doi:10.1016/S0022-2836(03)00176-1

J. Mol. Biol. (2003) 327, 1043–1051

Torque–speed Relationship of the Na1-driven Flagellar
Motor of Vibrio alginolyticus
Yoshiyuki Sowa1, Hiroyuki Hotta2, Michio Homma3 andAkihiko Ishijima2,4*
1

Department of BiophysicalEngineering, Osaka University
1-3, Machikaneyama, Toyonaka
Osaka 560-8531, Japan2
Department of Applied Physics
Graduate School of Engineering
Nagoya University
Furo-cho, Chikusa-ku
Nagoya, Aichi 464-8603, Japan
3

Division of Biological Science
Graduate School of Science
Nagoya University
Furo-cho, Chikusa-ku
Nagoya, Aichi 464-8602, JapanThe torque –speed relationship of the Naþ-driven flagellar motor of Vibrio
alginolyticus was investigated. The rotation rate of the motor wasmeasured by following the position of a bead, attached to a flagellar filament, using optical nanometry. In the presence of 50 mM NaCl, the generated torque was relatively constant (, 3800 pN nm) at lower speeds(speeds up to , 300 Hz) and then decreased steeply, similar to theHþ-driven flagellar motor of Escherichia coli. When the external NaCl concentration was varied, the generated torque of the flagellar motor waschanged over a wide range of speeds. This result could be reproduced
using a simple kinetic model, which takes into consideration the association and dissociation of Naþ onto the motor. These results imply thatfor a complete understanding of the mechanism of flagellar rotation it isessential to consider both the electrochemical gradient and the absoluteconcentration of the coupling ion.
q 2003 Elsevier Science Ltd. All rights reserved

4

PRESTO, Japan Scienceand Technology Corporation(JST), Japan*Corresponding author

Keywords: energy transduction; flagella; rotary motor; molecular machine;
VibrioIntroduction
The bacterial flagellar motor is a rotary molecular machine powered by the flow of ions acrossthe cytoplasmic membrane.1 Each motor rotates along (, 5 mm) helical filament that extends from acell body. The rotation of this filament is used forcell propulsion, which is the primary method of
locomotion. The influx of ions is driven by a transmembrane electrochemical gradient of a specificion, either Hþ (protonmotive force: pmf) or Naþ(sodium-motive force: smf).2 – 4 Hþ-driven typeflagellar motors are used in Bacillus subtilis,
Streptococcus or Escherichia coli, and Naþ-driven
type flagellar motors are found in alkalophilic
Bacillus or Vibrio species.
In Hþ-driven flagellar motors, approximately 50
proteins are required for the assembly and rotationof the flagella.5 Especially, MotA, MotB and FliG
are essential for the conversion of energy to torqueAbbreviations used: pmf, protonmotive force; smf,
sodium-motive force.
E-mail address of the corresponding author:
ishijima@nucc.cc.nagoya-u.ac.jp

and have been identified from genetic studies ofSalmonella typhimurium and E. coli.6,7 MotA and
MotB form a proton channel complex and function
as a component of a stator.6 The rotor consists of aseries of rings spanning the cell envelope, which
is attached via the flexible hook to the helicalfilament,8 and is surrounded by eight or 16 independent MotA/B complexes.9,10 FliG forms part of
the rotor and interacts with MotA to generate
torque for rotation. Analyses of site-directed mutagenesis have shown that the charged residues
crucial for torque generation are R90 and E98 in
MotA and R281, E288 and E289 in FliG.11,12 It isthought that these charged residues of MotA and
FliG engage in electrostatic interactions with eachother. When either R90 or E98 of MotA was converted to Glu or Lys, the motor was unable torotate.13
PomA and PomB in Naþ-driven flagellar motors
have sequences similar to MotA and MotB in theHþ-driven motors, respectively, and function as
sodium channels. PomA in V. alginolyticus conserves charged residues such as R88 and E96,
which correspond to R90 and E98 of MotA of
E. coli, respectively. In contrast to the Hþ-driven

0022-2836/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved

1044

Figure 1. Schematic of a measurement system of
Naþ-driven flagellar motor rotation in NMB136 cells. An
NMB cell was immobilized on the glass surface and abead was attached to its flagellar filament. The phasecontrast image of the bead was projected onto the faceof a quadrant-type photodiode (see the text for details).

motor, reversal of the charge of either R88 or E96
had no effect on flagellar rotation of the Naþdriven motors.14 Torque generation of Naþ-driven
motors requires the additional components MotX
and MotY.15,16 The components of the Hþ-driven
flagellar motor had no similarities with MotX and
MotY, except for the fact that the C terminus of
MotY has a peptidoglycan-binding motif.17 MotX
and MotY are located in the outer membrane18 andinteract with each other.19 MotX and MotY appear
to be components unique to the Naþ-driven
flagellar motors; however, their precise roles havenot been determined. Although there are differences between Hþ-driven and Naþ-driven motors,
it is thought that both motors operate by the sameunderlying mechanism because PomA defective
V. alginolyticus cells can regain flagellar rotation by
expressing MotA of Rhodobacter sphaeroides, which
has a Hþ-driven motor.20
Many models explaining the mechanism of thebacterial flagellar rotation have been proposed.21,22
To verify these models, the function of the rotary
motor must be investigated. The most common
method for investigating flagellar rotation has
been the tethered cell method.23 This method
involves attaching a single flagellar filament to a
microscope coverslip using antibodies against thefilament and observing the rotation of the cellbody. This is a very convenient method from theview-point that no specialized equipment isnecessary. However, the motor operates at approximately 10 Hz due to the unnatural high loads
imposed on the motor. Normal flagella rotate at
speeds greater than 100 Hz.24 In an extension of
the tethered cell method, the method of electrorotation has been applied to study flagellar motors.25
This method controlled the rotation rates of cells
by applying an external torque and clarified thetorque– speed relationship of E. coli.26 Anothereffective method of measuring flagellar rotation

