Confinement stabilizes a bacterial suspension into a spiral vortex

Flow-induced phase separation of active particles is controlled by boundary conditions

Active particles, including swimming microorganisms, autophoretic colloids, and droplets, are known to self-organize into ordered structures at fluid-solid boundaries. The entrainment of particles in the attractive parts of their spontaneous flows has been postulated as a possible mechanism underlying this phenomenon. Here, combining experiments, theory, and numerical simulations, we demonstrate the validity of this flow-induced ordering mechanism in a suspension of active emulsion droplets. We show that the mechanism can be controlled, with a variety of resultant ordered structures, by simply altering hydrodynamic boundary conditions. Thus, for flow in Hele-Shaw cells, metastable lines or stable traveling bands can be obtained by varying the cell height. Similarly, for flow bounded by a plane, dynamic crystallites are formed. At a no-slip wall, the crystallites are characterized by a continuous out-of-plane flux of particles that circulate and re-enter at the crystallite edges, thereby stabilizing them. At an interface where the tangential stress vanishes, the crystallites are strictly 2D, with no out-of-plane flux. We rationalize these experimental results by calculating, in each case, the slow viscous flow produced by the droplets and the long-ranged, many-body active forces and torques between them. The results of numerical simulations of motion under the action of the active forces and torques are in excellent agreement with experiments. Our work elucidates the mechanism of flow-induced phase separation in active fluids, particularly active colloidal suspensions, and demonstrates its control by boundaries, suggesting routes to geometric and topological phenomena in an active matter.

Boundaries Control Collective Dynamics of Inertial Self-Propelled Robots

Simple ingredients, such as well-defined interactions and couplings for the velocity and orientation of self-propelled objects, are sufficient to produce complex collective behavior in assemblies of such entities. Here, we use assemblies of rodlike robots made motile through self-vibration. When confined in circular arenas, dilute assemblies of these rods act as a gas. Increasing the surface fraction leads to a collective behavior near the boundaries: polar clusters emerge while, in the bulk, gaslike behavior is retained. The coexistence between a gas and surface clusters is a direct consequence of inertial effects as shown by our simulations. A theoretical model, based on surface mediated transport accounts for this coexistence and illustrates the exact role of the boundaries. Our study paves the way towards the control of collective behavior: By using deformable but free to move arenas, we demonstrate that the surface induced clusters can lead to directed motion, while the topology of the surface states can be controlled by biasing the motility of the particles.

Curvature-Guided Motility of Microalgae in Geometric Confinement

Microorganisms, such as bacteria and microalgae, often live in habitats consisting of a liquid phase and a plethora of interfaces. The precise ways in which these motile microbes behave in their confined environment remain unclear. Using experiments and Brownian dynamics simulations, we study the motility of a single Chlamydomonas microalga in an isolated microhabitat with controlled geometric properties. We demonstrate how the geometry of the habitat controls the cell's navigation in confinement. The probability of finding the cell swimming near the boundary increases with the wall curvature, as seen for both circular and elliptical chambers. The theory, utilizing an asymmetric dumbbell model of the cell and steric wall interactions, captures this curvature-guided navigation quantitatively with no free parameters.

Transition from turbulent to coherent flows in confined three-dimensional active fluids

Transport of fluid through a pipe is essential for the operation of macroscale machines and microfluidic devices. Conventional fluids only flow in response to external pressure. We demonstrate that an active isotropic fluid, composed of microtubules and molecular motors, autonomously flows through meter-long three-dimensional channels. We establish control over the magnitude, velocity profile, and direction of the self-organized flows and correlate these to the structure of the extensile microtubule bundles. The inherently three-dimensional transition from bulk-turbulent to confined-coherent flows occurs concomitantly with a transition in the bundle orientational order near the surface and is controlled by a scale-invariant criterion related to the channel profile. The nonequilibrium transition of confined isotropic active fluids can be used to engineer self-organized soft machines.

Complex self-propelled rings: a minimal model for cell motility

Collective behavior of active matter is observed for self-propelled particles, such as vibrated disks and active Brownian particles, as well as for cytoskeletal filaments in motile cells. Here, a system of quasi two-dimensional penetrable self-propelled rods inside rigid rings is used to construct a complex self-propelled particle. The rods interact sterically with each other and with a stationary or mobile ring via a separation-shifted Lennard-Jones potential. They either have a sliding attachment to the inside of the ring at one of their ends, or can move freely within the ring confinement. We study the inner structure and dynamics of the mobile self-propelled rings. We find that these complex particles cannot only be characterized as active Brownian particles, but can also exhibit cell-like motility: random walks, persistent motion, circling, and run-and-circle motion.

Self-Driven Droplet Powered By Active Nematics

Active matter defines a class of far-away-from-equilibrium systems comprising self-driven microparticles. Their anomalous physical properties could be applied in areas such as mixing or separation, micropumps, and self-healing materials. To realize such applications, a thorough understanding of the physical mechanisms as well as the development of methods to manipulate various active systems is required. Using a coarse-grained active liquid crystal model, we designed and investigated a single self-driven droplet which encapsulated a dense suspension comprising nonmotile but mobile active particles that generate extensile stresses. We showed that such droplets can be driven into motion and can have tunable mobilities owing to their internal collective motion, which is characterized by induced active flows and motile disclination defects. Furthermore, it was illustrated that the interplay among the internal directional flows, liquid crystalline structures, droplet size, and surface tension resulted in different types of locomotion and rotation.

