The development of concentration gradients in a suspension of chemotactic bacteria

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.

The impact of rheotaxis and flow on the aggregation of organisms

Dispersed populations often need to organize into groups. Chemical attractants provide one means of directing individuals into an aggregate, but whether these structures emerge can depend on various factors, such as there being a sufficiently large population or the response to the attractant being sufficiently sensitive. In an aquatic environment, fluid flow may heavily impact on population distribution and many aquatic organisms adopt a rheotaxis response when exposed to a current, orienting and swimming according to the flow field. Consequently, flow-induced transport could be substantially different for the population members and any aggregating signal they secrete. With the aim of investigating how flows and rheotaxis responses impact on an aquatic population's ability to form and maintain an aggregated profile, we develop and analyse a mathematical model that incorporates these factors. Through a systematic analysis into the effect of introducing rheotactic behaviour under various forms of environmental flow, we demonstrate that each of flow and rheotaxis can act beneficially or detrimentally on the ability to form and maintain a cluster. Synthesizing these findings, we test a hypothesis that density-dependent rheotaxis may be optimal for group formation and maintenance, in which individuals increase their rheotactic effort as they approach an aggregated state.

Numerical study of nano-biofilm stagnation flow from a nonlinear stretching/shrinking surface with variable nanofluid and bioconvection transport properties

A mathematical model is developed for stagnation point flow toward a stretching or shrinking sheet of liquid nano-biofilm containing spherical nano-particles and bioconvecting gyrotactic micro-organisms. Variable transport properties of the liquid (viscosity, thermal conductivity, nano-particle species diffusivity) and micro-organisms (species diffusivity) are considered. Buongiorno's two-component nanoscale model is deployed and spherical nanoparticles in a dilute nanofluid considered. Using a similarity transformation, the nonlinear systems of partial differential equations is converted into nonlinear ordinary differential equations. These resulting equations are solved numerically using a central space finite difference method in the CodeBlocks Fortran platform. Graphical plots for the distribution of reduced skin friction coefficient, reduced Nusselt number, reduced Sherwood number and the reduced local density of the motile microorganisms as well as the velocity, temperature, nanoparticle volume fraction and the density of motile microorganisms are presented for the influence of wall velocity power-law index (m), viscosity parameter [Formula: see text], thermal conductivity parameter (c4), nano-particle mass diffusivity (c6), micro-organism species diffusivity (c8), thermophoresis parameter [Formula: see text], Brownian motion parameter [Formula: see text], Lewis number [Formula: see text], bioconvection Schmidt number [Formula: see text], bioconvection constant (σ) and bioconvection Péclet number [Formula: see text]. Validation of the solutions via comparison related to previous simpler models is included. Further verification of the general model is conducted with the Adomian decomposition method (ADM). Extensive interpretation of the physics is included. Skin friction is elevated with viscosity parameter ([Formula: see text] whereas it is suppressed with greater Lewis number and thermophoresis parameter. Temperatures are elevated with increasing thermal conductivity parameter ([Formula: see text] whereas Nusselt numbers are reduced. Nano-particle volume fraction (concentration) is enhanced with increasing nano-particle mass diffusivity parameter ([Formula: see text]) whereas it is markedly reduced with greater Lewis number (Le) and Brownian motion parameter (Nb). With increasing stretching/shrinking velocity power-law exponent ([Formula: see text] skin friction is decreased whereas Nusselt number and Sherwood number are both elevated. Motile microorganism density is boosted strongly with increasing micro-organism diffusivity parameter ([Formula: see text]) and Brownian motion parameter (Nb) but reduced considerably with greater bioconvection Schmidt number (Sc) and bioconvection Péclet number (Pe). The simulations find applications in deposition processes in nano-bio-coating manufacturing processes.

