Global convergence of quorum-sensing networks

Kuramoto model with stochastic resetting and coupling through an external medium

Most studies of collective phenomena in oscillator networks focus on directly coupled systems, as exemplified by the classical Kuramoto model. However, there are a growing number of examples in which oscillators interact indirectly via a common external medium, including bacterial quorum sensing (QS) networks, pedestrians walking on a bridge, and centrally coupled lasers. In this paper, we analyze the effects of stochastic phase resetting on a Kuramoto model with indirect coupling. All the phases are simultaneously reset to their initial values at a random sequence of times generated from a Poisson process. On the other hand, the external environmental state is not reset. We first derive a continuity equation for the population density in the presence of resetting and show how the resulting density equation is itself subject to stochastic resetting. We then use an Ott-Antonsen (OA) Ansatz to reduce the infinite-dimensional system to a four-dimensional piecewise deterministic system with subsystem resetting. The latter is used to explore how synchronization depends on a cell density parameter. (In bacterial QS, this represents the ratio of the population cell volume and the extracellular volume.) At high densities, we recover the OA dynamics of the classical Kuramoto model with global resetting. On the other hand, at low densities, we show how subsystem resetting has a major effect on collective synchronization, ranging from noise-induced transitions to slow/fast dynamics.

Quorum-sensing induced transitions between bistable steady-states for a cell-bulk ODE-PDE model with lux intracellular kinetics

Intercellular signaling and communication are used by bacteria to regulate a variety of behaviors. In a type of cell-cell communication known as quorum sensing (QS), which is mediated by a diffusible signaling molecule called an autoinducer, bacteria can undergo sudden changes in their behavior at a colony-wide level when the density of cells exceeds a critical threshold. In mathematical models of QS behavior, these changes can include the switch-like emergence of intracellular oscillations through a Hopf bifurcation, or sudden transitions between bistable steady-states as a result of a saddle-node bifurcation of equilibria. As an example of this latter type of QS transition, we formulate and analyze a cell-bulk ODE-PDE model in a 2-D spatial domain that incorporates the prototypical LuxI/LuxR QS system for a collection of stationary bacterial cells, as modeled by small circular disks of a common radius with a cell membrane that is permeable only to the autoinducer. By using the method of matched asymptotic expansions, it is shown that the steady-state solutions for the cell-bulk model exhibit a saddle-node bifurcation structure. The linear stability of these branches of equilibria are determined from the analysis of a nonlinear matrix eigenvalue problem, called the globally coupled eigenvalue problem. The key role on QS behavior of a bulk degradation of the autoinducer field, which arises from either a Robin boundary condition on the domain boundary or from a constant bulk decay, is highlighted. With bulk degradation, it is shown analytically that the effect of coupling identical bacterial cells to the bulk autoinducer diffusion field is to create an effective bifurcation parameter that depends on the population of the colony, the bulk diffusivity, the membrane permeabilities, and the cell radius. QS transitions occur when this effective parameter passes through a saddle-node bifurcation point of the Lux ODE kinetics for an isolated cell. In the limit of a large but finite bulk diffusivity, it is shown that the cell-bulk system is well-approximated by a simpler ODE-DAE system. This reduced system, which is used to study the effect of cell location on QS behavior, is easily implemented for a large number of cells. Predictions from the asymptotic theory for QS transitions between bistable states are favorably compared with full numerical solutions of the cell-bulk ODE-PDE system.

Interspecies evolutionary dynamics mediated by public goods in bacterial quorum sensing

Bacterial quorum sensing is the communication that takes place between bacteria as they secrete certain molecules into the intercellular medium that later get absorbed by the secreting cells themselves and by others. Depending on cell density, this uptake has the potential to alter gene expression and thereby affect global properties of the community. We consider the case of multiple bacterial species coexisting, referring to each one of them as a genotype and adopting the usual denomination of the molecules they collectively secrete as public goods. A crucial problem in this setting is characterizing the coevolution of genotypes as some of them secrete public goods (and pay the associated metabolic costs) while others do not but may nevertheless benefit from the available public goods. We introduce a network model to describe genotype interaction and evolution when genotype fitness depends on the production and uptake of public goods. The model comprises a random graph to summarize the possible evolutionary pathways the genotypes may take as they interact genetically with one another, and a system of coupled differential equations to characterize the behavior of genotype abundance in time. We study some simple variations of the model analytically and more complex variations computationally. Our results point to a simple trade-off affecting the long-term survival of those genotypes that do produce public goods. This trade-off involves, on the producer side, the impact of producing and that of absorbing the public good. On the nonproducer side, it involves the impact of absorbing the public good as well, now compounded by the molecular compatibility between the producer and the nonproducer. Depending on how these factors turn out, producers may or may not survive.

