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

Chemotactic clusters in confined run-and-tumble bacteria: a numerical investigation

We present a simulation study of pattern formation in an ensemble of chemotactic run-and-tumble bacteria, focussing on the effect of spatial confinement, either within traps or inside a maze. These geometries are inspired by previous experiments probing pattern formation in chemotactic strains of E. coli under these conditions. Our main result is that a microscopic model of chemotactic run-and-tumble particles which themselves secrete a chemoattractant is able to reproduce the main experimental observations, namely the formation of bacterial aggregates within traps and in dead ends of a maze. Our simulations also demonstrate that stochasticity plays a key role and leads to a hysteretic response when the chemotactic sensitivity is varied. We compare the results of run-and-tumble particles with simulations performed with a simplified version of the model where the active particles are smooth swimmers which respond to chemotactic gradients by rotating towards the source of chemoattractant. This class of models leads again to aggregation, but with quantitative and qualitative differences in, for instance, the size and shape of clusters.

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

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

Collective motion of spherical bacteria

A large variety of motile bacterial species exhibit collective motions while inhabiting liquids or colonizing surfaces. These collective motions are often characterized by coherent dynamic clusters, where hundreds of cells move in correlated whirls and jets. Previously, all species that were known to form such motion had a rod-shaped structure, which enhances the order through steric and hydrodynamic interactions. Here we show that the spherical motile bacteria Serratia marcescens exhibit robust collective dynamics and correlated coherent motion while grown in suspensions. As cells migrate to the upper surface of a drop, they form a monolayer, and move collectively in whirls and jets. At all concentrations, the distribution of the bacterial speed was approximately Rayleigh with an average that depends on concentration in a non-monotonic way. Other dynamical parameters such as vorticity and correlation functions are also analyzed and compared to rod-shaped bacteria from the same strain. Our results demonstrate that self-propelled spherical objects do form complex ordered collective motion. This opens a door for a new perspective on the role of cell aspect ratio and alignment of cells with regards to collective motion in nature.

Active matter ratchets with an external drift

When active matter particles such as swimming bacteria are placed in an asymmetric array of funnels, it has been shown that a ratchet effect can occur even in the absence of an external drive. Here we examine active ratchets for two-dimensional arrays of funnels or L shapes where there is also an externally applied dc drive or drift. We show that for certain conditions the ratchet effect can be strongly enhanced and it is possible to have conditions under which run-and-tumble particles with one run length move in the opposite direction from particles with a different run length. For the arrays of L shapes, we find that the application of a drift force can enhance a transverse rectification in the direction perpendicular to the drift. When particle-particle steric interactions are included, we find that the ratchet effects can be either enhanced or suppressed depending on barrier geometry, particle run length, and particle density.

Correlation properties of collective motion in bacterial suspensions

The study of collective motion in bacterial suspensions has been of significant recent interest. To better understand the non-trivial spatio-temporal correlations emerging in the course of collective swimming in suspensions of motile bacteria, a simple model is employed: a bacterium is represented as a force dipole with size, through the use of a short-range repelling potential, and shape. The model emphasizes two fundamental mechanisms: dipolar hydrodynamic interactions and short-range bacterial collisions. Using direct particle simulations validated by a dedicated experiment, we show that changing the swimming speed or concentration alters the time scale of sustained collective motion, consistent with experiment. Also, the correlation length in the collective state is almost constant as concentration and swimming speed change even though increasing each greatly increases the input of energy to the system. We demonstrate that the particle shape is critical for the onset of collective effects. In addition, new experimental results are presented illustrating the onset of collective motion with an ultrasound technique. This work exemplifies the delicate balance between various physical mechanisms governing collective motion in bacterial suspensions and provides important insights into its mesoscopic nature.

