Active bacterial pattern formation in evaporating droplets

Abstract

Bacteria living on surfaces are often confined to droplets. When these droplets evaporate, the motion of the liquid-air interface and the associated internal capillary flow confine the bacteria. Here we study how \emph{E. coli} bacteria interact with this capillary confinement and agglomerate at the droplet's contact line. We identify three different types of bacterial pattern formation that depend on the bacterial activity and the environmental conditions imposed by the evaporating droplet. When the evaporation is fast, the bacteria are slow or the suspension is dilute, a uniform contact-line deposit forms. However, when the capillary confinement concentrates the bacteria at the contact line beyond a critical number density, localized collective motion spontaneously emerges. In that case, the bacteria induce a local stirring of the liquid that allows them to self-organize into periodic patterns and enables them to collectively escape from the contact line. At very high number densities, these periodic patterns get destabilized by bacterial turbulence in the bulk of the droplet resulting in the formation of mobile bacterial plumes at the contact line. Our results show how the subtle interplay between the bacteria and the capillary flow inside the droplet that surrounds them governs their dispersal.

Figures (3)

• Figure 1: Pattern formation at the contact line of evaporating droplets containing motile E. coli bacteria. (A) Top row: Side-view recordings the droplet's shape evolution. Bottom row: Bottom-view fluorescent microscopy images show the bacteria form radially inward pointing sweeping fingers that move along the contact line, split and merge. (B) Fluid velocity field inside the droplet measured with 1.6-$\mu$m colloidal tracer particles by Particle Image Velocimetry (PIV) and time-averaged over 4 s. The corresponding radial velocity profile $u$ is shown in the bottom panel. Points are the mean and the error bar is the standard deviation over the azimuthal coordinate. (C-E) Snapshots of the three deposit types observed: (C) for an initial bacterial number density $n_i= 8.3\times 10^8$ cells/mL, swimming speed $U_b= 17~\mu$m/s and humidity $H=0.85$ a uniform deposit forms. (D) For $n_i= 1.6\times 10^{9}$ cells/mL, $U_b=17~\mu$m/s and $H=0.53$ regularly spaced fingers form at the contact line. The azimuthal coordinate $\phi$ is indicated. (E) For $n_i=1.6\times 10^{10}$ cells/mL, $U_b=17~\mu$m/s and $H=0.53$sweeping fingers form. (F) Top: zoom of the finger pattern with the interpolated and smoothed finger edge $d(\phi$) overlaid as a solid red line, and the droplet's contact line as a dashed blue-white line. Bottom: the edge of the finger structure $d-d_0$, where $d_0$ is the main radial position of the inner edge of the deposit, as a function of $\phi$. In both panels the same two fingers are marked by arrows and labeled (i) and (ii). (G-I) The Fourier spectra of the finger edge $d(\phi)-d_0$. Spectra for different values of (G) $n_i$ at fixed $U_b=17~\mu$m/s and $H=0.53$, (H) $U_b$ at fixed $n_i=6.4\times 10^9$ cells/mL and $H=0.53$, and (I) $H$ at fixed $n_i=1.9\times 10^9$ cells/mL and $U_b=17~\mu$m/s. The fingers have a robust wavelength of $\sim 100 \pm 24~\mu$m irrespective of the initial bacterial number density, swimming speed or humidity.
• Figure 2: (A) Phase diagram of the three different contact-line patterns formed by motile E. coli bacteria: uniform (in red), fingers (in green), and sweeping fingers (in blue). On the $y$-axis, the bacterial swimming speed is rescaled by the evaporation-driven capillary flow speed $U_e$ and the ratio of bacterial length $L_b$ to droplet radius $R$, on the $x$-axis the bacterial number density is rescaled by the critical number density for collective motion $n_c$ (see main text). The results obtained for a non-motile strain, corresponding to $U_b=0$ (and hence $n_c$ is undefined), are denoted on a separate axis. The solid red line corresponds to the theoretical prediction Eq. (\ref{['eq:crit1']}) for the transition from a uniform to a finger-pattern, the dashed blue line denotes the transition from fingers to sweeping fingers according to Eq. (\ref{['eq:crit2']}), with free parameter $N=3$. (B-D) Examples of the corresponding patterns formed at $H\approx 0.4$: (B) a uniform pattern for $n_i\approx 5\times 10^7$ cells/mL. (C) A pattern with fingers at $n_i\approx 5\times 10^8$ cells/mL. (D) A pattern with sweeping fingers formed at $n_i\approx 5\times 10^9$ cells/mL. Corresponding movies can be found in the SI Appendix. The common scale bar for images (B-D) denotes 50 $\mu$m.
• Figure 3: Mechanism of finger nucleation and growth. Top: Single-frame projections of image sequences showing the different fates of a bacterium swimming towards the contact line: (A) The bacterium gets (temporarily) stuck at the contact line. (B) The bacterium re-orients and continues to swim parallel to contact line until it reaches an obstacle, or (C) The bacterium re-orients and swims away from the contact line. Such reorientation either happens at the contact line itself or at cell bodies of other bacteria that are stuck there. (D) The orientation of a bacterium that is stuck at the contact line changes over time due to the bacterium's wiggling motion, which could cause it to escape from the deposit. (E-F) Temporal evolution of bacterial deposits for (E) non-motile and (F) motile bacteria. Times $t/t_f$ are given in each panel, where $t_f$ is the droplet lifetime. The colors indicate the bacterial orientation, where green stands for perpendicular to the contact line and red for parallel. The bacterial orientation with respect to the contact line is obtained by fitting an ellipse to the cell body. Non-motile bacteria (E), once stuck, cannot escape the deposit and form a uniform pattern without any clusters or fingers. A deposit of motile bacteria (F) initially consists of clusters where bacteria nest into or escape from. Some of these clusters dissolve as bacteria escape, while others grow into fingers. (G) PIV measurements of motile bacteria around a finger. The velocity arrows represent a temporal average over 25 s. (H) Quantitative plot of the bacterial velocity in radial direction $u_{b,r}$ along a horizontal line (white dashed line in panel G) as function of the position, showing that bacteria escape the deposit at the fingers (resulting in $u_{b,r}<0$), while away from the fingers they move towards the deposit ($u_{b,r}>0$).