Dynamical quorum-sensing in oscillators coupled through an external medium

Mobility-induced order in active XY spins on a substrate

We elucidate that the nearly phase-ordered active XY spins in contact with a conserved, diffusing species on a substrate can be stable. For wide-ranging model parameters, it has stable uniform phases robust against noises. These are distinguished by generalized quasi-long-range (QLRO) orientational order logarithmically stronger or weaker than the well-known QLRO in equilibrium, together with miniscule (i.e., hyperuniform) or giant number fluctuations, respectively. This illustrates a direct correspondence between the two. The scaling of both phase and density fluctuations in the stable phase-ordered states is nonuniversal: they depend on the nonlinear dynamical couplings. For other parameters, it has no stable uniformly ordered phase. Our model, a theory for active spinners, provides a minimal framework for wide-ranging systems, e.g., active superfluids on substrates, synchronization of oscillators, active carpets of cilia and bacterial flagella, and active membranes.

Active XY model on a substrate: Density fluctuations and phase ordering

We explore the generic long-wavelength properties of an active XY model on a substrate, consisting of a collection of nearly phase-ordered active XY spins in contact with a diffusing, conserved species, as a representative system of active spinners with a conservation law. The spins rotate actively in response to the local density fluctuations and local phase differences, on a solid substrate. We investigate this system by Monte Carlo simulations of an agent-based model, which we set up, complemented by the hydrodynamic theory for the system. We demonstrate that this system can phase-synchronize without any hydrodynamic interactions. Our combined numerical and analytical studies show that this model, when stable, displays hitherto unstudied scaling behavior: As a consequence of the interplay between the mobility, active rotation, and number conservation, such a system can be stable over a wide range of the model parameters characterized by a novel correspondence between the phase and density fluctuations. In different regions of the phase space where the phase-ordered system is stable, it displays generalized quasi-long-range order (QLRO): It shows phase ordering which is generically either logarithmically stronger than the conventional QLRO found in its equilibrium limit, together with "miniscule number fluctuations," or logarithmically weaker than QLRO along with "giant number fluctuations," showing a novel one-to-one correspondence between phase ordering and density fluctuations in the ordered states. Intriguingly, these scaling exponents are found to depend explicitly on the model parameters. We further show that in other parameter regimes there are no stable, ordered phases. Instead, two distinct types of disordered states with short-range phase order are found, characterized by the presence or absence of stable clusters of finite sizes. In a surprising connection, the hydrodynamic theory for this model also describes the fluctuations in a Kardar-Parisi-Zhang (KPZ) surface with a conserved species on it, or an active fluid membrane with a finite tension, without momentum conservation and a conserved species living on it. This implies the existence of stable fluctuating surfaces that are only logarithmically smoother or rougher than the Edward-Wilkinson surface at two dimensions (2D) can exist, in contrast to the 2D pure KPZ-like "rough" surfaces.

Quenching of oscillations in a liquid metal via attenuated coupling

In this work, we report a quenching of oscillations observed upon coupling two chemomechanical oscillators. Each one of these oscillators consists of a drop of liquid metal submerged in an oxidizing solution. These pseudoidentical oscillators have been shown to exhibit both periodic and aperiodic oscillatory behavior. In the experiments performed on these oscillators, we find that coupling two such oscillators via an attenuated resistive coupling leads the coupled system towards an oscillation quenched state. To further comprehend these experimental observations, we numerically explore and verify the presence of similar oscillation quenching in a model of coupled Hindmarsh-Rose (HR) systems. A linear stability analysis of this HR system reveals that attenuated coupling induces a change in eigenvalues of the relevant Jacobian, leading to stable quenched oscillation states. Additionally, the analysis yields a threshold of attenuation for oscillation quenching that is consistent with the value observed in numerics. So this phenomenon, demonstrated through experiments, as well as simulations and analysis of a model system, suggests a powerful natural mechanism that can potentially suppress periodic and aperiodic oscillations in coupled nonlinear systems.

