Bacterial cell-body rotation driven by a single flagellar motor and by a bundle

Abstract

Using self-trapped Escherichia coli bacteria that have intact flagellar bundles on glass surfaces, we study statistical fluctuations of cell-body rotation in a steady (unstimulated) state. These fluctuations underline direction randomization of bacterial swimming trajectories and plays a fundamental role in bacterial chemotaxis. A parallel study is also conducted using a classical rotation assay in which cell-body rotation is driven by a single flagellar motor. These investigations allow us to draw the important conclusion that during periods of counterclockwise motor rotation, which contributes to a run, all flagellar motors are strongly correlated, but during the clockwise period, which contributes to a tumble, individual motors are uncorrelated in long times. Our observation is consistent with the physical picture that formation and maintenance of a coherent flagellar bundle is provided by a single dominant flagellum in the bundle.

Significance

Single flagellar motor switching behavior has long been studied to unravel aspects of Escherichia coli motility and chemotaxis. However, wild-type cells swim with a flagellar bundle driven by several motors. The complex interactions between motors on a swimming cell has received less attention. Here, we compare measurements from a single motor to measurements from multiple motors to develop a novel physical view that a single dominant flagellum is responsible for collection and dissociation of a flagellar bundle that underlines E. coli’s motility pattern.

Introduction

Peritrichously flagellated bacteria, such as Escherichia coli, perform chemotaxis by modulating the state of flagellar motor rotation; when all the motors are in the counterclockwise (CCW) direction, the flagellar filaments form a coherent bundle that pushes the cell body forward, and when one or a few motors is in the clockwise (CW) direction, the bundle falls apart, and the cell body twiddles. The former is called a run and the latter a tumble. When swimming in a favored direction, run intervals are extended, but tumbling intervals are shortened. The result is a biased random walk that leads the bacterium toward the sources of attractants and away from repellents. However, how a collection of motors works together to coordinate flagellar bundle formation and dissociation is not well understood.

In an attempt to address this question, we take advantage of a newly discovered effect of bacterial self-trapping near solid surfaces (1). We found recently that a freely swimming E. coli bacterium can be trapped by its own swimming force because of a counterflow that the motion generates near the surface. In a run state, a trapped cell has its flagella perpendicular to the long axis of the cell body and is aligned with the normal of the surface. Rotation of the flagella in the CCW direction causes cell-body rotation in the CW direction. Reversal of one or more flagella causes the cell body to slow down and in the extreme case makes the cell body rotate in the opposite direction (1). Importantly, the angular velocity probability density function (PDF) P(Ω) of a trapped bacterium is bimodal and can be described by a sum of two Gaussians, with one centered at a high frequency Ω_H and the other at a low frequency Ω_L. This allows experimenters to assign the state of bacterial swimming, either a run or a tumble, based on cell-body rotation speed Ω(t) at time t (1) (see Materials and methods).

The phenomenon of cell-body rotation observed in self-trapped bacteria is akin to the classical rotation assay (2), in which a short fragment of a flagellar filament is physically attached to a surface by anti-flagellum antibodies. A comparison between the trapped bacteria and tethered bacteria showed that the trapping assay can be used to study chemotaxis, with the result that the two assays are qualitatively identical (1). Because self-trapped bacteria have intact flagella that can form a coherent bundle, they provide a unique experimental system for studying dynamic fluctuations of flagellar bundles. In this study, we investigate cell-body rotation of self-trapped bacteria in the absence of an external stimulus and compare the result with an analogous measurement of the cell-body rotation due to a single motor.

Materials and methods

Our experiment used two bacterial strains: RP437 is commonly used for bacterial chemotaxis and motility studies. It is considered the wild-type (WT) in this experiment. XLWU100 is derived from RP437 by deleting fliC using a P1 transduction method (1). The function of fliC is restored by a plasmid pFD313 expressing a mutant FliC that displays a strong affinity to glass. The “sticky” filaments, after shortened by shearing, adhere to a glass surface. We found that among many surface-bound bacteria, some display cell-body rotation akin to what is seen in the classical rotation assay.

