Large-scale collective properties of self-propelled rods

Long-range nematic order and anomalous fluctuations in suspensions of swimming filamentous bacteria

We study the collective dynamics of elongated swimmers in a very thin fluid layer by devising long filamentous nontumbling bacteria. The strong confinement induces weak nematic alignment upon collision, which, for large enough density of cells, gives rise to global nematic order. This homogeneous but fluctuating phase, observed on the largest experimentally accessible scale of millimeters, exhibits the properties predicted by standard models for flocking, such as the Vicsek-style model of polar particles with nematic alignment: true long-range nematic order and nontrivial giant number fluctuations.

Emergent pattern formation in an interstitial biofilm

Collective behavior of bacterial colonies plays critical roles in adaptability, survivability, biofilm expansion and infection. We employ an individual-based model of an interstitial biofilm to study emergent pattern formation based on the assumptions that rod-shaped bacteria furrow through a viscous environment and excrete extracellular polymeric substances which bias their rate of motion. Because the bacteria furrow through their environment, the substratum stiffness is a key control parameter behind the formation of distinct morphological patterns. By systematically varying this property (which we quantify with a stiffness coefficient γ), we show that subtle changes in the substratum stiffness can give rise to a stable state characterized by a high degree of local order and long-range pattern formation. The ordered state exhibits characteristics typically associated with bacterial fitness advantages, even though it is induced by changes in environmental conditions rather than changes in biological parameters. Our findings are applicable to a broad range of biofilms and provide insights into the relationship between bacterial movement and their environment, and basic mechanisms behind self-organization of biophysical systems.

Large-Scale Patterns in a Minimal Cognitive Flocking Model: Incidental Leaders, Nematic Patterns, and Aggregates

We study a minimal cognitive flocking model, which assumes that the moving entities navigate using the available instantaneous visual information exclusively. The model consists of active particles, with no memory, that interact by a short-ranged, position-based, attractive force, which acts inside a vision cone (VC), and lack velocity-velocity alignment. We show that this active system can exhibit-due to the VC that breaks Newton's third law-various complex, large-scale, self-organized patterns. Depending on parameter values, we observe the emergence of aggregates or millinglike patterns, the formation of moving-locally polar-files with particles at the front of these structures acting as effective leaders, and the self-organization of particles into macroscopic nematic structures leading to long-ranged nematic order. Combining simulations and nonlinear field equations, we show that position-based active models, as the one analyzed here, represent a new class of active systems fundamentally different from other active systems, including velocity-alignment-based flocking systems. The reported results are of prime importance in the study, interpretation, and modeling of collective motion patterns in living and nonliving active systems.

Mesoscale pattern formation of self-propelled rods with velocity reversal

We study self-propelled particles with velocity reversal interacting by uniaxial (nematic) alignment within a coarse-grained hydrodynamic theory. Combining analytical and numerical continuation techniques, we show that the physics of this active system is essentially controlled by the reversal frequency. In particular, we find that elongated, high-density, ordered patterns, called bands, emerge via subcritical bifurcations from spatially homogeneous states. Our analysis reveals further that the interaction of bands is weakly attractive and, consequently, bands fuse upon collision in analogy with nonequilibrium nucleation processes. Moreover, we demonstrate that a renormalized positive line tension can be assigned to stable bands below a critical reversal rate, beyond which they are transversally unstable. In addition, we discuss the kinetic roughening of bands as well as their nonlinear dynamics close to the threshold of transversal instability. Altogether, the reduction of the multiparticle system onto the dynamics of bands provides a unified framework to understand the emergence and stability of nonequilibrium patterns in this self-propelled particle system. In this regard, our results constitute a proof of principle in favor of the hypothesis in microbiology that velocity reversal of gliding rod-shaped bacteria regulates the transitions between various self-organized patterns observed during the bacterial life cycle.

Generalized Swift-Hohenberg models for dense active suspensions

In describing the physics of living organisms, a mathematical theory that captures the generic ordering principles of intracellular and multicellular dynamics is essential for distinguishing between universal and system-specific features. Here, we compare two recently proposed nonlinear high-order continuum models for active polar and nematic suspensions, which aim to describe collective migration in dense cell assemblies and the ordering processes in ATP-driven microtubule-kinesin networks, respectively. We discuss the phase diagrams of the two models and relate their predictions to recent experiments. The satisfactory agreement with existing experimental data lends support to the hypothesis that non-equilibrium pattern formation phenomena in a wide range of active systems can be described within the same class of higher-order partial differential equations.

Symmetry-breaking phase transitions in highly concentrated semen

New experimental evidence of self-motion of a confined active suspension is presented. Depositing fresh semen sample in an annular shaped microfluidic chip leads to a spontaneous vortex state of the fluid at sufficiently large sperm concentration. The rotation occurs unpredictably clockwise or counterclockwise and is robust and stable. Furthermore, for highly active and concentrated semen, richer dynamics can occur such as self-sustained or damped rotation oscillations. Experimental results obtained with systematic dilution provide a clear evidence of a phase transition towards collective motion associated with local alignment of spermatozoa akin to the Vicsek model. A macroscopic theory based on previously derived self-organized hydrodynamics models is adapted to this context and provides predictions consistent with the observed stationary motion.