Scattering of biflagellate microswimmers from surfaces

We use a three-bead-spring model to investigate the dynamics of biflagellate microswimmers near a surface. While the primary dynamics and scattering are governed by geometric-dependent direct contact, the fluid flows generated by the swimmer locomotion are important in orienting it toward or away from the surface. Flagellar noise and in particular cell spinning about the main axis help a surface-trapped swimmer escape, whereas the time a swimmer spends at the surface depends on the incident angle. The dynamics results from a nuanced interplay of direct collisions, hydrodynamics, noise, and the swimmer geometry. We show that to correctly capture the dynamics of a biflagellate swimmer, minimal models need to resolve the shape asymmetry.

Geometry-driven collective ordering of bacterial vortices

Controlling the phases of matter is a challenge that spans from condensed materials to biological systems. Here, by imposing a geometric boundary condition, we study the controlled collective motion of Escherichia coli bacteria. A circular microwell isolates a rectified vortex from disordered vortices masked in the bulk. For a doublet of microwells, two vortices emerge but their spinning directions show transition from parallel to anti-parallel. A Vicsek-like model for confined self-propelled particles gives the point where the two spinning patterns occur in equal probability and one geometric quantity governs the transition as seen in experiments. This mechanism shapes rich patterns including chiral configurations in a quadruplet of microwells, thus revealing a design principle of active vortices.

Tuning bacterial hydrodynamics with magnetic fields

Magnetotactic bacteria are a group of motile prokaryotes that synthesize chains of lipid-bound, magnetic nanoparticles called magnetosomes. This study exploits their innate magnetism to investigate previously unexplored facets of bacterial hydrodynamics at surfaces. Through use of weak, uniform, external magnetic fields and local, micromagnetic surface patterns, the relative strength of hydrodynamic, magnetic, and flagellar force components is tuned through magnetic control of the bacteria's orientation. The resulting swimming behaviors provide a means to experimentally determine hydrodynamic parameters and offer a high degree of control over large numbers of living microscopic entities. The implications of this controlled motion for studies of bacterial motility near surfaces and for micro- and nanotechnology are discussed.

Population splitting of rodlike swimmers in Couette flow

We present a quantitative analysis on the response of a dilute active suspension of self-propelled rods (swimmers) in a planar channel subjected to an imposed shear flow. To best capture the salient features of the shear-induced effects, we consider the case of an imposed Couette flow, providing a constant shear rate across the channel. We argue that the steady-state behavior of swimmers can be understood in the light of a population splitting phenomenon, occurring as the shear rate exceeds a certain threshold, initiating the reversal of the swimming direction for a finite fraction of swimmers from down- to upstream or vice versa, depending on the swimmer position within the channel. Swimmers thus split into two distinct, statistically significant and oppositely swimming majority and minority populations. The onset of population splitting translates into a transition from a self-propulsion-dominated regime to a shear-dominated regime, corresponding to a unimodal-to-bimodal change in the probability distribution function of the swimmer orientation. We present a phase diagram in terms of the swim and flow Péclet numbers showing the separation of these two regimes by a discontinuous transition line. Our results shed further light on the behavior of swimmers in a shear flow and provide an explanation for the previously reported non-monotonic behavior of the mean, near-wall, parallel-to-flow orientation of swimmers with increasing shear strength.

Fluid viscoelasticity promotes collective swimming of sperm

From flocking birds to swarming insects, interactions of organisms large and small lead to the emergence of collective dynamics. Here, we report striking collective swimming of bovine sperm in dynamic clusters, enabled by the viscoelasticity of the fluid. Sperm oriented in the same direction within each cluster, and cluster size and cell-cell alignment strength increased with viscoelasticity of the fluid. In contrast, sperm swam randomly and individually in Newtonian (nonelastic) fluids of low and high viscosity. Analysis of the fluid motion surrounding individual swimming sperm indicated that sperm-fluid interaction was facilitated by the elastic component of the fluid. In humans, as well as cattle, sperm are naturally deposited at the entrance to the cervix and must swim through viscoelastic cervical mucus and other mucoid secretions to reach the site of fertilization. Collective swimming induced by elasticity may thus facilitate sperm migration and contribute to successful fertilization. We note that almost all biological fluids (e.g. mucus and blood) are viscoelastic in nature, and this finding highlights the importance of fluid elasticity in biological function.

Transitions between multiple dynamical states in a confined dense active-particle system

We study the collective motion of a dense suspension of active swimmers in a viscous fluid medium. The swimmers are modeled as soft spheres moving in a highly viscous fluid medium. The magnitude of the propelling thrust exerted by each particle is taken to be a constant and the direction is aligned to its velocity. Depending on the magnitude of the exerted thrust, several nonequilibrium steady states are observed. The transitions between the steady states are characterized using the total dissipation as a function of the magnitude of the thrust. The transitions between the nonequilibrium states are characterized by changes in exponent at low thrust values. At high thrust values, hysteretic transitions between ordered and disordered states are observed.