A Sequence of Developmental Events Occurs Underneath Growing Bacillus subtilis Pellicles

Biofilms are structured communities of bacteria that exhibit complex spatio-temporal dynamics. In liquid media, Bacillus subtilis produces an opaque floating biofilm, or a pellicle. Biofilms are generally associated with an interface, but the ability of Bacillus subtilis to swim means the bacteria are additionally able to reside within the liquid phase. However, due to imaging complications associated with the opacity of pellicles, the extent to which bacteria coexist within the liquid bulk as well as their behavior in the liquid is not well studied. We therefore develop a high-throughput imaging system to image underneath developing pellicles. Here we report a well-defined sequence of developmental events that occurs underneath a growing pellicle. Comparison with bacteria deficient in swimming and chemotaxis suggest that these properties enable collective bacterial swimming within the liquid phase which facilitate faster surface colonization. Furthermore, comparison to bacteria deficient in exopolymeric substances (EPS) suggest that the lack of a surface pellicle prevents further developmental steps from occurring within the liquid phase. Our results reveal a sequence of developmental events during pellicle growth, encompassing adhesion, conversion, growth, maturity, and detachment on the interface, which are synchronized with the bacteria in the liquid bulk increasing in density until the formation of a mature surface pellicle, after which the density of bacteria in the liquid drops.

Derivation of a bacterial nutrient-taxis system with doubly degenerate cross-diffusion as the parabolic limit of a velocity-jump process

This paper is devoted to the justification of the macroscopic, mean-field nutrient taxis system with doubly degenerate cross-diffusion proposed by Leyva et al. (Phys A 392:5644-5662, 2013) to model the complex spatio-temporal dynamics exhibited by the bacterium Bacillus subtilis during experiments run in vitro. This justification is based on a microscopic description of the movement of individual cells whose changes in velocity (in both speed and orientation) obey a velocity jump process governed by a transport equation of Boltzmann type. For that purpose, the asymptotic method introduced by Hillen and Othmer (SIAM J Appl Math 61:751-775, 2000; SIAM J Appl Math 62:1222-1250, 2002) is applied, which consists of the computation of the leading order term in a regular Hilbert expansion for the solution to the transport equation, under an appropriate parabolic scaling and a first order perturbation of the turning rate of Schnitzer type (Schnitzer in Phys Rev E 48:2553-2568, 1993). The resulting parabolic limit equation at leading order for the bacterial cell density recovers the degenerate nonlinear cross diffusion term and the associated chemotactic drift appearing in the original system of equations. Although the bacterium B. subtilis is used as a prototype, the method and results apply in more generality.

A numerical treatment of radiative nanofluid 3D flow containing gyrotactic microorganism with anisotropic slip, binary chemical reaction and activation energy

A numerical investigation of steady three dimensional nanofluid flow carrying effects of gyrotactic microorganism with anisotropic slip condition along a moving plate near a stagnation point is conducted. Additionally, influences of Arrhenius activation energy, joule heating accompanying binary chemical reaction and viscous dissipation are also taken into account. A system of nonlinear differential equations obtained from boundary layer partial differential equations is found by utilization of apposite transformations. RK fourth and fifth order technique of Maple software is engaged to acquire the solution of the mathematical model governing the presented fluid flow. A Comparison with previously done study is also made and a good agreement is achieved with existing results; hence reliable results are being presented. Evaluations are carried out for involved parameters graphically against velocity, temperature, concentration fields, microorganism distribution, density number, local Nusselt and Sherwood numbers. It is detected that microorganism distribution exhibit diminishing behavior for rising values of bio-convection Lewis and Peclet numbers.

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.

