Arrested phase separation in reproducing bacteria: a generic route to pattern formation?

We present a generic mechanism by which reproducing microorganisms, with a diffusivity that depends on the local population density, can form stable patterns. It is known that a decrease of swimming speed with density can promote separation into bulk phases of two coexisting densities; this is opposed by the logistic law for birth and death which allows only a single uniform density to be stable. The result of this contest is an arrested nonequilibrium phase separation in which dense droplets or rings become separated by less dense regions, with a characteristic steady-state length scale. Cell division mainly occurs in the dilute regions and cell death in the dense ones, with a continuous flux between these sustained by the diffusivity gradient. We formulate a mathematical model of this in a case involving run-and-tumble bacteria, and make connections with a wider class of mechanisms for density-dependent motility. No chemotaxis is assumed in the model, yet it predicts the formation of patterns strikingly similar to those believed to result from chemotactic behavior

Radial and spiral stream formation in Proteus mirabilis

The enteric bacterium Proteus mirabilis, which is a pathogen that forms biofilms in vivo, can swarm over hard surfaces and form concentric ring 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 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 cell-based 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. The spatial patterns generated from the model are in qualitative agreement with the experimental observations

Physical Mechanisms for Chemotactic Pattern Formation by Bacteria

AbstractThis paper formulates a theory for chemotactic pattern formation by the bacteria Escherichia coli in the presence of excreted attractant. In a chemotactically neutral background, through chemoattractant signaling, the bacteria organize into swarm rings and aggregates. The analysis invokes only those physical processes that are both justifiable by known biochemistry and necessary and sufficient for swarm ring migration and aggregate formation. Swarm rings migrate in the absence of an external chemoattractant gradient. The ring motion is caused by the depletion of a substrate that is necessary to produce attractant. Several scaling laws are proposed and are demonstrated to be consistent with experimental data. Aggregate formation corresponds to finite time singularities in which the bacterial density diverges at a point. Instabilities of swarm rings leading to aggregate formation occur via a mechanism similar to aggregate formation itself: when the mass density of the swarm ring exceeds a threshold, the ring collapses cylindrically and then destabilizes into aggregates. This sequence of events is demonstrated both in the theoretical model and in the experiments

Spatio-temporal patterns generated by Salmonella typhimurium

We present experimental results on the bacterium Salmonella typhimurium which show that cells of chemotactic strains aggregate in response to gradients of amino acids, attractants that they themselves excrete. Depending on the conditions under which cells are cultured, they form periodic arrays of continuous or perforated rings, which arise sequentially within a spreading bacterial lawn. Based on these experiments, we develop a biologically realistic cell-chemotaxis model to describe the self-organization of bacteria. Numerical and analytical investigations of the model mechanism show how the two types of observed geometric patterns can be generated by the interaction of the cells with chemoattractant they produce

Aggregation Patterns in Stressed Bacteria

We study the formation of spot patterns seen in a variety of bacterial species when the bacteria are subjected to oxidative stress due to hazardous byproducts of respiration. Our approach consists of coupling the cell density field to a chemoattractant concentration as well as to nutrient and waste fields. The latter serves as a triggering field for emission of chemoattractant. Important elements in the proposed model include the propagation of a front of motile bacteria radially outward form an initial site, a Turing instability of the uniformly dense state and a reduction of motility for cells sufficiently far behind the front. The wide variety of patterns seen in the experiments is explained as being due the variation of the details of the initiation of the chemoattractant emission as well as the transition to a non-motile phase.Comment: 4 pages, REVTeX with 4 postscript figures (uuencoded) Figures 1a and 1b are available from the authors; paper submitted to PRL

An equation-free computational approach for extracting population-level behavior from individual-based models of biological dispersal