Synchronisation of glycolytic activity in yeast cells

Glycolysis is the central metabolic pathway of almost every cell and organism. Under appropriate conditions, glycolytic oscillations may occur in individual cells as well as in entire cell populations or tissues. In many biological systems, glycolytic oscillations drive coherent oscillations of other metabolites, for instance in cardiomyocytes near anorexia, or in pancreas where they lead to a pulsatile release of insulin. Oscillations at the population or tissue level require the cells to synchronize their metabolism. We review the progress achieved in studying a model organism for glycolytic oscillations, namely yeast. Oscillations may occur on the level of individual cells as well as on the level of the cell population. In yeast, the cell-to-cell interaction is realized by diffusion-mediated intercellular communication via a messenger molecule. The present mini-review focuses on the synchronisation of glycolytic oscillations in yeast. Synchronisation is a quorum-sensing phenomenon because the collective oscillatory behaviour of a yeast cell population ceases when the cell density falls below a threshold. We review the question, under which conditions individual cells in a sparse population continue or cease to oscillate. Furthermore, we provide an overview of the pathway leading to the onset of synchronized oscillations. We also address the effects of spatial inhomogeneities (e.g., the formation of spatial clusters) on the collective dynamics, and also review the emergence of travelling waves of glycolytic activity. Finally, we briefly review the approaches used in numerical modelling of synchronized cell populations.

Collective behaviors of mean-field coupled Stuart-Landau limit-cycle oscillators under additional repulsive links

We study the collective behaviors of a large population of Stuart-Landau limit-cycle oscillators that coupled diffusively and equally with all of the others via the conjugate of the mean field, where the underlying interaction is shown to break the rotational symmetry of the coupled system. In the model, an ensemble of Stuart-Landau oscillators are in fact diffusively coupled via the mean field in the real parts, whereas additional repulsive links are present in the imaginary parts. All the oscillators are linked via the similar state variables, which distinctly differs from the conjugate coupling through dissimilar variables in the previous studies. We show that depending on the strength of coupling and the distribution of natural frequencies, the coupled system exhibits three qualitatively different types of collective stationary behaviors: amplitude death (AD), oscillation death (OD), and incoherent state. Our goal is to analytically characterize the onset of the above three typical macrostates by performing the rigorous linear stability analyses of the corresponding mean-field coupled system. We prove that AD is able to occur in the coupled system with identical frequencies, where the stable coupling interval of AD is found to be independent on the system's size N. When the natural frequencies are distributed according to a general density function, we obtain the analytic equations that govern the exact stability boundaries of AD, OD, and the incoherence for a coupled system in the thermodynamic limit N→∞. All the theoretical predictions are well confirmed via numerical simulations of the coupled system with a specific Lorentzian frequency distribution.

Arnold tongues in oscillator systems with nonuniform spatial driving

Nonlinear oscillator systems are ubiquitous in biology and physics, and their control is a practical problem in many experimental systems. Here we study this problem in the context of the two models of spatially coupled oscillators: the complex Ginzburg-Landau equation (CGLE) and a generalization of the CGLE in which oscillators are coupled through an external medium (emCGLE). We focus on external control drives that vary in both space and time. We find that the spatial distribution of the drive signal controls the frequency ranges over which oscillators synchronize to the drive and that boundary conditions strongly influence synchronization to external drives for the CGLE. Our calculations also show that the emCGLE has a low density regime in which a broad range of frequencies can be synchronized for low drive amplitudes. We study the bifurcation structure of these models and find that they are very similar to results for the driven Kuramoto model, a system with no spatial structure. We conclude by discussing qualitative implications of our results for controlling coupled oscillator systems such as the social amoebae Dictyostelium and populations of Belousov Zhabotinsky (BZ) catalytic particles using spatially structured external drives.

