Direct Measurement of Helical Cell Motion of the Spirochete Leptospira

Abstract

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.

Introduction

Spirochetes are motile bacteria that have flagella within the spiral or flat-wave cell body. Many species of spirochetes are clinically important. Treponema pallidum is a causative agent of the sexually transmitted disease syphilis (1). Borrelia burgdorferi causes Lyme disease, which is a vector-borne disease (2). Brachyspira species cause intestinal spirochetosis in humans and animals (3). Pathogenic Leptospira species cause leptospirosis, which is a global zoonosis. It has been reported that virulence of pathogenic bacteria including spirochetes correlates with their motility (4–6), therefore, understanding the motility mechanism should be valuable to prevent the infection. Spirochete flagella are present within the periplasmic space, between the outer membrane and the protoplasmic cylinder. They are called periplasmic flagella (PFs), and each PF is linked to a flagellar motor embedded in the cytoplasmic membrane (7). In Treponema, Borrelia, and Brachyspira, several PFs extend from each end of the cell and overlap in the middle of the cell. Rotation of the PFs causes wave propagation through the entire cell body, allowing the cell translation (7). In Leptospira, a single PF extends from each end of the cell, but these filaments are too short to overlap with each other at the center of the cell (7). Therefore, only the ends of the cell body are transformed by the PF rotation (8), and the motion of Leptospira can be readily distinguished from that of other spirochete species.

The motile forms of Leptospira are generally classified into two types: the swimming (translating) and rotating groups (9). The translating cells swim smoothly whereas the rotating cells show no net displacement and spin at one position. The morphology of Leptospira can dynamically change while the cells move. When a cell translates, the anterior is spiral-shaped (i.e., S-shaped) and the posterior is hook-shaped (i.e., H-shaped). In a nonswimming cell, both ends can be either H-shaped or S-shaped (9). Thus, the asymmetric shape is thought to be important for the efficient generation of thrust.

The motile mechanism of Leptospira has been investigated using genetic manipulations, microscopic observations, and theoretical models (8–11). Berg et al. (10), and Goldstein and Charon (11) proposed that two sources of thrust drive the motility of Leptospira: the anterior spiral end gyrated by the rotation of PF; and the coiled protoplasmic cylinder rotating in the opposite direction. The anterior S-shaped end is left-handed and gyrates counterclockwise (CCW), which produces backward motion of the spiral wave. In contrast, the protoplasmic cylinder is right-handed and rotates clockwise (CW). The posterior H-shaped end is approximately planar and rotates in the same direction as the S-end to allow the cell to translate without twisting (9). The S- and H-shapes of the two ends are thought to be determined by the shape of PF (8). A mathematical model has shown that the transition between the S- and H-shape can be caused by changes in the rotational direction of the flagellar motor and elastic interaction between the cell body and the flagellum (12). However, more quantitative measurements are required to gain deeper insights into the motion mechanism of Leptospira.

In this study, we analyzed the motion of saprophytic Leptospira biflexa by one-sided dark-field microscopy with a high-speed camera. One-sided illumination under a dark-field microscope allows visualization of helical objects, such as the flagellar filaments and spirochete cell bodies, as a series of bright spots because of the strong light scattering from parts of the helix that are illuminated perpendicularly (13,14). We measured the swimming speed and the gyration rates of the S-end and H-end and rotation rate of the protoplasmic cylinder around the helix axis of individual swimming Leptospira cells and analyzed force and torque of each part. In what follows, “gyration” is often called “rotation” for convenience.

Materials and Methods

Bacterial strains and media

A saprophytic species, Leptospira biflexa strain Patoc I, was used. Cells were grown in Korthof’s liquid medium at 30°C for 4 days. The cells were then resuspended in fresh Korthof’s medium, which was used as a motility medium. To increase viscosity, Ficoll (PM400; Amersham Biosciences, GE Healthcare, Uppsala, Sweden) was added to the motility medium.

One-sided dark-field microscopy

The cells were observed using a dark-field microscope (BX50; Olympus, Tokyo, Japan). For one-sided illumination, half of the light from a mercury lamp was shut out before reaching the dark-field condenser. The motion of the cells was recorded using a high-speed CMOS camera (Digimo, Tokyo, Japan) at a frame rate of 500 fps, and appropriate parts of the movie were captured on a computer. Individual swimming speeds and rotation rates were analyzed by the softwares ImageJ (National Institutes of Health, Bethesda, MD) and Excel (Microsoft, Redmond, WA) (see Fig. S1 in the Supporting Material).

