adhere the enzyme-particles to surfaces and have a tunable growth rate of the order of 10 µm min^-1 that depends on the size and spatial distribution of particles. This approach can be exploited to create enzyme-hydrogels or chemically patterned coatings for applications in biocatalytic flow reactors.

Segmentation clock dynamics is strongly synchronized in the forming somite

During vertebrate somitogenesis an inherent segmentation clock coordinates the spatiotemporal signaling to generate segmented structures that pattern the body axis. Using our experimental and quantitative approach, we study the cell movements and the genetic oscillations of her1 expression level at single-cell resolution simultaneously and scale up to the entire pre-somitic mesoderm (PSM) tissue. From the experimentally determined phases of PSM cellular oscillators, we deduced an in vivo frequency profile gradient along the anterior-posterior PSM axis and inferred precise mathematical relations between spatial cell-level period and tissue-level somitogenesis period. We also confirmed a gradient in the relative velocities of cellular oscillators along the axis. The phase order parameter within an ensemble of oscillators revealed the degree of synchronization in the tailbud and the posterior PSM being only partial, whereas synchronization can be almost complete in the presumptive somite region but with temporal oscillations. Collectively, the degree of synchronization itself, possibly regulated by cell movement and the synchronized temporal phase of the transiently expressed clock protein Her1, can be an additional control mechanism for making precise somite boundaries.

Size- and position-dependent bifurcations of chemical microoscillators in confined geometries

The present theoretical study deals with microparticles (beads) that contain an immobilized Belousov-Zhabotinsky (BZ) reaction catalyst. In the theoretical experiment, a BZ bead is immersed in a small water droplet that contains all of the BZ reaction reagents but no catalyst. Such heterogeneous reaction-diffusion BZ systems with the same BZ reactant concentrations demonstrate various dynamic modes, including steady state and low-amplitude, high-amplitude, and mixed-mode oscillations (MMOs). The emergence of such dynamics depends on the sizes of the bead and water droplet, as well as on the location of the bead inside the droplet. MMO emergence is explained by time-delayed positive feedback in combination with a canard phenomenon. If two identical BZ beads are immersed in the same droplet, many different dynamic modes including chaos are observed.

Phase transition to synchronization in generalized Kuramoto model with low-pass filter

A second-order continuous synchronization has been well documented for the classic Kuramoto model. Here we generalize the classic Kuramoto model by incorporating a low-pass filter (LPF) in the coupling, which serves as a simple form of indirect coupling through a common external dynamic environment. We uncover that a first-order explosive synchronization turns out to be a very generic phenomenon in this generalized Kuramoto model with LPF. We establish theoretical results by providing a rigorous analytical treatment, which is validated by conducting extensive numerical simulations. Our study provides a new root for the emergence of first-order explosive synchronization, which could substantially deepen the understanding of the underlying mechanism of a first-order phase transition towards synchronization in coupled dynamical networks.

Complex oscillation modes in the Belousov-Zhabotinsky reaction by weak diffusive coupling

We study the diffusive coupling of oscillating or excitable Belousov-Zhabotinsky reaction units arranged in a square lattice array and show that for certain sizes of the units and for certain distances between the units, complex oscillation modes of individual spots occur, which manifest themselves in multi-periodic, amplitude-modulated, and multi-mode oscillations. This experimental finding can be reproduced in simulations of the FitzHugh-Nagumo model mimicking the experimental setup, suggesting that it is a generic phenomenon in systems of coupled excitable units such as excitable cell tissues or coupled oscillators such as neurons. Further analysis let us conclude that the complex oscillation modes occur close to the transition from quiescent to coupling-induced oscillations states if this transition is taking place at weak coupling strength.

Selection of spatiotemporal patterns in arrays of spatially distributed oscillators indirectly coupled via a diffusive environment

Emergence of self-organized behaviors in diverse living systems often depends on population density. In these systems, cell-cell communications are usually mediated by the surrounding environment. Collective behaviors (e.g., synchrony and dynamical quorum sensing) of such systems with stirred environment have been extensively studied, but the spatiotemporal dynamics of the oscillators coupled via a diffusive environment (without stirring) is rather understudied. We here perform a computational study on the selection and competition of wave patterns in arrays of spatially distributed oscillators immersed in a diffusive medium. We find that population density plays a crucial role in the selection of wave patterns: (i) for a single spiral in the system, its rotation either inward or outward could be controlled by population density, and (ii) for spiral and target waves coexisting initially in the system, wave competition happens and population density decides which type of wave will finally survive. The latter phenomenon is further confirmed in a system whose individual element is excitable rather than self-sustained oscillatory. The mechanism underlying all these observations is attributed to the frequency competition. Our results in the excitable case may have implications on the experimental results.

