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Bacterial flagellar motor is one of the most complex and sophisticated nano machineries 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 where the kinetics such as a proton translocation speed limits the rotation rate, the dependency of the rotation rate on the number of stator units $N$ infers $D$; 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 {\it 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 $0.14 \pm 0.04$ from the correlation between the torque at high load and the rotation rate at low load., 12 pages, 6 figures. Title was changed from the previous version

The spirochete Leptospira has a coiled cell body and two periplasmic flagella (PFs) that reside beneath the outer sheath. PFs extend from each end of the cell body and are attached to the right-handed spiral protoplasmic cylinder (PC) via a connection with the flagellar motor embedded in the inner membrane. PFs bend each end of the cell body into left-handed spiral (S) or planar hook (H) shapes, allowing leptospiral cells to swim using combined anterior S-end and posterior H-end gyrations with PC rotations. As a plausible mechanism for motility, S- and H-end gyrations by PFs and PC rotations by PF countertorque imply mutual influences among the three parts. Here we show a correlation between H-end gyration and PC rotation from the time records of rotation rates and rotational directions of individual swimming cells. We then qualitatively explain the observed correlation using a simple rotation model based on the measurements of motility and intracellular arrangements of PFs revealed by cryo-electron microscopy and electron cryotomography.

Leptospira are spirochete bacteria distinguished by a short-pitch coiled body and intracellular flagella. Leptospira cells swim in liquid with an asymmetric morphology of the cell body; the anterior end has a long-pitch spiral shape (S-end) and the posterior end is hook-shaped (H-end). Although the S-end and the coiled cell body called the protoplasmic cylinder are thought to be responsible for propulsion together, most observations on the motion mechanism have remained qualitative. In this study, we analyzed the swimming speed and rotation rate of the S-end, protoplasmic cylinder, and H-end of individual Leptospira cells by one-sided dark-field microscopy. At various viscosities of media containing different concentrations of Ficoll, the rotation rate of the S-end and protoplasmic cylinder showed a clear correlation with the swimming speed, suggesting that these two helical parts play a central role in the motion of Leptospira. In contrast, the H-end rotation rate was unstable and showed much less correlation with the swimming speed. Forces produced by the rotation of the S-end and protoplasmic cylinder showed that these two helical parts contribute to propulsion at nearly equal magnitude. Torque generated by each part, also obtained from experimental motion parameters, indicated that the flagellar motor can generate torque >4000 pN nm, twice as large as that of Escherichia coli. Furthermore, the S-end torque was found to show a markedly larger fluctuation than the protoplasmic cylinder torque, suggesting that the unstable H-end rotation might be mechanically related to changes in the S-end rotation rate for torque balance of the entire cell. Variations in torque at the anterior and posterior ends of the Leptospira cell body could be transmitted from one end to the other through the cell body to coordinate the morphological transformations of the two ends for a rapid change in the swimming direction.

F(1)-ATPase is a rotary molecular motor powered by the torque generated by another rotary motor F(0) to synthesize ATP in vivo. Therefore elucidation of the behavior of F(1) under external torque is very important. Here, we applied controlled external torque by electrorotation and investigated the ATP-driven rotation for the first time. The rotation was accelerated by assisting torque and decelerated by hindering torque, but F(1) rarely showed rotations in the ATP synthesis direction. This is consistent with the prediction by models based on the assumption that the rotation is tightly coupled to ATP hydrolysis and synthesis. At low ATP concentrations (2 and 5 microM), 120 degrees stepwise rotation was observed. Due to the temperature rise during experiment, quantitative interpretation of the data is difficult, but we found that the apparent rate constant of ATP binding clearly decreased by hindering torque and increased by assisting torque.

The Na(+)-driven flagellar motor in Vibrio alginolyticus rotates very fast. Rotation of a single flagellum on a stuck cell was measured by laser darkfield microscopy with submillisecond temporal resolution. The rotation rate increased with increasing external concentration of NaCl, and reached 1000 r.p.s. at 300 mM NaCl. The Na+ influx through the motor should determine the rotation period (tau) and affect the speed stability. Fluctuation of the rotation period was analyzed at various rotation rates (from approximately 50 r.p.s. to approximately 1000 r.p.s.), which were changed by changing the external concentration of NaCl and the addition of a protonophore or a specific inhibitor. At high rotation rates (over 400 r.p.s.), the observed rotation was stable, and the standard deviation of tau (sigma tau) ranged from 7% to 16% of the average rotation period (tau). At low rotation rates (under 100 r.p.s), the rotation period tended to fluctuate, and the distributions of tau were non-Gaussian. The value of sigma tau ranged from 10 to 30% oftau. However, the observed minimum value of sigma tau at various rotation rates was approximately equal to the calculated standard deviation due to the rotational diffusion of the flagellar filament. These results suggest that the torque was stably generated at various Na+ influxes through the motor. We observed large fluctuations that cannot be explained by rotational diffusion. We discuss the factors that induce the large fluctuation.

Energetics of a rotary molecular motor F1-ATPase was studied by applying recent developments in nonequilibrium physics. Since molecular motors are engines that transduce chemical energy to mechanical motions, it is essential to focus on their energetics. Here, for a single F1-ATPase molecule, we have evaluated the amount of heat dissipation Qrot through its rotational degree of freedom as well as the work W against external load. Qrot was estimated using a new nonequilibrium equality connecting the heat dissipation to the violation of the fluctuation dissipation theorem. External torque was applied using the electrorotation method. We found a nontrivial energy balance relation that W+Qrot per 120° rotation was almost equal to the free energy change in a single ATP hydrolysis ∈″∈1/4 under various conditions. This implies that F1-ATPase focuses the free energy consumption toward rotations with an efficiency of nearly 100%. Moreover, we found that under a sufficiently strong torque in the opposite direction of ATP hydrolytic rotations, it rotated in the opposite direction, or the ATP synthetic direction, in a stepwise manner. The torque necessary for rotations in the synthetic direction times 120° was nearly equal to ∈″∈1/4 under various conditions except for conditions at sufficiently low ADP concentrations.

