Swarming dynamics in bacterial colonies

We determine and relate the characteristic velocity, length, and time scales for bacterial motion in swarming colonies of Paenibacillus dendritiformis growing on semi-solid agar substrates. The bacteria swim within a thin fluid layer, and they form long-lived jets and vortices. These coherent structures lead to anisotropy in velocity spatial correlations and to a two-step relaxation in velocity temporal correlations. The mean squared displacement of passive tracers exhibits a short-time regime with nearly ballistic transport and a diffusive long-time regime. We find that various definitions of the correlation length all lead to length scales that are, surprisingly, essentially independent of the mean bacterial speed, while the correlation time is linearly proportional to the ratio of the correlation length to the mean speed. Collections of self-propelled objects, such as vibrating rods [1, 2], flocking birds [3], and fish schools [4], often exhibit collective dynamics with extended spatiotemporal coherence. This biologically originated phenomenon has been studied from perspectives of nonequilibrium statistical mechanics and nonlinear dynamics. The studies have used discrete-particle dynamics based on simple local-interaction laws [5, 6], continuum ideas from liquid crystal physics [7], two-fluid models [8, 9], and hydrodynamics [10], and have simulated numerically idealized swimmers [11—13]. Coherent motions have been observed in suspensions of swimming bacteria. Dombrowski et al. [14] observed vortices and jets with a length scale ∼ 20 μm and a correlation time of 1—2 s in suspensions of Bacillus subtilis, where bioconvection was driven by chemotaxis and a RayleighTaylor instability. Wu and Libchaber [15] tracked particles suspended in a soap film containing motile Escherichia coli and found that the mean squared displacement ­ ∆r(∆t) ® of the passive tracers exhibited a short time regime with nearly ballistic transport and a long time regime with diffusive transport; they conjectured that the transition time from short to long time scales was set by the life time of the coherent structures. Sokolov et al [16] studied swarming of B. subtilis in a soap film and found that the correlation lengths varied smoothly and monotonically as the bacterial concentration increased. Collective dynamics has also been observed in bacterial colonies (e.g., B. subtilis and E. coli [17—22]), grown on an agar gel substrate. On the surface of a gel, swimming bacteria from a suspension differentiate into "swarmers", which are generally larger in size and have more flagella than the bacteria in suspension. Swarmers extract from an agar substrate nutrients and water, which mix with extracellular materials secreted by bacteria, such as polysaccharide, and form a thin slime layer. In this layer, densely packed swarmers move rapidly in coordinated motion. Although some experimental studies have been made to characterize the dynamics of swarming bacteria on an agar substrate [20—22], the relationship between the length, time, and velocity scales of the motion has not been systematically explored. We address this problem in a study of colonies of Paenibacillus dendritiformis bacteria. We determine the speeds of individual bacteria and the velocity field within colonies of the bacteria, and we obtain the relation between the bacterial mean speed and the correlation length and correlation time for the coherent swarming motion. Experiment. — Colonies of P. dendritiformis (Morphotype T ) [23, 24] are grown on substrates made of 1 liter solution of deionized water in which is dissolved 4 g of Bacto Peptone, 5 g of K2HPO4, 5 g of NaCl, and Difco Bacto agar. The agar mass varies from 10 g to 16 g, yielding agar concentrations 1.0-1.6%. We pour 12 ml of the solution into an 8.8 cm diameter Petri dish and main-