Evaluation of the Duty Ratio of the Bacterial Flagellar Motor by Dynamic Load Control

Abstract

The bacterial flagellar motor is one of the most complex and sophisticated nanomachineries in nature. A duty ratio D is a fraction of time that the stator and the rotor interact and is a fundamental property to characterize the motor but remains to be determined. It is known that the stator units of the motor bind to and dissociate from the motor dynamically to control the motor torque depending on the load on the motor. At low load, at which the kinetics such as proton translocation speed limits the rotation rate, the dependency of the rotation rate on the number of stator units N implies D: the dependency becomes larger for smaller D. Contradicting observations supporting both the small and large D have been reported. A dilemma is that it is difficult to explore a broad range of N at low load because the stator units easily dissociate, and N is limited to one or two at vanishing load. Here, we develop an electrorotation method to dynamically control the load on the flagellar motor of Salmonella with a calibrated magnitude of the torque. By instantly reducing the load for keeping N high, we observed that the speed at low load depends on N, implying a small duty ratio. We recovered the torque-speed curves of individual motors and evaluated the duty ratio to be 0.14 ± 0.04 from the correlation between the torque at high load and the rotation rate at low load.

Introduction

The bacterial flagellar motor is a large complex of proteins with a dimension of ∼50 nm (1, 2, 3). The motor rotates a flagellar filament longer than 5 μm at several hundred Hz unidirectionally and propels the cell body. For a tactic run-and-tumble response, the motor can reverse the rotation with a frequency controlled by an intracellular signal from the sensory protein on the membrane.

The motor implements an automatic torque control by a dynamic stator structure (4) (Fig. 1 a). The number of the stator units (N) changes depending on ion motive force (5), coupling ion (6), and load (5, 7). For example, when a load on the motor increased, more stator units are incorporated into the motor, and N increases.

Figure 1.

(a) The stator of the bacterium flagellar motor is a dynamic structure. Its stator units bind to and dissociate from the motor to control the motor torque. The number of stator units N depends on the load on the motor. (b) At high load, the rotation rate ω as well as the motor torque T_m changes with N. On the other hand, at low load, it has not yet been concluded whether they vary with N. We can use this dependency of ω and T_m on N to evaluate the duty ratio. Large dependency implies a small duty ratio, and small dependency implies a large duty ratio. (c) The duty ratio is also inferred from the reversed-resurrection experiment. As the load on the motor vanishes instantly, N decreases, and then the rotation rate may decrease (small duty ratio) or may keep a similar value (large duty ratio). The motor torque is calculated as T_m(ω) = γω − T_assist, where T_assist is the external assisting torque by the electrorotation method. Thus, a constant assisting torque by the electrorotation translates the equal-load line parallel. To see this figure in color, go online.

The duty ratio D is a fraction of time that the stator and the rotor interact and is a fundamental property to characterize the motor. D is inferred from the dependency of the rotation rate, ω, on N under a low-load condition (Fig. 1 b). At high load, ω depends on N independent of D (8, 9, 10). This is because the torque generation against load is the rate-limiting step at high load. The motor generates larger torque with more stator units, resulting in a faster rotation. On the other hand, under a low-load condition, the kinetics, such as a proton translocation speed, limits ω. If D is low, ω is expected to increase with N because of an increased chance of the stator-rotor interaction. If D is high, ω is less affected by N. However, contradicting observations were reported; some experiments implied that ω at vanishing load, referred to as zero-load speed ω₀ below, is independent of N (8, 11, 12), and others implied that ω₀ increases with N (9, 10).

For determining the dependency, it is necessary to measure ω₀ in a broad range of N. A dilemma is that the stator units readily dissociate from the motor at low load. In previous experiments, N was limited to one or two.

Here, for evaluating D, we used a calibrated electrorotation method to measure ω₀ of a tethered Salmonella enterica serovar Typhimurium cell while keeping N high. This was achieved by quickly reducing the load on the motor by dynamic control of the assisting electrorotation torque. If the load changes sufficiently faster than the stator dissociation dynamics, we could measure ω₀ while keeping N high.

