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

Chemotactic behavior for a self-phoretic Janus particle near a patch source of fuel

Many biological microswimmers are capable of chemotaxis, i.e., they can sense an ambient chemical gradient and adjust their mechanism of motility to move towards or away from the source of the gradient. Synthetic active colloids endowed with chemotactic behavior hold considerable promise for targeted drug delivery and the realization of programmable and reconfigurable materials. Here, we study the chemotactic behavior of a Janus particle, which converts "fuel" molecules, released at an axisymmetric chemical patch located on a planar wall, into "product" molecules at its catalytic cap and moves by self-phoresis induced by the product. The chemotactic behavior is characterized as a function of the interplay between the rates of release (at the patch) and the consumption (at the particle) of fuel, as well as of details of the phoretic response of the particle (i.e., its phoretic mobility). Among other results, we find that, under certain conditions, the particle is attracted to a stable "hovering state" in which it aligns its axis normal to the wall and rests (positions itself) at an activity-dependent distance above the center of the patch.

Bacterial bioconvection confers context-dependent growth benefits and is robust under varying metabolic and genetic conditions

Microbes often respond to environmental cues by adopting collective behaviors-like biofilms or swarming-that benefit the population. During "bioconvection," microbes gather in dense groups and plume downward through fluid environments, driving flow and mixing on the scale of millions of cells. Though bioconvection was observed a century ago, the effects of differing physical and chemical inputs and its potential selective advantages for different species of microbes remain largely unexplored. In Bacillus subtilis, vertical oxygen gradients that originate from air-liquid interfaces create cell-density inversions that drive bioconvection. Here, we develop Escherichia coli as a complementary model for the study of bioconvection. In the context of a still fluid, we found that motile and chemotactic genotypes of both E. coli and B. subtilis bioconvect and show increased growth compared to immotile or non-chemotactic genotypes, whereas in a well-mixed fluid, there is no growth advantage to motility or chemotaxis. We found that fluid depth, cell concentration, and carbon availability affect the emergence and timing of bioconvective patterns. Also, whereas B. subtilis requires oxygen gradients to bioconvect, E. coli deficient in aerotaxis (Δaer) or energy-taxis (Δtsr) still bioconvects, as do cultures that lack an air-liquid interface. Thus, in two distantly related microbes, bioconvection may confer context-dependent growth benefits, and E. coli bioconvection is robustly elicited by multiple types of chemotaxis. These results greatly expand the set of physical and metabolic conditions in which this striking collective behavior can be expected and demonstrate its potential to be a generic force for behavioral selection across ecological contexts. IMPORTANCE Individual microorganisms frequently move in response to gradients in their fluid environment, with corresponding metabolic benefits. At a population level, such movements can create density variations in a fluid that couple to gravity and drive large-scale convection and mixing called bioconvection. In this work, we provide evidence that this collective behavior confers a selective benefit on two distantly related species of bacteria. We develop new methods for quantifying this behavior and show that bioconvection in Escherichia coli is surprisingly robust to changes in cell concentration, fluid depth, interface conditions, metabolic sensing, and carbon availability. These results greatly expand the set of conditions known to elicit this collective behavior and indicate its potential to be a selective pressure across ecological contexts.

Kinetic modeling of the chemotactic process in run-and-tumble bacteria

The chemotactic process of run-and-tumble bacteria results from modulating the tumbling rate in response to changes in chemoattractant gradients felt by the bacteria. The response has a characteristic memory time and is subject to important fluctuations. These ingredients are considered in a kinetic description of chemotaxis, allowing the computation of the stationary mobility and the relaxation times needed to reach the steady state. For large memory times, these relaxation times become large, implying that finite-time measurements give rise to nonmonotonic currents as a function of the imposed chemoattractant gradient, contrary to the stationary regime where the response is monotonic. The case of an inhomogeneous signal is analyzed. Contrary to the usual Keller-Segel model, the response is nonlocal, and the bacterial profile is smoothed with a characteristic length that grows with the memory time. Finally, the case of traveling signals is considered, where appreciable differences appear compared to memoryless chemotactic descriptions.

