Large-scale collective properties of self-propelled rods

Revisiting the flocking transition using active spins

We consider an active Ising model in which spins both diffuse and align on lattice in one and two dimensions. The diffusion is biased so that plus or minus spins hop preferably to the left or to the right, which generates a flocking transition at low temperature and high density. We construct a coarse-grained description of the model that predicts this transition to be a first-order liquid-gas transition in the temperature-density ensemble, with a critical density sent to infinity. In this first-order phase transition, the magnetization is proportional to the liquid fraction and thus varies continuously throughout the phase diagram. Using microscopic simulations, we show that this theoretical prediction holds in 2D whereas the fluctuations alter the transition in 1D, preventing, for instance, any spontaneous symmetry breaking.

Order, intermittency, and pressure fluctuations in a system of proliferating rods

Nonmotile elongated bacteria confined in two-dimensional open microchannels can exhibit collective motion and form dense monolayers with nematic order if the cells proliferate, i.e., grow and divide. Using soft molecular dynamics simulations of a system of rods interacting through short range mechanical forces, we study the effects of the cell growth rate, the cell aspect ratio, and the sliding friction on nematic ordering and on pressure fluctuations in confined environments. Our results indicate that rods with aspect ratios >3.0 reach quasiperfect nematic states at low sliding friction. At higher frictions, the global nematic order parameter shows intermittent fluctuations due to sudden losses of order and the time intervals between these bursts are power-law distributed. The pressure transverse to the channel axis can vary abruptly in time and shows hysteresis due to lateral crowding effects. The longitudinal pressure field is on average correlated to nematic order, but it is locally very heterogeneous and its distribution follows an inverse power law, in sharp contrast with nonactive granular systems. We discuss some implications of these findings for tissue growth.

Collective cell streams in epithelial monolayers depend on cell adhesion

We report a spontaneously emerging, randomly oriented, collective streaming behavior within a monolayer culture of a human keratinocyte cell line, and explore the effect of modulating cell adhesions by perturbing the function of calcium-dependent cell adhesion molecules. We demonstrate that decreasing cell adhesion induces narrower and more anisotropic cell streams, reminiscent of decreasing the Taylor scale of turbulent liquids. To explain our empirical findings, we propose a cell-based model that represents the dual nature of cell-cell adhesions. Spring-like connections provide mechanical stability, while a cellular Potts model formalism represents surface-tension driven attachment. By changing the relevance and persistence of mechanical links between cells, we are able to explain the experimentally observed changes in emergent flow patterns.

Confinement stabilizes a bacterial suspension into a spiral vortex

Confining surfaces play crucial roles in dynamics, transport, and order in many physical systems, but their effects on active matter, a broad class of dynamically self-organizing systems, are poorly understood. We investigate here the influence of global confinement and surface curvature on collective motion by studying the flow and orientational order within small droplets of a dense bacterial suspension. The competition between radial confinement, self-propulsion, steric interactions, and hydrodynamics robustly induces an intriguing steady single-vortex state, in which cells align in inward spiraling patterns accompanied by a thin counterrotating boundary layer. A minimal continuum model is shown to be in good agreement with these observations.

Perspective: nanomotors without moving parts that propel themselves in solution

Self-propelled nanomotors use chemical energy to produce directed motion. Like many molecular motors they suffer strong perturbations from the environment in which they move as a result of thermal fluctuations and do not rely on inertia for their propulsion. Such tiny motors are the subject of considerable research because of their potential applications, and a variety of synthetic motors have been made and are being studied for this purpose. Chemically powered self-propelled nanomotors without moving parts that rely on asymmetric chemical reactions to effect directed motion are the focus of this article. The mechanisms they use for propulsion, how size and fuel sources influence their motion, how they cope with strong molecular fluctuations, and how they behave collectively are described. The practical applications of such nanomotors are largely unrealized and the subject of speculation. Since molecular motors are ubiquitous in biology and perform a myriad of complex tasks, the hope is that synthetic motors might be able to perform analogous tasks. They may have the potential to change our perspective on how chemical dynamics takes place in complex systems.

Optimal noise maximizes collective motion in heterogeneous media

We study the effect of spatial heterogeneity on the collective motion of self-propelled particles (SPPs). The heterogeneity is modeled as a random distribution of either static or diffusive obstacles, which the SPPs avoid while trying to align their movements. We find that such obstacles have a dramatic effect on the collective dynamics of usual SPP models. In particular, we report about the existence of an optimal (angular) noise amplitude that maximizes collective motion. We also show that while at low obstacle densities the system exhibits long-range order, in strongly heterogeneous media collective motion is quasi-long-range and exists only for noise values in between two critical values, with the system being disordered at both large and low noise amplitudes. Since most real systems have spatial heterogeneities, the finding of an optimal noise intensity has immediate practical and fundamental implications for the design and evolution of collective motion strategies.

Long-range ordering of vibrated polar disks

Vibrated polar disks have been used experimentally to investigate collective motion of driven particles, where fully ordered asymptotic regimes could not be reached. Here we present a model reproducing quantitatively the single, binary, and collective properties of this granular system. Using system sizes not accessible in the laboratory, we show in silico that true long-range order is possible in the experimental system. Exploring the model's parameter space, we find a phase diagram qualitatively different from that of dilute or pointlike particle systems.