Intraspecies variation in cotton border cell production: rhizosphere microbiome implications

PREMISE OF THE STUDY

Border cells, which separate from the root cap, can comprise >90% of carbon-based exudates released into the rhizosphere, but may not provide a general source of nutrients for soil microorganisms. Instead, this population of specialized cells appears to function in defense of the root tip by an extracellular trapping process similar to that of mammalian white blood cells. Border cell production is tightly regulated, and direct tests of their impact on crop production have been hindered by lack of intraspecies variation. •

METHODS

Border cell number, viability, and clumping were compared among 22 cotton cultivars. Slime layer "extracellular trap" production by border cells in response to copper chloride, an elicitor of plant defenses, was compared in two cultivars with divergent border cell production. Trapping of bacteria by border cells in these lines also was measured. •

KEY RESULTS

Emerging roots of some cultivars produced more than 20000 border cells per root, a 100% increase over previously reported values for this species. No differences in border cell morphology, viability, or clumping were found. Copper chloride-induced extracellular trap formation by border cells from a cultivar that produced 27921 ± 2111 cells per root was similar to that of cells from a cultivar with 10002 ± 614 cells, but bacterial trapping was reduced. •

CONCLUSIONS

Intraspecific variation in border cell production provides a tool to measure their impact on plant development in the laboratory, greenhouse, and field. Further research is needed to determine the basis for this variation, and its impact on rhizosphere community structure.

Periodic reversals in Paenibacillus dendritiformis swarming

Bacterial swarming is a type of motility characterized by a rapid and collective migration of bacteria on surfaces. Most swarming species form densely packed dynamic clusters in the form of whirls and jets, in which hundreds of rod-shaped rigid cells move in circular and straight patterns, respectively. Recent studies have suggested that short-range steric interactions may dominate hydrodynamic interactions and that geometrical factors, such as a cell's aspect ratio, play an important role in bacterial swarming. Typically, the aspect ratio for most swarming species is only up to 5, and a detailed understanding of the role of much larger aspect ratios remains an open challenge. Here we study the dynamics of Paenibacillus dendritiformis C morphotype, a very long, hyperflagellated, straight (rigid), rod-shaped bacterium with an aspect ratio of ~20. We find that instead of swarming in whirls and jets as observed in most species, including the shorter T morphotype of P. dendritiformis, the C morphotype moves in densely packed straight but thin long lines. Within these lines, all bacteria show periodic reversals, with a typical reversal time of 20 s, which is independent of their neighbors, the initial nutrient level, agar rigidity, surfactant addition, humidity level, temperature, nutrient chemotaxis, oxygen level, illumination intensity or gradient, and cell length. The evolutionary advantage of this unique back-and-forth surface translocation remains unclear.

Fluid dynamics of bacterial turbulence

Self-sustained turbulent structures have been observed in a wide range of living fluids, yet no quantitative theory exists to explain their properties. We report experiments on active turbulence in highly concentrated 3D suspensions of Bacillus subtilis and compare them with a minimal fourth-order vector-field theory for incompressible bacterial dynamics. Velocimetry of bacteria and surrounding fluid, determined by imaging cells and tracking colloidal tracers, yields consistent results for velocity statistics and correlations over 2 orders of magnitude in kinetic energy, revealing a decrease of fluid memory with increasing swimming activity and linear scaling between kinetic energy and enstrophy. The best-fit model allows for quantitative agreement with experimental data.

Collision of microswimmers in a viscous fluid

We investigate the effects of boundary conditions on the surface of self-propelled spherical swimmers moving in a viscous fluid with a low Reynolds number. We first show that collisions between the swimmers are impossible under the commonly used no-slip conditions. Next we demonstrate that collisions do occur if the more general Navier boundary conditions, allowing for a finite slip on the surface that produces drag, are imposed on the boundary of swimmers. The presence of a small inertia for each swimmer does not influence whether collisions occur between swimmers.

Emergent dynamics of laboratory insect swarms

Collective animal behaviour occurs at nearly every biological size scale, from single-celled organisms to the largest animals on earth. It has long been known that models with simple interaction rules can reproduce qualitative features of this complex behaviour. But determining whether these models accurately capture the biology requires data from real animals, which has historically been difficult to obtain. Here, we report three-dimensional, time-resolved measurements of the positions, velocities, and accelerations of individual insects in laboratory swarms of the midge Chironomus riparius. Even though the swarms do not show an overall polarisation, we find statistical evidence for local clusters of correlated motion. We also show that the swarms display an effective large-scale potential that keeps individuals bound together, and we characterize the shape of this potential. Our results provide quantitative data against which the emergent characteristics of animal aggregation models can be benchmarked.

Physical properties of collective motion in suspensions of bacteria

A suspension of microswimmers, the simplest realization of active matter, exhibits novel material properties: the emergence of collective motion, reduction in viscosity, increase in diffusivity, and extraction of useful energy. Bacterial dynamics in dilute suspensions suggest that hydrodynamic interactions and collisions between the swimmers lead to collective motion at higher concentrations. On the example of aerobic bacteria Bacillus subtilis, we report on spatial and temporal correlation functions measurements of collective state for various swimming speeds and concentrations. The experiments produced a puzzling result: while the energy injection rate is proportional to the swimming speed and concentration, the correlation length remains practically constant upon small speeds where random tumbling of bacteria dominates. It highlights two fundamental mechanisms: hydrodynamic interactions and collisions; for both of these mechanisms, the change of the swimming speed or concentration alters an overall time scale.

Emergent states in dense systems of active rods: from swarming to turbulence

Dense suspensions of self-propelled rod-like particles exhibit a fascinating variety of non-equilibrium phenomena. By means of computer simulations of a minimal model for rigid self-propelled colloidal rods with variable shape we explore the generic diagram of emerging states over a large range of rod densities and aspect ratios. The dynamics is studied using a simple numerical scheme for the overdamped noiseless frictional dynamics of a many-body system in which steric forces are dominant over hydrodynamic ones. The different emergent states are identified by various characteristic correlation functions and suitable order parameter fields. At low density and aspect ratio, a disordered phase with no coherent motion precedes a highly cooperative swarming state with giant number fluctuations at large aspect ratio. Conversely, at high densities weakly anisometric particles show a distinct jamming transition whereas slender particles form dynamic laning patterns. In between there is a large window corresponding to strongly vortical, turbulent flow. The different dynamical states should be verifiable in systems of swimming bacteria and artificial rod-like micro-swimmers.

Collective chemotactic dynamics in the presence of self-generated fluid flows

In microswimmer suspensions locomotion necessarily generates fluid motion, and it is known that such flows can lead to collective behavior from unbiased swimming. We examine the complementary problem of how chemotaxis is affected by self-generated flows. A kinetic theory coupling run-and-tumble chemotaxis to the flows of collective swimming shows separate branches of chemotactic and hydrodynamic instabilities for isotropic suspensions, the first driving aggregation, the second producing increased orientational order in suspensions of "pushers" and maximal disorder in suspensions of "pullers." Nonlinear simulations show that hydrodynamic interactions can limit and modify chemotactically driven aggregation dynamics. In puller suspensions the dynamics form aggregates that are mutually repelling due to the nontrivial flows. In pusher suspensions chemotactic aggregation can lead to destabilizing flows that fragment the regions of aggregation.