Torque Generated by the Naþ-driven Flagellar Motor

involves laser dark-field microscopy.27 In thismethod, a thin laser beam irradiates the flagellar
filament and the rotation rate under natural loadconditions can be measured from the intensitychange of the scattered light with high temporal
resolution. The average rotation rate using thismethod is reasonably stable; however, there are
abrupt slowdowns, pauses and reversals of therotation.
A method used to measure the displacement of
biological molecules with high temporal andspatial resolutions has been developed recently.
This method involves attaching a bead as a marker
to molecules and projecting the bead image ontoa photo-detector.28 This method has greatly
advanced the studies of motor proteins that perform sliding movements, such as kinesin andmyosin.29 – 31 This bead assay has also been applied
to studies of the flagellar motors. A spherical bead
was attached to the flagellum or hook and theposition of the bead was followed by photodetector.32 Careful experiments using a bead assay
confirmed the torque– speed relationship of theHþ-driven flagellar motor in E. coli.32,33
In this study, the flagellar rotation of the Naþdriven motor in V. alginolyticus was measured
using a bead assay. The torque– speed relationshipof the Naþ-driven flagellar motor is similar to thatof the Hþ-driven flagellar motor. This result
suggests that the mechanism of torque generationin both Naþ and Hþ-driven is similar. A kinetic
model applied to the data could reproduce theeffect of NaCl concentration in the medium on thegenerated torque.

Results
Measurement of flagellar rotation using abead assay
The rotation rates of Naþ-driven flagellar motors
in V. alginolyticus NMB136 have been measured byattaching a poly-L -lysine modified bead to aflagellar filament as a marker (Figure 1). NMB136
cells are useful for measurements of flagellar
rotation, because (1) these cells have a single polar
flagellar motor, no lateral flagella (Laf2), and (2) aflagellum rotates unidirectionally, i.e. without theclockwise direction (Che2).34 The polar flagellar
filament of V. alginolyticus is sheathed, and thecomponents of the sheath are thought to be similar
to that of the outer membrane.35 Because beads
have not been rarely observed attached to the cellbody under our experimental conditions and themembrane enclosing the filament is supposed to
be too fragile to rotate with the beads, it is likelythat the poly-L -lysine modified beads attached to
exposed areas on the filaments. To follow theposition of the beads with sub-millisecond andnanometer resolution, a phase-contrast image ofthe bead was projected onto the face of a quadrant-type photodiode.36 Currents from each of theTorque Generated by the Naþ-driven Flagellar Motor

1045

Figure 3. (a) The effect of NaCl concentration on theswimming speed of V. alginolyticus NMB136 cells. Eachpoint shows the average swimming speed for 50 cells
measured in several independent experiments in thepresence of varying NaCl concentrations. The Y-axis on
the right represents the flagellar rotation rate calculated
from the n:f ratio.37 (b) The relationship between the concentration of NaCl in the external medium and therotation rate of beads attached to the flagellar filaments.
Crosses ( £ ), filled diamonds (V) and open circles (W)show the average values of rotation rate when beads of
1.08, 0.85 and 0.60 mm diameter were attached to flagellar filaments, respectively. For each data point, at least
eight cells were measured.

Figure 2. Typical example of the data. (a) X and Y signals of the position of a bead attached to the flagellar filament are shown as a function of time. The size of thebead was 1.08 mm in diameter. The external NaCl concentration was 50 mM. The signals were passed througha low-pass filter of 1 kHz. (b) The FFT spectrum of thesignal from 1.0 second using the same data as in (a).
(c) A two-dimensional X– Y plot of the data shown in(a). Broken lines are the major and minor axes of theellipsoid.

four quadrants were converted to voltages, and theX signal and Y signal of the bead position were
calculated through a differential amplifier. Figure2(a) shows the typical time-course of the X and Y
signals of the bead position when a bead 1.08 mm
in diameter was attached to a flagellar filament.
Since the bead was rotating, the X and Y signals
appeared as sinusoidal curves with a 908 phaseshift. The FFT spectrum in the X signal (or Y signal)
had a single peak, and its frequency was 133 Hz,
so this has been taken as the rotation rate of theflagellar motor (Figure 2(b)). The experimental

conditions in this measurement system, such as
light intensity, had little effect on the flagellar
rotation. Most cells rotated their flagellar filaments
at a constant speed for at least ten minutes.
Cells that tilted up from the surface of a coverslip (as illustrated in Figure 1) were selected,
because this orientation allowed us to ignore thefriction between the beads and the coverslips. This
resulted in the two-dimensional X –Y plots of mostof the acquired data being ellipsoid in shape.
Then, the rotational radii of the beads were
measured as the major radius of the ellipsoid.
Figure 2(c) shows the X –Y plot of the beadpositions of a bead 1.08 mm in diameter. The
rotational radius of the bead was measured as
0.43 mm. This value is reasonable, because therotational radius is expected to be rb ^ r, where rb
is the radius of a bead and r is the helical radiusof the flagellar filament, 0.14 mm.37 However, therotational radii of beads were often larger than
expected. This was especially so for beads withdiameters of 1.65 mm and 1.08 mm, and is most
likely due to the eccentric rotation of the flagellar
filaments.

