Theory of periodic swarming of bacteria: application to Proteus mirabilis

Irreversibility and emergent structure in active matter

Active matter is rapidly becoming a key paradigm of out-of-equilibrium soft matter exhibiting complex collective phenomena, yet the thermodynamics of such systems remain poorly understood. In this article we study the dynamical irreversibility of large-scale active systems capable of motility-induced phase separation and polar alignment. We use a model with momenta in both translational and rotational degrees of freedom, revealing a hidden component not previously reported in the literature. Steady-state irreversibility is quantified at each point in the phase diagram which exhibits sharp discontinuities at phase transitions. Identification of the irreversibility in individual particles lays the groundwork for discussion of the thermodynamics of microfeatures, such as defects in the emergent structure. The interpretation of the time reversal symmetry in the dynamics of the particles is found to be crucial.

Considerations for Modeling Proteus mirabilis Swarming

In this chapter we provide some initial guidance to experimentalists on how they might go about creating mathematical representations of their systems under study. Because the interests and goals of different researchers can differ, we try to provide broad instruction on the creation and use of mathematical models. We provide a brief overview of some modeling that has been done with Proteus mirabilis colonies, and discuss the goals of modeling. We suggest ways that collaborative teams may communicate with one another more effectively, and how they can build more confidence in their model results.

Front structure and dynamics in dense colonies of motile bacteria: Role of active turbulence

We study the spreading of a bacterial colony undergoing turbulentlike collective motion. We present two minimalistic models to investigate the interplay between population growth and coherent structures arising from turbulence. Using direct numerical simulation of the proposed models we find that turbulence has two prominent effects on the spatial growth of the colony: (a) the front speed is enhanced, and (b) the front gets crumpled. Both these effects, which we highlight by using statistical tools, are markedly different in our two models. We also show that the crumpled front structure and the passive scalar fronts in random flows are related in certain regimes.

Nontoxic colloidal particles impede antibiotic resistance of swarming bacteria by disrupting collective motion and speed

A monolayer of swarming B. subtilis on semisolid agar is shown to display enhanced resistance against antibacterial drugs due to their collective behavior and motility. The dynamics of swarming motion, visualized in real time using time-lapse microscopy, prevents the bacteria from prolonged exposure to lethal drug concentrations. The elevated drug resistance is significantly reduced when the collective motion of bacteria is judiciously disrupted using nontoxic polystyrene colloidal particles immobilized on the agar surface. The colloidal particles block and hinder the motion of the cells, and force large swarming rafts to break up into smaller packs in order to maneuver across narrow spaces between densely packed particles. In this manner, cohesive rafts rapidly lose their collectivity, speed, and group dynamics, and the cells become vulnerable to the drugs. The antibiotic resistance capability of swarming B. subtilis is experimentally observed to be negatively correlated with the number density of colloidal particles on the engineered surface. This relationship is further tested using an improved self-propelled particle model that takes into account interparticle alignment and hard-core repulsion. This work has pertinent implications on the design of optimal methods to treat drug resistant bacteria commonly found in swarming colonies.

Observation of multicellular spinning behavior of Proteus mirabilis by atomic force microscopy and multifunctional microscopy

This study aimed to observe the multicellular spinning behavior of Proteus mirabilis by atomic force microscopy (AFM) and multifunctional microscopy in order to understand the mechanism underlying this spinning movement and its biological significance. Multifunctional microscopy with charge-coupled device (CCD) and real-time AFM showed changes in cell structure and shape of P. mirabilis during multicellular spinning movement. Specifically, the morphological characteristics of P. mirabilis, multicellular spinning dynamics, and unique movement were observed. Our findings indicate that the multicellular spinning behavior of P. mirabilis may be used to collect nutrients, perform colonization, and squeeze out competitors. The movement characteristics of P. mirabilis are vital to the organism's biological adaptability to the surrounding environment.