Force and torque on swimming Leptospira cell

What we describe here are forces and torques acting on a Leptospira cell body for its different motions, such as translation and rotation, in a liquid. Forces acting on the S-end (F_s) and the protoplasmic cylinder (F_pc) in a swimming Leptospira cell were calculated from the swimming speed and rotation rate with the formula (15)

$$F_s={\alpha }_{s}v+{\gamma }_s{\omega }_s,$$ (1)

$$F_{pc}={\alpha }_{pc}v+{\gamma }_{pc}{\omega }_{pc},$$ (2)

where v is the swimming speed, ω is the angular velocity, α and γ are the drag coefficients, and the subscripts s and pc indicate the S-end and the protoplasmic cylinder, respectively. The values ω_s and ω_pc were obtained from the gyration rate of the S-end and rotation rate of the protoplasmic cylinder, respectively. The drag coefficients are obtained from morphological parameters and viscosity of medium as follows (15–17):

$${\alpha }_s=-C_s\left(8{\pi }^2{r}_s^2+p_s^2\right),$$ (3)

$${\alpha }_{pc}=-C_{pc}\left(8{\pi }^2{r}_{pc}^2+p_{pc}^2\right),$$ (4)

$${\gamma }_s=2C_s\pi r_s^2{p}_s,$$ (5)

$${\gamma }_{pc}=-2C_{pc}\pi r_{pc}^2{p}_{pc},$$ (6)

$$C_s=\frac{2\pi \mu L_s}{\left(log\frac{2p_s}{a_s}-\frac{1}{2}\right)\left(4{\pi }^2{r}_s^2+p_s^2\right)},$$ (7)

$$C_{pc}=\frac{2\pi \mu L_{pc}}{\left(log\frac{2p_{pc}}{a_{pc}}-\frac{1}{2}\right)\left(4{\pi }^2{r}_{pc}^2+p_{pc}^2\right)},$$ (8)

where r and p are the radius and pitch of helix, respectively; and L and a are the length and radius of cell, respectively. As the viscosities of Ficoll solutions (μ), the values measured by Nakamura et al. (17) were used. Here, the helical angle and rotation direction of S-end were assumed to have opposite signs to those of the protoplasmic cylinder: left-handed and CCW rotation for S-end, and right-handed and CW rotation for the protoplasmic cylinder (see Fig. 2 later in article and see Movie S1 in the Supporting Material). Hereafter, F_s and F_pc are called “S-end force” and “protoplasmic cylinder force”, respectively.

Figure 2.

Helical handedness of the S-end and the protoplasmic cylinder of swimming Leptospira cells. Two different cells swimming in media containing 5% Ficoll are shown as examples. (Thin arrows) Directions of helical line segments visualized by one-sided dark-field illumination. (Thick arrows) Direction of illumination. The bright parts are below the helix axis of the cell body. The helical handedness of the S-end and the protoplasmic cylinder are opposite to each other, where the protoplasmic cylinder is right-handed as previously shown (7,19) and the S-end is left-handed.

Drag torques exerted on the S-end (T_s) and the protoplasmic cylinder (T_pc) were calculated as follows (15):

$$T_s={\gamma }_{s}v+{\beta }_s{\omega }_s,$$ (9)

$$T_{pc}={\gamma }_{pc}v+{\beta }_{pc}{\omega }_{pc}.$$ (10)

β is the drag coefficient, which is obtained as follows:

$${\beta }_s=C_s\left(2r_s^2{p}_s^2+4{\pi }^2{r}_s^4\right),$$ (11)

$${\beta }_{pc}=C_{pc}\left(2r_{pc}^2{p}_{pc}^2+4{\pi }^2{r}_{pc}^4\right).$$ (12)

Hereafter, T_s and T_pc values are called “S-end torque” and “PC torque”, respectively. The values used for calculations are listed in Table 1; see also Goldstein et al. (18).

Table 1.

