Swarming bacteria migrate by Lévy Walk

Individual swimming bacteria are known to bias their random trajectories in search of food and to optimize survival. The motion of bacteria within a swarm, wherein they migrate as a collective group over a solid surface, is fundamentally different as typical bacterial swarms show large-scale swirling and streaming motions involving millions to billions of cells. Here by tracking trajectories of fluorescently labelled individuals within such dense swarms, we find that the bacteria are performing super-diffusion, consistent with Lévy walks. Lévy walks are characterized by trajectories that have straight stretches for extended lengths whose variance is infinite. The evidence of super-diffusion consistent with Lévy walks in bacteria suggests that this strategy may have evolved considerably earlier than previously thought.

, plotted on a semi-log scale. A fit to a 2 nd order (quadratic) polynomial shows that the curve is convex, indicating that the fit to an exponential decay is poor.
Supplementary Figure 6 | The distribution of speeds within segments. The LW model assumes that the speed in each walking segment is constant. Variable speed is not expected to qualitatively change the predictions of the model as long as the distribution of speeds has a finite variance. To this end, we calculated the centralized, scaled (zero mean and unit variance) density of speeds in each trajectory. The figure shows the average density with the 20% longest trajectories, indicating a sharp cut-off at approximately twice the standard deviation. Figure 7 | Distribution of waiting times between turns analyzed using B. subtilis low magnification data. The slope of -2.55 is in agreement with the theory of LWs. Straight lines are least squares fits. Due to the reduced temporal and spatial resolution of the low magnification data (see Methods), rapid turds cannot be detected.

Supplementary Tables
Supplementary Table 1 Comparing WT B. subtilis cells and RFP labeled ones, with and without fluorescent illumination. Data is presented for both collective parameters (over 10 independent experiments), and individual ones (over 100 measurements). The table shows the mean±standard deviation. The microscopic mean speed is obtained using optical flow analysis of the swarm. WT

Supplementary Notes Supplementary Note 1: The velocity auto-correlation function for LWs.
Let v( ) t denote the velocity at time t of a particle following a LW. In the original LW model [1], particles move at constant speed between random reorientations. Without loss of generality, we assume unit speed. We assume that  has a finite average.
We define the velocity auto-correlation function as where · denotes averaging over all times t in independent samples of infinite trajectories.
Since following a reorientation event a particle completely losses its memory, we have that Using the total probability theorem, It is clear that . Assuming waiting times between reorientation events have a density ()  a power-law tail, See also [2].
In our experiments,

Supplementary Note 2: Comparing models of super diffusion -Lévy walk:
We simulated six different models showing super-diffusion: LW, fractional Brownian motion [3][4], generalized Langevin equations [3,[5][6], correlated and persistent random walks [3,[7][8][9][10][11][12], persistent random walks with variable persistence times [13] and Lévy flights [1,4]. In each simulation, 150 trajectories were sampled at a rate of 100 fps (similar to the high magnification data) and for 500 seconds (similar to the low magnification data). Parameters were chosen to match the MSD plot (Fig. 2), in particular the asymptotic slope (1.6) and the point of intersection with the y-axis (about -0.5 with the high magnification). Trajectories were analyzed using the same methods applied in generating Figs. 3 and 4. Thus, the experimental results can be compared with the different model predictions. All simulations were performed in Matlab.

The model:
The LW model is a Continuous Time Random Walk (CTRW) in which a particle moves with a constant speed v . At randomly drawn turning events 12 ,, the particle draws a new random orientation which is independent and uniformly distributed in [ ,]   . In the classical LW model [1], waiting times between turns, 1 ii    , are independent and identically distributed (IID) random variables (RV) with a density () t  that has a power-law tail, ( )tt    .
Simulation method: IID waiting times were drawn from a distribution with density Thus, a sequence of a particle's velocity and position at the turning events can be calculated exactly. Using linear interpolation (which is, in this case, exact), a discretized trajectory at uniform time intervals with 0.01  is the standard Wiener process) and 0 D  is a diffusion constant. Simulation method: See ref. [14].

