SI: Modeling torque versus speed, shot noise, and rotational diffusion of the bacterial
flagellar motor
1
0.5
Torque (normalized)
T = 15.8 ºC
0
1
0.5
0
T = 17.7 ºC
1
0.5
T = 22.7 ºC
0
0
100
200
Speed (Hz)
300
400
Fig. S 2: The torque-speed relationship at various temperature. Rotation speed measured with 0.25–0.36 µm diameter
beads attached to flagellar stubs for a wide range of viscosities, at three different temperatures [5]; circles: experimental
data, solid curves: model fits.
Δψ (mV)
-150
-100
-50
5
5.5
6
6.5
external pH
7
7.5
8
Fig. S 1: Electric potential difference versus external pH, as
obtained from the fit to data from [1] (Fig. 2a of the main text)
using ∆ψ = ∆p − (kB T /e) log(Hint /Hext ), and assuming that
the maximum torque (2500 pN.nm) is reached at ∆p = −150
mV. The apparent linear dependence of ∆ψ on external pH
is consistent with previous data [2, 3].
2
T = 15.8 ºC
1
T = 11.2 ºC
n=1
n=2
n=3
n=4
T = 11.2 ºC
1
0.5
0
100
200
1
300
T = 16.2 ºC
0.5
0
Torque (normalized)
Torque (normalized)
0.5
0
T = 16.2 ºC
T = 22.7 ºC
T = 22.6 ºC
0.5
0
T = 22.6 ºC
1
T = 17.7 ºC
1
1
0.5
0.5
0
0
100
200
Speed (Hz)
300
400
0
0
200
400 0
Speed (Hz)
200
Speed (Hz)
400
Fig. S 4: The torque-speed relationship with proton cooperativity n = 1, . . . , 4. Left panels: model fit to data from
[5]. Right panels: comparison between model prediction and
electrorotation data from [4] with the same parameter values
as in the left panels.
Fig. S 3: The torque-speed relationship at various temperatures. Comparison between data from [4] and model prediction (solid curves) with the same parameter values as in
Fig. S2 (no additional fitting parameters). Note that the temperature of the top panel (11.2o C) is below the range used to
fit parameters (15.8-22.7o C).
pH out pH in
2500
7.0
6.5
5.8
2000
Torque (pN.nm)
[1] Shuichi Nakamura, Nobunori Kami-ike, Jun ichi P Yokota,
Seishi Kudo, Tohru Minamino, and Keiichi Namba. Effect of intracellular ph on the torque-speed relationship
of bacterial proton-driven flagellar motor. J Mol Biol,
386(2):332–8, Feb 2009.
[2] Chien-Jung Lo, Mark C Leake, Teuta Pilizota, and
Richard M Berry. Nonequivalence of membrane voltage
and ion-gradient as driving forces for the bacterial flagellar motor at low load. Biophys J, 93(1):294–302, Jul 2007.
[3] Tohru Minamino, Yasuo Imae, Fumio Oosawa, Yuji
Kobayashi, and Kenji Oosawa. Effect of intracellular ph on
rotational speed of bacterial flagellar motors. J Bacteriol,
185(4):1190–4, Feb 2003.
[4] H C Berg and L Turner. Torque generated by the flagellar
motor of escherichia coli. Biophys J, 65(5):2201–16, Nov
1993.
[5] X Chen and H C Berg. Torque-speed relationship of
the flagellar rotary motor of escherichia coli. Biophys J,
78(2):1036–41, Feb 2000.
3000
7.4
6.7
6.1
1500
1000
500
0
−500
−1000
−1500
−300
−200
−100
0
100
Speed (Hz)
200
300
400
Fig. S 5: Model fits of the torque-speed relationships from [1]
with proton cooperativity n = 2.
[6] Christopher V Gabel and Howard C Berg. The speed
of the flagellar rotary motor of escherichia coli varies linearly with protonmotive force. Proc Natl Acad Sci USA,
100(15):8748–51, Jul 2003.
3
160
300
T = 24 °C
T = 16.2 °C
140
250
High Speed (Hz)
High speed (Hz)
120
200
150
100
100
80
60
40
50
0
0
20
1
2
3
4
Low speed (Hz)
5
6
Fig. S 6: Relationship between the high speed (low drag) and
low speed (high drag) regimes of the bacterial flagellar motor,
as the proton motive force varies from −150 to 0 mV, at temperature T = 24o C. The experimental data from individual
cells [6] are represented by symbols. Our fits with no proton
cooperativity (n = 1) are the solid curves.
0
0
1
2
3
4
Low Speed (Hz)
5
6
Fig. S 7: The same high speed versus low speed data as
Fig. 3, but model fits (solid curves) obtained with proton
cooperativity n = 2.