Dynamics of the bacterial flagellar motor with multiple stators - Supporting Information
aa r X i v : . [ q - b i o . S C ] J a n Supporting Information “Dynamics of the bacterial flagellar motor with multiple stators”
Giovanni Meacci and Yuhai Tu
I. TORQUE-SPEED CURVE MEASUREMENT
The measurement of the torque-speed curve is usually done by fixing the cell to a glassslide and tethering a polystyrene bead to the flagellar hook. An optical trap monitor therotational speed of the bead and the motor torque is calculated from τ = ( ξ L + ξ R ) ω ≈ ξ L ω ,where ξ R is the drag coefficient due to the internal friction in the motor, ξ L is the bead dragcoefficient, and ω is the angular velocity. The torque-speed curve is then obtained changing ξ L by varying the bed size [1] or the viscosity of the external medium. An alternative methodis to tether a cell to a glass coverslip by one of its shortened flagellar filament and exposethe cell to a rotating electric field [2]. Then the motor is broken spinning the cell backward.The difference between the cell body speeds before and after the broke of the motor, at thesame value for the external applied torque, is proportional to the motor torque. II. HOOK SPRING COMPLIANCE
Fig. S1 shows the compliance curve F (motor torque versus angular displacement θ − θ L )similar to the experimental one [3]. The following expression is used in our simulations: F ( x ) = κ x + ∆ x ( κ − κ ) log [1 + exp (cid:0) ( x − x ) / ∆ x (cid:1) ] , (1)with x ≡ θ − θ L , κ the spring constant a low load, κ ( ≫ κ ) the spring constant a highload. At both low and high loads the spring behaves linearly with different spring constants κ and κ respectively. ∆ x is the angular displacement interval of the non linear regioncentered around the turning point x . For the values of these parameters see Tab. S1. III. DISTRIBUTION FUNCTIONS OF t m AND t w Fig. S2 shows the distribution functions for t m and t w at different values of the load for thecase N=1. Fig. S2(a) shows the average waiting and moving times. The arrows labeled with1he letter (b), (c), and (d) indicate the points (i.e. the speed values) where the probabilitydistributions for t m and t w are shown in Fig. S2 (b), (c), and (d) respectively. The averagedwaiting-time decreases slightly with the speed, while the averaged moving-time decreases byfour orders of magnitude. The two time scales crossover at a speed around 170Hz, whichnaturally defines two regimes: i) h t w i ≪ h t m i and ii) h t w i ≫ h t m i . The crossover speedcorresponds roughly to ω n .Figs. S2(b), (c), and (d) show the differences in the distribution functions for t m and t w at high, medium, and low load respectively. The waiting times are exponentially distributedbecause the waiting time interval is determined by independent chemical transitions, i.e.,by Poisson processes. The average waiting time thus depends on the stator jump rate k ,which varies between two constants k + and k − (except for extreme high load where k = 0): h t w i ∝ /k . This explains why the averaged waiting time only weakly depends on the load.At low load, premature jumps are rare and the average angular movement δ m is determinedby a single stator jump and has a peaked distribution centered around δ /N . This explainsthe peaked distribution for t m at low load, as shown in Fig. S2(b). At medium load, both h t m i and the value of t m corresponding to the peak of the distribution increase, as shown inFig. S2(c). At high load, the t m distribution develops a flat region for shorter time intervals,as shown in Fig. S2(b). This is a consequence of the decreasing slope of the total potentialfelt by the rotor on the positive force side. As a result, fluctuations of the rotor angle θ dueto thermal noise increase, which leads to many short moving time intervals. IV. DEPENDENCE OF THE TORQUE PLATEAU REGION ON K + AND δ c In order to understand the origin of the torque plateau region, we have studied thedependence of ω n on the ratio r ≡ k + /k − and on the cutoff δ c for the case N=8 (similarresults have been obtained for different values of N). We define ω n as the speed value atwhich the torque decreases 10% from its value at stall. The size of the plateau regime ischaracterized by the following quantity:Σ = ω n /ω max . (2)Fig. S3(a) shows the torque-speed curves for N= 8 and for two different values of r : r = 0and r = 1 .
