Fluid velocity fluctuations in a suspension of swimming protists

Confinement, chaotic transport, and trapping of active swimmers in time-periodic flows

Microorganisms encounter complex unsteady flows, including algal blooms in marine settings, microbial infections in airways, and bioreactors for vaccine and biofuel production. Here, we study the transport of active swimmers in two-dimensional time-periodic flows using Langevin simulations and experiments with swimming bacteria. We find that long-term swimmer transport is controlled by two parameters, the pathlength of the unsteady flow and the normalized swimmer speed. The pathlength nonmonotonically controls swimmer dispersion dynamics, giving rise to three distinct dispersion regimes. Weak flows hinder swimmer transport by confining cells toward flow manifolds. As pathlength increases, chaotic transport along flow manifolds initiates, maximizing the number of unique flow cells traveled. Last, strong flows trap swimmers at the vortex core, suppressing dispersal. Experiments with Vibrio cholerae showed qualitative agreement with model dispersion patterns. Our results reveal that nontrivial chaotic transport can arise in simple unsteady flows and suggest a potentially optimal dispersal strategy for microswimmers in nature.

Bacteria hinder large-scale transport and enhance small-scale mixing in time-periodic flows

Understanding mixing and transport of passive scalars in active fluids is important to many natural (e.g., algal blooms) and industrial (e.g., biofuel, vaccine production) processes. Here, we study the mixing of a passive scalar (dye) in dilute suspensions of swimming Escherichia coli in experiments using a two-dimensional (2D) time-periodic flow and in a simple simulation. Results show that the presence of bacteria hinders large-scale transport and reduces overall mixing rate. Stretching fields, calculated from experimentally measured velocity fields, show that bacterial activity attenuates fluid stretching and lowers flow chaoticity. Simulations suggest that this attenuation may be attributed to a transient accumulation of bacteria along regions of high stretching. Spatial power spectra and correlation functions of dye-concentration fields show that the transport of scalar variance across scales is also hindered by bacterial activity, resulting in an increase in average size and lifetime of structures. On the other hand, at small scales, activity seems to enhance local mixing. One piece of evidence is that the probability distribution of the spatial concentration gradients is nearly symmetric with a vanishing skewness. Overall, our results show that the coupling between activity and flow can lead to nontrivial effects on mixing and transport.

Fine balance of chemotactic and hydrodynamic torques: When microswimmers orbit a pillar just once

We study the detention statistics of self-propelling droplet microswimmers attaching to microfluidic pillars. These droplets show negative autochemotaxis: they shed a persistent repulsive trail of spent fuel that biases them to detach from pillars in a specific size range after orbiting them just once. We have designed a microfluidic assay recording microswimmers in pillar arrays of varying diameter, derived detention statistics via digital image analysis, and interpreted these statistics via the Langevin dynamics of an active Brownian particle model. By comparing data from orbits with and without residual chemical field, we can independently estimate quantities such as hydrodynamic and chemorepulsive torques, chemical coupling constants and diffusion coefficients, as well as their dependence on environmental factors such as wall curvature. This type of analysis is generalizable to many kinds of microswimmers.

Pair dispersion in dilute suspension of active swimmers

Ensembles of biological and artificial microswimmers produce long-range velocity fields with strong nonequilibrium fluctuations, which result in a dramatic increase in diffusivity of embedded particles (tracers). While such enhanced diffusivity may point to enhanced mixing of the fluid, a rigorous quantification of the mixing efficiency requires analysis of pair dispersion of tracers, rather than simple one-particle diffusivity. Here, we calculate analytically the scale-dependent coefficient of relative diffusivity of passive tracers embedded in a dilute suspension of run-and-tumble microswimmers. Although each tracer is subject to strong fluctuations resulting in large absolute diffusivity, the small-scale relative dispersion is suppressed due to the correlations in fluid velocity which are relevant when the inter-tracer separation is below the persistence length of the swimmer's motion. Our results suggest that the reorientation of swimming direction plays an important role in biological mixing and should be accounted in the design of potential active matter devices capable of effective fluid mixing at microscale.

