Dynamics of the bacterial flagellar motor with multiple stators
aa r X i v : . [ q - b i o . S C ] J a n Dynamics of the bacterial flagellar motor with multiple stators
Giovanni Meacci and Yuhai Tu ∗ IBM T. J. Watson Research CenterP.O. Box 218, Yorktown Heights, NY 10598 ∗ Corresponding author (email: [email protected])
Abstract
The bacterial flagellar motor drives the rotation of flagellar filaments and enables manyspecies of bacteria to swim. Torque is generated by interaction of stator units, anchored tothe peptidoglycan cell wall, with the rotor. Recent experiments [Yuan, J. & Berg, H. C.(2008)
PNAS
Escherichia coli is propelled by the concerted ro-tational motion of its flagellar filaments [1, 2]. Each filament ( ∼ µ m long) is driven bya rotatory motor embedded in the cell wall, with a angular speed of the order of 100 Hz[2]. The motor has one rotor and multiple stators in a circular ring-like structure roughly45nm in diameter [3]. The stators are attached to the rigid peptidoglycan cell wall and thespinning of the rotor drives the flagellar filament through a short hook (see [3] for a 3Dreconstruction and Fig. 1(a) for a 2D sketch of the rotor-stator spatial arrangement). Therotor is composed of a ring of ∼
26 FliG proteins and each stator has four copies of proteinsMotA and two copies of proteins MotB, forming two proton-conducting transmembranechannels. A flow of protons (or, in some alkalophilic and marine
Vibrio species of bacteria,sodium ions), due to electrochemical gradients across the channels, causes conformationalchanges of the stator proteins that generate force on the rotor through electrostatic interac-tion between MotA and protein FliG [4]. The work per unit charge that a proton can do incrossing the cytoplasmic membrane through the proton channel is called the “proton-motiveforce” ( pmf ).At any given time, a stator is engaged with one of the 26 FliG monomers on the FliG ringas the duty ratio of the flagellar motor is close to unity[5]. Presumably, the passage of protonsswitches the stator to be engaged with the next FliG monomer on the FliG ring along thedirection of rotation, stretching the link between the stator and the rotor. The subsequentrelaxation process rotates the rotor and the attached load towards the new equilibriumposition. This can give rise to a step-like motion, characterized by advances of the rotorfollowed by waiting periods. The molecular details of the flagellar motor has been thesubject of intense research[2] and the step-like motion was recently demonstrated by directobservation [6] for a sodium-powered motor at very low pmf , but a general understandingof the stepping dynamics of a single flagellar motor is still lacking.The torque-speed dependence is the key characteristics of the motor[2, 7, 8]. The mea-sured torque-speed curve (see Supporting Information (SI)) for bacterial flagellar motorshows two distinctive regimes. From its maximum value τ max at stall (zero angular veloc-ity), the torque first falls slowly (by roughly 10%) as angular velocity increases at up toa large fraction ( ≈ E. coli at room temperature under physio-logically relevant conditions, the maximum angular velocity is ≈ t m associated with the (mechanical) rotation of the rotor and thewaiting-time interval t w determined by the (chemical) transition of the stator. We find that t m and t w depend differently on the load and their crossover provides a natural explanationfor the observed two regimes of the torque-speed curve. The fluctuation of the motor rotationare also studied in our model. We show that the sources of the motor speed fluctuation aretotally different in the high and low load regimes and that the number of steps per revolutioncan only be extracted from the analysis of motor speed fluctuation in the low load limit. I. MODEL
In Fig. 1(a), a schematic representation of a flagellar motor (rotor and stators) is shown.Each stator has two force-generating subunits symbolized by the light-blue and the redsprings. The two force units of a stator interact with the FliG ring (rotor) in a hand-over-hand fashion as illustrated in Fig. 1(b), analogous to the way kinesin proteins interact withmicrotubules [22, 23]. The switching of hands (force-generating unit) represents the energy-assisted transition when one hand releases its attachment and the other hand establishesits interaction with the FliG ring (rotor). The forces between the FliG ring and the statorsdrive the rotation of the rotor. In Fig. 1(c), the corresponding sequence of this hand-over-hand motion is shown in the energy landscape. The physical motion (solid arrow) ofthe rotor (green circle) is governed by its interaction potential with the engaged FliG. Thehand-switch transition (dotted arrow) corresponds to a shift of the potential energy in thedirection of motor rotation by angle δ and the subsequent motor motion is governed bythis new potential until the next switch. Microscopically, the shift angle could be differentfor the front hand and the back hand (with respect to the direction of the motor rotation);here for simplicity δ is a constant. Commensurate with the periodicity δ = 2 π/