Torque Generated by the Naþ-driven Flagellar Motor

1046

The effect of external NaCl concentrations on
flagellar rotation
Figure 3(a) shows the effect of external NaCl
concentration on the swimming speed of NMB136
cells. NMB136 cells have a single polar flagellar
motor, so the swimming speed should reflect therate at which the flagellum rotates. Studies using
laser dark-field microscopy reported that external
concentration of NaCl affected the swimming
speed and the flagellar rotation rate.38 The swimming speed and the flagellar rotation rate were
measured simultaneously in V. alginolyticus.37 The
relation between swimming speed (n) and flagellar
rotation ( f) can be defined by the equation:n ðmm=sÞ ¼ 0:113 ðmmÞ £ f ð1=sÞ
The flagellar rotation rate expected using the n:f
ratio is described in the Y-axis on the right of
Figure 3(a).
The flagellar rotation rates were measured using
a bead assay in the presence of different concentrations of NaCl (Figure 3(b)). As the swimming
speed, the rotation rate of the bead attached to theflagellar filament increased with the external NaCl
concentrations and reached a maximum speed at a
NaCl concentration of approximately 100 mM. The
profile was basically the same for all beads used.
The rotation rates at any NaCl concentration
decreased with an increase in the bead size. For
example, at high NaCl concentrations, the saturated or maximal rotation rates of beads 0.60 mm,
0.85 mm and 1.08 mm in diameter attached to theflagellar filament were 500 Hz, 250 Hz and 130 Hz,
respectively. This indicates that beads exerted theload on the motor. These results suggest that in
our measurement system, the rotation of the beadreflects flagellar rotation under various load conditions and that attaching a bead does not cause
any effects except loads to flagellar rotation.
The torque –speed relationship of Na1-driven
flagellar motors
In this study it was possible to estimate thegenerated torque of the Naþ-driven flagellar
motor. The generated torque of the motor (M) was
calculated from the rotational frictional drag coefficient ( fr) multiplied by the angular velocity (v),
M ¼ fr £ v:39 The rotational frictional drag coefficient was estimated from the sum of that resulting from the rotation of a bead ( fb) and a flagellar
filament ( ff) (for more details see Materials andMethods). To study the generated torque over awide range of speeds, the fr value was changed
from 0.5 pN nm s to 20 pN nm s by using beads ofdifferent sizes (0.46 – 1.65 mm in diameter).
Figure 4(a) shows the torque– speed relationshipof Naþ-driven flagellar motors at a NaCl concentration of 50 mM. Each point, except the open
diamond, represents a value of the generated
torque measured from one cell. The symbols
represent torque values from beads attached to the

Figure 4. (a) Torque – speed relationship at a NaCl
concentration of 50 mM. Symbols represent the beadsizes as follows: f, 1.65 mm; £ , 1.08 mm; V, 0.85 mm; K,
0.75 mm; W, 0.60 mm; A, 0.46 mm; and S, no beadattached. (b) The effect of NaCl concentration on thetorque –speed relationship. Symbols and error bars indicate the average values and standard deviation for eachsize of a bead, respectively. Circles (X), triangles (O) andsquares (B) are derived from experiments at NaCl
concentrations of 50, 10 and 3 mM, respectively. At theconcentrations of 10 and 3 mM NaCl, a bead 0.46 mm indiameter was not used. (c) The relationship between thegenerated torque and NaCl concentration when a bead
1.65 mm in diameter was attached to the flagellar filament. The horizontal scale is logarithmic.

flagellar filament having diameters of 1.65, 1.08,
0.85, 0.75, 0.60 and 0.46 mm. The open diamondplotted in Figure 4(a) was the generated torquewithout a bead attached to the filament and was
calculated by using data of the rotation rates offlagellar filaments (Figure 3(a)). The torque generated between 0 Hz and 300 Hz was relatively
constant at 3800 pN nm. In the high-speed region,
the generated torque declined steeply. The torque–
speed relationship of Naþ-driven flagellar motors
in V. alginolyticus is similar to that of Hþ-driven

Torque Generated by the Naþ-driven Flagellar Motor

motors in E. coli, although the flagellar rotationin V. alginolyticus is considerably faster than in
E. coli.33 Next, the torque–speed relationship was
measured at NaCl concentrations of 3, 10 and
50 mM (Figure 4(b)). These curves show similar
trends in the presence of any concentrations of
NaCl; that is, the generated torque was constant at
low speeds and decreased in the high-speed
region. However, both torque and speed decreased
with decreasing external NaCl concentrations. The
maximum torque, stall torque, is approximately
3800, 3000 and 2100 pN nm at NaCl concentrationsof 50, 10 and 3 mM, respectively. The zero-torquespeed estimated by extrapolating the values in thehigh-speed region was 710, 450 and 250 Hz in thepresence of 50, 10 and 3 mM NaCl, respectively.
The effect of the concentration of NaCl on theflagellar motor seems to be greater in the highspeed region. Figure 4(c) shows the relationshipbetween the generated torque and the external
NaCl concentrations when a bead 1.65 mm indiameter was attached to a flagellar filament. In
this condition, the load exerted on the motor isvery high, like that of the tethered cell. The torquegenerated by Naþ-driven flagellar motors underhigh load conditions is a linear function of thelogarithm of NaCl concentrations between 3 and50 mM.