Autonomous Actuation of Zero Modes in Mechanical Networks Far from Equilibrium

A zero mode, or floppy mode, is a nontrivial coupling of mechanical components yielding a degree of freedom with no resistance to deformation. Engineered zero modes have the potential to act as microscopic motors or memory devices, but this requires an internal actuation mechanism that can overcome unwanted fluctuations in other modes and the dissipation inherent in real systems. In this Letter, we show theoretically and experimentally that complex zero modes in mechanical networks can be selectively mobilized by nonequilibrium activity. We find that a correlated active bath actuates an infinitesimal zero mode while simultaneously suppressing fluctuations in higher modes compared to thermal fluctuations, which we experimentally mimic by high frequency shaking of a physical network. Furthermore, self-propulsive dynamics spontaneously mobilize finite mechanisms as exemplified by a self-propelled topological soliton. Nonequilibrium activity thus enables autonomous actuation of coordinated mechanisms engineered through network topology.

Analysis of a model microswimmer with applications to blebbing cells and mini-robots

Recent research has shown that motile cells can adapt their mode of propulsion depending on the environment in which they find themselves. One mode is swimming by blebbing or other shape changes, and in this paper we analyze a class of models for movement of cells by blebbing and of nano-robots in a viscous fluid at low Reynolds number. At the level of individuals, the shape changes comprise volume exchanges between connected spheres that can control their separation, which are simple enough that significant analytical results can be obtained. Our goal is to understand how the efficiency of movement depends on the amplitude and period of the volume exchanges when the spheres approach closely during a cycle. Previous analyses were predicated on wide separation, and we show that the speed increases significantly as the separation decreases due to the strong hydrodynamic interactions between spheres in close proximity. The scallop theorem asserts that at least two degrees of freedom are needed to produce net motion in a cyclic sequence of shape changes, and we show that these degrees can reside in different swimmers whose collective motion is studied. We also show that different combinations of mode sharing can lead to significant differences in the translation and performance of pairs of swimmers.

Intrinsic flow structure and multifractality in two-dimensional bacterial turbulence

The active interaction between the bacteria and fluid generates turbulent structures even at zero Reynolds number. The velocity of such a flow obtained experimentally has been quantitatively investigated based on streamline segment analysis. There is a clear transition at about 16 times the organism body length separating two different scale regimes, which may be attributed to the different influence of the viscous effect. Surprisingly the scaling extracted from the streamline segment indicates the existence of scale similarity even at the zero Reynolds number limit. Moreover, the multifractal feature can be quantitatively described via a lognormal formula with the Hurst number H=0.76 and the intermittency parameter μ=0.20, which is coincidentally in agreement with the three-dimensional hydrodynamic turbulence result. The direction of cascade is measured via the filter-space technique. An inverse energy cascade is confirmed. For the enstrophy, a forward cascade is observed when r/R≤3, and an inverse one is observed when r/R>3, where r and R are the separation distance and the bacteria body size, respectively. Additionally, the lognormal statistics is verified for the coarse-grained energy dissipation and enstrophy, which supports the lognormal formula to fit the measured scaling exponent.

Non-Gaussian limit fluctuations in active swimmer suspensions

We investigate the hydrodynamic fluctuations in suspensions of swimming microorganisms (Chlamydomonas) by observing the probe particles dispersed in the media. Short-term fluctuations of probe particles were superdiffusive and displayed heavily tailed non-Gaussian distributions. The analytical theory that explains the observed distribution was derived by summing the power-law-decaying hydrodynamic interactions from spatially distributed field sources (here, swimming microorganisms). The summing procedure, which we refer to as the physical limit operation, is applicable to a variety of physical fluctuations to which the classical central limiting theory does not apply. Extending the analytical formula to compare to experiments in active swimmer suspensions, we show that the non-Gaussian shape of the observed distribution obeys the analytic theory concomitantly with independently determined parameters such as the strength of force generations and the concentration of Chlamydomonas. Time evolution of the distributions collapsed to a single master curve, except for their extreme tails, for which our theory presents a qualitative explanation. Investigations thereof and the complete agreement with theoretical predictions revealed broad applicability of the formula to dispersions of active sources of fluctuations.

