Topological defects in multi-layered swarming bacteria.

Topological defects, which are singular points in a director field, play a major role in shaping active systems. Here, we experimentally study topological defects and the flow patterns around them, that are formed during the highly rapid dynamics of swarming bacteria. The results are compared to the predictions of two-dimensional active nematics. We show that, even though some of the assumptions underlying the theory do not hold, the swarm dynamics is in agreement with two-dimensional nematic theory. In particular, we look into the multi-layered structure of the swarm, which is an important feature of real, natural colonies, and find a strong coupling between layers. Our results suggest that the defect-charge density is hyperuniform, i.e., that long range density-fluctuations are suppressed.

Universal Relation between Entropy and Kinetics.

Relating thermodynamic and kinetic properties is a conceptual challenge with many practical benefits. Here, based on first principles, we derive a rigorous inequality relating the entropy and the dynamic propagator of particle configurations. It is universal and applicable to steady states arbitrarily far from thermodynamic equilibrium. Applying the general relation to diffusive dynamics yields a relation between the entropy and the (normal or anomalous) diffusion coefficient. The relation can be used to obtain useful bounds for the late-time diffusion coefficient from the calculated steady-state entropy or, conversely, to estimate the entropy based on measured diffusion coefficients. We demonstrate the validity and usefulness of the relation through several examples and discuss its broad range of applications, in particular, for systems far from equilibrium.

Detecting and characterizing phase transitions in active matter using entropy.

A major challenge in the study of active matter lies in quantitative characterization of phases and transitions between them. We show how the entropy of a collection of active objects can be used to classify regimes and spatial patterns in their collective behavior. Specifically, we estimate the contributions to the total entropy from correlations between the degrees of freedom of position and orientation. This analysis pin-points the flocking transition in the Vicsek model while clarifying the physical mechanism behind the transition. When applied to experiments on swarming Bacillus subtilis with different cell aspect ratios and overall bacterial area fractions, the entropy analysis reveals a rich phase diagram with transitions between qualitatively different swarm statistics. We discuss physical and biological implications of these findings.

Mutual Information between Order Book Layers.

The order book is a list of all current buy or sell orders for a given financial security. The rise of electronic stock exchanges introduced a debate about the relevance of the information it encapsulates of the activity of traders. Here, we approach this topic from a theoretical perspective, estimating the amount of mutual information between order book layers, i.e., different buy/sell layers, which are aggregated by buy/sell orders. We show that (i) layers are not independent (in the sense that the mutual information is statistically larger than zero), (ii) the mutual information between layers is small (compared to the joint entropy), and (iii) the mutual information between layers increases when comparing the uppermost layers to the deepest layers analyzed (i.e., further away from the market price). Our findings, and our method for estimating mutual information, are relevant to developing trading strategies that attempt to utilize the information content of the limit order book.

Turbulence-induced clustering in compressible active fluids.

We study a novel phase of active polar fluids, which is characterized by the continuous creation and destruction of dense clusters due to self-sustained turbulence. This state arises due to the interplay between self-advection of the aligned swimmers and their defect topology. The typical cluster size is determined by the characteristic vortex size. Our results are obtained by investigating a continuum model of compressible polar active fluids, which incorporates typical experimental observations in bacterial suspensions, in particular a non-monotone dependence of speed on density.

Estimating Differential Entropy using Recursive Copula Splitting.

A method for estimating the Shannon differential entropy of multidimensional random variables using independent samples is described. The method is based on decomposing the distribution into a product of marginal distributions and joint dependency, also known as the copula. The entropy of marginals is estimated using one-dimensional methods. The entropy of the copula, which always has a compact support, is estimated recursively by splitting the data along statistically dependent dimensions. The method can be applied both for distributions with compact and non-compact supports, which is imperative when the support is not known or of a mixed type (in different dimensions). At high dimensions (larger than 20), numerical examples demonstrate that our method is not only more accurate, but also significantly more efficient than existing approaches.

Chaotic Model for Lévy Walks in Swarming Bacteria.