Beyond convexity-Contraction and global convergence of gradient descent

This paper considers the analysis of continuous time gradient-based optimization algorithms through the lens of nonlinear contraction theory. It demonstrates that in the case of a time-invariant objective, most elementary results on gradient descent based on convexity can be replaced by much more general results based on contraction. In particular, gradient descent converges to a unique equilibrium if its dynamics are contracting in any metric, with convexity of the cost corresponding to the special case of contraction in the identity metric. More broadly, contraction analysis provides new insights for the case of geodesically-convex optimization, wherein non-convex problems in Euclidean space can be transformed to convex ones posed over a Riemannian manifold. In this case, natural gradient descent converges to a unique equilibrium if it is contracting in any metric, with geodesic convexity of the cost corresponding to contraction in the natural metric. New results using semi-contraction provide additional insights into the topology of the set of optimizers in the case when multiple optima exist. Furthermore, they show how semi-contraction may be combined with specific additional information to reach broad conclusions about a dynamical system. The contraction perspective also easily extends to time-varying optimization settings and allows one to recursively build large optimization structures out of simpler elements. Extensions to natural primal-dual optimization and game-theoretic contexts further illustrate the potential reach of these new perspectives.

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.

A Continuous-Time Analysis of Distributed Stochastic Gradient

We analyze the effect of synchronization on distributed stochastic gradient algorithms. By exploiting an analogy with dynamical models of biological quorum sensing, where synchronization between agents is induced through communication with a common signal, we quantify how synchronization can significantly reduce the magnitude of the noise felt by the individual distributed agents and their spatial mean. This noise reduction is in turn associated with a reduction in the smoothing of the loss function imposed by the stochastic gradient approximation. Through simulations on model nonconvex objectives, we demonstrate that coupling can stabilize higher noise levels and improve convergence. We provide a convergence analysis for strongly convex functions by deriving a bound on the expected deviation of the spatial mean of the agents from the global minimizer for an algorithm based on quorum sensing, the same algorithm with momentum, and the elastic averaging SGD (EASGD) algorithm. We discuss extensions to new algorithms that allow each agent to broadcast its current measure of success and shape the collective computation accordingly. We supplement our theoretical analysis with numerical experiments on convolutional neural networks trained on the CIFAR-10 data set, where we note a surprising regularizing property of EASGD even when applied to the non-distributed case. This observation suggests alternative second-order in time algorithms for nondistributed optimization that are competitive with momentum methods.

On distributed coordination in networks of cyber-physical systems

This paper is concerned with the study of the global emerging behavior in complex networks where each node can be modeled as a cyber-physical system. We recast the problem of characterizing the behavior of such systems as a stability problem and give two technical results to assess this property. We then illustrate the effectiveness of our approach by considering two testbed examples arising in applications where networks, arising from Internet of Things applications, need to be designed so as to fulfill a given task.

Mixed mode oscillations and phase locking in coupled FitzHugh-Nagumo model neurons

We study the dynamics of a low-dimensional system of coupled model neurons as a step towards understanding the vastly complex network of neurons in the brain. We analyze the bifurcation structure of a system of two model neurons with unidirectional coupling as a function of two physiologically relevant parameters: the external current input only to the first neuron and the strength of the coupling from the first to the second neuron. Leveraging a timescale separation, we prove necessary conditions for multiple timescale phenomena observed in the coupled system, including canard solutions and mixed mode oscillations. For a larger network of model neurons, we present a sufficient condition for phase locking when external inputs are heterogeneous. Finally, we generalize our results to directed trees of model neurons with heterogeneous inputs.

Modeling the Role of Feedback in the Adaptive Response of Bacterial Quorum Sensing

Bacterial quorum sensing (QS) is a form of intercellular communication that relies on the production and detection of diffusive signaling molecules called autoinducers. Such a mechanism allows the bacteria to track their cell density in order to regulate group behavior, such as biofilm formation and bioluminescence. In a number of bacterial QS systems, including V. harveyi, multiple signaling pathways are integrated into a single phosphorylation-dephosphorylation cycle. In this paper, we propose a weight control mechanism, in which QS uses feedback loops to 'decode' the integrated signals by actively changing the sensitivity in different pathways. We first use a slow/fast analysis to reduce a single-cell model to a planar dynamical system involving the concentrations of phosphorylated signaling protein LuxU and a small non-coding RNA. In addition to identifying the weight control mechanism, we show that adding a feedback loop can lead to a bistable QS response in certain parameter regimes. We then combine the slow/fast analysis with a contraction mapping theorem in order to reduce a population model to an effective single-cell model, and show how the weight control mechanism allows bacteria to have a finer discrimination of their social and physical environment.

