Effect of Cell Aspect Ratio on Swarming Bacteria

Swarming bacteria collectively migrate on surfaces using flagella, forming dynamic whirls and jets that consist of millions of individuals. Because some swarming bacteria elongate prior to actual motion, cell aspect ratio may play a significant role in the collective dynamics. Extensive research on self-propelled rodlike particles confirms that elongation promotes alignment, strongly affecting the dynamics. Here, we study experimentally the collective dynamics of variants of swarming Bacillus subtilis that differ in length. We show that the swarming statistics depends on the aspect ratio in a critical, fundamental fashion not predicted by theory. The fastest motion was obtained for the wild-type and variants that are similar in length. However, shorter and longer cells exhibit anomalous, non-Gaussian statistics and nonexponential decay of the autocorrelation function, indicating lower collective motility. These results suggest that the robust mechanisms to maintain aspect ratios may be important for efficient swarming motility. Wild-type cells are optimal in this sense.

Figure 1

Microscopic observations of swarming. (a) Phase contrast imaging (60X) of long Bacillus subtilis with a 8.0 aspect ratio (strain DS858) during swarming. (b) The instantaneous velocity (black arrows) and vorticity (red, clockwise; blue, counterclockwise) fields.

Figure 2

Optical flow analyses of the swarm edge. (a) The mean speed and mean absolute vorticity as a function of cell aspect ratio. The nonmonotonic dependence, with a maximum at aspect $\text{ratio}=4.9$ obtained for the WT (dashed line) shows the significance of the aspect ratio for effective swarming and the advantage of the $5:1$ value. (b) The scaled fourth moment (kurtosis) of the velocity and vorticity fields as a function of the aspect ratio. The minimum value of 3, indicating a Gaussian distribution, is obtained at aspect $\text{ratio}=4.9$ (WT, dashed line). (c) The exponent of the velocity field corresponding to a stretched exponential function of the form $z^{-1}\exp [-(v/v_0{)}^{\gamma }]$, with $\gamma =2$ for the WT (Gaussian) and lower values for shorter and longer strains. (d) The standard deviation of the velocity as a function of mean speed for the different strains is approximately linear, despite the different types of distribution. Numbers show the aspect ratio.

Figure 3

Temporal autocorrelations of the microscopic flow. (a) Example of temporal correlation functions for the velocity and vorticity obtained with a long bacterial strain (DS858, aspect ratio 8.0). (b) The amplitude $A$ obtained by fitting the temporal correlation function to a double exponential function $A{e}^{-t/{\tau }_1}+(1-A)e^{-t/{\tau }_2}$. (c) The stretching exponent $\beta$ obtained by fitting the temporal correlation function to a stretched exponential, $C(t)=\exp (-\alpha t^{\beta })$. (d) The observed decay time [at $C(t)=1/e$] as a function of aspect ratio. The left (right) $y$ axis shows the value for the correlation in velocity (vorticity).

Figure 4

Spatial correlation of the microscopic flow. (a) Example of spatial correlation functions for the velocity and vorticity obtained with a long bacterial strain (DS858, aspect ratio 8.0). (b) Semilog scale of the data in (a) suggesting exponential decay. All spatial correlation functions are approximately exponential. (c) The correlation length as a function of aspect ratio. (d) The location of the minimum of the graphs in (a).