We describe a new mechanism for Lévy walks, explaining the recently observed superdiffusion of swarming bacteria. The model hinges on several key physical properties of bacteria, such as an elongated cell shape, self-propulsion, and a collectively generated regular vortexlike flow. In particular, chaos and Lévy walking are a consequence of group dynamics. The model explains how cells can fine-tune the geometric properties of their trajectories. Experiments confirm the spectrum of these patterns in fluorescently labeled swarming Bacillus subtilis.

Effect of Cell Aspect Ratio on Swarming Bacteria.

Swarming bacteria collectively migrate on surfaces using flagella, forming dynamic whirls and jets that consist of millions of individuals. Because some swarming bacteria elongate prior to actual motion, cell aspect ratio may play a significant role in the collective dynamics. Extensive research on self-propelled rodlike particles confirms that elongation promotes alignment, strongly affecting the dynamics. Here, we study experimentally the collective dynamics of variants of swarming Bacillus subtilis that differ in length. We show that the swarming statistics depends on the aspect ratio in a critical, fundamental fashion not predicted by theory. The fastest motion was obtained for the wild-type and variants that are similar in length. However, shorter and longer cells exhibit anomalous, non-Gaussian statistics and nonexponential decay of the autocorrelation function, indicating lower collective motility. These results suggest that the robust mechanisms to maintain aspect ratios may be important for efficient swarming motility. Wild-type cells are optimal in this sense.

Kinetic order-disorder transitions in a pause-and-go swarming model with memory.

A two dimensional model of self-propelled particles combining both a pause-and-go movement pattern and memory is studied in simulations. It is shown, that in contrast to previously studied agent based models in two-dimensions, order and disorder are metastable states that can co-exist at some parameter range. In particular, this implies that the formation and decay of global order in swarms may be kinetic rather than a phase transition. Our results explain metastability recently observed in swarming locust and fish.

Anomalous Fluctuations in the Orientation and Velocity of Swarming Bacteria.

Simultaneous acquisition of phase-contrast light microscopy and fluorescently labeled bacteria, moving within a dense swarm, reveals the intricate interactions between cells and the collective flow around them. By comparing wild-type and immotile cells embedded in a dense wild-type swarm, the effect of the active thrust generated by the flagella can be singled out. It is shown that while the distribution of angles among cell velocity, cell orientation, and the local flow around it is Gaussian-like for immotile bacteria, wild-type cells exhibit anomalous non-Gaussian deviations and are able to move in trajectories perpendicular to the collective flow. Thus, cells can maneuver or switch between local streams and jets. A minimal model describing bacteria as hydrodynamic force dipoles shows that steric effects, hydrodynamics interactions, and local alignments all have to be taken into account to explain the observed dynamics. These findings shed light on the physical mechanisms underlying bacterial swarming and the balance between individual and collective dynamics.

Locust Collective Motion and Its Modeling.

Over the past decade, technological advances in experimental and animal tracking techniques have motivated a renewed theoretical interest in animal collective motion and, in particular, locust swarming. This review offers a comprehensive biological background followed by comparative analysis of recent models of locust collective motion, in particular locust marching, their settings, and underlying assumptions. We describe a wide range of recent modeling and simulation approaches, from discrete agent-based models of self-propelled particles to continuous models of integro-differential equations, aimed at describing and analyzing the fascinating phenomenon of locust collective motion. These modeling efforts have a dual role: The first views locusts as a quintessential example of animal collective motion. As such, they aim at abstraction and coarse-graining, often utilizing the tools of statistical physics. The second, which originates from a more biological perspective, views locust swarming as a scientific problem of its own exceptional merit. The main goal should, thus, be the analysis and prediction of natural swarm dynamics. We discuss the properties of swarm dynamics using the tools of statistical physics, as well as the implications for laboratory experiments and natural swarms. Finally, we stress the importance of a combined-interdisciplinary, biological-theoretical effort in successfully confronting the challenges that locusts pose at both the theoretical and practical levels.

On effective temperature in network models of collective behavior.

