Succeed escape: Flow shear promotes tumbling of Escherichia colinear a solid surface

Understanding how bacteria move close to a surface under various stimuli is crucial for a broad range of microbial processes including biofilm formation, bacterial transport and migration. While prior studies focus on interactions between single stimulus and bacterial suspension, we emphasize on compounding effects of flow shear and solid surfaces on bacterial motility, especially reorientation and tumble. We have applied microfluidics and digital holographic microscopy to capture a large number (>105) of 3D Escherichia coli trajectories near a surface under various flow shear. We find that near-surface flow shear promotes cell reorientation and mitigates the tumble suppression and re-orientation confinement found in a quiescent flow, and consequently enhances surface normal bacterial dispersion. Conditional sampling suggests that two complimentary hydrodynamic mechanisms, Jeffrey Orbit and shear-induced flagella unbundling, are responsible for the enhancement in bacterial tumble motility. These findings imply that flow shear may mitigate cell trapping and prevent biofilm initiation.

and was placed in a centrifuge at 2000 rpm to be washed. After carefully removing supernatant from the vial, the cells were suspended in a motility buffer (10 -2 M potassium phosphate buffer; pH = 7.5) containing 10 -4 M EDTA. This process was repeated twice to gain the best motility behavior of the cells. The prepared bacterial suspension was then transferred to a syringe to be injected in a microchannel. To create the shear flow inside the microchannel, bacterial solution was driven by a syringe pump (NE-1000, New Era pump System Inc.) to create constant flow rate inside the microchannel. The flow rates tested in the experiment were in the range of 0.2 − 100 / .

S1.2 Microfluidics
The microchannel (45 mm long, 200 μm deep, and 5 mm wide) used in the experiment was fabricated following soft lithography techniques 2 . The photoresist used for microchannel molds, was cast on the master and cured in the oven at 70°C for 6 hours. After curing, microchannel was bonded with bonded with a clean microscopic glass slide with anti-reflection (AR) MgF2 coating (<1% at 500-670nm) using ozone plasma activation. The bonding however temporary lasted for 10 hours during the entire experiments. We used the AR coated glass slide to reduce the coherent noise in holograms caused by the internal reflection among optics.

S1.3 Cell imaging and tracking
Imaging is performed using inline digital holography (DHM, [3][4][5][6][7][8][9][10] ) on an inverted microscope (TS-100, Nikon) with 40X magnification (Plan Fluor 40X Objective, NA=0.65, Nikon). Briefly, the optical setup for DHM includes a laser beam (7 mW He-Ne, 25LHR171-249, Mellosgriot-CVI), Neutral Density filter (ND = 2), a spatial filter, and a collimating lens. Digital holograms are recorded by a CCD camera (Imprex-4ML) at 15fps. The raw holograms are de-noised and enhanced using the correlation based de-noising technique described in details 11 . The field of view of the imaging is 400 × 400 μm 2 with reconstruction depth of 200 μm, and with a spatial resolution of 0.2 μm (lateral) and 0.5 μm (axial). After recording and reconstructing of the holograms we track the bacteria for 1 minute using in-house software to obtain bacterial position, velocity, and trajectories. To determine the angular motion of the cell during tumbles, we need to identify tumble events within trajectories. Two criteria are considered to determine a tumble event. The first criterion is a fast change in the swimming direction and it is satisfied when the angle between two consecutive swimming directions is more than 50 degree. The second criterion is an abrupt change in the swimming speed and it is satisfied when the swimming speed dropped more than one standard deviation below the mean swimming speed of each individual cell. The time between two consecutive tumbles was considered as a run time. The swimming direction before a tumble were computed over four consecutive positions of the cell prior to the tumble and the direction after the tumble were computed over four positions after the tumble. The tumble angle was the angle between these two direction considered as a tumbling angle. Note that the abovementioned motility analysis was only applied to bacterial trajectory with the flow advection velocity removed.

