Spatio-temporal patterns generated by Salmonella typhimurium

Multi-population dissolution in confined active fluids

Autonomous out-of-equilibrium agents or cells in suspension are ubiquitous in biology and engineering. Turning chemical energy into mechanical stress, they generate activity in their environment, which may trigger spontaneous large-scale dynamics. Often, these systems are composed of multiple populations that may reflect the coexistence of multiple species, differing phenotypes, or chemically varying agents in engineered settings. Here, we present a new method for modeling such multi-population active fluids subject to confinement. We use a continuum multi-scale mean-field approach to represent each phase by its first three orientational moments and couple their evolution with those of the suspending fluid. The resulting coupled system is solved using a parallel adaptive level-set-based solver for high computational efficiency and maximal flexibility in the confinement geometry. Motivated by recent experimental work, we employ our method to study the spatiotemporal dynamics of confined bacterial suspensions and swarms dominated by fluid hydrodynamic effects. Our in silico explorations reproduce observed emergent collective patterns, including features of active dissolution in two-population active-passive swarms, with results clearly suggesting that hydrodynamic effects dominate dissolution dynamics. Our work lays the foundation for a systematic characterization and study of collective phenomena in natural or synthetic multi-population systems such as bacteria colonies, bird flocks, fish schools, colloid swimmers, or programmable active matter.

Non-reciprocity across scales in active mixtures

In active matter, particles typically experience mediated interactions, which are not constrained by Newton's third law and are therefore generically non-reciprocal. Non-reciprocity leads to a rich set of emerging behaviors that are hard to account for starting from the microscopic scale, due to the absence of a generic theoretical framework out of equilibrium. Here we consider bacterial mixtures that interact via mediated, non-reciprocal interactions (NRI) like quorum-sensing and chemotaxis. By explicitly relating microscopic and macroscopic dynamics, we show that, under conditions that we derive explicitly, non-reciprocity may fade upon coarse-graining, leading to large-scale equilibrium descriptions. In turn, this allows us to account quantitatively, and without fitting parameters, for the rich behaviors observed in microscopic simulations including phase separation, demixing, and multi-phase coexistence. We also derive the condition under which non-reciprocity survives coarse-graining, leading to a wealth of dynamical patterns. Again, our analytical approach allows us to predict the phase diagram of the system starting from its microscopic description. All in all, our work demonstrates that the fate of non-reciprocity across scales is a subtle and important question.

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.

Pattern formation by bacteria-phage interactions

The interactions between bacteria and phages-viruses that infect bacteria-play critical roles in agriculture, ecology, and medicine; however, how these interactions influence the spatial organization of both bacteria and phages remain largely unexplored. Here, we address this gap in knowledge by developing a theoretical model of motile, proliferating bacteria that aggregate via motility-induced phase separation (MIPS) and encounter phage that infect and lyse the cells. We find that the non-reciprocal predator-prey interactions between phage and bacteria strongly alter spatial organization, in some cases giving rise to a rich array of finite-scale stationary and dynamic patterns in which bacteria and phage coexist. We establish principles describing the onset and characteristics of these diverse behaviors, thereby helping to provide a biophysical basis for understanding pattern formation in bacteria-phage systems, as well as in a broader range of active and living systems with similar predator-prey or other non-reciprocal interactions.

A biophysical threshold for biofilm formation

Bacteria are ubiquitous in our daily lives, either as motile planktonic cells or as immobilized surface-attached biofilms. These different phenotypic states play key roles in agriculture, environment, industry, and medicine; hence, it is critically important to be able to predict the conditions under which bacteria transition from one state to the other. Unfortunately, these transitions depend on a dizzyingly complex array of factors that are determined by the intrinsic properties of the individual cells as well as those of their surrounding environments, and are thus challenging to describe. To address this issue, here, we develop a generally-applicable biophysical model of the interplay between motility-mediated dispersal and biofilm formation under positive quorum sensing control. Using this model, we establish a universal rule predicting how the onset and extent of biofilm formation depend collectively on cell concentration and motility, nutrient diffusion and consumption, chemotactic sensing, and autoinducer production. Our work thus provides a key step toward quantitatively predicting and controlling biofilm formation in diverse and complex settings.

Influence of confinement on the spreading of bacterial populations

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

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

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.

Formation, collective motion, and merging of macroscopic bacterial aggregates

Chemotactic bacteria form emergent spatial patterns of variable cell density within cultures that are initially spatially uniform. These patterns are the result of chemical gradients that are created from the directed movement and metabolic activity of billions of cells. A recent study on pattern formation in wild bacterial isolates has revealed unique collective behaviors of the bacteria Enterobacter cloacae. As in other bacterial species, Enterobacter cloacae form macroscopic aggregates. Once formed, these bacterial clusters can migrate several millimeters, sometimes resulting in the merging of two or more clusters. To better understand these phenomena, we examine the formation and dynamics of thousands of bacterial clusters that form within a 22 cm square culture dish filled with soft agar over two days. At the macroscale, the aggregates display spatial order at short length scales, and the migration of cell clusters is superdiffusive, with a merging acceleration that is correlated with aggregate size. At the microscale, aggregates are composed of immotile cells surrounded by low density regions of motile cells. The collective movement of the aggregates is the result of an asymmetric flux of bacteria at the boundary. An agent-based model is developed to examine how these phenomena are the result of both chemotactic movement and a change in motility at high cell density. These results identify and characterize a new mechanism for collective bacterial motility driven by a transient, density-dependent change in motility.