Instabilities, defects, and defect ordering in an overdamped active nematic

We consider a phenomenological continuum theory for an extensile, overdamped active nematic liquid crystal, applicable in the dense regime. Constructed from general principles, the theory is universal, with parameters independent of any particular microscopic realization. We show that it exhibits two distinct instabilities, one of which arises due to shear forces, and the other due to active torques. Both lead to the proliferation of defects. We focus on the active torque bend instability and find three distinct nonequilibrium steady states including a defect-ordered nematic in which +½ disclinations develop polar ordering. We characterize the phenomenology of these phases and identify the relationship of this theoretical description to experimental realizations and other theoretical models of active nematics.

Active Curved Polymers Form Vortex Patterns on Membranes

Recent in vitro experiments with FtsZ polymers show self-organization into different dynamic patterns, including structures reminiscent of the bacterial Z ring. We model FtsZ polymers as active particles moving along chiral, circular paths by Brownian dynamics simulations and a Boltzmann approach. Our two conceptually different methods point to a generic phase behavior. At intermediate particle densities, we find self-organization into vortex structures including closed rings. Moreover, we show that the dynamics at the onset of pattern formation is described by a generalized complex Ginzburg-Landau equation.

Spontaneous chiral symmetry breaking in collective active motion

Chiral symmetry breaking is ubiquitous in biological systems, from DNA to bacterial suspensions. A key unresolved problem is how chiral structures may spontaneously emerge from achiral interactions. We study a simple model of active swimmers in three dimensions that effectively incorporates hydrodynamic interactions. We perform large-scale molecular dynamics simulations (up to 10(6) particles) and find long-lived metastable collective states that exhibit chiral organization although the interactions are achiral. We elucidate under which conditions these chiral states will emerge and grow to large scales. To explore the complex phase space available to the system, we perform nonequilibrium quenches on a one-dimensional Lebwohl-Lasher model with periodic boundary conditions to study the likelihood of formation of chiral structures.

Role of the active viscosity and self-propelling speed in channel flows of active polar liquid crystals

We study channel flows of active polar liquid crystals (APLCs) focusing on the role played by the active viscosity (β) and the self-propelling speed (ω) on the formation and long time evolution of spontaneous flows using a continuum model. First, we study the onset of spontaneous flows by carrying out a linear stability analysis on two special steady states subject to various physical boundary conditions. We identify a single parameter b1, proportional to a linear combination of the active viscosity and the self-propelling speed, and inversely proportional to a Frank elastic constant, the solvent viscosity, and the liquid crystal relaxation time. We show that the active viscosity and the self-propelling speed influence the onset of spontaneous flows through b1 in that for any fixed value of the bulk activity parameter ζ, large enough |b1| can suppress the spontaneous flow. We then follow spontaneous flows in long time to further investigate the role of β and ω on spatial-temporal structures in the nonlinear regime numerically. The numerical study demonstrates a strong correlation between the most unstable eigenfunction obtained from the linear analysis and the terminal steady state or the persistent, traveling wave structure, revealing the genesis of flow and orientational structures in the active matter system. In the nonlinear regime, a nonzero b1 facilitates the formation of traveling waves in the case of boundary anchoring (the Dirichlet boundary condition) so long as the linear stability analysis predicts an onset of spontaneous flows; in the case of the free boundary condition (the Neumann boundary condition), a stable, spatially homogeneous tilted state always emerges in the presence of two active effects. Finally, we note that various fully out-of-plane spatio-temporal structures can emerge in long time dynamics depending on the boundary condition as well as the initial state of the polarity vector field.

Hysteresis, reentrance, and glassy dynamics in systems of self-propelled rods

Nonequilibrium active matter made up of self-driven particles with short-range repulsive interactions is a useful minimal system to study active matter as the system exhibits collective motion and nonequilibrium order-disorder transitions. We studied high-aspect-ratio self-propelled rods over a wide range of packing fractions and driving to determine the nonequilibrium state diagram and dynamic properties. Flocking and nematic-laning states occupy much of the parameter space. In the flocking state, the average internal pressure is high and structural and mechanical relaxation times are long, suggesting that rods in flocks are in a translating glassy state despite overall flock motion. In contrast, the nematic-laning state shows fluidlike behavior. The flocking state occupies regions of the state diagram at both low and high packing fraction separated by nematic-laning at low driving and a history-dependent region at higher driving; the nematic-laning state transitions to the flocking state for both compression and expansion. We propose that the laning-flocking transitions are a type of glass transition that, in contrast to other glass-forming systems, can show fluidization as density increases. The fluid internal dynamics and ballistic transport of the nematic-laning state may promote collective dynamics of rod-shaped micro-organisms.

Nontoxic colloidal particles impede antibiotic resistance of swarming bacteria by disrupting collective motion and speed

A monolayer of swarming B. subtilis on semisolid agar is shown to display enhanced resistance against antibacterial drugs due to their collective behavior and motility. The dynamics of swarming motion, visualized in real time using time-lapse microscopy, prevents the bacteria from prolonged exposure to lethal drug concentrations. The elevated drug resistance is significantly reduced when the collective motion of bacteria is judiciously disrupted using nontoxic polystyrene colloidal particles immobilized on the agar surface. The colloidal particles block and hinder the motion of the cells, and force large swarming rafts to break up into smaller packs in order to maneuver across narrow spaces between densely packed particles. In this manner, cohesive rafts rapidly lose their collectivity, speed, and group dynamics, and the cells become vulnerable to the drugs. The antibiotic resistance capability of swarming B. subtilis is experimentally observed to be negatively correlated with the number density of colloidal particles on the engineered surface. This relationship is further tested using an improved self-propelled particle model that takes into account interparticle alignment and hard-core repulsion. This work has pertinent implications on the design of optimal methods to treat drug resistant bacteria commonly found in swarming colonies.