Sedimentation and polar order of active bottom-heavy particles

Self-propelled particles in an external gravitational field have been shown to display both an increased sedimentation length and polar order even without particle interactions. Here, we investigate self-propelled particles which additionally are bottom-heavy, that is they feel a torque aligning them to swim against the gravitational field. For bottom-heavy particles the gravitational field has the two opposite effects of i) sedimentation and ii) upward alignment of the particles' swimming direction. We perform a multipole expansion of the one-particle distribution of non-interacting particles with respect to orientation and derive expressions for sedimentation length and mean particle orientation which we check against Brownian Dynamics simulations. For large strength of gravity or small particle speeds and aligning torque, we observe sedimentation with increased sedimentation length compared with passive colloids but also active colloids without bottom-heaviness. Increasing, for example, swimming speed the sedimentation profile is inverted and the particles swim towards the top wall of the enclosing box. We find maximal orientational order at intermediate swimming speeds for both cases of particles with bottom-heaviness and those without. Ordering unsurprisingly is increased for the bottom-heavy particles, but this difference disappears at higher levels of activity and for very high activities ordering goes to zero in both cases.

Multi-scale inference of interaction rules in animal groups using Bayesian model selection

Inference of interaction rules of animals moving in groups usually relies on an analysis of large scale system behaviour. Models are tuned through repeated simulation until they match the observed behaviour. More recent work has used the fine scale motions of animals to validate and fit the rules of interaction of animals in groups. Here, we use a Bayesian methodology to compare a variety of models to the collective motion of glass prawns (Paratya australiensis). We show that these exhibit a stereotypical ‘phase transition’, whereby an increase in density leads to the onset of collective motion in one direction. We fit models to this data, which range from: a mean-field model where all prawns interact globally; to a spatial Markovian model where prawns are self-propelled particles influenced only by the current positions and directions of their neighbours; up to non-Markovian models where prawns have ‘memory’ of previous interactions, integrating their experiences over time when deciding to change behaviour. We show that the mean-field model fits the large scale behaviour of the system, but does not capture the observed locality of interactions. Traditional self-propelled particle models fail to capture the fine scale dynamics of the system. The most sophisticated model, the non-Markovian model, provides a good match to the data at both the fine scale and in terms of reproducing global dynamics, while maintaining a biologically plausible perceptual range. We conclude that prawns’ movements are influenced by not just the current direction of nearby conspecifics, but also those encountered in the recent past. Given the simplicity of prawns as a study system our research suggests that self-propelled particle models of collective motion should, if they are to be realistic at multiple biological scales, include memory of previous interactions and other non-Markovian effects.

The collective movement of animals in a group is an impressive phenomenon whereby large scale spatio-temporal patterns emerge from simple interactions between individuals. Theoretically, much of our understanding of animal group motion comes from models inspired by statistical physics. In these models, animals are treated as moving (self-propelled) particles that interact with each other according to simple rules. Recently, researchers have shown greater interest in using experimental data to verify which rules are actually implemented by a particular animal species. In our study, we present a rigorous selection between alternative models inspired by the literature for a system of glass prawns. We find that the classic theoretical models do not accurately predict either the fine scale or large scale behaviour of the system. Instead, individual animals appear to be interacting even when completely separated from each other. To resolve this we introduce a new class of models wherein prawns ‘remember‚ their previous interactions, integrating their experiences over time when deciding to change behaviour. These show that the fine scale and large scale behaviour of the prawns is consistent with interactions only between individuals who are close together.

Continuum modeling of clustering of myxobacteria

In this paper we develop a continuum theory of clustering in ensembles of self-propelled inelastically colliding rods with applications to collective dynamics of common gliding bacteria Myxococcus Xanthus. A multiphase hydrodynamic model that couples densities of oriented and isotropic phases is described. This model is used for the analysis of an instability that leads to spontaneous formation of directionally moving dense clusters within initially dilute isotropic "gas" of myxobacteria. Numerical simulations of this model confirm the existence of stationary dense moving clusters and also elucidate the properties of their collisions. The results are shown to be in a qualitative agreement with experiments.

Periodic and quasiperiodic motion of an elongated microswimmer in Poiseuille flow

We study the dynamics of a prolate spheroidal microswimmer in Poiseuille flow for different flow geometries. When moving between two parallel plates or in a cylindrical microchannel, the swimmer performs either periodic swinging or periodic tumbling motion. Although the trajectories of spherical and elongated swimmers are qualitatively similar, the swinging and tumbling frequency strongly depends on the aspect ratio of the swimmer. In channels with reduced symmetry the swimmers perform quasiperiodic motion which we demonstrate explicitly for swimming in a channel with elliptical cross-section.

Nonlinear field equations for aligning self-propelled rods

We derive a set of minimal and well-behaved nonlinear field equations describing the collective properties of self-propelled rods from a simple microscopic starting point, the Vicsek model with nematic alignment. Analysis of their linear and nonlinear dynamics shows good agreement with the original microscopic model. In particular, we derive an explicit expression for density-segregated, banded solutions, allowing us to develop a more complete analytic picture of the problem at the nonlinear level.