Parameters used for calculations

Parameters	Symbols	Values	Sources or references
S-end
Helix radius	rs	0.3 μm	Goldstein and Charon (9)
Helix pitch	ps	2.7 μm	Goldstein and Charon (9)
Cylinder radius	as	0.09 μm	Same value as helix radius of the protoplasmic cylindera
Length	Ls	3 μm	Kan and Wolgemuth (12)
Protoplasmic cylinder
Helix radius	rpc	0.09 μm	Goldstein et al. (18)
Helix pitch	ppc	0.7 μm	This study
Cylinder radius (= cell radius)	apc	0.07 μm	Goldstein et al. (18)
Length	Lpc	8 μm	This study
Viscosity of motility medium
Without Ficoll	μ	0.86 mPa × s	Nakamura et al. (17)
With 5% Ficoll		1.99 mPa × s	Nakamura et al. (17)
With 10% Ficoll		3.89 mPa × s	Nakamura et al. (17)
With 15% Ficoll		7.98 mPa × s	Nakamura et al. (17)

^aThe short-pitch coils within S-end were neglected, and the cylinder radius of S-end was assumed to be the same with the helix radius of the protoplasmic cylinder.

Results and Discussion

Motion analysis by one-sided dark-field microscopy

One-sided dark-field illumination was used to visualize Leptospira cells as sequential bright spots along the cell axis (Fig. 1 A). Translating Leptospira cells exhibited anterior S-ends and posterior H-ends (Fig. 1 B), in agreement with a previous report (9). When the cells swim by rotating their cell bodies, the sequential spots move backward from the anterior to the posterior end (see Movie S1). Kinematic and morphological parameters of individual cells were obtained from the swimming trajectory and movement of the periodic bright spots (see Fig. S1). The helical shape of the protoplasmic cylinder of Leptospiraceae is known to be right-handed (9,19). Multiple-exposure photographs of Leptospira illini showed that the S-end is left-handed and rotates counterclockwise (CCW) (9). The swimming cells analyzed in this study showed the S-end to be left-handed as well (Fig. 2). Furthermore, the waves of the S-end and the protoplasmic cylinder propagated from the anterior to posterior (see Movie S1). These data indicate that the rotational directions of the S-end and the protoplasmic cylinder are CCW and clockwise (CW), respectively, as previously reported for L. illini (9).

Figure 1.

Effect of one-sided dark-field illumination on a Leptospira cell. (A) Leptospira biflexa observed by conventional dark-field illumination (left) and one-sided dark-field illumination (right). (Arrow) Direction of one-sided illumination. The cells spontaneously adhered to glass surfaces were observed. (B) A swimming L. biflexa cell with an anterior spiral end (S-end), coiled protoplasmic cylinder (PC), and posterior hook-shaped end (H-end).

Kinematic parameters of swimming Leptospira

Fig. 3 shows motion data of Leptospira cells measured under various viscous conditions by increasing the Ficoll concentration. The swimming speed decreased as the Ficoll concentration increased (Fig. 3 A). The average rotation rate of the S-end and the protoplasmic cylinder also decreased more or less linearly with increasing the Ficoll concentration (Fig. 3 B, left and middle). Although it was difficult to accurately determine the rate and direction of the H-end rotation because it was quite unstable, frequently showing shaking motions and temporary or continuous paralysis, the H-end rotation rates measured in some swimming cells were affected little by change in Ficoll concentration up to 10% but significantly decreased at 15% Ficoll (Fig. 3 B, right). Fig. 3 C shows the relationships between the swimming speed and rotation rate of each part. It has been proposed that the S-end rotation is sufficient for propelling the cell in a low viscosity medium, whereas the protoplasmic cylinder contributes to the slipless translation in a gel-like medium (7,9,10). However, the swimming speed increased almost linearly with the rotation rates of the S-end and the protoplasmic cylinder (Fig. 3 C), suggesting that both of them are responsible for propulsion in Ficoll solution, which is a Newtonian fluid. The rotation rate of the H-end showed no correlation with the swimming speed (Fig. 3 C), suggesting that the H-end is not directly involved in thrust generation. Variety of structural characteristics of the H-end, such as the length of the hook and its angle relative to the long axis of the cell, could be one of the sources causing a large scatter in the rotation rate of the H-end.

Figure 3.

