Upcoming flow promotes the bundle formation of bacterial flagella

Abstract

Flagellated bacteria swim by rotating a bundle of helical flagella and commonly explore the surrounding environment in a “run-and-tumble” motility mode. Here, we show that the upcoming flow could impact the bacterial run-and-tumble behavior by affecting the formation and dispersal of the flagellar bundle. Using a dual optical tweezers setup to trap individual bacteria, we characterized the effects of the imposed fluid flow and cell body rotation on the run-and-tumble behavior. We found that the two factors affect the behavior differently, with the imposed fluid flow increasing the running time and decreasing the tumbling time and the cell body rotation decreasing the tumbling time only. Using numerical simulations, we computed the flagellar bundling time as a function of flow velocity, which agrees well with our experimental observations. The mechanical effects we characterized here provide novel, to our knowledge, ingredients for further studies of bacterial chemotaxis in complex environments such as dynamic fluid environments.

Significance

A hallmark of bacterial chemotaxis is the “run-and-tumble” behavior. It remains unclear whether and how external mechanical factors affect this behavior. Here, we used dual optical tweezers to trap individual bacteria horizontally and studied the run-and-tumble behavior under well-controlled mechanical conditions. We discovered that the mechanical factors such as imposed fluid flow and cell body rotation have a large impact on the run-and-tumble behavior through affecting flagellar bundle formation and dispersal. These mechanical effects provided novel, to our knowledge, insights for further studies of bacterial chemotaxis in complex environments.

Introduction

Flagellated bacteria, such as Escherichia coli, swim by rotating a bundle of helical flagellar filaments, each driven at its base by a bidirectional rotary motor embedded in the cell envelope (1). A typical E. coli cell contains, on average, four flagella attached at random sites (2). When all the cellular motors rotate counterclockwise (as viewed from distal end of the flagellum), their filaments bundle together and push the cell to run forward steadily (3). From time to time, the bundle falls apart when one or more of the motors switch their direction of rotation and the associated filaments come out of the bundle and go through polymorphic transformations, causing the cell to tumble erratically (2,4). Hence, the locomotion of a flagellated bacterium consists of its running-and-tumbling behaviors when its flagella are bundled together or dispersed (5).

E. coli exhibits running and tumbling to move toward nutrient-rich regions or escape from hazardous areas, resulting in chemotaxis (6). The run and tumble of E. coli can also be influenced by external mechanical factors such as light (7), temperature (8), osmotic pressure (9), and so on. Besides these, the underlying flow can also affect the swimming behavior of the E. coli cells by exerting hydrodynamic forces and torques on them, leading the cells to undergo rheotaxis (10). For example, it was observed that the cells under a near-wall shear flow were able to follow efficient routes to swim upward (11,12); in addition to the observed upstream locomotion, oscillatory rheotaxis of E. coli was also reported (13). These works have examined the role of the underlying flow on the swimming trajectories of a group of E. coli cells near a surface. Yet, it remains elusive how the underlying flow influences the swimming dynamics of an individual cell in an unbounded domain and whether the flow can affect its running and tumbling by reshaping its flagellar bundling and dispersal. To address this question, we hereby experimentally investigate the effect of the upcoming flow on the flagellar bundling dynamics and running-and-tumbling behavior of weakly confined individual E. coli cells with their position and orientation anchored optically.

We combined dual optical tweezers setup, fluorescence microscopy, and microfluidics to examine the run-and-tumble behavior of wild-type E. coli subjected to well-controlled upcoming flows. Details of the setup are presented in Materials and methods. After injecting E. coli cells to the chamber, the rod-shaped body of an individual cell was trapped by two optical traps at its two ends (Fig. 1 A) and was oriented horizontally (x axis). The fluid velocity was tuned by setting different pumping rates for the syringe pump and calibrated using 0.75-μm-diameter latex beads. The run-and-tumble durations were extracted from the optical trap signals (Fig. 1 B) (14), and the tumble bias (the fraction of time a cell spends tumbling) was calculated as the sum of tumble durations divided by the sum of run-and-tumble durations for each cell. The control experiment was performed to ensure that the laser-induced photodamage on the cell from the optical traps was negligible (Fig. S1).