Oscillatory activity regulation in an ensemble of autonomous mercury beating heart oscillators

Collective behavior of an ensemble of directly or indirectly coupled oscillators can be a function of population density. Experiments using autonomous mercury beating heart (MBH) oscillators coupled through their surroundings are employed, to study the existence of quorum-like (population dependent) phenomena. Two coupling mechanisms are used, namely, static and dynamic coupling. For the static coupling scheme, the transitions of a subset of the coupled oscillators occur from active (oscillatory) to inactive (quiescent) state and vice versa. A continuous variation of collective dynamics was observed as the population of the oscillators increased. For the dynamic coupling scheme, the time for which the coupled oscillators are active changes sharply as the population increases beyond a certain threshold.

Abrupt and gradual onset of synchronized oscillations due to dynamical quorum sensing in the single-cathode multi-anode nickel electrodissolution system

The nonlinear dynamics of an oscillatory Ni electrodissolution-hydrogen ion reduction system are explored in a multi-electrode anode-single cathode system. A mathematical analysis of the charge balance equations reveals that the coupling scheme is similar to dynamical quorum sensing, where the number of anode wires affects a parameter related to the population density. In a parameter region where the large population exhibits stationary behavior, with sufficiently strong coupling (with small individual resistances attached to the anode wires), synchronized oscillations emerge abruptly with decreasing the number of anodes. Therefore, an "inverse" dynamical quorum sensing takes place. With weak coupling the transition is gradual. The experiments are supported by numerical simulation of a kinetic model of the process. The results thus show that the description of nontrivial cathode-anode interactions in the form of dynamical quorum sensing provides an efficient way of analyzing the dynamical response of complex, interacting electrochemical reactions.

Influence of reaction-induced convection on quorum sensing in enzyme-loaded agarose beads

In theory, groups of enzyme-loaded particles producing an acid or base may show complex behavior including dynamical quorum sensing, the appearance of synchronized oscillations above a critical number or density of particles. Here, experiments were performed with the enzyme urease loaded into mm-sized agarose beads and placed in a solution of urea, resulting in an increase in pH. This behavior was found to be dependent upon the number of beads present in the array; however, reaction-induced convection occurred and plumes of high pH developed that extended to the walls of the reactor. The convection resulted in the motion of the mm-sized particles and conversion of the solution to high pH. Simulations in a simple model of the beads demonstrated the suppression of dynamical quorum sensing in the presence of flow.

Reaction processes among self-propelled particles

We study a system of self-propelled disks that perform run-and-tumble motion, where particles can adopt more than one internal state. One of those internal states can be transmitted to other particle if the particle carrying this state maintains physical contact with another particle for a finite period of time. We refer to this process as a reaction process and to the different internal states as particle species, making an analogy to chemical reactions. The studied system may fall into an absorbing phase, where due to the disappearance of one of the particle species no further reaction can occur, or may remain in an active phase where particles constantly react. By combining individual-based simulations and mean-field arguments, we study the dependency of the equilibrium densities of particle species on motility parameters, specifically the active speed v₀ and tumbling frequency λ. We find that the equilibrium densities of particle species exhibit two very distinct, non-trivial scaling regimes, with v₀ and λ depending on whether the system is in the so-called ballistic or diffusive regime. Our mean-field estimates lead to an effective renormalization of reaction rates that allow building the phase-diagram v₀-λ that separates the absorbing and active phases. We find an excellent agreement between numerical simulations and mean-field estimates. This study is a necessary step towards an understanding of phase transitions into absorbing states in active systems and sheds light on the spreading of information/signaling among moving elements.