Group chase and escape with some fast chasers

We study group chase and escape with some fast chasers. In our model chasers look for the nearest target and move to one of the nearest sites in order to catch the target. On the other hand, targets try to escape from the nearest chaser. When a chaser catches a target, the target is removed from the system and the number of targets decreases. The lifetime of targets, at which all targets caught, decreases as t^{α} with increasing the number of chasers. When there are no fast chasers and the total number of chasers is small, the exponent α is large. When the total number of chasers is large, α becomes small. There is an optimal number of chasers to minimize the cost used in order to catch all targets. However, when we add a few fast chasers, the region with the large α vanishes. The optimal number of chasers vanishes, and the cost monotonically increases with increasing the number of chasers.

Emergent states in dense systems of active rods: from swarming to turbulence

Dense suspensions of self-propelled rod-like particles exhibit a fascinating variety of non-equilibrium phenomena. By means of computer simulations of a minimal model for rigid self-propelled colloidal rods with variable shape we explore the generic diagram of emerging states over a large range of rod densities and aspect ratios. The dynamics is studied using a simple numerical scheme for the overdamped noiseless frictional dynamics of a many-body system in which steric forces are dominant over hydrodynamic ones. The different emergent states are identified by various characteristic correlation functions and suitable order parameter fields. At low density and aspect ratio, a disordered phase with no coherent motion precedes a highly cooperative swarming state with giant number fluctuations at large aspect ratio. Conversely, at high densities weakly anisometric particles show a distinct jamming transition whereas slender particles form dynamic laning patterns. In between there is a large window corresponding to strongly vortical, turbulent flow. The different dynamical states should be verifiable in systems of swimming bacteria and artificial rod-like micro-swimmers.

Competing ferromagnetic and nematic alignment in self-propelled polar particles

We study a Vicsek-style model of self-propelled particles where ferromagnetic and nematic alignment compete in both the usual "metric" version and in the "metric-free" case where a particle interacts with its Voronoi neighbors. We show that the phase diagram of this out-of-equilibrium XY model is similar to that of its equilibrium counterpart: The properties of the fully nematic model, studied before by Ginelli et al. [Phys. Rev. Lett. 104, 184502 (2010)], are thus robust to the introduction of a modest bias of interactions toward ferromagnetic alignment. The direct transitions between polar and nematic ordered phases are shown to be discontinuous in the metric case, and continuous, belonging to the Ising universality class, in the metric-free version.

Pattern-formation mechanisms in motility mutants of Myxococcus xanthus

Formation of spatial patterns of cells is a recurring theme in biology and often depends on regulated cell motility. Motility of the rod-shaped cells of the bacterium Myxococcus xanthus depends on two motility machineries, type IV pili (giving rise to S-motility) and the gliding motility apparatus (giving rise to A-motility). Cell motility is regulated by occasional reversals. Moving M. xanthus cells can organize into spreading colonies or spore-filled fruiting bodies, depending on their nutritional status. To ultimately understand these two pattern-formation processes and the contributions by the two motility machineries, as well as the cell reversal machinery, we analyse spatial self-organization in three M. xanthus strains: (i) a mutant that moves unidirectionally without reversing by the A-motility system only, (ii) a unidirectional mutant that is also equipped with the S-motility system, and (iii) the wild-type that, in addition to the two motility systems, occasionally reverses its direction of movement. The mutant moving by means of the A-engine illustrates that collective motion in the form of large moving clusters can arise in gliding bacteria owing to steric interactions of the rod-shaped cells, without the need of invoking any biochemical signal regulation. The two-engine strain mutant reveals that the same phenomenon emerges when both motility systems are present, and as long as cells exhibit unidirectional motion only. From the study of these two strains, we conclude that unidirectional cell motion induces the formation of large moving clusters at low and intermediate densities, while it results in vortex formation at very high densities. These findings are consistent with what is known from self-propelled rod models, which strongly suggests that the combined effect of self-propulsion and volume exclusion interactions is the pattern-formation mechanism leading to the observed phenomena. On the other hand, we learn that when cells occasionally reverse their moving direction, as observed in the wild-type, cells form small but strongly elongated clusters and self-organize into a mesh-like structure at high enough densities. These results have been obtained from a careful analysis of the cluster statistics of ensembles of cells, and analysed in the light of a coagulation Smoluchowski equation with fragmentation.

Deciphering interactions in moving animal groups

Collective motion phenomena in large groups of social organisms have long fascinated the observer, especially in cases, such as bird flocks or fish schools, where large-scale highly coordinated actions emerge in the absence of obvious leaders. However, the mechanisms involved in this self-organized behavior are still poorly understood, because the individual-level interactions underlying them remain elusive. Here, we demonstrate the power of a bottom-up methodology to build models for animal group motion from data gathered at the individual scale. Using video tracks of fish shoal in a tank, we show how a careful, incremental analysis at the local scale allows for the determination of the stimulus/response function governing an individual's moving decisions. We find in particular that both positional and orientational effects are present, act upon the fish turning speed, and depend on the swimming speed, yielding a novel schooling model whose parameters are all estimated from data. Our approach also leads to identify a density-dependent effect that results in a behavioral change for the largest groups considered. This suggests that, in confined environment, the behavioral state of fish and their reaction patterns change with group size. We debate the applicability, beyond the particular case studied here, of this novel framework for deciphering interactions in moving animal groups.