We developed the calibration method of the electrorotation established previously (13, 14, 15, 16) for applying it to the flagellar motor of the tethered cell. It provides not only the torque magnitude of the electrorotation torque but also the frictional coefficient felt by the motor in situ at the same time without any knowledge about the geometry of the probe-motor complex. This implements methodology to recover the precise torque characteristics of individual motors. Before going to the results, we briefly explain the calibration method.

Calibrated electrorotation method

The electrorotation method imposes an external torque to a microscopic dielectric object by inducing a high-frequency alternative current electric field. The method has been used to apply a dynamically controlled load on the motor of tethered bacterial cells (17, 18, 19, 20, 21, 22, 23) (Fig. 2). However, despite its potential advantages, the method is not widely used, mainly because of a lack of the torque calibration method. We previously established the calibration method and successfully applied it to an F₁-ATPase motor and revealed its torque characteristics (13, 14, 15, 16). Here, we further develop the method to reduce the experimental noise and apply it to the bacterial flagellar motor.

Figure 2.

Calibrated electrorotation method imposes torque with a calibrated magnitude on a tethered cell using high-frequency electric field. The method provides a dynamic torque control to measure the response of the motor, which enables us to measure the TS curves of individual motors. To see this figure in color, go online.

The electrorotation torque is proportional to the square of the applied voltage amplitude V₀. Therefore, the load on a motor can be dynamically and continuously controlled by modulating V₀. Because inertia is negligible in microscopic systems, the torque generated by the motor T_m(ω) at a rotation rate ω, the viscous torque loaded on the motor, the electrorotation torque, and the thermally fluctuating torque are balanced. The thermal fluctuation is negligible when averaged in a sufficiently long time. Thus, the motor torque becomes

$$T_{\text{m}}\left(\omega \right)=\gamma \omega -\alpha V_{\text{sq}}.$$ (1)

Here, γ is the rotational frictional coefficient and contains the contributions from the internal friction of the motor and the friction of the cell body and flagellar filament against the fluid. αV_sq is an external torque, and V_sq is $V_0^2$ or $-V_0^2$ depending on the sign of the phase shift of the applied alternative current voltages. The coefficient α depends on multiple factors, including the cell shape, the chamber geometry, and the dielectric properties of the cell body and the buffer. Therefore, it is not possible to determine α a priori. This has hampered the use of the electrorotation method to measure the motor torque. In previous studies (18, 19, 20, 21, 22, 23), “relative” external torque instead of the exact torque magnitude has been used to plot the torque characteristic.

The calibration is based on the fluctuation-response relation (FRR), which relates the thermal fluctuation to the response to external perturbation under a condition close to the equilibrium (24, 25). Because the motor’s rotation is limited to a low-frequency region and settled at equilibrium at high frequencies, the calibration is possible by comparing the rotational fluctuation and response of rotation against external perturbation at the high frequency. This relation connects the fluctuation $\tilde{C}\left(f\right)$ and the response to a small perturbation ${\tilde{R}}^{\prime }\left(f\right)$:

$$\tilde{C}\left(f\right)=2k_{\text{B}}T{\tilde{R}}^{\prime }\left(f\right)\text{.}$$ (2)

Here, $\tilde{C}\left(f\right)$ is the Fourier transform of the autocorrelation function of the rotational velocity $C\left(t\right)=\langle \omega \left(\tau +t\right)\omega \left(\tau \right)\rangle$ at a frequency f. For an equilibrium Brownian movement, γ is obtained by $\tilde{C}\left(f\right)=k_{\text{B}}T/\gamma$. On the other hand, ${\tilde{R}}^{\prime }\left(f\right)$ is the real part of the frequency response of ω(t), measured by applying a small sinusoidal torque $\tilde{R}\left(f\right)=w_{\text{r}}{e}^{i\varphi }/N_0$, where f and N₀ = αV_{sq, 0} are the frequency and amplitude of the sinusoidal torque. w_r and ϕ are the amplitude and phase of the velocity response. By measuring the fluctuation, $\tilde{C}\left(f_0\right)$, and the response, w_r and ϕ, of the rotation rate at sufficiently high frequency f₀, we can determine α₀ and γ using (Eq. 2):

$$\alpha =\frac{2w_{\text{r}}\text{cos}\varphi }{V_{\text{sq},0}}\gamma ,\gamma =\frac{k_{\text{B}}T}{\tilde{C}\left(f_0\right)}.$$ (3)

However, this calibration may suffer by the interaction between the cell body and the glass surface, which induces a periodic potential on the motor mechanics and yields peaks in the fluctuation spectra. We developed a method to eliminate this effect by using an extended FRR around a local mean velocity developed by Speck and Seifert (26). See the Materials and Methods for details.