Bacteria growing and swimming in soft agar often aggregate to form elaborate spatial patterns. Here we examine the patterns formed by the bacteria Enterobacter cloacae. An unusual behavior of these bacteria is the collective movement of cells after the initial aggregation into a tiny spot. Despite the majority of the cells within an aggregate being immotile at any point in the time, the flux of cells entering and leaving the aggregate, as motility is lost and regained in individual cells, led to a net, collective movement of the aggregate. These spots sometimes run into each other and combine. By looking at the cells within these spots under a microscope, we find that cells within each spot stop swimming. The process of switching back and forth between swimming and not swimming causes the movement and fusion of the spots. A numerical simulation shows that the migration and merging of these spots can be expected if the cells swim towards regions of space with high concentrations of attractant molecules and stop swimming in locations crowded with many cells. This work identifies a novel process through which populations of bacteria cooperate and control the movement of large groups of cells.

Effects of internal dynamics on chemotactic aggregation of bacteria

The effects of internal adaptation dynamics on the self-organized aggregation of chemotactic bacteria are investigated by Monte Carlo (MC) simulations based on a two-stream kinetic transport equation coupled with a reaction-diffusion equation of the chemoattractant that bacteria produce. A remarkable finding is a nonmonotonic behavior of the peak aggregation density with respect to the adaptation time; more specifically, aggregation is the most enhanced when the adaptation time is comparable to or moderately larger than the mean run time of bacteria. Another curious observation is the formation of a trapezoidal aggregation profile occurring at a very large adaptation time, where the biased motion of individual cells is rather hindered at the plateau regimes due to the boundedness of the tumbling frequency modulation. Asymptotic analysis of the kinetic transport system is also carried out, and a novel asymptotic equation is obtained at the large adaptation-time regime while the Keller-Segel type equations are obtained when the adaptation time is moderate. Numerical comparison of the asymptotic equations with MC results clarifies that trapezoidal aggregation is well described by the novel asymptotic equation, and the nonmonotonic behavior of the peak aggregation density is interpreted as the transient of the asymptotic solutions between different adaptation time regimes.

Chemotactic migration of bacteria in porous media

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

Field-theoretic model for chemotaxis in run and tumble particles

In this paper, we develop a field-theoretic description for run and tumble chemotaxis, based on a density-functional description of crystalline materials modified to capture orientational ordering. We show that this framework, with its in-built multiparticle interactions, soft-core repulsion, and elasticity, is ideal for describing continuum collective phases with particle resolution, but on diffusive timescales. We show that our model exhibits particle aggregation in an externally imposed constant attractant field, as is observed for phototactic or thermotactic agents. We also show that this model captures particle aggregation through self-chemotaxis, an important mechanism that aids quorum-dependent cellular interactions.

Approaching optimality in blow-up results for Keller–Segel systems with logistic-type dampening

Nonnegative solutions of the Neumann initial-boundary value problem for the chemotaxis system in smooth bounded domains$$\Omega \subset {\mathbb {R}}^n$$Ω⊂Rn,$$n \ge 1$$n≥1, are known to be global in time if$$\lambda \ge 0$$λ≥0,$$\mu > 0$$μ>0and$$\kappa > 2$$κ>2. In the present work, we show that the exponent$$\kappa = 2$$κ=2is actually critical in the four- and higher dimensional setting. More precisely, if$$\begin{aligned} \qquad n&\ge 4,&\quad \kappa \in (1, 2) \quad&\text {and} \quad \mu > 0 \\ \text {or}\qquad n&\ge 5,&\quad \kappa = 2 \quad&\text {and} \quad \mu \in \left( 0, \frac{n-4}{n}\right) , \end{aligned}$$n≥4,κ∈(1,2)andμ>0orn≥5,κ=2andμ∈0,n-4n,for balls$$\Omega \subset {\mathbb {R}}^n$$Ω⊂Rnand parameters$$\lambda \ge 0$$λ≥0,$$m_0 > 0$$m0>0, we construct a nonnegative initial datum$$u_0 \in C^0({{\overline{\Omega }}})$$u0∈C0(Ω¯)with$$\int \nolimits _\Omega u_0 = m_0$$∫Ωu0=m0for which the corresponding solution (u, v) of ($$\star $$⋆) blows up in finite time. Moreover, in 3D, we obtain finite-time blow-up for$$\kappa \in (1, \frac{3}{2})$$κ∈(1,32)(and$$\lambda \ge 0$$λ≥0,$$\mu > 0$$μ>0). As the corner stone of our analysis, for certain initial data, we prove that the mass accumulation function$$w(s, t) = \int \nolimits _0^{\root n \of {s}} \rho ^{n-1} u(\rho , t) \,\mathrm {d}\rho $$w(s,t)=∫0snρn-1u(ρ,t)dρfulfills the estimate$$w_s \le \frac{w}{s}$$ws≤ws. Using this information, we then obtain finite-time blow-up ofuby showing that for suitably chosen initial data,$$s_0$$s0and$$\gamma $$γ, the function$$\phi (t) = \int \nolimits _0^{s_0} s^{-\gamma } (s_0 - s) w(s, t)$$ϕ(t)=∫0s0s-γ(s0-s)w(s,t)cannot exist globally.