Under Diffusion Control: from Structuring Matter to Directional Motion

Self-organization in synthetic chemical systems is quickly developing into a powerful strategy for designing new functional materials. As self-organization requires the system to exist far from thermodynamic equilibrium, chemists have begun to go beyond the classical equilibrium self-assembly that is often applied in bottom-up supramolecular synthesis, and to learn about the surprising and unpredicted emergent properties of chemical systems that are characterized by a higher level of complexity and extended reactivity networks. The present review focuses on self-organization in reaction-diffusion systems. Selected examples show how the emergence of complex morphogenesis is feasible in synthetic systems leading to hierarchically and nanostructured matter. Starting from well-investigated oscillating reactions, recent developments extend diffusion-limited reactivity to supramolecular systems. The concept of dynamic instability is introduced and illustrated as an additional tool for the design of smart materials and actuators, with emphasis on the realization of motion even at the macroscopic scale. The formation of spatio-temporal patterns along diffusive chemical gradients is exploited as the main channel to realize symmetry breaking and therefore anisotropic and directional mechanical transformations. Finally, the interaction between external perturbations and chemical gradients is explored to give mechanistic insights in the design of materials responsive to external stimuli.

Switches induced by quorum sensing in a model of enzyme-loaded microparticles

Quorum sensing refers to the ability of bacteria and other single-celled organisms to respond to changes in cell density or number with population-wide changes in behaviour. Here, simulations were performed to investigate quorum sensing in groups of diffusively coupled enzyme microparticles using a well-characterized autocatalytic reaction which raises the pH of the medium: hydrolysis of urea by urease. The enzyme urease is found in both plants and microorganisms, and has been widely exploited in engineering processes. We demonstrate how increases in group size can be used to achieve a sigmoidal switch in pH at high enzyme loading, oscillations in pH at intermediate enzyme loading and a bistable, hysteretic switch at low enzyme loading. Thus, quorum sensing can be exploited to obtain different types of response in the same system, depending on the enzyme concentration. The implications for microorganisms in colonies are discussed, and the results could help in the design of synthetic quorum sensing for biotechnology applications such as drug delivery.

Microscopic agents programmed by DNA circuits

Information stored in synthetic nucleic acids sequences can be used in vitro to create complex reaction networks with precisely programmed chemical dynamics. Here, we scale up this approach to program networks of microscopic particles (agents) dispersed in an enzymatic solution. Agents may possess multiple stable states, thus maintaining a memory and communicate by emitting various orthogonal chemical signals, while also sensing the behaviour of neighbouring agents. Using this approach, we can produce collective behaviours involving thousands of agents, for example retrieving information over long distances or creating spatial patterns. Our systems recapitulate some fundamental mechanisms of distributed decision making and morphogenesis among living organisms and could find applications in cases where many individual clues need to be combined to reach a decision, for example in molecular diagnostics.

Boosting functionality of synthetic DNA circuits with tailored deactivation

Molecular programming takes advantage of synthetic nucleic acid biochemistry to assemble networks of reactions, in vitro, with the double goal of better understanding cellular regulation and providing information-processing capabilities to man-made chemical systems. The function of molecular circuits is deeply related to their topological structure, but dynamical features (rate laws) also play a critical role. Here we introduce a mechanism to tune the nonlinearities associated with individual nodes of a synthetic network. This mechanism is based on programming deactivation laws using dedicated saturable pathways. We demonstrate this approach through the conversion of a single-node homoeostatic network into a bistable and reversible switch. Furthermore, we prove its generality by adding new functions to the library of reported man-made molecular devices: a system with three addressable bits of memory, and the first DNA-encoded excitable circuit. Specific saturable deactivation pathways thus greatly enrich the functional capability of a given circuit topology.

Nonlinearity in synthetic molecular circuits is usually achieved by manipulation of network topology or of production kinetics. Here, the authors achieve bistability and other nonlinear behaviours by manipulating the individual degradation rate laws of circuit components using saturable pathways.

Synchronization of Coupled Oscillators on a Two-Dimensional Plane

The effect of the transfer rate of signal molecules on coupled chemical oscillators arranged on a two-dimensional plane was systematically investigated in this paper. A microreactor equipped with a surface acoustic wave (SAW) mixer was applied to adjust the transfer rate of the signal molecules in the microreactor. The SAW mixer with adjustable input powers provided a simple means to generate different mixing rates in the microreactor. A robust synchronization of the oscillators was found at an input radio frequency power of 20 dBm, with which the chemical waves were initiated at a fixed site of the oscillator system. With increasing input power, the frequency of the chemical waves was increased, which agreed well with the prediction given by the time-delayed phase oscillator model. Results from the finite element simulation agreed well with the experimental results.