Materials and Methods

Electrorotation method

Salmonella strain YSC2123, which lacks motA, motB, cheY, fimA, and fliC (204–292), was transformed with a plasmid encoding wild-type motA/motB (27) (referred to as a wild type) or not (statorless mutant).

Cells were grown in L-broth containing 100 μg/mL ampicillin for 5 h at 30°C with shaking; 0.0002, 0.002, or 0.2% arabinose was added and then incubated for 30 min at 30°C with shaking for protein expression (except the statorless mutant experiments in Fig. 5 a). L-broth was prepared as described previously (28). After replacement of L-broth with the observation buffer (10 mM 3-morpholinopropanesulfonic acid and 10 mM KCl adjusted to pH 7.0 with KOH), we partially sheared the sticky flagella filaments by passing the bacterial solution through a 25G needle 70 times.

Figure 5.

TS curves. (a) TS curves of statorless (ΔMotA/B) motors are shown. The TS curves of 19 cycles of eight motors are shown. The black circle is the average in the windows of 30 Hz. The error bars indicate the SD. (b) 76 TS curves of 46 wild-type motors were superposed. Each curve corresponds to a single ramp of a single motor, as in the case of Fig. 4a. The color indicates the torque at high load. Red color indicates a high torque. Inset: Averaged TS curves in groups divided according to the torque at high load T_{m, 0}. T_{m, 0} is calculated as the average of T_m in the first 0.4 s of the ramp. The torque was divided in windows of 120 pN nm. (c) The histogram of T_{m, 0} is plotted separately for different arabinose concentrations, which control the expression level of the stator units. To see this figure in color, go online.

The observation chamber has quadrupolar electrodes with a spacing of 50 μm on the surface of the bottom glass slide (Fig. 2). After a 10-μL droplet of the cell solution was placed at the center of the electrodes, a coverslip (Matsunami, Osaka, Japan) was placed with thin double-sided adhesive tape (10-μm thickness; Teraoka, Tokyo, Japan) as a spacer. Only the top of the double-sided adhesive tape was coated by a high-vacuum grease (Shin-Etsu, Tokyo, Japan) so that the top coverslip could be moved. This enabled us to move the cells tethered on the top glass slip into the center of the electrodes. After 3 min, the solution was replaced with 20 μL of blocking buffer (observation buffer containing 50 mg/mL Perfect Block (MobiTec, Göttingen, Germany)) was flowed into the chamber. Perfect Block serves as a blocking agent to suppress the interaction between the cell body and the glass surface. Then, 40 μL of the observation buffer was flowed into the chamber to wash Perfect Block free.

We observed the rotation of a tethered cell at room temperature (24°C) on a phase-contrast upright microscope (Olympus BX51WI; Tokyo, Japan) with a 60× objective lens (Olympus, NA = 1.42) at 4000 Hz using a high-speed complementary metal-oxide semiconductor camera (Basler, Ahrensburg, Germany), a high-intensity light-emitting diode (623 nm, 4.8 W; Thorlabs, Newton, NJ) for illumination, and a laboratory-made capturing software developed on LabVIEW 2014 (National Instruments, Austin, TX). The angular position of the cellular body was analyzed by an algorithm based on a principal component analysis of the cell image. This method reduces the instrumental noise compared to the centroid-based method.

A 5-MHz sinusoidal voltage with a phase shift of π/2 was induced on the four electrodes. The signal was generated by a function generator (nf, Japan) controlled by PC and divided by 180° phase distributors (Thamway, Shizuoka, Japan). They were amplified by four amplifiers (Analog Devices, Norwood, MA) and loaded on the electrodes. This generates an electric field rotating at 5 MHz in the center of the electrodes and induces a dipole moment rotating at 5 MHz on the cell body. Because there is a phase delay between the electric field and dipole moment, the cell body is subjected to a constant torque. The magnitude of torque is proportional to the square of the voltages’ amplitude V₀ and the volume of the cell body. We modulated V₀ by a signal generated by the multifunction board (National Instruments) equipped on PC. The camera and amplitude signal were synchronized at a time difference less than 1 μs.

