SUPPLEMENTARY INFORMATION

Characterization of hydrodynamic surface interactions of Escherichia coli cell bodies in shear flow

Tolga Kaya and Hur Kosera

Electrical Engineering Department, Yale University, PO Box 208284, New Haven, CT 06520-8284

a

Details of Experimental Setup

A syringe pump (Model No: KDS101 from KD Scientific, MA, USA) was used upstream of the microfluidic device to generate volumetric flow rates ranging from 100 μl/min to 3.5 ml. The specific volumetric flow rate achieved depended on the speed of the syringe pump and the diameter of the syringe used. For low volumetric flow rates, a 10 cc syringe was utilized; higher flow rates were achieved via a 60 cc syringe. During each experiment, this volumetric flow rate was manually confirmed by measuring it downstream of the microfluidic device through a careful injection of air bubbles into a larger tubing with a known internal diameter (1/8 inch). The travel speed of the air bubble times the cross-sectional area of that tubing would confirm the volumetric flow rate read from the syringe pump.

The images were captured via a 1.3 Megapixel (1280W x 1024H), black-and-white CCD digital camera (Model No: SI1300M-CL from Silicon Imaging, NY, USA) directly into computer memory at a maximum transfer rate of 50 MB/s. With each pixel’s intensity represented via 8 bits, the overall camera-computer system is capable of continuous real-time recording at over 30 frames/s at full resolution. Higher frame rates can be achieved with smaller regions of interest; for instance, 60 frames/s was achieved in experiments by simply reducing the width of the region of interest below half the full resolution width. During experiments, the exposure time was kept at a small fraction of the frame period in order to avoid motion artifacts in obtained images, even at the fastest shear flow rates recorded. Readout method for the camera was “progressive scan”; hence, interlacing effects were not relevant. Images were captured via PIXCI CL1 CameraLink interface board and XCAP image capturing/analysis software (also available from Silicon Imaging).

Verification of the Shear Rate

COMSOL was used to calculate the shear rate near the channel surface for different volumetric flow speeds (Fig. S1a). That shear rate was constant across the field of view and changed by less than 1.8% within the first 2.5 m over the bottom surface (Fig. S1b).

Fig. S1: Shear rate computation at a volumetric flow rate of 100 μl/min. (a) Shear rate () along the x-axis of the bottom surface within the field of view is 8.2 s-1 for the given flow rate. Inset: Computation of normalized flow (in the -y-direction) within the actual channel geometry (5 mm x 600 μm). (b) Only cells within the first 2.5 μm of the surface are selected for analysis; the shear rate in this range varies no more than 1.8% and can be assumed constant within experimental accuracy.

In order to verify the simulatedresults empirically, cells were pumped at a specific volumetric flow rate (60 μl/min) and the average drift velocity of a large number of cells at different focus positions over the center of the bottom surface was calculated.

Simulations showed that shear rate () linearly decreases as the height from the surface (z) increases (Fig. S2a). The average simulated shear rate (for a volumetric flow rate of 60 μl/min) in the first 32 μm of z is 4.745 s-1. The average shear rate deduced experimentally for the same range of z and the same volumetric flow rate within the microfluidic channel is 4.768 s-1 (Fig. S2b). The discrepancy between the simulated and the measured average  values is less than 0.5 %.

Fig. S2: Experimental verification of shear rate simulations. Volumetric flow rate in both the simulation and the experiment is 60 μl/min. (a) The shear rate in the channel linearly decreases with z across the entire height of the channel (Inset). The average shear rate in the first 36 μm above the surface is found to be 4.745 s-1. (b) The average drift velocity (vdrift) of cell bodies is measured for different focal planes across the height of the microfluidic channel. The focal planes start below the bottom surface of the channel (i.e., inside the glass slide). Within about  2 μm of the surface, only a portion of the depth of field of the 40x objective is inside the flow channel, corresponding to a bias. Beyond 4-5 μm above the surface, both the depth of field bias and wall effects on the drift velocity of cells become negligible, and the data can be used to compute the flow shear rate. The red, vertical arrow indicates the first focal plane that yields reliable data from which local shear rate could be calculated directly from vdrift. The average shear rate within the first 36 μm from the surface hence calculated is 4.768 s-1. Discrepancy between simulation and experiment is below 0.5 %.

