, Arrested phase separation in reproducing bacteria: a generic route to pattern formation? M. E. Cates, 1 D. Marenduzzo, 1 I. Pagonabarraga, 2 and J. Tailleur 1 $^{1}$SUPA, School of Physics and Astronomy, University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, UK $^{2}$Departament de F´ısica Fonamental, Universitat de Barcelona - Carrer Mart´ı Franqu´ es 1, 08028-Barcelona, Spain (Dated: October 26, 2018) 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. Microbial and cellular colonies are among the simplest examples of self-assembly in living organisms. In na- ture, bacteria are often found in concentrated biofilms, mats or other colony types, which can grow into spectac- ular patterns visible under the microscope [1, 2]. Also in the laboratory, bacteria such as E. coli and S. ty- phimirium form regular geometric patterns when they reproduce and grow on a Petri dish containing a gel such as agar. These patterns range from simple concentric rings to elaborate ordered or amorphous arrangements of dots [3–6]. Their formation results from collective be- haviour driven by interactions between the bacteria, such as chemotactic aggregation [6], competition for food [8] or changes in phenotypes according to density [11]. The question as to whether general mechanisms lie behind this diversity of microscopic pathways to patterning re- mains open. Unlike the self-assembly of colloidal particles, pattern formation in motile microorganisms and other living mat- ter is typically driven by non-equilibrium rather than thermodynamic forces. Indeed, the dynamics of both di- lute and concentrated bacterial fluids is already known to be vastly different from that of a suspensions of Brow- nian particles. For instance, suspensions of active, self- propelled particles, have been predicted to exhibit gi- ant density fluctuations [12, 13], which have been ob- served experimentally [14]. Similarly, an initially uni- form suspension of self-propelled particles performing a “run-and-tumble” motion like E. coli has recently been shown theoretically to separate into a bacteria-rich and a bacteria-poor phase, provided that the swimming speed decreases sufficiently rapidly with density [15]. This is akin to what happens in the spinodal decomposition of binary immiscible fluids, but has no counterpart in a sys- tem of Brownian particles interacting solely by density- dependent diffusivity. (The latter obey the fluctuation- dissipation theorem, ensuring that the equilibrium state Here we identify a very general mechanism that can lead to pattern formation in bacterial colonies and which may encompass a large class of experimental situations. This mechanism involves a density-dependent motility, giving rise to a phase separation which is then arrested, on a well defined characteristic length scale, by the birth and death dynamics of bacteria. For definiteness we will work this through for a particular model of bacte- Some aspects of bacterial patterning show features common to other nonequilibrium systems, and a cru- cial task is to identify the key ingredients that control their development. In many equilibrium and nonequi- librium phase transitions an initial instability creates density inhomogeneities; these coarsen, leading eventu- ally to macroscopic phase separation [20]. The situa- tion observed in bacterial assemblies often differs from this; long-lived patterns emerge with fixed characteristic length scales, suggesting that any underlying phase sep- aration is somehow arrested. The strong diversity of bio- logical functions met in experiments has led to an equally diverse range of proposed phenomenological models [5– 7, 9–11] to account for such effects. Most of them rely on the coupling of bacteria with external fields (food, chemoattractant, stimulant, etc.), and many involve a large number of parameters due to the complexity of the specific situation of interest. The most common mech- anism used to explain the bacterial patterns is chemo- taxis [6]: the propensity of bacteria to swim up/down gradients of chemoattractants/repellants. This explana- tion is so well established in the literature for at least two organisms ( E. coli and S. typhimuridium [6]) that obser- vation of similar patterns in other species might defensi- bly be taken as evidence for a chemotactic phenotype. is diffusivity-independent.) Other non-equilibrium ef- fects, such as ratchet physics, have also been observed and used either to rectify the density of bacteria [16–18] or to extract work from bacterial assemblies [19]. arXiv:1001.0057v1 [q-bio.PE] 4 Jan 2010