Discussion
Many models of the rotational mechanism of
the bacterial flagellar motor have been proposed
to explain the relationship between the input andoutput of the motor.40 – 45 The input of the flagellar
motor is considered to be the ion-motive force andthe ion fluxes, and the output of the motor ischaracterized by its torque and rotation rate. In
order to understand the mechanism of the motor,
it is essential to know the relationship between
the input and output in detail. Here, the rotation
rate and the generated torque as outputs of theNaþ-driven flagellar motor of V. alginolyticus were
measured using a bead assay and the results
obtained were compared to a simple kinetic
model. Naþ-driven motors are useful for modulating inputs of the motor as the rotation rate canbe altered by changing the external NaCl concentration. A bead assay is a very effective methodfor detecting the rotation rates of flagellar motors
using a conventional optical microscope equippedwith a quadrant-type photodiode. The sphericalbead is attached to a flagellar filament as a marker.
In our system, beads larger than 0.46 mm indiameter produced sufficient signal intensity tofollow the position of the bead. This assay will
also allow the switching mechanism of theflagellar motor to be investigated, since it measures
not only the rotation rate but also the rotational
direction.
The torque generated by the motor with a beadattached can be determined by the angular velocity

1047

of the motor times the viscous drag which can becalculated as the sum of the flagellar filament andthe bead. The torque– speed relationship of theNaþ-driven flagellar motor was estimated bychanging the viscous drag on the motor, i.e. using
beads of different sizes. At a NaCl concentrationof 50 mM, the torque was relatively constant at
3800 pN nm at low speed (speeds up to , 300 Hz).
This value corresponds to the maximum
torque values obtained for E. coli, (6300 pN nm46
or4500 pN nm47),
andforStreptococcus(3000 pN nm24). In the high-speed region, thegenerated torque decreased steeply. This patternobserved in the torque– speed curve of
V. alginolyticus was similar to that shown in
E. coli.26,33 These results suggest that the mechanism of torque generation in Naþ-driven flagellar
motors is similar to that in Hþ-driven motors,
although the charged residues essential for theinteraction between the stator and rotor seem tobe different.13,14
It is thought that the bacterial flagellar motor isrotated by the sum of the torque generated byindependently functioning force-generating units
in both Hþ and Naþ-driven motors.9,10,48 The
number of force-generating units of Naþ-driven
flagellar motors was estimated to be five to nineunits in alkalophilic Bacillus using a photoreactive
inhibitor of a sodium channel48 and five to eight
units in V. alginolyticus using the same reagent(our unpublished results). From the experimental
results (3000 pN nm at 500 Hz in the presence of
50 mM NaCl; Figure 4(b)), the maximum powerof each unit can be estimated to be at least
1.0 £ 106 pN nm s21 even when eight units are
involved in torque generation, where the outputof the power is defined as torque multiplied by
the angular velocity. This value is larger than thatobtained in E. coli, 1.75 £ 105 pN nm s21.32
When external NaCl concentrations werechanged, the torque– speed relationships in thepresence of any NaCl concentration showed similar trends (the torque was constant in the low
speed region and declined steeply in the highspeed region). The NaCl concentration had a great
influence on the values of the torque and rotation
rate obtained in the torque– speed curve (Figure4(b)). The effect of the concentrations of the coupling ion on torque –speed curves of V. alginolyticus
was different from those of E. coli. The curves forE. coli did not change even when the external
environment had a pH value in the range pH 4.7 –8.8.49 The difference in the effect of the couplingion on the torque –speed relationship between
V. alginolyticus and E. coli is possibly derived fromthe difference not in the mechanism of flagellar
motors but in the cell homeostasis when the concentration of the external coupling ion is varied.
In E. coli when the external pH is varied the cells
can maintain the cytoplasmic pH value, but thepmfs do not change considerably by adjusting
their membrane potential.50 In the cells that haveNaþ-driven motors, the membrane potential of theTorque Generated by the Naþ-driven Flagellar Motor1048
cells, DC is almost independent of the external Naþ
concentration.51,52 However, there are conflicting
reports on whether the cytoplasmic Naþ concentration of cells increases linearly with the externalNaþ concentration52 or remains constant at
30 mM.51 The exact effect that the external concentrations of NaCl have on the cytoplasmic Naþ concentration is unknown. However, the effect of theexternal NaCl concentration on the cytoplasmic
Naþ concentration could be predicted. Using
tethered cells of Streptococcus under high loadconditions, the generated torque of the flagellar
motor was a linear function of the pmf.53 We
applied this result to the Naþ-driven motors inour system; that is, the generated torque wasapplied as a linear function of:smf ¼ DC 2 59 log½Naþ out =½Naþ inwhere [Naþ]out and [Naþ]in are the external andcytoplasmic Naþ concentrations, respectively. At
high loads the generated torque was a linear function of the logarithm of the NaCl concentration(Figure 4(c)). Therefore, it is likely that the cytoplasmic Naþ concentrations remain stable evenwhen the external NaCl concentrations vary.
These experimental results were then simulatedusing a simple kinetic model in an attempt todemonstrate the effect of Naþ concentration on thetorque– speed curve.44,46 The Naþ-driven flagellar
motor was reported to have both external and cytoplasmic Naþ binding sites.54 Figure 4(b) shows thatthe effect of the concentration of NaCl on the flagellar motor is greater in the high-speed region.
These results suggest that the association and dissociation of Naþ on the binding sites limit the
kinetic cycle. Thus, the four-state model proposedby Iwazawa et al.46 was adopted with some modifications. This model takes into consideration theexternal and cytoplasmic concentrations of Naþ(Figure 5(a)). As in the study by Iwazawa et al.,46
we have also assumed the four-state model asfollows. (1) Each force-generating unit of the
motor has binding sites for Naþ on both sides ofthe membrane. (2) Binding sites are non-charged
carriers and move both in backward and forward
directions with a transmembrane rate constant, k0.
(3) Charged carriers move with a rate constant, kA
or kB, and participate in torque generation. When
Mo·Naþ is transformed to Mi·Naþ with a rate
constant kA, an external sodium ion is transported
into the cytoplasm. Consequently the rotor rotateswith a given angle, f, and generates torque. (4) As
for all reactions a reverse reaction can occur.
In this simulation, the step angle, f, wasassumed to be fixed under any condition. Whenthe rotor rotates one revolution, the influx of Naþ
through the multiple force generating units(nu ¼ 8 units) can be defined as 2pnu =f: In thisstudy, the net flux was taken to be 1000 ions per
revolution, the value measured for Streptococcus.55
Figure 5(b) shows the result of the simulation using
the values described above, when [Naþ]in and DC

