Collective behavior of penetrable self-propelled rods in two dimensions

Collective dynamics of intelligent active Brownian particles with visual perception and velocity alignment in 3D: spheres, rods, and worms

Many living systems, such as birds and fish, exhibit collective behaviors like flocking and swarming. Recently, an experimental system of active colloidal particles has been developed, where the motility of each particle is adjusted based on its visual detection of surrounding particles. These particles with visual-perception-dependent motility exhibit group formation and cohesion. Inspired by these behaviors, we investigate intelligent active Brownian particles (iABPs) equipped with visual perception and velocity alignment in three dimensions using computer simulations. The visual-perception-based self-steering describes the tendency of iABPs to move toward the center of mass of particles within their visual cones, while velocity alignment encourages alignment with neighboring particles. We examine how the behavior varies with the visual cone angle θ, self-propulsion speed (Péclet number Pe), and the interaction strengths of velocity alignment (Ωa) and visual-based self-steering (Ωv). Our findings show that spherical iABPs form dense clusters, worm-like clusters, milling behaviors, and dilute-gas phases, consistent with 2D studies. By reducing the simulation box size, we observe additional structures like band-like clusters and dense baitball formations. Additionally, rod-like iABPs form band-like, worm-like, radiating, and helical structures, while iABP worms exhibit band-like, streamlined, micellar-like and entangled structures. Many of these patterns resemble collective behaviors in nature, such as ant milling, fish baitballs, and worm clusters. Advances in synthetic techniques could enable nanorobots with similar capabilities, offering insights into multicellular systems through active matter.

Phase behavior and dynamics of active Brownian particles in an alignment field

Self-propelled particles that are subject to noise are a well-established generic model system for active matter. A homogeneous alignment field can be used to orient the direction of the self-propulsion velocity and to model systems like phoretic Janus particles with a magnetic dipole moment or magnetotactic bacteria in an external magnetic field. Computer simulations are used to predict the phase behavior and dynamics of self-propelled Brownian particles in a homogeneous alignment field in two dimensions. Phase boundaries of the gas-liquid coexistence region are calculated for various Péclet numbers, particle densities, and alignment field strengths. Critical points and exponents are calculated and, in agreement with previous simulations, do not seem to belong to the universality class of the 2D Ising model. Finally, the dynamics of spinodal decomposition for quenching the system from the one-phase to the two-phase coexistence region by increasing the Péclet number is characterized. Our results may help to identify parameters for optimal transport of active matter in complex environments.

A computer simulation study of a chiral active ring polymer

We investigate a ring polymer under the influence of chiral active Brownian forces in two dimensions using coarse-grained computer simulations. We observe a non-monotonic behavior of the radius of gyration of an active Brownian ring as a function of active force. However, the shrinkage of the ring in the intermediate strength of active forces becomes more pronounced in the presence of chiral active forces, and the shrinkage is monotonic at a given activity level as a function of the angular frequency controlling the direction of the active force. The distribution of radius of gyration, inter-monomer distance, and radial distribution suggest that the monomers come close to each other, eventually leading to the shrinkage of the ring. Moreover, the bond-correlation suggests that the chirality introduces a local folding of the monomers. Furthermore, using the diameter correlation function, we show that the ring performs tank-treading motion with a frequency following power-law relation with active force with exponent 3/2. The mean squared displacement of the monomers further assists the tank-treading dynamics by exhibiting oscillatory behavior.

The self-organized structure of glioma oncostreams and the disruptive role of passive cells

Oncostreams are self-organized structures formed by spindle-like, elongated, self-propelled cells recently described in glioblastomas and especially in gliosarcomas. Cells within these structures either move as large clusters in one main direction, flocks, or as linear, intermingling collections of cells advancing in opposite directions, streams. Round, passive cells are also observed, either inside or segregated from the oncostreams. Here we generalize a recently formulated particle-field approach to investigate the genesis and evolution of these structures, first showing that, in systems consisting only of identical self-propelled cells, both flocks and streams emerge as self-organized dynamic configurations. Flocks are the more stable configurations, while streams are transient and usually originate in collisions between flocks. Stream degradation is easier at low self-propulsion speeds. In systems consisting of both motile and passive cells, the latter block stream formation and accelerate their degradation and flock stabilization. Since the flock appears to be the most effective invasive structure, we thus argue that a phenotype mixture (motile and passive cells) may favor glioblastoma invasion. hlBy relating cellular properties to the observed outcome, our model shows that oncostreams are self-organized structures that result from the interplay between speed, shape, and steric repulsion.

Designing highly efficient interlocking interactions in anisotropic active particles

Cluster formation of microscopic swimmers is key to the formation of biofilms and colonies, efficient motion and nutrient uptake, but, in the absence of other interactions, requires high swimmer concentrations to occur. Here we experimentally and numerically show that cluster formation can be dramatically enhanced by an anisotropic swimmer shape. We analyze a class of model microswimmers with a shape that can be continuously tuned from spherical to bent and straight rods. In all cases, clustering can be described by Michaelis-Menten kinetics governed by a single scaling parameter that depends on particle density and shape only. We rationalize these shape-dependent dynamics from the interplay between interlocking probability and cluster stability. The bent rod shape promotes assembly in an interlocking fashion even at vanishingly low particle densities and we identify the most efficient shape to be a semicircle. Our work provides key insights into how shape can be used to rationally design out-of-equilibrium self-organization, key to creating active functional materials and processes that require two-component assembly with high fidelity.

Active osmoticlike pressure on permeable inclusions

We use a standard minimal active Brownian model to investigate the osmotic-like effective pressure generated by active fluids on fixed hollow inclusions. These inclusions are enclosed by a permeable (albeit nonflexible) membrane, and the interior and exterior regions of the inclusions have different particle motility strengths. We consider both rectangular and disklike inclusions and analyze the effects of various system parameters, such as excluded volume interaction between active particles, hardness of membrane, and active particle density, on the effective pressure produced on the enclosing membrane. We focus on the range of intermediate to high motility strengths and analyze the effective pressure in the steady state. Our findings for the active pressure produced in the interior and exterior regions of the inclusion indicate that the pressure is higher in the region with lower motility due to the relatively stronger accumulation of active particles.

