Physica D 234 (2007) 90–97 www.elsevier.com/locate/physd An (almost) solvable model for bacterial pattern formation B. Grammaticos ∗ , M. Badoual, M. Aubert IMNC, Universit´ e Paris VII-Paris XI, CNRS, UMR 8165, Bˆ at. 104, 91406 Orsay, France Received 22 January 2007; received in revised form 2 July 2007; accepted 4 July 2007 Available online 13 July 2007 Communicated by H. Levine Abstract We present a simple model for the description of ring-like concentric structures in bacterial colonies. We model the differences between Bacillus subtilis and Proteus mirabilis colonies by using a different dependence of the duration of the consolidation phase on the concentration of agar. We compare our results to experimental data from these two bacterial species colonies and obtain a good agreement. Based on this analysis, we formulate a hypothesis on the connection of the diffusion constant that appears in the model to the experimental agar concentration. c  2007 Elsevier B.V. All rights reserved. Keywords: Bacterial colonies; Pattern formation; Ring-like structures; Migration–consolidation 1. Introduction The study of pattern formation in bacterial colonies is of particular interest both from the biological and physical points of view (for a review see [15]). Since patterns are ubiquitous in biological systems, the study of very simple biological objects, like colonies of bacteria, provides an excellent starting point in our attempt to elucidate the complicated mechanisms which operate in biological pattern formation. For the physicist, these systems are equally interesting since they present a very rich behaviour, as far as patterns are concerned, while the number of control parameters stays manageable. In many cases, experiments involving bacterial colonies can be conducted like a physics experiment, where one has to ensure just the proper physical conditions in order to obtain a certain behaviour. It is thus quite natural that a very extensive literature exists on the matter (for reviews see [2,17,21,24]). In this paper, we shall focus on two kinds of bacteria which, under certain conditions, form concentric ring-like colonies: Proteus mirabilis and Bacillus subtilis. In the case of P. mirabilis, the ring-like patterns are the standard behaviour over a large range of parameters [19]. By contrast, for B. subtilis these patterns appear only for special conditions of nutrient (usually peptone) ∗ Corresponding author. E-mail address: grammaticos@univ-paris-diderot.fr (B. Grammaticos). abundance and substrate density (which is controlled by the agar concentration of the medium) [9,20,25]. As a matter of fact, in the case of B. subtilis colonies, the variety of possible patterns is impressive. In a nutrient-rich medium the observed patterns evolve with diminishing hardness from a very compact colony with a fractal-like border to concentric rings and to a disk-like, homogeneously spread colony. In a nutrient-poor medium, the colony grows branches which become more and more dense as the agar concentration diminishes. The modelling of pattern formation in bacterial colonies has been the object of numerous papers. Most of them are based on coupled reaction–diffusion-like equations [11] with vari- ous assumptions, like for instance, different populations [12], chemotaxis [3,12] or nonlinear diffusion of bacteria [18]. Dif- ferent approaches do also exist involving cellular automata [3] or hydrodynamics-based models [14]. Only one of these older approaches [13] has succeeded in reproducing concentric-ring colonies. Recently, models involving more microscopic aspects of swarming have appeared and manage to successfully reproduce ring patterns [1,6,8,16]. Mimura’s model [16] assumes the existence of two types of bacterial cells, active and inactive. Active cells become inactive when nutrient density becomes poor or cell density becomes low. Other models [1,6,8] are based on kinetic equations which describe the transition active–inactive (differentiation) as a function of bacterial 0167-2789/$ - see front matter c  2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physd.2007.07.002