Angular dependency

The periodic perturbation, caused by the interaction between the cell body and the glass surface and also the imperfect voltage balance, causes a periodic variation of the rotation rate with the frequencies equal to the integral multiple of the mean rotation rate. These can produce nonequilibrium fluctuations at more than 1000 Hz and affect the above torque calibration. These nonequilibrium fluctuations are superposed on the equilibrium Brownian spectrum and impede the torque calibration. We suppressed the nonequilibrium fluctuations dramatically by using a modified FRR derived by Speck and Seifert (25, 26): ${\tilde{C}}_{\nu }\left(f\right)=2k_{\text{B}}T{{\tilde{R}}^{\prime }}_{\nu }\left(f\right)$. Here, ${\tilde{C}}_{\nu }\left(f\right)$ and ${{\tilde{R}}^{\prime }}_{\nu }\left(f\right)$ are the fluctuation and response function of the rotation rate around the local mean velocity defined as ${\omega }_{\nu }\left(t\right)\equiv \omega \left(t\right)-\nu \left(\theta \left(t\right)\right)$. $\nu \left({\theta }_0\right)={\int }_0^{T}dt\delta \left(\theta \left(t\right)-{\theta }_0\right)\omega \left(t\right)$ is the local mean velocity at an angular position θ₀. δ(x) is the Dirac’s δ function. The idea behind this is that the FRR is restored for the rotation around the local mean velocity. The nonequilibrium rotations are embedded in the local mean velocity (25). Therefore, we can estimate γ and α₀ by eliminating the noise by any periodic potential due to, for example, the interaction with the glass surface. Throughout this work, we used ${\tilde{C}}_{\nu }\left(f\right)$ and ${{\tilde{R}}^{\prime }}_{\nu }\left(f\right)$ instead of $\tilde{C}\left(f\right)$ and ${\tilde{R}}^{\prime }\left(f\right)$ in (3) to eliminate the effect of the periodic potential for calculating γ and α₀.

Torque-speed curve

The experiments and analysis proceeded as follows. We applied an external torque varied at a constant ramp rate on the tethered cell superposed by a 1000-Hz small sinusoidal torque (Fig. 1 b). γ and α had a dependency on the electrorotation strength, possibly due to the pulling force toward the electric field. Therefore, we divided the trajectory into windows with a length of 512 frames with 128-frame shifts and calculated γ and α in each window. We averaged ${\tilde{C}}_{\nu }\left(f\right)$ in the ranges close to 1000 Hz, [700 Hz, 900 Hz] and [1100 Hz, 1300 Hz], to obtain γ. α was calculated at 1000 Hz by a discrete Fourier transform of the rotational trajectory. γ(t) and α(t) were smoothed by a linear fitting, and T_m was recovered by (1).

Results

Motor torque under low load

The motor of the Salmonella cell used here rotates only in the counterclockwise direction. For exploring the low-load region, we imposed an external torque that assists the rotation in the counterclockwise direction. A constant assisting torque was applied for 30 s and then switched off (Fig. 3 a). The magnitude of the torque was roughly tuned so that the load on the motor is close to zero by canceling the viscous load. Eq. 1 indicates that such a constant assisting torque shifts the equal-load line (Fig. 1 c, bottom).

Figure 3.

Response of the motor rotation to an assisting torque applied by the calibrated electrorotation method. (a) The temporal profile of the rotation rate is shown. The rotation rate increased steeply by an assisting torque (indicated by cyan) and then decreased in a stepwise manner, presumably because of the dissociation of the stator units. A steep drop of ω at around 10 s might be caused by a nonspecific interaction between the cell body and the glass surface. (b) The motor torque T_m profile calculated by Eq. 1 is shown. The horizontal lines are drawn as a guide for the eye with a spacing of 90 pN nm (left) or 120 pN nm (right). Inset: Load defined as T_m/ω. To see this figure in color, go online.