26 of theFliG ring, we should have 2 δ = mδ with a small integer m . In this paper we choose m = 1for simplicity.Due to the small Reynolds number, the dynamics of the rotor angle θ and the load angle θ L are over-damped and can be described by the following Langevin equations: ξ R dθdt = − ∂∂θ N X i =1 V ( θ − θ Si ) − F ( θ − θ L ) + p k B T ξ R α ( t ) , (1) ξ L dθ L dt = F ( θ − θ L ) + p k B T ξ L β ( t ) , (2)where ξ R and ξ L are the drag coefficients for the rotor and the load respectively, and N isthe total number of stators in the motor. V is the interaction potential between the rotor3nd the stator. V depends on the relative angular coordinates ∆ θ i = θ − θ Si , where θ Si isthe internal coordinate of the stator i . θ Si increases by δ when the stator switches hands.This discrete change in θ Si is called a jump of the stator in this paper. The load is coupledto the rotor via a nonlinear spring described by a function F , which can be determinedfrom the hook spring compliance measurement of ref. [24] (see Fig. S1 in SI). The lastterms in Eq. (1-2) are stochastic forces acting on the rotor and on the load, with k B theBoltzmann constant, T the absolute temperature, and α ( t ) and β ( t ) independent white noisefluctuations of unity intensity.The dynamics of the stator i is governed by the transition probability for the discretejump of its internal variable θ Si during the time interval t to t + ∆ t : P i ( θ Si → θ Si + δ ). In thispaper, P i is assumed to depend on the torque generated by the i ’th stator τ i ≡ − V ′ (∆ θ i ),which depends on the relative angle ∆ θ i : P i ( θ Si → θ Si + δ ) = r ( τ i )∆ t = k (∆ θ i )∆ t. (3)The specific form of the jumping rate r ( τ i ) (or k (∆ θ i )) is unknown. We assume it to bea decreasing function of τ i , with the stator stepping rate being higher when τ i is negative( τ i < , ∆ θ i >
0) than when τ i is positive ( τ i > , ∆ θ i < δ with a probability rate that is a function of its relative coordinate. For simplicity, weset the potential function V to be a V -shaped function: V (∆ θ ) = τ | ∆ θ | , and the torquefrom a single stator is τ with its sign depending on whether the stator is pulling (∆ θ < θ > k (∆ θ < − δ c ) = 0, k ( − δ c < ∆ θ <
0) = k + , k (∆ θ >
0) = k − ( > k + ) as illustrated inFig. 1(e). A cutoff angle δ c is introduced to prevent run-away stators. Quantitatively, weuse τ = 505pN-nm, ξ R = 0 . − , k + = 12000s − , k − = 2 k + , δ c = δ in thispaper unless otherwise stated. The load ξ L varies from 0 . − − . Simulationtime step ∆ t = 0 . − µs . II. RESULTSA. Two characteristic time scales and their different dependence on the motorspeed
In Fig. 2(a), a typical case of time dependence of the rotor angle θ ( t ) from our modelis shown. The motion of the rotor consists of two alternating phases: moving and waiting.The moving phase occurs when the net force on the motor is positive (in the direction ofmotion). The waiting phase is when the system reaches mechanical equilibrium (net forceequals zero) and the motions are driven by thermal fluctuation. The dynamics of the motorcan thus be characterized by the two time scales t m and t w . The waiting-time t w is thetime the rotor spends fluctuating around a equilibrium position, i.e., the bottom of the totalpotential V t ≡ P Ni V (∆ θ i ). Once in the waiting phase, the rotor can only start to have anet motion when a stator jumps to break the force balance and thus shift the equilibriumposition forward. The subsequent net motion of the rotor to reach the new equilibriumposition takes t m , which is defined as the moving-time. The definitions of t w and t m are4hown in Fig. 2(b).The dynamics of the motor depend on the load, higher load leading to slower speed. Westudy how the two scales t m and t w vary with the load or equivalently the speed of themotor (speed is chosen because of its direct measurability in experiments). We find thatthe two time intervals have very different dependence on the motor speed as shown in Fig.2(c). The waiting-time interval is determined by independent chemical transitions, i.e, by aPoisson process with rate k , so we have: h t w i ∝ h k − i . Since k varies between two constants k + and k − (except for extreme high load where k = 0), the averaged waiting-time has onlya weak dependence on motor speed as shown in Fig. 2(c). On the other hand, the averagemoving-time can be estimated as: h t m i ≈ δ m /ω m , with δ m the average angular movement, ω m ≡ τ m / ( ξ R + ξ L ) the average speed, and τ m ≡ h− V ′ t i m the average net torque in themoving phase. Increasing the load ξ L leads to a decrease of the speed ω m and an increaseof the moving-time. In addition, at lower speed, it is more likely for stator to jump in themiddle of a moving phase before the system reaches its force equilibrium. These prematurestator jumps effectively increase δ m and further increase h t m i . These two factors lead to astrong dependence of h t m i on the load (or the speed) as shown in Fig. 2(c). Besides thedifference in their average values, the distribution functions for t m and t w are also different(See Fig. S2 in SI for details). B. The two regimes of the torque-speed curve