Lamellar to Micellar Phases and Beyond: When Tactic Active Systems Admit Free Energy Functionals

We consider microscopic models of active particles whose velocities, rotational diffusivities, and tumbling rates depend on the gradient of a local field that is either externally imposed or depends on all particle positions. Despite the fundamental differences between active and passive dynamics at the microscopic scale, we show that a large class of such tactic active systems admit fluctuating hydrodynamics equivalent to those of interacting Brownian colloids in equilibrium. We exploit this mapping to show how taxis may lead to the lamellar and micellar phases observed for soft repulsive colloids. In the context of chemotaxis, we show how the competition between chemoattractant and chemorepellent may lead to a bona fide equilibrium liquid-gas phase separation in which a loss of thermodynamic stability of the fluid signals the onset of a chemotactic collapse.

Spatiotemporal pattern formation in E. coli biofilms explained by a simple physical energy balance

While the biofilm growth mode conveys notable thriving advantages to bacterial populations, the mechanisms of biofilm formation are still strongly debated. Here, we investigate the remarkable spontaneous formation of regular spatial patterns during the growth of an Escherichia coli biofilm. These patterns reported here appear with non-motile bacteria, which excludes both chemotactic origins and other motility-based ones. We demonstrate that a minimal physical model based on phase separation describes them well. To confirm the predictive capacity of our model, we tune the cell-cell and cell-surface interactions using cells expressing different surface appendages. We further explain how F pilus-bearing cells enroll their wild type kindred, poorly piliated, into their typical pattern when mixed together. This work supports the hypothesis that purely physicochemical processes, such as the interplay of cell-cell and cell-surface interactions, can drive the emergence of a highly organized spatial structure that is potentially decisive for community fate and for biological functions.

Connecting single-cell properties to collective behavior in multiple wild isolates of the Enterobacter cloacae complex

Some strains of motile bacteria self-organize to form spatial patterns of high and low cell density over length scales that can be observed by eye. One such collective behavior is the formation in semisolid agar media of a high cell density swarm band. We isolated 7 wild strains of the Enterobacter cloacae complex capable of forming this band and found its propagation speed can vary 2.5 fold across strains. To connect such variability in collective motility to strain properties, each strain’s single-cell motility and exponential growth rates were measured. The band speed did not significantly correlate with any individual strain property; however, a multilinear analysis revealed that the band speed was set by a combination of the run speed and tumbling frequency. Comparison of variability in closely-related wild isolates has the potential to reveal how changes in single-cell properties influence the collective behavior of populations.

The propagation of active-passive interfaces in bacterial swarms

Propagating interfaces are ubiquitous in nature, underlying instabilities and pattern formation in biology and material science. Physical principles governing interface growth are well understood in passive settings; however, our understanding of interfaces in active systems is still in its infancy. Here, we study the evolution of an active-passive interface using a model active matter system, bacterial swarms. We use ultra-violet light exposure to create compact domains of passive bacteria within Serratia marcescens swarms, thereby creating interfaces separating motile and immotile cells. Post-exposure, the boundary re-shapes and erodes due to self-emergent collective flows. We demonstrate that the active-passive boundary acts as a diffuse interface with mechanical properties set by the flow. Intriguingly, interfacial velocity couples to local swarm speed and interface curvature, raising the possibility that an active analogue to classic Gibbs-Thomson-Stefan conditions may control this boundary propagation.

Quantifying emergence and self-organisation of Enterobacter cloacae microbial communities

From microbial communities to cancer cells, many such complex collectives embody emergent and self-organising behaviour. Such behaviour drives cells to develop composite features such as formation of aggregates or expression of specific genes as a result of cell-cell interactions within a cell population. Currently, we lack universal mathematical tools for analysing the collective behaviour of biological swarms. To address this, we propose a multifractal inspired framework to measure the degree of emergence and self-organisation from scarce spatial (geometric) data and apply it to investigate the evolution of the spatial arrangement of Enterobacter cloacae aggregates. In a plate of semi-solid media, Enterobacter cloacae form a spatially extended pattern of high cell density aggregates. These aggregates nucleate from the site of inoculation and radiate outward to fill the entire plate. Multifractal analysis was used to characterise these patterns and calculate dynamics changes in emergence and self-organisation within the bacterial population. In particular, experimental results suggest that the new aggregates align their location with respect to the old ones leading to a decrease in emergence and increase in self-organisation.