Effect of local active fluctuations on structure and dynamics of flexible biopolymers

Active fluctuations play a significant role in the structure and dynamics of biopolymers (e.g. chromatin and cytoskeletal proteins) that are instrumental in the functioning of living cells. For a large range of experimentally accessible length and time scales, these polymers can be represented as flexible chains that are subjected to spatially and temporally varying fluctuating forces. In this work, we introduce a mathematical framework that correlates the spatial and temporal patterns of the fluctuations to different observables that describe the dynamics and conformations of the polymer. We demonstrate the power of this approach by analyzing the case of a point fluctuation on the polymer with an exponential decay of correlation in time with a finite time constant. Specifically, we identify the length and time scale over which the behavior of the polymer exhibits a significant departure from the behavior of a Rouse chain and the range of impact of the fluctuation along the chain. Furthermore, we show that the conformation of the polymer retains the memory of the active fluctuation from earlier times. Altogether, this work sets the basis for understanding and interpreting the role of spatio-temporal patterns of fluctuations in the dynamics, conformation, and functionality of biopolymers in living cells.

The crucial role of adhesion in the transmigration of active droplets through interstitial orifices

Active fluid droplets are a class of soft materials exhibiting autonomous motion sustained by an energy supply. Such systems have been shown to capture motility regimes typical of biological cells and are ideal candidates as building-block for the fabrication of soft biomimetic materials of interest in pharmacology, tissue engineering and lab on chip devices. While their behavior is well established in unconstrained environments, much less is known about their dynamics under strong confinement. Here, we numerically study the physics of a droplet of active polar fluid migrating within a microchannel hosting a constriction with adhesive properties, and report evidence of a striking variety of dynamic regimes and morphological features, whose properties crucially depend upon droplet speed and elasticity, degree of confinement within the constriction and adhesiveness to the pore. Our results suggest that non-uniform adhesion forces are instrumental in enabling the crossing through narrow orifices, in contrast to larger gaps where a careful balance between speed and elasticity is sufficient to guarantee the transition. These observations may be useful for improving the design of artificial micro-swimmers, of interest in material science and pharmaceutics, and potentially for cell sorting in microfluidic devices.

Morphologies and dynamics of free surfaces of crystals composed of active particles

Active matter exhibits various collective motions and nonequilibrium phases, such as crystals; however, their surface properties have been poorly explored. Here, we use Brownian dynamics simulations to investigate the surface morphology and dynamics of two-dimensional active crystals during and after growth. For crystal growth on a substrate, the position and roughness of the crystal surface reach steady states at different times. In the steady state, the surface exhibits superdiffusive behaviour at the short time, and the roughness is insensitive to the roughening process and particle activity. We observe two-stage and three-stage surface roughening at different Péclet numbers. The result of dynamic scaling analysis shows that the surface is similar to anomalous roughening, which is distinct from the normal roughening typically found in conventional passive systems. Capillary wave theory for a thermal equilibrium system can describe the active surface fluctuations only in the long-wavelength regime, indicating that active particles mainly drive the surface out of equilibrium locally. These similarities and differences between the active and passive crystal surfaces are essential for understanding active crystals and interfaces.

Emergent collective dynamics of pusher and puller squirmer rods: swarming, clustering, and turbulence

We study the interplay of steric and hydrodynamic interactions in suspensions of elongated microswimmers by simulating the full hydrodynamics of squirmer rods in the quasi two-dimensional geometry of a Hele-Shaw cell. To create pusher or puller-type squirmer rods, we concentrate the surface slip-velocity field more to the back or to the front of the rod and thereby are able to tune the rod's force-dipole strength. We study a wide range of aspect ratios and area fractions and provide corresponding state diagrams. The flow field of pusher-type squirmer rods destabilizes ordered structures and favors the disordered state at small area fractions and aspect ratios. Only when steric interactions become relevant, we observe a turbulent and dynamic cluster state, while for large aspect ratios a single swarm and jammed cluster occurs. The power spectrum of the turbulent state shows two distinct energy cascades at small and large wave numbers with power-law scaling and non-universal exponents. Pullers show a strong tendency to form swarms instead of the disordered state found for neutral and pusher rods. At large area fractions a dynamic cluster is observed and at larger aspect ratio a single swarm or jammed cluster occurs.

Effect of anisotropy on the formation of active particle films

Active colloids belong to a class of nonequilibrium systems where energy uptake, conversion, and dissipation occur at the level of individual colloidal particles, which can lead to particles' self-propelled motion and surprising collective behavior. Examples include coexistence of vapor- and liquid-like steady states for active particles with repulsive interactions only, phenomena known as motility-induced phase transitions. Similarly to motile unicellular organisms, active colloids tend to accumulate at confining surfaces forming dense adsorbed films. In this work, we study the structure and dynamics of aggregates of self-propelled particles near confining solid surfaces, focusing on the effects of the particle anisotropic interactions. We performed Langevin dynamics simulations of two complementary models for active particles: ellipsoidal particles interacting through the Gay-Berne potential and rodlike particles composed of several repulsive Lennard-Jones beads. We observe a nonmonotonic behavior of the structure of clusters formed along the confining surface as a function of the particle aspect ratio, with a film spreading when particles are near-spherical, compact clusters with hedgehog-like particle orientation for more elongated active particles, and a complex dynamical behavior for an intermediate aspect ratio. The stabilization time of cluster formation along the confining surface also displays a nonmonotonic dependence on the aspect ratio, with a local minimum at intermediate values. Additionally, we demonstrate that the hedgehog-like aggregates formed by Gay-Berne ellipsoids exhibit higher structural stability as compared to the ones formed by purely repulsive active rods, which are stable due to the particle activity only.

Dynamics of active polar ring polymers

The conformational and dynamical properties of isolated semiflexible active polar ring polymers are investigated analytically. A ring is modeled as a continuous Gaussian polymer exposed to tangential active forces. The analytical solution of the linear non-Hermitian equation of motion in terms of an eigenfunction expansion shows that ring conformations are independent of activity. In contrast, activity strongly affects the internal ring dynamics and yields characteristic time regimes, which are absent in passive rings. On intermediate timescales, flexible rings show an activity-enhanced diffusive regime, while semiflexible rings exhibit ballistic motion. Moreover, a second active time regime emerges on longer timescales, where rings display a snake-like motion, which is reminiscent to a tank-treading rotational dynamics in shear flow, dominated by the mode with the longest relaxation time.