In Fig. 3, the torque-speed curves calculated from our model for 8 different stator numbersare shown. Our model results closely resemble the observed torque-speed curves. There isa plateau regime with almost constant (10% decrease) torque from zero up to a large speed( ≈ Hz for N = 8), followed by a steep declining regime of the torque, all the way to zeroat a speed of roughly 300 Hz . By using the two time scales t m , t w , and noting that the nettorque is zero during the waiting phase of the motor, the time-averaged torque τ and speed ω can be estimated: τ ≈ h t m ih t m i + h t w i τ m , ω ≈ δ m h t m i + h t w i . (4)The two distinctive regimes in the torque-speed curve can be understood intuitively withinour model by the different dependence of h t m i and h t w i on the speed shown in the lastsection.In the low-speed (high-load) regime defined by h t m i ≫ h t w i , we have τ ≈ τ m and ω ≈ δ m / h t m i from Eq.(4). As discussed in the last section, for low speed a stator can jumpprematurely during the moving phase before the system reaches the bottom of the potentialwell. As a result, each stator spends most of its time generating positive torque τ . Therefore,in this high-load regime, while the speed changes significantly, the torque stays near itsmaximum value τ max = N τ , which is proportional to the number of stators.In the high-speed (low-load) regime defined by h t m i ≪ h t w i , we have τ ≈ τ m h t m i / h t w i and ω ≈ δ m / h t w i from Eq.(4). As shown in Fig. 2(c), for increasing speed h t m i decreasesquickly while h t w i remains roughly the same. This naturally explains the steep decrease ofthe torque τ with speed in the high-speed regime. Intuitively, in this high-speed regime, astator can be pushed into the negative torque region (∆ θ >
0) because the rotor rotatestoo fast for the premature jump to occur. As the stators spend large fractions of their timedragging the rotor, the torque of the motor decreases quickly.5he different dependence of the waiting and moving-time intervals on the speed notonly gives a clear general explanation for the two regimes of the torque-speed curve, it alsoexplains the sharpness of the transition between the two regimes. Since the dependence of h t m i on the speed is much steeper than that of h t w i (as shown in Fig. 2(c)), the crossoverbetween the two regimes takes place in a small region of the speed values, thus makingthe two regimes in the torque-speed curve well defined, as found in both experiments andsimulations of our model. C. Independence of the motor speed on the number of stators at near zero load
At near zero load, our model shows that the motor moves with a roughly constant speedthat is independent of the number of stators, as demonstrated in Fig. 3. Recent res-urrection experiments using gold nano-particle (extremely low load) indeed showed suchindependence[20]. The mechanism for this surprising behavior can be understood with ourmodel. In the low-load regime, the motor spends most its time in the waiting phase wherethe net torque is zero. In our model with symmetric potential V , this force equilibriumis achieved by having on average half of the stators pulling the rotor and the other halfdragging it. If we number the stators in Fig. 1(d) from left to right, the rotor’s equilibriumposition sits between the N/ N/ N/ N k − /
2. 2) The ( N/ N/ N/ k + . The average distance between the new and the old equilibriumpositions are δ m ( ≈ δ /N ) and δ m / N is due to the high duty ratio as first recognizedin [25, 26]. Similar step size reduction with N was recently observed in kinesin-1 motor[27].The maximum speed ω max near zero load is then: ω max ( N ) ≈ δ m N k − δ m k + ≈ k − δ k + /k − ) N − ] , (5)which only depends weakly on N , if k + /k − ≪
1. The estimated maximum speed ω max ∝ k − δ / ω max should be limited by the step size and the maximum steppingfrequency of an individual stator.We have studied the dependence of ω max on the ratio r ≡ k + /k − and N by numericalsimulations of our model. In Fig. 4(a), we show the torque-speed curves for N = 1 and N = 8 for two different values of r : r = 0 . r = 1 .