Chemotactic smoothing of collective migration

Collective migration-the directed, coordinated motion of many self-propelled agents-is a fascinating emergent behavior exhibited by active matter with functional implications for biological systems. However, how migration can persist when a population is confronted with perturbations is poorly understood. Here, we address this gap in knowledge through studies of bacteria that migrate via directed motion, or chemotaxis, in response to a self-generated nutrient gradient. We find that bacterial populations autonomously smooth out large-scale perturbations in their overall morphology, enabling the cells to continue to migrate together. This smoothing process arises from spatial variations in the ability of cells to sense and respond to the local nutrient gradient-revealing a population-scale consequence of the manner in which individual cells transduce external signals. Altogether, our work provides insights to predict, and potentially control, the collective migration and morphology of cellular populations and diverse other forms of active matter.

Interactions in active colloids

The past two decades have seen a remarkable progress in the development of synthetic colloidal agents which are capable of creating directed motion in an unbiased environment at the microscale. These self-propelling particles are often praised for their enormous potential to self-organize into dynamic nonequilibrium structures such as living clusters, synchronized super-rotor structures or self-propelling molecules featuring a complexity which is rarely found outside of the living world. However, the precise mechanisms underlying the formation and dynamics of many of these structures are still barely understood, which is likely to hinge on the gaps in our understanding of how active colloids interact. In particular, besides showing comparatively short-ranged interactions which are well known from passive colloids (Van der Waals, electrostatic etc), active colloids show novel hydrodynamic interactions as well as phoretic and substrate-mediated 'osmotic' cross-interactions which hinge on the action of the phoretic field gradients which are induced by the colloids on other colloids in the system. The present article discusses the complexity and the intriguing properties of these interactions which in general are long-ranged, non-instantaneous, non-pairwise and non-reciprocal and which may serve as key ingredients for the design of future nonequilibrium colloidal materials. Besides providing a brief overview on the state of the art of our understanding of these interactions a key aim of this review is to emphasize open key questions and corresponding open challenges.

Mathematical modeling of chemotaxis guided amoeboid cell swimming

Cells and microorganisms adopt various strategies to migrate in response to different environmental stimuli. To date, many modeling research has focused on the crawling-basedDictyostelium discoideum(Dd) cells migration induced by chemotaxis, yet recent experimental results reveal that even without adhesion or contact to a substrate, Dd cells can still swim to follow chemoattractant signals. In this paper, we develop a modeling framework to investigate the chemotaxis induced amoeboid cell swimming dynamics. A minimal swimming system consists of one deformable Dd amoeboid cell and a dilute suspension of bacteria, and the bacteria produce chemoattractant signals that attract the Dd cell. We use themathematical amoeba modelto generate Dd cell deformation and solve the resulting low Reynolds number flows, and use a moving mesh based finite volume method to solve the reaction-diffusion-convection equation. Using the computational model, we show that chemotaxis guides a swimming Dd cell to follow and catch bacteria, while on the other hand, bacterial rheotaxis may help the bacteria to escape from the predator Dd cell.

Evolution of chemotactic hitchhiking

Bacteria typically reside in heterogeneous environments with various chemogradients where motile cells can gain an advantage over nonmotile cells. Since motility is energetically costly, cells must optimize their swimming speed and behaviour to maximize their fitness. Here, we investigate how cheating strategies might evolve where slow or nonmotile microbes exploit faster ones by sticking together and hitching a ride. Starting with physical and biological first principles, we computationally study the effects of sticking on the evolution of motility in a controlled chemostat environment. We find that stickiness allows for slow cheaters to dominate when chemoattractants are dispersed at intermediate distances. In this case, slow microbes exploit faster ones until they consume the population, leading to a tragedy of commons. For long races, slow microbes do gain an initial advantage from sticking, but eventually fall behind. Here, fast microbes are more likely to stick to other fast microbes and co-operate to increase their own population. We therefore conclude that whether the nature of the hitchhiking interaction is parasitic or mutualistic, depends on the chemoattractant distribution.