The bacteria were grown according to the following protocol: an overnight culture was grown in tryptone broth (1% tryptone powder, 0.5% NaCl, and the antibiotic ampicillin at the concentration 50 μL/mL when needed) starting from a single colony, incubated at 33°C, and vigorously shaken at 200 rpm on a rotary shaker. The overnight culture was used to inoculate a fresh tryptone broth at 1:100 and incubated for 4 h at 33°C with shaking (3). The bacteria were washed twice in a motility buffer (10 mM KPO4, 0.1 mM ethylenediaminetetraacetic acid, 1 mM L-methionine, 67 mM NaCl, 10 mM sodium lactate) before measurement (4). Bacteria XLWU100 were sheared following the procedure described in (5). Briefly, the freshly grown bacteria were passed repeatedly through two 21-gauge syringe needles connected by a Tygon tube (Saint-Gobain, Malvern, PA) 5 cm long. The sheared cells were washed once more before measurement.

Our measurements were carried out on an inverted microscope (TE300; Nikon Instruments, Melville, NY) using a 100× phase contrast objective at room temperature (23°C). The observation chamber, depicted in Fig. 1 A, was custom made to examine bacterial motion in bulk fluid and near a planar surface. An untreated 25 mm round coverglass (#1.5) serves as the floor of the chamber. The wall of the chamber is a Delrin plastic ring that sandwiches a rubber gasket to the glass to create a sealed container. The top of the chamber is open to air, allowing the motility buffer to remain oxygenated.

Figure 1.

The experimental setup and methods. Shown in (A) is our observation chamber that is placed on an inverted microscope. The chamber is opened to air from top and filled with a thin layer of motility buffer (blue area) to enhance oxygenation. (B) and (C) display sample PDFs P(Ω) for single bacteria in the tethered and trapped assay, respectively. The inset in the whitespace of each plot displays a cartoon of a cell in the corresponding assay. For tethered cells (B), P(Ω) exhibits two well-separated peaks, and each can be fitted by a Gaussian function. The peaks of the distribution are nearly symmetrically located about Ω = 0. For trapped cells (C), the PDFs also display two peaks, and each can be described by a Gaussian function, as indicated by the blue and red lines. However, in this case there is an overlap between the two Gaussians. The two vertical lines specify the boundaries between the state of run (right peak centered at Ω_H) and the state of tumble (left peak centered at Ω_L). In Materials and methods, we discussed how these boundaries are set and how they are used to determine the transitions between the two rotation states. (D) and (E) display a typical time trace of the velocity Ω(t) (red line, right axis) superimposed over the corresponding the rotation state ω(t) (blue line, left axis) for a tethered cell (D) and a trapped cell (E). The horizontal line in (D) denotes Ω = 0, which was the threshold between ω = 1 and ω = 0. The horizontal red and green lines in (E) denote the boundaries between the states as in (C). In (D), the run state (ω = 1) contains quasiperiodic noise because of pitching of the cell body as it rotates. To see this figure in color, go online.

The chamber was prepared for measurement by cleaning the Delrin ring and gasket with 70% ethanol and placing a new coverglass. The entire chamber was then assembled and rinsed with motility buffer. A small volume (200 μL) of bacteria suspension was placed directly on the glass surface and allowed to sit at room temperature for 1–2 min before being rinsed several times with fresh motility buffer. This rinsing step does little to remove cells that are near the glass surface but greatly reduces the number of cells in the bulk fluid that would otherwise accumulate near the surface over the course of a measurement. The final volume of motility buffer left in the chamber after rinsing was ∼1.5 mL.

Individual rotating cells on the glass surface were found and recorded for several 12 s intervals. Recording was performed using a CCD camera (Model C9100-12; Hamamatsu, Hamamatsu City, Japan) operating at 50 Hz with a 5 ms exposure time. Each cell contributed an average of 17 trials, for a total time of ∼200 s. A total recording time of 3120 s for RP437 and 2916 s for XLWU100 was collected. Custom scripts written in MATLAB (The MathWorks, Natick, MA) extracted the angular velocity Ω(t) from the recordings and converted the velocity to the binary rotation state ω(t). Sample time traces ω(t), velocity traces Ω(t), and velocity PDFs P(Ω) are displayed in Fig. 1, B–E for both bacterial strains. Additional time traces for both bacterial strains can be found in Fig. S3.