Dynamical inversion of the energy landscape promotes non-equilibrium self-assembly of binary mixtures

When driven out of equilibrium, many diverse systems can form complex spatial and dynamical patterns, even in the absence of attractive interactions.

When driven out of equilibrium, many diverse systems can form complex spatial and dynamical patterns, even in the absence of attractive interactions. Using kinetic Monte Carlo simulations, we investigate the phase behavior of a binary system of particles of dissimilar size confined between semiflexible planar surfaces, in which the nanoconfinement introduces a non-local coupling between particles, which we model as an activation energy barrier to diffusion that decreases with the local fraction of the larger particle. The system autonomously reaches a cyclical non-equilibrium state characterized by the formation and dissolution of metastable micelle-like clusters with the small particles in the core and the large ones in the surrounding corona. The power spectrum of the fluctuations in the aggregation number exhibits 1/f noise reminiscent of self-organized critical systems. We suggest that the dynamical metastability of the micellar structures arises from an inversion of the energy landscape, in which the relaxation dynamics of one of the species induces a metastable phase for the other species.

Growth-mediated autochemotactic pattern formation in self-propelling bacteria

Bacteria, while developing a multicellular colony or biofilm, can undergo pattern formation by diverse intricate mechanisms. One such route is directional movement or chemotaxis toward or away from self-secreted or externally employed chemicals. In some bacteria, the self-produced signaling chemicals or autoinducers themselves act as chemoattractants or chemorepellents and thereby regulate the directional movements of the cells in the colony. In addition, bacteria follow a certain growth kinetics which is integrated in the process of colony development. Here, we study the interplay of bacterial growth dynamics, cell motility, and autochemotactic motion with respect to the self-secreted diffusive signaling chemicals in spatial pattern formation. Using a continuum model of motile bacteria, we show growth can act as a crucial tuning parameter in determining the spatiotemporal dynamics of a colony. In action of growth dynamics, while chemoattraction toward autoinducers creates arrested phase separation, pattern transitions and suppression can occur for a fixed chemorepulsive strength.

Spreading of nonmotile bacteria on a hard agar plate: Comparison between agent-based and stochastic simulations

We study spreading of a nonmotile bacteria colony on a hard agar plate by using agent-based and continuum models. We show that the spreading dynamics depends on the initial nutrient concentration, the motility, and the inherent demographic noise. Population fluctuations are inherent in an agent-based model, whereas for the continuum model we model them by using a stochastic Langevin equation. We show that the intrinsic population fluctuations coupled with nonlinear diffusivity lead to a transition from a diffusion limited aggregation type of morphology to an Eden-like morphology on decreasing the initial nutrient concentration.

Mathematical modeling and its analysis for instability of the immune system induced by chemotaxis

In this paper, we study how chemotaxis affects the immune system by proposing a minimal mathematical model, a reaction-diffusion-advection system, describing a cross-talk between antigens and immune cells via chemokines. We analyze the stability and instability arising in our chemotaxis model and find their conditions for different chemotactic strengths by using energy estimates, spectral analysis, and bootstrap argument. Numerical simulations are also performed to the model, by using the finite volume method in order to deal with the chemotaxis term, and the fractional step methods are used to solve the whole system. From the analytical and numerical results for our model, we explain not only the effective attraction of immune cells toward the site of infection but also hypersensitivity when chemotactic strength is greater than some threshold.

Spatial Control of Bacteria Using Screen Printing

Synthetic biology has led to advances in both our understanding and engineering of genetic circuits that affect spatial and temporal behaviors in living cells. A growing array of native and synthetic circuits such as oscillators, pattern generators, and cell-cell communication systems has been studied, which exhibit spatiotemporal properties. To better understand the design principles of these genetic circuits, there is a need for versatile and precise methods for patterning cell populations in various configurations. In this study, we develop a screen printing methodology to pattern bacteria on agar, glass, and paper surfaces. Initially, we tested three biocompatible resuspension media with appropriate rheological properties for screen printing. Using microscopy, we characterized the resolution and bleed of bacteria screen prints on agar and glass surfaces, obtaining resolutions as low as 188 μm. Next, we engineered bacterial strains producing visible chromoproteins analogous to the cyan, magenta, and yellow subtractive color system for the creation of multicolored bacteria images. Using this system, we printed distinct populations in overlapping or interlocking designs on both paper and agar substrates. These proof-of-principle experiments demonstrated how the screen printing method could be used to study microbial community interactions and pattern formation of biofilms at submillimeter length scales. Overall, our approach allows for rapid and precise prototyping of patterned bacteria species that will be useful in the understanding and engineering of spatiotemporal behaviors in microbial communities.

