Accumulation and depletion of E. coli in surfaces mediated by curvature

Can topography be used to control bacteria accumulation? We address this question in the model system of smooth-swimming and run-and-tumble Escherichia coli swimming near a sinusoidal surface, and show that the accumulation of bacteria is determined by the characteristic curvature of the surface. For low curvatures, cells swim along the surface due to steric alignment and are ejected from the surface when they reach the peak of the sinusoid. Increasing curvature enhances this effect and reduces the density of bacteria in the curved surface. However, for curvatures larger than ${\kappa }^{*}\approx 0.25\mu {\mathrm{m}}^{-1}$, bacteria become trapped in the valleys, where they can remain for long periods of time. Minimal simulations considering only steric interactions with the surface reproduce these results and give insights into the physical mechanisms defining the critical curvature, which is found to scale with the inverse of the bacterial length. We show that for curvatures larger than ${\kappa }^{*}$, the otherwise stable alignment with the wall becomes unstable while the stable orientation is now perpendicular to the wall, thus predicting accurately the onset of trapping at the valleys.

Figure 1

(a) Diagram of a section of the microfluidic device with constant amplitude and wavelength. (b) Snapshot obtained in a device with $A=3\mu \mathrm{m}$ and $\lambda =27\mu \mathrm{m}$. Color has been inverted, hence bacteria appear dark. The red lines represent the vertical walls of the channel and the scale bar is $10\mu \mathrm{m}$.

Figure 2

Color-inverted snapshot sequences obtained with fluorescence microscopy, showing smooth swimmers as they move in contact with curved walls of different amplitudes $A$ and wavelengths $\lambda$, with increasing curvature from top to bottom. Times indicated at the bottom of each frame are with respect to the first contact of the bacterium with the wall, except for (d), in which the cluster existed already at the beginning of the acquisition. The red lines mark the measured position of the walls. The scale bar in the top right represents $10\mu \mathrm{m}$. Corresponding movies S1, S2a, S3a, and S4 are available in the Supplemental Material [37].

Figure 3

Normalized linear density (a) and normalized average speed of swimmers in contact with the curved walls (b) as a function of $x$ for three cases with constant wavelength $\lambda =30\mu \mathrm{m}$ and varying amplitude $A$. Error bars represent the $95\%$ confidence interval for the mean. (c) Spatial average of the normalized concentration profile as a function of the measured amplitudes and wavelengths, $A$ and $\lambda$. The number of surface periods considered in the averages is indicated above each data point. (d) Valley to peak density ratio, $\varphi$, as a function of the measured amplitudes and wavelengths, $A$ and $\lambda$. The dashed black curves show ${\langle \mu \rangle }_x=1$ and $\varphi =1$, obtained from fitting the measurements to a second order polynomial in $A$ and $\lambda$. For $\varphi$, the fit is performed only with data points near the region of interest.

Figure 4

Examples of bacteria trajectories. The initial and final positions are represented by a star and a triangle, respectively. Due to the counterclockwise rotation at the surface, the orange trajectory leaves the curved surface at the right of the valley, so it curves out, while the opposite happens several times to the blue trajectory. The red lines represent the measured walls of the channel, and the scale bar is $10\mu \mathrm{m}$. The background is a superposition of color-inverted frames in a time interval of $5\mathrm{s}$. The corresponding movie S5 is available in the Supplemental Material [37].

Figure 5

(a) Average accumulation at the curved wall ${\langle \mu \rangle }_x$, (b) valley to peak density ratio $\varphi$, and (c) residence time $\tau$ as a function of the maximum curvature of the walls, $\kappa =4{\pi }^{2}A/{\lambda }^2$. As a reference, the theoretical values for vanishing curvature ${\langle \mu \rangle }_x=1$, $\varphi =1$ are shown with a star in (a) and (b). For ${\langle \mu \rangle }_x$ and $\varphi$, error bars represent the $95\%$ confidence interval for the average of the parameter. For the contact times, they represent the interval that contains $90\%$ of the measured contact times. Contact times for curvatures $\kappa >0.7\mu {\mathrm{m}}^{-1}$ are immeasurable due to the formation of clusters. The insets show a zoom of the small curvature region where the dashed lines are reference values of $\mu =1$, $\varphi =1$, and $\kappa ={\kappa }^{*}=0.25\mu {\mathrm{m}}^{-1}$.

Figure 6

Simulation snapshots, analog to Fig. 2, showing smooth swimmers moving in contact with the surface. Times indicated at the bottom of each frame are with respect to the first contact of the particle with the wall. The red lines mark the position of the walls and the scale bar is equivalent to $10\mu \mathrm{m}$. Corresponding movies S6–S9 are available in the Supplemental Material [37].