2. To quantify the dependenceof ω max on N , we define a quantity ∆ ≡ ω max (1) − ω max (8)) / ( ω max (1) + ω max (8)) tocharacterize the relative difference between the maximum speeds for motors with one andeight stators. As shown in Fig. 4(b), ω max is roughly independent of N , i.e., | ∆ | < . r ≤ .
5. However, ω max (1) becomes significantly bigger than ω max (8) for r ≥
1. Theobserved dependence of ∆ on r agrees well with the analytical estimate given by Eq.(5).6 . Motor speed fluctuation at different load levels and the estimate of step num-bers The measured motor speed fluctuates due to two main factors: the external noise such asthe Brownian noise and measurement noise, and the intrinsic probabilistic stepping dynamicsof the stators. Samuel and Berg[26] first investigated the speed fluctuations by studying thesmoothness of the periodic motor motion characterized by Γ ≡ n h T i / ( h T n i − h T n i ), where T n is the period for n revolutions. By measuring Γ in a resurrection experiment wherethe stator number is inferred from the discrete increments in average motor speed, it wasfound that Γ is proportional to the number of stators. The proportionality constant wasinterpreted as the number of steps per revolution. Here, we analyze the motor fluctuationby using our model to understand how different noise sources contribute to Γ and how Γbehaves differently at different load levels.For low loads, the motor spend most of its time in the waiting phase. The average motorstep size is δ m ≈ δ /N ( ≪ π ), there are n s ≡ π/δ m ≈ πN/δ steps in each revolution, andthe average periodicity is h T i = n s h t w i . Since the waiting-time intervals are uncorrelated,the variance of the n − revolution periodicity can be expressed as: h T n i − h T n i = nn s ( h t w i −h t w i ). Furthermore, because the waiting-time t w is determined by a Poisson process, itsvariance is equal to h t w i . Γ can thus be written as:Γ ≡ n h T i h T n i − h T n i ≈ n s h t w i h t w i − h t w i ≈ πδ N, (6)showing that Γ = γN is proportional to the stator number N , and the proportionalityconstant γ = 2 π/δ corresponds to the number of steps per stator per revolution. Thisbehavior is verified in our model by calculating Γ during a simulated resurrection process,where additional stators are added by a Poisson process with time constant t s = 400 s (Fig.5). For near zero load, the average speed is independent of the stator number in agreementwith [20] (see Fig. 5(a)). However, Γ increases by a fixed amount γ = 2 π/δ as a new statoris incorporated into the system as shown in Fig. 5(b), consistent with the analytical resultby Eq.(6). The behavior of Γ as shown in Fig. 5(b) represents a quantitative prediction ofour model that could be tested in resurrection experiments with extreme low load, such asin [20].For high load, the net torque is roughly constant τ ≈ N τ and the speed can be expressedas ω = τ / ( ξ L + ξ R ) ≈ N τ /ξ L , which explains the constant increment of speed for everyadditional stator (up to eight) seen in our model (Fig. 5(c)) as well as in the resurrectionexperiments by Blair and Berg[28]. For additional stators beyond a certain large numberof stators, the speed passes the knee in the speed-torque curve and our model predicts adecrease in the speed increment, which is consistent with the recent experiments by Reidet al [12] that showed the same decrease in speed increment as stator number goes upto N = 11. The dynamics of the load angle can be obtained by summing Eqs.(1-2) andtaking the limit ξ R /ξ L →
0. This leads to: ˙ θ L = ω + p K B T /ξ L β ( t ), which describesthe simple motion of the load with a constant speed ω perturbed by random noise. Fromthe equation for θ L , the periodicity and its variance can be determined: h T n i ≈ nπ/ω , h T n i − h T n i ≈ nπk B T / ( ξ L ω ). We can now express Γ as:Γ ≡ n h T i h T n i − h T n i ≈ πξ L k B T ω ≈ πτ k B T N. (7)7 is again proportional to N through its dependence on the speed ω . However, unlike inthe low-load regime, the proportionality constant γ = πτ / ( k B T ) has nothing to do with thenumber of steps per revolution. Instead, γ depends on the ratio between the intrinsic drivingforce ( τ ) and the external noise k B T , as the driving force overcomes the external noise tomake the motor moves smoothly. This behavior is verified in our model by calculating Γduring the resurrection simulation. As shown in Fig. 5(d), Γ goes up with the number ofstators but with a much larger proportionality constant γ , which quantitatively agrees withthe expression πτ / ( k B T ) from our analysis.Therefore, although the motor-speed fluctuation is always suppressed by higher numbersof stators, the mechanisms are different for different load levels. For low load, the smoothermotion for larger N is caused by the increase in step number per revolution. For high load,the smoother motion for larger N is caused by larger driving force (therefore larger speed)in comparison with the constant external noise. The difference in motor fluctuation betweenthe high and the low load regimes is confirmed by our simulation as shown in Fig. 6, wherethe proportional constant Γ is shown for different values of external noise strength k B T (Fig.6(a)) and different load (Fig. 6(b)). III. SUMMARY AND DISCUSSION
We have presented a mathematical description of the rotary flagellar motor driven byhand-over-hand power-thrusts of multiple stators attached to the motor. All key observedflagellar motor properties[2], including those from a recent resurrection experiment at nearzero load[20], can be explained consistently within our model. The crucial ingredient ofour model is that the hand-switching rate depends on the force between rotor and stator.This feature is known to be valid for other molecular motors, including kinesin[29] andmyosin[30]. Therefore our model should be generally applicable to the study of these linearmotors, especially in the case when there are multiple power-generating units attached tothe same track[27].For the flagellar motor, we find that its dynamics follows an alternating moving andwaiting pattern characterized by two time scales t m and t w . The mechanism underlyingthe observed torque-speed relationship and its dependence on the number of stators, isrevealed by studying the dependence of these two time scales on the load. For high load, h t w i ≪ h t m i , the motor spend most time moving (albeit slowly) with all the stators pullingthe motor in the same direction. So the torques generated by individual stators are additive,leading to a roughly constant torque τ max ≈ N τ , which persists up to the knee speed ω n .Microscopically, the existence of this torque-plateau regime is due to the premature statorjumps which prevent the stators from going into the negative torque region. Since the rate ofthe premature jumps is k (∆ θ < k + and larger cutoff δ c increase the knee-speed ω n (see Fig. S3 in SI for details). For low load, h t w i ≫ h t m i , the motor spends most time in thewaiting state. A waiting period ends when one of the dragging stators jumps to the pullingside or the pulling stator closest to the bottom of the potential well jumps. Therefore themaximum motor speed ω max is limited by the maximum jumping rate of the stators. ω max can be estimated from our model. Eq.(5) shows that ω max has only a weak dependence on N for small k + /k − , as confirmed by simulations of our model, and in agreement with recentresurrection experiments at near zero load[20]. Eq.(5) also explains the strong dependenceof ω max on N in a recent model by Xing et al[19]. The jumping probability used in [19]has a complicated profile and is maximum in the positive torque region. In our model, this8ould correspond to having k + /k − ≫
1, which is the opposite to what is required to achieveindependence of ω max on N .The robustness of our results were verified using different forms of the rotor-stator po-tential V and the force function F between the load and the rotor. In particular, we havestudied a smoothed symmetric potential with a parabolic bottom and an asymmetric po-tential V (similar to the one used in [19]) where the negative torque ( τ − ) is bigger than thepositive torque ( τ + ). We find that all of our general results remain the same (see SI and Fig.S4&S5 for details). For the asymmetric potential, the condition for ω max being independentof N is generalized to k + /k − ≪ τ + /τ − . From the analysis and direct simulation of our model,we do not find any significant dependence of the torque-speed curve characteristics on thespecifics of the force function F between the load and the rotor. In particular, contrary towhat was proposed in [19], there is no difference between the case of a viscous load thatinteract with the rotor through a soft spring and a viscous load without spring (see Fig. S6in SI for details).Besides the torque-speed curve which describes the time averaged behavior of the motor,we have also studied the speed fluctuation for individual flagellar motor. We find that thefluctuation is damped by the number of stators for all load levels. However, we show thatthe dominating source