2. For r = 0 the torque, after a small plateau, decreases linearly with the speed.2ncreasing r , ω n increases and the torque-speed curve increase its concavity. In Fig. S3(b),Σ increases from 0.3 to 0.6 as r increases from 0.05 to 1. This corresponds to the values of r that can produce a well defined shoulder and at the same time maintain the independenceof ω max on N, i.e. | ∆ | < δ c . Startingfrom the value used in the main text, i.e. δ c = δ , ω n increases with δ c . In particular, Σincreases significantly from δ c = δ to δ c = 3 δ , at which the plateau size ( ω n ) reaches avalue slightly bigger than 60% of the maximum speed (Fig. S3(d)).In conclusion the extension of the torque plateau region increases with the value of rate k + and the cutoff δ c , in consistent with our theory. V. ROBUSTNESS OF THE RESULTS AGAINST DIFFERENT ROTOR-STATORPOTENTIALS AND LOAD-ROTOR FORCES
In order to verify the independence of our results on the specifics of the rotor-statorpotential, we studied our model with asymmetric potentials and a smoothed symmetricpotential with a parabolic bottom (instead of the V-shaped bottom). In the asymmetriccase, the slope τ + of the left branch of the potential is much smaller than the slope τ − ofthe right branch, similar to the potential used in [4] (see the values used for τ + and τ − inTab. S1). Our model with the asymmetric potential yields qualitatively similar results aswith the symmetric potential used in the main text. In particular, the maximum speedsnear zero load are independent of the number of stators (see Fig. S4) provided the statorjumping rates satisfy: k − /k + ≫ τ − /τ + . Such a requirement can be understood intuitivelyin the following way. Given the condition τ − ≫ τ + , the force equilibrium (waiting phase) isachieved by having one stator spending part of its time dragging the rotor while all the otherstators are pulling the rotor. The waiting period ends when this dragging stator jumps witha rate that depends on the fractions of time it spends on the two sides of the potential, whichdepend on the ratio τ − /τ + . Therefore, the condition that the maximum speed is dominatedby k − (instead of k + ) has to be weighted by the ratio τ − /τ + .We have also studied a “semi-parabolic” potential V P (∆ θ ≡ θ − θ L ) (see insert in Fig. S5)3efined as: V P = τ [ | ∆ θ | − (∆ θ / | ∆ θ | > ∆ θ / ,τ ∆ θ / ∆ θ if | ∆ θ | < ∆ θ / , (3)where τ is the positive slope of the symmetric potential, and ∆ θ is the angular intervalof the parabolic region centered around the bottom of the potential. Correspondingly, thechemical rate is a continuum function of ∆ θ : k (∆ θ ) = k + + ( k − − k + ) / [1 + exp (cid:0) − (∆ θ ) / ∆ θ (cid:1) ] (4)Fig. S5 shows the torque-speed curves for N = 1 , , ...,
8. The curves show the same charac-teristics as for the V-shaped potential shown in the main text. The plateau region is a littlewider. This is due to the change of the slope near the potential bottom. A lower value ofthe slope slows down the motion of rotor, increasing the premature jump probability beforeit reaches the bottom.Next, we considered different forms of the force function F between the load and therotor. Fig. S6 shows torque-speed curves for two cases: with and without a spring betweenthe load and the rotor for different ratio k + /k − . The case with spring between the load andthe rotor is studied in the main text; the case without spring corresponds to infinite springconstant (rigid connection between load and rotor) with the following equation for the rotor: dθdt = − ξ R + ξ L ∂V∂θ + p k B T / ( ξ R + ξ L ) α ( t ) . (5)Contrary to the model proposed in [4], the concavity of the torque-speed curve does notdepends on the strength of the hook spring: the torque-curves are almost identical with andwithout spring. Instead, the concavity of the torque-speed curve depends on the ratio ofthe jump rates k + /k − , and also on the cutoff δ c as shown before in Fig. S3. In particular,as shown in Fig. S6, for k + = 0 the concavity is zero, and it increases as the ratio k + /k − increases. 4
1] Sowa, Y. Hotta, H. Homma, M. and Ishijima, A. (2003) Torque-speed Relationship of theNa + -driven Flagellar Motor of Vibrio alginolyticus . J. Mol. Biol. , 1043-1051.[2] Berg, H. C. and Turner, L. (1993) Torque generated by the flagellar motor of
Escherichia coli . Biophysical J. , 2201-2216.[3] Block, S. M. Blair, D. and Berg, H. C. (1989) Compliance of bacterial flagella measured withoptical tweezers. Nature , 514-517.[4] Xing, J. Bai, F. Berry, R. and Oster, G. (2006) Torque-speed relationship of the bacterialflagellar motor.
Proc. Nat. Acad. Sci. USA , 1260-1265.[5] Berg, H. C. (2003) The rotatory motor of bacterial flagella.
Annu. Rev. Biochem. , 19-54.[6] Thomas, D. R. Francis, N. R. Xu, C. and DeRosier, D. J. (2006) The Three-DimensionalStructure of the Flagellar Rotor from a Clockwise-Locked Mutant of Salmonella enterica SerovarTyphimurium. J. Bacteriol. , 7039-7048. A ng l e ( d e g ) FIG. 1: Compliance curve. The graph shows the angular displacement, θ − θ L , between the rotorand the load as a function of the rotor torque. The non linear spring behavior follows approximatelythe experimental measurement in reference [3]. The curve corresponds to the function F ( x ) − F (0)used in our simulations.