Study of dynamic heterogeneity of an active particle system

We have studied spatial and temporal dynamic heterogeneity (DH) in a system of hard-sphere particles, subjected to active forces with constant amplitude and random direction determined by rotational diffusion with correlation time τ. We have used a variety of observables to characterize the DH behavior, including the deviation from standard Stokes-Einstein (SE) relation, a non-Gaussian parameter α_{2}(Δt) for the distribution of particle displacement within a certain time interval Δt, a four-point susceptibility χ_{4}(Δt,ΔL) for the correlation in dynamics between any two points in space separated by distance ΔL within some time window Δt, and a vector spatial-temporal correlation function S_{vec}(R,Δt) for vector displacements within time interval Δt of particle pairs originally separated by R. By mapping the particle motion into a continuous-time random walk with constant jump length, we can obtain the average waiting time 〈t_{x}〉∝D_{s}^{-1} and persistence time 〈t_{p}〉∝η, with D_{s} the self-diffusion coefficient and η the shear viscosity, such that the observable λ=〈t_{p}〉/〈t_{x}〉∝D_{s}η can be calculated as a function of the control parameter τ to show how it deviates from its SE value λ_{0}. Interestingly, we find λ/λ_{0} shows a nonmonotonic behavior for large volume fraction φ_{a}, wherein λ/λ_{0} undergoes a minimum at a certain intermediate value of τ, indicating that both small and large particle activity may lead to strong DH. Such a reentrance phenomenon is further demonstrated in terms of the non-Gaussian parameters α_{2}, four-point susceptibility χ_{4}, and vector spatiotemporal correlation functions S_{vec}, respectively. Detail analysis shows that it is the competition between the dual roles of particle activity, namely, activity-induced higher effective temperature and activity-induced clustering, that leads to such nontrivial nonmonotonic behaviors. In addition, we find that DH may also show a maximum level at an intermediate value of φ_{a} if τ is large enough, implying that a more crowded system may be less heterogeneous than a less crowded one for a system with high particle activity.

Active matter logic for autonomous microfluidics

Chemically or optically powered active matter plays an increasingly important role in materials design, but its computational potential has yet to be explored systematically. The competition between energy consumption and dissipation imposes stringent physical constraints on the information transport in active flow networks, facilitating global optimization strategies that are not well understood. Here, we combine insights from recent microbial experiments with concepts from lattice-field theory and non-equilibrium statistical mechanics to introduce a generic theoretical framework for active matter logic. Highlighting conceptual differences with classical and quantum computation, we demonstrate how the inherent non-locality of incompressible active flow networks can be utilized to construct universal logical operations, Fredkin gates and memory storage in set-reset latches through the synchronized self-organization of many individual network components. Our work lays the conceptual foundation for developing autonomous microfluidic transport devices driven by bacterial fluids, active liquid crystals or chemically engineered motile colloids.

Spatial confinement of active microtubule networks induces large-scale rotational cytoplasmic flow

Collective behaviors of motile units through hydrodynamic interactions induce directed fluid flow on a larger length scale than individual units. In cells, active cytoskeletal systems composed of polar filaments and molecular motors drive fluid flow, a process known as cytoplasmic streaming. The motor-driven elongation of microtubule bundles generates turbulent-like flow in purified systems; however, it remains unclear whether and how microtubule bundles induce large-scale directed flow like the cytoplasmic streaming observed in cells. Here, we adopted Xenopus egg extracts as a model system of the cytoplasm and found that microtubule bundle elongation induces directed flow for which the length scale and timescale depend on the existence of geometrical constraints. At the lower activity of dynein, kinesins bundle and slide microtubules, organizing extensile microtubule bundles. In bulk extracts, the extensile bundles connected with each other and formed a random network, and vortex flows with a length scale comparable to the bundle length continually emerged and persisted for 1 min at multiple places. When the extracts were encapsulated in droplets, the extensile bundles pushed the droplet boundary. This pushing force initiated symmetry breaking of the randomly oriented bundle network, leading to bundles aligning into a rotating vortex structure. This vortex induced rotational cytoplasmic flows on the length scale and timescale that were 10- to 100-fold longer than the vortex flows emerging in bulk extracts. Our results suggest that microtubule systems use not only hydrodynamic interactions but also mechanical interactions to induce large-scale temporally stable cytoplasmic flow.

Spontaneous mirror-symmetry breaking induces inverse energy cascade in 3D active fluids

Classical turbulence theory assumes that energy transport in a 3D turbulent flow proceeds through a Richardson cascade whereby larger vortices successively decay into smaller ones. By contrast, an additional inverse cascade characterized by vortex growth exists in 2D fluids and gases, with profound implications for meteorological flows and fluid mixing. The possibility of a helicity-driven inverse cascade in 3D fluids had been rejected in the 1970s based on equilibrium-thermodynamic arguments. Recently, however, it was proposed that certain symmetry-breaking processes could potentially trigger a 3D inverse cascade, but no physical system exhibiting this phenomenon has been identified to date. Here, we present analytical and numerical evidence for the existence of an inverse energy cascade in an experimentally validated 3D active fluid model, describing microbial suspension flows that spontaneously break mirror symmetry. We show analytically that self-organized scale selection, a generic feature of many biological and engineered nonequilibrium fluids, can generate parity-violating Beltrami flows. Our simulations further demonstrate how active scale selection controls mirror-symmetry breaking and the emergence of a 3D inverse cascade.

