Fluid dynamics of bacterial turbulence

Dispersion of activity at an active-passive nematic interface

Efficient nutrient mixing is crucial for the survival of bacterial colonies and other living systems known as active nematics. However, the dynamics of this mixing is non-trivial as there is a coupling between nutrients concentration and velocity field. To address this question, we solve the hydrodynamic equation for active nematics to model the bacterial swarms coupled to an advection-diffusion equation for the activity field, which is proportional to the concentration of nutrients. At the interface between active and passive nematics the activity field is transported by the interfacial flows and in turn it modifies them through the generation of active stresses. We find that the dispersion of this conserved activity field is subdiffusive due to the emergence of a barrier of negative defects at the active-passive interface, which hinders the propagation of the motile positive defects.

Response of vesicle shapes to dense inner active matter

We consider experimentally the Takatori-Sahu model of vesicle shape fluctuations induced by enclosed active matter, a model till present tested only in the absence of collective motion because few enclosed bacteria were used to generate the desired active motion (S. C. Takatori and A. Sahu, Phys. Rev. Lett., 2020, 124, 158102). Using deformable giant unilamellar vesicles (GUVs) and phase contrast microscopy, we extract the mode-dependence of GUV shape fluctuations when hundreds of E. coli bacteria are contained within each GUV. In the microscope focal plane, patterns of collective bacteria flow include vortex flow, dipolar flow, and chaotic motion, all of which influence the GUV shapes. The Takatori-Sahu model generalizes well to this situation if one considers the moving element to be the experimentally-determined size of the collecively-moving flock.

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.

Chaos and irreversibility of a flexible filament in periodically driven Stokes flow

The flow of Newtonian fluid at low Reynolds number is, in general, regular and time-reversible due to absence of nonlinear effects. For example, if the fluid is sheared by its boundary motion that is subsequently reversed, then all the fluid elements return to their initial positions. Consequently, mixing in microchannels happens solely due to molecular diffusion and is very slow. Here, we show, numerically, that the introduction of a single, freely floating, flexible filament in a time-periodic linear shear flow can break reversibility and give rise to chaos due to elastic nonlinearities, if the bending rigidity of the filament is within a carefully chosen range. Within this range, not only the shape of the filament is spatiotemporally chaotic, but also the flow is an efficient mixer. Overall, we find five dynamical phases: the shape of a stiff filament is time-invariant-either straight or buckled; it undergoes a period-two bifurcation as the filament is made softer; becomes spatiotemporally chaotic for even softer filaments but, surprisingly, the chaos is suppressed if bending rigidity is decreased further.

Disordered Collective Motion in Dense Assemblies of Persistent Particles

We explore the emergence of nonequilibrium collective motion in disordered nonthermal active matter when persistent motion and crowding effects compete, using simulations of a two-dimensional model of size polydisperse self-propelled particles. In stark contrast with monodisperse systems, we find that polydispersity stabilizes a homogeneous active liquid at arbitrary large persistence times, characterized by remarkable velocity correlations and irregular turbulent flows. For all persistence values, the active fluid undergoes a nonequilibrium glass transition at large density. This is accompanied by collective motion, whose nature evolves from near-equilibrium spatially heterogeneous dynamics at small persistence, to a qualitatively different intermittent dynamics when persistence is large. This latter regime involves a complex time evolution of the correlated displacement field.

Nonlinear Phototaxis and Instabilities in Suspensions of Light-Seeking Algae

The mechanism by which living organisms seek optimal light conditions-phototaxis-is a fundamental process for motile photosynthetic microbes. It is involved in a broad array of natural processes and applications from bloom formation to the production of high-value chemicals in photobioreactors. Here, we show that a population of the model alga Chlamydomonas reinhardtii exhibits a highly sensitive nonlinear response to light and demonstrate that the self-organization of cells in a heterogeneous environment becomes unstable as the result of a coupling between bioconvective flows and phototaxis.

Bacterial active matter

Bacteria are among the oldest and most abundant species on Earth. Bacteria successfully colonize diverse habitats and play a significant role in the oxygen, carbon, and nitrogen cycles. They also form human and animal microbiota and may become sources of pathogens and a cause of many infectious diseases. Suspensions of motile bacteria constitute one of the most studied examples of active matter: a broad class of non-equilibrium systems converting energy from the environment (e.g., chemical energy of the nutrient) into mechanical motion. Concentrated bacterial suspensions, often termed active fluids, exhibit complex collective behavior, such as large-scale turbulent-like motion (so-called bacterial turbulence) and swarming. The activity of bacteria also affects the effective viscosity and diffusivity of the suspension. This work reports on the progress in bacterial active matter from the physics viewpoint. It covers the key experimental results, provides a critical assessment of major theoretical approaches, and addresses the effects of visco-elasticity, liquid crystallinity, and external confinement on collective behavior in bacterial suspensions.

Effective temperature and Einstein relation for particles in active matter flows

Active matter are a collection of units with intrinsic supply of energy that is utilized for self-propelled motion. Recent studies have confirmed that these active systems can exist in exotic phases, such as swarming, laning, jamming, and even turbulence, based on the size and density of the constituent units. An interesting question that naturally arises is whether one can identify an effective temperature for particles advected by such an active flow that is far from equilibrium. In this paper, we report using a continuum model of a dense bacterial suspension, an exact expression of the effective temperature for a distribution of interacting particles that are immersed in this suspension. We observe that this effective temperature is linear in particle diffusivity with the slope defining the particle mobility that is higher when the background fluid exhibits global polar ordering and lower when the fluid is in isotropic equilibrium. We believe our paper is a direct verification of the Einstein relation-the simplest fluctuation dissipation relation for interacting particles advected in an active matter flow.