Kinematic data of swimming Leptospira cells. (A) Swimming speed and (B) rotation rates of the S-end, the protoplasmic cylinder, and H-end measured at various Ficoll concentrations. The values of viscosity are listed in Table 1. (C) Relationship between the swimming speeds and rotation rates of the three parts of the cell body at various Ficoll concentrations. Each point corresponds to data obtained from individual cell. Different symbols represent data obtained in solutions of different Ficoll concentrations: (circle) 0, (square) 5, (triangle) 10, and (diamond) 15%. The correlation coefficients between the swimming speed and rotation rate are 0.56, 0.67, and 0.05 for the S-end, the protoplasmic cylinder, and H-end, respectively. (D) The v/f calculated from data shown in panel C. In panels A, B, and D, average values are shown, and error bars are the standard deviation. Statistical analysis (t-test) was carried out to indicate significant difference from the result of 0% Ficoll (^∗∗ for P < 0.01, ^∗ for P < 0.05).

The ratio of swimming speed to rotation rate of the cell body (v/f) indicates the progressing distance by one revolution of the helical cell body (14,15). Motion analysis of a swine intestinal spirochete, Brachyspira pilosicoli, showed that v/f increased in polyvinylpyrrolidone solutions but did not increase in Ficoll solutions (17). Also, Harman et al. (20) investigated the motion and morphological parameters of Bo. burgdorferi, showing that the cells translate without slippage in gelatin solutions. Fig. 3 D shows that v/f values for the S-end and the protoplasmic cylinder of Leptospira were ∼0.2 and 0.1 μm, respectively, over the measured range of Ficoll concentrations. These values correspond to ∼7 and 14% of the pitch of the S-end and the protoplasmic cylinder, respectively. These data indicate that the Leptospira cells translated with slippage in Ficoll solutions in agreement with previous reports (9): the protoplasmic cylinder propels the cell efficiently in a gel-like medium without much slippage but the cell advances little in a Ficoll solution. Cortez et al. (21) computed the relationship between translation and morphology of a rotating helix by simulating the motion of the protoplasmic cylinder, and showed that the pitch length significantly affects the progress distance of the helix per revolution. When the helical pitch is 0.7 μm, the same value as that of the Leptospira protoplasmic cylinder, 128 revolutions was required to progress 12 μm (21), corresponding to a v/f of ∼0.09 μm. This is ∼13% of the helical pitch and is almost the same as that of the protoplasmic cylinder obtained in our measurement.

Force and torque of S-end and the protoplasmic cylinder

Fig. 4 shows the force acting on the S-end and the protoplasmic cylinder calculated from data shown in Fig. 3 C and Table 1. Although pairwise plots for individual cells showed scattered results, the average value of the data under each load condition (Ficoll concentration) showed a linear relation with a slope of 1.19 as F_pc/F_s (Fig. 4). This suggests that the S-end and the protoplasmic cylinder rotations almost equally contribute to propulsion over the measured range of viscosity. According to Eqs. 1–8, F_pc/F_s is independent of viscosity, because the μ-value included in C_s and that in C_pc cancel each other out. This is consistent with the experimental result shown in Fig. 4.

Figure 4.

Forces acting on the S-end (F_s) and the protoplasmic cylinder (F_pc). (Open symbols) Data obtained from individual cells swimming in media at different Ficoll concentrations: (circle) 0, (square) 5, (triangle) 10, and (diamond) 15%. (Solid symbols) Average values for each Ficoll concentration. (Dashed line) Regression line fitted to the average values.

Fig. 5 A shows torque produced by the S-end and the protoplasmic cylinder as a function of their rotation rate. The average values of the S-end torque for each Ficoll concentration decreased linearly as the rotation rates increased. Because the rotational direction of the protoplasmic cylinder was opposite to that of the S-end, the PC torque is shown as negative values in Fig. 5 A. The absolute values of the PC torque also decreased linearly as the rotation rate increased. The largest torque measured in this study was ∼4000 pN nm, suggesting that the flagellar motor of L. biflexa can generate torque higher than 4000 pN nm, which is in good agreement with a theoretical estimation reported previously by Kan and Wolgemuth (12). Because the maximum torque of the Escherichia coli motor is 1000–2000 pN nm (22), the Leptospira motor can generate a greater-than-twofold torque of the E. coli motor.

Figure 5.