Geometric control of active collective motion

Recent experimental studies have shown that confinement can profoundly affect self-organization in semi-dilute active suspensions, leading to striking features such as the formation of steady and spontaneous vortices in circular domains and the emergence of unidirectional pumping motions in periodic racetrack geometries. Motivated by these findings, we analyze the two-dimensional dynamics in confined suspensions of active self-propelled swimmers using a mean-field kinetic theory where conservation equations for the particle configurations are coupled to the forced Navier-Stokes equations for the self-generated fluid flow. In circular domains, a systematic exploration of the parameter space casts light on three distinct states: equilibrium with no flow, stable vortex, and chaotic motion, and the transitions between these are explained and predicted quantitatively using a linearized theory. In periodic racetracks, similar transitions from equilibrium to net pumping to traveling waves to chaos are observed in agreement with experimental observations and are also explained theoretically. Our results underscore the subtle effects of geometry on the morphology and dynamics of emerging patterns in active suspensions and pave the way for the control of active collective motion in microfluidic devices.

On the Quantification of Cellular Velocity Fields

The application of flow visualization in biological systems is becoming increasingly common in studies ranging from intracellular transport to the movements of whole organisms. In cell biology, the standard method for measuring cell-scale flows and/or displacements has been particle image velocimetry (PIV); however, alternative methods exist, such as optical flow constraint. Here we review PIV and optical flow, focusing on the accuracy and efficiency of these methods in the context of cellular biophysics. Although optical flow is not as common, a relatively simple implementation of this method can outperform PIV and is easily augmented to extract additional biophysical/chemical information such as local vorticity or net polymerization rates from speckle microscopy.

Dynamic structure of active nematic shells

When a thin film of active, nematic microtubules and kinesin motor clusters is confined on the surface of a vesicle, four +1/2 topological defects oscillate in a periodic manner between tetrahedral and planar arrangements. Here a theoretical description of nematics, coupled to the relevant hydrodynamic equations, is presented here to explain the dynamics of active nematic shells. In extensile microtubule systems, the defects repel each other due to elasticity, and their collective motion leads to closed trajectories along the edges of a cube. That motion is accompanied by oscillations of their velocities, and the emergence and annihilation of vortices. When the activity increases, the system enters a chaotic regime. In contrast, for contractile systems, which are representative of some bacterial suspensions, a hitherto unknown static structure is predicted, where pairs of defects attract each other and flows arise spontaneously.

In active matter, chemical energy is transformed into mechanical motion; theoretical descriptions of nematic liquids are useful in understanding such phenomena. Here, Zhang et al. model the dynamics of active nematic liquid crystals confined onto a spherical shell in systems that mimic cell motion.

Spatiotemporal order and emergent edge currents in active spinner materials

Collections of interacting, self-propelled particles have been extensively studied as minimal models of many living and synthetic systems from bird flocks to active colloids. However, the influence of active rotations in the absence of self-propulsion (i.e., spinning without walking) remains less explored. Here, we numerically and theoretically investigate the behavior of ensembles of self-spinning dimers. We find that geometric frustration of dimer rotation by interactions yields spatiotemporal order and active melting with no equilibrium counterparts. At low density, the spinning dimers self-assemble into a triangular lattice with their orientations phase-locked into spatially periodic phases. The phase-locked patterns form dynamical analogs of the ground states of various spin models, transitioning from the three-state Potts antiferromagnet at low densities to the striped herringbone phase of planar quadrupoles at higher densities. As the density is raised further, the competition between active rotations and interactions leads to melting of the active spinner crystal. Emergent edge currents, whose direction is set by the chirality of the active spinning, arise as a nonequilibrium signature of the transition to the active spinner liquid and vanish when the system eventually undergoes kinetic arrest at very high densities. Our findings may be realized in systems ranging from liquid crystal and colloidal experiments to tabletop realizations using macroscopic chiral grains.

Negative stiffness and modulated states in active nematics

We examine the dynamics of an active nematic liquid crystal on a frictional substrate. When frictional damping dominates over viscous dissipation, we eliminate flow in favor of active stresses to obtain a minimal dynamical model for the nematic order parameter, with elastic constants renormalized by activity. The renormalized elastic constants can become negative at large activity, leading to the selection of spatially inhomogeneous patterns via a mechanism analogous to that responsible for modulated phases arising at an equilibrium Lifshitz point. Tuning activity and the degree of nematic order in the passive system, we obtain a linear stability phase diagram that exhibits a nonequilibrium tricritical point where ordered, modulated and disordered phases meet. Numerical solution of the nonlinear equations yields a succession of spatial structures of increasing complexity with increasing activity, including kink walls and active turbulence, as observed in experiments on microtubule bundles confined at an oil-water interface. Our work provides a minimal model for an overdamped active nematic that reproduces all the nonequilibrium structures seen in simulations of the full active nematic hydrodynamics and provides a framework for understanding some of the mechanisms for selection of the nonequilibrium patterns in the language of equilibrium critical phenomena.

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.

Shape and Displacement Fluctuations in Soft Vesicles Filled by Active Particles

We investigate numerically the dynamics of shape and displacement fluctuations of two-dimensional flexible vesicles filled with active particles. At low concentration most of the active particles accumulate at the boundary of the vesicle where positive particle number fluctuations are amplified by trapping, leading to the formation of pinched spots of high density, curvature and pressure. At high concentration the active particles cover the vesicle boundary almost uniformly, resulting in fairly homogeneous pressure and curvature, and nearly circular vesicle shape. The change between polarized and spherical shapes is driven by the number of active particles. The center-of-mass of the vesicle performs a persistent random walk with a long time diffusivity that is strongly enhanced for elongated active particles due to orientational correlations in their direction of propulsive motion. In our model shape-shifting induces directional sensing and the cell spontaneously migrate along the polarization direction.