Self-Organization of Embryonic Genetic Oscillators into Spatiotemporal Wave Patterns

In vertebrate embryos, somites, the precursor of vertebrae, form from the presomitic mesoderm (PSM), which is composed of cells displaying signaling oscillations. Cellular oscillatory activity leads to periodic wave patterns in the PSM. Here, we address the origin of such complex wave patterns. We employed an in vitro randomization and real-time imaging strategy to probe for the ability of cells to generate order from disorder. We found that, after randomization, PSM cells self-organized into several miniature emergent PSM structures (ePSM). Our results show an ordered macroscopic spatial arrangement of ePSM with evidence of an intrinsic length scale. Furthermore, cells actively synchronize oscillations in a Notch-signaling-dependent manner, re-establishing wave-like patterns of gene activity. We demonstrate that PSM cells self-organize by tuning oscillation dynamics in response to surrounding cells, leading to collective synchronization with an average frequency. These findings reveal emergent properties within an ensemble of coupled genetic oscillators.

Spiral wave chimeras in locally coupled oscillator systems

The recently discovered chimera state involves the coexistence of synchronized and desynchronized states for a group of identical oscillators. In this work, we show the existence of (inwardly) rotating spiral wave chimeras in the three-component reaction-diffusion systems where each element is locally coupled by diffusion. A transition from spiral waves with the smooth core to spiral wave chimeras is found as we change the local dynamics of the system or as we gradually increase the diffusion coefficient of the activator. Our findings on the spiral wave chimera in the reaction-diffusion systems suggest that spiral chimera states may be found in chemical and biological systems that can be modeled by a large population of oscillators indirectly coupled via a diffusive environment.

Extended source model for diffusive coupling

Motivated by the prevailing approach to diffusion coupling phenomena which considers point-like diffusing sources, we derived an analogous expression for the concentration rate of change of diffusively coupled extended containers. The proposed equation, together with expressions based on solutions to the diffusion equation, is intended to be applied to the numerical solution of systems exclusively composed of ordinary differential equations, however is able to account for effects due the finite size of the coupled sources.

pH Oscillator Stretched in Space but Frozen in Time

Chemical oscillations are studied using a continuous-flow microfluidic system transforming the time domain of chemical oscillators into a spatial domain. This system allows one (i) to monitor the dynamics of chemical oscillators with the accuracy of vigorously stirred batch reactors but with the ease and speed of CSTRs and (ii) to rapidly screen the phase space of chemical oscillators in just one experiment versus a traditional series of batch measurements.

Insights into collective cell behaviour from populations of coupled chemical oscillators

Biological systems such as yeast show coordinated activity driven by chemical communication between cells. Experiments with coupled chemical oscillators can provide insights into the collective behaviour.

Oregonator generalization as a minimal model of quorum sensing in Belousov–Zhabotinsky reaction with catalyst confinement in large populations of particles

The Oregonator demonstrates that quorum sensing in populations of Belousov–Zhabotinsky oscillators arises from modification of the stoichiometry by catalyst confinement.

Quorum sensing via static coupling demonstrated by Chua's circuits

Dynamical quorum sensing, the population based phenomenon, is believed to occur when the elements of a system interact via dynamic coupling. In the present work, we demonstrate an alternate scenario, involving static coupling, that could also lead to quorum sensing behavior. These static and dynamic coupling terms have already been employed by Konishi [Int. J. Bifurcation Chaos Appl. Sci. Eng. 17, 2781 (2007)]. In our context, the coupling is defined as static or dynamic, on the basis of the relative time scales at which the surrounding dynamics and the elements' dynamics evolve. According to this, if the variation in the surrounding dynamics happens on a much larger (fast) time scale than that at which the elements' dynamics are varying (such as seconds and μs), then the coupling is considered to be static, otherwise it is considered to be dynamic. A series of experiments have been performed starting from a system of three Chua's circuits to a system of 20 Chua's circuits to study two types of quorum transitions: the emergence and the extinction of global oscillations (period-1). The numerics involving up to 100 Chua's circuits validate the experimental observations.

