Enhanced mixing and spatial instability in concentrated bacterial suspensions

High-resolution optical coherence tomography is used to study the onset of a large-scale convective motion in free-standing thin films of adjustable thickness containing suspensions of swimming aerobic bacteria. Clear evidence is found that beyond a threshold film thickness there exists a transition from quasi-two-dimensional collective swimming to three-dimensional turbulent behavior. The latter state, qualitatively different from bioconvection in dilute bacterial suspensions, is characterized by enhanced diffusivities of oxygen and bacteria. These results emphasize the impact of self-organized bacterial locomotion on the onset of three-dimensional dynamics, and suggest key ingredients necessary to extend standard models of bioconvection to incorporate effects of large-scale collective motion.

Figure 1

(Color online) Schematic of the experimental apparatus. A thin liquid film containing bacteria spans two adjustable Pt wires and two dielectric fibers and is stretched by a movable platform controlled by a stepper motor. Bright-field images of the bottom surface of the film are obtained by a long working distance microscope objective mounted under the film and recorded by a high-speed charge-coupled device (CCD) camera (Spot Boost EMCCD 2100, Diagnostic Instruments Inc.). To obtain the local concentration of bacteria in the bulk of the film, we used a high-resolution, noninvasive, and optical coherence tomography to measure the profile of the optical scattering (which is approximately proportional to the bacterial concentration). From this, cross-sectional images were created by scanning the beam position laterally over the film or by horizontal displacement of the microscope platform with respect to OCT probe.

Figure 2

(Color online) (a) Pseudocolor representation of formation of spots in a type-I experiment during the migration of cells from the electrode, seen as black strip on the bottom (scale bar $200\mu \text{m}$). Color bar on the right indicates values of mean concentration $\overline{n}$ in $\text{cells}/{\text{cm}}^3$. See also movies 1 and 2 [20]. (b) and (c) Formation of spots after the thickness of the film was suddenly increased, using type-II technique. Initially uniform concentration of bacteria (2 s elapsed) (b); final stage after 60 s (c).

Figure 3

(Color online) Average spot size vs film thickness $d$ with corresponding error bars. Inset: temporal autocorrelation function of the bacterial concentration averaged over vertical coordinate, for thicknesses of $50\mu \text{m}$ (red ●) and $400\mu \text{m}$ (black $\times$).

Figure 4

(Color online) OCT images of bacterial concentration distribution inside film. (a) Formation of falling plumes. (b) Layer of immotile bacteria slowly decays (scale bar $100\mu \text{m}$); see also supplementary movies 3 and 4 [20]. (c) Concentration profile $n\left(z\right)$ vs vertical coordinate $z$ for different film thicknesses averaged over film area and time. $z=0$ corresponds to the top surface of the film and $z=d$ to the bottom surface. Error bars are on the order of $5–10\%$; instrumental errors are significantly smaller. Dashed line is theoretical prediction (4). (d) Three-dimensional scan of the bacterial concentration $n$ obtained by OCT for $d\approx 500\mu \text{m}$. The isosurface shown is for $n=\overline{n}/3$, where $\overline{n}$ is average concentration, and the colors indicate concentration distribution (red corresponds to maximum; blue to $n=\overline{n}/3$). See also movie 5 in [20].

Figure 5

(Color online) Bacterial concentration. (a) Vertical profiles of bacteria concentration average over 600 vertical scans (red line with circles) and in the depletion layer only (black line with diamonds) for $\overline{n}\approx 1.8\times {10}^{10}{\text{cm}}^{-3}$. Correspondingly, $\lambda \approx 200\mu \text{m}$ in the film and $\lambda \approx 50\mu \text{m}$ in the depletion layer; dashed lines are fits to Eq. (4). Inset: relative standard deviation of concentration $\delta n/n$ vs $z$. Inset also provides error bars for $n\left(z\right)$ profiles in (a). (b) Probability distribution $P\left(\xi \right)$ for the width of depletion layer $\xi$ for $d\approx 325\mu \text{m}$ film (circles) and $d\approx 475\mu \text{m}$ (diamonds). Lines show fits to the Gaussian law $P\left(\xi \right)\sim \text{exp}\left[-{\left(\xi -⟨\xi ⟩\right)}^2/{\sigma }^2\right]$. Inset: depletion layer thickness $\xi$ vs $d$ obtained from theoretical model. Symbols depict corresponding experimental results.

Figure 6

(Color online) Plumes. (a) Vertical concentration profiles in the plume (solid black line), in the vicinity of the plume (dashed green line), and near the potential location of plume formation (point-dashed blue line). (b) Grayscale representation of 2D probability distribution $P\left(n,z\right)$ for the concentration of bacteria $n$ and coordinate inside the film $z$, where black corresponds to higher probability. 25 sample points for concentration $\times 25$ points for thickness used, then smoothed. Average concentration $\overline{n}$ is $2.2\times {10}^{10}{\text{cm}}^{-3}$. Solid line depicts the mean concentration profile $⟨n\left(z\right)⟩$.