Figure 1.

(A) Experimental apparatus and cell configuration. The bacterium was trapped horizontally by two optical traps for various upcoming flow velocities U_∞. The resultant force of two optical traps F_trap is along the −e_x direction in the presence of a large U_∞, i.e., U_∞ > V_swim, where V_swim is the free-swimming speed of the bacterium. When U_∞ < V_swim, F_trap is along the +e_x direction. F_thrust is the propulsive force generated by the flagellar bundle, and F_drag is the viscous force acting on the cell body. (B) A typical optical trap signal of a motile cell along y and z directions. The gray scale depicts different body rotational frequencies for the run (high frequency) periods with lines on the top horizontal axis. The black line on the bottom axis delineates the tumbling (low frequency) duration. To see this figure in color, go online.

Materials and methods

Strains and plasmids

Strain HCB1668 used in this study is a Tn5 fliC null derivative of AW405, in which FliC(S353C) is expressed from the plasmid pBAD33 under control of the arabinose promoter. This strain swims and swarms vigorously (15). The plasmid pMT2 expresses CheY^D13KY106W under control of an arabinose-inducible promoter in the vector pBAD33. Strain HCB1736 (ΔCheY) carrying pMT2 was used for control experiments with the chemotaxis-defective mutant strain (16).

Cell culture

HCB1668 were firstly grown in 3 mL Luria Bertani broth (10 g bacto-tryptone, 5 g yeast extract, and 5 g NaCl per liter) with appropriate antibiotics (25 μg/mL kanamycin, 25 μg/mL chloramphenicol) for 16 h at 30°C with rotation (200 revolutions per minute (rpm)). We then cultured a total of 100 μL HCB1668 bacterial suspension in 10 mL of T-broth at 30°C with appropriate antibiotics and inducer (666 μM arabinose) for 4 h until the bacteria reached the midexponential growing phase. Appropriate antibiotics and L-arabinose (final concentration of 333 μM) were added to the T-broth culture of HCB1736 pMT2 to induce a suitable expression level of CheY^D13KY106W, so that the distribution of tumble bias without flow was similar to that of the wild-type cells. The suspension was then washed by different solutions to meet the corresponding requirements for the experiments.

The flow chamber

All coverslips used in experiments were sonicated successively in absolute ethyl alcohol, 1 M KOH solution, and distilled water, each for 20 min. A flow chamber was constructed by drilling two 0.75-mm-diameter holes through a glass slide, which formed the inlet and outlet of the chamber (17). A cleaned glass coverslip (24 × 60 mm²) was then put over the slide using three layers of double-sided sticky tape cut in a rectangular shape as the spacer, resulting in a flow chamber of 7 × 30 mm² in area and 300 μm in depth. A syringe pump (Pump-22; Harvard Apparatus, Holliston, MA) was used to draw the liquid and generate the upcoming flow at various velocities in the flow chamber by tuning the pumping rate. During the experiments, when the cells encountered an upcoming flow, diluted 0.75-μm-diameter polystyrene latex beads (2.69%, 07310; Polysciences, Warrington, PA) were used to calculate the flow rate. To minimize the hydrodynamic confinement of the chamber, we ensured that the cell was at least ∼60 μm away from the boundaries of the chamber. All experiments were carried out at 23°C.

Fluorescence measurements

The bacterial suspension was washed three times by centrifuging at 1200 rpm for 6 min to prevent the breakage of flagella and gently resuspended with fresh motility buffer (10 mM potassium phosphate, 0.1 mM EDTA, 1 mM methionine, 67 mM NaCl, 10 mM lactate (pH 7.5)) to label the flagella. In the final preparation, the suspension was concentrated 20-fold to 50 μL. Then, 5 μL solution (5 mg/mL in dimethyl sulfoxide) of a thiol-reactive dye (Alexa Fluor 568 C5 maleimide; Invitrogen-Molecular Probes, Thermo Fisher Scientific, Waltham, MA) was added to the suspension, and labeling was allowed to proceed for 60 min at room temperature with stirring by gyration at 80 rpm. After labeling, the cells were washed three times and diluted 300-fold with trapping medium. They were used immediately for the experiments or stored at 4°C for up to 2 h.

