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.

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.

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.

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.

Mixed-species bacterial swarms show an interplay of mixing and segregation across scales.

Bacterial swarms are a highly-researched example of natural active matter. In particular, the interplay between biological interactions and the physics underlying the swarming dynamics is of both biological and physical interest. In this paper, we study mixed swarms of Bacillus subtilis and Pseudomonas aeruginosa. We find intricate interactions between the species, showing both cooperation and segregation across different spatial and temporal scales. On one hand, even though axenic colonies grow on disparate time scale, an order of magnitude apart, the two-species swarm together, forming a single, combined colony. However, the rapidly moving populations are locally segregated, with different characteristic speeds and lengths (or cluster sizes) that depend on the ratio between the species. Comparison with controlled mutant strains suggest that both the physical and known biological differences in species characteristics may not be enough to explain the segregation between the species in the mixed swarm. We hypothesize that the heterogeneous spatial distribution is due to some mechanism that enables bacteria to recognize their own kind, whose precise origin we could not identify.

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.

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.

Biophysical aspects underlying the swarm to biofilm transition.

Bacteria organize in a variety of collective states, from swarming-rapid surface exploration, to biofilms-highly dense immobile communities attributed to stress resistance. It has been suggested that biofilm and swarming are oppositely controlled, making this transition particularly interesting for understanding the ability of bacterial colonies to adapt to challenging environments. Here, the swarm to biofilm transition is studied in Bacillus subtilis by analyzing the bacterial dynamics both on the individual and collective scales. We show that both biological and physical processes facilitate the transition. A few individual cells that initiate the biofilm program cause nucleation of large, approximately scale-free, stationary aggregates of trapped swarm cells. Around aggregates, cells continue swarming almost unobstructed, while inside, trapped cells are added to the biofilm. While our experimental findings rule out previously suggested purely physical effects as a trigger for biofilm formation, they show how physical processes, such as clustering and jamming, accelerate biofilm formation.

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.

Self-driven criticality in a stochastic epidemic model.

We present a generic epidemic model with stochastic parameters in which the dynamics self-organize to a critical state with suppressed exponential growth. More precisely, the dynamics evolve into a quasi-steady state, where the effective reproduction rate fluctuates close to the critical value 1 for a long period, as indeed observed for different epidemics. The main assumptions underlying the model are that the rate at which each individual becomes infected changes stochastically in time with a heavy-tailed steady state. The critical regime is characterized by an extremely long duration of the epidemic. Its stability is analyzed both numerically and analytically in different models.

Heterogeneous bacterial swarms with mixed lengths.

Heterogeneous systems of active matter exhibit a range of complex emergent dynamical patterns. In particular, it is difficult to predict the properties of the mixed system based on its constituents. These considerations are particularly significant for understanding realistic bacterial swarms, which typically develop heterogeneities even when grown from a single cell. Here, mixed swarms of cells with different aspect ratios are studied both experimentally and in simulations. In contrast with previous theory, there is no macroscopic phase segregation. However, locally, long cells act as nucleation cites, around which aggregates of short, rapidly moving cells can form, resulting in enhanced swarming speeds. On the other hand, high fractions of long cells form a bottleneck for efficient swarming. Our results suggest a physical advantage for the spontaneous heterogeneity of bacterial swarm populations.

Collective motion as a distinct behavioral state of the individual.

The collective motion of swarms depends on adaptations at the individual level. We explored these and their effects on swarm formation and maintenance in locusts. The walking kinematics of individual insects were monitored under laboratory settings, before, as well as during collective motion in a group, and again after separation from the group. It was found that taking part in collective motion induced in the individual unique behavioral kinematics, suggesting the existence of a distinct behavioral mode that we term a "collective-motion-state." This state, characterized by behavioral adaptation to the social context, is long lasting, not induced by crowding per se, but only by experiencing collective motion. Utilizing computational models, we show that this adaptability increases the robustness of the swarm. Overall, our findings suggest that collective motion is not only an emergent property of the group but also depends on a behavioral mode, rooted in endogenous mechanisms of the individual.

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.