For tethered XLWU100 cells, the rotation state was assigned as follows:

$$\omega \left(t_i\right)=\{\begin{array}{ll}1\hfill & \text{if }\text{Ω}\left(t_i\right)<0\hfill \\ 0\hfill & \text{if Ω}\left(t_i\right)\ge 0\hfill \end{array},$$ (1)

where Ω > 0 corresponds to CW rotation. Despite its simplicity, the above two-state algorithm captures, quantitatively, the identical information as the more complicated three-state algorithm detailed in Fig. S1, in which the third state characterizes brief “pauses” that were occasionally observed.

For trapped RP437 cells, ω reflects the state of a flagellar bundle rather than the rotation direction of a particular motor. Here, ω(t) was assigned using a two-threshold crossing algorithm:

$$\omega \left(t_i\right)=\{\begin{array}{ll}1\hfill & \text{if Ω}\left(t_i\right)\ge {\text{Ω}}_r,\hfill \\ 0\hfill & \text{if Ω}\left(t_i\right)\le {\text{Ω}}_t,\hfill \\ \omega \left(t_{i-1}\right)\hfill & {\text{if Ω}}_t<\text{Ω}\left(t_i\right)<{\text{Ω}}_r,\hfill \end{array}$$ (2)

where the subscripts r and t stand for run and tumble, respectively. The threshold values Ω_t and Ω_r for each cell are found as follows: the angular velocity PDF P(Ω) for the cell is first fitted by a bimodal distribution P_t(Ω) + P_r(Ω), where P_t(Ω) and P_r(Ω) are Gaussian functions. The fitting is delineated in Fig. 1 C, in which the purple line denotes the fit of the distribution and the red and blue lines are the individual Gaussians P_t(Ω) and P_r(Ω), respectively. The threshold value Ω_x, with x = r or t, on either side of the intersection of the two Gaussians is defined by P_x(Ω_x)/(P_t(Ω_x) + P_r(Ω_x)) = 0.75. The thresholds Ω_t and Ω_r are shown as vertical green and red lines, respectively, in Fig. 1 C. From the resulting binary traces, we extracted the durations (dwell times) of all CCW (1) and CW (0) events. Uncertainties associated with these assignments are discussed in SM. Events that bordered on a recording boundary were excluded. Note that even though the “run” state for trapped cells produces a CW cell-body rotation, we refer to these intervals as “CCW” because this is the motor rotation direction.

The algorithm in Eq. 2 could, in principle, add uncertainty to the duration of any event that began or ended with Ω between Ω_t and Ω_r, which could reduce the time resolution of our dwell-time PDFs. We analyzed all transitions and found that the uncertainty was on the order of our frame duration of 20 ms (see Supporting materials and methods, Section S2), which is unimportant for our analysis.

Results

Fig. 2 displays our measurement of the dwell-time PDFs P_Δ(Δ), where Δ denotes either CCW or CW interval durations. For tethered cells (XLWU100 in Fig. 2 A), P_Δ(Δ) decays monotonically for both CCW and CW intervals and is strongly nonexponential. The long-time decay, however, appears to follow a straight line on the semilog plot, indicating that asymptotically, P_Δ(Δ) may be approximated by an exponential function. We also noticed that the magnitudes of PDFs for the two intervals are nearly identical, which suggests that the motor spends about equal time spinning in CCW and CW directions. Specifically, the average CCW motor bias for the tethered cells (n = 14) was measured to be 64%. This is significantly lower than the observed run bias (>80%) typically seen in free-swimming cells (6). Such a difference can be explained by the fact that the rate of motor switching is strongly influenced by the load. As shown by Fahrner et al. (7), the average switching rates from CCW to CW (k⁺) and vice versa (k⁻) decrease with the load, and the dependence is much stronger for k⁻ than for k⁺. For XLWU100, cell-body rotation is driven by a single motor that significantly increases CW interval length, lowering the CCW bias, which is consistent with (7).