Activity-induced polar patterns of filaments gliding on a sphere

Active matter systems feature the ability to form collective patterns as observed in a plethora of living systems, from schools of fish to swimming bacteria. While many of these systems move in a wide, three-dimensional environment, several biological systems are confined by a curved topology. The role played by a non-Euclidean geometry on the self-organization of active systems is not yet fully understood, and few experimental systems are available to study it. Here, we introduce an experimental setup in which actin filaments glide on the inner surface of a spherical lipid vesicle, thus embedding them in a curved geometry. We show that filaments self-assemble into polar, elongated structures and that, when these match the size of the spherical geometry, both confinement and topological constraints become relevant for the emergent patterns, leading to the formation of polar vortices and jammed states. These results experimentally demonstrate that activity-induced complex patterns can be shaped by spherical confinement and topology.

Collective motion of cells modeled as ring polymers

In this article, we use a coarse-grained model of disjoint semi-flexible ring polymers to investigate computationally the spatiotemporal collective behavior of cell colonies. A ring polymer in this model is self-propelled by a motility force along the cell's polarity, which depends on its historical kinetics. Despite the repulsive interaction between the cells, a collective behavior sets in as a result of cells pushing against each other. This cooperative motion emerges as the amplitude of the motility force is increased and/or their areal density is increased. The degree of collectivity, characterized by the average cluster size, the velocity field order parameter, and the polarity field nematic order parameter, is found to increase with increasing the amplitude of the motility force and area coverage of the cells. Furthermore, the degree of alignment exhibited by the cell velocity field within a cluster is found to be stronger than that exhibited by the cell polarity. Comparison between the collective behavior of elongated cells and that of circular cells, at the same area coverage and motility force, shows that elongated cells exhibit a stronger collective behavior than circular cells, in agreement with earlier studies of self-propelled anisotropic particles. An investigation of two-cell collisions shows that while two clustered cells move in tandem, their polarities are misaligned. As such the cells push against each other while moving coherently.

Absolute negative mobility of active polymer chains in steady laminar flows

We investigate the transport of active polymer chains in steady laminar flows in the presence of thermal noise and an external constant force. In the model, the polymer chain is worm-like and is propelled by active forces along its tangent vectors. Compared with inertial Brownian particles, active polymer chains in steady laminar flows exhibit richer movement patterns due to their specific spatial structures. The simulation results show that the velocity-force relation is strongly dependent on the system parameters such as the chain length, bending rigidity, active force and so on. The polymer chain may move in some preferential movement directions and exhibits absolute negative mobility within appropriate parameter regimes, i.e., the polymer chain can move in a direction opposite to the external constant force. In particular, we can observe giant negative mobility in a broad range of parameter regimes.

Active nematic order and dynamic lane formation of microtubules driven by membrane-bound diffusing motors

Active nematics are ordered liquid crystalline fluids that exhibit spontaneous persistent flows and collective dynamics. The development of motile biopolymer systems inspired by nature has recently attracted considerable attention to out-of-equilibrium soft materials. We report the formation of an active nematic in which microtubules are propelled by kinesin motors coupled to a lipid membrane substrate. The system exhibits apolar order in a globally aligned nematic phase and locally ordered dynamic lanes. Use of a fluid substrate represents a significant advance for active matter as it allows for spatial re-organization of motors, which generate force, in response to the dynamics of the aligning microtubules. This self-organized feedback mechanism may have implications in vivo and for engineering efficient dynamic and reconfigurable materials.

Dynamic lane formation and long-range active nematic alignment are reported using a geometry in which kinesin motors are directly coupled to a lipid bilayer, allowing for in-plane motor diffusion during microtubule gliding. We use fluorescence microscopy to image protein distributions in and below the dense two-dimensional microtubule layer, revealing evidence of diffusion-enabled kinesin restructuring within the fluid membrane substrate as microtubules collectively glide above. We find that the lipid membrane acts to promote filament–filament alignment within the gliding layer, enhancing the formation of a globally aligned active nematic state. We also report the emergence of an intermediate, locally ordered state in which apolar dynamic lanes of nematically aligned microtubules migrate across the substrate. To understand this emergent behavior, we implement a continuum model obtained from coarse graining a collection of self-propelled rods, with propulsion set by the local motor kinetics. Tuning the microtubule and kinesin concentrations as well as active propulsion in these simulations reveals that increasing motor activity promotes dynamic nematic lane formation. Simulations and experiments show that, following fluid bilayer substrate mediated spatial motor restructuring, the total motor concentration becomes enriched below the microtubule lanes that they drive, with the feedback leading to more dynamic lanes. Our results have implications for membrane-coupled active nematics in vivo as well as for engineering dynamic and reconfigurable materials where the structural elements and power sources can dynamically colocalize, enabling efficient mechanical work.

Phase separation of self-propelled disks with ferromagnetic and nematic alignment

We present a comprehensive study of a model system of repulsive self-propelled disks in two dimensions with ferromagnetic and nematic velocity alignment interactions. We characterize the phase behavior of the system as a function of the alignment and self-propulsion strength, featuring orientational order for strong alignment and motility-induced phase separation (MIPS) at moderate alignment but high enough self-propulsion. We derive a microscopic theory for these systems yielding a closed set of hydrodynamic equations from which we perform a linear stability analysis of the homogenous disordered state. This analysis predicts MIPS in the presence of aligning torques. The nature of the continuum theory allows for an explicit quantitative comparison with particle-based simulations, which consistently shows that ferromagnetic alignment fosters phase separation, while nematic alignment does not alter either the nature or the location of the instability responsible for it. In the ferromagnetic case, such behavior is due to an increase of the imbalance of the number of particle collisions along different orientations, giving rise to the self-trapping of particles along their self-propulsion direction. On the contrary, the anisotropy of the pair correlation function, which encodes this self-trapping effect, is not significantly affected by nematic torques. Our work shows the predictive power of such microscopic theories to describe complex active matter systems with different interaction symmetries and sheds light on the impact of velocity-alignment interactions in motility-induced phase separation.