Collective behavior of self-propelled units is studied analytically within the Vectorial Network Model (VNM), a mean-field approximation of the well-known Vicsek model. We propose a dynamical systems framework to study the stochastic dynamics of the VNM in the presence of general additive noise. We establish that a single parameter, which is a linear function of the circular mean of the noise, controls the macroscopic phase of the system-ordered or disordered. By establishing a fluctuation-dissipation relation, we posit that this parameter can be regarded as an effective temperature of collective behavior. The exact critical temperature is obtained analytically for systems with small connectivity, equivalent to low-density ensembles of self-propelled units. Numerical simulations are conducted to demonstrate the applicability of this new notion of effective temperature to the Vicsek model. The identification of an effective temperature of collective behavior is an important step toward understanding order-disorder phase transitions, informing consistent coarse-graining techniques and explaining the physics underlying the emergence of collective phenomena.

Antibiotic-induced anomalous statistics of collective bacterial swarming.

Under sublethal antibiotics concentrations, the statistics of collectively swarming Bacillus subtilis transitions from normal to anomalous, with a heavy-tailed speed distribution and a two-step temporal correlation of velocities. The transition is due to changes in the properties of the bacterial motion and the formation of a motility-defective subpopulation that self-segregates into regions. As a result, both the colonial expansion and the growth rate are not affected by antibiotics. This phenomenon suggests a new strategy bacteria employ to fight antibiotic stress.

Lethal protein produced in response to competition between sibling bacterial colonies.

Sibling Paenibacillus dendritiformis bacterial colonies grown on low-nutrient agar medium mutually inhibit growth through secretion of a lethal factor. Analysis of secretions reveals the presence of subtilisin (a protease) and a 12 kDa protein, termed sibling lethal factor (Slf). Purified subtilisin promotes the growth and expansion of P. dendritiformis colonies, whereas Slf is lethal and lyses P. dendritiformis cells in culture. Slf is encoded by a gene belonging to a large family of bacterial genes of unknown function, and the gene is predicted to encode a protein of approximately 20 kDa, termed dendritiformis sibling bacteriocin. The 20 kDa recombinant protein was produced and found to be inactive, but exposure to subtilisin resulted in cleavage to the active, 12 kDa form. The experimental results, combined with mathematical modeling, show that subtilisin serves to regulate growth of the colony. Below a threshold concentration, subtilisin promotes colony growth and expansion. However, once it exceeds a threshold, as occurs at the interface between competing colonies, Slf is then secreted into the medium to rapidly reduce cell density by lysis of the bacterial cells. The presence of genes encoding homologs of dendritiformis sibling bacteriocin in other bacterial species suggests that this mechanism for self-regulation of colony growth might not be limited to P. dendritiformis.

Schooling of light reflecting fish.

One of the hallmarks of the collective movement of large schools of pelagic fish are waves of shimmering flashes that propagate across the school, usually following an attack by a predator. Such flashes arise when sunlight is reflected off the specular (mirror-like) skin that characterizes many pelagic fishes, where it is otherwise thought to offer a means for camouflage in open waters. While it has been suggested that these 'shimmering waves' are a visual manifestation of the synchronized escape response of the fish, the phenomenon has been regarded only as an artifact of esthetic curiosity. In this study we apply agent-based simulations and deep learning techniques to show that, in fact, shimmering waves contain information on the behavioral dynamics of the school. Our analyses are based on a model that combines basic rules of collective motion and the propagation of light beams in the ocean, as they hit and reflect off the moving fish. We use the resulting reflection patterns to infer the essential dynamics and inter-individual interactions which are necessary to generate shimmering waves. Moreover, we show that light flashes observed by the school members themselves may extend the range at which information can be communicated across the school. Assuming that fish pay heed to this information, for example by entering an apprehensive state of reduced response-time, our analysis suggests that it can speed up the propagation of information across the school. Further still, we use an artificial neural network to show that light flashes are, on their own, indicative of the state and dynamics of the school, and are sufficient to infer the direction of attack and the shape of the school with high accuracy.

Resolving entropy contributions in nonequilibrium transitions.