Hydrodynamic length-scale selection in microswimmer suspensions

A universal characteristic of mesoscale turbulence in active suspensions is the emergence of a typical vortex length scale, distinctly different from the scale invariance of turbulent high-Reynolds number flows. Collective length-scale selection has been observed in bacterial fluids, endothelial tissue, and active colloids, yet the physical origins of this phenomenon remain elusive. Here, we systematically derive an effective fourth-order field theory from a generic microscopic model that allows us to predict the typical vortex size in microswimmer suspensions. Building on a self-consistent closure condition, the derivation shows that the vortex length scale is determined by the competition between local alignment forces, rotational diffusion, and intermediate-range hydrodynamic interactions. Vortex structures found in simulations of the theory agree with recent measurements in Bacillus subtilis suspensions. Moreover, our approach yields an effective viscosity enhancement (reduction), as reported experimentally for puller (pusher) microorganisms.

Stochastic cycle selection in active flow networks

Active biological flow networks pervade nature and span a wide range of scales, from arterial blood vessels and bronchial mucus transport in humans to bacterial flow through porous media or plasmodial shuttle streaming in slime molds. Despite their ubiquity, little is known about the self-organization principles that govern flow statistics in such nonequilibrium networks. Here we connect concepts from lattice field theory, graph theory, and transition rate theory to understand how topology controls dynamics in a generic model for actively driven flow on a network. Our combined theoretical and numerical analysis identifies symmetry-based rules that make it possible to classify and predict the selection statistics of complex flow cycles from the network topology. The conceptual framework developed here is applicable to a broad class of biological and nonbiological far-from-equilibrium networks, including actively controlled information flows, and establishes a correspondence between active flow networks and generalized ice-type models.

Emergent ultra-long-range interactions between active particles in hybrid active-inactive systems

Particle-particle interactions determine the state of a system. Control over the range of such interactions as well as their magnitude has been an active area of research for decades due to the fundamental challenges it poses in science and technology. Very recently, effective interactions between active particles have gathered much attention as they can lead to out-of-equilibrium cooperative states such as flocking. Inspired by nature, where active living cells coexist with lifeless objects and structures, here we study the effective interactions that appear in systems composed of active and passive mixtures of colloids. Our systems are 2D colloidal monolayers composed primarily of passive (inactive) colloids, and a very small fraction of active (spinning) ferromagnetic colloids. We find an emergent ultra-long-range attractive interaction induced by the activity of the spinning particles and mediated by the elasticity of the passive medium. Interestingly, the appearance of such interaction depends on the spinning protocol and has a minimum actuation timescale below which no attraction is observed. Overall, these results clearly show that, in the presence of elastic components, active particles can interact across very long distances without any chemical modification of the environment. Such a mechanism might potentially be important for some biological systems and can be harnessed for newer developments in synthetic active soft materials.

Ferromagnetic and antiferromagnetic order in bacterial vortex lattices

Despite their inherent non-equilibrium nature1, living systems can self-organize in highly ordered collective states2,3 that share striking similarities with the thermodynamic equilibrium phases4,5 of conventional condensed matter and fluid systems. Examples range from the liquid-crystal-like arrangements of bacterial colonies6,7, microbial suspensions8,9 and tissues10 to the coherent macro-scale dynamics in schools of fish11 and flocks of birds12. Yet, the generic mathematical principles that govern the emergence of structure in such artificial13 and biological6-9,14 systems are elusive. It is not clear when, or even whether, well-established theoretical concepts describing universal thermostatistics of equilibrium systems can capture and classify ordered states of living matter. Here, we connect these two previously disparate regimes: Through microfluidic experiments and mathematical modelling, we demonstrate that lattices of hydrodynamically coupled bacterial vortices can spontaneously organize into distinct phases of ferro- and antiferromagnetic order. The preferred phase can be controlled by tuning the vortex coupling through changes of the inter-cavity gap widths. The emergence of opposing order regimes is tightly linked to the existence of geometry-induced edge currents15,16, reminiscent of those in quantum systems17-19. Our experimental observations can be rationalized in terms of a generic lattice field theory, suggesting that bacterial spin networks belong to the same universality class as a wide range of equilibrium systems.

Viscoelastic and elastomeric active matter: Linear instability and nonlinear dynamics

We consider a continuum model of active viscoelastic matter, whereby an active nematic liquid crystal is coupled to a minimal model of polymer dynamics with a viscoelastic relaxation time τ(C). To explore the resulting interplay between active and polymeric dynamics, we first generalize a linear stability analysis (from earlier studies without polymer) to derive criteria for the onset of spontaneous heterogeneous flows (strain rate) and/or deformations (strain). We find two modes of instability. The first is a viscous mode, associated with strain rate perturbations. It dominates for relatively small values of τ(C) and is a simple generalization of the instability known previously without polymer. The second is an elastomeric mode, associated with strain perturbations, which dominates at large τ(C) and persists even as τ(C)→∞. We explore the dynamical states to which these instabilities lead by means of direct numerical simulations. These reveal oscillatory shear-banded states in one dimension and activity-driven turbulence in two dimensions even in the elastomeric limit τ(C)→∞. Adding polymer can also have calming effects, increasing the net throughput of spontaneous flow along a channel in a type of drag reduction. The effect of including strong antagonistic coupling between the nematic and polymer is examined numerically, revealing a rich array of spontaneously flowing states.