Multicellular Self-Organization of P. aeruginosa due to Interactions with Secreted Trails

Guided movement in response to slowly diffusing polymeric trails provides a unique mechanism for self-organization of some microorganisms. To elucidate how this signaling route leads to microcolony formation, we experimentally probe the trajectory and orientation of Pseudomonas aeruginosa that propel themselves on a surface using type IV pili motility appendages, which preferentially attach to deposited exopolysaccharides. We construct a stochastic model by analyzing single-bacterium trajectories and show that the resulting theoretical prediction for the many-body behavior of the bacteria is in quantitative agreement with our experimental characterization of how cells explore the surface via a power-law strategy.

Travelling wave analysis in chemotaxis: case of starvation

In this paper we investigate the existence of travelling wave solutions for a chemotaxis model under the scenarios of zero growth and constant growth rate. We use Lie symmetry analysis to generate generalized travelling wave solutions, a wider class of solutions than that obtained from the standard ansatz. Unlike previous approaches, we allow for diffusivity and signal degradation. We study the influence of cell growth, diffusivity and signal degradation on the behaviour of the system. We apply realistic boundary conditions to explicitly provide biologically relevant solutions. Our results generalize known results.

Colony Expansion of Socially Motile Myxococcus xanthus Cells Is Driven by Growth, Motility, and Exopolysaccharide Production

Myxococcus xanthus, a model organism for studies of multicellular behavior in bacteria, moves exclusively on solid surfaces using two distinct but coordinated motility mechanisms. One of these, social (S) motility is powered by the extension and retraction of type IV pili and requires the presence of exopolysaccharides (EPS) produced by neighboring cells. As a result, S motility requires close cell-to-cell proximity and isolated cells do not translocate. Previous studies measuring S motility by observing the colony expansion of cells deposited on agar have shown that the expansion rate increases with initial cell density, but the biophysical mechanisms involved remain largely unknown. To understand the dynamics of S motility-driven colony expansion, we developed a reaction-diffusion model describing the effects of cell density, EPS deposition and nutrient exposure on the expansion rate. Our results show that at steady state the population expands as a traveling wave with a speed determined by the interplay of cell motility and growth, a well-known characteristic of Fisher’s equation. The model explains the density-dependence of the colony expansion by demonstrating the presence of a lag phase–a transient period of very slow expansion with a duration dependent on the initial cell density. We propose that at a low initial density, more time is required for the cells to accumulate enough EPS to activate S-motility resulting in a longer lag period. Furthermore, our model makes the novel prediction that following the lag phase the population expands at a constant rate independent of the cell density. These predictions were confirmed by S motility experiments capturing long-term expansion dynamics.

Collective motility is a key mechanism bacteria use to self-organize into multicellular structures and to adapt to various environments. An important example of such behavior is social (S) motility in the gram-negative bacterium Myxococcus xanthus. S-motile cells are restricted to movement in groups and do not move as individual cells. S-motility is powered by type IV pili (TFP)–multi-subunit filaments, which extrude from the cell poles, adhere to the substrate and retract, pulling the cell forward. TFP retraction or adhesion is suggested to be triggered by extracellular exopolysaccharides (EPS) deposited by cells on the substrate. As individual cells synthesize both pili and EPS, it is unclear why S-motile cells only exhibit group movement. Moreover, the experimentally observed initial cell-density dependence of S-motility remains unexplained. To understand these phenomena, we developed a mathematical model for the colony expansion of S-motile cells. Our model hypothesizes that the EPS level regulates the TFP activity that initiates collective cell movements. With this assumption, the model quantitatively matches the density-dependent expansion rate. Moreover, the model predicts two phases during colony expansion: an initial density-dependent lag phase with a slow expansion rate, followed by a faster expansion phase with a density-independent rate. These model predictions were confirmed by long-term colony expansion experiments.

Build to understand: synthetic approaches to biology

In this review we discuss how synthetic biology facilitates the task of investigating genetic circuits that are observed in naturally occurring biological systems. Specifically, we give examples where experimentation with synthetic gene circuits has been used to understand four fundamental mechanisms intrinsic to development and disease: multistability, stochastic gene expression, oscillations, and cell-cell communication. Within each area, we also discuss how mathematical modeling has been employed as an essential tool to guide the design of novel gene circuits and as a theoretical basis for exploring circuit topologies exhibiting robust behaviors in the presence of noise.

Collective Behavior of Quorum-Sensing Run-and-Tumble Particles under Confinement

We study a generic model for quorum-sensing bacteria in circular confinement. Every bacterium produces signaling molecules, the local concentration of which triggers a response when a certain threshold is reached. If this response lowers the motility, then an aggregation of bacteria occurs which differs fundamentally from standard motility-induced phase separation due to the long-ranged nature of the concentration of signal molecules. We analyze this phenomenon analytically and by numerical simulations employing two different protocols leading to stationary cluster and ring morphologies, respectively.

From noise to synthetic nucleoli: can synthetic biology achieve new insights?