Logarithmic sensing in Bacillus subtilis aerotaxis

Aerotaxis, the directed migration along oxygen gradients, allows many microorganisms to locate favorable oxygen concentrations. Despite oxygen's fundamental role for life, even key aspects of aerotaxis remain poorly understood. In Bacillus subtilis, for example, there is conflicting evidence of whether migration occurs to the maximal oxygen concentration available or to an optimal intermediate one, and how aerotaxis can be maintained over a broad range of conditions. Using precisely controlled oxygen gradients in a microfluidic device, spanning the full spectrum of conditions from quasi-anoxic to oxic (60 n mol/l-1 m mol/l), we resolved B. subtilis' 'oxygen preference conundrum' by demonstrating consistent migration towards maximum oxygen concentrations ('monotonic aerotaxis'). Surprisingly, the strength of aerotaxis was largely unchanged over three decades in oxygen concentration (131 n mol/l-196 μ mol/l). We discovered that in this range B. subtilis responds to the logarithm of the oxygen concentration gradient, a rescaling strategy called 'log-sensing' that affords organisms high sensitivity over a wide range of conditions. In these experiments, high-throughput single-cell imaging yielded the best signal-to-noise ratio of any microbial taxis study to date, enabling the robust identification of the first mathematical model for aerotaxis among a broad class of alternative models. The model passed the stringent test of predicting the transient aerotactic response despite being developed on steady-state data, and quantitatively captures both monotonic aerotaxis and log-sensing. Taken together, these results shed new light on the oxygen-seeking capabilities of B. subtilis and provide a blueprint for the quantitative investigation of the many other forms of microbial taxis.

Feeding ducks, bacterial chemotaxis, and the Gini index

Classic experiments on the distribution of ducks around separated food sources found consistency with the "ideal free" distribution in which the local population is proportional to the local supply rate. Motivated by this experiment and others, we examine the analogous problem in the microbial world: the distribution of chemotactic bacteria around multiple nearby food sources. In contrast to the optimization of uptake rate that may hold at the level of a single cell in a spatially varying nutrient field, nutrient consumption by a population of chemotactic cells will modify the nutrient field, and the uptake rate will generally vary throughout the population. Through a simple model we study the distribution of resource uptake in the presence of chemotaxis, consumption, and diffusion of both bacteria and nutrients. Borrowing from the field of theoretical economics, we explore how the Gini index can be used as a means to quantify the inequalities of uptake. The redistributive effect of chemotaxis can lead to a phenomenon we term "chemotactic levelling," and the influence of these results on population fitness are briefly considered.

Bacterial microsystems and microrobots

Microorganisms and specifically motile bacteria have been recently added to the list of micro-actuators typically considered for the implementation of microsystems and microrobots. Such trend has been motivated by the fact these microorganisms are self-powered actuators with overall sizes at the lower end of the micrometer range and which have proven to be extremely effective in low Reynolds number hydrodynamic regime of usually less than 10(-2). Furthermore, the various sensors or taxes in bacteria influencing their movements can also be exploited to perform tasks that were previously considered only for futuristic artificial microrobots. Bacterial implementations and related issues are not only reviewed, but this paper also proposes many techniques and approaches that can be considered as building blocks for the implementations of more sophisticated microsystems and microrobots.

Theoretical insights into bacterial chemotaxis

Research into understanding bacterial chemotactic systems has become a paradigm for Systems Biology. Experimental and theoretical researchers have worked hand-in-hand for over 40 years to understand the intricate behavior driving bacterial species, in particular how such small creatures, usually not more than 5 µm in length, detect and respond to small changes in their extracellular environment. In this review we highlight the importance that theoretical modeling has played in providing new insight and understanding into bacterial chemotaxis. We begin with an overview of the bacterial chemotaxis sensory response, before reviewing the role of theoretical modeling in understanding elements of the system on the single cell scale and features underpinning multiscale extensions to population models.

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.