Chemistry with spatial control using particles and streams()

Spatial control of chemical reactions, with micro- and nanometer scale resolution, has important consequences for one pot synthesis, engineering complex reactions, developmental biology, cellular biochemistry and emergent behavior. We review synthetic methods to engineer this spatial control using chemical diffusion from spherical particles, shells and polyhedra. We discuss systems that enable both isotropic and anisotropic chemical release from isolated and arrayed particles to create inhomogeneous and spatially patterned chemical fields. In addition to such finite chemical sources, we also discuss spatial control enabled with laminar flow in 2D and 3D microfluidic networks. Throughout the paper, we highlight applications of spatially controlled chemistry in chemical kinetics, reaction-diffusion systems, chemotaxis and morphogenesis.

Optimal cellular mobility for synchronization arising from the gradual recovery of intercellular interactions

Cell movement and intercellular signaling occur simultaneously during the development of tissues, but little is known about how movement affects signaling. Previous theoretical studies have shown that faster moving cells favor synchronization across a population of locally coupled genetic oscillators. An important assumption in these studies is that cells can immediately interact with their new neighbors after arriving at a new location. However, intercellular interactions in cellular systems may need some time to become fully established. How movement affects synchronization in this situation has not been examined. Here, we develop a coupled phase oscillator model in which we consider cell movement and the gradual recovery of intercellular coupling experienced by a cell after movement, characterized by a moving rate and a coupling recovery rate, respectively. We find (1) an optimal moving rate for synchronization and (2) a critical moving rate above which achieving synchronization is not possible. These results indicate that the extent to which movement enhances synchrony is limited by a gradual recovery of coupling. These findings suggest that the ratio of time scales of movement and signaling recovery is critical for information transfer between moving cells.

Active targeting in a random porous medium by chemical swarm robots with secondary chemical signaling

The multibody dynamics of a system of chemical swarm robots in a porous environment is investigated. The chemical swarm robots are modeled as brownian particles capable of delivering an encapsulated chemical payload toward a given target location and releasing it in response to an external stimulus. The presence of chemical signals (chemo-attractant) in the system plays a crucial role in coordinating the collective movement of the particles via chemotaxis. For a number of applications, such as distributed chemical processing and targeted drug delivery, the understanding of factors that govern the collective behavior of the particles, especially their ability to localize a given target, is of immense importance. A hybrid modeling methodology based on the combination of the brownian dynamics method and diffusion problem coupled through the chemotaxis phenomena is used to analyze the impact of a varying signaling threshold and the strength of chemotaxis on the ability of the chemical robots to fulfill their target localization mission. The results demonstrate that the selected performance criteria (the localization half time and the success rate) can be improved when an appropriate signaling process is chosen. Furthermore, for an optimum target localization strategy, the topological complexity of the porous environment needs to be reflected.

Distant synchronization through a passive medium

This paper deals with the phenomenon of synchronization of oscillatory ensembles interacting distantly through the passive medium. Main characteristics of such a kind of synchronization are studied. The results of this work can be applied to describe the synchronization of cardiac oscillatory cells separated by the passive fibroblasts. In this work the phenomenological models (Bonhoeffer-Van der Pol) of cardiac cells as well as biologically relevant (Luo-Rudy, Sachse) models are used. We also propose equivalent model of distant synchronization and derive on its basis an analytical scaling of the frequency of synchronous oscillations.

Dynamical structures in binary media of potassium-driven neurons

According to the conventional approach neural ensembles are modeled with fixed ionic concentrations in the extracellular environment. However, in some cases the extracellular concentration of potassium ions cannot be regarded as constant. Such cases represent specific chemical pathway for neurons to interact and can influence strongly the behavior of single neurons and of large ensembles. The released chemical agent diffuses in the external medium and lowers thresholds of individual excitable units. We address this problem by studying simplified excitable units given by a modified FitzHugh-Nagumo dynamics. In our model the neurons interact only chemically via the released and diffusing potassium in the surrounding nonactive medium and are permanently affected by noise. First, we study the dynamics of a single excitable unit embedded in the extracellular matter. That leads to a number of noise-induced effects such as self-modulation of firing rate in an individual neuron. After the consideration of two coupled neurons we consider the spatially extended situation. By holding parameters of the neuron fixed, various patterns appear ranging from spirals and traveling waves to oscillons and inverted structures depending on the parameters of the medium.