Modeling the role of water in Bacillus subtilis colonies

We propose a simple cellular automaton model for the description of the evolution of a colony of Bacillus subtilis. The originality of our model lies in the fact that the bacteria can move in a pool of liquid. We assume that each migrating bacterium is surrounded by an individual pool, and the overlap of the latter gives rise to a collective pool with a higher water level. The bacteria migrate collectively when the level of water is high enough. When the bacteria are far enough from each other, the level of water becomes locally too low to allow migration, and the bacteria switch to a proliferating state. The proliferation-to-migration switch is triggered by high levels of a substance produced by proliferating bacteria. We show that it is possible to reproduce in a fairly satisfactory way the various forms that make up the experimentally observed morphological diagram of B. subtilis. We propose a phenomenological relation between the size of the water pool used in our model and the agar concentration of the substrate on which the bacteria migrate. We also compare experimental results from cutting the central part of the colony with the results of our simulations.

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.

Bacterial Swarming: A Model System for Studying Dynamic Self-assembly

Bacterial swarming is an example of dynamic self-assembly in microbiology in which the collective interaction of a population of bacterial cells leads to emergent behavior. Swarming occurs when cells interact with surfaces, reprogram their physiology and behavior, and adapt to changes in their environment by coordinating their growth and motility with other cells in the colony. This review summarizes the salient biological and biophysical features of this system and describes our current understanding of swarming motility. We have organized this review into four sections: 1) The biophysics and mechanisms of bacterial motility in fluids and its relevance to swarming. 2) The role of cell/molecule, cell/surface, and cell/cell interactions during swarming. 3) The changes in physiology and behavior that accompany swarming motility. 4) A concluding discussion of several interesting, unanswered questions that is particularly relevant to soft matter scientists.

Living on a surface: swarming and biofilm formation

Swarming is the fastest known bacterial mode of surface translocation and enables the rapid colonization of a nutrient-rich environment and host tissues. This complex multicellular behavior requires the integration of chemical and physical signals, which leads to the physiological and morphological differentiation of the bacteria into swarmer cells. Here, we provide a review of recent advances in the study of the regulatory pathways that lead to swarming behavior of different model bacteria. It has now become clear that many of these pathways also affect the formation of biofilms, surface-attached bacterial colonies. Decision-making between rapidly colonizing a surface and biofilm formation is central to bacterial survival among competitors. In the second part of this article, we review recent developments in the understanding of the transition between motile and sessile lifestyles of bacteria.

Noise-induced transition from translational to rotational motion of swarms

We consider a model of active Brownian agents interacting via a harmonic attractive potential in a two-dimensional system in the presence of noise. By numerical simulations, we show that this model possesses a noise-induced transition characterized by the breakdown of translational motion and the onset of swarm rotation as the noise intensity is increased. Statistical properties of swarm dynamics in the weak noise limit are further analytically investigated.

Developmental waves in myxobacteria: A distinctive pattern formation mechanism

In early stages of their development, starving myxobacteria organize their motion to produce a periodic pattern of traveling cell density waves. These waves arise from coordination of individual cell reversals by contact signaling when they collide. Unlike waves generated by reaction-diffusion instabilities, which annihilate on collision, myxobacteria waves appear to pass through one another unaffected. Here we analyze a mathematical model of these waves developed earlier [Proc. Natl. Acad. Sci. USA 98, 14 913 (2001)]]. The mechanisms which generate and maintain the density waves are clearly revealed by tracing the reversal loci of individual cells. An evolution equation of reversal point density is derived in the weak-signaling limit. Linear stability analysis determines parameters favorable for the development of the waves. Numerical solutions demonstrate the stability of the fully developed nonlinear waves.

Rippling of myxobacteria

Myxobacteria colonies during their aggregation phase propagate complex waves over their surface. These waves are fundamentally different from the analogous phenomenon in diffusion-reaction systems or in populations of Dictyostelium discoideum where colliding waves annhilate. Myxobacterial waves appear to pass through one another, analogous to solitons. Moreover, individual bacteria oscillate back and forth, exhibiting no net mass transfer. A mathematical model can explain virtually all of the experimentally observed properties of these waves and draw several conclusions about the properties of the intercelular signaling system.