Biased-angle effect on diffusion dynamics and phase separation in anisotropic active particle system

A deep understanding for collective behavior in an active matter system with complex interactions has far-reaching impact in biology. In the present work, we adopt Langevin dynamics simulations to investigate diffusion dynamics and phase separation in an anisotropic active particle system with a tunable biased angle α defined as the deviation between the active force direction and anisotropic orientation. Our results demonstrate that the biased angle can induce super-rotational diffusion dynamics characterized by a power-law relationship between the mean square angle displacement (MSAD) and the time interval Δ t in the form of MSAD ∼ Δ t β with β > 1 and also result in non-trivial phase separation kinetics. As activity is dominant, nucleation time shows a non-monotonic dependence on the biased angle. Moreover, there arises a distinct transition of phase separation, from spinodal decomposition without apparent nucleation time to binodal decomposition with prominent nucleation delay. A significant inhibition effect occurs at right and obtuse angles, where the remarkable super-rotational diffusion prevents particle aggregation, leading to a slow nucleation process. As active force is competitive to anisotropic interactions, the system is almost homogeneous, while, intriguingly, we observe a re-entrant phase separation as a small acute angle is introduced. The prominent super-rotational diffusion under small angles provides an optimum condition for particle adsorption and cluster growth and, thus, accounts for the re-entrance of phase separation. A consistent scenario for the physical mechanism of our observations is achieved by properly considering the modulation of the biased angle on the interplay between activity and anisotropic interactions.

Topological fine structure of smectic grain boundaries and tetratic disclination lines within three-dimensional smectic liquid crystals

Observing and characterizing the complex ordering phenomena of liquid crystals subjected to external constraints constitutes an ongoing challenge for chemists and physicists alike. To elucidate the delicate balance appearing when the intrinsic positional order of smectic liquid crystals comes into play, we perform Monte-Carlo simulations of rod-like particles in a range of cavities with a cylindrical symmetry. Based on recent insights into the topology of smectic orientational grain boundaries in two dimensions, we analyze the emerging three-dimensional defect structures from the perspective of tetratic symmetry. Using an appropriate three-dimensional tetratic order parameter constructed from the Steinhardt order parameters, we show that those grain boundaries can be interpreted as a pair of tetratic disclination lines that are located on the edges of the nematic domain boundary. Thereby, we shed light on the fine structure of grain boundaries in three-dimensional confined smectics.

Influence of confinement on the spreading of bacterial populations

The spreading of bacterial populations is central to processes in agriculture, the environment, and medicine. However, existing models of spreading typically focus on cells in unconfined settings—despite the fact that many bacteria inhabit complex and crowded environments, such as soils, sediments, and biological tissues/gels, in which solid obstacles confine the cells and thereby strongly regulate population spreading. Here, we develop an extended version of the classic Keller-Segel model of bacterial spreading via motility that also incorporates cellular growth and division, and explicitly considers the influence of confinement in promoting both cell-solid and cell-cell collisions. Numerical simulations of this extended model demonstrate how confinement fundamentally alters the dynamics and morphology of spreading bacterial populations, in good agreement with recent experimental results. In particular, with increasing confinement, we find that cell-cell collisions increasingly hinder the initial formation and the long-time propagation speed of chemotactic pulses. Moreover, also with increasing confinement, we find that cellular growth and division plays an increasingly dominant role in driving population spreading—eventually leading to a transition from chemotactic spreading to growth-driven spreading via a slower, jammed front. This work thus provides a theoretical foundation for further investigations of the influence of confinement on bacterial spreading. More broadly, these results help to provide a framework to predict and control the dynamics of bacterial populations in complex and crowded environments.

The spreading of bacteria through their environments critically impacts our everyday lives; it can be harmful, underlying the progression of infections and spoilage of foods, or can be beneficial, enabling the delivery of therapeutics, sustaining plant growth, and remediating polluted terrain. In all these cases, bacteria typically inhabit crowded environments, such as soils, sediments, and biological tissues/gels, in which solid obstacles confine the cells and regulate their spreading. However, existing models of spreading typically focus on cells in unconfined settings, and thus are frequently not applicable to cells in more complex environments. Here, we address this gap in knowledge by extending the classic Keller-Segel model of bacterial spreading via motility to also incorporate cellular growth and division, and explicitly consider the influence of confinement. Through numerical simulations of this extended model, we show how confinement fundamentally alters the dynamics and morphology of spreading bacterial populations—in particular, driving a transition from chemotactic spreading of motile cells to growth-driven spreading via a slower, jammed front. These results provide a foundation for further investigations of the influence of confinement on bacterial spreading, both by yielding testable predictions for future experiments, and by providing guidelines to predict and control the dynamics of bacterial populations in complex and crowded environments.

