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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 bacterial flagellar motor generates torque by converting the energy of proton translocation through the transmembrane proton channel of the stator complex formed by MotA and MotB. The MotA/B complex is thought to be anchored to the peptidoglycan (PG) layer through the PG-binding domain of MotB to act as the stator. The stator units dynamically associate with and dissociate from the motor during flagellar motor rotation, and an electrostatic interaction between MotA and a rotor protein FliG is required for efficient stator assembly. However, the association and dissociation mechanism of the stator units still remains unclear. In this study, we analyzed the speed fluctuation of the flagellar motor of Salmonella enterica wild-type cells carrying a plasmid encoding a nonfunctional stator complex, MotA/B(D33N), which lost the proton conductivity. The wild-type motor rotated stably but the motor speed fluctuated considerably when the expression level of MotA/B(D33N) was relatively high compared to MotA/B. Rapid accelerations and decelerations were frequently observed. A quantitative analysis of the speed fluctuation and a model simulation suggested that the MotA/B(D33N) stator retains the ability to associate with the motor at a low affinity but dissociates more rapidly than the MotA/B stator. We propose that the stator dissociation process depends on proton translocation through the proton channel.

バクテリアは，らせん形をしたべん毛繊維を回転させて水中を遊泳する．べん毛の働きは，スクリューに当たる．べん毛の正確な形状を観測することは，その微小なサイズと高速回転故に容易ではなかった．本論文では，レーザー暗視野顕微鏡を用いて，べん毛のらせんパラメータを計測する方法を提案し，サルモネラ菌へ適用した結果を示す．らせん形のべん毛を，らせん軸に対して45°または90°の方向からレーザービームで一方向照明すると，べん毛の像が一周期に1個または2個の輝点の列となる．そのような像を，電子シャッターを備えた高感度CCDカメラを使って記録した．記録した輝点の間隔を読み取ることで，菌体から脱落したべん毛（回転していない）のらせんピッチと半径を求めることができた．同様にして菌体上で高速回転するべん毛に関しても，ピッチの値が求まり，その微小な変化を捉えることもできた．

The giant acceleration (GA) of diffusion is a universal phenomenon predicted by the theoretical analysis given by Reimann et al. [Phys. Rev. Lett. 87, 010602 (2001)]. Here we apply the theory of the GA of diffusion to a single-molecule experiment on a rotary motor protein, ${\mathrm{F}}_{1}$, which is a component of ${\mathrm{F}}_{\mathrm{o}}{\mathrm{F}}_{1}$ adenosine triphosphate synthase. We discuss the energetic properties of ${\mathrm{F}}_{1}$ and identify a high energy barrier of the rotary potential to be $20{k}_{B}T$, with the condition that the adenosine diphosphates are tightly bound to the ${\mathrm{F}}_{1}$ catalytic sites. To conclude, the GA of diffusion is useful for measuring energy barriers in nonequilibrium and single-molecule experiments.

The deformation of a flagellum of Vibrio alginolyticus, single-flagellate bacteria, is analyzed theoretically assuming the shape of the flagellum to be a circular helix. The viscous force exerted on the flagellum in aqueous fluid is estimated applying the resistive-force theory based on the Stokes flow. The moment of force in the flagellum are described in analytical expressions and also the curvature and the torsion of the deformed flagellum are expressed analytically according to the Kirchhoff rod model. The deformation of the flagellum is obtained numerically solving evolution equations which determine a space curve from the curvature and the torsion. Comparing variations of the pitch of helical flagella between the numerical solutions and the results of measurement, the flexural rigidity or the elastic bending coefficient for the flagellum of Vibrio alginolyticus is estimated.

There are some cases where the swimming speed of a bacterial cell in a polymer solution increases with the viscosity. This phenomenon is not yet explained by the traditional theories, and such phenomena never appear under macroscopic conditions. We showed it could be explained by the hypothesis in which the anisotropic effect of polymer was introduced into a traditional theory. To evaluate the hypothesis, it is necessary to measure the hydrodynamic force acting on an ultra small needle in both the directions parallel and perpendicular to its axis. Tobacco mosaic virus (TMV; needle-shaped, 18 nm in width, 400 nm in length) is a suitable sample for this purpose. It is, however, very difficult to measure TMV Brownian motion and estimate the force acting on it because TMV can not be observed by typical optical microscopy. In this paper, we introduce our recent study about the measurement of TMV Brownian motion.

This paper gives an overview of consecutive studies on the asymmetrical motion of Vibrio alginolyticus cells, which possess a single polar flagellum. Inertial forces are negligible because of the cell size and the motion is expected to be symmetrical. However, asymmetrical characteristics between forward and backward motions were observed. The asymmetry observed in trajectory, swimming speed, and residence time appears only when a cell swims close to a surface. In backward motion, a cell traces circular path, while in forward motion the cell moves in a straight line. The backward swimming speed is faster than the forward speed. Backward swimming cells tend to stay close to a surface longer than forward swimming cells do. An explanation for these asymmetrical characteristics is given based on the results of boundary element analyses of creeping flow around a cell model that consists of a cell body and a rotating flagellum. According to the explanation, the attitude of a cell relative to a surface produces the asymmetry. The studies presented here indicate that the fluid-dynamic interaction between bacterial cells and a surface produces the unexpected asymmetrical motion. This asymmetry may help cells search for preferable states on a surface or to attach to the surface.

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.

Incommensurate superlattice reflections have been observed by X-ray diffractometry in K2SeO4 and (NH4)2BeF4. A simple method is proposed based on the ω-scan technique. This is free from systematic errors, and is not affected by the precision of the lattice constant determination. The incommensurate superlattice parameter of K2SeO4 measured using this procedure is in agreement with that measured using neutron scattering. The incommensurate structure of (NH4)2BeF4 is elucidated for the first time by the present method.