Fig. 2: Rectification.

a Schematics of an optimal choice for feedback parameters resulting in maximum rectification towards the right. The white background corresponds to minimum light intensity I₀ with associated cell velocity modulus v₀ and the green areas to maximum light intensity I₁ and v₁. A circular spotlight of radius R is shifted by a distance \({{\Delta }}={{{\Delta }}}^{* }=\sqrt{{R}^{2}-{({v}_{1}\tau )}^{2}}\) (Eq. (3)) to the right of the cell's position at time t = 0. In red the different possible positions at time t = τ are represented showing that all cells swimming at an angle ∣θ∣ < π/2 will move with maximum speed v₁. b Drift speed of bacteria versus the displacement Δ for different radii R and velocities v₁, v₀. In all cases τ = R/2v₁. The dashed lines are the corresponding theoretical optima according to Eq. (3). c Evolution of the polar speed plot as a function of Δ. When Δ > R + v₁τ cells will always be outside of their spotlight and swim with an isotropic speed v₀ (yellow). For Δ < R − v₀τ cells will always be inside and swim with isotropic speed v₁ (blue). For intermediate values of Δ cells will have an angle dependent speed (red) swimming faster when moving towards right. d Polar speed plot for different values of Δ when \(\hat{{{{{{{{\bf{n}}}}}}}}}=\hat{{{{{{{{\bf{x}}}}}}}}}\), R = 2.7 μm, τ = 0.1 s, v₁ = 10 μm s⁻¹ and v₀ = 3.5 μm s⁻¹ (corresponding to the dark green line in b). Dashed line is the ideal speed profile for maximum flow (Eq. (1)).