Activity-Induced Enhancement of Superdiffusive Transport in Bacterial Turbulence

Superdiffusion processes significantly promote the transport of tiny passive particles within biological fluids. Activity, one of the essential measures for living matter, however, is less examined in terms of how and to what extent it can improve the diffusivity of the moving particles. Here, bacterial suspensions are confined within the microfluidic channel at the state of bacterial turbulence, and are tuned to different activity levels by oxygen consumption in control. Systematic measurements are conducted to determine the superdiffusion exponent, which characterizes the diffusivity strength of tracer particles, depending on the continuously injecting energy converted to motile activity from swimming individuals. Higher activity is quantified to drastically enhance the superdiffusion process of passive tracers in the short-time regime. Moreover, the number density of the swimming bacteria is controlled to contribute to the field activity, and then to strengthen the super-diffusivity of tracers, distinguished by regimes with and without collective motion of interacting bacteria. Finally, the non-slip surfaces of the microfluidic channel lower the superdiffusion of immersed tracers due to the resistance, with the small diffusivity differing from the counterpart in the bulk. The findings here suggest ways of controlled diffusion and transport of substances within the living system with different levels of nutrition and resources and boundary walls, leading to efficient mixing, drug delivery and intracellular communications.

Inverse Dirichlet weighting enables reliable training of physics informed neural networks

We characterize and remedy a failure mode that may arise from multi-scale dynamics with scale imbalances during training of deep neural networks, such as physics informed neural networks (PINNs). PINNs are popular machine-learning templates that allow for seamless integration of physical equation models with data. Their training amounts to solving an optimization problem over a weighted sum of data-fidelity and equation-fidelity objectives. Conflicts between objectives can arise from scale imbalances, heteroscedasticity in the data, stiffness of the physical equation, or from catastrophic interference during sequential training. We explain the training pathology arising from this and propose a simple yet effective inverse Dirichlet weighting strategy to alleviate the issue. We compare with Sobolev training of neural networks, providing the baseline of analytically ε-optimal training. We demonstrate the effectiveness of inverse Dirichlet weighting in various applications, including a multi-scale model of active turbulence, where we show orders of magnitude improvement in accuracy and convergence over conventional PINN training. For inverse modeling using sequential training, we find that inverse Dirichlet weighting protects a PINN against catastrophic forgetting.

Stochastic Hydrodynamics of Complex Fluids: Discretisation and Entropy Production

Many complex fluids can be described by continuum hydrodynamic field equations, to which noise must be added in order to capture thermal fluctuations. In almost all cases, the resulting coarse-grained stochastic partial differential equations carry a short-scale cutoff, which is also reflected in numerical discretisation schemes. We draw together our recent findings concerning the construction of such schemes and the interpretation of their continuum limits, focusing, for simplicity, on models with a purely diffusive scalar field, such as ‘Model B’ which describes phase separation in binary fluid mixtures. We address the requirement that the steady-state entropy production rate (EPR) must vanish for any stochastic hydrodynamic model in a thermal equilibrium. Only if this is achieved can the given discretisation scheme be relied upon to correctly calculate the nonvanishing EPR for ‘active field theories’ in which new terms are deliberately added to the fluctuating hydrodynamic equations that break detailed balance. To compute the correct probabilities of forward and time-reversed paths (whose ratio determines the EPR), we must make a careful treatment of so-called ‘spurious drift’ and other closely related terms that depend on the discretisation scheme. We show that such subtleties can arise not only in the temporal discretisation (as is well documented for stochastic ODEs with multiplicative noise) but also from spatial discretisation, even when noise is additive, as most active field theories assume. We then review how such noise can become multiplicative via off-diagonal couplings to additional fields that thermodynamically encode the underlying chemical processes responsible for activity. In this case, the spurious drift terms need careful accounting, not just to evaluate correctly the EPR but also to numerically implement the Langevin dynamics itself.

Understanding topological defects in fluidized dry active nematics

Dense assemblies of self-propelling rods (SPRs) may exhibit fascinating collective behaviors and anomalous physical properties that are far away from equilibrium. Using large-scale Brownian dynamics simulations, we investigate the dynamics of disclination defects in 2D fluidized swarming motions of dense dry SPRs (i.e., without hydrodynamic effects) that form notable local positional topological structures that are reminiscent of smectic order. We find the deformations of smectic-like rod layers can create unique polar structures that lead to slow translations and rotations of ±1/2-order defects, which are fundamentally different from the fast streaming defect motions observed in wet active matter. We measure and characterize the statistical properties of topological defects and reveal their connections with the coherent structures. Furthermore, we construct a bottom-up active-liquid-crystal model to analyze the instability of polar lanes, which effectively leads to defect formation between interlocked polar lanes and serves as the origin of the large-scale swarming motions.

Active Extensile Stress Promotes 3D Director Orientations and Flows

We use numerical simulations and linear stability analysis to study an active nematic layer where the director is allowed to point out of the plane. Our results highlight the difference between extensile and contractile systems. Contractile stress suppresses the flows perpendicular to the layer and favors in-plane orientations of the director. By contrast extensile stress promotes instabilities that can turn the director out of the plane, leaving behind a population of distinct, in-plane regions that continually elongate and divide. This supports extensile forces as a mechanism for the initial stages of layer formation in living systems, and we show that a planar drop with extensile (contractile) activity grows into three dimensions (remains in two dimensions). The results also explain the propensity of disclination lines in three dimensional active nematics to be of twist type in extensile or wedge type in contractile materials.