Photo-bioconvection: towards light control of flows in active suspensions

The persistent motility of individual constituents in microbial suspensions represents a prime example of the so-called active matter systems. Cells consume energy, exert forces and move, overall releasing the constraints of equilibrium statistical mechanics of passive elements and allowing for complex spatio-temporal patterns to emerge. Moreover, when subject to physico-chemical stimuli their collective behaviour often drives large-scale instabilities of a hydrodynamic nature, with implications for biomixing in natural environments and incipient industrial applications. In turn, our ability to exert external control of these driving stimuli could be used to govern the emerging patterns. Light, being easily manipulable and, at the same time, an important stimulus for a wide variety of microorganisms, is particularly well suited to this end. In this paper, we will discuss the current state, developments and some of the emerging advances in the fundamentals and applications of light-induced bioconvection with a focus on recent experimental realizations and modelling efforts. This article is part of the theme issue 'Stokes at 200 (part 2)'.

Active dipolar spheroids in shear flow and transverse field: Population splitting, cross-stream migration, and orientational pinning

We study the steady-state behavior of active, dipolar, Brownian spheroids in a planar channel subjected to an imposed Couette flow and an external transverse field, applied in the "downward" normal-to-flow direction. The field-induced torque on active spheroids (swimmers) is taken to be of magnetic form by assuming that they have a permanent magnetic dipole moment, pointing along their self-propulsion (swim) direction. Using a continuum approach, we show that a host of behaviors emerges over the parameter space spanned by the particle aspect ratio, self-propulsion and shear/field strengths, and the channel width. The cross-stream migration of the model swimmers is shown to involve a regime of linear response (quantified by a linear-response factor) in weak fields. For prolate swimmers, the weak-field behavior crosses over to a regime of full swimmer migration to the bottom half of the channel in strong fields. For oblate swimmers, a counterintuitive regime of reverse migration arises in intermediate fields, where a macroscopic fraction of swimmers reorient and swim to the top channel half at an acute "upward" angle relative to the field axis. The diverse behaviors reported here are analyzed based on the shear-induced population splitting (bimodality) of the swim orientation, giving two distinct, oppositely polarized, swimmer subpopulations (albeit very differently for prolate/oblate swimmers) in each channel half. In strong fields, swimmers of both types exhibit net upstream currents relative to the laboratory frame. The onsets of full migration and net upstream current depend on the aspect ratio, enabling efficient particle separation strategies in microfluidic setups.

Chemotactic behaviour of Escherichia coli at high cell density

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

Chemotaxis mediated interactions can stabilize the hydrodynamic instabilities in active suspensions

Ordered phases in active suspensions of polar swimmers are under long-wavelength hydrodynamic mediated instabilities. In this article, we show that chemical molecules dissolved in aqueous suspensions, as an unavoidable part of most wet active systems, can mediate long-range interactions and subsequently stabilize the polar phase. Chemoattractants in living suspensions and dissolved molecules in synthesized Janus suspensions are reminiscent of such chemical molecules. Communication between swimmers through the gradients of such chemicals is the foundation of this stabilization mechanism. To classify the stable states of such active systems, we investigate the detailed phase diagrams for two classes of systems with momentum conserving and non-conserving dynamics. Our linear stability analysis shows that the proposed stabilization mechanism can work for swimmers with different dynamical properties, e.g., pushers or pullers and with various static characteristics, e.g., spherical, oblate or prolate geometries.