We observed a steep rise of the rotation speed by the assisting torque and then a stepwise decrease (Fig. 3 a). The motor torque T_m calculated by Eq. 1 is shown in Fig. 3 b. Here, we used averaged values of γ and α in the periods before, during, and after the assist, respectively. The load on the motor was quantified as T_m/ω, which vanished during the assist (Fig. 3 b, inset).

During the assist, we observed a stepwise decrease in ω and T_m. After we turned off the assisting torque, the rotation stopped for a while. Then, we observed recovery of the rotation and succeeding stepwise increase in the rotation speed. This recovery phenomenon is supposed to be the so-called resurrection process, in which the stator units bind to the motor (4, 5, 6, 7, 8, 29, 30). On the other hand, the stepwise decrease in ω at a vanishing load is thought to be the dissociation of the stator units from the motor. These results suggest that the rotation speed at the vanishing load depends on the number of stator units N (Fig. 1 c), implying a small duty ratio. The negative torque observed during the assist is not expected in the large duty ratio (Fig. 1 c) and also supports a small duty ratio.

We observed a steep drop of T_m at the beginning of the assist. This is caused mainly by the reduction of the torque generated by each stator unit because of the load change. This is supported by the observation that the step size of the speed during the assist is smaller than that during the resurrection process. We do not deny the possibility that N also decreased instantly at the beginning of the assist.

Figure S1 shows the replicates of Fig. 3. We see that the rotation speed recovered to the speed level similar to that before the assist. We can estimate N from such resurrection traces by counting the number of steps, which should be theoretically equal to the number of speed drops during the assist. However, the statistical comparison between N and the number of drops during the assist was not possible in these experiments. This is because it is often difficult to resolve the steps during the assist because of the smaller step size, and we cannot resolve the drops at the instance we started to induce torque where we see a large torque drop due to the load change.

Torque-speed curve

For evaluating the value of the duty ratio, we measured the torque-speed (TS) curves of the motors under an assisting torque. We repeated a cycle consisting of a 7-s ramp from 0 to 300 V² and 3-s intervals alternatively (Fig. 4, b and c). The torque was modulated with a 1000-Hz sinusoidal torque with an amplitude of 10 V² for evaluating γ and α in situ, with which we can calculate the motor torque by Eq. 1.

Figure 4.

A typical TS curve of a wild-type motor. (a) The motor torque T_m(ω)calculated by Eq. 1 from the trajectory corresponding to a single ramp of a single motor is shown. (b) Ramp cycles of V_sq(t) are shown. We repeated a cycle of a 7-s ramp from 0 to 300 V² and a 3-s interval. A ramp in a single cycle is used to recover a TS curve. A 1-kHz sinusoidal perturbation with the amplitude of 10 V² is superposed for the torque calibration. (c) Rotational rate ω(t) during cycles is shown. (d) The power spectra of the rotational rate C_v(f) for 512-frame windows are shown. A steep peak (magnified in the right panel) corresponds to the response to the 1-kHz sinusoidal torque. (e) The frictional coefficient γ calculated from C_v(f) with the 512-frame windows with a 32-frame shift is shown. (f) The calibration coefficient α is shown. The solid curves in (e) and (f) are linear-fitting curves. To see this figure in color, go online.

The ramp duration of 7 s limits the duration under low load to a few seconds, which is shorter than the dynamics of the stator-unit dissociation (Fig. 3). We expect that the dissociation of the stator units during the ramp is significantly suppressed. However, we still observed a sudden speed change during the ramp for some curves, implying a binding or dissociation of the stator units (Fig. S2). For selecting the TS curves in which N is constant during the ramp, we chose the TS curves with only a slight change in the rotation speed during the ramp. The criterion is that the change in the rotation speed during 1 s before and after the ramp is less than 1.5 Hz. The TS curves containing a distinct up-and-down change, which is supposedly caused by a series of binding and dissociation of a stator unit, were also excluded from the analysis.

A typical TS curve is shown in Fig. 4 a. The TS curve reached the zero-load state, which is not reachable by a viscous load. In the most experiments, γ was smaller at higher V_sq (Fig. 4 e). This is because the electric field possibly induces not only a rotational torque but also a dielectric electrophoresis force that pulls the cell apart from the glass surface toward the electrodes. This reduces γ because of the smaller rotation radius and less surface effect (31). The fluctuation patterns of γ and α are similar because γ is multiplied to calculate α (Eq. 3).