An individual cell was trapped by a commercially available dual optical tweezers setup (NanoTracker; JPK Instruments, Berlin, Germany) (Fig. 1) (18). The light for the dual optical tweezers was provided by a 1064 nm laser through a 60× NA 1.0 water-immersion objective. The width of the well at focal plane is 1.2 μm. We used 200 mW of trapping power at the sample plane in each trap. Epifluorescence imaging of labeled filaments was achieved by excitation with a 75 W xenon lamp, using a fluorescence cube with an ET569/25× excitation filter, a T588lpxr dichroic mirror, and an ET615/40 m emission filter (Chroma Technologies, Brattleboro, VT). Fluorescence images were taken at 100 frames per second focusing on selected flagella using a scientific complementary metal oxide semiconductor camera (Dhyana 400BSI v2.0; Tuscsen, Fuzhou, China). The duration of each video was more than 20 s. A fluorescent image of bacterial flagella is shown in Fig. 2. Fluorescently labeled flagella were counted on cells stuck by their bodies to the coverslip for the measurement of flagellar number. For brightfield imaging, the chamber was faintly illuminated using a light-emitting diode white light source with a colored glass filter (FGL610S; Thorlabs), which allowed the passage of light with a wavelength larger than 610 nm. The brightfield image of the cell body was recorded with a CMOS camera (DCC1545M-GL; Thorlabs, Newton, NJ), and the typical image for the measurement of body length is shown in Fig. S2.

Figure 2.

The fluorescence images of the bacterial flagella during (A) running and (B) tumbling. The dotted line outlines the cell body. To see this figure in color, go online.

Using ImageJ (https://imagej.nih.gov/ij/), AVI files were converted to TIF stacks, which were analyzed to determine the durations when the flagella were unbundled. The probability of the unbundled state was defined as the summed duration when the flagella were unbundled divided by the total sampling time.

Run-and-tumble measurements

The bacterial suspension was washed three times by centrifuging at 1200 rpm for 6 min with trapping medium (70 mM NaCl, 0.1 mM methionine, 100 mM Tris-Cl, 2% wt/vol glucose, 80 μg/mL glucose oxidase, and 13 μg/mL catalase) (14). Although immobilized by the optical traps, cell displayed motile behavior. This behavior was detected by imaging light from both orthogonally polarized trapping beams onto two separate position-sensitive photodetectors. The output of the position-sensitive photodetector is the optical trap signal, which was taken at a sampling rate of 1000 Hz. The rotational frequencies of flagella bundle Ω_f and cell body Ω_b were obtained by analyzing power spectra from the signal, and a typical result is shown in Fig. S3. To determine the run-and-tumble duration, the optical trap signals in the y and z directions were low-pass filtered to 100 Hz and analyzed using custom scripts in MATLAB (The MathWorks, Natick, MA), following the algorithms described previously (14).

Control experiments

We measured the tumble bias distributions of the nonchemotactic mutant (with wild-type CheY replaced by CheY^D13KY106W) with and without flow (Fig. S4), and they were similar to those of the chemotactic strain with and without flow, respectively (Fig. 5 A), confirming that the flow-induced change of tumble bias distributions was not due to possible chemotactic effect. Control experiments were also performed to verify that the imposed flow on the cells did not change their energetic states that may affect their run-and-tumble behaviors (Fig. S5). The effect of shear rate on the tumble bias distribution was characterized. The tumble bias distribution without shear rate was measured at distance of ∼150 μm from the coverslip, which was in the center of the Poiseuille flow profile. The tumble bias distribution with shear rate (∼0.6 s⁻¹) was measured at a distance of ∼60 μm from the coverslip. The two distributions are very similar (Fig. S6), demonstrating that the flagellar dynamics is not affected by the shear rate used in the experiments.

Figure 5.

Statistics of run-and-tumble behavior for E. coli cells with and without flow. (A) Tumble bias distributions. The red lines are Gaussian fits. PDFs of (B) run duration and (C) tumble duration are given. The numbers of counted cells with and without flow were 126 and 86, respectively. The flow velocity was on average 25 μm/s, similar to the typical swimming speed of E. coli (27). A two-tailed Student’s t-test was performed for comparison between samples, and significance is denoted by the asterisks (^∗∗∗p < 0.001). Error bars represent standard error of the mean. To see this figure in color, go online.