S1.4 Shear flow measurement by micro-Particle Image Velocimetry (PIV) and Particle
Tracking Velocimetry (PTV) To estimate flow velocity within the channel, we performed µPIV analysis to the reconstructed holographic bacterial images at different depths of the channel to obtain mean velocity distribution.
Concerns may arise whether bacteria cells would be an appropriate flow tracer since they are motile. In the absence of mechanical stimuli, the bacteria move randomly hence their spatially averaged motion must vanish. The coherent motion among those cells at the same depth in the microchannel yields the advection generated by the flow. Additional particle tracking procedure (PTV) are applied to near surface cell motion to further improve the near surface flow estimation.
In the following paragraph, we present the PIV analysis including constructing PIV images from a stack of reconstructed holographic images. We then show PTV analysis for near surface flow measurement.
µPIV analysis: Cross-correlation based PIV analysis is applied to a sequence of reconstructed 3D particle fields to obtain mean advection flow profile along the depth direction (y axis). To determine the flow velocity at a given y position, PIV image pair, ( , ; , 1 ) and ( , ; , 2 ) where 1 and 2 are separated by a small time delay, is constructed using in-focus images of bacteria located at that depth using the following ensemble averaging procedure: where ( , ; , ) is the xz-plane light scattering of the i-th bacterium at its in-focus depth, , and reconstructed at time frame, ; and is the thickness of the layer within which all in-focus cell images will be integrated. The "ensemble averaging" procedure employed here is a simple minimization operation along the y axis (denoted as the subscript, , under the operator, min, in Eqn. S1), since the in-focus bacterial image are darker than their surroundings. Note that to allow high the measurement accuracy in a wall bounded shear flow, the integration thickness, , must be kept as small as possible, e.g.
where is the subset of the entire recordings and + is the same randomly selected subset but with a specific time delay, . Note that the integration is still performed using minimization along the time axis. Fig. S1a represents an example of 1 located in the middle of the channel, y=100μm, with the bin size of = 2.5 . The image is integrated over 50 randomly selected 3D DHM particle fields out of 20,000 recordings. To perform cross-correlation PIV analysis, PIV images obtained in Eqn. (S2) must be inverted (as illustrated in Green or Red color fields in Fig. S2). is shown as red dots in Fig. S1b. Once an image pair is obtained, the 2D mean velocity distribution ( and in the x and z directions respectively) at the given depth, y, is determined using FFT-based crosscorrelation PIV analysis. Since the flow in microfluidics is often assumed homogeneous along the streamwise (x) direction and the dimension of the sample area in the spanwise (z) direction is much smaller that the channel width, the spatial variations in the obtained velocity field is expected to be small. For simplicity, only averaged streamwise velocity component, ( ), will be retained.
The abovementioned technique will be applied to the image pairs at various depth locations, to obtain the velocity profile, ( ), along the y direction.
Estimating flow velocity accurately using motile bacteria as tracer particles requires advection flow to be coherent and bacterial motility to be random in directions. However, due to the potential bias error introduced by averaging over insufficient data points (e.g. angular distribution of bacteria is not uniformly distributed over the range of [0, 2 ]), the relative error can be reduced where the local advection velocity, , is much larger than the swimming speed, . In bulk, the flow advection is larger than the swimming speed of the cells, spatial averaging yields accurate results. In near-surface region, the swimming speed of the cell is larger than the flow velocity; the bias error is often overwhelming. To resolve the problem, we further perform Particle Tracking , , ) and subsequently particle velocity, ( ).

(iv) Repeat
Step i-iii until all bacteria in the near surface regions are processed.
Once particle velocity is obtained for a time step, the near surface velocity profile, ⃗ ⃗ ( ), is recomputed by binning s with a size of 1μm at the interval of 1μm and averaging over each bin  The best fits of ( ) measurements yield the characteristic tumble frequencies at different shear rate (Fig. 2).
Direct estimation using the survival distribution: When the PDs are fluctuating due to lack of sufficiently large data point when constructing the statistics, such as conditional sampled mean run time based on the swimming orientation to the spanwise (z) direction ( Fig. 5a and Fig. S5). In this case, direct estimation using the prior method will yield large errors. Instead, we estimate the efolding time scales using the survival probability derived directly from the PD, The e-folding time scales can be obtained by fitting Eqn. S5 with the exponential function, − .
The accumulative probability in Eqn. S5 provide a smooth distribution for accurate estimation. Brief recap: Noted that both methods yield similar distributions, i.e. exponentially decrease as the run time increases (Fig. 5a and Fig. S4). It is also noted that a larger characteristic tumble frequency, , has a faster decay rate. Elucidated clearly in Fig. 5a and Fig. S5, at any flow shear, when E. coli swims in the cross-flow (z) direction, the motile cell is more likely than those swim in the streamwise (x) direction. The tumbling motility decreases as the swimming direction with respect to z axis increases.

S2.1 Hypothesis on cell re-orientation
Our experimental results clearly show that when wild type E. coli (tumble capable strain, AW405) swims over a solid surface in the presence of flow shear, the suppressed tumble motility 11 is strongly mitigated and the characteristic tumble frequency, , is increased with the near surface flow shear, S (Fig. 2). It is found that this effect depends highly on cell orientation, i.e. when E.
coli swims normal to the flow (i.e. at acute angle to the z axis), the characteristics tumble frequency, , is larger than that when it swims in the direction of flow (Fig. 5a, Figs S5a and S5b).
It is also shown clearly in Fig. 5b that at low flow shear cases ( = 0.06, 3 −1 ), s increase linearly with respect to the magnitude of |cos ( )|, where is the swimming angle of each run to z axis; whereas at high flow shear case ( = 30.9 −1 ), we observe a noticeable deviation from linear distribution at the alignment towards the flow direction, i.e. | | = 0.
As discussed in main text, two mechanisms are considered capable of explaining this shearinduced re-orientation enhancement: In the swimming direction normal to the flow, the near surface flow shear ( > 0) helps the flagellar unbundling and improve tumble motility; while in the swimming direction in the flow direction, the sufficiently large near surface flow shear (above a critical shear) will cause non-spherical particle to re-orient passively via Jeffrey Orbit and rapidly enough to cause a large sudden change in the swimming direction, of which it can be mistakenly considered as a "pseudo" tumble. We have demonstrated adequately in the main text that the shear-induce tumble mechanism is indeed a key alternative to the well accepted passive mechanism (e.g. Jeffrey Orbits). In the section, we will explore that at sufficiently high flow shear, the passive mechanism (Jeffrey Orbit) will play increasingly significant role in cell re-orientation.

