Chaotic Model for Lévy Walks in Swarming Bacteria

We describe a new mechanism for Lévy walks, explaining the recently observed superdiffusion of swarming bacteria. The model hinges on several key physical properties of bacteria, such as an elongated cell shape, self-propulsion, and a collectively generated regular vortexlike flow. In particular, chaos and Lévy walking are a consequence of group dynamics. The model explains how cells can fine-tune the geometric properties of their trajectories. Experiments confirm the spectrum of these patterns in fluorescently labeled swarming Bacillus subtilis.

Figure 1

Dynamics of individual bacteria within a dense swarm. (a) Optical microscopy image. (b) The collective flow of the swarm reveals a swirling dynamical flow pattern. (c) A low density of fluorescently labeled cells observed within the dense swarm. (d) Example trajectories obtained in experiments show superdiffusive behavior and fit a Weierstrassian Lévy walk model well. (e) Simulated trajectories show a Lévy walk pattern generated by chaotic dynamics.

Figure 2

Simulations with biologically realistic parameters. (a) A sample trajectory of a nonspherical particle (red) in a double-periodic flow (the black arrows). (b) The mean-squared displacement shows superdiffusive behavior with an exponent of 1.65. (c) The density of displacements $p(\mathrm{\Delta }t,\mathrm{\Delta }x)$, scaled by a factor of $\mathrm{\Delta }{t}^{1/\beta }$ ($\beta =1.22$), fits a Lévy stable distribution with exponent $3-\beta$ well. (d) The complement of the cumulative frequency distribution for the observed step lengths in simulated trajectories (∘), the best-fit WRW (green), and the best-fit Lévy walk model (red). The WRW fit implies a power-law exponent $\mu =1.45$.

Figure 3

Simulated trajectories show chaos. (a) Ensemble average power spectrum (the black line) and a best-fit stretched exponential (the red line). (b) A numerical approximation of the Lyapunov exponent indicates chaotic dynamics. (c),(d) Poincaré return maps showing successive recordings of a particle position $(x,y)$ mod 2 at (c) $\theta =0$ and (d) $y=0$.

Figure 4

Experimental results with swarming B. subtilis. (a) The complement of the cumulative frequency distribution for the observed step lengths in individual cells (°) and the best-fit WRW (Weierstrassian Lévy walk) (the green line), Lévy (power-law) walk model (the red line), and Brownian (exponential) walk (the blue line). (b) Ensemble average power spectrum (the black line) and the best-fit stretched exponential (the red line).

Figure 5

The dependence of the Lévy exponent $\mu$ on the aspect ratio $\lambda$ for different speeds $V$. For $\mu >3$, steps have a finite variance, implying normal diffusion. For a range of physically realistic speeds, the lowest Lévy exponent (the most superdiffusive) is obtained at an aspect ratio of around 5, similar to swarming Bacillus subtilis.