Figure 2.

Dwell-time PDFs for E. coli XLWU100 (A) and RP437 (B). For tethered bacteria (XLWU100), the dwell-time PDFs P_Δ(Δ) are similar for the CCW (red circles) and CW (orange squares). Both PDFs decay rapidly for small time intervals and have a broad tail, which is approximately exponential as delineated by the straight lines on the semilog plot. A total of 14 tethered cells were recorded, and their average CCW motor bias was 64%. For self-trapped bacteria (RP437) that have an intact flagellar bundle, there is a distinct difference in the PDFs for CCW (blue circles) and CW (green squares) intervals. Whereas for the CCW intervals, the PDF appears similar to the tethered cells in (A), the PDF for the CW intervals has a very different functional form, particularly for large time intervals. As delineated by the green solid line, this regime is not exponential but can be approximately mimicked by a Gaussian. A total of 13 trapped cells were used, and their average run bias was 65%. The insets of (A) and (B) are the survival probability for XLWU100 and RP437, respectively. In both cases, P(t) decays more slowly than P_Δ(Δ), indicating a different time dependence. To see this figure in color, go online.

The dwell-time PDFs presented in Fig. 2 B show that the switching behavior for swimming cells (RP437) is quite different than that for single motors (XLWU100 in Fig. 2 A). Although the CCW dwell-time distributions are similar for the two measurements, the CW is significantly different for long events (those greater than ∼0.3 s). In fact, the long-time trend for all curves except for the CW trapped cells appear exponential, as delineated by the straight lines in Fig. 2, A and B. The CW dwell time for trapped cells decays much more rapidly and can be fitted approximately by a Gaussian function, which is shown as a green line in Fig. 2 B. We discuss this difference and a plausible interpretation in the Discussion.

Although the above observation gives some clues about the underlying physics, the presentation using P_Δ(Δ) is not particularly informative or useful for analyzing and understanding the mechanisms involved in the transitions from a run to a tumble and vice versa. In our opinion, the quantity of biophysical significance is the transition rate k and how this rate changes with external conditions. The dwell-time distribution is related to this transition rate, but in a complicated way. An important clue for understanding our data is that motor switching kinetics clearly involve multiple timescales. The phenomenon therefore could be treated as a nonstationary Poisson process characterized by a time-dependent rate k(t). In such a process, the survival probability P(t) has a simpler mathematical form P(t) = exp(−I(t)) than the dwell-time PDF P_Δ(Δ) = k(Δ)exp(−I(Δ)), where I(t) is the integrated transition rate given by I(t) $\equiv {\int }_0^t$k(t′)dt′. Given the transition rate, P(t) and P_Δ(Δ) are related by a time derivative, P_Δ(Δ) = −∂P(t)/∂t|_Δ. In light of this observation, we seek to present our data using the survival probability P(t) instead. To do so, we numerically integrate the measured P_Δ(Δ) to obtain P(t), and the results are plotted in the insets of Fig. 2. An added benefit of converting to the survival probability is that P(t) now appears much smoother than the original data.

To gain more information about the switching mechanism, we next proceed to extract the integrated transition rate I(t) from the survival probability using I(t) = −lnP(t). This quantity is related to k(t) by an integration and therefore can tell us how k(t) behaves as a function of time t. The results are plotted in Fig. 3, A and B for XLWU100 and RP437, respectively. For a stationary Poisson process, I(t) is linear in time, but none of the curves in Fig. 3, A or B behave in this way. The general trend is that k(t), which is proportional to the slope of I(t), is quite large for small times, then decreases, and finally reaches some constant value for large times. Moreover, for both bacteria, CW transition rates (orange and green squares) are higher than the CCW rates (red and blue circles), which is consistent with the normal swimming behavior of E. coli cells. We also note that whereas the integrated switching rates of tethered cells in Fig. 3 A are similar for the two motor directions, both in their functional forms and magnitudes, I(t) for WT bacteria (Fig. 3 B) are considerably different for the two directions. Specifically, I(t), and therefore k(t), for CW (or tumble) state is significantly higher than its counterpart for CCW (or run) state, suggesting that switching dynamics for cell-body rotation driven by a single motor and by a flagellar bundle are quite different. We wish to explain this difference in the Discussion below.