Computational model

Because bacteria swim at sufficiently low Reynolds numbers (<10⁻³), the fluid inertia can be safely neglected, which allows us to model the flow in the Stokes regime. As illustrated in Fig. 3, the cell body is modeled as a spheroid given the rod-like cell shape of E. coli. The cell body and flagella are assumed to be rigid, and each flagellum is attached to the body by flexible hook. The flagellum is modeled as a slender cylinder with cross-sectional radius r around the centerline, which is a left-handed helix with a short tapering described by Higdon (19).

$$\mathbf{x}=\left[s,R\left(1-e^{-k_E^2{s}^2}\right)\text{cos}ks,R\left(1-e^{-k_E^2{s}^2}\right)\text{sin}ks\right],s\in \left[0,L_{\text{f}}\right],$$ (1)

where L_f is axial length of the helix, k is the wavenumber, and k_E is a constant determining how quickly the helix grows to its maximal amplitude R. The flagellum is not directly attached to the cell body, but with a small separation of hook length L_h to ensure numerical stability.

Figure 3.

Simulation model of a bacterium with four flagella. To see this figure in color, go online.

In the Stokes flow of a Newtonian fluid with dynamic viscosity μ, the disturbed velocity field surrounding a bacterium exposed to an incident flow u_∞ can be expressed as the boundary integral equation (20),

$$\mathbf{u}\left(\mathbf{x}\right)={\mathbf{u}}_{\infty }\left(\mathbf{x}\right)-\frac{1}{8\pi \mu }\underset{S}{\int }\mathbf{G}\left(\mathbf{x},\hat{\mathbf{x}}\right)\cdot \mathbf{f}\left(\hat{\mathbf{x}}\right)\text{d}S,$$ (2)

where the tensor G is the free-space Green’s function for Stokes flow and f is the traction force distributed on the fluid boundary S, which consists of cell body surface S_b and flagellum surface $S_{\text{f}}^j$ (j = 1, …, N_flag), where N_flag is the number of flagella.

The cell body is constrained to rotate about the e_x axis, namely, V_b = V_b e_x, whereas the j-th flagellum is free to rotate in space with angular velocity ${\mathbf{V}}_{\text{f}}^j$ relative to the cell body. The velocity at position x on the surface of the swimmer can be written as

$$\mathbf{u}\left(\mathbf{x}\right)=\{\begin{array}{l}{\mathbf{V}}_{\text{b}}\times \left(\mathbf{x}-{\mathbf{x}}_c\right),\mathbf{x}\in S_{\text{b}},\\ {\mathbf{V}}_{\text{b}}\times \left(\mathbf{x}-{\mathbf{x}}_c\right)+{\mathbf{V}}_{\text{f}}^j\times \left(\mathbf{x}-{\mathbf{x}}_a^j\right),\mathbf{x}\in S_{\text{f}}^j\forall j,\end{array}$$ (3)

where x_c denotes the geometrical body center and ${\mathbf{x}}_a^j$ is the attachment point of the j-th flagellum.

Assuming that the bacterium is neutrally buoyant, the only external force comes from the optical tweezers that fixes the cell’s position. The torque due to the optical tweezers was negligible as our measured body-rotation rate for trapped bacteria was consistent with that observed in experiments with freely swimming cells (5). By neglecting the torque contribution from the optical tweezers along the axis e_x, the torque balance conditions are

$$0={\mathbf{e}}_x\cdot \underset{S}{\int }\left(\hat{\mathbf{x}}-{\mathbf{x}}_c\right)\times \mathbf{f}\left(\hat{\mathbf{x}}\right)\text{d}S$$ (4a)

and

$$\tau {\mathbf{e}}_{\text{f}}^j+{\mathbf{T}}^j=-\underset{S_{\text{f}}^j}{\int }\left(\hat{\mathbf{x}}-{\mathbf{x}}_a^j\right)\times \mathbf{f}\left(\hat{\mathbf{x}}\right)\text{d}S\forall j,$$ (4b)

where the former describes the torque balance on the bacterium in the e_x direction and the latter indicates the same on each flagellum about its attachment point; here, τ is the magnitude of the motor torque, ${\mathbf{e}}_{\text{f}}^j$ is the unit direction vector of the j-th flagellum axis, and the torque T^j is the repulsive torque to prevent the overlapping of the flagellum and cell body. The repulsive torque is normal to the plane spanned by the helix axis and the surface normal at ${\mathbf{x}}_a^j$, and the angle between the two vectors is denoted as tilt angle θ^j. The magnitude of repulsive torque T^j is computed as a smooth step function like θ^j to reduce unphysical oscillations (21).