For validating the experimental procedure and the analysis, we applied the method to a statorless motor. This mutant motor lacks the stator units MotA and MotB and does not generate torque. However, it still has a rotor and flagellar filament and exhibits a rotational Brownian motion. The fluctuation is not free but confined around a certain angular position. The confinement potential is weak, and the motor can be readily rotated under an external torque. We found that the motor torque was zero within the error (Fig. 5 a), validating our method. The SDs were typically less than 100 pN nm/rad in a broad range of ω, which indicates the accuracy of the method. The deviation was larger for higher external torque. This is because, with higher external torque, T_m is obtained by subtracting a large value of αV_sq from a large value of γω (1), causing a large statistical error. A small bump was observed around $\omega \simeq 50$ Hz. This was caused by the above-described confinement potential specific to the statorless mutant and does not affect the torque measurement of the wild-type motors with stator units.

Evaluation of duty ratio

For evaluating the duty ratio, we compared the TS curves of multiple motors. In Fig. 5 b, 76 TS curves of 46 wild-type motors are superposed. We used cells with different expression levels of the stator units for sampling a broad range of N by controlling an arabinose concentration (Fig. 5 c; see Materials and Methods). We observed a broad distribution of the motor torque at high load T_{m, 0} calculated by averaging T_m in the first 0.4 s of the ramp and a tendency for T_{m, 0} to become smaller with a lower arabinose concentration. This means that we could sample a broad range of N because T_{m, 0} ∝ N is expected (8).

The curve shapes were similar among the TS curves, producing a broad distribution of the zero-load speed ω₀, supporting a small duty ratio D. For characterizing the TS curves, we divided them into groups according to the value of T_{m, 0} (Fig. 5 b, inset). Note that we did not specify N in this study, and therefore there is no one-to-one correspondence between the group and N. We found that ω₀ averaged in each group had significantly distinct values.

We see a clear positive correlation between ω₀ and T_m, 0 (Fig. 6). A simple model neglecting the interactions between the stators (12) predicts a relation

$${\omega }_0={\omega }_{0,\text{max}}\left[1-{\left(1-D\right)}^N\right].$$ (4)

Figure 6.

The correlation between the zero-load speed ω₀ and the motor torque at high load T_{m, 0}. See Fig. 5b for the definition of T_{m, 0}. A theoretical curve ${\omega }_0={\omega }_{0,\text{max}}\left[1-{\left(1-D\right)}^{T_{\text{m},0}/s}\right]$ was fitted (solid curve) with fitting parameters, the maximal speed ω_{0, max}, and the duty ratio D (12). s is a torque generated by a single stator at high load. We roughly estimated s from the resurrection trace (Fig. 3b). For s = 130 pN nm, the fitted values are ω_{0, max} = 306 ± 54 Hz and D = 0.14 ± 0.04. To see this figure in color, go online.

ω_{0, max} is the maximal rotation rate at vanishing load when N is sufficiently large. We evaluated D based on (4) assuming a proportional relation T_{m, 0} = Ns. The proportional coefficient s corresponds to a torque generated by a single stator at high load. A rough estimation from the resurrection trace (Fig. 3 b) gave s = 130 pN nm. By fitting (4) with fitting parameters D and ω_0, max, we obtained D = 0.14 ± 0.04 and ω_{0, max} = 306 ± 54 Hz. That is, each stator unit interacts with the rotor for a time fraction of 14%. The fitted value of D varies to 0.11 ± 0.03 and 0.16 ± 0.05 for s = 100 and 160 pN nm, respectively.

Discussion

Contradicting observations have been reported for the dependency of the zero-load speed ω₀ on the number of the stator units N at a vanishing load for the Escherichia coli flagellar motor. Early experiments with the H⁺ motor of E. coli implied that ω₀ does not depend on N (8, 11), suggesting a high duty ratio close to one (32, 33). Although they did not assume that N decreases at the zero-load region (5, 7), the independence of ω₀ from N was supported by a recent experiment with the E. coli motor (12). On the other hand, another group reported that ω₀ of the chimeric Na⁺ motor and also the H⁺ motor of E. coli depends on N (9, 10). A theoretical model based on a small duty ratio of each motor explains this result (34). An experimental challenge regarding this apparent contradiction is that N is limited to one or two under a vanishing load, and it is difficult to observe the N dependency of the rotation rate.