Swarms of insects, schools of fish and flocks of birds display an impressive variety of collective patterns that emerge from local interactions among group members. These puzzling phenomena raise a variety of questions about the behavioral rules that govern the coordination of individuals' motions and the emergence of large-scale patterns. While numerous models have been proposed, there is still a strong need for detailed experimental studies to foster the biological understanding of such collective motion. Here, we use data recorded on fish barred flagtails moving in groups of increasing sizes in a water tank to demonstrate the power of an incremental methodology for building a fish behavior model completely based on interactions with the physical environment and neighboring fish. In contrast to previous works, our model revealed an implicit balancing of neighbors position and orientation on the turning speed of fish, an unexpected transition between shoaling and schooling induced by a change in the swimming speed, and a group-size effect which results in a decrease of social interactions among fish as density increases. An important feature of this model lies in its ability to allow a large palette of adaptive patterns with a great economy of means.

Meso-scale turbulence in living fluids

Turbulence is ubiquitous, from oceanic currents to small-scale biological and quantum systems. Self-sustained turbulent motion in microbial suspensions presents an intriguing example of collective dynamical behavior among the simplest forms of life and is important for fluid mixing and molecular transport on the microscale. The mathematical characterization of turbulence phenomena in active nonequilibrium fluids proves even more difficult than for conventional liquids or gases. It is not known which features of turbulent phases in living matter are universal or system-specific or which generalizations of the Navier-Stokes equations are able to describe them adequately. Here, we combine experiments, particle simulations, and continuum theory to identify the statistical properties of self-sustained meso-scale turbulence in active systems. To study how dimensionality and boundary conditions affect collective bacterial dynamics, we measured energy spectra and structure functions in dense Bacillus subtilis suspensions in quasi-2D and 3D geometries. Our experimental results for the bacterial flow statistics agree well with predictions from a minimal model for self-propelled rods, suggesting that at high concentrations the collective motion of the bacteria is dominated by short-range interactions. To provide a basis for future theoretical studies, we propose a minimal continuum model for incompressible bacterial flow. A detailed numerical analysis of the 2D case shows that this theory can reproduce many of the experimentally observed features of self-sustained active turbulence.

Nucleation-induced transition to collective motion in active systems

While the existence of polar ordered states in active systems is well established, the dynamics of the self-assembly processes are still elusive. We study a lattice gas model of self-propelled elongated particles interacting through excluded volume and alignment interactions, which shows a phase transition from an isotropic to a polar ordered state. By analyzing the ordering process we find that the transition is driven by the formation of a critical nucleation cluster and a subsequent coarsening process. Moreover, the time to establish a polar ordered state shows a power-law divergence.

Self-regulation in self-propelled nematic fluids

We consider the hydrodynamic theory of an active fluid of self-propelled particles with nematic aligning interactions. This class of materials has polar symmetry at the microscopic level, but forms macrostates of nematic symmetry. We highlight three key features of the dynamics. First, as in polar active fluids, the control parameter for the order-disorder transition, namely the density, is dynamically convected by the order parameter via active currents. The resulting dynamical self-regulation of the order parameter is a generic property of active fluids and destabilizes the uniform nematic state near the mean-field transition. Secondly, curvature-driven currents render the system unstable deep in the nematic state, as found previously. Finally, and unique to self-propelled nematics, nematic order induces local polar order that in turn leads to the growth of density fluctuations. We propose this as a possible mechanism for the smectic order of polar clusters seen in numerical simulations.

Continuous theory of active matter systems with metric-free interactions

We derive a hydrodynamic description of metric-free active matter: starting from self-propelled particles aligning with neighbors defined by "topological" rules, not metric zones-a situation advocated recently to be relevant for bird flocks, fish schools, and crowds-we use a kinetic approach to obtain well-controlled nonlinear field equations. We show that the density-independent collision rate per particle characteristic of topological interactions suppresses the linear instability of the homogeneous ordered phase and the nonlinear density segregation generically present near threshold in metric models, in agreement with microscopic simulations.

Multifractal dynamics of turbulent flows in swimming bacterial suspensions

We experimentally investigate the self-propelled two-dimensional turbulent flows of Escherichia coli suspensions in thin liquid films at two different cell concentrations. It is found that the flow has fluctuating vortices with a broad range of scales and intensities through the nonlinear interaction of the swimming bacteria. Increasing cell concentration increases the total propelling power and the nonlinear interaction. It causes the generation of vortices with larger scale, lower frequency, and higher intensity. It also widens the histograms of the flow velocity and the velocity increment between two spatially separated points with more stretched non-Gaussian tails. From the scaling analysis of the structure function S(q)(r) of the qth moment of the velocity increment between two points with spatial separation r, nonlinear relations between the scaling exponent ζ(q) of S(q)(r) and q are found for both cell concentrations, which manifests the multifractal dynamics. The multifractality can be enhanced by increasing cell concentration.