Nanofluid bioconvection in water-based suspensions containing nanoparticles and oxytactic microorganisms: oscillatory instability

The aim of this article is to propose a novel type of a nanofluid that contains both nanoparticles and motile (oxytactic) microorganisms. The benefits of adding motile microorganisms to the suspension include enhanced mass transfer, microscale mixing, and anticipated improved stability of the nanofluid. In order to understand the behavior of such a suspension at the fundamental level, this article investigates its stability when it occupies a shallow horizontal layer. The oscillatory mode of nanofluid bioconvection may be induced by the interaction of three competing agencies: oxytactic microorganisms, heating or cooling from the bottom, and top or bottom-heavy nanoparticle distribution. The model includes equations expressing conservation of total mass, momentum, thermal energy, nanoparticles, microorganisms, and oxygen. Physical mechanisms responsible for the slip velocity between the nanoparticles and the base fluid, such as Brownian motion and thermophoresis, are accounted for in the model. An approximate analytical solution of the eigenvalue problem is obtained using the Galerkin method. The obtained solution provides important physical insights into the behavior of this system; it also explains when the oscillatory mode of instability is possible in such system.

Fluid particle diffusion in a semidilute suspension of model micro-organisms

We calculate non-Brownian fluid particle diffusion in a semidilute suspension of swimming micro-organisms. Each micro-organism is modeled as a spherical squirmer, and their motions in an infinite suspension otherwise at rest are computed by the Stokesian-dynamics method. In calculating the fluid particle motions, we propose a numerical method based on a combination of the boundary element technique and Stokesian dynamics. We present details of the numerical method and examine its accuracy. The limitation of semidiluteness is required to ensure accuracy of the fluid particle velocity calculation. In the case of a suspension of non-bottom-heavy squirmers the spreading of fluid particles becomes diffusive in a shorter time than that of the squirmers, and the diffusivity of fluid particles is smaller than that of squirmers. It is confirmed that the probability density distribution of fluid particles also shows diffusive properties. The effect of tracer particle size is investigated by inserting some inert spheres of the same radius as the squirmers, instead of fluid particles, into the suspension. The diffusivity for inert spheres is not less than one tenth of that for fluid particles, even though the particle size is totally different. Scaling analysis indicates that the diffusivity of fluid particles and inert spheres becomes proportional to the volume fraction of squirmers in the semidilute regime provided that there is no more than a small recirculation region around a squirmer, which is confirmed numerically. In the case of a suspension of bottom-heavy squirmers, horizontal diffusivity decreases considerably even with small values of the bottom heaviness, which indicates the importance of bottom heaviness in the diffusion phenomena. We believe that these fundamental findings will enhance our understanding of the basic mechanics of a suspension of swimming micro-organisms.

Suspension biomechanics of swimming microbes

Micro-organisms play a vital role in many biological, medical and engineering phenomena. Some recent research efforts have demonstrated the importance of biomechanics in understanding certain aspects of micro-organism behaviours such as locomotion and collective motions of cells. In particular, spatio-temporal coherent structures found in a bacterial suspension have been the focus of many research studies over the last few years. Recent studies have shown that macroscopic properties of a suspension, such as rheology and diffusion, are strongly affected by meso-scale flow structures generated by swimming microbes. Since the meso-scale flow structures are strongly affected by the interactions between microbes, a bottom-up strategy, i.e. from a cellular level to a continuum suspension level, represents the natural approach to the study of a suspension of swimming microbes. In this paper, we first provide a summary of existing biomechanical research on interactions between a pair of swimming micro-organisms, as a two-body interaction is the simplest many-body interaction. We show that interactions between two nearby swimming micro-organisms are described well by existing mathematical models. Then, collective motions formed by a group of swimming micro-organisms are discussed. We show that some collective motions of micro-organisms, such as coherent structures of bacterial suspensions, are satisfactorily explained by fluid dynamics. Lastly, we discuss how macroscopic suspension properties are changed by the microscopic characteristics of the cell suspension. The fundamental knowledge we present will be useful in obtaining a better understanding of the behaviour of micro-organisms.