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.

Collective chemotaxis and segregation of active bacterial colonies

Still recently, bacterial fluid suspensions have motivated a lot of works, both experimental and theoretical, with the objective to understand their collective dynamics from universal and simple rules. Since some species are active, most of these works concern the strong interactions that these bacteria exert on a forced flow leading to instabilities, chaos and turbulence. Here, we investigate the self-organization of expanding bacterial colonies under chemotaxis, proliferation and eventually active-reaction. We propose a simple model to understand and quantify the physical properties of these living organisms which either give cohesion or on the contrary dispersion to the colony. Taking into account the diffusion and capture of morphogens complicates the model since it induces a bacterial density gradient coupled to bacterial density fluctuations and dynamics. Nevertheless under some specific conditions, it is possible to investigate the pattern formation as a usual viscous fingering instability. This explains the similarity and differences of patterns according to the physical bacterial suspension properties and explain the factors which favor compactness or branching.

Ratchet transport powered by chiral active particles

We numerically investigate the ratchet transport of mixtures of active and passive particles in a transversal asymmetric channel. A big passive particle is immersed in a ‘sea’ of active particles. Due to the chirality of active particles, the longitudinal directed transport is induced by the transversal asymmetry. For the active particles, the chirality completely determines the direction of the ratchet transport, the counterclockwise and clockwise particles move to the opposite directions and can be separated. However, for the passive particle, the transport behavior becomes complicated, the direction is determined by competitions among the chirality, the self-propulsion speed, and the packing fraction. Interestingly, within certain parameters, the passive particle moves to the left, while active particles move to the right. In addition, there exist optimal parameters (the chirality, the height of the barrier, the self-propulsion speed and the packing fraction) at which the rectified efficiency takes its maximal value. Our findings could be used for the experimental pursuit of the ratchet transport powered by chiral active particles.

Coherent Motion of Monolayer Sheets under Confinement and Its Pathological Implications

Coherent angular rotation of epithelial cells is thought to contribute to many vital physiological processes including tissue morphogenesis and glandular formation. However, factors regulating this motion, and the implications of this motion if perturbed, remain incompletely understood. In the current study, we address these questions using a cell-center based model in which cells are polarized, motile, and interact with the neighboring cells via harmonic forces. We demonstrate that, a simple evolution rule in which the polarization of any cell tends to orient with its velocity vector can induce coherent motion in geometrically confined environments. In addition to recapitulating coherent rotational motion observed in experiments, our results also show the presence of radial movements and tissue behavior that can vary between solid-like and fluid-like. We show that the pattern of coherent motion is dictated by the combination of different physical parameters including number density, cell motility, system size, bulk cell stiffness and stiffness of cell-cell adhesions. We further observe that perturbations in the form of cell division can induce a reversal in the direction of motion when cell division occurs synchronously. Moreover, when the confinement is removed, we see that the existing coherent motion leads to cell scattering, with bulk cell stiffness and stiffness of cell-cell contacts dictating the invasion pattern. In summary, our study provides an in-depth understanding of the origin of coherent rotation in confined tissues, and extracts useful insights into the influence of various physical parameters on the pattern of such movements.

Epithelial and endothelial cells that line various cavities and the vasculature in our bodies, are tightly connected to each other and exist as sheets. Upon confinement in two-dimensional geometries, these cells exhibit rotational motion, which has also been observed in vivo and implicated in physiological processes. However, how this rotational motion is achieved remains unclear. We show that a simple rule wherein preferred direction of motion (i.e., polarization) of cells tends to align with the direction of their velocity is sufficient to induce such coherent movement in confined geometries. We also show that the number of cells within the confinement, the size of the tissue, cell motility and physical properties of the cell and cell-cell connections regulate this coherent motion, and the pattern of invasion when the confinement is relaxed.

Motility of active fluid drops on surfaces

Drops of active liquid crystal have recently shown the ability to self-propel, which was associated with topological defects in the orientation of active filaments [Sanchez et al., Nature 491, 431 (2013)]. Here, we study the onset and different aspects of motility of a three-dimensional drop of active fluid on a planar surface. We analyze theoretically how motility is affected by orientation profiles with defects of various types and locations, by the shape of the drop, and by surface friction at the substrate. In the scope of a thin drop approximation, we derive exact expressions for the flow in the drop that is generated by a given orientation profile. The flow has a natural decomposition into terms that depend entirely on the geometrical properties of the orientation profile, i.e., its bend and splay, and a term coupling the orientation to the shape of the drop. We find that asymmetric splay or bend generates a directed bulk flow and enables the drop to move, with maximal speeds achieved when the splay or bend is induced by a topological defect in the interior of the drop. In motile drops the direction and speed of self-propulsion is controlled by friction at the substrate.

Shape matters: Near-field fluid mechanics dominate the collective motions of ellipsoidal squirmers

Microswimmers show a variety of collective motions. Despite extensive study, questions remain regarding the role of near-field fluid mechanics in collective motion. In this paper, we describe precisely the Stokes flow around hydrodynamically interacting ellipsoidal squirmers in a monolayer suspension. The results showed that various collective motions, such as ordering, aggregation, and whirls, are dominated by the swimming mode and the aspect ratio. The collective motions are mainly induced by near-field fluid mechanics, despite Stokes flow propagation over a long range. These results emphasize the importance of particle shape in collective motion.