Anomalous diffusion in a monolayer of lightweight spheres fluidized in air flow

This paper presents statistical analyses of random motions in a single layer of fluidized lightweight spherical particles. Foam polystyrene spheres were driven by an upward airflow through the sieve mesh, and their two-dimensional motion was acquired using image analysis. In the bulk region, the particle velocity distributions changed from Gaussian to heavy-tailed distribution as the bulk packing fraction ϕ_{b} was increased. The mean square displacement of the particles exhibited transition to subdiffusion at much lower ϕ_{b} than observed in previous studies using similar setup but with heavier particles. A slight superdiffusion and significant growth of the correlation length in the two-body velocity correlation was observed at further large ϕ_{b}. The effect of the wall on the dynamics of the particles was also investigated, and the anisotropy of the granular temperature was found to be a useful index to discriminate between the wall region and the bulk. The turbulence statistics in the wake of a particle indicated a strong wall-normal asymmetry of aerodynamic forcing as the "thermal" agitation in the wall region.

Noncentral forces mediated between two inclusions in a bath of active Brownian rods

Using Brownian Dynamics simulations, we study effective interactions mediated between two identical and impermeable disks (inclusions) immersed in a bath of identical, active (self-propelled), Brownian rods in two spatial dimensions, by assuming that the self-propulsion axis of the rods may generally deviate from their longitudinal axis. When the self-propulsion is transverse (perpendicular to the rod axis), the accumulation of active rods around the inclusions is significantly enhanced, causing a more expansive steric layering (ring formation) of the rods around the inclusions, as compared with the reference case of longitudinally self-propelling rods. As a result, the transversally self-propelling rods also mediate a significantly longer ranged effective interaction between the inclusions. The bath-mediated interaction arises due to the overlaps between the active-rod rings formed around the inclusions, as they are brought into small separations. When the self-propulsion axis is tilted relative to the rod axis, we find an asymmetric imbalance of active-rod accumulation around the inclusion dimer. This leads to a noncentral interaction, featuring an anti-parallel pair of transverse force components and, hence, a bath-mediated torque on the dimer.

Multi-particle collision dynamics with a non-ideal equation of state. II. Collective dynamics of elongated squirmer rods

Simulations of flow fields around microscopic objects typically require methods that both solve the Navier-Stokes equations and also include thermal fluctuations. One such method popular in the field of soft-matter physics is the particle-based simulation method of multi-particle collision dynamics (MPCD). However, in contrast to the typically incompressible real fluid, the fluid of the traditional MPCD methods obeys the ideal-gas equation of state. This can be problematic because most fluid properties strongly depend on the fluid density. In a recent article, we proposed an extended MPCD algorithm and derived its non-ideal equation of state and an expression for the viscosity. In the present work, we demonstrate its accuracy and efficiency for the simulations of the flow fields of single squirmers and of the collective dynamics of squirmer rods. We use two exemplary squirmer-rod systems for which we compare the outcome of the extended MPCD method to the well-established MPCD version with an Andersen thermostat. First, we explicitly demonstrate the reduced compressibility of the MPCD fluid in a cluster of squirmer rods. Second, for shorter rods, we show the interesting result that in simulations with the extended MPCD method, dynamic swarms are more pronounced and have a higher polar order. Finally, we present a thorough study of the state diagram of squirmer rods moving in the center plane of a Hele-Shaw geometry. From a small to large aspect ratio and density, we observe a disordered state, dynamic swarms, a single swarm, and a jammed cluster, which we characterize accordingly.

Cluster and conquer: the morphodynamics of invasion of a compliant substrate by active rods

The colonisation of a soft passive material by motile cells such as bacteria is common in biology. The resulting colonies of the invading cells are often observed to exhibit intricate patterns whose morphology and dynamics can depend on a number of factors, particularly the mechanical properties of the substrate and the motility of the individual cells. We use simulations of a minimal 2D model of self-propelled rods moving through a passive compliant medium consisting of particles that offer elastic resistance before being plastically displaced from their equilibrium positions. It is observed that the clustering of active (self-propelled) particles is crucial for understanding the morphodynamics of colonisation. Clustering enables motile colonies to spread faster than they would have as isolated particles. The colonisation rate depends non-monotonically on substrate stiffness with a distinct maximum at a non-zero value of substrate stiffness. This is observed to be due to a change in the morphology of clusters. Furrow networks created by the active particles have a fractal-like structure whose dimension varies systematically with substrate stiffness but is less sensitive to particle activity. The power-law growth exponent of the furrowed area is smaller than unity, suggesting that, to sustain such extensive furrow networks, colonies must regulate their overall growth rate.

A Review on the Some Issues of Multiphase Flow with Self-Driven Particles

Multiphase flow with self-driven particles is ubiquitous and complex. Exploring the flow properties has both important academic meaning and engineering value. This review emphasizes some recent studies on multiphase flow with self-driven particles: the hydrodynamic interactions between self-propelled/self-rotary particles and passive particles; the aggregation, phase separation and sedimentation of squirmers; the influence of rheological properties on its motion; and the kinematic characteristics of axisymmetric squirmers. Finally, some open problems, challenges, and future directions are highlighted.

Pattern formation and polarity sorting of driven actin filaments on lipid membranes

Pattern formation processes in active systems give rise to a plethora of collective structures. Predicting how the emergent structures depend on the microscopic interactions between the moving agents remains a challenge. By introducing a high-density actin gliding assay on a fluid membrane, we demonstrate the emergence of polar structures in a regime of nematic binary interactions dominated by steric repulsion. The transition from a microscopic nematic symmetry to a macroscopic polar structure is linked to microscopic polarity sorting mechanisms, including accumulation in wedge-like topological defects. Our results should be instrumental for a better understanding of pattern formation and polarity sorting processes in active matter.