Partial synchronisation of glycolytic oscillations in yeast cell populations

The transition between synchronized and asynchronous behaviour of immobilized yeast cells of the strain Saccharomyces carlsbergensis was investigated by monitoring the autofluorescence of the coenzyme NADH. In populations of intermediate cell densities the individual cells remained oscillatory, whereas on the level of the cell population both a partially synchronized and an asynchronous state were accessible for experimental studies. In the partially synchronized state, the mean oscillatory frequency was larger than that of the cells in the asynchronous state. This suggests that synchronisation occurred due to entrainment by the cells that oscillated more rapidly. This is typical for synchronisation due to phase advancement. Furthermore, the synchronisation of the frequency of the glycolytic oscillations preceded the synchronisation of their phases. However, the cells did not synchronize completely, as the distribution of the oscillatory frequencies only narrowed but did not collapse to a unique frequency. Cells belonging to spatially denser clusters showed a slightly enhanced local synchronisation during the episode of partial synchronisation. Neither the clusters nor a transition from partially synchronized glycolytic oscillations to travelling glycolytic waves did substantially affect the degree of partial synchronisation. Chimera states, i.e., the coexistence of a synchronized and an asynchronous part of the population, could not be found.

Symmetry breakings in two populations of oscillators coupled via diffusive environments: Chimera and heterosynchrony

We consider two diffusively coupled populations of identical oscillators, where the oscillators in each population are coupled with a common dynamic environment. Existence and stability of a variety of stationary states are analyzed on the basis of the Ott-Antonsen reduction method, which reveals that the chimera state occurs under the diffusive coupling scheme. Furthermore, we find an exotic symmetry-breaking behavior, the so-called the heterosynchronous state, in which each population exhibits in-phase coherence, while the order parameters of two populations rotate at different phase velocities. The chimera and heterosynchronous states emerge from bistabilities of distinct states for decoupled population and occur as a unique continuation for weak diffusive couplings. The heterosynchronous state is caused by an indirect coupling scheme via dynamic environments and could occur for a finite-size system as well, even for the system that consists of one oscillator per population.

Amplitude chimera and chimera death induced by external agents in two-layer networks

We report the emergence of stable amplitude chimeras and chimera death in a two-layer network where one layer has an ensemble of identical nonlinear oscillators interacting directly through local coupling and indirectly through dynamic agents that form the second layer. The nonlocality in the interaction among the dynamic agents in the second layer induces different types of chimera-related dynamical states in the first layer. The amplitude chimeras developed in them are found to be extremely stable, while chimera death states are prevalent for increased coupling strengths. The results presented are for a system of coupled Stuart-Landau oscillators and can, in general, represent systems with short-range interactions coupled to another set of systems with long-range interactions. In this case, by tuning the range of interactions among the oscillators or the coupling strength between two types of systems, we can control the nature of chimera states and the system can also be restored to homogeneous steady states. The dynamic agents interacting nonlocally with long-range interactions can be considered as a dynamic environment or a medium interacting with the system. We indicate how the second layer can act as a reinforcement mechanism on the first layer under various possible interactions for desirable effects.

A Novel Time-Based Modulation Scheme in Time-Asynchronous Channels for Molecular Communications

In this paper, a novel time-based modulation scheme is proposed in the time-asynchronous channel for diffusion-based molecular communication systems with drift. Based on this modulation scheme, we demonstrate that the sample variance of information molecules' arrival time approximately follows a noncentral chi-squared distribution. According to its conditional probability density function (PDF), the asynchronous receiver designs are deduced based on the maximum likelihood (ML) detection, with or without background noise in the channel environment. Since the proposed schemes can be applied to the case of transmitting multiple information molecules, simulation results reveal that the bit error ratio (BER) performance improves with the increase of the number of released information molecules per bit. Furthermore, when the background noise is not negligible, our proposed asynchronous scheme outperforms the asynchronous modulation techniques based on encoding information on the time between two consecutive release of information molecules.

