doi:10.1016/j.jmb.2007.12.023

J. Mol. Biol. (2008) 376, 1251–1259

Available online at www.sciencedirect.com

Torque–Speed Relationships of Na + -driven Chimeric
Flagellar Motors in Escherichia coli
Yuichi Inoue 1 , Chien-Jung Lo 2 , Hajime Fukuoka 1 , Hiroto Takahashi 1 ,Yoshiyuki Sowa 2 , Teuta Pilizota 2 , George H. Wadhams 3 ,Michio Homma 4 , Richard M. Berry 2 and Akihiko Ishijima 1 ⁎1Institute of Multidisciplinary
Research for Advanced
Materials, Tohoku University,
Katahira, Aoba-ku,
Sendai 980-8577, Japan
2

Clarendon Laboratory,
Department of Physics,
University of Oxford,
Parks Road, Oxford OX1 3PU,UK
3

Department of Biochemistry,
University of Oxford,
South Parks Road,
Oxford OX1 3QU, UK
4
Division of Biological Science,
Graduate School of Science,
Nagoya University, Furo-cho,
Chikusa-ku, Nagoya 464-8602,
Japan

Received 12 October 2007;
received in revised form
10 December 2007;
accepted 11 December 2007
Available online
15 December 2007
Edited by J. Karn

The bacterial flagellar motor is a rotary motor in the cell envelope of bacteriathat couples ion flow across the cytoplasmic membrane to torque generation
by independent stators anchored to the cell wall. The recent observation of
stepwise rotation of a Na+-driven chimeric motor in Escherichia coli promises
to reveal the mechanism of the motor in unprecedented detail. We measured
torque–speed relationships of this chimeric motor using back focal planeinterferometry of polystyrene beads attached to flagellar filaments in thepresence of high sodium-motive force (85 mM Na+). With full expression of
stator proteins the torque–speed curve had the same shape as those of wildtype E. coli and Vibrio alginolyticus motors: the torque is approximately
constant (at ∼ 2200 pN nm) from stall up to a “knee” speed of ∼ 420 Hz, andthen falls linearly with speed, extrapolating to zero torque at ∼ 910 Hz.
Motors containing one to five stators generated ∼ 200 pN nm per stator at
speeds up to ∼100 Hz/stator; the knee speed in 4- and 5-stator motors is not
significantly slower than in the fully induced motor. This is consistent withthe hypothesis that the absolute torque depends on stator number, but the
speed dependence does not. In motors with point mutations in either of two
critical conserved charged residues in the cytoplasmic domain of PomA,
R88A and R232E, the zero-torque speed was reduced to ∼ 400 Hz. The
torque at low speed was unchanged by mutation R88A but was reduced to∼1500 pN nm by R232E. These results, interpreted using a simple kinetic
model, indicate that the basic mechanism of torque generation is the same
regardless of stator type and coupling ion and that the electrostaticinteraction between stator and rotor proteins is related to the torque–speed
relationship.
© 2007 Elsevier Ltd. All rights reserved.

Keywords: rotary motor; flagellar motor; torque generation; torque–speed
relationship; chimeric motor

Introduction
The bacterial flagellar motor is a rotary molecularmachine that allows many species of bacteria to
swim.1,2 The motor is powered by the flow of ions,
H+ in Escherichia coli and Na+ in marine Vibrio
species, driven by an inward-directed electrochemical gradient across the cytoplasmic membrane.
*Corresponding author. E-mail address:
ishijima@tagen.tohoku.ac.jp
Abbreviation used: smf, sodium-motive force.

Rotation is driven by an ensemble of up to ∼11
torque-generating stators3 containing the proteins
MotA and MotB in the H+-driven motor of E. coli and
PomA and PomB in the Na+-driven motor of Vibrio
alginolyticus.1,2 The torque–speed relationship is an
important test of models of the motor mechanism as
it allows quantitative comparisons between predictions of models and experimental data. It has beenmeasured using electrorotation 4 , 5 or opticaltweezers6 to apply external torque to tethered cells
and by varying the viscous load either in swimming
cells,7 or with polystyrene beads attached to truncated flagellar stubs.8,9 A common feature of the0022-2836/$ - see front matter © 2007 Elsevier Ltd. All rights reserved.

1252
torque–speed relationships of E. coli and V. alginolyticus is a sharp knee between a “plateau” of nearly
constant torque at low speeds and a much steeper
linear decrease of torque with speed at higher speeds.
At room temperature the knee is at ∼175 Hz in E. coli
and at ∼450 Hz in V. alginolyticus, and the zero-torquespeeds are ∼350 and ∼700 Hz, respectively. In E. coli,
the knee speed and zero-torque speed are temperature dependent, but the plateau torque is not.4,8Plateau torques appear to vary widely between
species, from ∼ 3800 pN nm in V. alginolyticus,
through ∼1300 pN nm in E. coli (in the most recent
estimate3), down to ∼350 pN nm in Caulobacter
crescentus.7 Torque–speed curves have also beenmeasured for E. coli motors containing low numbersof stators.10 Single-stator motors operated in theplateau regime up to the highest speeds measured,∼100 Hz, and the knee speed in motors with two to
five stators appeared to be the same as in motors witha full stator complement. In C. crescentus, motors
operated in the plateau regime up to the highest
speeds measured, ∼320 Hz.7 The knee typicallycorresponds to the speed at which the output powerof the motor is maximum. Different values of knee
speed and plateau torque in different species may
reflect different modes of swimming or different ionmotive forces, whereas the common shape of thetorque–speed curve probably indicates that the basic
mechanism of the motor is the same in all species.
A recent breakthrough in experimental investigations of the flagellar motor was the observation of 26
angular steps per revolution in the rotation of a Na+driven chimeric motor in E. coli with a stator composed of PomA from V. alginolyticus and the PomB/
MotB fusion protein, PotB.11 The ability to control
the speed of the chimeric motor via the sodiummotive force (smf), which is less tightly constrainedin E. coli than the proton-motive force, is likely tomake it an important tool in future investigations of
the mechanism of the flagellar motor. Therefore, it isimportant to characterize its mechanical properties,
such as the torque–speed relationship.
Electrostatic interactions that are important for
motor rotation in E. coli have been demonstrated
between two conserved charged residues in thecytoplasmic domain of MotA (Arg90 and Glu98) andconserved charged residues in the rotor protein
FliG.12 The corresponding residues in PomA (Arg88
and Glu96) are less important in V. alginolyticus,13,14
but regain their importance when PomA is part of thechimeric PomA/PotB stator in E. coli.15 Furthermore,
PomA contains additional charged residues, such as
Arg215 and Arg232 in the cytoplasmic domain,which appear to be important for rotation of Na+driven motors.15 Measuring the torque–speed relationships of a range of mutants in these cri132#?>ticalresidues will allow detailed inferences to be madeabout their effect on the rotor–stator interactions that
underlie torque generation in the flagellar motor.
In this study we measured torque–speed relationships of the Na+-driven chimeric PomA/PotB motorin E. coli. The torque–speed relationship with a fullcomplement of stators was the same shape as those

Torque and Speed of Chimeric Flagellar Motorsof wild-type H+- and Na+-driven motors, with aplateau torque close to that of the E. coli motor andknee and zero-torque speeds close to those of the V.
alginolyticus motor. Torque–speed relationships with
low numbers of stators were consistent withprevious measurements of the E. coli motor. We
also obtained the first torque–speed curves ofmotors containing mutations in critical charged
residues, demonstrating that these mutations affectthe knee and zero-torque speeds and the plateau
torque in different ways.

Results
Torque–speed relationship with full induction of
stator proteins
We measured the speed of polystyrene beads
attached to truncated flagellar filament stubs usingback focal plane interferometry as described elsewhere.10 The torque generated by the motor canbe estimated as M = ( fb + ff)ω, where fb and ff arerotational frictional drag coefficients of the bead andthe filament stub, respectively, and ω is angular velocity. fb can be estimated as fb = 8πηrb3 + 6πηrbrr 2 =gbη,10 where rb and rr are the radius and therotational eccentricity of the bead, η is the viscosityof the medium and gb collects all parameters otherthan η that affect fb. Assuming no change oforientation of bead and filament with motor
speed,3 gb is a constant for each bead on a particularmotor. Because the length and orientation of thefilament stub is unknown, we need to estimate ff toobtain an accurate estimate of M. Figure 1a shows
the speed versus time of a 0.36-μm bead attached to achimeric motor with full induction of stator proteins,in motility buffer with concentrations 0–13% (w/v)
of the viscous agent Ficoll.8 As the Ficoll concentration increased, the speed decreased from ∼ 600 to∼ 120 Hz. The speed recovered when the medium
was exchanged back to motility buffer without
Ficoll, indicating that the motor was not permanently altered by the medium exchanges. Figure 1b
shows preliminary estimates of the torque versus
speed using the equation M = fbω, which underestimates M due to the neglect of ff. The filled squareshows the preliminary estimate from fifteen 1-μm
polystyrene beads and no Ficoll; the other points arefrom five different 0.36-μm beads and various Ficoll
concentrations as in Fig. 1a. All estimates for a given
bead underestimate the true torque by the same
factor, R = fb/( fb + ff) = gb/( gb + gf), where ff = gfη andthe geometric factor for filament drag, gf, is assumed
to be constant. A value of gf can be obtained byequating the plateau torques measured at low speedby 1-μm beads without Ficoll and by 0.36-μm beads
with 13% Ficoll:
ðgb1 þ gf ÞD1 N1 ¼ ðgb2 þ gf ÞD2 N2 ;

ð1Þwhere gb1 and gb2 are the geometric factors of 1- and0.36-μm beads, η1 (= 9.86 × 10 − 4 Pa s) and η2

1253

Torque and Speed of Chimeric Flagellar Motors

Fig. 1. Torque–speed relationships with fully induced stators. (a) Speed versus time of a single chimeric motor
measured using a 0.36-μm (diameter) bead in media containing 0–13% (w/v) of the viscous agent Ficoll. (b) Preliminary
torque estimates versus speed, neglecting the contribution of filament drag, using fifteen 1-μm beads without Ficoll(squares, mean ± SD) and 5 individual 0.36-μm beads in 0%, 2%, 3%, 4%, 5%, 6%, 7%, 9% and 13% Ficoll (one symbol foreach bead). (c) Corrected torque versus speed, using the method described in the text. The average filament dragcoefficient without Ficoll was estimated as 0.80 ± 0.20 pN nm s/rad (mean ± SD). The torque–speed curve was fitted by two
straight lines with an intersection at 421 Hz. (d) Comparison of the torque–speed relationships of different flagellar
motors, estimated using four different bead sizes without Ficoll. Measured speeds and estimated torques of chimericmotors (circles) and the H+-driven E. coli motor (squares), using the average filament drag coefficient from (c) (mean ± SD).
Continuous line: the fitted line for the chimeric motor in (c). Dotted line: published torque–speed relationship,8 assuming
the zero-speed torque is similar to the torque with 1-μm beads. Dashed line: published torque–speed relationship for
V. alginolyticus motors.9 Beads (10–18) of each size were measured for each motor.

(= 8.18 × 10− 3 Pa s) are viscosities at 0% and 13%
Ficoll, and ω1 and ω2 are speeds measured with 1μm beads without Ficoll and 0.36-μm beads with
13% Ficoll, respectively. Rearranging Eq. (1) gives
gf ¼

gb2 D2 N2  gb1 D1 N1:
D1 N1  D2 N2

ð2Þ

We used the data shown in Fig. 1a and b to estimate
gf separately for each of the five 0.36-μm beads
measured with Ficoll [ω1 = 84.0 ± 6.5 Hz, gb1 = (4.09 ±0.49) × 109 nm3, five separate pairs of values of ω2
and gb2). The average value was gf = (1.29 ±0.32) × 108
nm3, comparable to gb2 values, and the corresponding R values for the 0.36-μm beads ranged from 0.59
to 0.74 with average 0.64 ± 0.06 (mean ± SD). For
1-μm beads we found R = 0.97 ± 0.01; the filamentdrag is very small compared to the drag of 1-μm
beads. Figure 1c shows corrected torques equal tothe preliminary torques divided by R, using thesame symbols as in Fig. 1b. The corrected torque–
speed relationship was fitted by two straight lines,

following previous reports of the flagellar motor
torque–speed relationship, 8,9 with a maximum
torque of ∼ 2200 pN nm, a zero-torque speed of∼ 910 Hz and a knee speed of ∼ 420 Hz.
Rotational speed of the chimeric motor was alsomeasured with four different sizes of beads without Ficoll. In Fig. 1d, torque was calculated by M =( fb + ff)ω, using the average value of gf (above) toestimate ff = 0.80 ± 0.20 (pN nm s/rad) for all beadsizes. The torque–speed relationship measured inthis way (circles) was consistent with the fitted lineof Fig. 1c (continuous line), validating this simpler
method. Further validation was obtained by comparing the torque–speed curve of the wild-type
E. coli motor measured using the same method(Fig. 1d, squares) to published results (dotted line).8
Torque–speed relationships with low numbersof stators
Figure 2 shows the speed versus time of motors
attached to beads of diameter 1, 0.55 and 0.36 μm

1254

Torque and Speed of Chimeric Flagellar Motorsthat each generates the same torque at speeds below
the knee. With 0.36-μm beads, the speed increase
was almost linear for 1–5 levels (filled squares), but asaturating fit was required to include the measured
fully induced speed at level 11 (open square).
Assuming that the number of stators in a fully
induced motor does not depend on bead size, thisresult is consistent with the measured reduction in
motor torque at speeds above the knee at ∼ 420 Hz.
Figure 3b shows torque–speed relationships ofFig. 2. Resurrection experiments. Speed versus time(left) and speed histograms (right) for individual 1- (a),0.55- (b) and 0.36-μm (c) beads attached to chimericmotors after induced expression of stator proteins.
Continuous lines are multiple Gaussian fits.

(Fig. 2a, b and c, respectively), following theinduction of stator proteins. Speed increments
correspond to successive incorporation of torquegenerating stators, a process known as “resurrection.”3,10,16 Speed levels during resurrection wereobtained by fitting a multiple Gaussian distribution
to speed histograms (Fig. 2a–c, right) and plotting
against level number in Fig. 3a. With 0.55- and 1-μm
beads, the speed increased linearly with speed level(Fig. 3a, filled triangles and circles), extrapolating to
the fully induced speed at approximately level 11(open triangle and circle). This confirms that thechimeric motor, like the H+-driven wild-type motor,
can contain up to ∼ 11 torque-generating units3 andFig. 3. Torque–speed relationships with low numbersof stators. (a) Speed versus level number from resurrection
traces (filled symbols). Speeds (mean ± SD) were frommultiple-Gaussian fitting of speed histograms (Fig. 2,
right) of 6–15 resurrections with each bead size 1 μm
(circles), 0.55 μm (triangles), and 0.36 μm (squares).
Speeds of fully induced motors with the same beadsizes (open symbols) are shown at level 11, whichassumes that the number of torque-generating units atfull induction is ∼ 11. Dashed lines are linear fits except
for 0.36-μm beads, where the Michaelis–Menten equation
is used. (b) Torque–speed relationships of motors with
1–5 stators (filled squares, open squares, filled triangles,
open triangles and diamonds, respectively) calculatedusing the data of (a) and the filament drag coefficient in
Fig. 1c. The torque–speed relation for fully induced
motors is reproduced from Fig. 1d for comparison (filled
circles). Dashed lines are drawn at intervals of 200 pN nm
to guide the eye.

Torque and Speed of Chimeric Flagellar Motors

1255

motors with one to five stators (filled squares, open
squares, filled triangles, open triangles and diamonds, respectively) calculated using the data ofFig. 3a and the method of Fig. 1d. The torque–
speed relation for fully induced motors is reproduced from Fig. 1d for comparison (filled circles).
Torque increased in proportion to the number ofstators, and for each stator number torque wasapproximately constant over the range of measured
speeds, up to ∼ 100 Hz per stator. Note in particularthat the knee speed in four- and five-stator motors
is not significantly slower than in the fully induced
motor.
Torque–speed relationships of motors with
PomA mutations
Conserved arginines at positions 88 and 232 in
the cytoplasmic domain of PomA have been shown
to be important for torque generation in the chimeric motor.15 Their approximate locations areillustrated in Fig. 4a, which also indicates putativeinteractions with charged residues on FliG. R88
corresponds to the conserved residue R90 in E. coli
MotA; R232 is a charged residue near the C-terminus of PomA with no corresponding residue inE. coli. Chimeric motors with the point mutations
R88A or R232E in PomA were reported as totally
deficient in swarming on soft agar and weakly
swimming in liquid media.15 We repeated theseexperiments using the host strain RP6894 in which
motAmotB is deleted and flagellar filament androtor proteins are wild type.17 Our results were thesame as those reported previously except we foundthat R88A showed normal rather than weakswimming (Fig. 4b), perhaps as a consequence of
expressing FliG from the chromosome rather thanfrom a plasmid as in the previous work.15 Figure 4cshows torque–speed curves for chimeric motors
with fully expressed PomA, either wild type(circles) or with mutations R88A (triangles) or
R232E (squares). The method and filament dragcoefficients are the same as for Fig. 1d, but the hoststrain is different. Sticky flagellar filaments wereexpressed from plasmid pYS11 in host strain YS34
in the experiments of Figs. 1–3, but from thechromosome in host strain JHC36 in the experiments of Fig. 4c. The torque–speed relationship of
the wild-type PomA motor reconstituted in JHC36(Fig. 4c, circles) was indistinguishable from that of
the same motor in YS34 (Fig. 1d), demonstratingthat this change of host strain does not alter the
motor. At low speeds the mutation R232E reduces
the torque to ∼65%, whereas R88A causes littlechange. At high speeds both mutations reduce thetorque substantially and the zero-torque speed isreduced from ∼ 910 to ∼ 400 Hz. These results
indicate that both mutations slow the mechanism of
the motor and that R232 has an additional effect inreducing the torque at low speeds. The lines inFig. 4c are torque–speed curves predicted by asimple kinetic model for the wild-type and mutant
motors (see Discussion).

Fig. 4. Chimeric motors with point mutations in PomA.
(a) A sketch of the topology of PomA and PotB, showing thelocations of conserved charged residues R88 and R232 in
PomA and putative interactions with the rotor protein FliG.
(b) Swimming of cells with PomA point mutations.
Chimeric stators with no mutation, R88A, or R232E in
PomA were expressed in RP6894, a strain with normal
flagellar filament and rotor. Fractions of cells swimming(mean ± SD, left axis) are shown as bars, speeds ofswimming cells (mean ± SD, right axis) as circles. (c)Torque–speed relationships. The same PomA variants asin b were expressed in JHC36, a strain with sticky flagellar
filaments expressed from the chromosome. Torque-speed
curves were measured using the method and filament dragcoefficients of Fig. 1d. Circles, chimeric motor; triangles,
chimeric motor with R88A; squares, chimeric motor with
R232E. Speeds were measured in 10–20 cells from two to
three separate cultures. Fitted lines are torque–speed curves
predicted by a simple kinetic model (Fig. 5 and Table 1). For
PomA-R88A the three parameter sets of Table 1 gave
indistinguishable fits, so only the fit for the parameters in
the rightmost column is shown (dashed line). For PomAR232E, fits with reduced numbers of ions (N, dotted line) or
reduced membrane potential (ΔΨ, dashed line) are shown.

1256

Discussion
The torque–speed relationship of the chimericmotor has the same shape as those of the wild-type
motors of E. coli and V. alginolyticus (Fig. 1d),suggesting that all three motors share the same
basic mechanism for torque generation. In particular, our Ficoll experiments demonstrate that the
Na+-driven chimera shows the same sharp knee
(Fig. 1c) between the plateau and high-speed
domains as the H+-driven E. coli motor, a featurethat is not easily predicted by models of the motor
mechanism.5,18 The Na+-driven motors are faster,with knee and zero-torque speeds approximately
double those of the H+-driven motor, which isfurther slowed when H+ is replaced by deuterium. 19
These observations suggest that the rate-limiting
step in the mechanochemical cycle of torque generation at low load is related to ion transfer.
The plateau torque of the chimera was ∼ 2200 pN
nm, compared to ∼3800 and ∼1500 pN nm for
V. alginolyticus and E. coli motors, respectively (Fig.
1d). The values for the chimera and E. coli motors are
consistent with previous results using 1-μm beads.3
Considering the balance between input energy(number of ions × ion-motive force) and output
work (2π × torque), the minimum number of ionsrequired per revolution of the E. coli motor waspreviously estimated as 38 ± 11 per stator.3 A similarcalculation using the measured smf in E. coli(− 187 mV20) yields a similar result for the chimericmotor (42 ± 4 per stator). The plateau torque reported for V. alginolyticus is 1.7 times larger than thatof the chimeric motor, which is difficult to explain.
Excluding the possibility of experimental error, wemight speculate that the diameter of the motor islarger, which would increase both the number ofstators and the number of ions used per stator per
revolution (presumably proportional to the periodicity of the rotor). Increasing both of these numbers,
and also the smf, by a factor of 1.2 would account for
the larger torque compared to E. coli. The chimericmotor has been observed to take 26 steps per
revolution when driven by a presumed single stator
at low smf.11 Thus, it is just possible within the limits
of experimental error that a single ion might drive asingle step of this size in the chimera and E. coli
motors, but the V. alginolyticus motor would needeither to take smaller steps or to use more ions per
step. If all three types of motors do indeed use two
or more ions per step, then the increased torque ofthe V. alginolyticus motor could be explained bygreater efficiency, although it seems unlikely that theE. coli motor would be less than half as efficient asthat of V. alginolyticus. The additional motor
proteins, MotX and MotY, in V. alginolyticus might
have some role to play, but these questions will only
be resolved with further experimental investigation.
Our measurements of torque and speed in resurrection experiments are consistent with previous
results using the E. coli motor:10 the shape of thetorque–speed curve appears to be independent ofstator number. Thus, it appears that in the chimera,

Torque and Speed of Chimeric Flagellar Motors

as previously inferred for the wild-type motor, thestators work with a high duty ratio and there
exists a rate-limiting step in the mechanochemical
cycle of each stator that cannot be speeded up by
the torque transmitted from other stators via therotor.
Our measurements of torque–speed curves ofsingle flagellar motors with mutations in criticalcharged residues of PomA represent an improvement over the swarming and swimming assays thatare used to discover and classify these mutations.15
Swarm assays are dependent upon growth andchemotaxis21 and the relationship between swimming and motor rotation is complicated by theformation of flagellar bundles in E. coli.22 The
mutations R88A and R232E leave the shape ofthe torque–speed curve essentially unchanged,
although the slope in the plateau region may be
steeper, and the ratio of knee- to zero-torque speeds
larger, than with wild-type PomA. Both mutations
reduce the zero-torque speed to ∼400 Hz, indicatingthat they slow the rates of transitions in themechanochemical cycle. R232E also reduces theplateau torque to about ∼ 65%. Several explanations
for this are possible. Perhaps the simplest is that the
number of stators in a fully induced motor is
reduced by this mutation, indicating a role for
R232 in stator assembly. Independent evidence for aconnection between torque generation and statorassembly already exists: stators detach from the
motor when it is stopped by de-energization.11,23,24
However, our preliminary experiments with expressing low levels of stators suggest that the motor canbe equipped with 11 R232E chimeric stators and the
speed per stator is reduced by R232E mutation (datanot shown). Thus, alternatives other than stator
number may be required for the explanations. It is
possible that the coupling ratio (fewer ions per
revolution) or efficiency (ion flow without torquegeneration) could be altered, for example, byslippage between the rotor and stator, or the smf
could be reduced because ion leakage through the
mutant: expression of chimeric stators has beenshown to increase the internal Na+ concentration in
E. coli,25 indicating that the ion flux through achimeric motor is sufficient to alter the smf in E. coli.
Further experiments will be needed to distinguishbetween these alternatives.
We used a simple kinetic model of the motor
mechanism to calculate torque–speed curves forcomparison to the data. The kinetic scheme of themodel, which was originally proposed by Iwazawa
et al.26 and was used to fit the V. alginolyticus data,9 isreproduced in Fig. 5. Rate constants k1–k4 describe
Na+ exchange with the external and cytoplasmic
media, k0 a conformational change that changes
which side of the membrane accesses the Na+binding site, and kA, kB the equivalent conformational change with Na+ bound. The model assumes
that the conformational change with Na+ bound isentirely responsible for tight coupling between
charge translocation across the membrane androtation, so that only the rate constants kA and kB

1257

Torque and Speed of Chimeric Flagellar Motors

Fig. 5. A simple kinetic model of the mechanochemical
cycle of torque generation9,26 was used to understand theeffect of mutation on the torque–speed relationship of thechimeric motor. Parameters used to fit the data of Fig. 4c
are shown in Table 1. See the text for details.

depend upon membrane voltage (ΔΨ) and torque(M). For simplicity, these rate constants are defined
as kA = k0 exp[−(2πM/N + eΔΨ)/2kT] and kB = k0 exp[(2πM/N + eΔΨ)/2kT], where k is Boltzmann's constant, e is the elementary charge, T is the absolutetemperature and N is the number of ions coupled toeach revolution of the motor. These expressions
automatically satisfy detailed balance, which further
requires (k1k4)/(k2k3) = 1. Thus, the model has five
free parameters (N, k0, and any three out of k1–k4) inaddition to M, ΔΨ and the internal and external Na+
concentrations (Table 1).
Using the parameter values listed in Table 1, themodel predicts torque–speed curves that capture theconcave-down shape of the experimental results andthe differences between the different PomA variants(Fig. 4c). Tight-coupled models of this type have notbeen able to predict the sharp knee in the torque–
speed curve,5 but by virtue of their simplicity compared to more detailed models18 they are a useful
tool to understand the broad features of the torque–
speed relationship. The reduced speed of motors
with mutations can be fitted in three different ways:

by reducing rates of Na+ binding and dissociation(k1–k4) 15-fold (right column for PomA-R88A in
Table 1, dashed line in Fig. 4c), by reducing the rate ofconformational changes (k0) 2.5-fold (middle column), or by reducing all rates (k0–k4) 2.3-fold (leftcolumn). Thus, the electrostatic interaction between
PomA and FliG affects the rate-limiting step intorque generation at low load, but we cannot
distinguish between changes in ion binding and therates of conformational changes without further
measurements of torque–speed curves with different
sodium concentrations and membrane voltages. The
reduced plateau torque of the R232E mutation was
fitted by decreasing either N by ∼ 37% (dotted line)or ΔΨ by ∼ 36% (dashed line): the model cannot
distinguish between the possible mechanisms forthis reduction, although our preliminary experiments favour the reduction of ΔΨ, as discussed
above. Further torque–speed measurements of mutant chimeric motors will allow a detailed investigation of the electrostatic interactions between statorand rotor proteins that are essential for torquegeneration in the flagellar motor.

Materials and Methods
Chemicals
Ficoll 400 (dialyzed and lyophilized) was from Sigma
Chemical Co. (St. Louis, MO). Polystyrene latex beads(diameter 1.053 ± 0.010, 0.771 ± 0.025, 0.548 ± 0.016 and0.356 ± 0.014 μm; 2.6% solids) were from Polysciences(Warrington, PA). All other chemicals were reagent grade.
Motility buffer was 10 mM potassium phosphate, 85 mM
NaCl and 0.1 mM ethylenediaminetetraacetic acid, pH 7.0.
Bacterial strains
Chimeric motors with no mutations on PomA wereexpressed in E. coli strain YS34 (ΔcheY, fliC∷Tn10, ΔpilA,
ΔmotAmotB) with plasmids pYS11 ( fliCsticky, ampicillin
resistance) and pYS13 (pomApotB, IPTG inducible,
chrolamphenicol resistance) as described.11 H+-driven

Table 1
Parameter

PomA

PomA-R88A

PomA-R232E[Na+]out (mM)
[Na+]in (mM)ΔΨ (mV)N
k0 (s− 1)
k1 = k3 (M− 1 s− 1)
k2 = k4 (s− 1)

85a14b− 140c520d
1.2 × 105
6.0 × 107e
6.0 × 105e

85
14
− 140
520
4.8 × 104
6.0 × 107
6.0 × 105k0 ÷ 2.585
144

5.2 × 10
2.6 × 107
2.6 × 105
(k0–k4) ÷ 2.3

5

1.2 × 10
4.0 × 106
4.0 × 104
(k1–k4) ÷ 15

−140
330

5

1.2 × 10
4.0 × 106
4.0 × 104
Reduce N− 90
520

Reduce ΔΨ

Parameters for data fitting with the four-state model (see the text; Figs. 4c and 5). The three sets of rate constants (k0–k4) used for the R88A
mutation gave very similar torque–speed curves (see the text; Fig. 4c). Further reducing either N or ΔΨ gave similar quality fits to thetorque–speed curve for the R232E mutation.
aSodium concentration of motility buffer.
bFrom Lo et al.25
c
From Lo et al.20d
The number of ions is determined by the stall torque in a tight-coupled model such as this, which has 100% efficiency at stall.
e
As used for V. alginolyticus.9

1258
motors were expressed in E. coli strain HCB1271(fliC∷Tn10, ΔpilA′-KnR, motA448) with plasmids
pDFB36 (motA, IPTG inducible, ampicillin resistance)and pDF313Cm (fliCsticky, chrolamphenicol resistance) as
described.10 Chimeric motors with PomA mutations15
were expressed in E. coli strain JHC36 (this work). The
fliCsticky region of strain PS2589 (a gift from Jeffrey Stock,
Princeton University) chromosome was amplified using
primers that annealed approximately 500 bp upstream anddownstream of fliCsticky. This PCR fragment was digested
with SphI/SacI, ligated into the suicide vector pDS132,27
sequenced and transformed into electrocompetent strain
YS34. Genomic replacement of fliC with fliCsticky was carriedout as described27 and gene replacement confirmed bySouthern blot analysis and PCR. The resultant strain was
designated JHC36. Chimeric stators were expressed in JHC36
from plasmids pTH200, pTH201 and pTH202, which are
pMMB206 derivatives carrying PomA/PotB, PomA-R88A/
PotB,and PomA-R232E/PotB, respectively, inducible by
IPTG. For swimming assays, chimeric stators were expressedin strain RP6894 (ΔmotAmotB, normal flagellar filaments,17
from plasmids pTH200, pTH201 and pTH202.

Torque and Speed of Chimeric Flagellar Motors

wavelength 632.8 nm (He–Ne laser, 05LHP-111, Melles
Griot, USA) or 1064 nm (diode-pumped solid-state laser,
L04-1000S-1064, Elforlight, Northants, UK). Rotational
speed was obtained from power spectra of the beadposition as described.3 All experiments were performed at
23 °C.
Torque was calculated as described in the text, using thereported viscosity of a Ficoll solution.8 The rotational
eccentricity of a bead was estimated from the quadrant
photodiode signal, calibrated using beads stuck to thesurface and moved with a piezoelectric stage (P-517P3CL,
Physik Instrumente, Germany).
Tight-coupled model
The speed of the motor at given torque, Na+ concentrations and ΔΨ was predicted using the four-state model ofFig. 5 as described.9,26

Cell culture and stator expression

Acknowledgements

Cells were grown from frozen stocks with shaking in T
broth (1% Bacto tryptone, 0.5% NaCl) containing 25 μg/ml
chrolamphenicol and 50 μg/ml ampicillin at 31 °C for 5 h.
For full induction of chimeric stator proteins, cells weregrown in the presence of 0.05 mM IPTG. For full inductionof wild-type H+ stator proteins, cells were treated with
5 mM IPTG at 25 °C for 3 h after growth at 25 °C to OD590 of∼ 0.4. IPTG concentrations and growth times were selected
to give the maximum rotation rate within the ranges tested(0.005–5 mM, 2–8 h). For resurrection experiments cells
were grown with 0–0.005 mM IPTG, and resurrections
were observed typically within 1 h after application of
0.05–0.9 mM IPTG in motility buffer containing 10% T
broth.

We thank Prof. David Blair for the gift of the strain
RP6894 and Prof. Judith Armitage and Dr. Jennifer
Chandler for help with the construction of strain
JHC36. This work was supported by Grants-in-Aid
for Scientific Research from the Japan Society for the
Promotion of Science (A.I.). C.-J.L. thanks Swire
Group/ORS for financial support. Y. S. thanks
Uehara Memorial Foundation and Murata Overseas
Scholarship Foundation for financial support. The
research of R.B. was supported by the EPSRC andBBSRC, the Interdisciplinary Research Collaboration in Bionanotechnology and the EU FP6 programme, and of G.W. by the Interdisciplinary
Research Collaboration in Bionanotechnology.

Swimming cells
After application of 20 mM serine, which was used tosuppress clockwise rotation, cells in T broth wereobserved for 1 min under a phase-contrast microscopeand recorded on a videotape for later analysis. Cells thatmoved more than 5 μm in 1 s were counted as swimming
cells. Cells which moved less than 5 μm in 1 s (due toBrownian motion) were counted as nonswimming cells.
Some nonswimming cells showed behavior distinct from
Brownian motion, such as very slow swimming due tofrequent changes in swimming direction (N 1 s− 1), rotationof the cell body with the filament stuck to the glass surface,or complete immobilization stuck to the glass surface.
Such cells were ∼ 10% of the population for all three motor
types and were excluded in this analysis. Swimming
fraction was calculated as number of swimming cells
divided by the total number of cells that were not excluded
from the analysis. Swimming fractions, mean ± SD, were
estimated from three different cultures. Swimming speeds,
mean ± SD, were calculated from 60 cells.
Torque–speed measurements
Polystyrene beads were attached to truncated flagellaand their position was measured with back focal planeinterferometry as described,10 except using a laser ofReferences
1. Blair, D. F. (2003). Flagellar movement driven byproton translocation. FEBS Lett. 545, 86–95.
2. Yorimitsu, T. & Homma, M. (2001). Na+ -driven
flagellar motor of Vibrio. Biochim. Biophys. Acta, 1505,
82–93.
3. Reid, S. W., Leake, M. C., Chandler, J. H., Lo, C. J.,
Armitage, J. P. & Berry, R. M. (2006). The maximum
number of torque-generating units in the flagellar
motor of Escherichia coli is at least 11. Proc. Natl Acad.
Sci. USA, 103, 8066–8071.
4. Berg, H. C. & Turner, L. (1993). Torque generated by
the flagellar motor of Escherichia coli. Biophys. J. 65,
2201–2216.
5. Berry, R. M. & Berg, H. C. (1999). Torque generated by
the flagellar motor of Escherichia coli while driven
backward. Biophys. J. 76, 580–587.
6. Berry, R. M. & Berg, H. C. (1997). Absence of a barrier
to backwards rotation of the bacterial flagellar motor
demonstrated with optical tweezers. Proc. Natl Acad.
Sci. USA, 94, 14433–14437.
7. Li, G. & Tang, J. X. (2006). Low flagellar motor torqueand high swimming efficiency of Caulobacter crescentus
swarmer cells. Biophys. J. 91, 2726–2734.
8. Chen, X. & Berg, H. C. (2000). Torque–speed relation-

ship of the flagellar rotary motor of Escherichia coli.
Biophys. J. 78, 1036–1041.
9. Sowa, Y., Hotta, H., Homma, M. & Ishijima, A. (2003).
Torque–speed relationship of the Na+-driven flagellar motor of Vibrio alginolyticus. H.