Collective motion of active matter is ubiquitously observed, ranging from propelled colloids to flocks of bird, and often features the formation of complex structures composed of agents moving coherently. However, it remains extremely challenging to predict emergent patterns from the binary interaction between agents, especially as only a limited number of interaction regimes have been experimentally observed so far. Here, we introduce an actin gliding assay coupled to a supported lipid bilayer, whose fluidity forces the interaction between self-propelled filaments to be dominated by steric repulsion. This results in filaments stopping upon binary collisions and eventually aligning nematically. Such a binary interaction rule results at high densities in the emergence of dynamic collectively moving structures including clusters, vortices, and streams of filaments. Despite the microscopic interaction having a nematic symmetry, the emergent structures are found to be polar, with filaments collectively moving in the same direction. This is due to polar biases introduced by the stopping upon collision, both on the individual filaments scale as well as on the scale of collective structures. In this context, positive half-charged topological defects turn out to be a most efficient trapping and polarity sorting conformation.

Collective motion of driven semiflexible filaments tuned by soft repulsion and stiffness

In active matter systems, self-propelled particles can self-organize to undergo collective motion, leading to persistent dynamical behavior out of equilibrium. In cells, cytoskeletal filaments and motor proteins form complex structures important for cell mechanics, motility, and division. Collective dynamics of cytoskeletal systems can be reconstituted using filament gliding experiments, in which cytoskeletal filaments are propelled by surface-bound motor proteins. These experiments have observed diverse dynamical states, including flocks, polar streams, swirling vortices, and single-filament spirals. Recent experiments with microtubules and kinesin motor proteins found that the collective behavior of gliding filaments can be tuned by altering the concentration of the crowding macromolecule methylcellulose in solution. Increasing the methylcellulose concentration reduced filament crossing, promoted alignment, and led to a transition from active, isotropically oriented filaments to locally aligned polar streams. This emergence of collective motion is typically explained as an increase in alignment interactions by Vicsek-type models of active polar particles. However, it is not yet understood how steric interactions and bending stiffness modify the collective behavior of active semiflexible filaments. Here we use simulations of driven filaments with tunable soft repulsion and rigidity in order to better understand how the interplay between filament flexibility and steric effects can lead to different active dynamic states. We find that increasing filament stiffness decreases the probability of filament alignment, yet increases collective motion and long-range order, in contrast to the assumptions of a Vicsek-type model. We identify swirling flocks, polar streams, buckling bands, and spirals, and describe the physics that govern transitions between these states. In addition to repulsion and driving, tuning filament stiffness can promote collective behavior, and controls the transition between active isotropic filaments, locally aligned flocks, and polar streams.

A particle-field approach bridges phase separation and collective motion in active matter

Whereas self-propelled hard discs undergo motility-induced phase separation, self-propelled rods exhibit a variety of nonequilibrium phenomena, including clustering, collective motion, and spatio-temporal chaos. In this work, we present a theoretical framework representing active particles by continuum fields. This concept combines the simplicity of alignment-based models, enabling analytical studies, and realistic models that incorporate the shape of self-propelled objects explicitly. By varying particle shape from circular to ellipsoidal, we show how nonequilibrium stresses acting among self-propelled rods destabilize motility-induced phase separation and facilitate orientational ordering, thereby connecting the realms of scalar and vectorial active matter. Though the interaction potential is strictly apolar, both, polar and nematic order may emerge and even coexist. Accordingly, the symmetry of ordered states is a dynamical property in active matter. The presented framework may represent various systems including bacterial colonies, cytoskeletal extracts, or shaken granular media.

Interacting self-propelled particles exhibit phase separation or collective motion depending on particle shape. A unified theory connecting these paradigms represents a major challenge in active matter, which the authors address here by modeling active particles as continuum fields.

Effective interactions mediated between two permeable disks in an active fluid

We study steady-state properties of a bath of active Brownian particles (ABPs) in two dimensions in the presence of two fixed, permeable (hollow) disklike inclusions, whose interior and exterior regions can exhibit mismatching motility (self-propulsion) strengths for the ABPs. We show that such a discontinuous motility field strongly affects spatial distribution of ABPs and thus also the effective interaction mediated between the inclusions through the active bath. Such net interactions arise from soft interfacial repulsions between ABPs that sterically interact with and/or pass through permeable membranes assumed to enclose the inclusions. Both regimes of repulsion and attractive (albeit with different mechanisms) are reported and summarized in overall phase diagrams.

An attraction-repulsion transition of force on two asymmetric wedges induced by active particles

Effective interaction between two asymmetric wedges immersed in a two-dimensional active bath is investigated by computer simulations. The attraction–repulsion transition of effective force between two asymmetric wedges is subjected to the relative position of two wedges, the wedge-to-wedge distance, the active particle density, as well as the apex angle of two wedges. By exchanging the position of the two asymmetric wedges in an active bath, firstly a simple attraction–repulsion transition of effective force occurs, completely different from passive Brownian particles. Secondly the transition of effective force is symmetric for the long-range distance between two asymmetric wedges, while it is asymmetric for the short-range case. Our investigations may provide new possibilities to govern the motion and assembly of microscopic objects by taking advantage of the self-driven behaviour of active particles.

Mean field approach of dynamical pattern formation in underdamped active matter with short-ranged alignment and distant anti-alignment interactions

Many active matter systems, especially on the microscopic scale, are well approximated as overdamped, meaning that any inertial momentum is immediately dissipated by the environment. On the other hand, especially for macroscopic active systems but also for many mesoscopic systems the time scale of translational inertial motion can become large enough to be relevant for the dynamics. This raises the question how collective dynamics and the resulting states in active matter are influenced by inertia. Therefore, we propose a coarse-grained continuum model for underdamped active matter based on a mean field description for passive systems. Furthermore, as an example, we apply the model to a system with interactions that support an alignment on short distances and an anti-alignment on longer length scales as known in the context of pattern formation due to orientational interactions. Our numerical calculations of the under- and overdamped dynamics both predict a structured laning state. However, activity induced convective flows that are only present in the underdamped model destabilize this state when the anti-alignment is weakened, leading to a collective motion state which does not occur in the overdamped limit. A turbulent transition regime between the two states can be characterized by strong density fluctuations and the absence of global ordering.