Orientational relaxation time of bottom-heavy squirmers in a semi-dilute suspension

One of the important quantities to characterize unsteady behaviour of a cell suspension is the orientational relaxation time, which is the time scale for a micro-organism to re-orientate to its preferred direction from disorientated conditions. In this paper, a swimming micro-organism is modelled as a squirming sphere with prescribed tangential surface velocity, in which the centre of mass of the sphere is displaced from the geometric centre (bottom-heaviness). The orientational relaxation time of bottom-heavy squirmers in a suspension is investigated both analytically and numerically. The three-dimensional movement of 64 identical squirmers in a fluid otherwise at rest, contained in a cube with periodic boundary conditions, is dynamically computed, for random initial positions and orientations. The effects of volume fraction of squirmers, the bottom-heaviness and the squirming mode on the relaxation time are discussed. The results for a semi-dilute suspension show that both the mean stresslet strength and the orientational relaxation time decrease from those for a dilute suspension. We also observe a stress overshoot in some cases. The mechanism for this is different from that for a visco-elastic fluid, and is explained by the change with time of the orientation of squirmers.

Analysis of cell growth and diffusion in a scaffold for cartilage tissue engineering

Developments in tissue engineering over the past decade have offered promising future for the repair and reconstruction of damaged tissues. To regenerate three dimensional and weight-bearing implants, advances in biomaterials and manufacturing technologies prompted cell cultivations with natural or artificial scaffolds, in which cells are allowed to proliferate, migrate, and differentiate in vitro. In this article, we develop a mathematical model for cell growth in a porous scaffold. By treating the cell-scaffold construct as a porous medium, a continuum model is set up based on basic principles of mass conservation. In addition to cell growth kinetics, we incorporate cell diffusion in the model to describe the effects of cell random walks. Computational results are compared to experimental data found in the literature. With this model, we are able to investigate cell motility, heterogeneous cell distributions, and non-uniform seeding for tissue engineering applications. Results show that random walks tend to enhance uniform cell spreads in space, which in turn increases the probabilities for cells to acquire nutrients; therefore random walks are likely to be a positive contribution to the overall cell growth on scaffolds.

Bioluminescent monitoring of turbulent bioconvection

Under adjusted experimental conditions, open-to-air cultures of lux gene-engineered Ralstonia eutropha (wholecell biosensors of copper) exhibit bioconvection, which accounts for fluctuating bioluminescence. The power spectrum of bioluminescence intensity fluctuations recorded from a cylindrical sample 9 mm in diameter and approximately 10 mm in height is characterized by a dominant low-frequency oscillation (with a characteristic period of approximately 8-12 min), which is occasionally accompanied by a few weaker oscillations. The corresponding spectral peaks emerge on a high-noise background. The spectra of bioluminescence intensity fluctuations qualitatively resemble the spectra of temperature or fluid velocity fluctuations in an appropriate turbulent thermal convection system. It has been suggested that in a bioconvective system, like in thermal convection systems, the emergence of oscillation reflects the large-scale convective circulation that spans the height of the cylindrical cell. The velocity of large-scale bioconvective circulation was estimated to be 37-48 microm/s. The occasional emergence of weaker-than-dominant oscillations was explained through the coexistence and interaction of the large-scale circulation with, presumably, a gene-expression-related cyclic process (with a characteristic period of approximately 25-50 min).

Bacterial swimming and oxygen transport near contact lines

Aerobic bacteria often live in thin fluid layers near solid-air-water contact lines, in which the biology of chemotaxis, metabolism, and cell-cell signaling is intimately connected to the physics of buoyancy, diffusion, and mixing. Using the geometry of a sessile drop, we demonstrate in suspensions of Bacillus subtilis the self-organized generation of a persistent hydrodynamic vortex that traps cells near the contact line. Arising from upward oxygentaxis and downward gravitational forcing, these dynamics are related to the Boycott effect in sedimentation and are explained quantitatively by a mathematical model consisting of oxygen diffusion and consumption, chemotaxis, and viscous fluid dynamics. The vortex is shown to advectively enhance uptake of oxygen into the suspension, and the wedge geometry leads to a singularity in the chemotactic dynamics near the contact line.