Synthetic Chemotaxis and Collective Behavior in Active Matter

The ability to navigate in chemical gradients, called chemotaxis, is crucial for the survival of microorganisms. It allows them to find food and to escape from toxins. Many microorganisms can produce the chemicals to which they respond themselves and use chemotaxis for signaling, which can be seen as a basic form of communication, allowing ensembles of microorganisms to coordinate their behavior, for example, during embryogenesis, biofilm formation, or cellular aggregation. For example, Dictyostelium cells use signaling as a survival strategy: when starving, they produce certain chemicals toward which other cells show taxis. This leads to aggregation of the cells resulting in a multicellular aggregate that can sustain long starvation periods. Remarkably, the past decade has led to the development of synthetic microswimmers, which can self-propel through a solvent, analogously to bacteria and other microorganisms. The mechanism underlying the self-propulsion of synthetic microswimmers like camphor boats, droplet swimmers, and in particular autophoretic Janus colloids involves the production of certain chemicals. As we will discuss in this Account, the same chemicals (phoretic fields) involved in the self-propulsion of a (Janus) microswimmer also act on other ones and bias their swimming direction toward (or away from) the producing microswimmer. Synthetic microswimmers therefore provide a synthetic analogue to motile microorganisms interacting by taxis toward (or away from) self-produced chemical fields. In this Account, we review recent progress in the theoretical description of synthetic chemotaxis mainly based on simulations and field theoretical descriptions. We will begin with single motile particles leaving chemical trails behind with which they interact themselves, leading to effects like self-trapping or self-avoidance. Besides these self-interactions, in ensembles of synthetic motile particles each particle also responds to the chemicals produced by other particles, inducing chemical (or phoretic) cross-interactions. When these interactions are attractive, they commonly lead to clusters, even at low particle density. These clusters may either proceed toward macrophase separation, resembling Dictyostelium aggregation, or, as shown very recently, lead to dynamic clusters of self-limited size (dynamic clustering) as seen in experiments in autophoretic Janus colloids. Besides the classical case where chemical interactions are attractive, this Account discusses, as its main focus, repulsive chemical interactions, which can create a new and less known avenue to pattern formation in active systems leading to a variety of pattern, including clusters which are surrounded by shells of chemicals, traveling waves and more complex continuously reshaping patterns. In all these cases "synthetic signalling" can crucially determine the collective behavior of synthetic microswimmer ensembles and can be used as a design principle to create patterns in motile active particles.

Combined influence of hydrodynamics and chemotaxis in the distribution of microorganisms around spherical nutrient sources

We study how the interaction between hydrodynamics and chemotaxis affects the colonization of nutrient sources by microorganisms. We use an individual-based model and perform probabilistic simulations to ascertain the impact of important environmental and motility characteristics on the spatial distribution of microorganisms around a spherical nutrient source. In general, we unveil four distinct regimes based on the distribution of the microorganisms: (i) strong surface colonization, (ii) rotary-diffusion-induced "off-surface" accumulation, (iii) a depletion zone in the spatial distribution, and (iv) no appreciable aggregation, with their occurrence being contingent on the relative strengths of hydrodynamic and chemotactic effects. More specifically, we show that the extent of surface colonization first increases, then reaches a plateau, and finally decreases as the nutrient availability is increased. We also show that surface colonization reduces monotonically as the mean run length of the chemotactic microorganisms increases. Our study provides insight into the interplay of two important mechanisms governing microorganism behavior near nutrient sources, isolates each of their effects, and thus offers greater predictability of this nontrivial phenomenon.

Modeling of active swimmer suspensions and their interactions with the environment

In this article, we review mathematical models used to study the behaviour of suspensions of micro-swimmers and the accompanying biophysical phenomena, with specific focus on stimulus response. The methods discussed encompass a range of interactions exhibited by the micro-swimmers; including passive hydrodynamic (gyrotaxis) and gravitational (gravitaxis) effects, and active responses to chemical cues (chemotaxis) and light intensities (phototaxis). We introduce the simplest models first, and then build towards more sophisticated recent developments, in the process, identifying the limitations of the former and the new results obtained by the latter. We comment on the accuracy/validity of the models adopted, based on the agreement between theoretical results and experimental observations. We conclude by identifying some of the open problems and associated challenges faced by researchers in the realm of active suspensions.