The 2020 motile active matter roadmap

Activity and autonomous motion are fundamental in living and engineering systems. This has stimulated the new field of 'active matter' in recent years, which focuses on the physical aspects of propulsion mechanisms, and on motility-induced emergent collective behavior of a larger number of identical agents. The scale of agents ranges from nanomotors and microswimmers, to cells, fish, birds, and people. Inspired by biological microswimmers, various designs of autonomous synthetic nano- and micromachines have been proposed. Such machines provide the basis for multifunctional, highly responsive, intelligent (artificial) active materials, which exhibit emergent behavior and the ability to perform tasks in response to external stimuli. A major challenge for understanding and designing active matter is their inherent nonequilibrium nature due to persistent energy consumption, which invalidates equilibrium concepts such as free energy, detailed balance, and time-reversal symmetry. Unraveling, predicting, and controlling the behavior of active matter is a truly interdisciplinary endeavor at the interface of biology, chemistry, ecology, engineering, mathematics, and physics. The vast complexity of phenomena and mechanisms involved in the self-organization and dynamics of motile active matter comprises a major challenge. Hence, to advance, and eventually reach a comprehensive understanding, this important research area requires a concerted, synergetic approach of the various disciplines. The 2020 motile active matter roadmap of Journal of Physics: Condensed Matter addresses the current state of the art of the field and provides guidance for both students as well as established scientists in their efforts to advance this fascinating area.

From scalar to polar active matter: Connecting simulations with mean-field theory

We study numerically the phase behavior of self-propelled elliptical particles interacting through the "hard" repulsive Gay-Berne potential at infinite Péclet number. Changing a single parameter, the aspect ratio, allows us to continuously go from discoid active Brownian particles to elongated polar rods. Discoids show phase separation, which changes to a cluster state of polar domains, which then form polar bands as the aspect ratio is increased. From the simulations, we identify and extract the two effective parameters entering the mean-field description: the force imbalance coefficient and the effective coupling to the local polarization. These two coefficients are sufficient to obtain a complete and consistent picture, unifying the paradigms of scalar and polar active matter.

Conformation and dynamics of a self-avoiding active flexible polymer

We investigate conformations and dynamics of a polymer considering its monomers to be active Brownian particles. This active polymer shows very intriguing physical behavior which is absent in an active Rouse chain. The chain initially shrinks with active force, which starts swelling on further increase in force. The shrinkage followed by swelling is attributed purely to excluded-volume interactions among the monomers. In the swelling regime, the chain shows a crossover from the self-avoiding behavior to the Rouse behavior with scaling exponent ν_{a}≈1/2 for end-to-end distance. The nonmonotonicity in the structure is analyzed through various physical quantities; specifically, radial distribution function of monomers, scattering time, as well as various energy calculations. The chain relaxes faster than the Rouse chain in the intermediate force regime, with a crossover in variation of relaxation time at large active force as given by a power law τ_{r}∼Pe^{-4/3} (Pe is Péclet number).

Gliding filament system giving both global orientational order and clusters in collective motion

Emergence and collapse of coherent motions of self-propelled particles are affected more by particle motions and interactions than by their material or biological details. In the reconstructed systems of biofilaments and molecular motors, several types of collective motion including a global-order pattern emerge due to the alignment interaction. Meanwhile, earlier studies show that the alignment interaction of a binary collision of biofilaments is too weak to form the global order. The multiple collision is revealed to be important to achieve global order, but it is still unclear what kind of multifilament collision is actually involved. In this study, we demonstrate that not only alignment but also crossing of two filaments is essential to produce an effective multiple-particle interaction and the global order. We design the reconstructed system of biofilaments and molecular motors to vary a probability of the crossing of biofilaments on a collision and thus control the effect of volume exclusion. In this system, biofilaments glide along their polar strands on the turf of molecular motors and can align themselves nematically when they collide with each other. Our experiments show the counterintuitive result, in which the global order is achieved only when the crossing is allowed. When the crossing is prohibited, the cluster pattern emerges instead. We also investigate the numerical model in which we can change the strength of the volume exclusion effect and find that the global orientational order and clusters emerge with weak and strong volume exclusion effects, respectively. With those results and simple theory, we conclude that not only alignment but also finite crossing probability are necessary for the effective multiple-particles interaction forming the global order. Additionally, we describe the chiral symmetry breaking of a microtubule motion which causes a rotation of global alignment.

A simulation study of aggregation mediated by production of cohesive molecules

Mechanical interactions between biological cells can be mediated by secreted products. Here, we investigate how such a scenario could affect the cells' collective behaviour. We show that if the concentration field of secreted products around a cell can be considered to be in steady state, this scenario can be mapped onto an effective attractive interaction that depends on the local cell density. Using a field-theory approach, this density-dependent attraction gives rise to a cubic term in the Landau-Ginzburg free energy density. In continuum field simulations this can lead to "nucleation-like" appearance of homogeneous clusters in the spinodal phase separation regime. Implementing the density-dependent cohesive attraction in Brownian dynamics simulations of a particle-based model gives rise to similar "spinodal nucleation" phase separation behaviour.

Chemotactic behaviour of Escherichia coli at high cell density

At high cell density, swimming bacteria exhibit collective motility patterns, self-organized through physical interactions of a however still debated nature. Although high-density behaviours are frequent in natural situations, it remained unknown how collective motion affects chemotaxis, the main physiological function of motility, which enables bacteria to follow environmental gradients in their habitats. Here, we systematically investigate this question in the model organism Escherichia coli, varying cell density, cell length, and suspension confinement. The characteristics of the collective motion indicate that hydrodynamic interactions between swimmers made the primary contribution to its emergence. We observe that the chemotactic drift is moderately enhanced at intermediate cell densities, peaks, and is then strongly suppressed at higher densities. Numerical simulations reveal that this suppression occurs because the collective motion disturbs the choreography necessary for chemotactic sensing. We suggest that this physical hindrance imposes a fundamental constraint on high-density behaviours of motile bacteria, including swarming and the formation of multicellular aggregates and biofilms.