Analytical Limit Distributions from Random Power-Law Interactions

Nature is full of power-law interactions, e.g., gravity, electrostatics, and hydrodynamics. When sources of such fields are randomly distributed in space, the superposed interaction, which is what we observe, is naively expected to follow a Gauss or Lévy distribution. Here, we present an analytic expression for the actual distributions that converge to novel limits that are in between these already-known limit distributions, depending on physical parameters, such as the concentration of field sources and the size of the probe used to measure the interactions. By comparing with numerical simulations, the origin of non-Gauss and non-Lévy distributions are theoretically articulated.

Intermittency measurement in two-dimensional bacterial turbulence

In this paper, an experimental velocity database of a bacterial collective motion, e.g., Bacillus subtilis, in turbulent phase with volume filling fraction 84% provided by Professor Goldstein at Cambridge University (UK), was analyzed to emphasize the scaling behavior of this active turbulence system. This was accomplished by performing a Hilbert-based methodology analysis to retrieve the scaling property without the β-limitation. A dual-power-law behavior separated by the viscosity scale ℓ_{ν} was observed for the qth-order Hilbert moment L_{q}(k). This dual-power-law belongs to an inverse-cascade since the scaling range is above the injection scale R, e.g., the bacterial body length. The measured scaling exponents ζ(q) of both the small-scale (k>k_{ν}) and large-scale (k<k_{ν}) motions are convex, showing the multifractality. A log-normal formula was put forward to characterize the multifractal intensity. The measured intermittency parameters are μ_{S}=0.26 and μ_{L}=0.17, respectively, for the small- and large-scale motions. It implies that the former cascade is more intermittent than the latter one, which is also confirmed by the corresponding singularity spectrum f(α) versus α. Comparison with the conventional two-dimensional Ekman-Navier-Stokes equation, a continuum model indicates that the origin of the multifractality could be a result of some additional nonlinear interaction terms, which deservers a more careful investigation.

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.

Green Algae as Model Organisms for Biological Fluid Dynamics

In the past decade the volvocine green algae, spanning from the unicellular Chlamydomonas to multicellular Volvox, have emerged as model organisms for a number of problems in biological fluid dynamics. These include flagellar propulsion, nutrient uptake by swimming organisms, hydrodynamic interactions mediated by walls, collective dynamics and transport within suspensions of microswimmers, the mechanism of phototaxis, and the stochastic dynamics of flagellar synchronization. Green algae are well suited to the study of such problems because of their range of sizes (from 10 μm to several millimetres), their geometric regularity, the ease with which they can be cultured and the availability of many mutants that allow for connections between molecular details and organism-level behavior. This review summarizes these recent developments and highlights promising future directions in the study of biological fluid dynamics, especially in the context of evolutionary biology, that can take advantage of these remarkable organisms.

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

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

Fluid mixing by curved trajectories of microswimmers

We consider the tracer diffusion D(rr) that arises from the run-and-tumble motion of low Reynolds number swimmers, such as bacteria. Assuming a dilute suspension, where the bacteria move in uncorrelated runs of length λ, we obtain an exact expression for D(rr) for dipolar swimmers in three dimensions, hence explaining the surprising result that this is independent of λ. We compare D(rr) to the contribution to tracer diffusion from entrainment.

Active temperature and velocity correlations produced by a swimmer suspension

The agitation produced in a fluid by a suspension of microswimmers in the low Reynolds number limit is studied. In this limit, swimmers are modeled as force dipoles all with equal strength. The agitation is characterized by the active temperature defined, as in kinetic theory, as the mean square velocity, and by the equal-time spatial correlations. Considering the phase in which the swimmers are homogeneously and isotropically distributed in the fluid, it is shown that the active temperature and velocity correlations depend on a single scalar correlation function of the dipole-dipole correlation function. By making a simple medium-range order model, in which the dipole-dipole correlation function is characterized by a single correlation length k(0)(-1) it is possible to make quantitative predictions. It is found that the active temperature depends on the system size, scaling as L(4-d) at large correlation lengths L<<k(0)(-1), while in the opposite limit it saturates in three dimensions and diverges logarithmically with the system size in two dimensions. In three dimensions the velocity correlations decay as 1/r for small correlation lengths, while at large correlation lengths the transverse correlation function becomes negative at maximum separation r~L/2, an effect that disappears as the system increases in size.