Ising-like Critical Behavior of Vortex Lattices in an Active Fluid

Turbulent vortex structures emerging in bacterial active fluids can be organized into regular vortex lattices by weak geometrical constraints such as obstacles. Here we show, using a continuum-theoretical approach, that the formation and destruction of these patterns exhibit features of a continuous second-order equilibrium phase transition, including long-range correlations, divergent susceptibility, and critical slowing down. The emerging vorticity field can be mapped onto a two-dimensional (2D) Ising model with antiferromagnetic nearest-neighbor interactions by coarse graining. The resulting effective temperature is found to be proportional to the strength of the nonlinear advection in the continuum model.

Active matter dynamics in confined microfluidic environments

The field of active matter is a nascent area of research in soft condensed matter physics, which is drawing on the expertise of researchers from diverse disciplines. Small scale active particles-both inorganic and biological-display non-trivial emergent dynamics and interactions that could help us understand complex biological processes and phenomena. Recently, using microfluidic technologies, several research groups have performed important experimental and theoretical studies to understand the behavior of self-propelled particles and molecular active matter within confined environments-to glean a fundamental understanding of the cellular processes occurring under ultra-low Reynolds number conditions. In this chapter, we would like to review applications of microfluidics in active matter research, highlighting a few important theoretical and experimental investigations. We will conclude the discussion with a note on the future of this field mentioning a few open questions that are at the forefront of our minds.

Cooperation in a fluid swarm of fuel-free micro-swimmers

While motile bacteria display rich dynamics in dense colonies, the phoretic nature of artificial micro-swimmers restricts their activity when crowded. Here we introduce a new class of synthetic micro-swimmers that are driven solely by light. By coupling a light absorbing particle to a fluid droplet we produce a colloidal chimera that transforms optical power into propulsive thermo-capillary action. The swimmers' internal drive allows them to operate for a long duration (days) and remain active when crowded, forming a high density fluid phase. We find that above a critical concentration, swimmers form a long lived crowded state that displays internal dynamics. When passive particles are introduced, the dense swimmer phase can re-arrange to spontaneously corral the passive particles. We derive a geometrical, depletion-like condition for corralling by identifying the role the passive particles play in controlling the effective concentration of the micro-swimmers.

Learning developmental mode dynamics from single-cell trajectories

Embryogenesis is a multiscale process during which developmental symmetry breaking transitions give rise to complex multicellular organisms. Recent advances in high-resolution live-cell microscopy provide unprecedented insights into the collective cell dynamics at various stages of embryonic development. This rapid experimental progress poses the theoretical challenge of translating high-dimensional imaging data into predictive low-dimensional models that capture the essential ordering principles governing developmental cell migration in complex geometries. Here, we combine mode decomposition ideas that have proved successful in condensed matter physics and turbulence theory with recent advances in sparse dynamical systems inference to realize a computational framework for learning quantitative continuum models from single-cell imaging data. Considering pan-embryo cell migration during early gastrulation in zebrafish as a widely studied example, we show how cell trajectory data on a curved surface can be coarse-grained and compressed with suitable harmonic basis functions. The resulting low-dimensional representation of the collective cell dynamics enables a compact characterization of developmental symmetry breaking and the direct inference of an interpretable hydrodynamic model, which reveals similarities between pan-embryo cell migration and active Brownian particle dynamics on curved surfaces. Due to its generic conceptual foundation, we expect that mode-based model learning can help advance the quantitative biophysical understanding of a wide range of developmental structure formation processes.

Role of Advective Inertia in Active Nematic Turbulence

Suspensions of active agents with nematic interactions exhibit complex spatiotemporal dynamics such as mesoscale turbulence. Since the Reynolds number of microscopic flows is very small on the scale of individual agents, inertial effects are typically excluded in continuum theories of active nematic turbulence. Whether active stresses can collectively excite inertial flows is currently unclear. To address this question, we investigate a two-dimensional continuum theory for active nematic turbulence. In particular, we compare mesoscale turbulence with and without the effects of advective inertia. We find that inertial effects can influence the flow already close to the onset of the turbulent state and, moreover, give rise to large-scale fluid motion for strong active driving. A detailed analysis of the kinetic energy budget reveals an energy transfer to large scales mediated by inertial advection. While this transfer is small in comparison to energy injection and dissipation, its effects accumulate over time. The inclusion of friction, which is typically present in experiments, can compensate for this effect. The findings suggest that the inclusion of inertia and friction may be necessary for dynamically consistent theories of active nematic turbulence.

Density fluctuations and energy spectra of 3D bacterial suspensions

Giant number fluctuations are often considered as a hallmark of the emergent nonequilibrium dynamics of active fluids. However, these anomalous density fluctuations have only been reported experimentally in two-dimensional dry active systems heretofore. Here, we investigate density fluctuations of bulk Escherichia coli suspensions, a paradigm of three-dimensional (3D) wet active fluids. Our experiments demonstrate the existence and quantify the scaling relation of giant number fluctuations in 3D bacterial suspensions. Surprisingly, the anomalous scaling persists at small scales in low-concentration suspensions well before the transition to active turbulence, reflecting the long-range nature of hydrodynamic interactions of 3D wet active fluids. To illustrate the origin of the density fluctuations, we measure the energy spectra of suspension flows and explore the density-energy coupling in both the steady and transient states of active turbulence. A scale-invariant density-independent correlation between density fluctuations and energy spectra is uncovered across a wide range of length scales. In addition, our experiments show that the energy spectra of bacterial turbulence exhibit the scaling of 3D active nematic fluids, challenging the common view of dense bacterial suspensions as active polar fluids.

Flow transitions and length scales of a channel-confined active nematic

We perform lattice Boltzmann simulations of an active nematic fluid confined in a two-dimensional channel to study the range of flow states that are stabilised by the confinement: unidirectional flow, oscillatory flow, the dancing state, localised active turbulence and fully-developed active turbulence. We analyse the flows in Fourier space, and measure a range of different length scales which describe the flows. We argue that the different states occur as a result of flow instabilities inherent to the system. As a consequence the characteristic length scale for oscillatory flow, the dancing state and localised active turbulence is set by the channel width. Fully-developed active turbulence occurs only when the channel width is larger than the intrinsic, active length scale of the bulk fluid. The results clarify why the activity number is a control parameter for the flow transitions.

Multiple functions of flagellar motility and chemotaxis in bacterial physiology

Most swimming bacteria are capable of following gradients of nutrients, signaling molecules and other environmental factors that affect bacterial physiology. This tactic behavior became one of the most-studied model systems for signal transduction and quantitative biology, and underlying molecular mechanisms are well characterized in Escherichia coli and several other model bacteria. In this review, we focus primarily on less understood aspect of bacterial chemotaxis, namely its physiological relevance for individual bacterial cells and for bacterial populations. As evident from multiple recent studies, even for the same bacterial species flagellar motility and chemotaxis might serve multiple roles, depending on the physiological and environmental conditions. Among these, finding sources of nutrients and more generally locating niches that are optimal for growth appear to be one of the major functions of bacterial chemotaxis, which could explain many chemoeffector preferences as well as flagellar gene regulation. Chemotaxis might also generally enhance efficiency of environmental colonization by motile bacteria, which involves intricate interplay between individual and collective behaviors and trade-offs between growth and motility. Finally, motility and chemotaxis play multiple roles in collective behaviors of bacteria including swarming, biofilm formation and autoaggregation, as well as in their interactions with animal and plant hosts.

Accumulation of Tetrahymena pyriformis on Interfaces

The behavior of ciliates has been studied for many years through environmental biology and the ethology of microorganisms, and recent hydrodynamic studies of microswimmers have greatly advanced our understanding of the behavioral dynamics at the single-cell level. However, the association between single-cell dynamics captured by microscopic observation and pattern dynamics obtained by macroscopic observation is not always obvious. Hence, to bridge the gap between the two, there is a need for experimental results on swarming dynamics at the mesoscopic scale. In this study, we investigated the spatial population dynamics of the ciliate, Tetrahymena pyriformis, based on quantitative data analysis. We combined the image processing of 3D micrographs and machine learning to obtain the positional data of individual cells of T. pyriformis and examined their statistical properties based on spatio-temporal data. According to the 3D spatial distribution of cells and their temporal evolution, cells accumulated both on the solid wall at the bottom surface and underneath the air–liquid interface at the top. Furthermore, we quantitatively clarified the difference in accumulation levels between the bulk and the interface by creating a simple behavioral model that incorporated quantitative accumulation coefficients in its solution. The accumulation coefficients can be compared under different conditions and between different species.

Emergent spiral vortex of confined biased active particles

Confinement is known to have profound effects on the collective dynamics of many active systems. Here, we investigate a modeled active system in circular confinement consisting of biased active particles, where the direction of active force deviates a biased angle from the principle orientation of the anisotropic interaction. We find that such particles can spontaneously form a spiral vortex with two concentric and counter-rotating regions near the boundary. The emerged vortex can be measured by the vortex order parameter which shows nonmonotonic dependencies on both the biased angle and the strength of the anisotropic interaction. Our work can provide an understanding of such dynamic behaviors and enable different strategies for designing ordered collective behaviors.

Effects of confinement on the dynamics and correlation scales in kinesin-microtubule active fluids

We study the influence of solid boundaries on dynamics and structure of kinesin-driven microtubule active fluids as the height of the container, H, increases from hundreds of micrometers to several millimeters. By three-dimensional tracking of passive tracers dispersed in the active fluid, we observe that the activity level, characterized by velocity fluctuations, increases as system size increases and retains a small-scale isotropy. Concomitantly, as the confinement level decreases, the velocity-velocity temporal correlation develops a strong positive correlation at longer times, suggesting the establishment of a "memory". We estimate the characteristic size of the flow structures from the spatial correlation function and find that, as the confinement becomes weaker, the correlation length, l_{c}, saturates at approximately 400 microns. This saturation suggests an intrinsic length scale which, along with the small-scale isotropy, demonstrates the multiscale nature of this kinesin-driven bundled microtubule active system.

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.

Anomalous Diffusion and Lévy Walks Distinguish Active from Inertial Turbulence

Bacterial swarms display intriguing dynamical states like active turbulence. Now, using a hydrodynamic model, we show that such dense active suspensions manifest superdiffusion, via Lévy walks, which masquerades as a crossover from ballistic to diffusive scaling in measurements of mean-squared displacements, and is tied to the emergence of hitherto undetected oscillatory streaks in the flow. Thus, while laying the theoretical framework of an emergent advantageous strategy in the collective behavior of microorganisms, our Letter underlines the essential differences between active and inertial turbulence.