Depletion force induced collective motion of microtubules driven by kinesin

Collective motion is a fascinating example of coordinated behavior of self-propelled objects, which is often associated with the formation of large scale patterns. Nowadays, the in vitro gliding assay is being considered a model system to experimentally investigate various aspects of group behavior and pattern formation by self-propelled objects. In the in vitro gliding assay, cytoskeletal filaments F-actin or microtubules are driven by the surface immobilized associated biomolecular motors myosin or dynein respectively. Although the F-actin/myosin or microtubule/dynein system was found to be promising in understanding the collective motion and pattern formation by self-propelled objects, the most widely used biomolecular motor system microtubule/kinesin could not be successfully employed so far in this regard. Failure in exhibiting collective motion by kinesin driven microtubules is attributed to the intrinsic properties of kinesin, which was speculated to affect the behavior of individual gliding microtubules and mutual interactions among them. In this work, for the first time, we have demonstrated the collective motion of kinesin driven microtubules by regulating the mutual interaction among the gliding microtubules, by employing a depletion force among them. Proper regulation of the mutual interaction among the gliding microtubules through the employment of the depletion force was found to allow the exhibition of collective motion and stream pattern formation by the microtubules. This work offers a universal means for demonstrating the collective motion using the in vitro gliding assay of biomolecular motor systems and will help obtain a meticulous understanding of the fascinating coordinated behavior and pattern formation by self-propelled objects.

Mesoscopic turbulence and local order in Janus particles self-propelling under an ac electric field

To elucidate mechanisms of mesoscopic turbulence exhibited by active particles, we experimentally study turbulent states of nonliving self-propelled particles. We realize an experimental system with dense suspensions of asymmetrical colloidal particles (Janus particles) self-propelling on a two-dimensional surface under an ac electric field. Velocity fields of the Janus particles in the crowded situation can be regarded as a sort of turbulence because it contains many vortices and their velocities change abruptly. Correlation functions of their velocity field reveal the coexistence of polar alignment and antiparallel alignment interactions, which is considered to trigger mesoscopic turbulence. Probability distributions of local order parameters for polar and nematic orders indicate the formation of local clusters with particles moving in the same direction. A broad peak in the energy spectrum of the velocity field appears at the spatial scales where the polar alignment and the cluster formation are observed. Energy is injected at the particle scale and conserved quantities such as energy could be cascading toward the larger clusters.

Comparison between Smoluchowski and Boltzmann approaches for self-propelled rods

Considering systems of self-propelled polar particles with nematic interactions ("rods"), we compare the continuum equations describing the evolution of polar and nematic order parameters, derived either from Smoluchowski or Boltzmann equations. Our main goal is to understand the discrepancies between the continuum equations obtained so far in both frameworks. We first show that, in the simple case of point-like particles with only alignment interactions, the continuum equations obtained have the same structure in both cases. We further study, in the Smoluchowski framework, the case where an interaction force is added on top of the aligning torque. This clarifies the origin of the additional terms obtained in previous works. Our observations lead us to emphasize the need for a more involved closure scheme than the standard normal form of the distribution when dealing with active systems.

Kinetic attractor phase diagrams of active nematic suspensions: the dilute regime

Large-scale simulations by the authors of the kinetic-hydrodynamic equations for active polar nematics revealed a variety of spatio-temporal attractors, including steady and unsteady, banded (1d) and cellular (2d) spatial patterns. These particle scale activation-induced attractors arise at dilute nanorod volume fractions where the passive equilibrium phase is isotropic, whereas all previous model simulations have focused on the semi-dilute, nematic equilibrium regime and mostly on low-moment orientation tensor and polarity vector models. Here we extend our previous results to complete attractor phase diagrams for active nematics, with and without an explicit polar potential, to map out novel spatial and dynamic transitions, and to identify some new attractors, over the parameter space of dilute nanorod volume fraction and nanorod activation strength. The particle-scale activation parameter corresponds experimentally to a tunable force dipole strength (so-called pushers with propulsion from the rod tail) generated by active rod macromolecules, e.g., catalysis with the solvent phase, ATP-induced propulsion, or light-activated propulsion. The simulations allow 2d spatial variations in all flow and orientational variables and full spherical orientational degrees of freedom; the attractors correspond to numerical integration of a coupled system of 125 nonlinear PDEs in 2d plus time. The phase diagrams with and without the polar interaction potential are remarkably similar, implying that polar interactions among the rodlike particles are not essential to long-range spatial and temporal correlations in flow, polarity, and nematic order. As a general rule, above a threshold, low volume fractions induce 1d banded patterns, whereas higher yet still dilute volume fractions yield 2d patterns. Again as a general rule, varying activation strength at fixed volume fraction induces novel dynamic transitions. First, stationary patterns saturate the instability of the isotropic state, consisting of discrete 1d banded or 2d cellular patterns depending on nanorod volume fraction. Increasing activation strength further induces a sequence of attractor bifurcations, including oscillations superimposed on the 1d and 2d stationary patterns, a uniform translational motion of 1d and 2d oscillating patterns, and periodic switching between 1d and 2d patterns. These results imply that active macromolecular suspensions are capable of long-range spatial and dynamic organization at isotropic equilibrium concentrations, provided particle-scale activation is sufficiently strong.

