Motility of Escherichia coli in a quasi-two-dimensional porous medium

Bacterial migration through confined spaces is critical for several phenomena, such as biofilm formation, bacterial transport in soils, and bacterial therapy against cancer. In the present work, E. coli (strain K12-MG1655 WT) motility was characterized by recording and analyzing individual bacterium trajectories in a simulated quasi-two-dimensional porous medium. The porous medium was simulated by enclosing, between slide and cover slip, a bacterial-culture sample mixed with uniform 2.98-$\mu \text{m}$-diameter spherical latex particles. The porosity of the medium was controlled by changing the latex particle concentration. By statistically analyzing several trajectory parameters (instantaneous velocity, turn angle, mean squared displacement, etc.), and contrasting with the results of a random-walk model developed ad hoc, we were able to quantify the effects that different obstacle concentrations have upon bacterial motility.

Figure 1

(a) Typical E. coli trajectory at very low obstacle area fraction ($\varphi =0.01$). Run and tumble starting points are respectively indicated with green and red marks. (b) Speed trace for the trajectory in (a). (c) Experimental residence-time PDFs (dots) and best fitting exponential distributions (solid lines) for runs (green) and tumbles (red). (d) Experimental longitudinal velocity PDFs for runs (green dots) and tumbles (red dots), and the corresponding best fitting Gaussian distributions (green and red solid lines). (e) Same as in (d), but for transverse velocity.

Figure 2

Statistical characteristics of trajectories recorded at very low obstacle concentrations ($\varphi =0.01$)—symbols—and of simulated trajectories under the same conditions—lines. (a) Probability density function (PDF) for the velocity component along the $x$ axis. (b) PDF for the velocity component along the $y$ axis. (c) Speed probability density function. (d) PDF for the turn angle between consecutive steps. (e) Mean squared displacement (MSD) from all bacterial trajectories.

Figure 3

Experimental bacterial trajectories under low and high obstacle-area-fraction conditions: (a) $\varphi =0.01$ and (b) $\varphi =0.39$. Obstacles are represented as white circles.

Figure 4

Probability density functions for the bacterium horizontal ($v_x$) and vertical ($v_y$) velocity components, calculated from experimental (symbols) and simulated (solid lines) trajectories, at different obstacle area fractions ($\varphi$): blue, $\varphi =0.01$; cyan, $\varphi =0.1$; green, $\varphi =0.2$; light brown, $\varphi =0.29$; red, $\varphi =0.39$.

Figure 5

(a) Speed probability density functions (PDFs) computed from experimental (symbols) and simulated (solid lines) trajectories at various obstacle area fractions ($\varphi$): blue, $\varphi =0.01$; cyan, $\varphi =0.1$; green, $\varphi =0.2$; light brown, $\varphi =0.29$; red, $\varphi =0.39$. (b) Turn-angle PDFs computed from experimental (symbols) and simulated (solid lines) trajectories at various obstacle area fractions. The color code is the same as in (a).

Figure 6

Mean speed (averaged over time and over all trajectories in every squared micron) maps computed from experimental—(a) and (b)—and simulated—(c) and (d)—trajectories at different obstacle concentrations (same experiments as those in Fig. 1). The color code is given in the adjacent bar. Spherical beads are represented as white circles.

Figure 7

Plots of mean squared displacement (MSD) vs time, computed from experimental (symbols) and simulated (solid lines) trajectories, at different obstacle area fractions ($\varphi$).

Figure 8

Plots of mean bacterial speed (a), duration of the MSD superdiffusive phase (b), standard deviation across the mean speed maps (c), and extrapolation of the MSD value at $t=0$ (d) vs the obstacle area fraction ($\varphi$) for different values of the run-to-tumble transition probability after a bacterium-obstacle collision ($P_{R-T}$) and for an effective obstacle radius $r_{\mathrm{eff}}=2\mu \text{m}$.

Figure 9

Plots of mean bacterial speed (a), duration of the MSD superdiffusive phase (b), standard deviation across the mean speed maps (c), and extrapolation of the MSD value at $t=0$ (d) vs the obstacle area fraction ($\varphi$) for different values of the obstacle effective radius ($r_{\mathrm{eff}}$) and for a run-to-tumble transition probability after a bacterium-obstacle collision $P_{R-T}=0.4$.

Figure 10

Plots of mean bacterial speed (a), duration of the MSD superdiffusive phase (b), standard deviation across the mean speed maps (c), and extrapolation of the MSD value at $t=0$ (d) vs the obstacle area fraction ($\varphi$) for two different types of bacterium-obstacle collisions (for $P_{R-T}=0.4$ and $r_{\mathrm{eff}}=2\mu \text{m}$): “arch” and tangential deflections.