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.

The Clock-Free Asynchronous Receiver Design for Molecular Timing Channels in Diffusion-Based Molecular Communications

In diffusion-based molecular communications, time synchronization is a major reason for the increase of system structure complexity. In this paper, we consider the asynchronous receiver design for molecular communications with information symbols conveyed in the time of released molecules. The main contribution of this paper is that we develop a detector called clock-free asynchronous receiver design (CFARD), in which the receiver recovers the information symbols without measuring the arrival time of molecules. The theoretical analysis indicates that compared with the synchronous receiver designs, the proposed scheme considerably lowers the structure complexity for information demodulation, which is of great significance to the feasibility of nano-scale molecular communications systems with the limitation of energy and size. The numerical results show that in the comparison of bit error ratio (BER) performance, the proposed asynchronous receiver design outperforms the synchronous linear average filter (LAF) detector and approaches to the synchronous maximum likelihood (ML) detector and first arrival (FA) detector.

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.

Mapping quorum sensing onto neural networks to understand collective decision making in heterogeneous microbial communities

Microbial communities frequently communicate via quorum sensing (QS), where cells produce, secrete, and respond to a threshold level of an autoinducer (AI) molecule, thereby modulating gene expression. However, the biology of QS remains incompletely understood in heterogeneous communities, where variant bacterial strains possess distinct QS systems that produce chemically unique AIs. AI molecules bind to 'cognate' receptors, but also to 'non-cognate' receptors found in other strains, resulting in inter-strain crosstalk. Understanding these interactions is a prerequisite for deciphering the consequences of crosstalk in real ecosystems, where multiple AIs are regularly present in the same environment. As a step towards this goal, we map crosstalk in a heterogeneous community of variant QS strains onto an artificial neural network model. This formulation allows us to systematically analyze how crosstalk regulates the community's capacity for flexible decision making, as quantified by the Boltzmann entropy of all QS gene expression states of the system. In a mean-field limit of complete cross-inhibition between variant strains, the model is exactly solvable, allowing for an analytical formula for the number of variants that maximize capacity as a function of signal kinetics and activation parameters. An analysis of previous experimental results on the Staphylococcus aureus two-component Agr system indicates that the observed combination of variant numbers, gene expression rates and threshold concentrations lies near this critical regime of parameter space where capacity peaks. The results are suggestive of a potential evolutionary driving force for diversification in certain QS systems.

Revoking amplitude and oscillation deaths by low-pass filter in coupled oscillators

When in an ensemble of oscillatory units the interaction occurs through a diffusion-like manner, the intrinsic oscillations can be quenched through two structurally different scenarios: amplitude death (AD) and oscillation death (OD). Unveiling the underlying principles of stable rhythmic activity against AD and OD is a challenging issue of substantial practical significance. Here, by developing a low-pass filter (LPF) to track the output signals of the local system in the coupling, we show that it can revoke both AD and OD, and even the AD to OD transition, thereby giving rise to oscillations in coupled nonlinear oscillators under diverse death scenarios. The effectiveness of the local LPF is proven to be valid in an arbitrary network of coupled oscillators with distributed propagation delays. The constructive role of the local LPF in revoking deaths provides a potential dynamic mechanism of sustaining a reliable rhythmicity in real-world systems.

Spatiotemporal dynamics of the Kuramoto-Sakaguchi model with time-dependent connectivity

We study the dynamics of the paradigmatic Kuramoto-Sakaguchi model of identical coupled phase oscillators with various kinds of time-dependent connectivity using Eulerian discretization. We explore the parameter spaces for various types of collective states using the phase plots of the two statistical quantities, namely, the strength of incoherence and the discontinuity measure. In the quasistatic limit of the changing of coupling range, we observe how the system relaxes from one state to another and identify a few interesting collective dynamical states along the way. Under a sinusoidal change of the coupling range, the global order parameter characterizing the degree of synchronization in the system is shown to undergo a hysteresis with the coupling range. We also study the low-dimensional spatiotemporal dynamics of the local order parameter in the continuum limit using the recently developed Ott-Antonsen ansatz and justify some of our numerical results. In particular, we identify an intrinsic time scale of the Kuramoto system and show that the simulations exhibit two distinct kinds of qualitative behavior in two cases when the time scale associated with the switching of the coupling radius is very large compared to the intrinsic time scale and when it is comparable to the intrinsic time scale.