of the motor fluctuation is different depending on the load. For lowload, the speed fluctuation is dominated by the discrete stochastic stepping events whereasfor the high load, it is controlled by the external noise, such as Brownian fluctuations orpossibly measurement noises. The original measurements on motor fluctuation by Samueland Berg[26] were done in the high-load regime as evidenced by the discrete increment ofspeed in their resurrection experiment. Therefore the strength of the fluctuations obtainedthere is probably more reflective of the strength of external noise than the number of stepsper revolution. It would be interesting to perform the fluctuation analysis in the low-loadregime as achieved in [20] to determine the steps number per revolution and compare withthe recent direct observation of the steps[6].Simple relations between the macroscopic observables ( τ max , ω n , ω max ) and the micro-scopic variables of the system ( τ , k + , k − ) are established by analysis of our model. Theserelations can be used to predict the microscopic parameters quantitatively from the torque-speed measurements. They can also be used to study the dependence of the flagellar motorproperties on other relevant external parameters such as the pmf , the temperature, andsolvent isotope effects. For example, since changing of pmf gives rise to self-similar torque-speed curves[32], we conclude from our model that larger pmf not only increases the chem-ical transition rates k ’s, it also increases the stator-rotor interaction strength τ . Changingtemperature or replacing H + with D + (solvent isotope effect) should affect the chemicaltransition rates. These changes in k + and k − lead to changes in the knee speed ω n and themaximum speed ω max in our model without changing the maximum torque at stall, whichis consistent with previous experimental observations[9, 31].Backward stator jumps with θ S → θ S − δ can be incorporated in our model to studythe relatively rare motor back-steps[6]. The back-jumps are neglected in this paper as theirprobabilities are much smaller than those for the forward jump in the region of relativeangles (∆ θ > − δ c ) relevant for our study here. However, we expect the back-jumps tobecome dominant for ∆ θ < − δ c where the forward jumps are prohibited. Since the landingpoints of these back-jumps are still on the positive side of the potential with positive torque τ , inclusion of back-jumps in our model for ∆ θ < − δ c can naturally explain the observedtorque continuity near stall when the motor is driven backwards by an optical tweezer[10].9n our model, the step size depends inversely on the stator number N . This behavioris a general consequence of duty ratio being unity and independent stepping of the stators,as pointed out by Samuel and Berg[26]. This N − dependence of the step size seems to beinconsistent with an “apparent independence” of step size on N claimed in [6]. However, acareful study of the experimental data reveals that the N dependence of the step size cannot be ruled out, because the step size distribution was measured for a varying population ofstators, whose number was neither controlled nor measured precisely in [6]. An unambiguousway to determine whether the step size depends on N is to measure the step size for differentfixed N or at least to measure N simultaneously. Such experiment was done recently forkinesin-1[27] and showed that step size for N = 2 is half of that for N = 1.Our model works for the clockwise (CW) as well as for the counterclockwise (CCW)rotation. It was recently suggested that the switching between the CW and CCW state ofthe motor is a non-equilibrium process and the energy needed to drive the motor switchcould be provided by the same pmf that drives the mechanical motion of the motor [33].The possible link between the switching process and the rotational motion of the motor issupported by experimental observations [11] showing that the average switching frequencydepends on the proton flux. It is therefore highly desirable to develop an integrated modelto describe both the mechanical part of the flagellar motor, associated with the rotationalmotion, with the signaling part, associated with the switching process. More experimentalinformation on the components of the motor (M-ring/C-ring/MotAB) and how they interactwith each other are needed to achieve this goal. IV. ACKNOWLEDGMENTS
This work is partially supported by a NSF grant (CCF-0635134) to YT.10