Synthetic biology aims to re-organise and control biological components to make functional devices. Along the way, the iterative process of designing and testing gene circuits has the potential to yield many insights into the functioning of the underlying chassis of cells. Thus, synthetic biology is converging with disciplines such as systems biology and even classical cell biology, to give a new level of predictability to gene expression, cell metabolism and cellular signalling networks. This review gives an overview of the contributions that synthetic biology has made in understanding gene expression, in terms of cell heterogeneity (noise), the coupling of growth and energy usage to expression, and spatiotemporal considerations. We mainly compare progress in bacterial and mammalian systems, which have some of the most-developed engineering frameworks. Overall, one view of synthetic biology can be neatly summarised as "creating in order to understand."

The effects of chemical interactions and culture history on the colonization of structured habitats by competing bacterial populations

Background

Bacterial habitats, such as soil and the gut, are structured at the micrometer scale. Important aspects of microbial life in such spatial ecosystems are migration and colonization. Here we explore the colonization of a structured ecosystem by two neutrally labeled strains of Escherichia coli. Using time-lapse microscopy we studied the colonization of one-dimensional arrays of habitat patches linked by connectors, which were invaded by the two E. coli strains from opposite sides.

Results

The two strains colonize a habitat from opposite sides by a series of traveling waves followed by an expansion front. When population waves collide, they branch into a continuing traveling wave, a reflected wave and a stationary population. When the two strains invade the landscape from opposite sides, they remain segregated in space and often one population will displace the other from most of the habitat. However, when the strains are co-cultured before entering the habitats, they colonize the habitat together and do not separate spatially. Using physically separated, but diffusionally coupled, habitats we show that colonization waves and expansion fronts interact trough diffusible molecules, and not by direct competition for space. Furthermore, we found that colonization outcome is influenced by a culture’s history, as the culture with the longest doubling time in bulk conditions tends to take over the largest fraction of the habitat. Finally, we observed that population distributions in parallel habitats located on the same device and inoculated with cells from the same overnight culture are significantly more similar to each other than to patterns in identical habitats located on different devices inoculated with cells from different overnight cultures, even tough all cultures were started from the same −80°C frozen stock.

Conclusions

We found that the colonization of spatially structure habitats by two interacting populations can lead to the formation of complex, but reproducible, spatiotemporal patterns. Furthermore, we showed that chemical interactions between two populations cause them to remain spatially segregated while they compete for habitat space. Finally, we observed that growth properties in bulk conditions correlate with the outcome of habitat colonization. Together, our data show the crucial roles of chemical interactions between populations and a culture’s history in determining the outcome of habitat colonization.

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.

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.

Macroscopic equations for bacterial chemotaxis: integration of detailed biochemistry of cell signaling

Chemotaxis of single cells has been extensively studied and a great deal on intracellular signaling and cell movement is known. However, systematic methods to embed such information into continuum PDE models for cell population dynamics are still in their infancy. In this paper, we consider chemotaxis of run-and-tumble bacteria and derive continuum models that take into account of the detailed biochemistry of intracellular signaling. We analytically show that the macroscopic bacterial density can be approximated by the Patlak-Keller-Segel equation in response to signals that change slowly in space and time. We derive, for the first time, general formulas that represent the chemotactic sensitivity in terms of detailed descriptions of single-cell signaling dynamics in arbitrary space dimensions. These general formulas are useful in explaining relations of single cell behavior and population dynamics. As an example, we apply the theory to chemotaxis of bacterium Escherichia coli and show how the structure and kinetics of the intracellular signaling network determine the sensing properties of E. coli populations. Numerical comparison of the derived PDEs and the underlying cell-based models show quantitative agreements for signals that change slowly, and qualitative agreements for signals that change extremely fast. The general theory we develop here is readily applicable to chemotaxis of other run-and-tumble bacteria, or collective behavior of other individuals that move using a similar strategy.

Chemotactic sensing towards ambient and secreted attractant drives collective behaviour of E. coli

We simulate the dynamics of a suspension of bacterial swimmers, which chemotactically sense gradients in either ambient or self-secreted attractants (e.g. nutrient or aspartate respectively), or in both. Unlike previous mean field models based on a set of continuum partial differential equations, our model resolves single swimmers and therefore incorporates stochasticity and effects due to fluctuations in the bacterial density field. The algorithm we use is simple enough that we can follow the evolution of colonies of up to over a million bacteria for timescales relevant to pattern formation for E. coli growing in semisolid medium such as agar, or in confined geometries. Our results confirm previous mean field results that the patterns observed experimentally can be reproduced with a model incorporating chemoattractant secretion, chemotaxis (towards gradients in the chemoattractant field), and bacterial reproduction. They also suggest that further experiments with bacterial strains chemotactically moving up both nutrient and secreted attractant field may yield yet more dynamical patterns.