Supplementary Note 4: Comparing models of super diffusion -Generalised Langevin equation:
The model: Generalised Langevin equation (GLE) is a non-Markovian generalization of the wellknown Langevin dynamics [6,15]. The generalization is obtained by introducing a singular power-law memory kernel. Formally, in 1D the GLE is given by a 2 nd order SDE,  is a memory kernel, ( , ) V x t is an external potential (zero in our case) and () t  is a stochastic driving force with zero mean and covariance The relation between the memory kernel and the covariance of noise is called a fluctuation dissipation theorem, which is a property of thermodynamic equilibrium. We note that the system of swarming bacteria is not in thermodynamic equilibrium due to the constant injection of energy by the bacterial self-propulsion mechanism.
If the spectrum of () t  , has a power law growth at low frequencies, ( )~r    , then trajectories have a MSD that grows asymptotically as 2/ (1 ) r t r D  . Hence, with 12 r , the dynamics is super-diffusive.
Simulation method: See ref. [15]. See Supplementary Figure 9a for example of sample trajectories.
Simulation parameters: 1.6 r  and 40 D  , chosen to fit Fig. 2. Results: See Supplementary Figure 9. The asymptotic slope of the MSD curve on a log-log plot is, as expected by the theoretical prediction. The optimal scaling of displacements was 1.12, which is lower than the experimental results. Since the GLE noise is a Gaussian process, displacements have a normal distribution, as observed in Supplementary Figure 9c The model: Correlated random walk (CRW) is a Markovian discrete random walk process in which increments are correlated. In 2D, t  is the simulation time step and the velocity at step norm v . The angles between consecutive increments, are IID with a given (even) distribution in [ ,]   [16][17].

Simulation method:
i   is drawn from the von-Misses distribution with zero mean and concentration parameter  . See Supplementary Figure 10a for example of sample trajectories.

Simulation parameters:
14 v  and 1/ t   , which implies a persistence time of about 1 sec, a typical run length for swimming bacteria. Figure 10. In agreement with theoretical predictions [17], the slope of the MSD curve on a log-log plot changes from 2 for short time scales (ballistic motion) to 1 on long scales (normal diffusion). The transition to a diffusive behavior occurs at around 2 t  secs. This is in contrast to the experimental observations showing anomalous diffusion up to 40 secs. The optimal scaling of displacements was 1.6, which is in agreement with the experimental results. However, the scaling of different time steps does not all fall nicely onto a single master curve. Since the CRW is a Gaussian process, displacements have a normal distribution, as observed in Supplementary Figure 10c. The velocity auto-correlation function (Supplementary Figure 10d) decays exponentially. In addition, the distribution of waiting times (Supplementary Figure 10e) is also exponential, unlike the power-law behavior in observed experiments.

Supplementary Note 6: Comparing models of super diffusion -Persistent random walk:
The model: Persistent random walk (PRW) is a CTRW in which a particle moves with a constant speed v . At randomly drawn turning events 12 ,, the particle draws a new random orientation which is independent and uniformly distributed in [ ,]   . However, unlike in the LW model, waiting times between turns, 1 ii    , are IID exponential RVs [18].
Simulation method: At every simulation step, a particle has probability P t  to make a random turn, where P is the persistence length and t  is the simulation time step. See Supplementary Figure 11a for example of sample trajectories.

Simulation parameters:
14 v  and 1 1sec P   , which implies a persistence time of 1 sec, a typical run length for swimming bacteria. Figure 11. Results are similar to CRW. In agreement with theoretical predictions [17], the slope of the MSD curve on a log-log plot changes from 2 for short time scales (ballistic motion) to 1 on long scales (normal diffusion). The transition to a diffusive behavior occurs at around 2 t  secs. This is in contrast to the experimental observations showing anomalous diffusion up to 40 secs. The optimal scaling of displacements was 1.3, which is slightly lower than the experimental results. Moreover, the scaling of different time steps does not all fall nicely onto a single master curve. Since the PRW is a Gaussian process, displacements have a normal distribution, as observed in Supplementary Figure 11c