Self-organized pattern formation of a bacteria colony modeled by a reaction diffusion system and nucleation theory

Self-organized pattern formation is observed in bacterial colony growth. The recently reported knotted-branching pattern of the Bacillus circulans colony consists of the trajectories of aggregates which grow, move, and reproduce simultaneously. We modeled these processes by combining a reaction diffusion system of nutrient dynamics, nucleation theory for aggregate generation, and individual based dynamics of motion and growth of aggregates. The branching pattern produced by computer simulation shows great similarity with experiments. Response to the initial nutrient concentration is also consistent with the experiments.

Vascular assembly in natural and engineered tissues

With the advent of molecular embryology and exploitation of genetic models systems, many genes necessary for normal blood vessel formation during early development have been identified. These genes include soluble effectors and their receptors, as well as components of cell-cell junctions and mediators of cell-matrix interactions. In vitro model systems (2-D and 3-D) to study paracrine and autocrine interactions of vascular cells and their progenitors have also been created. These systems are being combined to study the behavior of genetically altered cells to dissect and define the cellular role(s) of specific genes and gene families in directing the migration, proliferation, and differentiation needed for blood vessel assembly. It is clear that a complex spatial and temporal interplay of signals, including both genetic and environmental, modulates the assembly process. The development of real-time imaging and image analysis will enable us to gain further insights into this process. Collaborative efforts among vascular biologists, biomedical engineers, mathematicians, and physicists will allow us to bridge the gap between understanding vessel assembly in vivo and assembling vessels ex vivo.

Excitation of rotational modes in two-dimensional systems of driven Brownian particles

Models of active Brownian motion in two-dimensional (2D) systems developed earlier are investigated with respect to the influence of linear attracting forces and external noise. Our consideration is restricted to the case that the driving is rather weak and that the forces show only weak deviations from radial symmetry. In this case an analytical study of the bifurcations of the system is possible. We show that in the presence of external linear forces with only small deviations from radial symmetry, the system develops rotational excitations with left-right symmetry, corresponding to limit cycles in the 4D phase space, the corresponding distribution has the form of a hoop or a tire in the 4D space. In the last part we apply the theory to swarms of Brownian particles that are held together by weak and attracting forces, which lead to cluster formation. Since near the center the potential is at least approximately parabolic and near to the radial symmetry, the swarm develops rotational modes of motion with left-right symmetry.

Growth pulsations in symmetric dendritic crystallization in thin polymer blend films

The crystallization of polymeric and metallic materials normally occurs under conditions far from equilibrium, leading to patterns that grow as propagating waves into the surrounding unstable fluid medium. The Mullins-Sekerka instability causes these wave fronts to break up into dendritic arms, and we anticipate that the normal modes of the dendrite tips have a significant influence on pattern growth. To check this possibility, we focus on the dendritic growth of polyethylene oxide in a thin-film geometry. This crystalline polymer is mixed with an amorphous polymer (polymethyl-methacrylate) to "tune" the morphology and clay was added to nucleate the crystallization. The tips of the main dendrite trunks pulsate during growth and the sidebranches, which grow orthogonally to the trunk, pulsate out of phase so that the tip dynamics is governed by a limit cycle. The pulsation period P increases sharply with decreasing film thickness L and then vanishes below a critical value L(c) approximately 80 nm. A change of dendrite morphology accompanies this transition.

Pattern formation and traveling waves in myxobacteria: theory and modeling

Recent experiments have provided new quantitative measurements of the rippling phenomenon in fields of developing myxobacteria cells. These measurements have enabled us to develop a mathematical model for the ripple phenomenon on the basis of the biochemistry of the C-signaling system, whereby individuals signal by direct cell contact. The model quantitatively reproduces all of the experimental observations and illustrates how intracellular dynamics, contact-mediated intercellular communication, and cell motility can coordinate to produce collective behavior. This pattern of waves is qualitatively different from that observed in other social organisms, especially Dictyostelium discoideum, which depend on diffusible morphogens.