Mechanism for Collective Cell Alignment in Myxococcus xanthus Bacteria

Myxococcus xanthus cells self-organize into aligned groups, clusters, at various stages of their lifecycle. Formation of these clusters is crucial for the complex dynamic multi-cellular behavior of these bacteria. However, the mechanism underlying the cell alignment and clustering is not fully understood. Motivated by studies of clustering in self-propelled rods, we hypothesized that M. xanthus cells can align and form clusters through pure mechanical interactions among cells and between cells and substrate. We test this hypothesis using an agent-based simulation framework in which each agent is based on the biophysical model of an individual M. xanthus cell. We show that model agents, under realistic cell flexibility values, can align and form cell clusters but only when periodic reversals of cell directions are suppressed. However, by extending our model to introduce the observed ability of cells to deposit and follow slime trails, we show that effective trail-following leads to clusters in reversing cells. Furthermore, we conclude that mechanical cell alignment combined with slime-trail-following is sufficient to explain the distinct clustering behaviors observed for wild-type and non-reversing M. xanthus mutants in recent experiments. Our results are robust to variation in model parameters, match the experimentally observed trends and can be applied to understand surface motility patterns of other bacterial species.

Self-propelled rods exhibit a phase-separated state characterized by the presence of active stresses and the ejection of polar clusters

We study collections of self-propelled rods (SPR) moving in two dimensions for packing fractions less than or equal to 0.3. We find that in the thermodynamical limit the SPR undergo a phase transition between a disordered gas and a novel phase-separated system state. Interestingly, (global) orientational order patterns-contrary to what has been suggested-vanish in this limit. In the found novel state, the SPR self-organize into a highly dynamical, high-density, compact region-which we call aggregate-which is surrounded by a disordered gas. Active stresses build inside aggregates as a result of the combined effect of local orientational order and active forces. This leads to the most distinctive feature of these aggregates: constant ejection of polar clusters of SPR. This novel phase-separated state represents a novel state of matter characterized by large fluctuations in volume and shape, related to mass ejection, and exhibits positional as well as orientational local order. SPR systems display new physics unseen in other active matter systems.

Anomalous segregation dynamics of self-propelled particles

A number of novel experimental and theoretical results have recently been obtained on active soft matter, demonstrating the various interesting universal and anomalous features of this kind of driven systems. Here we consider the adhesion difference-driven segregation of actively moving units, a fundamental but still poorly explored aspect of collective motility. In particular, we propose a model in which particles have a tendency to adhere through a mechanism which makes them both stay in touch and synchronize their direction of motion - but the interaction is limited to particles of the same kind. The calculations corresponding to the related differential equations can be made in parallel, thus a powerful GPU card allows large scale simulations. We find that in a very large system of particles, interacting without explicit alignment rule, three basic segregation regimes seem to exist as a function of time: i) at the beginning the time dependence of the correlation length is analogous to that predicted by the Cahn-Hillard theory, ii) next rapid segregation occurs characterized with a separation of the different kinds of units being faster than any previously suggested speed, finally, iii) the growth of the characteristic sizes in the system slows down due to a new regime in which self-confined, rotating, splitting and re-joining clusters appear. Our results can explain recent observations of segregating tissue cells in vitro.

Physics of microswimmers--single particle motion and collective behavior: a review

Locomotion and transport of microorganisms in fluids is an essential aspect of life. Search for food, orientation toward light, spreading of off-spring, and the formation of colonies are only possible due to locomotion. Swimming at the microscale occurs at low Reynolds numbers, where fluid friction and viscosity dominates over inertia. Here, evolution achieved propulsion mechanisms, which overcome and even exploit drag. Prominent propulsion mechanisms are rotating helical flagella, exploited by many bacteria, and snake-like or whip-like motion of eukaryotic flagella, utilized by sperm and algae. For artificial microswimmers, alternative concepts to convert chemical energy or heat into directed motion can be employed, which are potentially more efficient. The dynamics of microswimmers comprises many facets, which are all required to achieve locomotion. In this article, we review the physics of locomotion of biological and synthetic microswimmers, and the collective behavior of their assemblies. Starting from individual microswimmers, we describe the various propulsion mechanism of biological and synthetic systems and address the hydrodynamic aspects of swimming. This comprises synchronization and the concerted beating of flagella and cilia. In addition, the swimming behavior next to surfaces is examined. Finally, collective and cooperate phenomena of various types of isotropic and anisotropic swimmers with and without hydrodynamic interactions are discussed.

Collective motion of self-propelled particles with memory

We show that memory, in the form of underdamped angular dynamics, is a crucial ingredient for the collective properties of self-propelled particles. Using Vicsek-style models with an Ornstein-Uhlenbeck process acting on angular velocity, we uncover a rich variety of collective phases not observed in usual overdamped systems, including vortex lattices and active foams. In a model with strictly nematic interactions the smectic arrangement of Vicsek waves giving rise to global polar order is observed. We also provide a calculation of the effective interaction between vortices in the case where a telegraphic noise process is at play, explaining thus the emergence and structure of the vortex lattices observed here and in motility assay experiments.

