Collective motion and density fluctuations in bacterial colonies

Abstract

Flocking birds, fish schools, and insect swarms are familiar examples of collective motion that plays a role in a range of problems, such as spreading of diseases. Models have provided a qualitative understanding of the collective motion, but progress has been hindered by the lack of detailed experimental data. Here we report simultaneous measurements of the positions, velocities, and orientations as a function of time for up to a thousand wild-type Bacillus subtilis bacteria in a colony. The bacteria spontaneously form closely packed dynamic clusters within which they move cooperatively. The number of bacteria in a cluster exhibits a power-law distribution truncated by an exponential tail. The probability of finding clusters with large numbers of bacteria grows markedly as the bacterial density increases. The number of bacteria per unit area exhibits fluctuations far larger than those for populations in thermal equilibrium. Such "giant number fluctuations" have been found in models and in experiments on inert systems but not observed previously in a biological system. Our results demonstrate that bacteria are an excellent system to study the general phenomenon of collective motion.

Fig. 1.

Instantaneous configurations at two densities with the average total number of bacteria in the imaging window N_total = 343 (A) and N_total = 718 (B). Velocity vectors are overlayed on the raw images of bacteria. The length of the arrows corresponds to bacterial speed, and nearby bacteria with arrows of the same color belong to the same dynamic cluster. (A Inset) A laboratory coordinate frame () and a local frame () are defined. The cell body and flagella of the ith bacterium are shown by solid and dashed lines, respectively (the flagella are not visible in our images); the local frame is centered at the center of mass , with and pointing at the transverse and longitudinal axes respectively. See Movies S1 and S2 for temporal evolution of the system.

Fig. 2.

Two-dimensional pair correlation (A and D), orientational correlation (B and E), and velocity correlation (C and F) functions at two bacterial densities: N_total = 343 (A–C) and N_total = 718 (D–F). The dashed gray lines in A–F near the origin mark the border of the excluded volume. Transverse profiles, g(x,y = 0), are shown in G–I, where black and red lines correspond to N_total = 343 and N_total = 718, respectively.

Fig. 3.

Statistical properties of dynamic clusters. (A) Cluster-size-dependent pair correlation g(x,y,n), (B) orientational correlation C_θ(x,y,n), and (C) velocity correlation C_v(x,y,n) functions. In A, the black line is the contour for g(x,y,n) = 0.1, which intersects the x and y axes at (λ_x,0) and (0,λ_y). Longitudinal λ_y and transverse λ_x lengthscales are plotted in D as functions of the cluster size n. The black dashed lines in D are fits of . In E, values of velocity and orientational correlation at (0,λ_y) are plotted against n. In D and E, data in blue and red are from N_total = 718 and N_total = 513, respectively. More cluster-size-dependent correlation functions computed at various conditions can be found in Figs. S2 and S3B.

Fig. 4.

Size distribution of bacterial clusters for three bacterial densities: N_total = 343 (squares), N_total = 539 (circles), and N_total = 718 (triangles). Solid lines are fits to Eq. 1. (Inset) All data collapse onto a master curve by rescaling and are plotted in a log-log frame with a solid line showing the rescaled fit.

Fig. 5.

Anomalous density fluctuations in collectively moving bacteria. (A) Total number of bacteria in the field of view as a function of time. Two snapshots, corresponding to minimal and maximal instantaneous bacteria density, are shown as insets. (B) The magnitude of the density fluctuations (quantified by the ratio of ΔN to ) against the mean bacterial number N, for interrogation areas of various sizes. Results from three conditions are shown: N_total = 343 (squares), N_total = 539 (circles), and N_total = 718 (triangles). The solid line in B has a slope of 0.25. To obtain the data in B, we define a series of interrogation areas centered at the imaging window with increasing sizes from A_i = 5.4 × 5.4 μm² to 90 × 90 μm². We then construct a temporal record of the number of bacteria in each interrogation area A_i (similar to the one in A). From these temporal records, we compute the standard deviation ΔN(A_i) and the mean N(A_i) for each A_i.