Nonequilibrium condensation and coarsening of field-driven dipolar colloids

In colloidal suspensions, self-organization processes can be easily fueled by external fields. Here we consider monolayers of particles with permanent dipole moments that are driven by rotating external fields. In recent experiments, it has been shown that the particles in such systems self-organize into two-dimensional clusters. Here we report results from a computer simulation study of these pattern forming systems. Specifically, we employ Langevin dynamics simulations, Brownian dynamics simulations that include hydrodynamic interactions, and Wang-Landau Monte Carlo simulations of soft spheres interacting via dipolar potentials. We investigate at which field strengths and frequencies clusters form and explore the influence of hydrodynamic interactions. We also examine the phase behavior of the equilibrium system resulting from a time average of the colloidal interactions in the rotating field. In this way we demonstrate that the clustering described in the driven system arises from a first-order phase transition between a vapor and a condensed phase.

Pattern formation in self-propelled particles with density-dependent motility

We study the behavior of interacting self-propelled particles, whose self-propulsion speed decreases with their local density. By combining direct simulations of the microscopic model with an analysis of the hydrodynamic equations obtained by explicitly coarse graining the model, we show that interactions lead generically to the formation of a host of patterns, including moving clumps, active lanes, and asters. This general mechanism could explain many of the patterns seen in recent experiments and simulations.

Athermal phase separation of self-propelled particles with no alignment

We study numerically and analytically a model of self-propelled polar disks on a substrate in two dimensions. The particles interact via isotropic repulsive forces and are subject to rotational noise, but there is no aligning interaction. As a result, the system does not exhibit an ordered state. The isotropic fluid phase separates well below close packing and exhibits the large number fluctuations and clustering found ubiquitously in active systems. Our work shows that this behavior is a generic property of systems that are driven out of equilibrium locally, as for instance by self-propulsion.

Spatial structures and giant number fluctuations in models of active matter

The large scale fluctuations of the ordered state in active matter systems are usually characterized by studying the "giant number fluctuations" of particles in any finite volume, as compared to the expectations from the central limit theorem. However, in ordering systems, the fluctuations in density ordering are often captured through their structure functions deviating from Porod's law. In this Letter we study the relationship between giant number fluctuations and structure functions for different models of active matter as well as other nonequilibrium systems. A unified picture emerges, with different models falling in four distinct classes depending on the nature of their structure functions. For one class, we show that experimentalists may find Porod's law violation, by measuring subleading corrections to the number fluctuations.

Chasing and escaping by three groups of species

We study group chasing and escaping between three species. In our model, one species acts as a group of chasers for another species and acts as a group of targets for the third species. When a particle is caught by a target, the particle becomes a new chaser. Although the ratio of three species is changed, the total number of particles is conserved. When particles move randomly, the numbers of the three species change periodically but no species seems to become extinct. If particles escape from the nearest chaser and chase the nearest target, the extinction of a species occurs. The extinction induces that of the second species and finally only one species survives.

Dynamical self-regulation in self-propelled particle flows

We study a continuum model of overdamped self-propelled particles with aligning interactions in two dimensions. Combining analytical theory and computations, we map out the phase diagram for the parameter space covered by the model. We find that the system self-organizes into two robust structures in different regions of parameter space: solitary waves composed of ordered swarms moving through a low density disordered background, and stationary radially symmetric asters. The self-regulating nature of the flow yields phase separation, ubiquitous in this class of systems, and controls the formation of solitary waves. Self-propulsion and the associated active convection play a crucial role in aster formation.

Collective dynamics of model microorganisms with chemotactic signaling

Various microorganisms use chemotaxis for signaling among individuals-a common strategy for communication that is responsible for the formation of microcolonies. We model the microorganisms as autochemotactic active random walkers and describe them by an appropriate Langevin dynamics. It consists of rotational diffusion of the walker's velocity direction and a deterministic torque that aligns the velocity direction along the gradient of a self-generated chemical field. To account for finite size, each microorganism is treated as a soft disk. Its velocity is modified when it overlaps with other walkers according to a linear force-velocity relation and a harmonic repulsion force. We analyze two-walker collisions by presenting typical trajectories and by determining a state diagram that distinguishes between free walker, metastable, and bounded cluster states. We mention an analogy to Kramer's escape problem. Finally, we investigate relevant properties of many-walker systems and describe characteristics of cluster formation in unbounded geometry and in confinement.