Inferring entropy from structure.

The thermodynamic definition of entropy can be extended to nonequilibrium systems based on its relation to information. To apply this definition in practice requires access to the physical system's microstates, which may be prohibitively inefficient to sample or difficult to obtain experimentally. It is beneficial, therefore, to relate the entropy to other integrated properties which are accessible out of equilibrium. We focus on the structure factor, which describes the spatial correlations of density fluctuations and can be directly measured by scattering. The information gained by a given structure factor regarding an otherwise unknown system provides an upper bound for the system's entropy. We find that the maximum-entropy model corresponds to an equilibrium system with an effective pair interaction. Approximate closed-form relations for the effective pair potential and the resulting entropy in terms of the structure factor are obtained. As examples, the relations are used to estimate the entropy of an exactly solvable model and two simulated systems out of equilibrium. The focus is on low-dimensional examples, where our method, as well as a recently proposed compression-based one, can be tested against a rigorous direct-sampling technique. The entropy inferred from the structure factor is found to be consistent with the other methods, superior for larger system sizes, and accurate in identifying global transitions. Our approach allows for extensions of the theory to more complex systems and to higher-order correlations.

A statistical physics view of swarming bacteria.

Bacterial swarming is a collective mode of motion in which cells migrate rapidly over surfaces, forming dynamic patterns of whirls and jets. This review presents a physical point of view of swarming bacteria, with an emphasis on the statistical properties of the swarm dynamics as observed in experiments. The basic physical principles underlying the swarm and their relation to contemporary theories of collective motion and active matter are reviewed and discussed in the context of the biological properties of swarming cells. We suggest a paradigm according to which bacteria have optimized some of their physical properties as a strategy for rapid surface translocation. In other words, cells take advantage of favorable physics, enabling efficient expansion that enhances survival under harsh conditions.

The 3D architecture of a bacterial swarm has implications for antibiotic tolerance.

Swarming bacteria are an example of a complex, active biological system, where high cell density and super-diffusive cell mobility confer survival advantages to the group as a whole. Previous studies on the dynamics of the swarm have been limited to easily observable regions at the advancing edge of the swarm where cells are restricted to a plane. In this study, using defocused epifluorescence video imaging, we have tracked the motion of fluorescently labeled individuals within the interior of a densely packed three-dimensional (3D) region of a swarm. Our analysis reveals a novel 3D architecture, where bacteria are constrained by inter-particle interactions, sandwiched between two distinct boundary conditions. We find that secreted biosurfactants keep bacteria away from the swarm-air upper boundary, and added antibiotics at the lower swarm-surface boundary lead to their migration away from this boundary. Formation of the antibiotic-avoidance zone is dependent on a functional chemotaxis signaling system, in the absence of which the swarm loses its high tolerance to the antibiotics.

Modeling cooperating micro-organisms in antibiotic environment.

Recent experiments with the bacteria Paenibacillus vortex reveal a remarkable strategy enabling it to cope with antibiotics by cooperating with a different bacterium-Escherichia coli. While P. vortex is a highly effective swarmer, it is sensitive to the antibiotic ampicillin. On the other hand, E. coli can degrade ampicillin but is non-motile when grown on high agar percentages. The two bacterial species form a shared colony in which E. coli is transported by P. vortex and E. coli detoxifies the ampicillin. The paper presents a simplified model, consisting of coupled reaction-diffusion equations, describing the development of ring patterns in the shared colony. Our results demonstrate some of the possible cooperative movement strategies bacteria utilize in order to survive harsh conditions. In addition, we explore the behavior of mixed colonies under new conditions such as antibiotic gradients, synchronization between colonies and possible dynamics of a 3-species system including P. vortex, E. coli and a carbon producing algae that provides nutrients under illuminated, nutrient poor conditions. The derived model was able to simulate an asymmetric relationship between two or three micro-organisms where cooperation is required for survival. Computationally, in order to avoid numerical artifacts due to symmetries within the discretizing grid, the model was solved using a second order Vectorizable Random Lattices method, which is developed as a finite volume scheme on a random grid.