Information transmission in microbial and fungal communication: from classical to quantum

Microbes have their own communication systems. Secretion and reception of chemical signaling molecules and ion-channels mediated electrical signaling mechanism are yet observed two special ways of information transmission in microbial community. In this article, we address the aspects of various crucial machineries which set the backbone of microbial cell-to-cell communication process such as quorum sensing mechanism (bacterial and fungal), quorum sensing regulated biofilm formation, gene expression, virulence, swarming, quorum quenching, role of noise in quorum sensing, mathematical models (therapy model, evolutionary model, molecular mechanism model and many more), synthetic bacterial communication, bacterial ion-channels, bacterial nanowires and electrical communication. In particular, we highlight bacterial collective behavior with classical and quantum mechanical approaches (including quantum information). Moreover, we shed a new light to introduce the concept of quantum synthetic biology and possible cellular quantum Turing test.

Dynamical modes of two almost identical chemical oscillators connected via both pulsatile and diffusive coupling

The dynamics of two almost identical chemical oscillators with mixed diffusive and pulsatile coupling are systematically studied.

Engineered Control of Genetic Variability Reveals Interplay among Quorum Sensing, Feedback Regulation, and Biochemical Noise

Stochastic fluctuations in gene expression trigger both beneficial and harmful consequences for cell behavior. Therefore, achieving a desired mean protein expression level while minimizing noise is of interest in many applications, including robust protein production systems in industrial biotechnology. Here, we consider a synthetic gene circuit combining intracellular negative feedback and cell-to-cell communication based on quorum sensing. Accounting for both intrinsic and extrinsic noise, stochastic simulations allow us to analyze the capability of the circuit to reduce noise strength as a function of its parameters. We obtain mean expression levels and noise strengths for all species under different scenarios, showing good agreement with system-wide available experimental data of protein abundance and noise in Escherichia coli. Our in silico experiments, validated by preliminary in vivo results, reveal significant noise attenuation in gene expression through the interplay between quorum sensing and negative feedback and highlight the differential role that they play in regard to intrinsic and extrinsic noise.

Population Model of Quorum Sensing with Multiple Parallel Pathways

Quorum sensing (QS) is a bacterial communication mechanism that uses signal-receptor binding to regulate gene expression based on cell density, resulting in group behaviors such as biofilm formation, bioluminescence and stress response. In certain bacterial species such as Vibrio harveyi, several parallel QS signaling pathways drive a single phosphorylation-dephosphorylation cycle, which in turn regulates QS target genes. In this paper, we investigate the possible role of parallel signaling pathways by developing a mathematical model of QS in V. harveyi at both the single-cell and population levels. First we explore how signal integration may be achieved at the single-cell level, and how different model parameters influence the process. We then consider two examples of signal integration at the population level: a one-population model responding to two environmental cues (cell density and mass transfer), and a two-population model with distinct cell densities. In each case, we use contraction analysis to reduce the population model to an effective single-cell model.

How to desynchronize quorum-sensing networks

In this paper we investigate how so-called quorum-sensing networks can be desynchronized. Such networks, which arise in many important application fields, such as systems biology, are characterized by the fact that direct communication between network nodes is superimposed to communication with a shared, environmental variable. In particular, we provide a new sufficient condition ensuring that the trajectories of these quorum-sensing networks diverge from their synchronous evolution. Then, we apply our result to study two applications.

Ultrasensitivity and noise amplification in a model of V. harveyi quorum sensing

We analyze ultrasensitivity in a model of Vibrio harveyi quorum sensing. We consider a feedforward model consisting of two biochemical networks per cell. The first represents the interchange of a signaling molecule (autoinducer) between the cell cytoplasm and an extracellular domain and the binding of intracellular autoinducer to cognate receptors. The unbound and bound receptors within each cell act as kinases and phosphotases, respectively, which then drive a second biochemical network consisting of a phosphorylation-dephosphorylation cycle. We ignore subsequent signaling pathways associated with gene regulation and the possible modification in the production rate of an autoinducer (positive feedback). We show how the resulting quorum sensing system exhibits ultrasensitivity with respect to changes in cell density. We also demonstrate how quorum sensing can protect against the noise amplification of fast environmental fluctuations in comparison to a single isolated cell.

The sympathy of two pendulum clocks: beyond Huygens' observations

This paper introduces a modern version of the classical Huygens’ experiment on synchronization of pendulum clocks. The version presented here consists of two monumental pendulum clocks—ad hoc designed and fabricated—which are coupled through a wooden structure. It is demonstrated that the coupled clocks exhibit ‘sympathetic’ motion, i.e. the pendula of the clocks oscillate in consonance and in the same direction. Interestingly, when the clocks are synchronized, the common oscillation frequency decreases, i.e. the clocks become slow and inaccurate. In order to rigorously explain these findings, a mathematical model for the coupled clocks is obtained by using well-established physical and mechanical laws and likewise, a theoretical analysis is conducted. Ultimately, the sympathy of two monumental pendulum clocks, interacting via a flexible coupling structure, is experimentally, numerically, and analytically demonstrated.