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Proc. Nat. Acad. Sci. USA , 11737-11741. IG. 1: A Model for the flagellar rotary motor. (a) Schematic illustration of the rotor-statorsspatial arrangement. The rotor contains 26 FliG proteins and there are multiple stators, each withtwo subunits (red and light blue springs). (b) A sequence of three rotor-stator configurations (fromtop to bottom) illustrating the hand-over-hand interaction between the two subunits of a statorand the FliG proteins in the rotor. (c) The same sequence as in (b) is shown in the potentiallandscape. The solid arrow represents the physical rotation of the rotor angle ( θ ) down a given(V-shaped) potential, the dotted arrow represents the chemical change (switching of hands) thatshifts the potential. (d) The full motor model with multiple stators in the angle space. Each statoris represented by its internal angle θ S . The rotor is pulled forward by the stators in front it anddragged back by the stators behind it. The stator angle can only change by jumping forward withrate k that depends on the relative angle ∆ θ = θ − θ S . The form of k (∆ θ ) used in this paper isgiven in (e), which shows the dragging stators have a higher jump rate k − > k + and a cutoff angle − δ c where k (∆ θ < − δ c ) = 0. IG. 2: The motor dynamics and its dependence on the load. (a) Rotor angle θ versus timefor ∼ / N = 1. The angular unit is the FliG periodicity δ . The insert isenlarged in (b). (b) Zoom of the time series in (a) showing two complete steps. Solid line showsthe stator position θ S . A jump in θ S marks the start of a moving phase for the rotor and thewaiting phase starts when the rotor catches up with the stator. The definitions of the moving-time t m and the waiting-time t w are shown. (c) The average waiting-time h t w i (dashed line) and theaverage moving-time h t m i (solid line) over 500 revolution as a function of the rotational speed forN= 1. h t w i decreases slowly with increasing speed from 1ms to 0.1ms while h t m i decreases muchfaster from roughly 50ms to 0.005ms.
50 100 150 200 250 300 350Speed (Hz)01000200030004000 M o t o r T o r qu e τ ( p N - n m ) τ / N ( p N - n m ) N=8N=7N=6N=5N=4N=3N=2N=1
FIG. 3: The torque-speed ( τ − ω ) curves for different stator numbers ( N = 1 to N = 8) from ourmodel. Two regimes of the τ − ω curves, i.e., constant τ up to a large knee speed ω n and fastdecrease of τ to zero at the maximum speed ω max , are evident for all stator numbers. The torqueper stator τ /N versus the speed ω is shown in the insert. The torque at stall scales with N whilethe maximum speed is independent of N . IG. 4: Dependence of the maximum speed at zero load on N as a function of r ≡ k + /k − . (a)Torque-Speed curves for r = 0 . r = 1 . N = 1 (blacklines) and N = 8 (red lines). We change r by varying k + and keeping k − constant. For r = 0 . ω max (1) and ω max (8) at zero load are roughly the same, while they differsignificantly for r = 1 .
2. (b) The dependence of ∆ = 2( ω max (1) − ω max (8)) / ( ω max (1) + ω max (8))on r. The red line represents our analytical predictions from Eq.(5). The shaded region shows the ∼
12% experimental error.
800 1600 2400 32000100200300 S p ee d ( H z ) Γ (a) (c)(b) (d) FIG. 5: The motor speed and its fluctuation in a simulation of the resurrection process for lowand high loads. (a) Speeds as a function of time after successive stators are added at low load( ξ L = 0 . − ). The simulation shows no dependence of the speed on the number ofstator (labeled with a number form 1 to 8). (b) The smoothness parameter Γ with n= 5 calculatedfrom the time series shown in (a). Γ value is calculated with a moving-time window of 1 . s ( ∼ ξ L = 8pN-nm-s-rad − )showing the roughly lineardependence of the motor speed ω on N . (d) Γ at high load from the time series shown in (c) usinga window of roughly 10 s corresponding to a total of 100 revolutions. (b) and (d) show that Γincreases with N at both high and low load. T / T Γ Theory at high loadTheory at low loadHigh loadLow load
Speed (Hz)0100200300400500 (a) (b)
FIG. 6: The speed fluctuation and its dependence on the load (with N = 1). (a) The dependenceof Γ on the external noise strength defined as T /T , where T is the room temperature. Γ dependsstrongly on T for high load (crosses) while it is a constant determined by the step size at load load(dots), in agreement with our theoretical results (lines).(b) The dependence of Γ on speed ω for T = T . Typical error bar (SD) is shown at high load; the error at low load is comparable to thesymbol size.. Typical error bar (SD) is shown at high load; the error at low load is comparable to thesymbol size.