Active bath-induced localization and collapse of passive semiflexible polymers

The conformational and dynamical properties of a passive polymer embedded in a bath of active Brownian particles (ABPs) are studied by Langevin dynamics simulations. Various activities and ABP concentrations below and above the critical values for motility-induced phase separation (MIPS) are considered. In a homogeneous ABP fluid, the embedded polymer swells with increasing bath activity, with stronger swelling for larger densities. The polymer dynamics is enhanced, with the diffusion coefficient increasing by a power-law with increasing activity, where the exponent depends on the ABP concentration. For ABP concentrations in the MIPS regime, we observe a localization of the polymer in the low-density ABP phase associated with polymer collapse for moderate activities and a reswelling for high activities accompanied by a preferred localization in the high-density ABP phase. Localization and reswelling are independent of the polymer stiffness, with stiff polymers behaving similarly to flexible polymers. The polymer collapse is associated with a slowdown of its dynamics and a significantly smaller center-of-mass diffusion coefficient. In general, the polymer dynamics can only partially be described by an effective (bath) temperature. Moreover, the properties of a polymer embedded in a homogeneous active bath deviate quantitatively from those of a polymer composed of active monomers, i.e., linear chains of ABPs; however, such a polymer exhibits qualitatively similar activity-dependent features.

Collective excitations in active fluids: Microflows and breakdown in spectral equipartition of kinetic energy

The effect of particle activity on collective excitations in active fluids of microflyers is studied. With an in silico study, we observed an oscillating breakdown of equipartition (uniform spectral distribution) of kinetic energy in reciprocal space. The phenomenon is related to short-range velocity-velocity correlations that were realized without forming of long-lived mesoscale vortices in the system. This stands in contrast to well-known mesoscale turbulence operating in active nematic systems (bacterial or artificial) and reveals the features of collective dynamics in active fluids, which should be important for structural transitions and glassy dynamics in active matter.

Flow coupling between active and passive fluids across water-oil interfaces

Active fluid droplets surrounded by oil can spontaneously develop circulatory flows. However, the dynamics of the surrounding oil and their influence on the active fluid remain poorly understood. To investigate interactions between the active fluid and the passive oil across their interface, kinesin-driven microtubule-based active fluid droplets were immersed in oil and compressed into a cylinder-like shape. The droplet geometry supported intradroplet circulatory flows, but the circulation was suppressed when the thickness of the oil layer surrounding the droplet decreased. Experiments with tracers and network structure analyses and continuum models based on the dynamics of self-elongating rods demonstrated that the flow transition resulted from flow coupling across the interface between active fluid and oil, with a millimeter-scale coupling length. In addition, two novel millifluidic devices were developed that could trigger and suppress intradroplet circulatory flows in real time: one by changing the thickness of the surrounding oil layer and the other by locally deforming the droplet. This work highlights the role of interfacial dynamics in the active fluid droplet system and shows that circulatory flows within droplets can be affected by millimeter-scale flow coupling across the interface between the active fluid and the oil.

Collective effects in confined active Brownian particles

We investigate a two-dimensional system of active particles confined to a narrow annular domain. Despite the absence of explicit interactions among the velocities or the active forces of different particles, the system displays a transition from a disordered and stuck state to an ordered state of global collective motion where the particles rotate persistently clockwise or anticlockwise. We describe this behavior by introducing a suitable order parameter, the velocity polarization, measuring the global alignment of the particles' velocities along the tangential direction of the ring. We also measure the spatial velocity correlation function and its correlation length to characterize the two states. In the rotating phase, the velocity correlation displays an algebraic decay that is analytically predicted together with its correlation length, while in the stuck regime, the velocity correlation decays exponentially with a correlation length that increases with the persistence time. In the first case, the correlation (and, in particular, its correlation length) does not depend on the active force but the system size only. The global collective motion, an effect caused by the interplay between finite-size, periodicity, and persistent active forces, disappears as the size of the ring becomes infinite, suggesting that this phenomenon does not correspond to a phase transition in the usual thermodynamic sense.

Enhanced diffusion of a tracer particle in a lattice model of a crowded active system

Living systems at the subcellular, cellular, and multicellular levels are often crowded systems that contain active particles. The active motion of these particles can also propel passive particles, which typically results in enhanced effective diffusion of the passive particles. Here we study the diffusion of a passive tracer particle in such a dense system of active crowders using a minimal lattice model incorporating particles pushing each other. We show that the model exhibits several regimes of motility and quantify the enhanced diffusion as a function of density and activity of the active crowders. Moreover, we demonstrate an interplay of tracer diffusion and clustering of active particles, which suppresses the enhanced diffusion. Simulations of mixtures of passive and active crowders show that a rather small fraction of active particles is sufficient for the observation of enhanced diffusion.

Defect Spirograph: Dynamical Behavior of Defects in Spatially Patterned Active Nematics

Topological defects in active liquid crystals can be confined by introducing gradients of activity. Here, we examine the dynamical behavior of two defects confined by a sharp gradient of activity that separates an active circular region and a surrounding passive nematic material. Continuum simulations are used to explain how the interplay among energy injection into the system, hydrodynamic interactions, and frictional forces governs the dynamics of topologically required self-propelling +1/2 defects. Our findings are rationalized in terms of a phase diagram for the dynamical response of defects in terms of activity and frictional damping strength. Different regions of the underlying phase diagram correspond to distinct dynamical modes, namely immobile defects, steady rotation of defects, bouncing defects, bouncing-cruising defects, dancing defects, and multiple defects with irregular dynamics. These dynamic states raise the prospect of generating synchronized defect arrays for microfluidic applications.