Keeping speed and distance for aligned motion

The cohesive collective motion (flocking, swarming) of autonomous agents is ubiquitously observed and exploited in both natural and man-made settings, thus, minimal models for its description are essential. In a model with continuous space and time we find that if two particles arrive symmetrically in a plane at a large angle, then (i) radial repulsion and (ii) linear self-propelling toward a fixed preferred speed are sufficient for them to depart at a smaller angle. For this local gain of momentum explicit velocity alignment is not necessary, nor are adhesion or attraction, inelasticity or anisotropy of the particles, or nonlinear drag. With many particles obeying these microscopic rules of motion we find that their spatial confinement to a square with periodic boundaries (which is an indirect form of attraction) leads to stable macroscopic ordering. As a function of the strength of added noise we see--at finite system sizes--a critical slowing down close to the order-disorder boundary and a discontinuous transition. After varying the density of particles at constant system size and varying the size of the system with constant particle density we predict that in the infinite system size (or density) limit the hysteresis loop disappears and the transition becomes continuous. We note that animals, humans, drones, etc., tend to move asynchronously and are often more responsive to motion than positions. Thus, for them velocity-based continuous models can provide higher precision than coordinate-based models. An additional characteristic and realistic feature of the model is that convergence to the ordered state is fastest at a finite density, which is in contrast to models applying (discontinuous) explicit velocity alignments and discretized time. To summarize, we find that the investigated model can provide a minimal description of flocking.

Tunable dynamics of microtubule-based active isotropic gels

We investigate the dynamics of an active gel of bundled microtubules (MTs) that is driven by clusters of kinesin molecular motors. Upon the addition of ATP, the coordinated action of thousands of molecular motors drives the gel to a highly dynamical turbulent-like state that persists for hours and is only limited by the stability of constituent proteins and the availability of the chemical fuel. We characterize how enhanced transport and emergent macroscopic flows of active gels depend on relevant molecular parameters, including ATP, kinesin motor and depletant concentrations, MT volume fraction, as well as the stoichiometry of the constituent motor clusters. Our results show that the dynamical and structural properties of MT-based active gels are highly tunable. They also indicate existence of an optimal concentration of molecular motors that maximize far-from-equilibrium activity of active isotropic MT gels.

Dynamics of a homogeneous active dumbbell system

We analyze the dynamics of a two-dimensional system of interacting active dumbbells. We characterize the mean-square displacement, linear response function, and deviation from the equilibrium fluctuation-dissipation theorem as a function of activity strength, packing fraction, and temperature for parameters such that the system is in its homogeneous phase. While the diffusion constant in the last diffusive regime naturally increases with activity and decreases with packing fraction, we exhibit an intriguing nonmonotonic dependence on the activity of the ratio between the finite-density and the single-particle diffusion constants. At fixed packing fraction, the time-integrated linear response function depends nonmonotonically on activity strength. The effective temperature extracted from the ratio between the integrated linear response and the mean-square displacement in the last diffusive regime is always higher than the ambient temperature, increases with increasing activity, and, for small active force, monotonically increases with density while for sufficiently high activity it first increases and next decreases with the packing fraction. We ascribe this peculiar effect to the existence of finite-size clusters for sufficiently high activity and density at the fixed (low) temperatures at which we worked. The crossover occurs at lower activity or density the lower the external temperature. The finite-density effective temperature is higher (lower) than the single dumbbell one below (above) a crossover value of the Péclet number.

Phase separation and emergent structures in an active nematic fluid

We consider a phenomenological continuum theory for an active nematic fluid and show that there exists a universal, model-independent instability which renders the homogeneous nematic state unstable to order fluctuations. Using numerical and analytic tools we show that, in the vicinity of a critical point, this instability leads to a phase-separated state in which the ordered regions form bands in which the direction of nematic order is perpendicular to the direction of the density gradient. We argue that the underlying mechanism that leads to this phase separation is a universal feature of active fluids of different symmetries.

Collision dynamics of traveling bands in systems of deformable self-propelled particles

We study the soliton-like character of traveling bands in systems of interacting deformable self-propelled particles in two dimensions. The collision dynamics of the model in which migration velocity increases with increasing local density is investigated numerically by changing the relaxation rate of deformations. The bands are unstable upon head-on collisions for larger relaxation rates. This clearly indicates that deformability plays a crucial role of the soliton-like behavior.

Interfacial mechanisms in active emulsions

Active emulsions, i.e., emulsions whose droplets perform self-propelled motion, are of tremendous interest for mimicking collective phenomena in biological populations such as phytoplankton and bacterial colonies, but also for experimentally studying rheology, pattern formation, and phase transitions in systems far from thermal equilibrium. For fuelling such systems, molecular processes involving the surfactants which stabilize the emulsions are a straightforward concept. We outline and compare two different types of reactions, one which chemically modifies the surfactant molecules, the other which transfers them into a different colloidal state. While in the first case symmetry breaking follows a standard linear instability, the second case turns out to be more complex. Depending on the dissolution pathway, there is either an intrinsically nonlinear instability, or no symmetry breaking at all (and hence no locomotion).