Emergent properties of bacterial chemotaxis pathway

The chemotaxis pathway of Escherichia coli is the most studied sensory system in prokaryotes. The highly conserved general architecture of this pathway consists of two modules which mediate signal transduction and adaptation. The signal transduction module detects and amplifies changes in environmental conditions and rapidly transmits these signals to control bacterial swimming behavior. The adaptation module gradually resets the activity and sensitivity of the first module after initial stimulation and thereby enables the temporal comparisons necessary for bacterial chemotaxis. Recent experimental and theoretical work has unraveled multiple quantitative features emerging from the interplay between these two modules. This has laid the groundwork for rationalization of these emerging properties in the context of the evolutionary optimization of the chemotactic behavior.

Stability and dynamics of anisotropically tumbling chemotactic swimmers

Microswimmers such as bacteria perform random walks known as run-and-tumbles to move up chemoattractant gradients and as a result aggregate with others. It is also known that such micro-swimmers can self-organize into macroscopic patterns due to interactions with neighboring cells through the fluidic environment they live in. While the pattern formation resulting from chemotactic and hydrodynamic interactions separately and together have been previously investigated, the effect of the anisotropy in the tumbles of microswimmers has been unexplored. Here we show through linear analysis and full nonlinear simulations that the slight anisotropy in the individual swimmer tumbles can alter the collective pattern formation in nontrivial ways. We show that tumbling anisotropy diminishes the magnitude of the chemotactic aggregates but may result in more such aggregation peaks.

A model for collective dynamics in ant raids

Ant raiding, the process of identifying and returning food to the nest or bivouac, is a fascinating example of collective motion in nature. During such raids ants lay pheromones to form trails for others to find a food source. In this work a coupled PDE/ODE model is introduced to study ant dynamics and pheromone concentration. The key idea is the introduction of two forms of ant dynamics: foraging and returning, each governed by different environmental and social cues. The model accounts for all aspects of the raiding cycle including local collisional interactions, the laying of pheromone along a trail, and the transition from one class of ants to another. Through analysis of an order parameter measuring the orientational order in the system, the model shows self-organization into a collective state consisting of lanes of ants moving in opposite directions as well as the transition back to the individual state once the food source is depleted matching prior experimental results. This indicates that in the absence of direct communication ants naturally form an efficient method for transporting food to the nest/bivouac. The model exhibits a continuous kinetic phase transition in the order parameter as a function of certain system parameters. The associated critical exponents are found, shedding light on the behavior of the system near the transition.

The swim force as a body force

Net (as opposed to random) motion of active matter results from an average swim (or propulsive) force. It is shown that the average swim force acts like a body force - an internal body force. As a result, the particle-pressure exerted on a container wall is the sum of the swim pressure [Takatori et al., Phys. Rev. Lett., 2014, 113, 028103] and the 'weight' of the active particles. A continuum description is possible when variations occur on scales larger than the run length of the active particles and gives a Boltzmann-like distribution from a balance of the swim force and the swim pressure. Active particles may also display 'action at a distance' and accumulate adjacent to (or be depleted from) a boundary without any external forces. In the momentum balance for the suspension - the mixture of active particles plus fluid - only external body forces appear.

Anisotropic particle in viscous shear flow: Navier slip, reciprocal symmetry, and Jeffery orbit

The hydrodynamic reciprocal theorem for Stokes flows is generalized to incorporate the Navier slip boundary condition, which can be derived from Onsager's variational principle of least energy dissipation. The hydrodynamic reciprocal relations and the Jeffery orbit, both of which arise from the motion of a slippery anisotropic particle in a simple viscous shear flow, are investigated theoretically and numerically using the fluid particle dynamics method [Phys. Rev. Lett. 85, 1338 (2000)]. For a slippery elliptical particle in a linear shear flow, the hydrodynamic reciprocal relations between the rotational torque and the shear stress are studied and related to the Jeffery orbit, showing that the boundary slip can effectively enhance the anisotropy of the particle. Physically, by replacing the no-slip boundary condition with the Navier slip condition at the particle surface, the cross coupling between the rotational torque and the shear stress is enhanced, as manifested through a dimensionless parameter in both of the hydrodynamic reciprocal relations and the Jeffery orbit. In addition, simulations for a circular particle patterned with portions of no-slip and Navier slip are carried out, showing that the particle possesses an effective anisotropy and follows the Jeffery orbit as well. This effective anisotropy can be tuned by changing the ratio of no-slip portion to slip potion. The connection of the present work to nematic liquid crystals' constitutive relations is discussed.