Collective sensing and collective responses in quorum-sensing bacteria

Bacteria often face fluctuating environments, and in response many species have evolved complex decision-making mechanisms to match their behaviour to the prevailing conditions. Some environmental cues provide direct and reliable information (such as nutrient concentrations) and can be responded to individually. Other environmental parameters are harder to infer and require a collective mechanism of sensing. In addition, some environmental challenges are best faced by a group of cells rather than an individual. In this review, we discuss how bacteria sense and overcome environmental challenges as a group using collective mechanisms of sensing, known as 'quorum sensing' (QS). QS is characterized by the release and detection of small molecules, potentially allowing individuals to infer environmental parameters such as density and mass transfer. While a great deal of the molecular mechanisms of QS have been described, there is still controversy over its functional role. We discuss what QS senses and how, what it controls and why, and how social dilemmas shape its evolution. Finally, there is a growing focus on the use of QS inhibitors as antibacterial chemotherapy. We discuss the claim that such a strategy could overcome the evolution of resistance. By linking existing theoretical approaches to data, we hope this review will spur greater collaboration between experimental and theoretical researchers.

Co-adaptation and the emergence of structure

Co-adaptation (or co-evolution), the parallel feedback process by which agents continuously adapt to the changes induced by the adaptive actions of other agents, is a ubiquitous feature of complex adaptive systems, from eco-systems to economies. We wish to understand which general features of complex systems necessarily follow from the (meta)-dynamics of co-adaptation, and which features depend on the details of particular systems. To begin this project, we present a model of co-adaptation ("The Stigmergy Game") which is designed to be as a priori featureless as possible, in order to help isolate and understand the naked consequences of co-adaptation. In the model, heterogeneous, co-adapting agents, observe, interact with and change the state of an environment. Agents do not, ab initio, directly interact with each other. Agents adapt by choosing among a set of random "strategies," particular to each agent. Each strategy is a complete specification of an agent's actions and payoffs. A priori, all environmental states are equally likely and all strategies have payoffs that sum to zero, so without co-adaptation agents would on average have zero "wealth". Nevertheless, the dynamics of co-adaptation generates a structured environment in which only a subset of environmental states appear with high probability (niches) and in which agents accrue positive wealth. Furthermore, although there are no direct agent-agent interactions, there are induced non-trivial inter-agent interactions mediated by the environment. As a function of the population size and the number of possible environmental states, the system can be in one of three dynamical regions. Implications for a basic understanding of complex adaptive systems are discussed.

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.

Dynamical quorum-sensing in oscillators coupled through an external medium

Many biological and physical systems exhibit population-density dependent transitions to synchronized oscillations in a process often termed "dynamical quorum sensing". Synchronization frequently arises through chemical communication via signaling molecules distributed through an external medium. We study a simple theoretical model for dynamical quorum sensing: a heterogenous population of limit-cycle oscillators diffusively coupled through a common medium. We show that this model exhibits a rich phase diagram with four qualitatively distinct physical mechanisms that can lead to a loss of coherent population-level oscillations, including a novel mechanism arising from effective time-delays introduced by the external medium. We derive a single pair of analytic equations that allow us to calculate phase boundaries as a function of population density and show that the model reproduces many of the qualitative features of recent experiments on BZ catalytic particles as well as synthetically engineered bacteria.

Synchronization and quorum sensing in an ensemble of indirectly coupled chaotic oscillators

The fact that the elements in some realistic systems are influenced by each other indirectly through a common environment has stimulated a new surge of studies on the collective behavior of coupled oscillators. Most of the previous studies, however, consider only the case of coupled periodic oscillators, and it remains unknown whether and to what extent the findings can be applied to the case of coupled chaotic oscillators. Here, using the population density and coupling strength as the tuning parameters, we explore the synchronization and quorum sensing behaviors in an ensemble of chaotic oscillators coupled through a common medium, in which some interesting phenomena are observed, including the appearance of the phase synchronization in the process of progressive synchronization, the various periodic oscillations close to the quorum sensing transition, and the crossover of the critical population density at the transition. These phenomena, which have not been reported for indirectly coupled periodic oscillators, reveal a corner of the rich dynamics inherent in indirectly coupled chaotic oscillators, and are believed to have important implications to the performance and functionality of some realistic systems.

Symmetries, stability, and control in nonlinear systems and networks

This paper discusses the interplay of symmetries and stability in the analysis and control of nonlinear dynamical systems and networks. Specifically, it combines standard results on symmetries and equivariance with recent convergence analysis tools based on nonlinear contraction theory and virtual dynamical systems. This synergy between structural properties (symmetries) and convergence properties (contraction) is illustrated in the contexts of network motifs arising, for example, in genetic networks, from invariance to environmental symmetries, and from imposing different patterns of synchrony in a network.