Sustainability of Transient Kinetic Regimes and Origins of Death

It is generally recognized that a distinguishing feature of life is its peculiar capability to avoid equilibration. The origin of this capability and its evolution along the timeline of abiogenesis is not yet understood. We propose to study an analog of this phenomenon that could emerge in non-biological systems. To this end, we introduce the concept of sustainability of transient kinetic regimes. This concept is illustrated via investigation of cooperative effects in an extended system of compartmentalized chemical oscillators under batch and semi-batch conditions. The computational study of a model system shows robust enhancement of lifetimes of the decaying oscillations which translates into the evolution of the survival function of the transient non-equilibrium regime. This model does not rely on any form of replication. Rather, it explores the role of a structured effective environment as a contributor to the system-bath interactions that define non-equilibrium regimes. We implicate the noise produced by the effective environment of a compartmentalized oscillator as the cause of the lifetime extension.

Modeling of the ComRS Signaling Pathway Reveals the Limiting Factors Controlling Competence in Streptococcus thermophilus

In streptococci, entry in competence is dictated by ComX abundance. In Streptococcus thermophilus, production of ComX is transient and tightly regulated during growth: it is positively regulated by the cell-cell communication system ComRS during the activation phase and negatively regulated during the shut-off phase by unidentified late competence gene(s). Interestingly, most S. thermophilus strains are not or weakly transformable in permissive growth conditions (i.e., chemically defined medium, CDM), suggesting that some players of the ComRS regulatory pathway are limiting. Here, we combined mathematical modeling and experimental approaches to identify the components of the ComRS system which are critical for both dynamics and amplitude of ComX production in S. thermophilus. We built a deterministic, population-scaled model of the time-course regulation of specific ComX production in CDM growth conditions. Strains LMD-9 and LMG18311 were respectively selected as representative of highly and weakly transformable strains. Results from in silico simulations and in vivo luciferase activities show that ComR concentration is the main limiting factor for the level of comX expression and controls the kinetics of spontaneous competence induction in strain LMD-9. In addition, the model predicts that the poor transformability of strain LMG18311 results from a 10-fold lower comR expression level compared to strain LMD-9. In agreement, comR overexpression in both strains was shown to induce higher competence levels with deregulated kinetics patterns during growth. In conclusion, we propose that the level of ComR production is one important factor that could explain competence heterogeneity among S. thermophilus strains.

Oscillation suppression in indirectly coupled limit cycle oscillators

We study the phenomena of oscillation quenching in a system of limit cycle oscillators which are coupled indirectly via a dynamic environment. The dynamics of the environment is assumed to decay exponentially with some decay parameter. We show that for appropriate coupling strength, the decay parameter of the environment plays a crucial role in the emergent dynamics such as amplitude death (AD) and oscillation death (OD). The critical curves for the regions of oscillation quenching as a function of coupling strength and decay parameter of the environment are obtained analytically using linear stability analysis and are found to be consistent with the numerics.