We directly observed a stepwise reduction of the rotation rate at the low-load region, which supposedly corresponds to a change in the stator-unit number N. This is evidence to support that ω₀ depends on N, suggesting a small duty ratio for the Salmonella flagellar motor. The previous report on the dependency of ω₀ on N was limited to N ≤ 2 (10). The electrorotation method enabled us to explore a broad range of N by instantaneously changing the load.

By ramping external torque at a constant rate, we recovered the TS curves of individual motors (Fig. 4). The TS curves reached the zero-load regions, which have been estimated by extrapolation in the methods using a viscous load such as the beads assay (3, 8, 35, 36, 37, 38, 39) except that Nord et al. used a hybrid motor containing both the Na⁺ stator and H⁺ stator to evaluate the TS curve beyond the zero load in the beads assay. We observed that the rotation rate does not converge to a single value in the absence of the load but had a variation (Fig. 5 b). The duty ratio was determined to be 0.14 ± 0.04 from the correlation between the motor torque under a high load and ω₀ (Fig. 3). This defines how we model the rotation mechanism of the flagellar motor. A small duty ratio might be effective for a high-speed movement, whereas a high duty ratio might be effective for a large-torque generation and a long processivity. It would be intriguing to evaluate the duty ratio of the motor of different species living in different environments.

Assuming that torque at high load T_{m, 0} is proportional to N, the torque generated by each stator is estimated to be ∼130 pN nm from the resurrection traces (Figs. 3 and S1). The relation between ω₀ and T_{m, 0} in Fig. 6 implies ω₀ = 43 Hz (N = 1) and 80 Hz (N = 2). These values are much smaller than the reported values for the H⁺ motor of E. coli (10): ω₀ = 120 Hz (N = 1) and 240 Hz (N = 2). This discrepancy may indicate a possibility that the Salmonella and E. coli motors have different kinetics with a different rate-limiting step at the vanishing load. Therefore, our result with the Salmonella motor does not necessarily disprove the large duty ratio for the E. coli motor.

These results were obtained by fully taking advantage of the electrorotation method for observing the dynamic response to an instant load change. A precise measurement of the TS characteristics and dynamic load control are central to elucidate the torque generation mechanism. A precise shape of the TS curve implies, for example, whether the torque is generated by a power-stroke type or a ratchet-type mechanism (40). On the other hand, the measurement of the motor response to a dynamic load modulation would become a powerful tool to investigate the dynamic stator assembly. The beads assay has been used successfully to reveal the motor properties. It is simple and effective to know the average behavior of the motor under each condition. However, the TS curve is blurred by averaging the torque characteristics of multiple motors in the beads assay. Also, the method is not convenient for a dynamical response measurement, whereas there are some attempts to control the load dynamically by changing the viscosity of a buffer (35, 41, 42) or attaching a probe to the flagellum during rotation (7, 12).

A future study specifying N of each TS curve would be helpful to scrutinize the stator dynamics. This may be possible by a combination of fluorescently labeled stator units and the calibrated electrorotation. Note that the geometry of the tethered cell has an advantage for total internal reflection fluorescent microscopy because of the short distance between the motor and the bottom glass surface (30). Thus, the methodology demonstrated here would add a new dimension to the study of bacterial flagellar motors.

Finally, we mention that the estimation of γ based on the fluctuation does not require preknowledge of the geometry of the system and can be applied to broad systems, including the beads assay of the bacterial flagellar motor, though precise noise-free fluctuation spectrum is necessary for this. See Fig. S3 for the comparison between γ calculated from the fluctuation and that estimated using the geometric parameters. This method should be helpful to provide more reliable values of motor torque and understand the torque-generating mechanism.

Author Contributions

K.S., S.N., and S.T. designed research, performed experiments, contributed analytic tools, and analyzed data. K.S., S.N., S.K., and S.T. wrote the article.

Editor: Hiroyuki Noji.