Figure 5. (a) A kinetic model for the cyclic reaction in
the bacterial flagellar motor. When a Naþ in the external
medium flows into the cytoplasmic medium, the procedure is as follows. (i) A Naþ in the external medium
binds to site Mo, which faces the external medium, withthe rate constant, k1, and forms Mo·Naþ; (ii) the charged
carrier, Mo·Naþ, moves across the membrane with therate constant, kA, and transforms into Mi·Naþ; (iii) Mi·Naþreleases Naþ with the rate constant, k4, and to form Mi;(iv) Mi returns to the initial state Mo with the rate constant, k0. As for all reactions, a reverse reaction may also
occur. The transition across the cytoplasmic membraneof a non-charged carrier is determined by the absolute
rate constant, k0. The transition rate constants of charged
carriers, kA and kB, are given by kA ¼ k0 exp{ 2 ðMf=nu þeDCÞ=2kT}andkB ¼ k0 exp{ðMf=nu þ eDCÞ=2kT};
respectively, where k is the Boltzmann constant, e is the
charge of an ion, T is the absolute temperature, M is thegenerated torque, nu is the number of torque generators,
DC is the membrane potential and f is the angle moved
in one cycle. In this simulation, it has been assumed
that the kinetic cycle is in a steady-state. Steady-state
occupancy probabilities ([Mo], [Mi], [Mo·Naþ], [Mi·Naþ])were calculated from four equations assuming that thetime change to each state is zero. The net rate of
the mechano-chemical cycle (kC) is given by kC ¼
kA ½Mo ·Naþ  2 kB ½Mi ·Naþ ; and the rotation rate of theflagellar motor is calculated from kC f=2p: See the text,
Iwazawa et al.46 and Berry & Berg44 for details. (b) Simulation of the dependence of the external NaCl concentration on the torque– speed relationship. Symbols are
the same as used for Figure 4(b). Parameters used toreproduce the experimental results are as follows;
k0 ¼ 5.0 £ 105 s21,
k1 ¼ k3 ¼ 6.0 £ 107 M21 s21
andk2 ¼ k4 ¼ 6.0 £ 105 s21.

are 30 mM and 2 150 mV, respectively. This model
can explain the essential features of the effect of
NaCl concentration on the torque– speed relationship using the appropriate parameters of the rate
constants. Since the values of all rate constantsin this simulation, translation, association andTorque Generated by the Naþ-driven Flagellar Motordissociation of Naþ, are in a similar range, it islikely that all kinetic steps could be rate limitingin this cycle. In other words, the steps of the association and dissociation of the coupling ions are
essential processes. The simplified model that considered smf as an input could not reproduce theeffect of the external Naþ concentration on thetorque– speed relationship in our results, because
this model predicts that the external Naþ concentration seldom had an affect in the high-speed
region. Thus, it is necessary not only to consider
the simple gradients but also the absolute concentration of the coupling ion when the models of
the flagellar rotation are proposed, as predicted by
Yoshida et al.54 and Minamino et al.56 However, inthe high-speed region, although the generated
torque decreased steeply in our experimental
results, a vertical slope in the torque –speed curve
was observed in the four-state model. Resultsfrom electrorotation studies show that the larger
rotation rates were observed when the larger
torque was applied to the flagellar motor in acounter-clockwise direction.26 In our simulations,even when an external force is applied to theflagellar motor, forcing it to rotate in a counterclockwise direction, the rotation rate does notincrease more than the zero-torque speed. Two
possibilities can explain this problem. One is thatall steps in the four-state model depend on theterm of the generated torque, as shown by Berry& Berg.44 The slope of torque –speed curve in thehigh-speed region is more sensitive to the weak
torque-dependent transitions. In the four-statemodel, if all transitions contribute to torque generation, the change in slope of the generated torque
will be shallower in the high-speed region. The
second possibility is that the angle moved in one
cycle, f, should be changeable according to thegenerated torque. If the step angle, f, generated in
the high-speed region is larger than in the lowspeed one, as observed in muscle contraction,57
then the torque–speed curve obtained here and in
the study of electrorotation could also be explainedby the four-state kinetic model. Thus, it is very
important when considering the mechanism of theflagellar motor rotation to determine the rotational
angle of the single processes and the effect of theload being exerted on the motor. From now on,
we will need to develop a measurement systemwith high enough sensitivity to detect such finemovements.

