Rectification of Swimming Bacteria and Self-Driven Particle Systems by Arrays of Asymmetric Barriers

We show that the recent experimental observation of the rectification of swimming bacteria in a system with an array of asymmetric barriers occurs due to the ballistic component of the bacteria trajectories introduced by the bacterial “motor.” Each bacterium selects a random direction for motion and then moves in this direction for a fixed period of time before randomly changing its orientation and moving in a new direction. In the limit where the bacteria undergo only Brownian motion on the size scale of the barriers, rectification does not occur. We examine the effects of steric interactions between the bacteria and observe a clogging effect upon increasing the bacteria density.

Figure 1

Images of the simulation system with $\theta =60°$. Black dots: bacteria positions; heavy black lines: barrier locations. (a) Initial system configuration with uniform bacteria density for $N_b=980$ and $l_b=40$. The density in the upper chamber is denoted ${\rho }^{(1)}$, and the density in the lower chamber is ${\rho }^{(2)}$. (b) The same system after $100\tau$ simulation steps showing the higher bacteria density in the upper chamber ${\rho }^{(1)}>{\rho }^{(2)}$.

Figure 2

Dots: Position of one bacterium in the system. Lines: Trajectory of the bacterium over a fixed time period. (a) A system with only a motor force showing ballistic motion and a random reorientation each time the bacterium has moved a distance $l_b=10$. Here $F^T=0$. (b) A system with small $l_b=1$ and $F^T=0$. (c) A system with both ballistic motion and thermal motion, where $l_b=10$ and $F^T=10$.

Figure 3

(a) The time dependence of the bacteria density ${\rho }^{(1)}/{\rho }_b$ (upper line) and ${\rho }^{(2)}/{\rho }_b$ (lower line), normalized by the overall bacteria density ${\rho }_b$, for the same system in Fig. 1 with $l_b=180$, showing the density buildup over time in the top half of the sample. Dotted lines: ${\rho }^{(1)}/{\rho }_b$ and ${\rho }^{(2)}/{\rho }_b$ for the same system but with $l_b=0.01$ so that the bacteria undergo Brownian motion on the size scale of the barrier and no rectification occurs. (b) $r={\rho }^{(1)}/{\rho }^{(2)}$ vs $l_b$ for a sample with $\theta =60°$ measured after ${10}^3\tau$ simulation steps.

Figure 4

(a) Schematic trajectories for (dark line) a bacterium moving under motor forces only and being entrained by the wall and (light line) a bacterium experiencing only Brownian forces. The barrier spacing $l_S$, opening size $l_o$, funnel width $l_w$, and angle $\varphi$ between the bacterium trajectory and the funnel barrier wall are also indicated. (b) $r={\rho }^{(1)}/{\rho }^{(2)}$ vs $\theta$ for a system with $l_b=20$, $F^T=0$, $f_s=0$, and $L=99$. Inset: $r$ vs $\theta$ for the same system with fixed $l_o=3.25$ and varied $L$. (c) $r$ vs $F^T$ for a system with $l_b=20$, $\theta =60°$, $L=99$, and $f_s=0$. Inset: $r$ vs $L_B$ for a system with $l_b=20$, $F^T=0$, $f_s=0$, fixed $l_o=3.25$, and varied $L$. (d) $r$ vs $l_o$ for a system with $l_b=20$, $\theta =60°$, $L_B=5.0$, $f_s=0$, and varied $L$. Inset: $r$ vs overall bacteria density ${\rho }_b$ for a system with steric interactions, $f_s=150$, $\theta =60°$, $f^T=0$, $L=99$, and $l_b=20$.