Coordinate transformation

Fig. S3a describes the relative coordinate system (one that drifts together with an ellipsoid’s center) that Jeffery defined in his original calculations of closed orbits [1].

Fig. S3: Difference in relative coordinate systems. (a) Jeffery’s original calculations define a spherical coordinate system in which the vorticity is along the -z’-axis and the spheroidal particle drifts along the x’-axis. (b) In our experimental setup, we depict the x’-axis along the vorticity direction and the –y’-axis as the drift direction. In this manner, the x’y’-plane is parallel to the xy-plane (i.e., the bottom surface of the flow channel). We define as the acute angle between the x’y’-plane and the axis of symmetry of the bacterium, and as the acute angle between the x’-axis and the projection of the symmetry axis onto the x’y’ plane (positive counterclockwise as shown).

The particle’s orientation at any point in time is described by two spherical coordinate angles, and , as shown in Fig. S3a. The angular motion of a prolate spheroid in a Newtonian fluid is then described by

(S1)

(S2)

In this Letter, we have chosen a different coordinate system – one that enables the definition of an orientation (or "yaw") angle and a “pitch” angle , both of which are easily deducible from our microscopy observations. Fig. S3b depicts the definitions of these angles with respect to and as used by Jeffery. The x’y’-plane in this coordinate system is parallel to the xy-plane (bottom surface of the flow channel).

Using Fig. S3b, it is rather straightforward to express all angles in terms of the Cartesian coordinate positions of the bacterium’s tip

(S3)

(S4)

(S5)

(S6)

so that one could express and in terms of and

(S7)

(S8)

Once and are determined through integrating (S1) and (S2), and can be simply calculated for all time using (S7) and (S8). The model fits depicted in Fig. 3 of the main text have been calculated in this fashion.

Values of f(rp)

The slopes of the linear fits depicted in Fig. 4c of the main text correspond to f(rp) values of equation (2) for different rp’s. Fig. S4 shows these f(rp) values. For the purposes of providing an analytical expression, we fit a third order polynomial to the plotted data and present the best fit (R-squared value 0.999) in Fig. S4 as well. The resulting fitted polynomial is

(S9)

Fig. S4: Values of f(rp) correspond to the slopes of the linear fits to the data depicted in Fig. 4c, in which re/rp is plotted vs. w-1.5. Here, a third-order polynomial, given by (S9), is also shown as a best fit to the f(rp) values.

Check against selection bias

Since a tracked bacterium eventually drifts out of the field of view (extent: Δy) with drift velocity v, its orbital period (T) can be inferred only when the drift time (Δt) is longer than T/2 (i.e., ). Otherwise, bacteria with longer aspect ratios (and longer corresponding periods) would not be included in the data set, introducing a selection bias. Preventing this selection bias requires that Tγ is below a certain limit () for a given w. Considering that largest w is 2.5 μm and Δy is 126.4 μm, this limit for Tγ is 101 for the bacteria in the data set.

Fig. S5 illustrates that the Tγ values measured for all 13284 bacteria are well below this limit and that there is no tendency for saturation in Tγ as a function of rp. This indicates that, for the cell body aspect ratios present in our bacteria cultures and the height (w) values that we have included in the data set, there is no size-dependent selection bias.

Fig. S5: Tγ vs. cell body aspect ratio for all 13284 tracked bacteria. All Tγ values are well below the bias threshold and there is no saturation behavior in Tγ as a function of rp.

References

[1] G.B. Jeffery, Proc. R. Soc. London A 102, 161 (1922)