Materials and Methods
Bacterial strains and the experimental media
A smooth-swimming mutant of V. alginolyticus, strain
NMB136, was used in this study.34 Cells were cultured
to the late logarithmic phase at 30 8C in VPG medium(1% (w/v) polypeptone, 0.4% (w/v) K2HPO4, 3% (w/v)
NaCl, 0.5% (w/v) glycerol). The swimming speed andthe flagellar rotation rates of the cells were measured in
TMN medium (50 mM Tris – HCl (pH 7.5), 5 mM MgCl2,

1049
5 mM glucose, 300 mM NaCl þ KCl). Measurements
were performed after the cells were suspended in the
TMN medium with the appropriate experimental NaCl
concentration for at least 30 minutes.
Preparation of poly-L -lysine modified beads andimmobilization of cells
Carboxylated beads were incubated in 5 mg/ml poly(P-7890; Sigma Chemical Co.) in 0.4 M KOH
solution on ice for three hours. The beads were centrifuged and washed repeatedly in TMN medium (0 mM
NaCl) and stored on ice. The rotational frictional dragcoefficient of the motor was changed by using beadsof different sizes. Beads with diameters of, 1.65, 1.08,
0.75 (Polysciences), 0.85, 0.60 (Bangs Laboratories) and
0.46 mm (Molecular Probes) were used. The cell suspension was mixed with diluted poly-L -lysine modified
beads and introduced into the flow cell. To detect flagellar rotation, cells, which had a single bead attached to
the filament and were immobilized to the surface of acleaned coverslip, were selected.
L -lysine

Measurement of flagellar rotation
An inverted microscope (IX70; Olympus, Japan) witha phase-contrast objective (UplanApo £ 60, NA 1.4;
Olympus, Japan) was used for following flagellar
rotation. The microscope was placed on an air-dampingtable (TDIS-189LA; Herz, Japan). The microscope specimen could be moved by between ,1 nm and 10 mm
using a hydraulic micromanipulator (MHW-103;
Narishige, Japan) with a handmade piezo-electric stage
moved by piezo stacks (AE0203D08; TOKIN, Japan).
The phase-contrast image of the bead, which was
attached to a flagellar filament, was projected onto theface of a quadrant-type photodiode (S-6242; Hamamatsu
Photonics, Japan), which was placed in the image planeof the camera port of the microscope, coupled to a differential amplifier (OP711; Sentec, Japan). The beadpositions were recorded at a sampling rate of 48 kHz bya DAT recorder (RD-125T; TEAC, Japan), and analysedoffline by using DADiSP software (DSP Development
Corp.). The acquired data were passed through a digitalrecursive Chebyshev type 1 low-pass filter of adequate
bandwidth according to the rotation rate of a bead. The
rotation rate of a motor was computed from a peak in
the FFT spectrum of a bead position (X or Y channel),and the rotational radius of a bead was measured froma major radius of the ellipsoid in the X – Y plot. Conversion voltage to displacement was calibrated by using
a piezo-electric stage. All experiments were performed
at 25 8C.
Calculation of the generated torque
The generated torque of the motor (M) was calculated
from its rotational frictional drag coefficient ( fr) multiplied by the angular velocity (v), M ¼ fr £ v. fr was estimated from the sum of the rotational frictional dragcoefficient resulting from the rotation of the bead ( fb)and the flagellar filament ( ff). fb and ff were calculated
as:38,39
fb ¼ 8phr3b þ 6phrb r2r
ff ¼ 2pLr2 ð2p2 þ 4p2 r2 Þh={ðp2 þ 4p2 r2 Þðlnð2p=dÞ 2 0:5Þ}Here, rb is the radius of a bead, rr is the rotational

1050
radius of a bead, L ¼ 5.02 mm is the length of a filament,
d ¼ 0.016 mm is the radius of a filament, r ¼ 0.14 mm is
the helical radius, p ¼ 1.58 mm is the helical pitch. L, d, r
and p of V. alginolyticus are values measured with
TEM.37 h ¼ 9.6 £ 1024 Pa s is the viscosity of the motility
medium.

Acknowledgements
We thank Toshio Yanagida for kindly arranging
for Y.S. to work on this project; Fumio Oosawa,
Richard Berry, Tohru Minamino and Keiichi
Namba for discussions; Jan West and Yoshiharu
Ishii for critically reading the manuscript. This
work was supported, in part, by grants-in-aid
for scientific researches from the Ministry of
Education, Science and Culture of Japan (to M.H.
and A.I.) and from the JSPS Research Fellowshipfor Young Scientists (to Y.S.).