A model of filamentous cyanobacteria leading to reticulate pattern formation

The filamentous cyanobacterium, Pseudanabaena, has been shown to produce reticulate patterns that are thought to be the result of its gliding motility. Similar fossilized structures found in the geological record constitute some of the earliest signs of life on Earth. It is difficult to tie these fossils, which are billions of years old, directly to the specific microorganisms that built them. Identifying the physicochemical conditions and microorganism properties that lead microbial mats to form macroscopic structures can lead to a better understanding of the conditions on Earth at the dawn of life. In this article, a cell-based model is used to simulate the formation of reticulate patterns in cultures of Pseudanabaena. A minimal system of long and flexible trichomes capable of gliding motility is shown to be sufficient to produce stable patterns consisting of a network of streams. Varying model parameters indicate that systems with little to no cohesion, high trichome density and persistent movement are conducive to reticulate pattern formation, in conformance with experimental observations.

Large-scale chaos and fluctuations in active nematics

We show that dry active nematics, e.g., collections of shaken elongated granular particles, exhibit large-scale spatiotemporal chaos made of interacting dense, ordered, bandlike structures in a parameter region including the linear onset of nematic order. These results are obtained from the study of both the well-known (deterministic) hydrodynamic equations describing these systems and of the self-propelled particle model they were derived from. We prove, in particular, that the chaos stems from the generic instability of the band solution of the hydrodynamic equations. Revisiting the status of the strong fluctuations and long-range correlations in the particle model, we show that the giant number fluctuations observed in the chaotic phase are a trivial consequence of density segregation. However anomalous, curvature-driven number fluctuations are present in the homogeneous quasiordered nematic phase and characterized by a nontrivial scaling exponent.

Topological structure dynamics revealing collective evolution in active nematics

Topological defects frequently emerge in active matter like bacterial colonies, cytoskeleton extracts on substrates, self-propelled granular or colloidal layers and so on, but their dynamical properties and the relations to large-scale organization and fluctuations in these active systems are seldom touched. Here we reveal, through a simple model for active nematics using self-driven hard elliptic rods, that the excitation, annihilation and transportation of topological defects differ markedly from those in non-active media. These dynamical processes exhibit strong irreversibility in active nematics in the absence of detailed balance. Moreover, topological defects are the key factors in organizing large-scale dynamic structures and collective flows, resulting in multi-spatial temporal effects. These findings allow us to control the self-organization of active matter through topological structures.

Topological defects are observed in a range of active systems, but their dynamical properties are largely unknown. Here, the authors use a simulation of self-propelled hard-rods to generate topological defects in active nematics, finding that their anomalous dynamics may lead to large-scale collective motions.

Fluid flows created by swimming bacteria drive self-organization in confined suspensions

Concentrated suspensions of swimming microorganisms and other forms of active matter are known to display complex, self-organized spatiotemporal patterns on scales that are large compared with those of the individual motile units. Despite intensive experimental and theoretical study, it has remained unclear the extent to which the hydrodynamic flows generated by swimming cells, rather than purely steric interactions between them, drive the self-organization. Here we use the recent discovery of a spiral-vortex state in confined suspensions of Bacillus subtilis to study this issue in detail. Those experiments showed that if the radius of confinement in a thin cylindrical chamber is below a critical value, the suspension will spontaneously form a steady single-vortex state encircled by a counter-rotating cell boundary layer, with spiral cell orientation within the vortex. Left unclear, however, was the flagellar orientation, and hence the cell swimming direction, within the spiral vortex. Here, using a fast simulation method that captures oriented cell-cell and cell-fluid interactions in a minimal model of discrete particle systems, we predict the striking, counterintuitive result that in the presence of collectively generated fluid motion, the cells within the spiral vortex actually swim upstream against those flows. This prediction is then confirmed by the experiments reported here, which include measurements of flagella bundle orientation and cell tracking in the self-organized state. These results highlight the complex interplay between cell orientation and hydrodynamic flows in concentrated suspensions of microorganisms.

Clustering and heterogeneous dynamics in a kinetic Monte Carlo model of self-propelled hard disks

We introduce a kinetic Monte Carlo model for self-propelled hard disks to capture with minimal ingredients the interplay between thermal fluctuations, excluded volume, and self-propulsion in large assemblies of active particles. We analyze in detail the resulting (density, self-propulsion) nonequilibrium phase diagram over a broad range of parameters. We find that purely repulsive hard disks spontaneously aggregate into fractal clusters as self-propulsion is increased and rationalize the evolution of the average cluster size by developing a kinetic model of reversible aggregation. As density is increased, the nonequilibrium clusters percolate to form a ramified structure reminiscent of a physical gel. We show that the addition of a finite amount of noise is needed to trigger a nonequilibrium phase separation, showing that demixing in active Brownian particles results from a delicate balance between noise, interparticle interactions, and self-propulsion. We show that self-propulsion has a profound influence on the dynamics of the active fluid. We find that the diffusion constant has a nonmonotonic behavior as self-propulsion is increased at finite density and that activity produces strong deviations from Fickian diffusion that persist over large time scales and length scales, suggesting that systems of active particles generically behave as dynamically heterogeneous systems.