Self-Sustained Density Oscillations of Swimming Bacteria Confined in Microchambers

We numerically study the dynamics of run-and-tumble particles confined in two chambers connected by thin channels. Two dominant dynamical behaviors emerge: (i) an oscillatory pumping state, in which particles periodically fill the two vessels, and (ii) a circulating flow state, dynamically maintaining a near constant population level in the containers when connected by two channels. We demonstrate that the oscillatory behavior arises from the combination of a narrow channel, preventing bacteria reorientation, and a density-dependent motility inside the chambers.

Films of bacteria at interfaces: three stages of behaviour

We report an investigation of the formation of films by bacteria at an oil-water interface using a combination of particle tracking and pendant drop elastometry. The films display a remarkably varied series of dynamical and mechanical properties as they evolve over the course of minutes to hours following the creation of an initially pristine interface. At the earliest stage of formation, which we interrogate using dispersions of colloidal probes, the interface is populated with motile bacteria. Interactions with the bacteria dominate the colloidal motion, and the interface displays canonical features of active matter in a quasi-two-dimensional context. This active stage gives way to a viscoelastic transition, presumably driven by the accumulation at the interface of polysaccharides and surfactants produced by the bacteria, which instill the interface with the hallmarks of soft glassy rheology that we characterize with microrheology. Eventually, the viscoelastic film becomes fully elastic with the capability to support wrinkling upon compression, and we investigate this final stage with the pendant drop measurements. We characterize quantitatively the dynamic and mechanical properties of the films during each of these three stages - active, viscoelastic, and elastic - and comment on their possible significance for the interfacial bacterial colony. This work also brings to the forefront the important role that interfacial mechanics may play in bacterial suspensions with free surfaces.

Shape control and compartmentalization in active colloidal cells

Small autonomous machines like biological cells or soft robots can convert energy input into control of function and form. It is desired that this behavior emerges spontaneously and can be easily switched over time. For this purpose we introduce an active matter system that is loosely inspired by biology and which we term an active colloidal cell. The active colloidal cell consists of a boundary and a fluid interior, both of which are built from identical rotating spinners whose activity creates convective flows. Similarly to biological cell motility, which is driven by cytoskeletal components spread throughout the entire volume of the cell, active colloidal cells are characterized by highly distributed energy conversion. We demonstrate that we can control the shape of the active colloidal cell and drive compartmentalization by varying the details of the boundary (hard vs. flexible) and the character of the spinners (passive vs. active). We report buckling of the boundary controlled by the pattern of boundary activity, as well as formation of core-shell and inverted Janus phase-separated configurations within the active cell interior. As the cell size is increased, the inverted Janus configuration spontaneously breaks its mirror symmetry. The result is a bubble-crescent configuration, which alternates between two degenerate states over time and exhibits collective migration of the fluid along the boundary. Our results are obtained using microscopic, non-momentum-conserving Langevin dynamics simulations and verified via a phase-field continuum model coupled to a Navier-Stokes equation.

A physical perspective on cytoplasmic streaming

Organisms show a remarkable range of sizes, yet the dimensions of a single cell rarely exceed 100 µm. While the physical and biological origins of this constraint remain poorly understood, exceptions to this rule give valuable insights. A well-known counterexample is the aquatic plant Chara, whose cells can exceed 10 cm in length and 1 mm in diameter. Two spiralling bands of molecular motors at the cell periphery drive the cellular fluid up and down at speeds up to 100 µm s(-1), motion that has been hypothesized to mitigate the slowness of metabolite transport on these scales and to aid in homeostasis. This is the most organized instance of a broad class of continuous motions known as 'cytoplasmic streaming', found in a wide range of eukaryotic organisms-algae, plants, amoebae, nematodes and flies-often in unusually large cells. In this overview of the physics of this phenomenon, we examine the interplay between streaming, transport and cell size and discuss the possible role of self-organization phenomena in establishing the observed patterns of streaming.

Coherent motions in confluent cell monolayer sheets

Cell migration plays a pivotal role in many physiologically important processes such as embryogenesis, wound-healing, immune defense, and cancer metastasis. Although much effort has been directed toward motility of individual cells, the mechanisms underpinning collective cell migration remain poorly understood. Here we develop a collective motility model that incorporates cell mechanics and persistent random motions of individual cells to study coherent migratory motions in epithelial-like monolayers. This model, in absence of any external chemical signals, is able to explain coordinate rotational motion seen in systems ranging from two adherent cells to multicellular assemblies. We show that the competition between the active persistent force and random polarization fluctuation is responsible for the robust rotation. Passive mechanical coupling between cells is necessary but active chemical signaling between cells is not. The predicted angular motions also depend on the geometrical shape of the underlying substrate: cells exhibit collective rotation on circular substrates, but display linear back-and-forth motion on long and narrow substrates.

Emergent vortices in populations of colloidal rollers

Coherent vortical motion has been reported in a wide variety of populations including living organisms (bacteria, fishes, human crowds) and synthetic active matter (shaken grains, mixtures of biopolymers), yet a unified description of the formation and structure of this pattern remains lacking. Here we report the self-organization of motile colloids into a macroscopic steadily rotating vortex. Combining physical experiments and numerical simulations, we elucidate this collective behaviour. We demonstrate that the emergent-vortex structure lives on the verge of a phase separation, and single out the very constituents responsible for this state of polar active matter. Building on this observation, we establish a continuum theory and lay out a strong foundation for the description of vortical collective motion in a broad class of motile populations constrained by geometrical boundaries.