The movement of many organisms can be described as a random walk at either or both the individual and population level. The rules for this random walk are based on complex biological processes and it may be difficult to develop a tractable, quantitatively-accurate, individual-level model. However, important problems in areas ranging from ecology to medicine involve large collections of individuals, and a further intellectual challenge is to model population-level behavior based on a detailed individual-level model. Because of the large number of interacting individuals and because the individual-level model is complex, classical direct Monte Carlo simulations can be very slow, and often of little practical use. In this case, an equation-free approach may provide effective methods for the analysis and simulation of individual-based models. In this paper we analyze equation-free coarse projective integration. For analytical purposes, we start with known partial differential equations describing biological random walks and we study the projective integration of these equations. In particular, we illustrate how to accelerate explicit numerical methods for solving these equations. Then we present illustrative kinetic Monte Carlo simulations of these random walks and show a decrease in computational time by as much as a factor of a thousand can be obtained by exploiting the ideas developed by analysis of the closed form PDEs. The illustrative biological example here is chemotaxis, but it could be any random walker which biases its movement in response to environmental cues.Comment: 30 pages, submitted to Physica

Population Dynamics and Non-Hermitian Localization

We review localization with non-Hermitian time evolution as applied to simple models of population biology with spatially varying growth profiles and convection. Convection leads to a constant imaginary vector potential in the Schroedinger-like operator which appears in linearized growth models. We illustrate the basic ideas by reviewing how convection affects the evolution of a population influenced by a simple square well growth profile. Results from discrete lattice growth models in both one and two dimensions are presented. A set of similarity transformations which lead to exact results for the spectrum and winding numbers of eigenfunctions for random growth rates in one dimension is described in detail. We discuss the influence of boundary conditions, and argue that periodic boundary conditions lead to results which are in fact typical of a broad class of growth problems with convection.Comment: 19 pages, 11 figure

The Two Fluid Drop Snap-off Problem: Experiments and Theory

We address the dynamics of a drop with viscosity $\lambda \eta$ breaking up inside another fluid of viscosity $\eta$. For $\lambda=1$, a scaling theory predicts the time evolution of the drop shape near the point of snap-off which is in excellent agreement with experiment and previous simulations of Lister and Stone. We also investigate the $\lambda$ dependence of the shape and breaking rate.Comment: 4 pages, 3 figure

Range expansion with mutation and selection: dynamical phase transition in a two-species Eden model

The colonization of unoccupied territory by invading species, known as range expansion, is a spatially heterogeneous non-equilibrium growth process. We introduce a two-species Eden growth model to analyze the interplay between uni-directional (irreversible) mutations and selection at the expanding front. While the evolutionary dynamics leads to coalescence of both wild-type and mutant clusters, the non-homogeneous advance of the colony results in a rough front. We show that roughening and domain dynamics are strongly coupled, resulting in qualitatively altered bulk and front properties. For beneficial mutations the front is quickly taken over by mutants and growth proceeds Eden-like. In contrast, if mutants grow slower than wild-types, there is an antagonism between selection pressure against mutants and growth by the merging of mutant domains with an ensuing absorbing state phase transition to an all-mutant front. We find that surface roughening has a marked effect on the critical properties of the absorbing state phase transition. While reference models, which keep the expanding front flat, exhibit directed percolation critical behavior, the exponents of the two-species Eden model strongly deviate from it. In turn, the mutation-selection process induces an increased surface roughness with exponents distinct from that of the classical Eden model

Optically trapped bacteria pairs reveal discrete motile response to control aggregation upon cell–cell approach

Aggregation of bacteria plays a key role in the formation of many biofilms. The critical first step is cell–cell approach, and yet the ability of bacteria to control the likelihood of aggregation during this primary phase is unknown. Here, we use optical tweezers to measure the force between isolated Bacillus subtilis cells during approach. As we move the bacteria towards each other, cell motility (bacterial swimming) initiates the generation of repulsive forces at bacterial separations of ~3 μm. Moreover, the motile response displays spatial sensitivity with greater cell–cell repulsion evident as inter-bacterial distances decrease. To examine the environmental influence on the inter-bacterial forces, we perform the experiment with bacteria suspended in Tryptic Soy Broth, NaCl solution and deionised water. Our experiments demonstrate that repulsive forces are strongest in systems that inhibit biofilm formation (Tryptic Soy Broth), while attractive forces are weak and rare, even in systems where biofilms develop (NaCl solution). These results reveal that bacteria are able to control the likelihood of aggregation during the approach phase through a discretely modulated motile response. Clearly, the force-generating motility we observe during approach promotes biofilm prevention, rather than biofilm formation