Large-scale vortex lattice emerging from collectively moving microtubules

Spontaneous collective motion, as in some flocks of bird and schools of fish, is an example of an emergent phenomenon. Such phenomena are at present of great interest and physicists have put forward a number of theoretical results that so far lack experimental verification. In animal behaviour studies, large-scale data collection is now technologically possible, but data are still scarce and arise from observations rather than controlled experiments. Multicellular biological systems, such as bacterial colonies or tissues, allow more control, but may have many hidden variables and interactions, hindering proper tests of theoretical ideas. However, in systems on the subcellular scale such tests may be possible, particularly in in vitro experiments with only few purified components. Motility assays, in which protein filaments are driven by molecular motors grafted to a substrate in the presence of ATP, can show collective motion for high densities of motors and attached filaments. This was demonstrated recently for the actomyosin system, but a complete understanding of the mechanisms at work is still lacking. Here we report experiments in which microtubules are propelled by surface-bound dyneins. In this system it is possible to study the local interaction: we find that colliding microtubules align with each other with high probability. At high densities, this alignment results in self-organization of the microtubules, which are on average 15 µm long, into vortices with diameters of around 400 µm. Inside the vortices, the microtubules circulate both clockwise and anticlockwise. On longer timescales, the vortices form a lattice structure. The emergence of these structures, as verified by a mathematical model, is the result of the smooth, reptation-like motion of single microtubules in combination with local interactions (the nematic alignment due to collisions)--there is no need for long-range interactions. Apart from its potential relevance to cortical arrays in plant cells and other biological situations, our study provides evidence for the existence of previously unsuspected universality classes of collective motion phenomena.

Collective motion and nonequilibrium cluster formation in colonies of gliding bacteria

We characterize cell motion in experiments and show that the transition to collective motion in colonies of gliding bacterial cells confined to a monolayer appears through the organization of cells into larger moving clusters. Collective motion by nonequilibrium cluster formation is detected for a critical cell packing fraction around 17%. This transition is characterized by a scale-free power-law cluster-size distribution, with an exponent 0.88±0.07, and the appearance of giant number fluctuations. Our findings are in quantitative agreement with simulations of self-propelled rods. This suggests that the interplay of self-propulsion and the rod shape of bacteria is sufficient to induce collective motion.

Collective motion of binary self-propelled particle mixtures

In this study, we investigate the phenomenon of collective motion in binary mixtures of self-propelled particles. More precisely, we consider two particle species, each of which consisting of pointlike objects that propel with a velocity of constant magnitude. Within each species, the particles try to achieve polar alignment of their velocity vectors, whereas we analyze the cases of preferred polar, antiparallel, as well as perpendicular alignment between particles of different species. Our focus is on the effect that the interplay between the two species has on the threshold densities for the onset of collective motion and on the nature of the solutions above onset. For this purpose, we start from suitable Langevin equations in the particle picture, from which we derive mean field equations of the Fokker-Planck type and finally macroscopic continuum field equations. We perform particle simulations of the Langevin equations and linear stability analyses of the Fokker-Planck and macroscopic continuum equations, and we numerically solve the Fokker-Planck equations. Both spatially homogeneous and inhomogeneous solutions are investigated, where the latter correspond to stripelike flocks of collectively moving particles. In general, the interaction between the two species reduces the threshold density for the onset of collective motion of each species. However, this interaction also reduces the spatial organization in the stripelike flocks. The case that shows the most interesting behavior is the one of preferred perpendicular alignment between different species. There a competition between polar and truly nematic orientational ordering of the velocity vectors takes place within each particle species. Finally, depending on the alignment rule for particles of different species and within certain ranges of particle densities, identical and inverted spatial density profiles can be found for the two particle species. The system under investigation is confined to two spatial dimensions.

Collective dynamics of self-propelled sphere-dimer motors

The collective dynamics of ensembles of chemically powered sphere dimer motors is investigated. Sphere dimers are self-propelled nanomotors built from linked catalytic and noncatalytic spheres. They consume fuel in the environment and utilize the resulting self-generated concentration gradients to produce directed motion along their internuclear axes. In collections of such motors, the individual motors interact through forces that arise from concentration gradients, hydrodynamic coupling, and direct intermolecular forces. Under nonequilibrium conditions it is found that the sphere dimer motors self-assemble into transient aggregates with distinctive structural correlations and exhibit swarming where the aggregates propagate through the system. The mean square displacement of a dimer motor in the ensemble displays short-time ballistic and long-time diffusive regimes and, for ensembles containing many motors, an increasingly prominent intermediate regime. The self-diffusion coefficient of a motor in a many-motor system behaves differently from that of an isolated motor, and the decay of orientational correlations is a nonmonotonic function of the number of motors. The results presented here illustrate the phenomena to be expected in applications, such as cargo transport, where many motors may act in consort.

Multi-scale inference of interaction rules in animal groups using Bayesian model selection

Inference of interaction rules of animals moving in groups usually relies on an analysis of large scale system behaviour. Models are tuned through repeated simulation until they match the observed behaviour. More recent work has used the fine scale motions of animals to validate and fit the rules of interaction of animals in groups. Here, we use a Bayesian methodology to compare a variety of models to the collective motion of glass prawns (Paratya australiensis). We show that these exhibit a stereotypical ‘phase transition’, whereby an increase in density leads to the onset of collective motion in one direction. We fit models to this data, which range from: a mean-field model where all prawns interact globally; to a spatial Markovian model where prawns are self-propelled particles influenced only by the current positions and directions of their neighbours; up to non-Markovian models where prawns have ‘memory’ of previous interactions, integrating their experiences over time when deciding to change behaviour. We show that the mean-field model fits the large scale behaviour of the system, but does not capture fine scale rules of interaction, which are primarily mediated by physical contact. Conversely, the Markovian self-propelled particle model captures the fine scale rules of interaction but fails to reproduce global dynamics. The most sophisticated model, the non-Markovian model, provides a good match to the data at both the fine scale and in terms of reproducing global dynamics. We conclude that prawns' movements are influenced by not just the current direction of nearby conspecifics, but also those encountered in the recent past. Given the simplicity of prawns as a study system our research suggests that self-propelled particle models of collective motion should, if they are to be realistic at multiple biological scales, include memory of previous interactions and other non-Markovian effects.