S2.2 Simulation of reorientation based on Jeffrey Orbit of passive particle
We use a mathematical model for passive The linear and angular kinematics of a smooth swimming E. coli can be modeled as the following set of linear dynamic equations: where v is the swimming speed, and S is surface shear on the bottom surface of the microfluidics with the channel height of ; the parameter, G, is the geometry factor (Eq. S11) based on the aspect ratio of the cell, 16 (Table S1). The black dashed line in the figure indicates the minimum reorientation to be considered as a "pseudo" tumble (50°). The time lapse between two adjacent "pseudo" tumbles (time lapse of the re-orientation below 50°) is defined as mean run time and the inverse is considered as the characteristic tumble frequency, . When there is no "pseudo" tumbles detected, i.e. the cell reorientation over 1/15 s is below 50°, will be zero. For instance, the "pseudo" tumble frequency is measured as 0.962 s -1 in the example provided in Fig. S7. The example demonstrates anecdotally that when the flow shear is sufficiently large, the cell can indeed be reoriented owning to Jeffrey Orbit motion.  Table S1: Conditions used in the simulation shown in Fig. S5.

S2.3 Phase diagrams of "pseudo" tumbles
The results of simulation example show that Jeffrey Orbit motion of a prolong cell body (including flagella) at high flow shear can reorient itself with large angle to be considered as a "pseudo" tumble ( Fig. S7). It has been shown by Jeffrey (1952) and many others that angular motion of the passive particle in a shear flow depends deterministically on flow shear, , aspect ratio of particle, , and initial cell orientation, (0). In this section, we apply the developed simulations above b a Fig. S8| (a) The phase distribution of the characteristic tumbling frequency, , with respect to initial c cell orientation, and ; (b) Mean tumbling frequency, , averaged over those initial orientation vectores lying on a cone surface with as its major axis. The simulation is conducted at a constant flow shear of = 30 −1 and cell aspect ratio of 10. well as a complimenting mechanism at higher flow shears.

S2.4. Further analysis to support shear induced unbundling mechanism
In this subsection, we provide further analysis to support our assertion that the flow shear acts in favor of bacterial flagellar unbundling and subsequently improve the tumbling motility near a solid surface, whereas it is otherwise suppressed in the quiescent condition 11 . Instead of performing full scale modeling, we will conduct analysis to answer the following two questions: (i) What is the magnitude of the augmented unbundling stresses on flagella bundle during a shear accelerated unbundling event in comparison to those otherwise suppressed by hydrodynamic hindrance mechanism?
(ii) What is the critical shear rate at which the effect of shear accelerated unbundling and that of hydrodynamic hindrance mechanism balances each other?
Molaei et al. (2014) 11 have presented a conceptual model that a solid surface lengthens the timescale of flagella unbundling process, and subsequently causes an unsuccessful execution of a tumble, which effectively combines two consecutive runs into a long run. In that model, we use a drift velocity concept to model the process (S.3.2 in Ref 11 ). In a nutshell, since the unbundling process can be estimated by the relative motion of flagella to its surrounding flow, namely drift and can be modeled and linearized to 1 ( ) −1 , which is the first term of Eqn. S19 in SI of Ref 11 , where is the distance to the surface and Lc the total length of the cell including the flagella. For the given physiological parameters of an AW405 bacterium, the constant is estimated as 1 ≈ 0.4 (following the same procedure described in S.3.4 in SI of Ref 11 ). The reduction of the drift velocity can then be found in the simple form as In a quiescent condition, the relative flow around a rotating flagellum, , , is simply the rotational speed of the filament, i.e. , = , where is the angular rotation of the filament and is the effective radius of the filament. Note that is the radius of a helical filament (e.g. Critical Shear: By equating Eqn. S14 and S15, one can obtain a critical shear rate, , at which the effect of flow shear on unbundling balances that of surface hydrodynamic hindrance: It shows that depends on the distance to the surface. One can further integrate Eqn. S16  (Fig. 5b).