Design of nematic liquid crystals to control microscale dynamics

The dynamics of small particles, both living such as swimming bacteria and inanimate, such as colloidal spheres, has fascinated scientists for centuries. If one could learn how to control and streamline their chaotic motion, that would open technological opportunities in the transformation of stored or environmental energy into systematic motion, with applications in micro-robotics, transport of matter, guided morphogenesis. This review presents an approach to command microscale dynamics by replacing an isotropic medium with a liquid crystal. Orientational order and associated properties, such as elasticity, surface anchoring, and bulk anisotropy, enable new dynamic effects, ranging from the appearance and propagation of particle-like solitary waves to self-locomotion of an active droplet. By using photoalignment, the liquid crystal can be patterned into predesigned structures. In the presence of the electric field, these patterns enable the transport of solid and fluid particles through nonlinear electrokinetics rooted in anisotropy of conductivity and permittivity. Director patterns command the dynamics of swimming bacteria, guiding their trajectories, polarity of swimming, and distribution in space. This guidance is of a higher level of complexity than a simple following of the director by rod-like microorganisms. Namely, the director gradients mediate hydrodynamic interactions of bacteria to produce an active force and collective polar modes of swimming. The patterned director could also be engraved in a liquid crystal elastomer. When an elastomer coating is activated by heat or light, these patterns produce a deterministic surface topography. The director gradients define an activation force that shapes the elastomer in a manner similar to the active stresses triggering flows in active nematics. The patterned elastomer substrates could be used to define the orientation of cells in living tissues. The liquid-crystal guidance holds a major promise in achieving the goal of commanding microscale active flows.

Spontaneous directional flow of active magnetic particles

We predict the emergence of large-scale polar order and spontaneous directional flows in a class of self-propelled autonomous particles that interact via passive repulsion between off-center sites. The coupling of active motion with the passive torque acting about the particle centers results in hybrid active-passive interactions responsible for a macroscopic phase transition from an isotropic state to a polar-aligned state in systems of particles with front interaction sites. We employ a continuum kinetic theory to explain that the emergence of long-ranged orientational order, which occurs in unbounded domains at finite densities, can be externally activated independently of the self-propulsion mechanism and drives a macroscopic particle flow in a direction selected by symmetry breaking.

Chemotactic migration of bacteria in porous media

Chemotactic migration of bacteria-their ability to direct multicellular motion along chemical gradients-is central to processes in agriculture, the environment, and medicine. However, current understanding of migration is based on studies performed in bulk liquid, despite the fact that many bacteria inhabit tight porous media such as soils, sediments, and biological gels. Here, we directly visualize the chemotactic migration of Escherichia coli populations in well-defined 3D porous media in the absence of any other imposed external forcing (e.g., flow). We find that pore-scale confinement is a strong regulator of migration. Strikingly, cells use a different primary mechanism to direct their motion in confinement than in bulk liquid. Furthermore, confinement markedly alters the dynamics and morphology of the migrating population-features that can be described by a continuum model, but only when standard motility parameters are substantially altered from their bulk liquid values to reflect the influence of pore-scale confinement. Our work thus provides a framework to predict and control the migration of bacteria, and active matter in general, in complex environments.

Persistence length regulates emergent dynamics in active roller ensembles

Active colloidal fluids, biological and synthetic, often demonstrate complex self-organization and the emergence of collective behavior. Spontaneous formation of multiple vortices has been recently observed in a variety of active matter systems, however, the generation and tunability of the active vortices not controlled by geometrical confinement remain challenging. Here, we exploit the persistence length of individual particles in ensembles of active rollers to tune the formation of vortices and to orchestrate their characteristic sizes. We use two systems and employ two different approaches exploiting shape anisotropy or polarization memory of individual units for control of the persistence length. We characterize the dynamics of emergent multi-vortex states and reveal a direct link between the behavior of the persistence length and properties of the emergent vortices. We further demonstrate common features between the two systems including anti-ferromagnetic ordering of the neighboring vortices and active turbulent behavior with a characteristic energy cascade in the particles velocity field energy spectra. Our findings provide insights into the onset of spatiotemporal coherence in active roller systems and suggest a control knob for manipulation of dynamic self-assembly in active colloidal ensembles.

Spatial velocity correlations in inertial systems of active Brownian particles

Recently, it has been discovered that systems of active Brownian particles (APB) at high density organise their velocities into coherent domains showing large spatial structures in the velocity field. This collective behavior occurs spontaneously, i.e. is not caused by any specific interparticle force favoring the alignment of the velocities. This phenomenon was investigated in the absence of thermal noise and in the overdamped regime where inertial forces could be neglected. In this work, we demonstrate through numerical simulations and theoretical analysis that velocity alignment is a robust property of ABP and persists even in the presence of inertial forces and thermal fluctuations. We also show that a single dimensionless parameter, such as the Péclet number customarily employed in the description of self-propelled particles, is not sufficient to fully characterize this phenomenon either in the regimes of large viscosity or small mass. Indeed, the size of the velocity domains, measured through the correlation length of the spatial velocity correlation, remains constant when the swim velocity increases and decreases as the rotational diffusion becomes larger. We find that, contrary to the common belief, the spatial velocity correlation not only depends on inertia but is also non-symmetrically affected by mass and inverse viscosity variations. We conclude that in self-propelled systems, at variance with passive systems, variations in the inertial time (mass over solvent viscosity) and mass act as independent control parameters. Finally, we highlight the non-thermal nature of the spatial velocity correlations that are fairly insensitive both to solvent and active temperatures.