The role of neighbours selection on cohesion and order of swarms

We introduce a multi-agent model for exploring how selection of neighbours determines some aspects of order and cohesion in swarms. The model algorithm states that every agents' motion seeks for an optimal distance from the nearest topological neighbour encompassed in a limited attention field. Despite the great simplicity of the implementation, varying the amplitude of the attention landscape, swarms pass from cohesive and regular structures towards fragmented and irregular configurations. Interestingly, this movement rule is an ideal candidate for implementing the selfish herd hypothesis which explains aggregation of alarmed group of social animals.

Gait synchronization in Caenorhabditis elegans

Collective motion is observed in swarms of swimmers of various sizes, ranging from self-propelled nanoparticles to fish. The mechanisms that govern interactions among individuals are debated, and vary from one species to another. Although the interactions among relatively large animals, such as fish, are controlled by their nervous systems, the interactions among microorganisms, which lack nervous systems, are controlled through physical and chemical pathways. Little is known, however, regarding the mechanism of collective movements in microscopic organisms with nervous systems. To attempt to remedy this, we studied collective swimming behavior in the nematode Caenorhabditis elegans, a microorganism with a compact nervous system. We evaluated the contributions of hydrodynamic forces, contact forces, and mechanosensory input to the interactions among individuals. We devised an experiment to examine pair interactions as a function of the distance between the animals and observed that gait synchronization occurred only when the animals were in close proximity, independent of genes required for mechanosensation. Our measurements and simulations indicate that steric hindrance is the dominant factor responsible for motion synchronization in C. elegans, and that hydrodynamic interactions and genotype do not play a significant role. We infer that a similar mechanism may apply to other microscopic swimming organisms and self-propelled particles.

Transport powered by bacterial turbulence

We demonstrate that collective turbulentlike motion in a bacterial bath can power and steer the directed transport of mesoscopic carriers through the suspension. In our experiments and simulations, a microwedgelike "bulldozer" draws energy from a bacterial bath of varied density. We obtain that an optimal transport speed is achieved in the turbulent state of the bacterial suspension. This apparent rectification of random motion of bacteria is caused by polar ordered bacteria inside the cusp region of the carrier, which is shielded from the outside turbulent fluctuations.

Synchronization and liquid crystalline order in soft active fluids

We introduce a phenomenological theory for a new class of soft active fluids with the ability to synchronize. Our theoretical framework describes the macroscopic behavior of a collection of interacting anisotropic elements with cyclic internal dynamics and a periodic phase variable. This system can (i) spontaneously undergo a transition to a state with macroscopic orientational order, with the elements aligned, a liquid crystal, (ii) attain another broken symmetry state characterized by synchronization of their phase variables, or (iii) a combination of both types of order. We derive the equations describing a spatially homogeneous system and also study the hydrodynamic fluctuations of the soft modes in some of the ordered states. We find that synchronization can promote or inhibit the transition to a state with orientational order, and vice versa. We provide an explicit microscopic realization: a suspension of microswimmers driven by cyclic strokes.

Dynamical maximum entropy approach to flocking

We derive a new method to infer from data the out-of-equilibrium alignment dynamics of collectively moving animal groups, by considering the maximum entropy model distribution consistent with temporal and spatial correlations of flight direction. When bird neighborhoods evolve rapidly, this dynamical inference correctly learns the parameters of the model, while a static one relying only on the spatial correlations fails. When neighbors change slowly and the detailed balance is satisfied, we recover the static procedure. We demonstrate the validity of the method on simulated data. The approach is applicable to other systems of active matter.

Dynamics of self-organized rotating spiral-coils in bacterial swarms

Self-propelled particles (SPP) exhibit complex collective motions, mimicking autonomous behaviors that are often seen in the natural world, but essentially are generated by simple mutual interactions. Previous research on SPP systems focuses on collective behaviors of a uniform population. However, very little is known about the evolution of individual particles under the same global influence. Here we show self-organized rotating spiral coils in a two-dimensional (2D) active system. By using swarming bacteria Vibrio alginolyticus as an ideal experimental realization of a well-controlled 2D self-propelled system, we study the interaction between ultra-long cells and short background active cells. The self-propulsion of long cells and their interactions with neighboring short cells leads to a self-organized, stable spiral rotational state in 2D. We find four types of spiral coils with two main features: the rotating direction (clockwise or counter-clockwise) and the central structure (single or double spiral). The body length of the spiral coils falls between 32 and 296 μm and their rotational speed is within a range from 2.22 to 22.96 rad s(-1). The dynamics of these spiral coils involves folding and unfolding processes, which require local velocity changes of the long bacterium. This phenomenon can be qualitatively replicated by a Brownian dynamics simulation using a simple rule of the propulsion thrust, imitating the reorientation of bacterial flagella. Apart from the physical and biological interests in swarming cells, the formation of self-organized spiral coils could be useful for the next generation of microfabrication.

Self-organized vortices of circling self-propelled particles and curved active flagella

Self-propelled pointlike particles move along circular trajectories when their translocation velocity is constant and the angular velocity related to their orientation vector is also constant. We investigate the collective behavior of ensembles of such circle swimmers by Brownian dynamics simulations. If the particles interact via a "velocity-trajectory coordination" rule within neighboring particles, a self-organized vortex pattern emerges. This vortex pattern is characterized by its particle-density correlation function Gρ, the density correlation function Gc of trajectory centers, and an order parameter S representing the degree of the aggregation of the particles. Here we systematically vary the system parameters, such as the particle density and the interaction range, in order to reveal the transition of the system from a light-vortex-dominated to heavy-vortex-dominated state, where vortices contain mainly a single and many self-propelled particles, respectively. We also study a semidilute solution of curved, sinusoidal-beating flagella, as an example of circling self-propelled particles with explicit propulsion mechanism and excluded-volume interactions. Our simulation results are compared with previous experimental results for the vortices in sea-urchin sperm solutions near a wall. The properties of the vortices in simulations and experiments are found to agree quantitatively.