Clustering and anisotropic correlated percolation in polar flocks

We study clustering and percolation phenomena in the Vicsek model, taken here in its capacity of prototypical model for dry aligning active matter. Our results show that the order-disorder transition is not related in any way to a percolation transition, contrary to some earlier claims. We study geometric percolation in each of the phases at play, but we mostly focus on the ordered Toner-Tu phase, where we find that the long-range correlations of density fluctuations give rise to an anisotropic percolation transition.

Translational and rotational diffusion of rod shaped molecules by molecular dynamics simulations

The results of molecular dynamics simulations of the dynamical evolution of assemblies of linear rigid rods of variable aspect ratio, a, and number density, ρ, in the isotropic phase are reported. The rods consist of m equally spaced sites interacting with the Weeks-Chandler-Andersen repulsive pair potential, where 2 < m < 16. With increasing m, features specific to long rods, such as anisotropic self-diffusion, become apparent. There is also an increasing separation between the characteristic relaxation times of the torque, angular velocity, and reorientational time correlation functions with increasing density. The latter is exponential at high densities even for dimers. The isotropic translational diffusion coefficient, D_i, and rotational diffusion coefficient, D_r, are reported as a function of m and ρ or volume fraction, ξ. The mD_i data scale with ξ throughout much of the simulated range, while the rotational diffusion coefficients scale approximately as m³D_r against ρ at low densities but as ∼m⁶D_r at high ρ, consistent with theories of colloidal and noncolloidal rod-containing liquids. The crossover density between the two regimes is parameterized in analytic form. The probability distribution functions for displacements and angular jumps in a given time show evidence of non-Gaussian behavior with increasing density. The shear viscosity and D_i scale approximately as m and m^-1, respectively, in the semidilute regime, which is consistent with a Stokes-Einstein-like relationship. At high concentrations, a frustrated or glassy structure formed in which the rods were randomly oriented.

Effects of spatial dimensionality and steric interactions on microtubule-motor self-organization

Active networks composed of filaments and motor proteins can self-organize into a variety of architectures. Computer simulations in two or three spatial dimensions and including or omitting steric interactions between filaments can be used to model active networks. Here we examine how these modelling choices affect the state space of network self-organization. We compare the networks generated by different models of a system of dynamic microtubules and microtubule-crosslinking motors. We find that a thin 3D model that includes steric interactions between filaments is the most versatile, capturing a variety of network states observed in recent experiments. In contrast, 2D models either with or without steric interactions which prohibit microtubule crossings can produce some, but not all, observed network states. Our results provide guidelines for the most appropriate choice of model for the study of different network types and elucidate mechanisms of active network organization.

Collective oscillation in dense suspension of self-propelled chiral rods

Active particles capable of self-propulsion commonly exhibit rich collective dynamics and have attracted increasing attention due to their applications in biology, robotics, social transport, and biomedicine. However, it remains unclear how the geometric features of active particles affect their collective behaviors. In this paper, we explore the collective dynamics of L-shaped active rods. We show that a dense suspension of self-propelled L-shaped rods exhibits fascinating non-equilibrium oscillatory dynamic clustering. A new oscillation phase can form due to distinct collisions and aggregation mechanisms arising from the L-shaped chirality of elements. A generic diagram of emerging states is provided over a wide range of geometric parameters. Our findings show that the comparative strength between the periodic separation and proximity effect from chirality and the alignment effect from elongated geometry drive the formation and transition of dynamic patterns. This chirality-triggered oscillation phase suggests a new route to understand active matter and paves a way for emerging applications.

Chronology of motor-mediated microtubule streaming

We introduce a filament-based simulation model for coarse-grained, effective motor-mediated interaction between microtubule pairs to study the time-scales that compose cytoplasmic streaming. We characterise microtubule dynamics in two-dimensional systems by chronologically arranging five distinct processes of varying duration that make up streaming, from microtubule pairs to collective dynamics. The structures found were polarity sorted due to the propulsion of antialigned microtubules. This also gave rise to the formation of large polar-aligned domains, and streaming at the domain boundaries. Correlation functions, mean squared displacements, and velocity distributions reveal a cascade of processes ultimately leading to microtubule streaming and advection, spanning multiple microtubule lengths. The characteristic times for the processes extend over three orders of magnitude from fast single-microtubule processes to slow collective processes. Our approach can be used to directly test the importance of molecular components, such as motors and crosslinking proteins between microtubules, on the collective dynamics at cellular scale.

Two types of exclusion interactions for self-propelled objects and collective motion induced by their combination

Exclusive interactions between self-driven objects may play crucial roles in their collective behavior, e.g., in collective migration of living cells. Here, such collective behavior is studied based on a simple but sufficient model taking account the exclusion effects, which incorporate the following two distinct kinds of exclusion interactions in two dimensions: The first is the mechanical exclusion wherein two objects mechanically repel each other when they overlap. The second is the scattering exclusion, wherein the directions along which each object tries to move are modulated to avoid overlapping. We propose a theoretical model based on two principles: (1) Each object maintains its own polarity with a fixed strength and attempts to move into the polarity direction and (2) objects interact with each other through the abovementioned exclusions. Based on this model, we look at the difference of consequences and combinatory effects of these two kinds of exclusions. Furthermore, we calculate the polar order of polarity directions without an external directional bias. Our results suggest that the combination of these two kinds of exclusions leads to effectively inelastic scattering of two objects, which eventually gives rise to global polar ordering. We also find that the traveling band can arise by this mechanism of alignment at the intermediate density, as generally seen in collective motion with polar alignment and investigated in various earlier works. Characteristics of transitions among disordered, traveling band, and homogeneously ordered states of the presented model are investigated, and their similarities and differences with those given by the explicit alignment interaction are discussed.