Confined populations of interacting motile particles often display collective motion in the form of large-scale vortices, such as fish groups and bacteria colonies. Bricard et al. study a model system with self-propelled colloidal rollers and identify the constituents responsible for emergent vortices.

Emergence and Persistence of Collective Cell Migration on Small Circular Micropatterns

The spontaneous formation of vortices is a hallmark of collective cellular activity. Here, we study the onset and persistence of coherent angular motion as a function of the number of cells N confined in circular micropatterns. We find that the persistence of coherent angular motion increases with N but exhibits a pronounced discontinuity accompanied by a geometric rearrangement of cells to a configuration containing a central cell. Computer simulations based on a generalized Potts model reproduce the emergence of vortex states and show in agreement with experiment that their stability depends on the interplay of the spatial arrangement and internal polarization of neighboring cells. Hence, the distinct migrational states in finite size ensembles reveal significant insight into the local interaction rules guiding collective migration.

Chirality separation of mixed chiral microswimmers in a periodic channel

Dynamics and separation of mixed chiral microswimmers are numerically investigated in a channel with regular arrays of rigid half-circle obstacles. For zero shear flow, transport behaviors are the same for different chiral particles: the average velocity decreases with increase of the rotational diffusion coefficient, the direction of the transport can be reversed by tuning the angular velocity, and there exists an optimal value of the packing fraction at which the average velocity takes its maximal value. However, when the shear flow is considered, different chiral particles show different behaviors. By suitably tailoring parameters, particles with different chiralities can move in different directions and can be separated. In addition, we also proposed a space separation method by introducing a constant load, where counterclockwise and clockwise particles stay in different regions of the channel.

Collective dynamics in a binary mixture of hydrodynamically coupled microrotors

We study, numerically, the collective dynamics of self-rotating nonaligning particles by considering a monolayer of spheres driven by constant clockwise or counterclockwise torques. We show that hydrodynamic interactions alter the emergence of large-scale dynamical patterns compared to those observed in dry systems. In dilute suspensions, the flow stirred by the rotors induces clustering of opposite-spin rotors, while at higher densities same-spin rotors phase separate. Above a critical rotor density, dynamic hexagonal crystals form. Our findings underscore the importance of inclusion of the many-body, long-range hydrodynamic interactions in predicting the phase behavior of active particles.

Geometric capture and escape of a microswimmer colliding with an obstacle

Motivated by recent experiments, we consider the hydrodynamic capture of a microswimmer near a stationary spherical obstacle. Simulations of model equations show that a swimmer approaching a small spherical colloid is simply scattered. In contrast, when the colloid is larger than a critical size it acts as a passive trap: the swimmer is hydrodynamically captured along closed trajectories and endlessly orbits around the colloidal sphere. In order to gain physical insight into this hydrodynamic scattering problem, we address it analytically. We provide expressions for the critical trapping radius, the depth of the "basin of attraction," and the scattering angle, which show excellent agreement with our numerical findings. We also demonstrate and rationalize the strong impact of swimming-flow symmetries on the trapping efficiency. Finally, we give the swimmer an opportunity to escape the colloidal traps by considering the effects of Brownian, or active, diffusion. We show that in some cases the trapping time is governed by an Ornstein-Uhlenbeck process, which results in a trapping time distribution that is well-approximated as inverse-Gaussian. The predictions again compare very favorably with the numerical simulations. We envision applications of the theory to bioremediation, microorganism sorting techniques, and the study of bacterial populations in heterogeneous or porous environments.

Giant adsorption of microswimmers: Duality of shape asymmetry and wall curvature

The effect of shape asymmetry of microswimmers on their adsorption capacity at confining channel walls is studied by a simple dumbbell model. For a shape polarity of a forward-swimming cone, like the stroke-averaged shape of a sperm, extremely long wall retention times are found, caused by a nonvanishing component of the propulsion force pointing steadily into the wall, which grows exponentially with the self-propulsion velocity and the shape asymmetry. A direct duality relation between shape asymmetry and wall curvature is proposed and verified. Our results are relevant for the design microswimmer with controlled wall-adhesion properties. In addition, we confirm that pressure in active systems is strongly sensitive to the details of the particle-wall interactions.

Circularly confined microswimmers exhibit multiple global patterns

Geometric confinement plays an important role in the dynamics of natural and synthetic microswimmers from bacterial cells to self-propelled particles in high-throughput microfluidic devices. However, little is known about the effects of geometric confinement on the emergent global patterns in such self-propelled systems. Recent experiments on bacterial cells report that, depending on the cell concentration, cells either spontaneously organize into vortical motion in thin cylindrical and spherical droplets or aggregate at the inner boundary of the droplets. Our goal in this paper is to investigate, in the context of an idealized physical model, the interplay between geometric confinement and level of flagellar activity on the emergent collective patterns. We show that decreasing flagellar activity induces a hydrodynamically triggered transition in confined microswimmers from swirling to global circulation (vortex) to boundary aggregation and clustering. These results highlight that the complex interplay between confinement, flagellar activity, and hydrodynamic flows in concentrated suspensions of microswimmers could lead to a plethora of global patterns that are difficult to predict from geometric consideration alone.