Physical Properties of Collective Motion in Suspensions of Bacteria

A suspension of microswimmers, the simplest realization of active matter, exhibits novel material properties: the emergence of collective motion, reduction in viscosity, increase in diffusivity, and extraction of useful energy. Bacterial dynamics in dilute suspensions suggest that hydrodynamic interactions and collisions between the swimmers lead to collective motion at higher concentrations. On the example of aerobic bacteria Bacillus subtilis, we report on spatial and temporal correlation functions measurements of collective state for various swimming speeds and concentrations. The experiments produced a puzzling result: while the energy injection rate is proportional to the swimming speed and concentration, the correlation length remains practically constant upon small speeds where random tumbling of bacteria dominates. It highlights two fundamental mechanisms: hydrodynamic interactions and collisions; for both of these mechanisms, the change of the swimming speed or concentration alters an overall time scale.

Figure 1

(a) Dependence of average swimming speed of bacteria at high concentration (collective regime) on the concentration of oxygen; blue heavy dots are experimental measurements, and dashed line is a guide for the eye. (b) Correlation length as a function of swimming speed for different speed change rates. The arrows indicate the direction of the change in swimming speed. Dashed line depicts size of PIV interrogation window.

Figure 2

Select images illustrating instant flow pattern at the bottom of the film for different concentrations of oxygen (or average swimming speeds); red arrows depict the direction and magnitude of bacterial flow, and colors show the vorticity magnitude of bacterial flow. Concentration of bacteria $c=30c_0$. The corresponding swimming speeds are indicated in Fig. 1b by letters a–d, see also Supplemental Material [29], movies 1 and 2.

Figure 3

(a) Velocity’s time autocorrelation functions at different average swimming speed values $V$. Time is scaled by mean free time $t_0=1/cV{l}^2$. Concentration of bacteria is $30c_0$ (b) Normalized correlation time $T$ (scaled by $t_0=1/cV{l}^2$) for different values of swimming speed. (c) Velocity’s time autocorrelation functions scaled by $t_0$ for different concentrations. Average swimming speed $\approx 39\mu \mathrm{m}/\mathrm{s}$. Collective behavior disappears at concentration $\approx 12c_0\sim {10}^9{\mathrm{cm}}^{-3}$ (d) Correlation time $T$ (scaled by $t_0=1/cV{l}^2$) and spatial correlation length $L$ (scaled by size of bacterium $l$) for different values of bacterial concentration. Open symbols mark concentration $9.5c_0$ where no collective motion is observed. Vertical optical coherence tomography (OCT) scans of the bottom part of the film for concentrations $c=15c_0$ (e) and $c=30c_0$ (f). Bioconvection on the panel (f) is manifested by bright plumes of increased bacterial concentration. Scale bar is $225\mu \mathrm{m}$.

Figure 4

Tumbling of Bacillus subtilis at low concentrations. (a,b): Tracks of individual bacteria for different oxygen concentrations; color represents average swimming speed for the last 1.2 s according to color bar on the right. At low oxygen concentration (a), the tracks resemble random walk. Increasing oxygen concentration (b) results in an increase of straight-swimming fraction. (c) Magnitude of angular velocity vs swimming speed averaged over all bacteria in the field of view with 0.5-s time interval; two oxygen cycles are shown. Red heavy dots depict experimental values, dashed blue line shows running average. The direction is shown by black arrows. (d) Probability distribution of rotational diffusion $D$ obtained from $2\times {10}^5$ independent tracks in the range of swimming speeds. Two dashed black lines indicate boundaries with the probability above 50% for each speed value. See also Supplemental Material [29], movies 3 and 4.