Effect of motility on the transport of bacteria populations through a porous medium

The role of activity on the hydrodynamic dispersion of bacteria in a model porous medium is studied by tracking thousands of bacteria in a microfluidic chip containing randomly placed pillars. We first evaluate the spreading dynamics of two populations of motile and nonmotile bacteria injected at different flow rates. In both cases, we observe that the mean and the variance of the distances covered by the bacteria vary linearly with time and flow velocity, a result qualitatively consistent with the standard geometric dispersion picture. However, quantitatively, the motile bacteria display a systematic retardation effect when compared to the nonmotile ones. Furthermore, the shape of the traveled distance distribution in the flow direction differs significantly for both the motile and the nonmotile strains, hence probing a markedly different exploration process. For the nonmotile bacteria, the distribution is Gaussian, whereas for the motile ones, the distribution displays a positive skewness and spreads exponentially downstream akin to a $\mathrm{\Gamma }$ distribution. The detailed microscopic study of the trajectories reveals two salient effects characterizing the exploration process of motile bacteria: (1) the emergence of an “active” retention effect due to an extended exploration of the pore surfaces and (2) an enhanced spreading at the forefront due to the transport of bacteria along “fast tracks” where they acquire a velocity larger than the local flow velocity. We finally discuss the practical applications of these effects on the large-scale macroscopic transfer and contamination processes caused by microbes in natural environments.

Figure 1

(a) Schematic view of the overall microfluidic chip. (b) Colored lines representing trajectories of motile bacteria in the porous space. (c, d) Superimpositions of the velocity field obtained with passive tracers and E. coli trajectories (black solid lines) for nonmotile (c) and motile (d) bacteria during $\mathrm{\Delta }t$ = 10 s at a flow rate $U$ = 36 $\mu \text{m}/\text{s}$. White circles represent the pillars (average diameter $d$ = 35 $\mu \mathrm{m}$). Images are recorded with a $10\times$ objective. The field of view is 1 mm $\times$ 0.5 mm.

Figure 2

(a, c) Distributions $P(\mathrm{\Delta }x,\mathrm{\Delta }t)$ of the distances $\mathrm{\Delta }x$, traveled along the flow direction at a time $\mathrm{\Delta }t=3$ s, for motile (a) and nonmotile (c) bacteria at different flow velocities. (b, d) Normalized distributions $P^{*}\left(\xi \right)$ where $\xi =\frac{\mathrm{\Delta }x-\overline{\mathrm{\Delta }x}}{{\sigma }_x}$ for the motile (b) and the nonmotile (d) bacteria. For each flow velocity, five different $\mathrm{\Delta }t$ are represented ($\mathrm{\Delta }t=$ 1, 3, 5, 7, and 9 s). Solid red line: (b) a Gaussian distribution of zero mean and unit variance and (d) a best fit from a $\mathrm{\Gamma }$ distribution.

Figure 3

(a) Mean positions of the bacterial population $\overline{\mathrm{\Delta }x}$, as a function of the mean distance traveled by the fluid $U\mathrm{\Delta }t$, for both motile (M) and nonmotile (NM) bacteria and for different flow rates. (b) Mean transport velocity of the motile bacteria $U_M$ as a function of the mean flow velocity $U$. The dotted line represents the $y=x$ function.

Figure 4

(a, b) Variation of the dimensionless quadratic displacement $\frac{{\sigma }_x^2}{d^2}$ as a function of the normalized time $t^{*}=\frac{U\mathrm{\Delta }t}{d}$ for different average flow velocities respectively for motile (a) and nonmotile (b) E. coli. Solid lines: adjustments by the function $f\left(t^{*}\right)=\alpha [\beta t^{*}-(1-e^{-\beta t^{*}})]$. Insets represent the dimensionless quadratic displacement as a function of $t^{*2}$ for less than one pore size.

Figure 5

(a) Longitudinal dispersion coefficient $D_L$ as a function of $U$ for motile and nonmotile bacteria. Inset: Dispersivity ${\zeta }_L^{NM}=\frac{D_L^{NM}}{U}$ and ${\zeta }_L^M=\frac{D_L^M-D_0^M}{U}$ as a function of $U$ for nonmotile and motile bacteria, respectively. The dotted line represents the constant function $y=$ 50 $\mu \mathrm{m}$. (b) Transverse dispersion coefficient $D_T$ of motile bacteria. The solid line represents a linear regression of the data yielding ${\zeta }_T^M=1.5\pm 0.1$ $\mu \mathrm{m}$.

Figure 6

Schematics of a bacterium trajectory and the corresponding parameters as defined in the text. ${\tau }_{\mathrm{up}}$ is the period of time spent with a negative $v_x$, ${\tau }_g$ is the period of time spent on the grain vicinity, i.e., when the distance bacteria/obstacle is smaller than $\delta$ = 1.5 $\mu \mathrm{m}$, and ${\tau }_f$ is the period of time spent in the fluid when the distance bacteria/obstacle is larger than $\delta$ = 1.5 $\mu \mathrm{m}$.

Figure 7

(a) Average time spent by the bacteria swimming in the vicinity of an obstacle ${\tau }_g$ normalized by the total time ${\tau }_g+{\tau }_f$, where ${\tau }_f$ is the average time spent in the fluid as a function of $U$. (b) Mean number of occurrences a bacterium gets in contact with an obstacle per unit of time $\tau =d/U$.

Figure 8

Average duration of the upstream displacement ${\tau }_{\mathrm{up}}$ normalized by the total time ${\tau }_{\mathrm{up}}+{\tau }_{\mathrm{down}}$, where ${\tau }_{\mathrm{down}}$ is the average duration of the downstream ($v_x>0$) displacement as a function of $U$.

Figure 9

Overlap of two close trajectories of a motile (orange circles) and a nonmotile (blue squares) bacteria with the local fluid velocity. The inset shows the speed of the bacteria $v_x$ as a function of the longitudinal position $x$. Experiments are performed at $U=$ 72 $\mu \text{m}/\text{s}$.

Figure 10

(a) Speed of the fastest detected bacteria as a function of the mean flow velocity $U$. (b) Average duration, ${\tau }_{\mathrm{fast}}$, of the sequences during which a bacteria is moving with a speed $v_x$ larger than twice the average flow velocity, normalized by ${\tau }_{\mathrm{fast}}+{\tau }_{\mathrm{slow}}$, where ${\tau }_{\mathrm{slow}}$ is the averaged duration of the sequences during which $v_x<2U$.