Identification of the YfgF MASE1 domain as a modulator of bacterial responses to aspartate

Complex 3′-5′-cyclic diguanylic acid (c-di-GMP) responsive regulatory networks that are modulated by the action of multiple diguanylate cyclases (DGC; GGDEF domain proteins) and phosphodiesterases (PDE; EAL domain proteins) have evolved in many bacteria. YfgF proteins possess a membrane-anchoring domain (MASE1), a catalytically inactive GGDEF domain and a catalytically active EAL domain. Here, sustained expression of the Salmonella enterica spp. Enterica ser. Enteritidis YfgF protein is shown to mediate inhibition of the formation of the aspartate chemotactic ring on motility agar under aerobic conditions. This phenomenon was c-di-GMP-independent because it occurred in a Salmonella strain that lacked the ability to synthesize c-di-GMP and also when PDE activity was abolished by site-directed mutagenesis of the EAL domain. YfgF-mediated inhibition of aspartate chemotactic ring formation was impaired in the altered redox environment generated by exogenous p-benzoquinone. This ability of YfgF to inhibit the response to aspartate required a motif, ²¹³Lys-Lys-Glu²¹⁵, in the predicted cytoplasmic loop between trans-membrane regions 5 and 6 of the MASE1 domain. Thus, for the first time the function of a MASE1 domain as a redox-responsive regulator of bacterial responses to aspartate has been shown.

Excitation and adaptation in bacteria-a model signal transduction system that controls taxis and spatial pattern formation

The machinery for transduction of chemotactic stimuli in the bacterium E. coli is one of the most completely characterized signal transduction systems, and because of its relative simplicity, quantitative analysis of this system is possible. Here we discuss models which reproduce many of the important behaviors of the system. The important characteristics of the signal transduction system are excitation and adaptation, and the latter implies that the transduction system can function as a "derivative sensor" with respect to the ligand concentration in that the DC component of a signal is ultimately ignored if it is not too large. This temporal sensing mechanism provides the bacterium with a memory of its passage through spatially- or temporally-varying signal fields, and adaptation is essential for successful chemotaxis. We also discuss some of the spatial patterns observed in populations and indicate how cell-level behavior can be embedded in population-level descriptions.

Modeling spatiotemporal dynamics of bacterial populations

Quantitative modeling of spatiotemporal dynamics of cells facilitates understanding and engineering of biological systems. Using a synthetic bacterial ecosystem as a workbench, we present the approach to mathematically simulate the spatiotemporal population dynamics of the ecosystem. A description of ecosystem's genetic construction and model development is firstly given. Parameter estimation and computational approach for the derived partial differential equations (PDEs) are then given. Spatiotemporal pattern formation is computed by numerically solving the PDE model. Biodiversity of the ecosystem and its impacts by cellular seeding distance and motility are computed according to the cell distribution patterns.

PATTERN FORMATION IN CHEMOTAXIC REACTION-DIFFUSION SYSTEMS

In this study, we investigate two-dimensional patterns generated by chemotaxis reaction-diffusion systems. We numerically examine the Keller–Segel models with the volume-filling aggregation term and the receptor aggregation term in two dimensions. Spotted, striped and reversed spotted patterns are obtained as stable motionless equilibrium patterns. The relative stability of these patterns is studied numerically on the basis of the derived free energy. The intuitive understanding of these generated patterns and the relation with three-dimensional patterns are also discussed.

Collective dynamics of model microorganisms with chemotactic signaling

Various microorganisms use chemotaxis for signaling among individuals-a common strategy for communication that is responsible for the formation of microcolonies. We model the microorganisms as autochemotactic active random walkers and describe them by an appropriate Langevin dynamics. It consists of rotational diffusion of the walker's velocity direction and a deterministic torque that aligns the velocity direction along the gradient of a self-generated chemical field. To account for finite size, each microorganism is treated as a soft disk. Its velocity is modified when it overlaps with other walkers according to a linear force-velocity relation and a harmonic repulsion force. We analyze two-walker collisions by presenting typical trajectories and by determining a state diagram that distinguishes between free walker, metastable, and bounded cluster states. We mention an analogy to Kramer's escape problem. Finally, we investigate relevant properties of many-walker systems and describe characteristics of cluster formation in unbounded geometry and in confinement.

Coupling between switching regulation and torque generation in bacterial flagellar motor

The bacterial flagellar motor plays a crucial role in both bacterial locomotion and chemotaxis. Recent experiments reveal that the switching dynamics of the motor depend on the rotation speed of the motor, and thus the motor torque, nonmonotonically. Here we present a unified mathematical model which treats motor torque generation based on experimental torque-speed curves and the torque-dependent switching based on the conformational spread model. The model successfully reproduces the observed switching rate as a function of the rotation speed, and provides a generic physical explanation independent of most details. A stator affects the switching dynamics through two mechanisms: accelerating the conformational flipping rate of individual rotor-switching units, which contributes most when the stator works at a high torque and thus a low speed; and influencing a larger number of rotor-switching units within unit time, whose contribution is the greatest when the motor rotates at a high speed. Consequently, the switching rate shows a maximum at intermediate speed, where the above two mechanisms find an optimal output. The load-switching relation may serve as a mechanism for sensing the physical environment, similar to the chemotaxis mechanism for sensing the chemical environment. It may also coordinate the switch dynamics of motors within the same cell.

Diffusive transport without detailed balance in motile bacteria: does microbiology need statistical physics?