$$T^j=T_m\left(1-\frac{1}{1+{\left({\theta }^j/{\theta }_0\right)}^{k_s}}\right),$$ (5)

where T_m is the maximal value, k_s is the steepness of the increase, and θ₀ is the midpoint of the increase. Furthermore, a short-range steric force among flagella has been applied to avoid flagellar intersection, which requires calculating the minimal distance between the centerlines of flagella (22). The details of the implementation are elaborated in previous research by Adhyapak and Stark (23).

We use the boundary element method (BEM) to simulate the bundle formation of bacteria with four flagella, similar to the work of Shum and Gaffney (24) or Kanehl and Ishikawa (25). The surface of the cell body and flagella are discretized into quadratically interpolated triangular elements, and Gauss-Legendre quadrature is used to integrate quantities over each element (26). A typical simulation uses 504 elements on each flagellum and 128 elements on the cell body, which is determined by mesh convergence verification. The angular velocities (V_b, ${\mathbf{V}}_{\text{f}}^j$) and the surface tractions f are unknowns, which are solved by linear algebraic solvers like generalized minimal residual method (GMRES) after combining Eqs. 2, 3, and (4a), (4b). The time stepping is performed using the two-step Adams-Bashforth method. The minor semiaxis of the cell body 0.4 μm, the flagellar motor torque 190 pN ⋅ nm, and fluid viscosity 1 × 10⁻³ Pa ⋅ s are chosen to be the characteristic scales of the system. The dimensionless values of the model bacterium used in simulation are listed in Table S1, in which geometrical parameters of a bacterium are specified according to values measured in (5). In this simulation, the upcoming flow is set to u_∞ = U_∞e_x.

In simulation, the bundling index γ is defined as (25)

$$\gamma =\frac{1}{N_{\text{flag}}}\left|\sum _j{\mathbf{e}}_{\text{f}}^j\right|,$$ (6)

with flagella number N_flag = 4. This index expresses the mutual orientation of all flagella, which becomes 1 for all flagella spreading to a same direction. We fix the initial configuration of flagella such that two flagella axes are on the e_x-e_z plane and the other two are on the e_x-e_y plane, with an angle φ₀ between ${\mathbf{e}}_{\text{f}}^j$ and e_x. According to the average flagella spreading angle observed in the experiment, we set φ₀ to 22° in the simulation. Fig. 4 shows the time evolution of the bundling index γ for different imposed flow velocities. In our simulation, we assume that the flagella form a bundle when γ exceeds 0.995. Although the flagellar number used in the simulation (N_flag = 4) is less than that of the measured strain (∼6–7 in Fig. S10), the bundling behavior is qualitatively independent of the flagellar number (25).

Figure 4.

Time evolution of the bundling index γ for various upcoming flow velocities. To see this figure in color, go online.

We validate the BEM by comparing our simulations of bacterial swimming speed to that calculated by Higdon (19). In this case, the cell body has arbitrary translational and angular velocity, which can be achieved by a little modification to the kinematic conditions. Shown in Fig. S7, our simulation results are in good agreement with that of Higdon (19).