Torque-speed relationship of the S-end and the protoplasmic cylinder and pairwise comparison of torque. (A) Relationship between the torque and rotation rate of the S-end and the protoplasmic cylinder. Data are labeled with same symbols as used in Fig. 4. (Small symbols) Data of the S-end (open) and the protoplasmic cylinder (solid). (Large circles) Average values of the S-end (solid) and the protoplasmic cylinder (open) for each Ficoll concentration. (Dashed lines) Regression lines fitted to the average values. (B) Pairwise comparison between T_s and T_pc. Data points are the same as those shown in panel A. (Horizontal lines) Average values of T_pc for different Ficoll concentrations: (thick line) 0, (thin line) 5, (thick dashed line) 10, and (thin dashed line) 15%.

Because the structure of the Leptospira flagellar motor revealed by electron cryotomography looks quite similar to those of externally flagellated bacteria (23), the rotation mechanism should, in principle, be the same. In E. coli, Salmonella, and Vibrio, which all have extracellular flagella, rotation assays of the flagellar motor have shown that torque of the motor rotating CCW is almost constant, decreasing very slowly up to a certain rotation speed under high-to-medium load conditions and then sharply decreases to zero under a low-load condition (22,24). In contrast, the torque-speed relationship of the E. coli motor rotating CW shows a linear decrease in torque as the rotation rate increases (25). The torque-speed relationship of the S-end is likely to reflect the characteristic of a single motor, albeit indirectly. Because we did not observe the constant torque region in high-load regime, the torque-speed relationship of the S-end is similar to that of the CW-biased motors of externally flagellated bacteria. However, because the transmission of the motor torque to the S-end gyration is complex, it may not be appropriate to consider this torque-speed behavior to be the actual characteristic of the Leptospira motor except for the lower limit of the maximum torque mentioned above. If the morphological parameter of the gyrating S-end can be determined accurately, the motor torque might be estimated as reported previously by Kan and Wolgemuth (12). Because the PC torque shown in Fig. 5 A could be the sum of torque generated by the two flagellar motors at the two cell ends, it is more complex and difficult to decompose its torque-speed relationship into that of individual motors.

Mechanical coordination between the two cell ends

The asymmetric cell shape of Leptospira with the S-shaped anterior and the H-shaped posterior ends appears to be essential for unidirectional swimming, as shown in Fig. 1 B. Transition of the S-end to the hook shape occurs when the cell changes its swimming direction, and it has been reported that the transition occurs within 0.1 s (9). We too observed that the change in the swimming direction was completed within several hundred milliseconds (data not shown). Morphological changes of the two cell ends are thought to be caused by the changes in the direction of motor rotation (12). Polymorphic transition of supercoiled flagella caused by the motor reversal can transform the cell ends (26). The reversal of the Leptospira motor rotation is believed to occur through the chemotactic signal transduction in a similar manner to those found in E. coli and Salmonella (27). Because the flagellar motors of spirochete exist at both ends of the long cell body (at least 10 μm for Leptospira), it was thought that it may require several seconds for signal proteins to diffuse from one end of the cell to the other (28).

However, recent structural study of a spirochete by electron cryotomography revealed that chemoreceptors and flagellar motors are in close proximity to each other at the two cell poles (23,29). This may partly explain the rapid switching of swimming direction by coordinated morphological changes of the two cell ends. Still, however, there must be some other mechanisms that transmit the signal more quickly from one end of the cell to the other to directly coordinate their morphological changes. Membrane potential is known to be involved in chemosensory signal transduction in Spirochaeta aurantia (30). Therefore, some electrogenic mechanism could trigger the coordinated morphological transition of the two ends of the Leptospira cell, although no evidence has been reported.

Here, we suggest another possible mechanism, based on the following fluctuation analysis, that the dynamics of the two ends of the cell body are mechanically coordinated with each other: The dispersion of the S-end torque data was found to be much greater than that of the PC torque data (Fig. 5 B). The standard deviations (σ) normalized by the average torque of the S-end (σ_s/T_s) and the protoplasmic cylinder (σ_pc/T_pc) were ∼0.4 and 0.2, respectively (Table 2). This indicates that the rotation of some other parts should fluctuate largely to compensate the S-end fluctuation for torque balance of the entire cell. As shown in Fig. 3 C, the H-end does not contribute to translation, and its rotation was found to be very unstable, suggesting that the change in the H-end rotation rate occurs in response to the change in the S-end rotation rate, while the protoplasmic cylinder rotation remains relatively stable.