Microbiology is the science of microbes, particularly bacteria. Many bacteria are motile: they are capable of self-propulsion. Among these, a significant class execute so-called run-and-tumble motion: they follow a fairly straight path for a certain distance, then abruptly change direction before repeating the process. This dynamics has something in common with Brownian motion (it is diffusive at large scales), and also something in contrast. Specifically, motility parameters such as the run speed and tumble rate depend on the local environment and hence can vary in space. When they do so, even if a steady state is reached, this is not generally invariant under time reversal: the principle of detailed balance, which restores the microscopic time-reversal symmetry of systems in thermal equilibrium, is mesoscopically absent in motile bacteria. This lack of detailed balance (allowed by the flux of chemical energy that drives motility) creates pitfalls for the unwary modeller. Here I review some statistical-mechanical models for bacterial motility, presenting them as a paradigm for exploring diffusion without detailed balance. I also discuss the extent to which statistical physics is useful in understanding real or potential microbiological experiments.

Towards an integrated experimental-theoretical approach for assessing the mechanistic basis of hair and feather morphogenesis

In his seminal 1952 paper, 'The Chemical Basis of Morphogenesis', Alan Turing lays down a milestone in the application of theoretical approaches to understand complex biological processes. His deceptively simple demonstration that a system of reacting and diffusing chemicals could, under certain conditions, generate spatial patterning out of homogeneity provided an elegant solution to the problem of how one of nature's most intricate events occurs: the emergence of structure and form in the developing embryo. The molecular revolution that has taken place during the six decades following this landmark publication has now placed this generation of theoreticians and biologists in an excellent position to rigorously test the theory and, encouragingly, a number of systems have emerged that appear to conform to some of Turing's fundamental ideas. In this paper, we describe the history and more recent integration between experiment and theory in one of the key models for understanding pattern formation: the emergence of feathers and hair in the skins of birds and mammals.

Radial and spiral stream formation in Proteus mirabilis colonies

The enteric bacterium Proteus mirabilis, which is a pathogen that forms biofilms in vivo, can swarm over hard surfaces and form a variety of spatial patterns in colonies. Colony formation involves two distinct cell types: swarmer cells that dominate near the surface and the leading edge, and swimmer cells that prefer a less viscous medium, but the mechanisms underlying pattern formation are not understood. New experimental investigations reported here show that swimmer cells in the center of the colony stream inward toward the inoculation site and in the process form many complex patterns, including radial and spiral streams, in addition to previously-reported concentric rings. These new observations suggest that swimmers are motile and that indirect interactions between them are essential in the pattern formation. To explain these observations we develop a hybrid model comprising cell-based and continuum components that incorporates a chemotactic response of swimmers to a chemical they produce. The model predicts that formation of radial streams can be explained as the modulation of the local attractant concentration by the cells, and that the chirality of the spiral streams results from a swimming bias of the cells near the surface of the substrate. The spatial patterns generated from the model are in qualitative agreement with the experimental observations.

Bacteria frequently colonize surfaces and grow as biofilm communities embedded in a gel-like polysaccharide matrix, and when this occurs on catheters, heart valves and other medical implants, it can lead to serious, hard-to-treat infections. Proteus mirabilis is an enteric bacterium that forms biofilms on urinary catheters, but in laboratory experiments it can swarm over hard surfaces and form a variety of spatial patterns. Understanding these patterns is a first step toward understanding biofilm formation, and here we describe new experimental results and mathematical models of pattern formation in Proteus. The experiments show that swimmer cells in the center of the colony stream inward toward the inoculation site and in the process form many complex patterns, including radial and spiral streams, in addition to concentric rings. To explain these observations we develop a model that incorporates a chemotactic response of swimmers to a chemical they produce. The model predicts that formation of radial streams can be explained as the modulation of the local attractant concentration by the cells, and that the chirality of the spiral streams can be predicted by incorporating a swimming bias of the cells near the surface of the substrate.

Migration of chemotactic bacteria in soft agar: role of gel concentration

We study the migration of chemotactic wild-type Escherichia coli populations in semisolid (soft) agar in the concentration range C = 0.15-0.5% (w/v). For C≲0.35%, expanding bacterial colonies display characteristic chemotactic rings. At C = 0.35%, however, bacteria migrate as broad circular bands rather than sharp rings. These are growth/diffusion waves arising because of suppression of chemotaxis by the agar and have not been previously reported experimentally to our knowledge. For C = 0.4-0.5%, expanding colonies do not span the depth of the agar and develop pronounced front instabilities. The migration front speed is weakly dependent on agar concentration at C < 0.25%, but decreases sharply above this value. We discuss these observations in terms of an extended Keller-Segel model for which we derived novel transport parameter expressions accounting for perturbations of the chemotactic response by collisions with the agar. The model makes it possible to fit the observed front speed decay in the range C = 0.15-0.35%, and its solutions qualitatively reproduce the observed transition from chemotactic to growth/diffusion bands. We discuss the implications of our results for the study of bacteria in porous media and for the design of improved bacteriological chemotaxis assays.