Chemoconvection patterns in the methylene-blue-glucose system: weakly nonlinear analysis

The oxidation of solutions of glucose with methylene-blue as a catalyst in basic media can induce hydrodynamic overturning instabilities, termed chemoconvection in recognition of their similarity to convective instabilities. The phenomenon is due to gluconic acid, the marginally dense product of the reaction, which gradually builds an unstable density profile. Experiments indicate that dominant pattern wavenumbers initially increase before gradually decreasing or can even oscillate for long times. Here, we perform a weakly nonlinear analysis for an established model of the system with simple kinetics, and show that the resulting amplitude equation is analogous to that obtained in convection with insulating walls. We show that the amplitude description predicts that dominant pattern wavenumbers should decrease in the long term, but does not reproduce the aforementioned increasing wavenumber behavior in the initial stages of pattern development. We hypothesize that this is due to horizontally homogeneous steady states not being attained before pattern onset. We show that the behavior can be explained using a combination of pseudo-steady-state linear and steady-state weakly nonlinear theories. The results obtained are in qualitative agreement with the analysis of experiments.

Self-concentration and large-scale coherence in bacterial dynamics

Suspensions of aerobic bacteria often develop flows from the interplay of chemotaxis and buoyancy. We find in sessile drops that flows related to those in the Boycott effect of sedimentation carry bioconvective plumes down the slanted meniscus and concentrate cells at the drop edge, while in pendant drops such self-concentration occurs at the bottom. On scales much larger than a cell, concentrated regions in both geometries exhibit transient, reconstituting, high-speed jets straddled by vortex streets. A mechanism for large-scale coherence is proposed based on hydrodynamic interactions between swimming cells.

Model of bacterial band formation in aerotaxis

Aerotaxis is a particular form of "energy taxis". It is based on a largely elusive signal transduction machinery. In aerotaxis, oxygen dissolved in water plays the role of both attractant (at moderate concentrations) and repellent (at high and low concentrations). Cells swimming from favorable oxygen concentrations into regions with unfavorable concentrations increase the frequency of reversals, turn back into the favorable domain, and become effectively trapped there. At the same time, bacteria consume oxygen, creating an oxygen gradient. This behavior leads to a pattern formation phenomenon: bacteria self-organize into a dense band at a certain distance from the air-water interface. We incorporate experimental observations of the aerotactic bacterium, Azospirillum brasilense, into a mathematical model. The model consists of a system of differential equations describing swimming bacterial cells and diffusing oxygen. The cells' frequency of reversals depends on the concentration of oxygen and its time derivative while oxygen is depleted by the bacteria. We suggest a hypothetical model of energy sensing mediated by aerotactic receptors Aer and Tsr. Computer simulations and analysis of the model equations allow comparisons of theoretical and experimental results and provide insight into the mechanisms of bacterial pattern formation and underlying signal transduction machinery. We make testable predictions about position and density of the bacterial band.

Consumption, supply and transport: self-organization without direct communication

Swimming bacteria of the species Bacillus subtilis require and consume oxygen. In static liquid cultures the cells' swimming behaviour leads them to accumulate up oxygen concentration gradients generated by consumption and supply. Since the density of bacterial cells exceeds that of the fluid in which they live, fluid regions where cells have accumulated are denser than depleted regions. These density variations cause convection. The fluid motion is dynamically maintained by the swimming of the cells toward regions of attraction: the air-fluid interface and the fluctuating advecting attractors, gradients of oxygen concentration that are embedded in the convecting fluid. Because of the fluid dynamical conservation laws, these complex physical and biological factors generate patterns ordered over distances > 10000 bacterial cell diameters. The convection enhances long-range transport and mixing of oxygen, cells and extracellular products by orders of magnitude. Thus, through the interplay of physical and biological factors, a population of undifferentiated selfish cells creates functional dynamic patterns. Populations of bacteria that have organised themselves into regularly patterned regions of vigorous convection and varying cell concentration interact with their environment as if they were one purposeful, coherent multicellular individual. The mathematical and experimental ingredients of these remarkable phenomena are presented here.