Formation and collision of traveling bands in interacting deformable self-propelled particles

We study the collective dynamics of interacting deformable self-propelled particles whose migration velocity increases with increasing local density. In two-dimensional numerical simulations of this system, the local density dependence on migration velocity leads to traveling bands similar to those previously reported for Vicsek-type models. We show that a pair of straight bands moving in opposite directions survives a head-on collision. Although traveling bands also appear in systems of constant migration velocity subjected to random noise, they are found to be unstable in a head-on collision.

Flocking with minimal cooperativity: the panic model

We present a two-dimensional lattice model of self-propelled spins that can change direction only upon collision with another spin. We show that even with ballistic motion and minimal cooperativity, these spins display robust flocking behavior at nearly all densities, forming long bands of stripes. The structural transition in this system is not a thermodynamic phase transition, but it can still be characterized by an order parameter, and we demonstrate that if this parameter is studied as a dynamical variable rather than a steady-state observable, we can extract a detailed picture of how the flocking mechanism varies with density.

Collective behavior of penetrable self-propelled rods in two dimensions

Collective behavior of self-propelled particles is observed on a microscale for swimmers such as sperm and bacteria as well as for protein filaments in motility assays. The properties of such systems depend both on their dimensionality and the interactions between their particles. We introduce a model for self-propelled rods in two dimensions that interact via a separation-shifted Lennard-Jones potential. Due to the finite potential barrier, the rods are able to cross. This model allows us to efficiently simulate systems of self-propelled rods that effectively move in two dimensions but can occasionally escape to the third dimension in order to pass each other. Our quasi-two-dimensional self-propelled particles describe a class of active systems that encompasses microswimmers close to a wall and filaments propelled on a substrate. Using Monte Carlo simulations, we first determine the isotropic-nematic transition for passive rods. Using Brownian dynamics simulations, we characterize cluster formation of self-propelled rods as a function of propulsion strength, noise, and energy barrier. Contrary to rods with an infinite potential barrier, an increase of the propulsion strength does not only favor alignment but also effectively decreases the potential barrier that prevents crossing of rods. We thus find a clustering window with a maximum cluster size at medium propulsion strengths.

Structure and dynamics of self-assembling colloidal monolayers in oscillating magnetic fields

Many fascinating phenomena such as large-scale collective flows, enhanced fluid mixing, and pattern formation have been observed in so-called active fluids, which are composed of particles that can absorb energy and dissipate it into the fluid medium. For active particles immersed in liquids, fluid-mediated viscous stresses can play an important role on the emergence of collective behavior. Here, we experimentally investigate their role in the dynamics of self-assembling magnetically driven colloidal particles which can rapidly form organized hexagonal structures. We find that viscous stresses reduce hexagonal ordering, generate smaller clusters, and significantly decrease the rate of cluster formation, all while holding the system at constant number density. Furthermore, we show that time and length scales of cluster formation depend on the Mason number (Mn), or ratio of viscous to magnetic forces, scaling as t∝Mn and L∝Mn(-1/2). Our results suggest that viscous stresses hinder collective behavior in a self-assembling colloidal system.

Loss of collective motion in swarming bacteria undergoing stress

The collective motion of Bacillus subtilis in the presence of a photosensitizer is disrupted by reactive oxygen species when exposed to light of sufficient dosages and is partially recovered when light irradiation is suspended. The transition from a highly collective to a more random motion is modeled using an improved self-propelled model with alignment rule. The increment in noise level describes the enhanced uncertainty in the motion of swarming bacteria under stress as observed experimentally.

Diffusion, subdiffusion, and trapping of active particles in heterogeneous media

We study the transport properties of a system of active particles moving at constant speed in a heterogeneous two-dimensional space. The spatial heterogeneity is modeled by a random distribution of obstacles, which the active particles avoid. Obstacle avoidance is characterized by the particle turning speed γ. We show, through simulations and analytical calculations, that the mean square displacement of particles exhibits two regimes as function of the density of obstacles ρ(o) and γ. We find that at low values of γ, particle motion is diffusive and characterized by a diffusion coefficient that displays a minimum at an intermediate obstacle density ρ(o). We observe that in high obstacle density regions and for large γ values, spontaneous trapping of active particles occurs. We show that such trapping leads to genuine subdiffusive motion of the active particles. We indicate how these findings can be used to fabricate a filter of active particles.

Invasion-wave-induced first-order phase transition in systems of active particles

An instability near the transition to collective motion of self-propelled particles is studied numerically by Enskog-like kinetic theory. While hydrodynamics breaks down, the kinetic approach leads to steep solitonlike waves. These supersonic waves show hysteresis and lead to an abrupt jump of the global order parameter if the noise level is changed. Thus they provide a mean-field mechanism to change the second-order character of the phase transition to first order. The shape of the wave is shown to follow a scaling law and to quantitatively agree with agent-based simulations.