Figure 3.

Integrated switching rates I(t) for E. coli XLWU100 and RP437. All measurements using tethered XLWU100 (A) and self-trapped RP437 (B) show that the integrated switching rates I(t) are not linear in time, suggesting a nonstationary Poisson process. Moreover, because I(t) = ${\int }_{\delta t}^t$k(t′)dt′ and CCW bias Φ ≥ 1/2, it is expected that the CW switching rates (orange and green squares) are larger than the CCW switching rate (red and blue circles). The solid lines in the figure are fits using k(t) = k₀(1 + t₀/t), with k₀ and t₀ being the adjustable parameters (see Discussion in the main text). The inset in (B) displays the short-time data in more detail. For a quantitative comparison, we plot the CCW (run) intervals and CW (tumble) intervals for the two bacteria (XLWU100 and RP437) in (C) and (D), respectively. For the CCW intervals (C), we found that the integrated switching rates I(t) for a single motor and for the flagellar bundle are nearly identical. On the other hand, for the CW (tumble) intervals shown in (D), the integrated switching rate I(t) for the bundle (green squares) is very different from a single motor (orange squares). In short times, I(t) is lower than that of a single motor, but for t ≥ 0.5 s, I(t) rises sharply. The inset in (D) depicts the short-time behavior of RP437 (green squares) and XLWU100 (orange squares); here, I(t) for RP437 has been multiplied by a factor of 1.7 to make the two curves overlap with each other. The similar functional forms of the two curves in this region suggest the underlying physics are the same, i.e., the transition is governed by the single motor statistics. To see this figure in color, go online.

Discussion

Based on the above qualitative analysis, we conjectured the following time-dependent switching rate: k(t) = k₀(1 + t₀/t). This function is singular for t → 0⁺ and becomes a constant in long times t ≫ t₀, where t₀ is a crossover time. This gives the integrated switching rate,

$$I\left(t\right)\left(\equiv \underset{\delta t}{\overset{t}{\int }}k\left(t^{\prime }\right)d{t}^{\prime }\right)=k_0\left(\left(t-\delta t\right)+t_0\text{ln}\left(t/\delta t\right)\right),$$ (3)

where δt is a cutoff time that regulates the singularity at t = 0. As shown by the solid lines in Fig. 3, A and B, this simple mathematical form describes our experimental data reasonably well particularly for both states of tethered cells (Fig. 3 A) and for CCW rotation of WT cells (Fig. 3 B). In those cases, the typical long-time transition rate k₀ is ∼0.5 s⁻¹, and the crossover time t₀ is about a second. In all cases, the cutoff δt is small, i.e., comparable to the inverse video frame rate, as it should be. This simple mathematical form, however, only provides a marginally acceptable fit to CW rotation of WT cells, particularly for short times, as shown in the inset of Fig. 3 B. As we shall discuss below, the reason for this is because a tumble for WT cells is a mixture of different states governed by additional biophysical processes. For convenience, the numerical values for k₀, t₀, and δt are tabulated in Table 1. Aside from the CW state of WT cells, the reasonably good agreement between the proposed transition rate k(t) and the observation suggests that an incipient state is more likely to “fail” in early times than in late times. This provides a useful clue about how the flagellar motor switch may operate, and it should be a touchstone for any mathematical model that attempts to describe the E. coli flagellar motor switch.

Table 1.

Adjustable parameters used in fitting I(t)

Strain	XLWU100			RP437
Parameter	k0 (s−1)	t0 (s)	δt (s)	k0 (s−1)	t0 (s)	δt (s)
CCW	0.54 ± 0.08	0.9 ± 0.2	0.04 ± 0.02	0.40 ± 0.05	1.7 ± 0.4	0.06 ± 0.02
CW	0.58 ± 0.08	1.0 ± 0.2	0.02 ± 0.01	2.29 ± 0.3	0.2 ± 0.1	0.05 ± 0.02

I(t) fits displayed in Fig. 3.

