Morphological Diagram of Bacterial Colony Pattern Simulated Using a Revised Mimura–Sakaguchi–Matsushita Model

Abstract

In this Note we report the attempt of improving the Mimura–Sakaguchi–Matsushita (MSM) model of bacterial colony formation by revising the nonlinear diffusion term in the reaction diffusion equation. Using this revised MSM model, all five typical colony patterns are reproduced and the experimental morphological diagram of colony patterns are completely simulated. During the last few decades, much attention has been paid to self-assembly pattern formation in various fields. Among them, pattern formation in biological systems is the most sophisticated and interesting. Due to the complexity of biological systems, recoganizing the mechanism of pattern formation in biological systems is difficult. Therefore, investigation of simple biological objects, which is dominated by purely physical conditions, is a good choice. Here we focus on bacterial colony formation, which is one of the simplest biological objects. It is known that some kinds of bacteria such as Bacillus subtilis (B. subtilis) exhibit various colony patterns on an agar medium by changing the incubation conditions, i.e., the concentration of agar Ca and that of nutrient Cn in the medium. Cn determines the rate of division of active bacteria, and Ca controls the motivebility of bacteria. Ohgiwari et al. completed an experimental morphological diagram (see Fig. 1). It can be seen from region A (high Ca and low Cn) of Fig. 1 that the colony pattern exhibits tip-splitting growth with diffusion-limitted aggregation (DLA)-like ramified structures. Keeping a high Ca and increasing Cn (region B),the pattern becomes round with Eden-like rough interface. If Cn is high and Ca is low (region D), the pattern looks macroscopically disklike. The colony growth in region D is consistent with the solution of the two-dimentional Fisher equation. In region E, which is between regions A and D, the pattern shows highly branched structures similar to the dense-branching morphology (DBM). When Cn is high and Ca takes an appropriate intermediate value (region C), colony exhibits concentricring patterns which are produced alternately in the active state and inactive state of the bacteria. In order to explain these observations, various reaction diffusion models have been proposed. The MSM model is one of them, which is based on the assumption that there exist two types of bacteria; active ones that move, grow and divide, and inactive ones that do nothing at all. Using the MSM model with linear diffusion patterns A, C, D and E are reproduced. Using MSM model with nonlinear diffusion, pattern B is also generated. In the MSM model, the relation among the population density of active bacteria u, the density of the inactive bacteria w, and the nutrient concentration v is expressed by a set of equations as