The collective movement of animals in a group is an impressive phenomenon whereby large scale spatio-temporal patterns emerge from simple interactions between individuals. Theoretically, much of our understanding of animal group motion comes from models inspired by statistical physics. In these models, animals are treated as moving (self-propelled) particles that interact with each other according to simple rules. Recently, researchers have shown greater interest in using experimental data to verify which rules are actually implemented by a particular animal species. In our study, we present a rigorous selection between alternative models inspired by the literature for a system of glass prawns. We find that the classic theoretical models can accurately capture either the fine-scale behaviour or the large-scale collective patterns of movement of the prawns. However, none are able to reproduce both levels of description at the same time. To resolve this conflict we introduce a new class of models wherein prawns ‘remember’, their previous interactions, integrating their experiences over time when deciding to change behaviour. These outperform the traditional models in predicting when individual prawns will change their direction of motion and restore consistency between the fine-scale rules of interaction and the global behaviour of the group.

Shearing self-propelled suspensions: arrest of coarsening and suppression of giant density fluctuations

We study the effect of a linear shear flow on a collection of interacting active, self-propelled particles modeled via the Vicsek model. The imposed flow has a dramatic effect on the behavior of the model. We find that in the presence of shear there is no order-disorder transition, and that coarsening of the domains is arrested. Shear also suppresses the so-called giant density fluctuations that are observed in the quiescent limit.

Active colloidal suspensions exhibit polar order under gravity

Recently, the steady sedimentation profile of a dilute suspension of chemically powered colloids was studied experimentally [J. Palacci et al., Phys. Rev. Lett. 105, 088304 (2010)]. It was found that the sedimentation length increases quadratically with the swimming speed of the active Brownian particles. Here we investigate theoretically the sedimentation of self-propelled particles undergoing translational and rotational diffusion. We find that the measured increase of the sedimentation length is coupled to a partial alignment of the suspension with the mean swimming direction oriented against the gravitational field. We suggest realistic parameter values to observe this polar order. Furthermore, we find that the dynamics of the active suspension can be derived from a generalized free energy functional.

Fluid dynamics and noise in bacterial cell-cell and cell-surface scattering

Bacterial processes ranging from gene expression to motility and biofilm formation are constantly challenged by internal and external noise. While the importance of stochastic fluctuations has been appreciated for chemotaxis, it is currently believed that deterministic long-range fluid dynamical effects govern cell-cell and cell-surface scattering-the elementary events that lead to swarming and collective swimming in active suspensions and to the formation of biofilms. Here, we report direct measurements of the bacterial flow field generated by individual swimming Escherichia coli both far from and near to a solid surface. These experiments allowed us to examine the relative importance of fluid dynamics and rotational diffusion for bacteria. For cell-cell interactions it is shown that thermal and intrinsic stochasticity drown the effects of long-range fluid dynamics, implying that physical interactions between bacteria are determined by steric collisions and near-field lubrication forces. This dominance of short-range forces closely links collective motion in bacterial suspensions to self-organization in driven granular systems, assemblages of biofilaments, and animal flocks. For the scattering of bacteria with surfaces, long-range fluid dynamical interactions are also shown to be negligible before collisions; however, once the bacterium swims along the surface within a few microns after an aligning collision, hydrodynamic effects can contribute to the experimentally observed, long residence times. Because these results are based on purely mechanical properties, they apply to a wide range of microorganisms.

Dynamics of swimming bacteria: transition to directional order at high concentration

At high cell concentrations, bacterial suspensions are known to develop a state of collective swimming (the "zooming bionematic phase," or ZBN) characterized by transient, recurring regions of coordinated motion greatly exceeding the size of individual cells. Recent theoretical studies of semidilute suspensions have suggested that long-range hydrodynamic interactions between swimming cells are responsible for long-wavelength instabilities that lead to these patterns, while models appropriate for higher concentrations have suggested that steric interactions between elongated cells play an important role in the self-organization. Using particle imaging velocimetry in well-defined microgeometries, we examine the statistical properties of the transition to the ZBN in suspensions of Bacillus subtilis, with particular emphasis on the distribution of cell swimming speeds and its correlation with orientational order. This analysis reveals a nonmonotonic relationship between mean cell swimming speed and cell concentration, with a minimum occurring near the transition to the ZBN. Regions of high orientational order in the ZBN phase have locally high swimming speeds, while orientationally disordered regions have lower speeds. A model for steric interactions in concentrated suspensions and previous observations on the kinetics of flagellar rebundling associated with changes in swimming direction are used to explain this observation. The necessity of incorporating steric effects on cell swimming in theoretical models is emphasized.