Emergence of coherent structures and large-scale flows in motile suspensions

The emergence of coherent structures, large-scale flows and correlated dynamics in suspensions of motile particles such as swimming micro-organisms or artificial microswimmers is studied using direct particle simulations. A detailed model is proposed for a slender rod-like particle that propels itself in a viscous fluid by exerting a prescribed tangential stress on its surface, and a method is devised for the efficient calculation of hydrodynamic interactions in large-scale suspensions of such particles using slender-body theory and a smooth particle-mesh Ewald algorithm. Simulations are performed with periodic boundary conditions for various system sizes and suspension volume fractions, and demonstrate a transition to large-scale correlated motions in suspensions of rear-actuated swimmers, or Pushers, above a critical volume fraction or system size. This transition, which is not observed in suspensions of head-actuated swimmers, or Pullers, is seen most clearly in particle velocity and passive tracer statistics. These observations are consistent with predictions from our previous mean-field kinetic theory, one of which states that instabilities will arise in uniform isotropic suspensions of Pushers when the product of the linear system size with the suspension volume fraction exceeds a given threshold. We also find that the collective dynamics of Pushers result in giant number fluctuations, local alignment of swimmers and strongly mixing flows. Suspensions of Pullers, which evince no large-scale dynamics, nonetheless display interesting deviations from the random isotropic state.

Interplay between energetics and dynamics in bacterial motility

We study how self-propelled organisms administer their energetic resources in order to optimize space exploration. Noting the existence of two very different time scales, we use a quasistatic approximation to analyze the relation between bacterial dynamics and changes in the energy stored by a bacterium. We then find both steady-state and time-dependent solutions for the bacterial speed and stored energy. The model also predicts the volume of the region that a bacterium may visit in a resource-depleted medium.

Enhancement of biomixing by swimming algal cells in two-dimensional films

Fluid mixing in active suspensions of microorganisms is important to ecological phenomena and presents a fascinating stochastic process. We investigate the mixing produced by swimming unicellular algal cells (Chlamydomonas) in quasi-two-dimensional liquid films by simultaneously tracking the motion of the cells and that of microscopic passive tracer particles advected by the fluid. The reduced spatial dimension of the system leads to long-range flows and a surprisingly strong dependence of tracer transport on the concentration of swimmers, which is explored over a wide range. The mean square displacements are well described by a stochastic Langevin model, which is used to parameterize the mixing. The effective diffusion coefficient D grows rapidly with the swimmer concentration Φ as D ∼ Φ(3/2), as a result of the increasing frequency of tracer-swimmer interactions and the long-range hydrodynamic disturbances created by the swimmers. Conditional sampling of the tracer data based on the instantaneous swimmer position shows that the rapid growth of the diffusivity enhancement with concentration must be due to particle interactions with multiple swimmers simultaneously. Finally, the anomalous probability distributions of tracer displacements become Gaussian at high concentration, but manifest strong power-law tails at low concentration, while the tracer displacements always grow diffusively in time.

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

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

Lévy fluctuations and mixing in dilute suspensions of algae and bacteria

Swimming micro-organisms rely on effective mixing strategies to achieve efficient nutrient influx. Recent experiments, probing the mixing capability of unicellular biflagellates, revealed that passive tracer particles exhibit anomalous non-Gaussian diffusion when immersed in a dilute suspension of self-motile Chlamydomonas reinhardtii algae. Qualitatively, this observation can be explained by the fact that the algae induce a fluid flow that may occasionally accelerate the colloidal tracers to relatively large velocities. A satisfactory quantitative theory of enhanced mixing in dilute active suspensions, however, is lacking at present. In particular, it is unclear how non-Gaussian signatures in the tracers' position distribution are linked to the self-propulsion mechanism of a micro-organism. Here, we develop a systematic theoretical description of anomalous tracer diffusion in active suspensions, based on a simplified tracer-swimmer interaction model that captures the typical distance scaling of a microswimmer's flow field. We show that the experimentally observed non-Gaussian tails are generic and arise owing to a combination of truncated Lévy statistics for the velocity field and algebraically decaying time correlations in the fluid. Our analytical considerations are illustrated through extensive simulations, implemented on graphics processing units to achieve the large sample sizes required for analysing the tails of the tracer distributions.