References
1. Berg, H. C. & Anderson, R. A. (1973). Bacteria swim
by rotating their flagellar filaments. Nature, 245,
380– 382.
2. Manson, M. D., Tedesco, P., Berg, H. C., Harold, F. M.
& van der Drift, C. (1977). A protonmotive force
drives bacterial flagella. Proc. Natl Acad. Sci. USA,74, 3060– 3064.
3. Matsuura, S., Shioi, J. & Imae, Y. (1977). Motility in
Bacillus subtilis driven by an artificial protonmotive
force. FEBS Letters, 82, 187–190.
4. Hirota, N., Kitada, M. & Imae, Y. (1981). Flagellar
motors of alkalophilic Bacillus are powered by anelectrochemical potential gradient of Naþ. FEBS
Letters, 132, 278– 280.
5. Macnab, R. M. (1996). Flagella and motility. In
Escherichia coli and Salmonella (Neidhardt, F. C., ed.),pp. 123– 145, American Society for Microbiology,
Washington, DC.
6. Blair, D. F. (1995). How bacteria sense and swim.
Annu. Rev. Microbiol. 49, 489– 522.
7. Lloyd, S. A., Tang, H., Wang, X., Billings, S. & Blair,
D. F. (1996). Torque generation in the flagellar motorof Escherichia coli: evidence of a direct role for FliG
but not for FliM or FliN. J. Bacteriol. 178, 223– 231.
8. Francis, N. R., Sosinsky, G. E., Thomas, D. &Derosier, D. J. (1994). Isolation, characterization andstructure of bacterial flagellar motors containing theswitch complex. J. Mol. Biol. 235, 1261– 1270.
9. Block, S. M. & Berg, H. C. (1984). Successive incorporation of force-generating units in the bacterial
rotary motor. Nature, 309, 470– 472.
10. Blair, D. F. & Berg, H. C. (1988). Restoration of torquein defective flagellar motors. Science, 242, 1678– 1681.
11. Lloyd, S. A. & Blair, D. F. (1997). Charged residues of
the rotor protein FliG essential for torque generationin the flagellar motor of Escherichia coli. J. Mol. Biol.
266, 733–744.
12. Zhou, J. D. & Blair, D. F. (1997). Residues of the cytoplasmic domain of MotA essential for torque generation in the bacterial flagellar motor. J. Mol. Biol.
273, 428– 439.

Torque Generated by the Naþ-driven Flagellar Motor

13. Zhou, J. D., Lloyd, S. A. & Blair, D. F. (1998). Electrostatic interactions between rotor and stator in thebacterial flagellar motor. Proc. Natl Acad. Sci. USA,95, 6436– 6441.
14. Yorimitsu, T., Sowa, Y., Ishijima, A., Yakushi, T. &Homma, M. (2002). The systematic substitutions
around the conserved charged residues of thecytoplasmic loop of Naþ-driven flagellar motor
component PomA. J. Mol. Biol. 320, 403– 413.
15. McCarter, L. L. (2001). Polar flagellar motility of theVibrionaceae. Microbiol. Mol. Biol. Rev. 65, 445– 462.
16. Yorimitsu, T. & Homma, M. (2001). Naþ-driven
flagellar motor of Vibrio. Biochim. Biophys. Acta, 1505,
82 –93.
17. Okunishi, I., Kawagishi, I. & Homma, M. (1996).
Cloning and characterization of motY, a gene codingforacomponentofthesodium-driven
flagellar motor in Vibrio alginolyticus. J. Bacteriol. 178,
2409– 2415.
18. Okabe, M., Yakushi, T., Kojima, M. & Homma, M.
(2002). MotX and MotY, specific components of thesodium-driven flagellar motor, colocalize to theouter membrane in Vibrio alginolyticus. Mol. Microbiol.
46, 125– 134.
19. Okabe, M., Yakushi, T., Asai, Y. & Homma, M. (2001).
Cloning and characterization of motX, a Vibrio
alginolyticus sodium-driven flagellar motor gene.
J. Biochem. 130, 879– 884.
20. Asai, Y., Kawagishi, I., Sockett, E. & Homma, M.
(1999). Hybrid motor with the Hþ- and Naþ-driven
components can rotate Vibrio polar flagella by using
sodium ions. J. Bacteriol. 181, 6322– 6338.
21. Caplan, S. R. & Karaivanov, M. (1993). The bacterial
flagellar motor. Int. Rev. Cytol. 147, 97– 164.
22. Berry, R. M. & Armitage, J. P. (1999). The bacterial
flagella motor. Advan. Microb. Physiol. 41, 291–337.
23. Silverman, M. & Simon, M. I. (1974). Flagellar
rotation and the mechanism of bacterial motility.
Nature, 249, 73 – 74.
24. Lowe, G., Meister, M. & Berg, H. C. (1987). Rapid
rotation of flagellar bundles in swimming bacteria.
Nature, 325, 637– 640.
25. Washizu, M., Kurahashi, Y., Iochi, H., Kurosawa, O.,
Aizawa, S., Kudo, S. et al. (1993). Dielectrophoretic
measurement of bacterial motor characteristics. IEEE
Trans. Ind. Appl. 29, 286– 294.
26. Berg, H. C. & Turner, L. (1993). Torque generated by
the flagellar motor of Escherichia coli. Biophys. J. 65,
2201– 2216.
27. Kudo, S., Magariyama, Y. & Aizawa, S.-I. (1990).
Abrupt changes in flagellar rotation observed bylaser dark-field microscopy. Nature, 346, 677– 680.
28. Kamimura, S. (1987). Direct measurement of nanometric displacement under an optical microscope.
Appl. Optics, 26, 3425– 3427.
29. Svoboda, K., Schmidt, C. F., Schnapp, B. J. & Block,
S. M. (1993). Direct observation of kinesin steppingby optical trapping interferometry. Nature, 365,
721– 727.
30. Finer, J. T., Simmons, R. M. & Spudich, J. A. (1994).
Single myosin molecule mechanics: piconewton
forces and nanometre steps. Nature, 368, 113 –119.
31. Ishijima, A., Kojima, H., Funatsu, T., Tokunaga, M.,
Higuchi, H., Tanaka, H. & Yanagida, T. (1998).
Simultaneous observation of individual ATPaseand mechanical events by a single myosin
molecule during interaction with actin. Cell, 92,
161– 171.

