Spatio - temporal Patterns in Colonies of Rod - shaped Bacteria

In incubation experiments of bacterial colonies of Proteus Mirabilis, macroscopic spatio-temporal patterns, such as turbulent and unidirectional spiral patterns, appear in colonies. Considering only kinetic propeties of rod-shaped bacteria, we propose a phenomenological model for the directional and positional distributions. As the average density increases, homogeneous states bifurcate sub-critically into nonuniform states exhibiting localized collective motion, and spiral patterns appear for sufficiently large density. These patterns result from interactions between the local bacteria densities and the order parameter representing collective motion. Our model can be described by reduced equations using a perturbative method for large density. The unidirectionality of sprial rotation is also discussed. In systems of self-propelled particles, it is known that collective motion appears spontaneously with an increase in population. Collective motion has been reported to be observed as macroscopic spatio-temporal patterns in incubation experiments of bacterial colonies employing Proteus Mirabilis. 1)–3) We propose a mathematical model to investigate the mechanism of collective motion that causes such patterns. Proteus is a rod-shaped bacterium with many flagella. In dense states, swarms of proteus move like log rafts on the surface of a culture medium. Spatio-temporal patterns appear in a colony which has spread thinly on the surface of ager media. 1)–3) They are typically turbulent, target and spiral patterns, which resemble those observed in reaction-diffusion systems. However, what we observe as such patterns is the directional order of bacteria visualized by exploiting certain optical properties. The bacteria density is approximately uniform in a colony. In addition, only counterclockwise spirals appear in colonies of Proteus, and this is believed to be caused by some biological factor, such as the rotation direction of the flagella. We infer that a lack of nutrient and the growth of population plays no role in these phenomena because such patterns appear in nutrient-rich cultures, and because the density increases slowly. In this paper, we treat the spatial average of bacteria density, ρ, as a control parameter and consider only kinetic properties. We focus on the numerical and analytical results for a 2-dimensional model here. We express the density of bacteria moving in the direction of the angle θ at the position x ≡ (x, y) and time t as n(θ, x, t), and define the local bacteria density ρ(x, t) ≡ n and the complex order parameter W (x, t) ≡ ne iθ ≡ |W |e iΘ using the directional average ···· ≡ (1/2π) ∫ …