Excitable patterns in active nematics

We analyze a model of mutually propelled filaments suspended in a two-dimensional solvent. The system undergoes a mean-field isotropic-nematic transition for large enough filament concentrations, and the nematic order parameter is allowed to vary in space and time. We show that the interplay between nonuniform nematic order, activity, and flow results in spatially modulated relaxation oscillations, similar to those seen in excitable media. In this regime the dynamics consists of nearly stationary periods separated by "bursts" of activity in which the system is elastically distorted and solvent is pumped throughout. At even higher activity, the dynamics becomes chaotic.

Traffic jams, gliders, and bands in the quest for collective motion of self-propelled particles

We study a simple swarming model on a two-dimensional lattice where the self-propelled particles exhibit a tendency to align ferromagnetically. Volume exclusion effects are present: particles can only hop to a neighboring node if the node is empty. Here we show that such effects lead to a surprisingly rich variety of self-organized spatial patterns. As particles exhibit an increasingly higher tendency to align to neighbors, they first self-segregate into disordered particle aggregates. Aggregates turn into traffic jams. Traffic jams evolve toward gliders, triangular high density regions that migrate in a well-defined direction. Maximum order is achieved by the formation of elongated high density regions--bands--that transverse the entire system. Numerical evidence suggests that below the percolation density the phase transition associated with orientational order is of first order, while at full occupancy it is of second order.

Polar patterns of driven filaments

The emergence of collective motion exhibited by systems ranging from flocks of animals to self-propelled microorganisms to the cytoskeleton is a ubiquitous and fascinating self-organization phenomenon. Similarities between these systems, such as the inherent polarity of the constituents, a density-dependent transition to ordered phases or the existence of very large density fluctuations, suggest universal principles underlying pattern formation. This idea is followed by theoretical models at all levels of description: micro- or mesoscopic models directly map local forces and interactions using only a few, preferably simple, interaction rules, and more macroscopic approaches in the hydrodynamic limit rely on the systems' generic symmetries. All these models characteristically have a broad parameter space with a manifold of possible patterns, most of which have not yet been experimentally verified. The complexity of interactions and the limited parameter control of existing experimental systems are major obstacles to our understanding of the underlying ordering principles. Here we demonstrate the emergence of collective motion in a high-density motility assay that consists of highly concentrated actin filaments propelled by immobilized molecular motors in a planar geometry. Above a critical density, the filaments self-organize to form coherently moving structures with persistent density modulations, such as clusters, swirls and interconnected bands. These polar nematic structures are long lived and can span length scales orders of magnitudes larger than their constituents. Our experimental approach, which offers control of all relevant system parameters, complemented by agent-based simulations, allows backtracking of the assembly and disassembly pathways to the underlying local interactions. We identify weak and local alignment interactions to be essential for the observed formation of patterns and their dynamics. The presented minimal polar-pattern-forming system may thus provide new insight into emerging order in the broad class of active fluids and self-propelled particles.

Persistent random motion: uncovering cell migration dynamics

In this paper we study analytically the stick-slip models recently introduced to explain the stochastic migration of free cells. We show that persistent motion of cells of many different types is compatible with stochastic reorientation models which admit an analytical mesoscopic treatment. This is proved by examining and discussing experimental data compiled from different sources in the literature, and by fitting some of these results too. We are able to explain many of the 'apparently complex' migration patterns obtained recently from cell tracking data, like power-law dependences in the mean square displacement or non-Gaussian behavior for the kurtosis and the velocity distributions, which depart from the predictions of the classical Ornstein-Uhlenbeck process.

Swarm behavior of self-propelled rods and swimming flagella

Systems of self-propelled particles are known for their tendency to aggregate and to display swarm behavior. We investigate two model systems: self-propelled rods interacting via volume exclusion and sinusoidally beating flagella embedded in a fluid with hydrodynamic interactions. In the flagella system, beating frequencies are gaussian distributed with a nonzero average. These systems are studied by brownian-dynamics simulations and by mesoscale hydrodynamics simulations, respectively. The clustering behavior is analyzed as the particle density and the environmental or internal noise are varied. By distinguishing three types of cluster-size probability density functions, we obtain a phase diagram of different swarm behaviors. The properties of clusters such as their configuration, lifetime, and average size are analyzed. We find that the swarm behavior of the two systems, characterized by several effective power laws, is very similar. However, a more careful analysis reveals several differences. Clusters of self-propelled rods form due to partially blocked forward motion and are therefore typically wedge shaped. At higher rod density and low noise, a giant mobile cluster appears, in which most rods are mostly oriented toward the center. In contrast, flagella become hydrodynamically synchronized and attract each other; their clusters are therefore more elongated. Furthermore, the lifetime of flagella clusters decays more quickly with cluster size than of rod clusters.

Collective motion of vibrated polar disks

We experimentally study a monolayer of vibrated disks with a built-in polar asymmetry which enables them to move quasibalistically on a large persistence length. Alignment occurs during collisions as a result of self-propulsion and hard core repulsion. Varying the amplitude of the vibration, we observe the onset of large-scale collective motion and the existence of giant number fluctuations with a scaling exponent in agreement with the predicted theoretical value.