We next compare in Fig. 3, C and D the integrated switching rates I(t) between the two bacteria. For CCW intervals (Fig. 3 C), we found that I(t) is nearly identical for RP437 and XLWU100, indicating that the transition rate k(t) of the bundle is very similar to that of a single motor. We posit that this can happen if while in the CCW state or a run, the flagellar bundle is a highly coherent entity driven by a single dominant filament. Perhaps because of its special location on a cell body, this dominant flagellum determines the cell-body rotation, which in turn enforces the cohesiveness of the bundle. This hypothetical model is depicted in Fig. 4. In this model, the torque on the cell body generated by the bundle is largely unaffected by the switching of nondominant flagella. Also, because the switching rate k is a function of the load or the torque on the motor (7), it follows that the torque exerted on the dominant flagellum in a CCW state is similar to that of a single tethered motor.

Figure 4.

Transitions between rotation states aided by a dominant flagellum. In our hypothetical model, a dominant (red) flagellum is “strategically” positioned so that the cell-body rotation axis (dash-dot vertical line) is defined. Other flagella are secondary because of either their unfavorable positions, such as the green one, or being shorter, such as the blue and purple ones. In the run state, the flagellar bundle is maintained by the CW cell-body rotation as indicated by the red arrow. The transition to a tumble is initiated by a change of rotation direction of the dominant (red) flagellum from CCW to CW. This causes the cell body to slow down or even rotate in the opposite direction, resulting in the dissociation of the bundle. To account for our observation displayed in Fig. 3D, the transition from a tumble to a run state undergoes a transient period in which the dominant flagellum drives the cell body to rotate in the CW direction, causing the flagella to coalesce into a bundle. Here, a mini (tumble) initiated by a brief reversal, from CCW to CW rotation, of the dominant flagellum is not shown. To see this figure in color, go online.

The situation is different for the CW (tumble) state (Fig. 3 D), in which it is seen that whereas I(t) is smaller for the WT cells than XLWU100 for t < 0.5 s, it is significantly greater than that of XLWU100 for t > 0.5 s. Inspection of the data (green squares) in Fig. 3 D reveals two different tumbling behaviors for WT cells; one occurs in short times and the other in long times. For short times t < 0.2 s, I(t) for both bacteria has the same functional form but a different amplitude, which is demonstrated in the inset by a rescaling of the curve for RP437 that makes the two curves collapse. This suggests that very brief (<0.2 s) CW events of the dominant filament can slightly perturb the flagella bundle, causing a change in the cell-body rotation speed that is large enough to be detected in our measurement. We refer to these short perturbations in the run state as “mini-tumbles.” Our measurement also shows that the magnitude of the switching rate in this episode is suppressed compared to a tethered motor, suggesting that the load on this single flagellum is higher than that of a tethered motor (7). This is plausible because the motor would be turning the filament CW, i.e., against the direction of the rest of the flagella in the bundle. It should be mentioned that a short tumble due to only a single motor switching from CCW to CW has been previous proposed and observed using a novel fluorescence labeling technique (8). Herein, we show that such a behavior can be measured in a quantitative manner.