Results

To explore the effect of flow on bacterial run-and-tumble behavior, we measured the probability distributions of tumble bias for E. coli with and without flow, finding that they were dramatically different (Fig. 5 A). The peaks and standard deviations with fitted Gaussian function are 0.21 ± 0.15 and 0.38 ± 0.22, respectively. Under flow, the distribution of tumble bias clearly shifts left, suggesting that the upcoming flow lowers the tumble bias and enables the bacteria to swim more smoothly. We then show the probability density function (PDF) for the run-and-tumble duration with and without flow in Fig. 5, B and C, respectively. The data indicate that the underlying flow extends the run duration and shortens the tumble duration of the swimming cells, thus decreasing the tumble bias. We further tuned the flow velocity from 3 to 57 μm/s and observed that increasing the flow rate would reduce the tumble duration and increase the run duration as shown in Fig. 6. The anticorrelation between the flow velocity and tumble duration indicated by Fig. 6 A was also captured by our numerical simulations (see below). Several control experiments were conducted to verify that the flow itself was responsible for the change of tumble bias. Details are presented in Materials and methods.

Figure 6.

(A) Tumble duration, (B) run duration, and (C) tumble bias of the E. coli cells versus the upcoming flow velocity U_∞. The red curve in (A) represents the numerical prediction for the tumble duration. The number of cells measured was 618. Error bars represent standard error of the mean. To see this figure in color, go online.

We speculated that the upcoming flow influenced the bacterial run-and-tumble behavior by affecting the formation and dispersal of their flagellar bundles. To demonstrate this hypothesis, we visualized the flagella of E. coli labeled by fluorescence to directly capture their bundling and unbundling processes (see Fig. 2 for typical fluorescent images). We calculated the probability of unbundled flagella as the summed duration of unbundled state divided by the total duration. The probability distributions of the unbundled state with and without flow (Fig. S8 A) closely resemble those of tumble bias (Fig. 5 A), providing the nearly one-to-one correspondence between cell tumbling and flagellar unbundling. PDFs of the bundled and unbundled states (Fig. S8, B and C) are also aligned with those of run-and-tumble durations whether the upcoming flow is applied or not. This alignment demonstrates that the flow alters the run-and-tumble behavior by influencing the formation and maintenance of the flagellar bundles. We further measured the probability of flagella being on the ends of the cell, which was defined as the proportional duration when the maximal angle among those between all pairs of flagella was an obtuse angle. The probability in the presence of an upcoming flow (0.11 ± 0.02) is significantly below that without the flow (0.26 ± 0.04), implying that the flow would bring the flagella together to facilitate the bundle formation.

The tumble (and flagella unbundled) duration decreases with the upcoming flow velocity, suggesting that the upcoming flow promotes bundle formation. To characterize this, we numerically simulated the process of flagellar bundle formation at various upcoming flow velocities using the BEM. Details of the numerical method are presented in Materials and methods. The tumble duration can be separated into two durations. One duration is that the bundle falls apart and the bundle formation has not started yet, which is equal to the clockwise interval of the flagellar motor and is independent of the upcoming flow. The other is the bundle formation time that was captured in the simulation. The comparison between the experiment and the simulation is presented in Fig. 6 A, with the clockwise interval of the motor set as 0.3 s.

We then characterized the experimentally observed role of flow in stabilizing the flagellar bundle. Regarding the flagellar bundle as a rigid helix (4,28,29), we calculated the hydrodynamic torque due to the upcoming flow,

$$M_{\text{flow}}=RL\text{sin}\psi \text{cos}\psi \left({\xi }_{\perp }-{\xi }_{\parallel }\right)U_{\infty },$$ (7)

where R is the radius of bundle helical coil, L is the contour length of the bundle, ψ is the pitch angle of the helix, and ${\xi }_{\perp }$ and ${\xi }_{\parallel }$ are perpendicular and tangential drag coefficient, respectively. The hydrodynamic torque due to the upcoming flow is proportional to the flow velocity, and the direction of the hydrodynamic torque is in the same direction as the torque exerted by the flagellar motor. According to the previous simulation (30), the bundle becomes tighter (and thus more stable) as the torque increases. Therefore, the upcoming flow stabilizes the flagellar bundle and extends the run duration.

