Swarming bacteria migrate by Lévy Walk

Abstract

Individual swimming bacteria are known to bias their random trajectories in search of food and to optimize survival. The motion of bacteria within a swarm, wherein they migrate as a collective group over a solid surface, is fundamentally different as typical bacterial swarms show large-scale swirling and streaming motions involving millions to billions of cells. Here by tracking trajectories of fluorescently labelled individuals within such dense swarms, we find that the bacteria are performing super-diffusion, consistent with Lévy walks. Lévy walks are characterized by trajectories that have straight stretches for extended lengths whose variance is infinite. The evidence of super-diffusion consistent with Lévy walks in bacteria suggests that this strategy may have evolved considerably earlier than previously thought.

Figure 1. Tracking individual bacteria within a dense swarm.

(a–b) Phase contrast imaging of a wild type B. subtilis swarming colony: at high (a) and low (b) magnifications (region of interest is marked with an arrow in (b)). (c) Fluorescent microscopy showing the fluorescently labelled bacteria only, at high magnification. (d–e) Example trajectories of individual bacteria inside the swarm at high (d) and low (e) magnifications. Left/Red: B. subtilis and Blue/Right: S. marcescens.

Figure 2. Mean square displacement of single bacteria.

A slope of 1.6 is obtained for all bacteria; red lines show results with B. subtilis and blue lines with S. marcescens. The black line is the average of all bacteria. Data obtained with (a) high and (b) low magnifications.

Figure 3. Statistics of cell displacements.

Analysis of trajectories using the low magnification. (a) The distribution of cell displacements along each axis between a fixed number of frames. Following a scaling of Δt^1/1.33, all distributions approximately fit a Lévy stable distribution with parameter 1.33. (b) The probability density function (PDF) of cell directions (with respect to an arbitrary lateral axis of the frame) is uniform for all time intervals. Data were taken at high magnification. (c) Persistence in the direction of motion. The direction of motion remains fairly constant for times that are significantly longer than the characteristic run times in bacteria (∼1 s). Experiments with S. marcescens yield similar results (data not shown). (d) The velocity auto-correlation function decays algebraically with slope of −0.41, in agreement with the theory of LWs.

Figure 4. Bacterial trajectories have Lévy walk characteristics.

Analysis using the high magnification. (a) A typical trajectory of a B. subtilis bacterium with turning points marked in red. Trajectory was obtained at the high magnification. (b) The length ΔL of walking segments as a function of its duration Δt, indicating an approximately constant speed. A similar plot was obtained for S. marcescens cells with the same average speed. (c) Distribution of waiting times between turns showing a power-law decay with a slope of −2.5, in agreement with the theory of LWs. Filled circles, squares and diamonds show results with B. subtilis; empty circles, squares and diamonds show S. marcescens. The slope is independent of the cutoff ω. Straight lines are least squares fits. At short and long times (indicated by a vertical dashed line), the fit to a power-law deteriorates.