The bacterial flagellar motor is the only molecular rotary machine found in living organisms, converting the protonmotive force, i.e., the membrane voltage and proton gradients across the cell membrane, into the mechanical force of rotation (torque). We have developed a method for holding a bacterial cell at the tip of a glass micropipette and applying electric pulses through the micropipette. This method has enabled us to observe the dynamical responses of flagellar rotation to electric pulses that change the membrane voltage transiently and repeatedly. We have observed that acceleration and deceleration of motor rotation are induced by application of these electric pulses. The change in the rotation rate occurred within 5 ms after pulse application.

BACTERIA such as Escherichia coli and Salmonella typhlmurium swim by rotating their flagella1,2, each of which consists of an external helical filament and a rotary motor embedded in the cell surface (see ref. 3 for a review). The function of the flagellar motor has been examined mainly by tethering the flagellar filament to a glass slide and observing the resultant rotation of the cell body2. But under these conditions the motor operates at a very low speed (about lOr.p.s.) owing to the unnaturally high load conditions inherent in this technique. Lowe et al.4 analysed the frequency of light scattered from swimming cells to estimate the average rotation speed of flagellar bundles of E. coli as about 270 r.p.s. To analyse motor function in more detail, however, measurement of high-speed rotation of a single flagellum (at low load) with a temporal resolution better than 1 ms is needed. We have now developed a new method—laser dark-field microscopy—which fulfils these requirements. We find that although the average rotation speed of S. typhimurium flagella is rather stable, there are occasional abrupt slowdowns, pauses and reversals (accomplished within 1 ms). These changes were frequently observed in mutants defective in one of the motor components (called the switch complex), suggesting that this component is important not only in switching rotational direction but also in torque generation or regulation.

Swimming speed (v) and flagellar-bundle rotation rate (f) of Salmonella typhimurium, which has peritrichous flagella, were simultaneously measured by laser dark-field microscopy (LDM). Clear periodic changes in the LDM signals from a rotating bundle indicated in-phase rotation of the flagella in the bundle. A roughly linear relation between v and f was observed, though the data points were widely distributed. The ratio of v to f (v-f ratio), which indicates the propulsive distance during one flagellar rotation, was 0.27 microm (11% of the flagellar pitch) on average. The experimental v-f ratio was twice as large as the calculated one on the assumption that a cell had a single flagellum. A flagellar bundle was considered to propel a cell more efficiently than a single flagellum.

Rotational characteristics of Na+-driven flagellar motor in the presence and absence of coupling ion were analyzed by electrorotation method. The motor rotated spontaneously in the presence of Na+, and the rotation accelerated or decelerated following the direction of the applied external torque. The spontaneous motor rotation was inhibited by removal of external Na+, however, the motor could be forcibly rotated by relatively small external torque applied by the electrorotation apparatus. The observed characteristic of the motor was completely different from that of ATP-driven motor systems, which form rigor bond when their energy source, ATP, is absent. The internal resistance of the flagellar motor increased significantly when the coupling ion could not access the inside of the motor, suggesting that the interaction between the rotor and the stator is changed by the binding of the coupling ion to the internal sites of the motor.

Bacterial swimming speed is sometimes known to increase with viscosity. This phenomenon is peculiar to bacterial motion. Berg and Turner (Nature. 278:349–351, 1979) indicated that the phenomenon was caused by a loose, quasi-rigid network formed by polymer molecules that were added to increase viscosity. We mathematically developed their concept by introducing two apparent viscosities and obtained results similar to the experimental data reported before. Addition of polymer improved the propulsion efficiency, which surpasses the decline in flagellar rotation rate, and the swimming speed increased with viscosity.

Bacterial cells possess ultra-small motors on their surfaces with which to rotate their flagellar filaments. The motor utilizes the electrochemical energy stored in the proton gradient across the cytoplasmic membrane, and can rotate at more than 200 r.p.s. without a load. It can rotate in both clockwise and counterclockwise directions and switch the rotational direction in 1 msec. Its rotator is made of about 10 kinds of proteins and is about 30 nm in diameter. To analyze the motor function in detail, the authors have developed a laser dark-field microscopy technique by which high-speed rotation of a single flagellum can be measured. They have also succeeded in controlling the rotation speed by applying an external electric pulse to a bacterial cell that is held at the tip of a micropipette. >

Two methods of measuring bacterial motor characteristics using AC field effects in a microfabricated electrode system are presented. One is the measurement of the external force-velocity characteristics (F-v) of swimming bacteria. Electrostatic orientation of bacteria parallel to the field lines is used to guide the bacterial locomotion along a line. Dielectrophoresis is used to apply an external force, either forward or backward, to the swimming bacteria, and the velocity of locomotion is measured to obtain the F-v curve. The other approach is to measure torque-speed characteristics (T-w) of the motor. Electrorotation is used to apply external torque to the tethered cells, and by changing the applied torque and measuring the rotation speed, the T- omega curve is obtained. The results show that the motor generates approximately constant torque in the measured range of omega (0-100 Hz), regardless of the direction of rotation. >