100 200 300Speed (Hz)0.010.1110 < t w > , < t m > ( m s ) t w , t m (ms) m (t m )P w (t w )0.01 0.1 t w , t m (ms) m (t m )P w (t w )0.01 0.1 1 t w , t m (ms) m (t m )P w (t w )(a) (b)(c) (d)(b) (d)(c) FIG. 2: Waiting-time and moving-time statistic. (a) The average waiting-time (dashed line) andmoving-time (solid line) averaged over 500 revolutions as a function of the rotational speed forN= 1. With increasing speed, the averaged waiting-time slowly decreases from 1ms to 0.1ms whilethe average moving-time decreases much faster over four orders of magnitude, from roughly 50msto 0.005ms. The vertical arrows labeled with the letters (b), (c), and (d), indicate the points wherethe t w and t m distributions are shown in the corresponding figures (b), (c), and (d). The value ofthe load is 14, 1, and 0.1pN-nm-s-rad − , for (b), (c), and (d) respectively. Probability distributions P m for t m are shown by solid lines and P w for t w are shown by dashed lines. The waiting-timesare exponentially distributed in all load range. The moving-times show peaked distributions. Thepeak at high load is partially hidden by a flat region for small t w , as a consequence of the rotorfluctuations. ω / ω max τ ( p N - n m ) r=0r=1.4 r Σ Σ Σ τ ( p N - n m ) δ c / δ Σ (a) (b)(c) (d) FIG. 3: Dependence of the knee speed ω n on r ≡ k + /k − and on the cutoff δ c for the symmetricpotential with a parabolic bottom. (a) Motor torque as a function of the normalized angular speedfor two different values of r : r = 0 (black solid line), and r = 1 . r = 0. (b) Σ ≡ ω n /ω max as afunction of r . (c) Torque-speed curves for different values of the cutoff δ c . Black, red, green, blue,and violet lines correspond to values of δ c of 1, 2, 3, 6, 100 δ respectively. Σ and Σ correspond tothe values of Σ for δ c = δ and δ c = 100 δ respectively. (d) Σ as a function of the ratio δ c /δ shownas the black solid line. The red dashed line corresponds to the asymptotic value Σ( δ c → ∞ ).
100 200 300Speed (Hz)0100200 τ / Ν ( p N - n m ) M o t o r T o r qu e τ ( p N - n m ) N=1N=2N=3N=4N=5N=6N=7N=8
FIG. 4: Torque-speed curves for an asymmetric potential. The torque-speed relationship is verysimilar to the symmetric potential case. The figure shows the motor torque τ as a function ofrotational speed for N= 1 , , ...,
8. For a given value of N the torque is almost constant up to theknee before it decreases roughly linearly. At low speed, the motor torque increases linearly with N.At high speed near zero load all curves collapse to the same maximum speed. These characteristicsare apparent in the insert, where the torque per stator τ /N versus ω are shown.
100 200 300 400Speed (Hz)010002000300040005000 M o t o r t o r qu e τ ( p N - n m ) -0.4 -0.2 0 0.2 0.4( θ−θ S )/ δ E n e r gy ( k B T ) N=1N=2N=3N=4N=5N=6N=7N=8
FIG. 5: Torque-speed curves for the symmetric potential with a smooth, parabolic bottom. Thebehavior is very similar to those observed in the symmetric and asymmetric V-shaped potentialcases. The figure shows the motor torque τ as a function of rotational speed for N= 1 , , ..., ω .The insert shows the “semi-parabolic” potential (solid red curve) and the symmetric V-shapedpotential (dashed black curve) for comparison. τ ( p N - n m ) τ ( p N - n m ) τ ( p N - n m ) SpringWithout spring k + =k_/2k + =k_k + =0 (a)(b)(c) FIG. 6: Independence of the torque-speed characteristics on the load-rotor spring. Torques-speedcurves ( N = 8) with spring (dotted lines) and without spring (squared lines) show almost identicalbehavior. Instead, the concavity of the torque-speed curve depends on the stator jump rate ratio r = k + /k − : (a) r = 0, (b) r = 0 .
5, and (c) r = 1. uantity Value Comment ξ R − Estimated from [2] ξ L ≈ (0.002-50)pN-nm-s-rad − - δ π /26 From Ref. [6] δ δ /2 - T τ τ + K B T/0.95 δ Typical value τ − K B T/0.05 δ Typical value k + (symm. case) 12000 s − Fitting data k − (symm. case) 2 k + From theory k + (asymm. case) 10000 s − Fitting data k − (asymm. case) 20 k + From theory κ − From Ref. [3] κ − From Ref. [3]( θ − θ L ) π/ θ − θ L ) 2 π /7 From Ref. [3]∆ θ δ -TABLE I: Parameters used in the calculation.-TABLE I: Parameters used in the calculation.