Modeling oscillations and spiral waves in Dictyostelium populations

Unicellular organisms exhibit elaborate collective behaviors in response to environmental cues. These behaviors are controlled by complex biochemical networks within individual cells and coordinated through cell-to-cell communication. Describing these behaviors requires new mathematical models that can bridge scales-from biochemical networks within individual cells to spatially structured cellular populations. Here we present a family of "multiscale" models for the emergence of spiral waves in the social amoeba Dictyostelium discoideum. Our models exploit new experimental advances that allow for the direct measurement and manipulation of the small signaling molecule cyclic adenosine monophosphate (cAMP) used by Dictyostelium cells to coordinate behavior in cellular populations. Inspired by recent experiments, we model the Dictyostelium signaling network as an excitable system coupled to various preprocessing modules. We use this family of models to study spatially unstructured populations of "fixed" cells by constructing phase diagrams that relate the properties of population-level oscillations to parameters in the underlying biochemical network. We then briefly discuss an extension of our model that includes spatial structure and show how this naturally gives rise to spiral waves. Our models exhibit a wide range of novel phenomena. including a density-dependent frequency change, bistability, and dynamic death due to slow cAMP dynamics. Our modeling approach provides a powerful tool for bridging scales in modeling of Dictyostelium populations.

From intracellular signaling to population oscillations: bridging size- and time-scales in collective behavior

Collective behavior in cellular populations is coordinated by biochemical signaling networks within individual cells. Connecting the dynamics of these intracellular networks to the population phenomena they control poses a considerable challenge because of network complexity and our limited knowledge of kinetic parameters. However, from physical systems, we know that behavioral changes in the individual constituents of a collectively behaving system occur in a limited number of well-defined classes, and these can be described using simple models. Here, we apply such an approach to the emergence of collective oscillations in cellular populations of the social amoeba Dictyostelium discoideum. Through direct tests of our model with quantitative in vivo measurements of single-cell and population signaling dynamics, we show how a simple model can effectively describe a complex molecular signaling network at multiple size and temporal scales. The model predicts novel noise-driven single-cell and population-level signaling phenomena that we then experimentally observe. Our results suggest that like physical systems, collective behavior in biology may be universal and described using simple mathematical models.

Synchronization of degrade-and-fire oscillations via a common activator

The development of synthetic gene oscillators has not only demonstrated our ability to forward engineer reliable circuits in living cells, but it has also proven to be an excellent testing ground for the statistical behavior of coupled noisy oscillators. Previous experimental studies demonstrated that a shared positive feedback can reliably synchronize such oscillators, though the theoretical mechanism was not studied in detail. In the present work, we examine an experimentally motivated stochastic model for coupled degrade-and-fire gene oscillators, where a core delayed negative feedback establishes oscillations within each cell, and a shared delayed positive feedback couples all cells. We use analytic and numerical techniques to investigate conditions for one cluster and multicluster synchrony. A nonzero delay in the shared positive feedback, as expected for the experimental systems, is found to be important for synchrony to occur.

Secreting and sensing the same molecule allows cells to achieve versatile social behaviors

Cells that secrete and sense the same signaling molecule are ubiquitous. To uncover the functional capabilities of the core "secrete-and-sense" circuit motif shared by these cells, we engineered yeast to secrete and sense the mating pheromone. Perturbing each circuit element revealed parameters that control the degree to which the cell communicated with itself versus with its neighbors. This tunable interplay of self-communication and neighbor communication enables cells to span a diverse repertoire of cellular behaviors. These include a cell being asocial by responding only to itself and social through quorum sensing, and an isogenic population of cells splitting into social and asocial subpopulations. A mathematical model explained these behaviors. The versatility of the secrete-and-sense circuit motif may explain its recurrence across species.

Kuramoto model with coupling through an external medium

Synchronization of coupled oscillators is often described using the Kuramoto model. Here, we study a generalization of the Kuramoto model where oscillators communicate with each other through an external medium. This generalized model exhibits interesting new phenomena such as bistability between synchronization and incoherence and a qualitatively new form of synchronization where the external medium exhibits small-amplitude oscillations. We conclude by discussing the relationship of the model to other variations of the Kuramoto model including the Kuramoto model with a bimodal frequency distribution and the Millennium bridge problem.