Torque Generated by the Naþ-driven Flagellar Motor

32. Ryu, W. S., Berry, R. M. & Berg, H. C. (2000). Torquegenerating units of the flagellar motor of Escherichia
coli have a high duty ratio. Nature, 403, 444– 447.
33. Chen, X. & Berg, H. C. (2000). Torque– speed
relationship of the flagellar rotary motor ofEscherichia coli. Biophys. J. 78, 1036– 1041.
34. Kojima, S., Atsumi, T., Muramoto, K., Kudo, S.,
Kawagishi, I. & Homma, M. (1997). Vibrio alginolyticus
mutants resistant to phenamil, a specific inhibitorof the sodium-driven flagellar motor. J. Mol. Biol.
265, 310–318.
35. Follett, E. A. C. & Gordon, J. (1963). An electron
microscope study of Vibrio flagellar. J. Gen. Microbiol.
32, 235– 239.
36. Kamimura, S. & Kamiya, R. (1992). High-frequency
vibration in flagellar axonemes with amplitudes
reflecting the size of tubulin. J. Cell Biol. 116,
1443–1454.
37. Magariyama, Y., Sugiyama, S., Muramoto, K.,
Kawagishi, I., Imae, Y. & Kudo, S. (1995). Simultaneous measurement of bacterial flagellar rotation
rate and swimming speed. Biophys. J. 69, 2154– 2162.
38. Muramoto, K., Kawagishi, I., Kudo, S., Magariyama,Y., Imae, Y. & Homma, M. (1995). High-speed
rotation and speed stability of sodium-driven
flagellar motor in Vibrio alginolyticus. J. Mol. Biol.
251, 50 – 58.
39. Berg, H. C. (1993). Random Walks in Biology, Princeton
University Press, Princeton, NJ.
40. Oosawa, F. & Hayashi, S. (1983). Coupling between
flagellar motor rotation and proton flux in bacteria.
J. Phys. Soc. Jpn, 52, 4019– 4028.
41. Läuger, P. (1988). Torque and rotation rate of thebacterial flagellar motor. Biophys. J. 53, 53 – 65.
42. Meister, M., Caplan, S. R. & Berg, H. C. (1989).
Dynamics of a tightly coupled mechanism forflagellar rotation. Biophys. J. 55, 905– 914.
43. Berry, R. M. (1993). Torque and switching in thebacterial flagellar motor. An electrostatic model. Biophys. J. 64, 961– 973.
44. Berry, R. M. & Berg, H. C. (1999). Torque generated
by the flagellar motor of Escherichia coli while driven
backward. Biophys. J. 31, 580– 587.
45. Atsumi, T. (2001). An ultrasonic motor model forbacterial flagellar motors. J. Theor. Biol. 213, 31 –51.

1051

46. Iwazawa, J., Imae, Y. & Kobayasi, S. (1993). Study ofthe torque of the bacterial flagellar motor using arotating electric field. Biophys. J. 64, 925– 933.
47. Berry, R. M. & Berg, H. C. (1997). Absence of abarrier to backwards rotation of the bacterial flagellar
motor demonstrated with optical tweezers. Proc. Natl
Acad. Sci. USA, 94, 14433– 14437.
48. Muramoto, K., Sugiyama, S., Cragoe, E. J. & Imae, Y.
(1994). Successive inactivation of the force-generating units of sodium-driven bacterial flagellar motors
by a photoreactive amiloride analog. J. Biol. Chem.
269, 3374– 3380.
49. Chen, X. & Berg, H. C. (2000). Solvent-isotope andpH effects on flagellar rotation in Escherichia coli.
Biophys. J. 78, 2280– 2284.
50. Harold, F. M. & Maloney, P. C. (1996). Energy
transduction by ion currents. In Escherichia coli and
Salmonella (Neidhardt, F. C., ed.), pp. 283– 306,
American Society for Microbiology, Washington, DC.
51. Hirota, N. & Imae, Y. (1983). Naþ-driven flagellar
motors of an alkalophilic Bacillus strain YN-1. J. Biol.
Chem. 258, 10577– 10581.
52. Liu, J. Z., Dapice, M. & Khan, S. (1990). Ion selectivity of the Vibrio alginolyticus flagellar motor.
J. Bacteriol. 172, 55236– 55244.
53. Manson, M. D., Tedesco, P. & Berg, H. C. (1980).
Energetics of flagellar rotation in bacteria. J. Mol.
Biol. 138, 541–561.
54. Yoshida, S., Sugiyama, S., Hojo, Y., Tokuda, H. &Imae, Y. (1990). Intracellular Naþ kinetically interferes with the rotation of the Naþ-driven flagellar
motors of Vibrio alginolyticus. J. Biol. Chem. 256,
20346 – 20350.
55. Meister, M., Lowe, G. & Berg, H. C. (1987). The
proton flux through the bacterial flagellar motor.
Cell, 49, 643– 650.
56. Minamino, T., Imae, Y., Oosawa, F., Kobayashi, Y. &Oosawa, K. (2003). Effect of intracellular pH on
rotational speed of bacterial flagellar motors.
J. Bacteriol. 185, 1190– 1194.
57. Yanagida, T., Harada, Y. & Ishijima, A. (1993). Nanomanipulation of actomyosin molecular motors invitro: a new working principle. Trends Biochem. Sci.
18, 319–324.

Edited by J. Karn
(Received 21 October 2002; received in revised form 27 January 2003; accepted 30 January 2003)