Clusters, asters, and collective oscillations in chemotactic colloids

The creation of synthetic systems that emulate the defining properties of living matter, such as motility, gradient-sensing, signaling, and replication, is a grand challenge of biomimetics. Such imitations of life crucially contain active components that transform chemical energy into directed motion. These artificial realizations of motility point in the direction of a new paradigm in engineering, through the design of emergent behavior by manipulating properties at the scale of the individual components. Catalytic colloidal swimmers are a particularly promising example of such systems. Here we present a comprehensive theoretical description of gradient-sensing of an individual swimmer, leading controllably to chemotactic or anti-chemotactic behavior, and use it to construct a framework for studying their collective behavior. We find that both the positional and the orientational degrees of freedom of the active colloids can exhibit condensation, signaling formation of clusters and asters. The kinetics of catalysis introduces a natural control parameter for the range of the interaction mediated by the diffusing chemical species. For various regimes in parameter space in the long-ranged limit our system displays precise analogs to gravitational collapse, plasma oscillations, and electrostatic screening. We present prescriptions for how to tune the surface properties of the colloids during fabrication to achieve each type of behavior.

Collective surfing of chemically active particles

We study theoretically the collective dynamics of immotile particles bound to a 2D surface atop a 3D fluid layer. These particles are chemically active and produce a chemical concentration field that creates surface-tension gradients along the surface. The resultant Marangoni stresses create flows that carry the particles, possibly concentrating them. For a 3D diffusion-dominated concentration field and Stokesian fluid we show that the surface dynamics of active particle density can be determined using nonlocal 2D surface operators. Remarkably, we also show that for both deep or shallow fluid layers this surface dynamics reduces to the 2D Keller-Segel model for the collective chemotactic aggregation of slime mold colonies. Mathematical analysis has established that the Keller-Segel model can yield finite-time, finite-mass concentration singularities. We show that such singular behavior occurs in our finite-depth system, and study the associated 3D flow structures.

Spiral and never-settling patterns in active systems

We present a combined numerical and analytical study of pattern formation in an active system where particles align, possess a density-dependent motility, and are subject to a logistic reaction. The model can describe suspensions of reproducing bacteria, as well as polymerizing actomyosin gels in vitro or in vivo. In the disordered phase, we find that motility suppression and growth compete to yield stable or blinking patterns, which, when dense enough, acquire internal orientational ordering to give asters or spirals. We predict these may be observed within chemotactic aggregates in bacterial fluids. In the ordered phase, the reaction term leads to previously unobserved never-settling patterns which can provide a simple framework to understand the formation of motile and spiral patterns in intracellular actin systems.

Catalytically powered dynamic assembly of rod-shaped nanomotors and passive tracer particles

Nano- and microscale motors powered by catalytic reactions exhibit collective behavior such as swarming, predator-prey interactions, and chemotaxis that resemble those of biological microorganisms. A quantitative understanding of the catalytically generated forces between particles that lead to these behaviors has so far been lacking. Observations and numerical simulations of pairwise interactions between gold-platinum nanorods in hydrogen peroxide solutions show that attractive and repulsive interactions arise from the catalytically generated electric field. Electrokinetic effects drive the assembly of staggered doublets and triplets of nanorods that are moving in the same direction. None of these behaviors are observed with nanorods composed of a single metal. The motors also collect tracer microparticles at their head or tail, depending on the charge of the particles, actively assembling them into close-packed rafts and aggregates of rafts. These motor-tracer particle interactions can also be understood in terms of the catalytically generated electric field around the ends of the nanorod motors.