Meso-scale turbulence in living fluids

Turbulence is ubiquitous, from oceanic currents to small-scale biological and quantum systems. Self-sustained turbulent motion in microbial suspensions presents an intriguing example of collective dynamical behavior among the simplest forms of life and is important for fluid mixing and molecular transport on the microscale. The mathematical characterization of turbulence phenomena in active nonequilibrium fluids proves even more difficult than for conventional liquids or gases. It is not known which features of turbulent phases in living matter are universal or system-specific or which generalizations of the Navier-Stokes equations are able to describe them adequately. Here, we combine experiments, particle simulations, and continuum theory to identify the statistical properties of self-sustained meso-scale turbulence in active systems. To study how dimensionality and boundary conditions affect collective bacterial dynamics, we measured energy spectra and structure functions in dense Bacillus subtilis suspensions in quasi-2D and 3D geometries. Our experimental results for the bacterial flow statistics agree well with predictions from a minimal model for self-propelled rods, suggesting that at high concentrations the collective motion of the bacteria is dominated by short-range interactions. To provide a basis for future theoretical studies, we propose a minimal continuum model for incompressible bacterial flow. A detailed numerical analysis of the 2D case shows that this theory can reproduce many of the experimentally observed features of self-sustained active turbulence.

Origin of polar order in dense suspensions of phototactic micro-swimmers

A main question for the study of collective motion in living organisms is the origin of orientational polar order, i.e., how organisms align and what are the benefits of such collective behaviour. In the case of micro-organisms swimming at a low Reynolds number, steric repulsion and long-range hydrodynamic interactions are not sufficient to explain a homogeneous polar order state in which the direction of motion is aligned. An external symmetry-breaking guiding field such as a mechanism of taxis appears necessary to understand this phonemonon. We have investigated the onset of polar order in the velocity field induced by phototaxis in a suspension of a motile micro-organism, the algae Chlamydomonas reinhardtii, for density values above the limit provided by the hydrodynamic approximation of a force dipole model. We show that polar order originates from a combination of both the external guiding field intensity and the population density. In particular, we show evidence for a linear dependence of a phototactic guiding field on cell density to determine the polar order for dense suspensions and demonstrate the existence of a density threshold for the origin of polar order. This threshold represents the density value below which cells undergoing phototaxis are not able to maintain a homogeneous polar order state and marks the transition to ordered collective motion. Such a transition is driven by a noise dominated phototactic reorientation where the noise is modelled as a normal distribution with a variance that is inversely proportional to the guiding field strength. Finally, we discuss the role of density in dense suspensions of phototactic micro-swimmers.

Modeling a self-propelled autochemotactic walker

We develop a minimal model for the stochastic dynamics of microorganisms where individuals communicate via autochemotaxis. This means that microorganisms, such as bacteria, amoebae, or cells, follow the gradient of a chemical that they produce themselves to attract or repel each other. A microorganism is represented as a self-propelled particle or walker with constant speed while its velocity direction diffuses on the unit circle. We study the autochemotactic response of a single self-propelled walker whose dynamics is non-Markovian. We show that its long-time dynamics is always diffusive by deriving analytic expressions for its diffusion coefficient in the weak- and strong-coupling case. We confirm our findings by numerical simulations.

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.

Fluid dynamics and noise in bacterial cell-cell and cell-surface scattering

Bacterial processes ranging from gene expression to motility and biofilm formation are constantly challenged by internal and external noise. While the importance of stochastic fluctuations has been appreciated for chemotaxis, it is currently believed that deterministic long-range fluid dynamical effects govern cell-cell and cell-surface scattering-the elementary events that lead to swarming and collective swimming in active suspensions and to the formation of biofilms. Here, we report direct measurements of the bacterial flow field generated by individual swimming Escherichia coli both far from and near to a solid surface. These experiments allowed us to examine the relative importance of fluid dynamics and rotational diffusion for bacteria. For cell-cell interactions it is shown that thermal and intrinsic stochasticity drown the effects of long-range fluid dynamics, implying that physical interactions between bacteria are determined by steric collisions and near-field lubrication forces. This dominance of short-range forces closely links collective motion in bacterial suspensions to self-organization in driven granular systems, assemblages of biofilaments, and animal flocks. For the scattering of bacteria with surfaces, long-range fluid dynamical interactions are also shown to be negligible before collisions; however, once the bacterium swims along the surface within a few microns after an aligning collision, hydrodynamic effects can contribute to the experimentally observed, long residence times. Because these results are based on purely mechanical properties, they apply to a wide range of microorganisms.