Table 2.

Torques of S-end and protoplasmic cylinder

	S-end		Protoplasmic cylinder
	Ts ± σs	σs/Ts	Tpc ± σpc	σpc/Tpc
No Ficoll (n = 16)	894 ± 390	0.44	−623 ± 94	0.15
5% Ficoll (n = 23)	1680 ± 708	0.42	−1099 ± 174	0.16
10% Ficoll (n = 16)	2614 ± 1363	0.52	−1816 ± 439	0.24
15% Ficoll (n = 7)	3603 ± 1578	0.44	−2842 ± 496	0.17

In the chemotaxis of Bo. burgdorferi, direct interactions between the PFs extending from both cell ends are thought to be involved in the coordination of the motor rotations because the PFs overlap in the central region of the cell body (31). A mathematical model also showed that the overlap of the PFs is required for wave propagation along the cell body (32). In contrast, the PFs of Leptospira are too short to overlap. The ratio of the bending moduli of the PF to that of the cell cylinder has been theoretically estimated to be ∼0.15 for Leptospira (12), in contrast to the ratio of 2–6 reported for Bo. burgdorferi (33). These indicate that the relative stiffness of the Leptospira cell cylinder is 10–40 times larger than that of Borrelia. Thus, instead of the mechanical transmission through PFs, a change in torque at one end might be directly transmitted to the other end through the cell body. If one end of the cell transforms its morphology from the S- to the H-shape, the other end could change the rotational direction to maintain the torque balance, and consequently, the transformation from the H- to S-shape might occur in the other end. To examine and confirm this hypothesis, we will need to carry out more detailed analysis of whether changes in the kinematic parameters (rotational direction and torque) and morphological changes of both ends (transition between the S- and H-shapes) coordinate with each other.

In this study, we measured kinematic parameters of Leptospira swimming in various viscous conditions and discussed the force and torque using a simple theoretical model based on the resistive force theory. Although these results would be helpful for gaining insights into the mechanism of Leptospira motility, more-precise measurements and mathematical analyses are required. For microorganisms swimming in a liquid at a constant speed, forces acting on the motions are all viscous drag forces, and the inertial forces can be ignored (34). Here, we showed that the sum of F_s and F_pc was ∼1.5 pN in the medium without Ficoll (Fig. 4). Assuming the H-end as a half circle, the force acting on the H-end is estimated to be ∼−0.1 pN (16,35). Hence, the net force acting on the entire Leptospira cell (F_s + F_pc + F_h) obtained by this model is not zero. The values of F_s and F_pc might have been overestimated due to the approximations adopted here. For example, any hydrodynamic interactions between parts of Leptospira were not considered in the resistive force theory (14–17,34), and the architecture of the Leptospira cell may be oversimplified (Table 1).

We could not determine the rotational direction of the H-end for all cells. However, the H-end often seemed to rotate in the opposite direction to the S-end rotation (see Movie S1). Assuming that the rotational directions of the H-ends are CW in all cells, the magnitude of T_pc is ∼90% of that of T_s + T_h (see Fig. S2 A). In contrast, assuming that the H-ends rotate CCW, T_pc is ∼60% of T_s + T_h (see Fig. S2 B). These suggest that most cells are possibly rotating their H-ends CW and that torque balance is achieved in the counterrotation of the H-end against the S-end. It has been proposed that the spirochete flagellar motor at one end of the cell body should rotate in the opposite direction to the one at the other end for the cell to move without twisting (9,36). Furthermore, asymmetric rotation of the Leptospira motors is thought to be required to transform the anterior end into the S-shape and the posterior ends into the H-shape (12). In this study, it was difficult to offer a reasonable explanation of how the S- and H-ends can rotate in opposite directions to each other. However, because it has also been shown that the outer membrane of Leptospira is fluid, and because antibody-coated latex beads adhered on the Leptospira cells are free to move (37), the counterrotation of the S- and H-ends would be possible without twisting of the cell body just by changing the morphology of the cell envelope. In any case, more-accurate measurements of cell dynamics and further improvement of the theoretical model are required to promote better understanding of the physics in Leptospira motion.