We derive a functional for the entropy contributed by any microscopic degrees of freedom as arising from their measurable pair correlations. Applicable both in and out of equilibrium, this functional yields the maximum entropy which a system can have given a certain correlation function. When applied to different correlations, the method allows us to identify the degrees of freedom governing a certain physical regime, thus capturing and characterizing dynamic transitions. The formalism applies also to systems whose translational invariance is broken by external forces and whose number of particles may vary. We apply it to experimental results for jammed bidisperse emulsions, capturing the crossover of this nonequilibrium system from crystalline to disordered hyperuniform structures as a function of mixture composition. We discover that the cross-correlations between the positions and sizes of droplets in the emulsion play the central role in the formation of the disordered hyperuniform states. We discuss implications of the approach for entropy estimation out of equilibrium and for characterizing transitions in disordered systems.

Pair correlation function based on Voronoi topology.

The pair correlation function (PCF) has proven an effective tool for analyzing many physical systems due to its simplicity and its applicability for simulated and experimental data. However, as an averaged quantity, the PCF can fail to capture subtle structural differences in particle arrangements, even when those differences can have a major impact on system properties. Here, we use Voronoi topology to introduce a discrete version of the PCF that highlights local interparticle topological configurations. The advantages of the Voronoi PCF are demonstrated in several examples including crystalline, hyperuniform, and active systems showing clustering and giant number fluctuations.

Smart swarms of bacteria-inspired agents with performance adaptable interactions.

Collective navigation and swarming have been studied in animal groups, such as fish schools, bird flocks, bacteria, and slime molds. Computer modeling has shown that collective behavior of simple agents can result from simple interactions between the agents, which include short range repulsion, intermediate range alignment, and long range attraction. Here we study collective navigation of bacteria-inspired smart agents in complex terrains, with adaptive interactions that depend on performance. More specifically, each agent adjusts its interactions with the other agents according to its local environment--by decreasing the peers' influence while navigating in a beneficial direction, and increasing it otherwise. We show that inclusion of such performance dependent adaptable interactions significantly improves the collective swarming performance, leading to highly efficient navigation, especially in complex terrains. Notably, to afford such adaptable interactions, each modeled agent requires only simple computational capabilities with short-term memory, which can easily be implemented in simple swarming robots.

Light flashes and the geometry of specular fish schools.

The risk of predation presents a difficult challenge in environments that offer no physical shelter, such as the open waters of the world's seas. In the absence of hiding places, many marine fishes turn to two main anti-predator strategies: aggregation and camouflage, which, mostly, have been studied separately. Here, we consider both aspects together and examine the visual imprint of fish schools of different sizes and geometries, given that camouflage is attained by specular (mirror-like) skin texture. To do so, we developed ray-tracing simulations that model the passage of sunbeams as they go through an optically realistic aquatic environment and reflect off the skins of the fish. We find that due to frequent high-intensity specular reflections (light flashes), the marginal increase in detectability with increasing school size is significantly higher than previously estimated under the assumption of diffusive reflection. However, we also find that by increasing density and alignment the fish can mitigate the detectability of individuals, albeit at the expense of the detectability of the school as a whole. Our findings provide a new perspective on documented responses to threat by schooling pelagic fishes and underscore the importance of the optical signature of animals in structuring their behaviour.

Physical characteristics of mixed-species swarming colonies.

In nature, bacterial collectives typically consist of multiple species, which are interacting both biochemically and physically. Nonetheless, past studies on the physical properties of swarming bacteria were focused on axenic (single-species) populations. In bacterial swarming, intricate interactions between the individuals lead to clusters, rapid jets, and vortices that depend on cell characteristics such as speed and length. In this work, we show the first results of rapidly swarming mixed-species populations of Bacillus subtilis and Serratia marcescens, two model swarm species that are known to swarm well in axenic situations. In mixed liquid cultures, both species have higher reproduction rates. We show that the mixed population swarms together well and that the fraction between the species determines all dynamical scales-from the microscopic (e.g., speed distribution), mesoscopic (vortex size), and macroscopic (colony structure and size). Understanding mixed-species swarms is essential for a comprehensive understanding of the bacterial swarming phenomenon and its biological and evolutionary implications.