We now turn our attention to the long-time behavior of the CW state for WT cells. Different from the minis, a long tumble is more persistent and yet characterized by a greater transition rate. We posit that this state corresponds to a higher degree of “disorder” in the flagellar bundle; in the extreme case, one can assume that all motors are in the CW direction as depicted in Fig. 4. To make a transition to form a coherent bundle, various scenarios are feasible: 1) Suppose that the transition is a serial process, i.e., one flagellum after another makes a transition from CW to CCW, which could be the case that each motor is triggered by a CheY-p concentration wave inside a cell as observed in (9). In such a scenario, the rate of transition to a coherent bundle is dictated by the motor with the slowest transition rate, and one expects that the overall transition rate will be even smaller. But this is not observed in our experiment. Alternatively, 2) suppose that the transition is a parallel process in which individual flagella are competing with each other to restore a coherent bundle by cell-body rotation (10). In such a case, the overall transition is dominated by the one flagellum that has the greatest effect on cell-body rotation, causing the body to displace over a large enough angle, say θ_C. This is consistent with our observation and suggests again the existence of a dominant flagellum in a bacterium. The mean transition time is then given by $\overline{t}={\theta }_C/\dot{\theta }$, where $\dot{\theta }$ is the mean angular speed of the cell body in a run state. Considering this transition as a first-passage problem (11), $\overline{t}$ is expected to be smeared over a range of time Δt because of noise in the rotation speed. Thus, the transition time t has a distribution that peaks at $\overline{t}$. For simplicity, we tried a Gaussian distribution centered at $\overline{t}$, but this scheme did not work well. Instead, we found that the hyperbolic secant function given by p(t) ∝ sech$\left(\frac{t-\overline{t}}{\text{Δ}t}\right)$ provided a reasonable fit. Unlike the Gaussian function, this function decays exponentially for large and small times. Using k(t) = k₁p(t), where k₁ sets an overall scale of the transition rate, we found that the long-time behavior of the CW interval of RP437 can be well described by this form. A fitting procedure yields k₁ $\simeq$ 2.3 s⁻¹, $\overline{t}\simeq$ 0.7 s, and Δt $\simeq$ 0.56 s. The result is displayed by the solid green line in Fig. 3 D.

Conclusions

In conclusion, E. coli cell-body rotation driven by a single tethered flagellar motor and by a flagellar bundle is investigated. The study allows us to address a long-standing question concerning the formation of a flagellar bundle that initiates a run and bundle dissociation that triggers a tumble. We found that for a tethered motor (XLWU100), the transition rate for cell-body rotation, either CW or CCW, can be described to a good approximation by the simple equation k(t) = k₀(1 + t₀/t). Remarkably, for WT cells (RP437), the transition rate k(t) for the CCW motor rotation, or a run, can also be described by the same equation with a similar magnitude. This suggests that the flagellar bundle in this state is highly coherent and maintained by a single dominant flagellum. On the other hand, a tumble in WT RP437 cells is characterized by two distinctive states: one for short times with the transition rate that scales as k(t) ∝ 1/t, which is similar to the short-time behavior of a single motor, and one for long times with transition rate peaked at $\overline{t}\simeq$ 0.7 s.

Correlations between rotation of different flagellar motors in E. coli have been systematically studied (9,12, 13, 14). In the more recent work (14), the investigators found that two motors separated by a micron or so are correlated, with the correlation peaked at 50%. The effect was attributed to wave-like CheY-p concentration fluctuations inside a cell (9). This is a significant finding because it suggests a mechanism for motor synchronization. In a steady state without stimulation such as in our study, this type of synchronization, however, is not effective because for all motors, 4 or 5 of them, to be synchronized, the successful rate (∝ 0.5^3–4) is only 12% or less. The significantly elevated transition rate of 2.3 s⁻¹ for the long time seen in our experiment is also inconsistent with transitions taking sequential steps for individual motors, i.e., switching from CW to CCW one motor after another (9,14). However, our observation is consistent with a parallel process, i.e., a winner-take-all type of process, which again suggests the existence of a dominant flagellum that facilitates coalescence of a bundle via cell-body rotation as illustrated in Fig. 4. When cells are stimulated, on the other hand, higher correlation can be achieved, and to reach a successful switch with 50% probability, the correlation between a pair of motors must reach 80% or higher. Taken together, we believed that to synchronize all motors in a steady state, mechanisms other than the sequential process must be involved as pointed out in (15), and what we found in this work, namely synchronization via a dominant flagellum, is an attractive candidate. In the presence of external stimuli, it could be that both mechanisms are at work for efficient chemotaxis.

There has been some discussion about mechanisms of a functional motor switch (16, 17, 18, 19). The important ingredients are strong coupling between protomers FliM that comprise the motor switch box and the existence of hysteresis in the motor switch. However, it remains unclear whether these ingredients could predict the transition rates, k(t), observed in this experiment. This part of the analysis is highly desirable but is beyond the scope of this discussion. We wish to address this issue in a separate publication.

Author contributions

C.N.D. and X.-L.W. designed research, performed research, analyzed data, and wrote the manuscript.

Editor: Ryota IINO.