Scale-Invariant Correlations in Dynamic Bacterial Clusters

In Bacillus subtilis colonies, motile bacteria move collectively, spontaneously forming dynamic clusters. These bacterial clusters share similarities with other systems exhibiting polarized collective motion, such as bird flocks or fish schools. Here we study experimentally how velocity and orientation fluctuations within clusters are spatially correlated. For a range of cell density and cluster size, the correlation length is shown to be 30% of the spatial size of clusters, and the correlation functions collapse onto a master curve after rescaling the separation with correlation length. Our results demonstrate that correlations of velocity and orientation fluctuations are scale invariant in dynamic bacterial clusters.

Figure 1

(a) Instantaneous velocity vectors of bacteria in four clusters, overlaid on an experimental image where the white rods are individual bacteria; the insert shows schematically the position ${\overrightarrow{r}}_{i,I}$ and orientation, ${\overrightarrow{\theta }}_{i,I}$ of the $i$th bacterium in the $I$th cluster. (b) Fluctuations from the mean motion direction, mean orientation, and mean velocity are shown by arrows for the red, green, and blue clusters. For the cyan cluster in (a), positive and negative speed fluctuations from the mean are shown by red and blue ellipses, respectively. Each cluster has regions of correlated fluctuations with opposite signs (cf. left and right halves of the green cluster). The blue, green, red, and cyan clusters in (a) contain 106, 57, 98, and 48 bacteria, respectively, with corresponding polarizations 0.98, 0.97, 0.91, and 0.94, and cluster sizes 43.2, 33.7, 40.2, and $25.7\mu \mathrm{m}$. The temporally averaged number of bacteria in the imaging window ($1000\times 1000\mathrm{\text{pixels}}$) is $N_{\mathrm{total}}=720$. The temporal evolution of these clusters is shown in the Supplemental Material [27].

Figure 2

(a) Cluster size as a function of the number of bacteria in a cluster. The dashed line is a fit to the data, $L=2.89n^{0.6}$, where $L$ is averaged in bins of width $\Delta n=6$ and the error bars represent the standard deviation in each bin. (b) Number of clusters as a function of their size $L$, for different average numbers of bacteria in a fixed field of view. Neighboring bacteria are identified as belonging to the same cluster if they are closer together than $R=3.6\mu \mathrm{m}$ and move in the same direction within $\alpha =20°$, except for the blue triangles where $R=4.1\mu \mathrm{m}$ and $\alpha =15°$. The results for $N_{\mathrm{total}}=720$ for two sets of parameters $R$ and $\alpha$ differ only slightly.

Figure 3

Correlation functions of motion direction, (a) and (b); velocity, (c) and (d); speed, (e) and (f); and orientation, (g) and (h). The correlation functions are plotted as a function of $r$ on the left and as a function of $r/\xi$ on the right, where the correlation length $\xi$ was determined from the zero crossings of the left-hand correlation functions, for three cluster sizes ($L=18.5\mu \mathrm{m}$, $36.5\mu \mathrm{m}$, and $54.5\mu \mathrm{m}$, from left to right in each left-hand panel). The dashed lines in the right-hand plots are stretched-exponential fits, described in the Supplemental Material [27]. The data correspond to bacterial densities $N_{\mathrm{total}}=343$, 623, and 720, and cluster sizes $L$ from $12.5\mu \mathrm{m}$ to $50.8\mu \mathrm{m}$.

Figure 4

Motion direction (${\xi }^d$, ▵), velocity (${\xi }^u$, □), speed (${\xi }^s$, ○), and orientation (${\xi }^{\varphi }$, ◊) correlation length are plotted as functions of cluster spatial size, $L$. Results from five densities are shown: $N_{\mathrm{total}}=343$ (black), $N_{\mathrm{total}}=480$ (magenta), $N_{\mathrm{total}}=623$ (red), $N_{\mathrm{total}}=674$ (yellow) and $N_{\mathrm{total}}=720$ (blue). All quantities are linearly proportional to $L$, as demonstrated by green dashed lines: $\xi =0.3L$. Correlation lengths depend weakly on the parameters ($R$ and $\alpha$) used to identify clusters as shown in the Supplemental Material [27].