Rectification and confinement of photokinetic bacteria in an optical feedback loop

Active particles can self-propel by exploiting locally available energy resources. When powered by light, these resources can be distributed with high resolution allowing spatio-temporal modulation of motility. Here we show that the random walks of light-driven bacteria are rectified when they swim in a structured light field that is obtained by a simple geometric transformation of a previous system snapshot. The obtained currents achieve an optimal value that we establish by general theoretical arguments. This optical feedback is used to gather and confine bacteria in high-density and high-activity regions that can be dynamically relocated and reconfigured. Moving away from the boundaries of these optically confined states, the density decays to zero in a few tens of micrometers, exhibiting steep exponential tails that suppress cell escape and ensure long-term stability. Our method is general and scalable, providing a versatile tool to produce localized and tunable active baths for microengineering applications and systematic studies of non-equilibrium phenomena in active systems.


Supplementary Movies
With the article there are three videoclips that supplement the figures in the main text.
• Supplementary Movie 1 Photokinetic bacteria showing directed transport towards the rightn =x. Dark-field microscope images of bacteria (red) is superimposed to the projected light pattern (green and black) with R = 3.8 µm ∆ = 5.1 µm and τ = 0.2s The black background corresponds to minimum light intensity I 0 with associated cell velocity modulus v 0 and the green spots to maximum light intensity I 1 and v 1 . White traces represent bacterial trajectories and are obtained by superimposing images acquired in the previous 2 s. The corresponding process is illustrated in Fig. 1 of the article. Playback speed is 4x.
• Supplementary Movie 2 Confinement of bacteria in circles of different radii R c . Feedback loop parameters are ∆ = 1.8 µm, R = 2µm, τ = 0.1s, v 1 = 10µm s −1 and v 0 = 5µm s −1 . In the end, the feedback is turned off. The corresponding process is illustrated in Fig. 4a of the article. Playback speed is 10x and 40x, as indicated in the video.

• Supplementary Movie 3 Confinement of bacteria by an optical feedback loop with parameters
In the end, the feedback is turned off. The corresponding process is illustrated in Fig. 4b of the article. Playback speed is 100x and 10x, as indicated in the video.
• Supplementary Movie 4 Splitting and merging of optically confined clouds of motile bacteria. We split an optically confined region of highly motile bacteria into two separate clouds and then merge them together. The corresponding process is illustrated in Fig. 5 of the article. Playback speed is 40x.

Supplementary Note 1: RnT model with feedback
To check in detail the theoretical predictions we simulate the "Run and Tumble" (RnT) model with direction-dependent speed. In these simulations RnT particles move in a squared box extending between ±L/2 both along x and y (periodic boundary conditions apply). In the case where the feedback is applied only along the x-axis, particles at |x| > a move at speed v 0 when pointing away from x = 0 (while moving at speed v 1 > v 0 when pointing towards x = 0). If |x| < a particles always move at maximum speed v 1 . We use the experimental values for the speeds v 0 = 5 µm s −1 , v 1 = 10 µm s −1 , and for the tumbling rate λ = 0.75 Hz. Since these parameters correspond to a decay length ℓ ≈ 20 µm we set the simulation box size to L = 750 µm (≫ ℓ).

FIG. 2. The angular probability density χ(θ) obtained in simulations is represented by the full thick line and this is well described by the theoretical prediction
After reaching the stationary state the density profile shows a clear exponential form for |x| > a as shown Fig. 3 of the main text (i.e. ρ(x) ∝ e −k|x−a| ) as predicted by the theory. Here we show (see Fig. 2) that also the probability density χ(θ) of the particles orientation θ (full line) agrees with theory. To do this computed only for particles that are at least one persistence length away from the border (i.e. for |x| > a + v 1 /λ). We recall that the angular theoretical distribution is given by: which is plotted as a dashed curve in In Fig. 2 (where v(θ) = v 0 if |θ| < π/2 and v(θ) = v 1 otherwise). As described in the main text the value of k is obtained by imposing normalization of the angular distribution: where we introduced the variable h = k/λ. After some simple manipulation we get: which can be rewritten as: Now we numerically find that for typical parameters value the denominator is a slowly varying function of θ and a very good estimate for h can be found by simply imposing that the integral of the numerator vanishes: