On Torque and Tumbling in Swimming Escherichia coli䌤
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JOURNAL OF BACTERIOLOGY, Mar. 2007, p. 1756–1764 Vol. 189, No. 5
0021-9193/07/$08.00⫹0 doi:10.1128/JB.01501-06
Copyright © 2007, American Society for Microbiology. All Rights Reserved.
On Torque and Tumbling in Swimming Escherichia coli䌤†
Nicholas C. Darnton,1 Linda Turner,1 Svetlana Rojevsky,1 and Howard C. Berg1,2*
Rowland Institute at Harvard, Cambridge, Massachusetts 02142,1 and Department of Molecular and
Cellular Biology, Harvard University, Cambridge, Massachusetts 021382
Received 24 September 2006/Accepted 12 December 2006
Bacteria swim by rotating long thin helical filaments, each driven at its base by a reversible rotary motor.
When the motors of peritrichous cells turn counterclockwise (CCW), their filaments form bundles that drive
the cells forward. We imaged fluorescently labeled cells of Escherichia coli with a high-speed charge-coupled-
Downloaded from http://jb.asm.org/ on December 28, 2020 by guest
device camera (500 frames/s) and measured swimming speeds, rotation rates of cell bodies, and rotation rates
of flagellar bundles. Using cells stuck to glass, we studied individual filaments, stopping their rotation by
exposing the cells to high-intensity light. From these measurements we calculated approximate values for
bundle torque and thrust and body torque and drag, and we estimated the filament stiffness. For both
immobilized and swimming cells, the motor torque, as estimated using resistive force theory, was significantly
lower than the motor torque reported previously. Also, a bundle of several flagella produced little more torque
than a single flagellum produced. Motors driving individual filaments frequently changed directions of
rotation. Usually, but not always, this led to a change in the handedness of the filament, which went through
a sequence of polymorphic transformations, from normal to semicoiled to curly 1 and then, when the motor
again spun CCW, back to normal. Motor reversals were necessary, although not always sufficient, to cause
changes in filament chirality. Polymorphic transformations among helices having the same handedness
occurred without changes in the sign of the applied torque.
The peritrichous bacterium Escherichia coli executes a ran- a right-handed filament outside the bundle turning CW, both
dom walk: an alternating sequence of runs (relatively long pushing the cell body forward. When the reversed motor
intervals during which the cell swims smoothly) and tumbles switches back to CCW rotation, the single filament regains its
(relatively short intervals during which the cell changes course) normal conformation and rejoins the bundle. However, more
(8). A cell is propelled by several helical flagellar filaments, exotic things can happen; for example, several filaments can
each attached by a hook (a universal joint) to a reversible undergo polymorphic transformations, and bundles can go di-
rotary motor (7). During runs, the filaments coalesce into a rectly from normal to curly 1 or from normal to a mixture of
bundle that pushes the cell forward (24). When viewed from normal and semicoiled or curly 1 (30). For recent reviews of
behind the cell, the bundle rotates counterclockwise (CCW), bacterial motility and chemotaxis, see references 4 and 31, and
and, to balance the torque, the cell body rotates clockwise for recent reviews of the flagellar rotary motor, see references
(CW). Tumbles are initiated by CW motor rotation (21). Based 1, 6, and 11.
on studies of Salmonella using dark-field microscopy, it was A limitation in our previous study of swimming behavior
thought that the motors change direction synchronously, caus- (30) was the fact that images were recorded at 60 Hz, a rate
ing the bundle to fly apart (24, 25). Based on studies using lower than the rate of filament rotation, so rotation frequen-
fluorescence microscopy, it became apparent that different fil- cies could not be measured and directions of rotation were
aments can change directions at different times and that a inferred from filament shape and cell motion. Here, to better
tumble can result from a change in direction of as few as one understand swimming in a dilute aqueous buffer or in a buffer
filament (30). During a tumble, the reversed filament comes containing methylcellulose, we recorded the motion of fluores-
out of the bundle and transforms from normal (a left-handed cently labeled cells at 500 Hz. Methylcellulose was used be-
helix with a pitch of 2.3 m and a diameter of 0.4 m) to cause it was included in early tracking experiments (8) as a
semicoiled (a right-handed helix with half the normal pitch but viscous agent to suppress Brownian motion and make cells
normal amplitude) and then to curly 1 (a right-handed helix easier to follow; however, it did not alter the run-tumble sta-
with half the normal pitch and half the normal amplitude). The tistics (our unpublished data). Using frame-by-frame analysis,
change in direction of the cell’s track generated by the tumble we measured the swimming speed, the rate of rotation of the
occurs during the transformation from normal to semicoiled, cell body, and the rate of rotation of the flagellar bundle. We
so at the beginning of the subsequent run, the cell swims for a also measured the rate of rotation of single filaments on cells
time with left-handed filaments in a bundle turning CCW and stuck to glass and in buffer. We compared the shapes of normal
filaments when they were spinning to their shapes when they
were stalled. We estimated values for motor torque and for
* Corresponding author. Mailing address: Department of Molecular filament stiffness.
and Cellular Biology, Harvard University, 16 Divinity Ave., Cam-
bridge, MA 02138. Phone: (617) 495-0924. Fax: (617) 496-1114.
E-mail: hberg@mcb.harvard.edu.
MATERIALS AND METHODS
† Supplemental material for this article may be found at http:
//jb.asm.org/. Labeling cells. E. coli strain AW405 (3) was grown as described previously
䌤
Published ahead of print on 22 December 2006. (30). All subsequent steps were carried out at room temperature (23°C). Bacteria
1756VOL. 189, 2007 TORQUE AND TUMBLING IN SWIMMING E. COLI 1757
TABLE 1. Data for cells with normal bundles swimming in MB⫹ or in MB⫹ with 0.18% methylcellulosea
Body rotation Bundle rotation Motor rotation Cell speed
Medium
rate (Hz) rate (Hz) rate (Hz) (m/s)
MB⫹ 24 ⫾ 12 (53) 130 ⫾ 40 (73) 163 ⫾ 43 (53) 25 ⫾ 8 (73)
MB⫹ with methylcellulose 23 ⫾ 11 (58) 72 ⫾ 28 (94) 92 ⫾ 31 (58) 33 ⫾ 11 (94)
a
The values are means ⫾ standard deviations. The numbers in parentheses are numbers of cells.
were washed twice by centrifugation (2,000 ⫻ g, 10 min) and gentle resuspension PA). Images of single flagellar filaments were fitted to helical curves using
with motility buffer (MB) (0.01 M potassium phosphate, 0.067 M NaCl, 10⫺4 M custom code written in MATLAB (The MathWorks, Natick, MA). The shape
EDTA; pH 7.0) and once with MB at pH 7.5. In the final preparation (0.5 ml), analysis involved fitting a recorded image to a helical curve defined by eight
the bacteria were concentrated 20-fold to 0.5 ml. One package of Cy3 mono- parameters, including three physical parameters (helix pitch [p], diameter [d],
functional succinimidyl ester (PA23001; Amersham Pharmacia Biotech, Newark, and contour length [L]), three rotation parameters (␣, , and ␥), and two
Downloaded from http://jb.asm.org/ on December 28, 2020 by guest
NJ) and 25 l of 1.0 M sodium bicarbonate were added to the bacterial suspen- displacement parameters (⌬x and ⌬y). We chose images containing flagella that
sion. Labeling was performed for 90 min with stirring by gyration at 100 rpm. were practically coplanar with the image plane, so the tilt out of that plane ()
Excess dye was removed by washing the bacteria with MB⫹ (motility buffer could be ignored. The proximal end of the filament was often indistinguishable
containing 0.002% Tween 20 [Sigma-Aldrich, St. Louis, MO] and 0.5% glucose).
from the bright cell body, so we fixed the contour length by eye before fitting.
Tween was added to prevent labeled cells from sticking to glass but was omitted
Together, these factors reduced the total number of parameters from eight to six.
in experiments in which cells were stuck to glass. In some experiments, MB⫹ was
Starting from a canonical form (a helix with a pitch of 2.3 m and a diameter of
supplemented with 0.18% hydroxypropylmethylcellulose (3,500 to 5,600 cP;
0.50 m aligned with the x axis), we allowed sequential rotations ␣ and ␥ around
H7509 lot 90K0802; Sigma-Aldrich, St. Louis, MO). Bulk viscosities (0.93 and
3.07 cP for MB⫹ and MB⫹ with 0.18% methylcellulose, respectively) were the x and z axes, followed by translation (⌬x and ⌬y) to bring the curve into
determined at 23°C with a Cannon-Ubbeholde viscometer, as described previ- approximate register with the recorded image. Since the microscope viewed
ously (9). “from above” (along the z axis), we actually observed the projection of the
Preparing slides. The suspension of labeled bacteria was diluted between 25- rotated, translated helix in the xy plane. The rotation (␣) changed the helix’s
and 50-fold with MB⫹. About 50 l of labeled bacteria was sealed within a thin phase, and ␥ rotated the helix within the plane of view. We represented the
ring of Apiezon M grease (Fisher Scientific, Pittsburgh, PA) between a coverslip helical curve by 100 equally spaced points and performed the rotations and
(22 by 44 mm) and a microscope slide. The coverslip was seated carefully to translation numerically. Conceptually, the best fit is the curve that passes through
eliminate air bubbles and then squeezed to form a chamber about 50 m thick. the most, brightest pixels of an image. We linearly interpolated between the
Samples were used immediately and for up to about 1 h. We have no evidence measured pixel values to estimate the picture brightness at each of the 100 points
that the preparations became anaerobic, but glucose was added to allow the cells along the curve and maximized the sum of the 100 values, which represented the
to swim without oxygen. In any event, the cells remained vigorously motile for an total brightness “captured” by the curve. The maximization was carried out by
hour or more, and their swimming speeds were similar to those observed else- using a MATLAB routine, starting with the initial approximate fit, sequentially
where (e.g., by tracking [23]). freeing each parameter, and refitting. Values for L were calculated from the axial
Acquiring images. Bacteria were observed at room temperature (23°C) with a length (z) of the flagellum (measured by hand) and the fitted pitch and diameter
according to the formula L ⫽ z 冑1 ⫹ 2共d/p兲2.
Nikon Diaphot 200 epifluorescence microscope using a phase-contrast objective
(Nikon PlanApo 60/1.4 oil DM) and a 4⫻ or 5⫻ camera relay lens. Images were
acquired with a high-speed (500-Hz) black and white charge-coupled-device
camera modified for low-light conditions (HSC 500x2; J C Labs, La Honda, CA).
Illumination was provided by an argon ion laser (Stabilite 2017; Spectra-Physics,
Mountain View, CA) run at 514 nm, using a fluorescence cube with a D514/10x
excitation filter, a 527 DCLP dichroic mirror, and an E535LP emission filter
(C7408; Chroma Technologies, Brattleboro, VT). The vertical sync pulse from
the camera was used to synchronize rotation of a slotted wheel that generated
⬃0.2-ms exposures (one exposure per frame). The microscope was configured in
the standard epifluorescence mode, with the illumination restricted to a circle
about 40 m in diameter matching the camera’s field of view. The laser power at
the back focal plane of the objective was 100 to 300 mW. Cells were faintly
illuminated in phase contrast, using a tungsten filament light source, making it
possible to visualize cell bodies and to focus prior to laser illumination. Images
were captured at a rate of 500 frames/s directly to a personal computer equipped
with an I-60 analog video capture board and IDEA software (both obtained from
Foresight Imaging, Lowell, MA). Images were acquired for 1 s. After a few initial
frames of phase-contrast illumination, the laser was switched on, guaranteeing
that the start of high-intensity exposure was known. This procedure was used to
minimize laser damage to cells during image acquisition, since intense light,
especially at short wavelengths, is known to interfere with motor function (32).
In order to image stationary filaments on stuck bacteria, cells were exposed to
continuous laser illumination; filaments stopped rotating within a few seconds.
Analyzing images. Using ImageJ (http://rsb.info.nih.gov/ij/), AVI files were
converted to TIF stacks, and the motion of a cell was followed over a convenient
number of frames. If the cell body had a distinctive mark or pattern of flagella- FIG. 1. Consecutive images (500 video frames/s) of a cell swimming
tion, its rotation rate was determined by counting the number of video frames for toward the bottom of the field, propelled by a normal flagellar bundle.
one revolution of that reference point. Filament rotation rates for bundles of The position of an individual helical wavecrest is indicated by white ar-
swimming bacteria or single filaments of stuck bacteria were determined either rows. As the wave propagates away from the cell body, a second crest
by counting the number of frames required for the distal tip to complete one (gray arrow in frame 5) appears at the original position of the first crest,
revolution or by following an individual wavecrest until it propagated one pitch identifying a complete CCW revolution of the filament. Frame numbers
length (see Fig. 1, 4, and 5). All measurements of rotation rates were obtained can be converted to elapsed time by multiplying by 0.002 s. Details of this
within the few first video frames of laser illumination. Distances were calibrated motion are seen more clearly in the movie file “500 Hz swimming.avi” in
from recorded images of an objective micrometer (Fischer Scientific, Pittsburgh, the supplemental material.1758 DARNTON ET AL. J. BACTERIOL.
Downloaded from http://jb.asm.org/ on December 28, 2020 by guest
FIG. 2. Every fourth image for the cell shown in Fig. 1. The arrow in
frame 4 indicates where a filament arises from the bacterium’s surface and
joins the bundle. After one-half revolution of the cell body, the bundle
appears on the opposite side of the cell (frame 24); after one full revolu-
tion, it reappears on the original side of the cell (frame 44). Details of this
motion are seen more clearly in the movie file “500 Hz swimming.avi” in
the supplemental material. FIG. 3. Swimming speed (A) and body rotation rate (B) as a func-
tion of the bundle rotation rate in MB⫹ (E) or MB⫹ with 0.18%
methylcellulose (F). The slopes of the linear regression lines are as
RESULTS follows: 0.180 m for the dashed line and 0.418 m for the solid line
in panel A; and 0.171 for the dashed line and 0.311 for the solid line in
Rotation rates and swimming speeds for a sample of 50 to panel B.
100 cells, swimming in MB⫹ and MB⫹ with 0.18% methylcel-
lulose, are shown in Table 1. Figure 1 shows a typical swim-
ming cell to illustrate our measurement technique. CCW ro- dle. That is, the cells moved 8% or 18% as fast as they would
tation of a normal left-handed bundle appeared as a wave have moved if the flagella had bored through the medium
propagating away from the cell body. The wave moved one without slip, i.e., like a corkscrew through a cork. For some
wavelength between frames 0 and 5 (a time span of 0.01 s), bacteria, we also determined the counterrotation rate of the
indicating that the bundle rotation rate was ⬃100 Hz. Figure 2 cell body, which is plotted as a function of the bundle rotation
shows every fourth frame for the same cell; the cell body rate in Fig. 3B. Again, the relationships were approximately
completed one revolution between frames 4 and 44 (a time linear; the cell bodies rotated 0.171 and 0.311 times as fast as
span of 0.08 s), indicating that the body rotation rate was ⬃12.5 the flagellar bundles in MB⫹ and in MB⫹ with methylcellu-
Hz. In frame 4, the cell body angled toward the lower left lose, respectively.
corner of the frame and the bundle appeared to its left; in For a subset of the cells in Table 1, we generated a more
frame 24, the cell body angled toward the lower right corner of complete data set that also included body length, bundle
the frame and the bundle appeared to its right; in frame 44, the length, and body wobble angle (Table 2). We examined the
orientations were the same as those in frame 4. The flagellar extended data set for correlations between dynamic parame-
bundle and the cell body must turn in opposite directions, since ters (cell and bundle rotation rates, swimming speed, and body
bundle and body torques balance (5), so the flagellar motors wobble) and also between dynamic parameters and cell geom-
were spinning at ⬃112.5 Hz, the sum of the bundle and body etry (bundle length, cell width, and cell length). One might
rates. This cell swam at a speed of ⬃25 m/s. expect that bundle length would correlate with either swim-
In Fig. 3A, the swimming speeds shown in Table 1 are ming speed or the bundle rotation rate, but we found no such
plotted as a function of bundle rotation rates for cells in MB⫹ relationship. Other than the dependence on the rotation rate
and MB⫹ with 0.18% methylcellulose. In both media the re- (Fig. 3), the only additional important factor affecting swim-
lationship was approximately linear, with an average speed-to- ming speed was the body wobble angle, which was anticorre-
rate ratio, called the v-f ratio by Magariyama et al. (27), of lated with speed for cells swimming both in buffer and, less
0.180 m in MB⫹ and 0.418 m in methylcellulose. This significantly, in methylcellulose. Only the bundle and motor
indicates that bacteria translated about 8% and 18% of the rotation rates and, to a lesser extent, body wobble were af-
flagellar pitch, respectively, per revolution of the flagellar bun- fected by the addition of methylcellulose. The correlationsVOL. 189, 2007 TORQUE AND TUMBLING IN SWIMMING E. COLI 1759
TABLE 2. Data for cells with normal bundles swimming in MB⫹ or in MB⫹ with 0.18% methylcellulosea
Body length Body width Body wobble Body rotation Bundle length Bundle rotation Motor rotation Cell speed
Medium
(m) (m) angle (°)b rate (Hz) (m)c rate (Hz) rate (Hz)d (m/s)
MB⫹ 2.5 ⫾ 0.6 0.88 ⫾ 0.09 46 ⫾ 24 23 ⫾ 8 8.3 ⫾ 2.0 131 ⫾ 31 154 ⫾ 30 29 ⫾ 6
MB⫹ with methylcellulose 2.0 ⫾ 0.4 0.86 ⫾ 0.08 36 ⫾ 17 21 ⫾ 11 10.0 ⫾ 1.5 67 ⫾ 24 87 ⫾ 31 31 ⫾ 10
a
The values are the means ⫾ standard deviations for 32 cells in each medium.
b
The angle swept out by the axis of the cell body as it rolls about the bundle axis.
c
The distance between the back end of the cell body and the distal end of the bundle.
d
Since the cell body and bundle rotate in opposite directions, the motor rotation rate is the sum of the body and bundle rotation rates.
between rotation rates and swimming speeds were significantly
stronger for cells in methylcellulose than for cells in buffer.
Figure 4 shows 12 consecutive frames from a movie of a
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normal filament rotating in isolation on a stuck cell. In frames
0 through 3, the filament completed one CCW revolution,
indicating that the rate was ⬃167 Hz. The filament stopped
between frames 4 and 5 and then rotated in the opposite
direction, completing one CW revolution between frames 6
and 11 (⬃100 Hz). We presumed that between frames 4 and 5
the motor changed direction and the hook unwound and then
rewound in the opposite sense. This is an example of a filament
that remained left-handed while being spun CW. Such events
occurred infrequently, about once in 100 reversals. Although
we observed several instances of CW-rotating filaments, in
most cases the filament moved out of the focal plane, making
its rotation rate difficult to measure.
Under our buffer conditions, the normal, left-handed form is
the only stable filament geometry at rest. To cause the filament
to change to another form, in particular to a right-handed
form, force must be applied to it. Based on consideration of the
signs of the torque involved, only CW rotation of a left-handed
FIG. 4. Consecutive images (500 video frames/s) of a stuck cell
spinning a single flagellar filament. The position of an individual he-
lical wavecrest is indicated by white arrows. As the wave propagates
away from the cell body, a second crest (gray arrow in frame 3) appears FIG. 5. Consecutive images (500 video frames/s) of a stuck cell spin-
at the original position of the first crest, identifying a complete CCW ning a single flagellar filament. The position of an individual helical
revolution of the filament. Frames 4 and 5 are identical; the filament wavecrest is indicated by white arrows as the wave propagates away from
has stopped rotating. The white arrows in frames 6 to 11 indicate the the cell body (frames 0 to 4). In frames 5 to 9 filament rotation stops. In
retrograde motion of a helical wavecrest toward the cell body. As the frames 10 to 14, the distal end of the filament remains stopped, while a
wave propagates toward the cell body, a second crest (gray arrow in short-pitch region of the transformed filament, indicated by a gray arrow,
frame 11) appears at the original position of the first crest, identifying appears in frame 14. The proximal region is now inclined toward the left
a complete CW revolution of the filament. Details of this motion are of the cell’s longitudinal axis (compare frames 1 and 14). Details of this
seen more clearly in the movie file “500 Hz reversal 1.avi” in the motion are seen more clearly in the movie file “500 Hz reversal 2.avi” in
supplemental material. the supplemental material.1760 DARNTON ET AL. J. BACTERIOL.
filament would “untwist” it toward the right-handed forms.
Thus, motor reversal is required (although not sufficient, as
shown in Fig. 4) to cause any polymorphic transformation of
the normal form. Under our conditions, the right-handed
forms are not stable at rest; they can be maintained only by the
application of torque from CW rotation of the motor. We have
never seen a right-handed, CW-rotating filament spontane-
ously revert to the normal form, although we presume that this
would occur, even without a motor reversal, if the applied
torque dropped significantly below the normal, fully energized
level. When the torque changes sign, as it does upon motor
reversal, the filament always goes back to normal. Motor re-
versal is required (and is sufficient) to cause helicity-changing
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polymorphic transformation of the right-handed forms. Cer-
tain mutations in the hook-associated protein at the base of the
filament can upset this balance. For instance, in sag mutants
(mutants unable to swim in 0.28% agar but otherwise normal),
CCW rotation can drive a normal filament to the left-handed
straight form and CW rotation can drive a curly 1 filament to
the right-handed straight form (18).
FIG. 6. Typical single-frame images overlaid with a projection of
Every reversal observed included a pause of at least one
the best-fit helical form. The same flagellar filament is shown in the
video frame between sequences of rotation; we have never two panels; it is stopped in panel A and moving in panel B. Since the
seen an entire filament rotating CCW in one frame and CW in length of the flagellum was not relevant for our purposes, we some-
the next frame. It is possible for the distal end of a filament to times fit to slightly less than the full-length filament, as in panel A. For
scale, the pitch is 2.3 m.
stop rotating while a polymorphic transformation occurs in its
proximal end, as shown in Fig. 5. Initially, such a filament
rotated CCW at about 125 Hz, completing one revolution DISCUSSION
between frames 0 and 4, as indicated by the progression of the
arrow toward the distal tip of the filament. In frames 5 through Following Magariyama et al. (27), we applied resistive force
9 the rotation appears to stop, and the most proximal portion theory (20) to the single-filament data in Table 3, with a swim-
of the filament changes its inclination with respect to the cell ming speed (v) of 0. We used a filament angular velocity () of
2 ⫻ 111 Hz, a helix radius (r) of 0.2 m, a helix pitch (P) of
body, moving slightly to the left. In frames 10 through 14, the
2.22 m, a filament radius () of 0.012 m, and a filament
distal end of the filament remains stopped, while a short-pitch
contour length (L) of 7.1 m, obtaining a filament torque of
region of transformed filament appears (indicated by an arrow
370 ⫾ 100 pN nm. Motors run at nearly constant torque up to
in frame 14); compare the proximal filament positions in
frequencies of about 175 Hz (15), so it is puzzling that this
frames 1 and 14. All helices with shorter-than-normal pitch and value is ⬎10-fold less than the stall torque for the flagellar
a small radius are right-handed (13); therefore, the change in motor measured with optical tweezers (10), ⬃4,600 pN nm.
helicity that we observed must have been caused by a period of This discrepancy led us to examine more recent estimates for
CW rotation of the motor. The total length of this pause (eight motor torque obtained by spinning latex beads on flagellar
frames, or 0.016 s) is consistent with the winding up of the stubs. Working within the low-speed, high-torque limit with
flagellar hook and the accumulation of added twist in the spheres whose diameters ranged from 1.0 to 2.1 m, Fahrner
transformed segment. In subsequent frames the filament re- et al. (19) obtained rotation speeds ranging from 78 to 8.6 Hz.
sumed CCW rotation (not shown). These measurements yielded a mean torque of 1,370 ⫾ 50 pN
Table 3 shows the results of measurement of 24 normal nm, in agreement with the value of 1,260 pN nm obtained
filaments rotating in isolation on cells that were stuck to a glass recently using rotating 1-m beads (28), which we believe to be
surface. As shown by these data and the fits illustrated in Fig. closer to the mark; however, this value is still substantially
6, the shapes of spinning and stopped filaments were indistin- larger than 370 pN nm. Thus, either the resistive force theory
guishable. predicts a torque that is too low, or a substantial burden is
imposed by rotation of the filament near a glass surface. Ac-
cording to resistive force theory, the drag coefficient of an iso-
lated, translating helix is inversely proportional to ln(2p/) ⫺ 0.5
TABLE 3. Helical parameters for normal filaments on stuck bacteriaa (27). When the helix is placed close to a surface, hydrodynamic
shielding by the surface changes this expression to ln(2l/),
Contour length Rotation
Movement Pitch (m) Diam (m)
(m) rate (Hz)
where l is the distance to the surface (22). A 4-fold or 12-fold
increase in the drag coefficient, which would bring the single-
CCW 2.22 ⫾ 0.20 0.39 ⫾ 0.05 7.1 ⫾ 1.8 111 ⫾ 20 filament torques into agreement with the previously described
Stopped 2.28 ⫾ 0.15 0.42 ⫾ 0.05 7.1 ⫾ 1.7 0
torque (1,370 pN nm or 4,600 pN nm), corresponds to a prox-
a
The values are the means ⫾ standard deviations for 24 filaments. imity of 0.02 m or 0.01 m. These distances are rather smallVOL. 189, 2007 TORQUE AND TUMBLING IN SWIMMING E. COLI 1761
(approximately 1/10 the radius of the helix), but not impossibly than the single-filament speed, so the total torque supplied by
so. all four motors driving the bundle is only 30% higher than the
The filament is sufficiently stiff that we were not able to single-motor torque; i.e., each motor operates at about 32% of
detect differences in the shape of a normal filament when it was the single-motor torque. For a fully assembled motor operating
spinning or stopped, as shown in Table 3 and Fig. 6. Based on in a fully energized cell, one would not expect to see such a
a simple elastic model of the filament (16), the axial force (F) dramatic torque reduction unless the motor were operating at
and torque (⌫) required to deform a filament with natural, around 300 Hz, well above the “knee” frequency (15). We
unstressed pitch (p0) and radius (r0) to a new pitch (p) and believe that the motors in a swimming cell do, in fact, deliver
radius (r) are close to peak torque but that the effective drag of the bundle is
much larger than the calculation described above suggests.
p0 r ⫺ pr0 Either the bundle has an effective hydrodynamic radius that is
F ⫽ 42EI
r共p0 ⫹ 42r02兲 冑p2 ⫹ 42r2
2 30 times larger than the single-filament radius (much looser
than has been imagined [24]), or the filaments in multiply
pp0 ⫺ p02 ⫹ 42共r ⫺ r0兲r0
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flagellated cells generate substantial internal drag. Even if the
⌫ ⫽ 2EI ,
共p02 ⫹ 42r02兲 冑p2 ⫹ 42r2 filaments were in very close contact (average separation of one
filament radius, 12 nm), they would dissipate little extra power
where EI is the flagellar stiffness. Since the forces are gener- (7), but such dissipation could be accomplished by flagella
ated or dissipated uniformly along the length of the rotating dragging over the surface of the cell. A cell with a single
filament, F, on average, is half of the thrust generated by the filament can always orient itself so that the flagellum rotates
filament, and ⌫, on average, is half of the torque applied by the clear of the body, but any additional filaments, which usually
motor. If we take the natural pitch and radius from the data for arise from points far from the axis of rotation, generally have
the stopped filaments (Table 3) and account for uncertainties to cross the cell body during rotation. Unlike the drag between
by allowing a range of axial forces and torques (0.25 pN ⬍ F ⬍ two thin filaments, the drag against a large surface can be
0.85 pN and ⫺1,500 pN nm ⬍ ⌫ ⬍ ⫺300 pN nm) and a substantial, so added torque contributed by additional flagella
three-standard-deviation range for the CCW form parameters might be dissipated against the cell body.
(2.19 m ⬍ p ⬍ 2.37 m and 0.17 m ⬍ r ⬍ 0.23 m), a If they do not allow the cell to swim faster, why does a cell
self-consistent set of numbers requires that the flagellar stiff- have multiple flagella? One possible explanation is that having
ness be greater than 5.5 pN m2. This is reasonably consistent “extra” flagella allows cells to maintain motility while dividing
with the measured stiffness, 3.5 pN m2 (16). quickly. There is a lag of several generations between turning
The hook is known to be more flexible than the filament; in on flagellar synthesis and completing the first new flagellum
fact, it changes its twist by about one full turn during a motor (2). If cells did not have a reservoir of flagella when they start
reversal (12). The transformation from normal to semicoiled a growth spurt (e.g., when they encounter a newly rich me-
involves supertwisting by about 3 rad/m or about 1.25 turns dium), cell division during this lag period would produce many
per pitch (13); at a motor speed between 300 and 100 Hz, unflagellated, nonmotile cells. Another possibility, assuming
transformation of a single pitch would require between 0.008 that a cell with a single flagellum swims poorly unless that
and 0.022 s. Thus, the first few rotations of the motor can be flagellum is at a cell pole, is that inserting several flagella at
absorbed by the hook plus a polymorphic change of the prox- random points on the cell surface is easier than building a
imal end of the filament, without requiring the distal end to specific motor mount at one pole. Yet another possibility is
rotate much at all, consistent with Fig. 5. If the CW interval is that having multiple, distributed flagella allows cells to change
short enough, when the motor again turns CCW, the polymor- directions more efficiently when they tumble, i.e., to try a new
phed sections simply propagate back down the filament and direction at random (8) rather than just back up (29), which
are reabsorbed into the hook. In a swimming bacterium such a searches some but not all (17) environments more efficiently.
brief motor reversal would not interfere with rotation of the We believe that the last factor, namely, the connection be-
bundle or alter the cell’s trajectory and would probably be tween the presence of multiple, distributed flagella and search-
undetectable with current microscopic techniques. ing efficiency, is an essential component of bacterial taxis, so
Why is the single-filament rotation rate (111 Hz) (Table 3) we hope to understand the tumbling process in E. coli in detail.
so similar to the bundle rotation rate (130 Hz) (Table 1)? In Since the flagellar bundle has the largest hydrodynamic size, its
our previous study of fluorescent flagellar filaments (30), cells orientation determines the direction of cell motion. Any motor
of the same strain grown in the same way produced an average reversal (CCW to CW) results in deflection of the cell from this
of 3.4 filaments per cell. This is consistent with our observa- trajectory, unless the motor happens to be located in line with
tions of these swimming cells, where we could usually distin- the bundle axis. In a previous study (30), we found that normal-
guish at least three separate filaments in a bundle. At a mean to-semicoiled transformation of a filament resulted in deflec-
motor rate of 166 Hz (Table 1), all these flagella should be tion of the cell body during tumbles (4). Using the high-speed
running in a constant-torque regimen (15). Consider four fil- camera, we were able to confirm these events. A motor reversal
aments forming a compact bundle. If interactions between (CCW to CW) causes the filament to pause and then change its
these filaments can be ignored, the hydrodynamic properties of direction of rotation. This deflects the cell body and unwinds
the bundle should be similar to those of a single filament with the filament from the bundle. The small initial deflection of the
roughly twice the diameter. The viscous load depends only cell body is reversed as the filament transforms to the right-
logarithmically on this diameter, so it should be roughly 15% handed semicoiled form, changing the thrust that the filament
larger. Additionally, the bundle speed is about 15% higher exerts on the cell body. The tumble usually ends with the1762 DARNTON ET AL. J. BACTERIOL.
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FIG. 7. Idealized sequence of events in a tumble caused by the reversal of a single motor. The upper timeline indicates the direction of motor
rotation of the filament causing the tumble, and the lower timeline indicates the behavior as judged by motion of the cell body. From left to right:
1, a bacterium swimming along its original trajectory with all left-handed normal filaments; 2, a motor reversal (CCW to CW) causing the filament
to start unbundling and the cell body to deflect slightly; 3, initiation of the transformation of the filament from the left-handed normal form to the
right-handed semicoiled form and the beginning of a large deflection of the cell body opposite the previous small deflection; 4, complete
transformation of the filament to the semicoiled form and reorientation of the cell along a new trajectory; 5, movement of the cell along the new
trajectory, propelled by a normal bundle turning CCW and a semicoiled filament turning CW which has partially transformed to the right-handed
curly 1 form; 6, complete conversion of the filament to the curly 1 form, which is flexible enough to twist loosely around the bundle; 7, the motor
reversing again (CW to CCW), causing the curly 1 form to revert to normal; and 8, after the filament has rejoined the bundle.
conversion of the semicoiled form to the curly 1 form, followed Fbody ⫽ v(A1sin2 ⫹ A2cos2) and a torque resisting the rota-
later by a motor reversal (CW to CCW), causing the filament tion of magnitude ⌫body ⫽ ⍀[(D1 ⫹ m2A1) sin2 ⫹ D2cos2].
to transform back to its normal form and rejoin the bundle, as With viscosity , eccentricity e [e ⫽ (a2 ⫺ b2)1/2/a], and E ⫽
shown in Fig. 7. Although this is our best reconstruction of the ln[(1 ⫹ e)/(1 ⫺ e)], the values of the coefficients are:
canonical tumble, other endings are possible. For example, if
A1 ⫽ 32ae3/[(3e2 ⫺ 1)E ⫹ 2e]
the second motor reversal (CW to CCW) occurs while the
A2 ⫽ 16ae3/[(1 ⫹ e2)E ⫺ 2e]
filament is still in the semicoiled form, the filament transforms
D1 ⫽ 32ab2e3(2 ⫺ e2)/3(1 ⫺ e2)[(1 ⫹ e2)E ⫺ 2e]
directly from semicoiled back to normal, skipping the curly
D2 ⫽ 32ab2e3/3[2e ⫺ (1 ⫺ e2)E]
form entirely.
We applied resistive force theory (20, 27) to the data ob- For each cell, we obtained two independent measurements
tained with free-swimming cells and found that the torque of torque and force; one measurement was based on resistive
required to spin the filaments is roughly the same as the torque
required to spin the cell body. Assuming the same helix radius
and pitch as before (0.2 m and 2.22 m), but treating the
bundle as a single a filament having twice the radius (0.024
m), for the 32 cells in Table 2 we obtained a bundle torque
(⌫bundle) of 650 ⫾ 220 pN nm, a bundle thrust (Fbundle) of
0.41 ⫾ 0.23 pN, a body torque (⌫body) of 840 ⫾ 360 pN nm, and
a body drag (Fbody) of 0.32 ⫾ 0.08 pN. Chattopadhyay et al.
(14) used an optical trap to measure the propulsion matrix,
which connected bundle torque and bundle thrust to swimming
speed and bundle angular velocity, as ⌫bundle ⫽ ⫺Bv ⫹ D and
Fbundle ⫽ ⫺Av ⫹ B. Using the values of Chattopadhyay et al.
for A, B, and D with our measured swimming speed and
bundle rate gives a ⌫bundle value of 550 pN nm and an Fbundle
value of 0.28 pN, in agreement with our values for these pa-
rameters. In our calculations, the body was assumed to be a
prolate ellipsoid with the length and width shown in Table 2,
rotating about the bundle axis at angular velocity ⍀ at distance
m from the body center along the cell major axis, with the axes
forming an angle () equal to half the wobble angle, as shown
in Fig. 8. The expression for the viscous drag of the cell body FIG. 8. Cell body in the shape of a prolate ellipsoid having length
2a and width 2b swimming at velocity v along the bundle axis, with the
averaged about the bundle axis, adapted from a solution kindly center of its body at distance m from, and at angle with respect to, the
provided by Tobias Löcsei and John Rallison of Cambridge bundle axis, and rolling about that axis at angular velocity ⍀. is half
University, yields a force resisting the translation of magnitude the body wobble.VOL. 189, 2007 TORQUE AND TUMBLING IN SWIMMING E. COLI 1763
Table 1 shows that this does not occur when viscosity is tripled
by adding methylcellulose. Only the cells’ bundle and motor
rotation rates are substantially decreased; the body rotation
rate is unaffected, and the cell speed actually increases. This
qualitatively agrees with the predictions of an anisotropic vis-
cosity model of swimming in methylcellulose (26).
In summary, assuming the validity of resistive force theory
and neglecting interactions with nearby surfaces, we estimated
the torque generated by an isolated filament to be ⬃400 pN
nm, a value substantially lower than current estimates of motor
torque. Filaments are quite stiff; changes in shape between
spinning filaments and stationary filaments were not detected.
The torque generated by a flagellar bundle is surprisingly
small, ⬃700 pN nm. Evidently, a substantial fraction of the
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torque supplied by the several motors that drive a bundle is
dissipated through internal friction within the bundle or be-
tween the bundle and the cell wall. However, the torque and
thrust generated by the bundle are balanced, as they should be,
by the drag computed for the cell body. Even though additional
filaments in a bundle might not add much to a cell’s speed, they
are useful for reorientation during tumbling. CW rotation of-
ten, although not always, triggers a polymorphic transforma-
tion to a right-handed filament form. This transformation plays
an important role in generating changes in the direction of
swimming.
FIG. 9. (A) Plot of bundle torque (⌫bundle) versus torque on the cell ACKNOWLEDGMENTS
body (⌫body). (B) Plot of propulsive force produced by the bundle
(Fpropulsion) versus total drag (Fbody ⫹ Fself-drag), calculated for 32 cells We thank William S. Ryu for computer expertise and Peter Chupity
swimming in MB⫹. The dashed lines are least-squares linear fits; the at J C Labs for modifying his camera design for low-light operation.
best-fit slopes are 0.82 (A) and 1.04 (B), compared with the dotted 45° Their support and encouragement were greatly appreciated in the
line indicating perfect agreement. One could break the bundle torque initial phase of this project.
in panel A into two components and plot torques analogous to forces, This work was supported by the Rowland Institute at Harvard and
as shown in panel B; however, the rotary self-drag (Bv) is so small that by grants AI016478 and AI065540 from the National Institutes of
this would not substantially change panel A. Health.
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