Imaging the emergence of bacterial turbulence: Phase diagram and transition kinetics

We experimentally study the emergence of collective bacterial swimming, a phenomenon often referred to as bacterial turbulence. A phase diagram of the flow of 3D Escherichia coli suspensions spanned by bacterial concentration, the swimming speed of bacteria, and the number fraction of active swimmers is systematically mapped, which shows quantitative agreement with kinetic theories and demonstrates the dominant role of hydrodynamic interactions in bacterial collective swimming. We trigger bacterial turbulence by suddenly increasing the swimming speed of light-powered bacteria and image the transition to the turbulence in real time. Our experiments identify two unusual kinetic pathways, i.e., the one-step transition with long incubation periods near the phase boundary and the two-step transition driven by long-wavelength instabilities deep inside the turbulent phase. Our study provides not only a quantitative verification of existing theories but also insights into interparticle interactions and transition kinetics of bacterial turbulence.

Hydrodynamic collision between a microswimmer and a passive particle in a micro-channel

Microswimmers interacting with passive particles in confinement are common in many systems, e.g., spermatozoa encountering other cells or debris in the female reproductive tract or active particles interacting with polymers and tracers in microfluidic channels. The behaviour of such systems is driven by simultaneous, three way hydrodynamic interactions between the microswimmer, the passive particle and the microchannel walls. Therefore, in this work we investigate the hydrodynamic collision between a model microswimmer and a passive particle using three different methods: (i) the point particle approach, (ii) analytical calculations based on method of reflections, and (iii) lattice Boltzmann numerical simulations. We show that the hydrodynamic collision is essentially an asymmetric process - the trajectory of the microswimmer is altered only in an intermediate stage while the passive particle undergoes a three stage displacement with a net displacement towards or away from the microchannel walls. The path of the passive particle is a simple consequence of the velocity field generated by the swimmer: an open triangle in bulk fluid and a loop-like trajectory in confinement. We demonstrate the generality of our findings and conclude that the net displacement of the passive particle due to collision may be capitalised in order to develop applications such as size separation of colloidal particles and deposition of particles in the microchannel interiors.

Demixing of two species via reciprocally concentration-dependent diffusivity

We propose a model for demixing of two species by assuming a density-dependent effective diffusion coefficient of the particles. Both sorts of microswimmers diffuse as active overdamped Brownian particles with a noise intensity that is determined by the surrounding density of the respective other species within a sensing radius r_{s}. A higher concentration of the first (second) sort will enlarge the diffusion and, in consequence, the intensity of the noise experienced by the second (first) sort. Numerical and analytical investigations of steady states of the macroscopic equations prove the demixing of particles due to this reciprocally concentration-dependent diffusivity. An ambiguity of the numerical integration scheme for the purely local model (r_{s}→0) is resolved by considering nonvanishing sensing radii in a nonlocal model with r_{s}>0.

Memory effects, arches and polar defect ordering at the cross-over from wet to dry active nematics

We use analytic arguments and numerical solutions of the continuum, active nematohydrodynamic equations to study how friction alters the behaviour of active nematics. Concentrating on the case where there is nematic ordering in the passive limit, we show that, as the friction is increased, memory effects become more prominent and +1/2 topological defects leave increasingly persistent trails in the director field as they pass. The trails are preferential sites for defect formation and they tend to impose polar order on any new +1/2 defects. In the absence of noise and for high friction, it becomes very difficult to create defects, but trails formed by any defects present at the beginning of the simulations persist and organise into parallel arch-like patterns in the director field. We show aligned arches of equal width are approximate steady state solutions of the equations of motion which co-exist with the nematic state. We compare our results to other models in the literature, in particular dry systems with no hydrodynamics, where trails, arches and polar defect ordering have also been observed.

Heterogeneous bacterial swarms with mixed lengths

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

Confinement Controls the Bend Instability of Three-Dimensional Active Liquid Crystals

Spontaneous growth of long-wavelength deformations is a defining feature of active liquid crystals. We investigate the effect of confinement on the instability of 3D active liquid crystals in the isotropic phase composed of extensile microtubule bundles and kinesin molecular motors. When shear aligned, such fluids exhibit finite-wavelength self-amplifying bend deformations. By systematically changing the channel size we elucidate how the instability wavelength and its growth rate depend on the channel dimensions. Experimental findings are qualitatively consistent with a minimal hydrodynamic model, where the fastest growing deformation is set by a balance of active driving and elastic relaxation. Our results demonstrate that confinement determines the structure and dynamics of active fluids on all experimentally accessible length scales.

Role of Friction in Multidefect Ordering

We use continuum simulations to study the impact of friction on the ordering of defects in an active nematic. Even in a frictionless system, +1/2 defects tend to align side by side and orient antiparallel reflecting their propensity to form, and circulate with, flow vortices. Increasing friction enhances the effectiveness of the defect-defect interactions, and defects form dynamically evolving, large-scale, positionally, and orientationally ordered structures, which can be explained as a competition between hexagonal packing, preferred by the -1/2 defects, and rectangular packing, preferred by the +1/2 defects.