We then fixed the upcoming flow velocity U_∞ ≈ 25 μm/s and examined how the rotational frequency Ω_b of the cell body influenced the run-and-tumble behavior of E. coli. We measured the behavior of individual bacterium using a population of cells with various body lengths because of cell-to-cell heterogeneity. We sorted the data into four groups according to the body length. In Fig. 7 A, we observe that the tumble bias increases with the body length l. Showing the rotational frequency of the cell body Ω_b and flagellar bundle Ω_f vs. l in Fig. 7 B, we find that Ω_f is mostly independent of l, implying that the torque generated by the flagellar motor does not critically depend on the body length. In contrast, the body’s rotational frequency Ω_b decreases with the body length as the rotational viscous drag coefficient of the cell body increases with the body length (nearly proportional). The relationship between the body rotational frequency and body length can be well fitted with an inversely proportional function, as expected (Fig. S9). This implies that Ω_b might result in the variation of tumble bias in Fig. 7 A. We then show in Fig. 7, C and D the tumble duration and run duration as a function of the body’s rotational frequency Ω_b, respectively. We observe that the tumble duration decreases with Ω_b, implying that the body rotation promotes flagellar bundle formation, although the run duration does not change significantly with Ω_b, suggesting that the body rotation does not stabilize the bundle or make it tighter. The results in Fig. 7, C and D contribute to the negative correlation between the tumble bias and ${\text{Ω}}_{\text{b}}$ in Fig. 7 A.

Figure 7.

(A) Tumble bias versus the body length l and rotational frequency Ω_b of the body. (B) Rotational frequency of the body Ω_b and of the flagellar bundle Ω_f vs. l. (C) Tumble duration and (D) run duration versus Ω_b. The number of cells measured was 241. Error bars are standard error of the mean. To see this figure in color, go online.

Longer cells may have more flagella on average, which might modify the tumble duration and result in a dependence of tumble bias on body length. To explore this, we measured the mean number of flagella per cell and found that it increased from 5.9 to 6.7 when the mean body length increased from 2.56 to 5.34 μm (Fig. S10). Considering the small increase in flagellar number and the fact that E. coli swimming is robust against variations in flagellar number (31), we estimate the effect of variation in flagellar number (because of change in body length) on the tumble bias, finding that it is negligible. Details of the estimation are presented in Supporting materials and methods.

The Pearson coefficients for the pairs of variables in Figs. 6 and 7 are presented in Table S2.

Discussion

In summary, we have combined experiments and simulations to study the impacts of upcoming flow on bacterial run-and-tumble dynamics. During the experiments, the individual cell was trapped away from the surface and subjected to an upcoming flow against the cell. We found that the tumble bias and the tumble duration decreased as the flow velocity increased, whereas the run duration increased with the flow velocity. The dependence of the tumble duration on the flow velocity was well captured by the BEM simulation, verifying that the upcoming flow could promote the flagellar bundle formation. The hydrodynamic torque was computed to explain the correlation between the run duration and the flow velocity, indicating that the upcoming flow could stabilize the bacterial flagellar bundle.

Here, we comment on the differences between the flow condition of a freely swimming bacterium and our experimental condition imposing an upcoming flow speed equaling to the bacterial run speed. A bacterium establishes relative flow between its body and the surrounding fluid by freely swimming. The swimming speed is not constant because of the “run-and-tumble” behavior; the swimming speed reaches maximum when it runs and decreases when it tumbles. In our assay, the relative flow is constant no matter whether the bacterium runs or tumbles as long as the pumping rate is fixed.

In this work, we demonstrate that the upcoming flow could extend the run duration by stabilizing the flagellar bundle and shorten the tumble duration by reducing the flagella rebundling time. The result we found has a large impact on bacterial run-and-tumble behavior, indicating that the influence of external fluid environment on bacterial motility cannot be neglected. The effects of upcoming flow we characterized here should be considered in future studies of bacterial motility in dynamic fluid environments.

We also found that the tumble bias and tumble duration decreased with the body’s rotational frequency, whereas the run duration was insensitive to the body rotation. This observation suggests that faster body rotation can promote the bundle formation but does not stabilize the bundle or make it tighter. Our experiments thus have confirmed the theoretical and numerical findings of the accelerated bundling process because of a stronger body rotation (23,25,32). The effect of body rotation on the run-and-tumble behavior (due to different body lengths) we characterized here would be useful in future studies of heterogeneity in bacterial chemotaxis because of the difference in cell body length.

Author contributions

J.Y., L.Z., and R.Z. designed the research. G.L. performed the measurements. Z.L. carried out all simulations. All authors wrote the article.

Editor: Dimitrios Vavylonis.