Clustering of microswimmers: interplay of shape and hydrodynamics

The spatiotemporal dynamics in systems of active self-propelled particles is controlled by the propulsion mechanism in combination with various direct interactions, such as steric repulsion and hydrodynamics. These direct interactions are typically anisotropic, and come in different "flavors", such as spherical and elongated particle shapes, pusher and puller flow fields, etc. The combination of the various aspects is expected to lead to new emergent behavior. However, it is a priori not evident whether shape and hydrodynamics act synergistically or antagonistically to generate motility-induced clustering (MIC) and phase separation (MIPS). We employ a model of prolate spheroidal microswimmers-called squirmers-in quasi-two-dimensional confinement to address this issue by mesoscale hydrodynamic simulations. For comparison, non-hydrodynamic active Brownian particles (ABPs) are considered to elucidate the contribution of hydrodynamic interactions. For spherical particles, the comparison between ABPs and hydrodynamic-squirmer ensembles reveals a suppression of MIPS due to hydrodynamic interactions. Yet, our analysis shows that dynamic clusters exist, with a broad size distribution. The fundamental difference between ABPs and squirmers is attributed to an increased reorientation of squirmers by hydrodynamic torques during their collisions. In contrast, for elongated squirmers, hydrodynamics interactions enhance MIPS. The transition to a phase-separated state strongly depends on the nature of the swimmer's flow field-with an increased tendency toward MIPS for pullers, and a reduced tendency for pushers. Thus, hydrodynamic interactions show opposing effects on MIPS for spherical and elongated microswimmers, and details of the propulsion mechanism of biological microswimmers may be very important to determine their collective behavior.

Collective behavior of self-propelled rods with quorum sensing

Active agents-like phoretic particles, bacteria, sperm, and cytoskeletal filaments in motility assays-show a large variety of motility-induced collective behaviors, such as aggregation, clustering, and phase separation. The behavior of dense suspensions of engineered phoretic particles and of bacteria during biofilm formation is determined by two qualitatively different physical mechanisms: (i) volume exclusion (short-range steric repulsion) and (ii) quorum sensing (longer-range reduced propulsion due to alteration of the local chemical environment). To systematically characterize such systems, we study semi-penetrable self-propelled rods in two dimensions, with a propulsion force that decreases with increasing local rod density, by employing Brownian dynamics simulations. Volume exclusion and quorum sensing both lead to phase separation; however, the structure of the systems and the rod dynamics vastly differ. Quorum sensing enhances the polarity of the clusters, induces perpendicularity of rods at the cluster borders, and enhances cluster formation. For systems where the rods essentially become passive at high densities, formation of asters and stripes is observed. Systems of rods with larger aspect ratios show more ordered structures compared to those with smaller aspect ratios, due to their stronger alignment, with almost circular asters for strongly density-dependent propulsion force. With increasing range of the quorum-sensing interaction, the local density decreases, asters become less stable, and polar hedgehog clusters and clusters with domains appear.

Active Brownian Filamentous Polymers under Shear Flow

The conformational and rheological properties of active filaments/polymers exposed to shear flow are studied analytically. Using the continuous Gaussian semiflexible polymer model extended by the activity, we derive analytical expressions for the dependence of the deformation, orientation, relaxation times, and viscosity on the persistence length, shear rate, and activity. The model yields a Weissenberg-number dependent shear-induced deformation, alignment, and shear thinning behavior, similarly to the passive counterpart. Thereby, the model shows an intimate coupling between activity and shear flow. As a consequence, activity enhances the shear-induced polymer deformation for flexible polymers. For semiflexible polymers/filaments, a nonmonotonic deformation is obtained because of the activity-induced shrinkage at moderate and swelling at large activities. Independent of stiffness, activity-induced swelling facilitates and enhances alignment and shear thinning compared to a passive polymer. In the asymptotic limit of large activities, a polymer length- and stiffness-independent behavior is obtained, with universal shear-rate dependencies for the conformations, dynamics, and rheology.

Isotropic-nematic transition of self-propelled rods in three dimensions

Using overdamped Brownian dynamics simulations we investigate the isotropic-nematic (IN) transition of self-propelled rods in three spatial dimensions. For two well-known model systems (Gay-Berne potential and hard spherocylinders) we find that turning on activity moves to higher densities the phase boundary separating an isotropic phase from a (nonpolar) nematic phase. This active IN phase boundary is distinct from the boundary between isotropic and polar-cluster states previously reported in two-dimensional simulation studies and, unlike the latter, is not sensitive to the system size. We thus identify a generic feature of anisotropic active particles in three dimensions.

Beating of grafted chains induced by active Brownian particles

We study the interplay between active Brownian particles (ABPs) and a "hairy" surface in two-dimensional geometry. We find that the increase of propelling force leads to and enhances inhomogeneous accumulation of ABPs inside the brush region. Oscillation of chain bundles (beating like cilia) is found in company with the formation and disassembly of a dynamic cluster of ABPs at large propelling forces. Meanwhile chains are stretched and pushed down due to the effective shear force by ABPs. The decrease of the average brush thickness with propelling force reflects the growth of the beating amplitude of chain bundles. Furthermore, the beating phenomenon is investigated in a simple single-chain system. We find that the chain swings regularly with a major oscillatory period, which increases with chain length and decreases with the increase of propelling force. We build a theory to describe the phenomenon and the predictions on the relationship between the period and amplitude for various chain lengths, and propelling forces agree very well with simulation data.

Collective dynamics of self-propelled semiflexible filaments

The collective behavior of active semiflexible filaments is studied with a model of tangentially driven self-propelled worm-like chains. The combination of excluded-volume interactions and self-propulsion leads to several distinct dynamic phases as a function of bending rigidity, activity, and aspect ratio of individual filaments. We consider first the case of intermediate filament density. For high-aspect-ratio filaments, we identify a transition with increasing propulsion from a state of free-swimming filaments to a state of spiraled filaments with nearly frozen translational motion. For lower aspect ratios, this gas-of-spirals phase is suppressed with growing density due to filament collisions; instead, filaments form clusters similar to self-propelled rods. As activity increases, finite bending rigidity strongly effects the dynamics and phase behavior. Flexible filaments form small and transient clusters, while stiffer filaments organize into giant clusters, similarly to self-propelled rods, but with a reentrant phase behavior from giant to smaller clusters as activity becomes large enough to bend the filaments. For high filament densities, we identify a nearly frozen jamming state at low activities, a nematic laning state at intermediate activities, and an active-turbulence state at high activities. The latter state is characterized by a power-law decay of the energy spectrum as a function of wave number. The resulting phase diagrams encapsulate tunable non-equilibrium steady states that can be used in the organization of living matter.