Our model explains stepping of the bacterial flagellar motor by interpreting its rotation as a viscously damped random walk driven by a constant torque and by a heterogeneous contact potential caused by the physical irregularities of the rotor. In this picture, steps are recognized as barrier-crossing events between adjacent minima of a tilted and corrugated energy potential. Corrugations are caused by contact between the stators and the protein arrays (FliG, among others) making up the rotor structure. Recently a more accurate picture of this structure has emerged, thanks notably to electron microscopy studies 
Our model predicts a
periodicity of the potential, so that the absolute angular position of the rotor with respect to the stator is an underlying determinant for step statistics, and this prediction is found to be consistent with the available experimental data. In particular, our model offers an explanation for the experimental observation that backward steps are smaller than forward steps on average.
Another prediction of the model is that rotor speed grows sublinearly as a function of torque. At low torques rotation is slow because of trapping in local minima, whereas at high torques the barriers between minima are lowered and eventually eliminated. Additionally, we predict that at low torques rotor diffusion is hindered by barriers, while at high torques the variability of the potential actually enhances diffusion. Although in principle other sources of fluctuations, such as ion translocation, could impact rotor diffusion, we showed that in the relevant regimes simple diffusion can account for nearly all of the observed variance in cycle time 
. In order to verify these predictions experimentally, one would need to simultaneously measure the rotor speed and the proton (or
) motive force, believed to be proportional to torque at low speeds, in the regime where torque and contact forces are comparable. (Note that in the stepping data 
we have analyzed, torque could vary during the course of the experiment as the result of changes in the number of stator units.) Interestingly, both a stepping behavior and a sublinear speed vs.
torque relationship were reported in experiments on flagellar motors in Streptococcus 
. Cells were starved, and then energized or de-energized to control PMF. Motion was found to have a rotational symmetry of
, which corresponds to one revolution of the one-start helix of an axial component (11 in two revolutions). When energized (increasing PMF), cells displayed a sublinear speed vs.
PMF dependence in agreement with our prediction. However, when cells were de-energized (decreasing PMF), this relation became linear. Such history dependence could occur if the stator elements hindering rotation are pushed away as PMF increases, leaving rotation unhindered during PMF decrease.
Our model is consistent with other experimental results on the bacterial flagellar motor. Because the model relies on the assumption that the energy from ion translocation is reversibly stored in protein springs 
, it implies a near-perfect efficiency of the motor at low torques 
. The same mechanism can account for both clockwise and counterclockwise motor rotation 
—these two cases simply corresponding to the springs being stretched in opposite directions. If the contact potential stays the same when the motor changes direction, our model predicts that backward steps will occur preferentially at the same absolute angles irrespectively of the direction of rotation. The observation that the duty ratio is very close to one even with a single torque-generating unit 
can be encompassed in our model by assuming that each torque-generating unit comprises at least two springs. The advantage of a spring mechanism over other mechanisms is that it naturally entails
efficiency, at least at lower speeds. When the system is in thermodynamic equilibrium, which is the case near stall where kinetic rates are much faster than spring relaxation, the average energy provided to the springs by the passage of one proton is exactly
equal to the potential energy difference
between the exterior and the interior of the cell. This is simply a consequence of reversibility. The key point is that all the energy stored in protein springs is eventually used to move the rotor. This stands in contrast to mechanisms driven by irreversible conformational changes, where some energy is typically wasted because the energy required for the conformational change is less than the energy provided by the source, e.g. ATP hydrolysis for myosin motors. The utilization of springs for the reversible storage of the energy suggests a general mechanism underlying the operation of high-efficiency molecular machines.
Our study has focused on the adiabatic regime, where springs are near equilibrium with respect to the PMF, as this is the relevant regime for the stepping experiments. However, our model predicts that when the proton flux becomes kinetically limited, the springs will fail to restretch completely, causing the torque to drop. This observation can explain the observed “knee” in the motor's torque-speed relationship 
, when coupled to an explicit model of proton translocation 
In our analysis we have neglected one effect that is not crucial for our analysis, but which may prove important for inferring the detailed nature of the contact potential. Specifically, we have assumed that equilibration of the elastic linkage between the motor and the load is rapid compared to the waiting time between steps. For a torsion constant 
between the rotor and the load, and a drag coefficient
, the relaxation time is
. In contrast, the typical waiting time between steps ranges from
, depending on experimental conditions. If the elastic linkage was too soft, the polystyrene bead would respond to the motion of the rotor with a delay
, and steps would be smoothed out. This does not seem to occur in the experiment.
Another effect, which we have considered (see Rotor diffusion) but did not include in our simulations, is the relaxation of MotA/B protein springs as a rotor step occurs. For example, when the rotor moves forward, the torque decreases because the protein springs relax. Usually these springs are restretched so quickly by ion translocation that the transient decrease of torque can be neglected. However, during a barrier crossing event the rotor motion might be so fast that protons are not able keep up. This would result in a temporary drop in torque and make barrier crossing more difficult. A similar argument applies to backward steps. We have already shown that at the “mean-field” level, where rotation speed and ion flux are time-averaged, this negative feedback has only a small effect. However, the instantaneous rotation speed during a step can be much larger than its average. How fast can a proton translocate through the motor? The maximum flux of protons through a single motor unit can be estimated by considering the maximum rotation speed before the torque starts dropping (the “knee” of the torque-speed relationship 
). For a single motor unit in natural conditions, this speed is about 150 Hz for a torque of 
. The power generated by the motor is then
. Each proton provides at most
, so that the number of protons per second is at least
. The timescale of proton passage is therefore less than
. A single rotor step corresponds to the passage of 
, so restretching the protein springs should take less than
, which is below the current experimental time resolution. For comparison, an instanton calculation 
reveals that the typical time for crossing a barrier is bounded from below by
In our analysis we have also neglected the effect of the “shot noise” arising from the discrete nature of the proton flux. This shot noise leads to fluctuations in torque, which could in principle affect the stepping behavior as well as the rotor diffusivity. While we have neglected this source of noise on the basis that it is averaged out by the presence of multiple stators, its influence can be significant at very low loads 
Other molecular motors have shown stepping behavior, including the actin-myosin motor 
, the dynein-microtubule motor 
, and kinesin 
. In these ATP-powered motors, which are less powerful than the bacterial flagellar motor by orders of magnitude, stepping is a built-in and essential part of motor operation. By contrast, we have argued that in the bacterial flagellar motor the observed stepping arises solely from steric hindrance.
Our work leaves open a number of questions. It would be interesting to infer the precise form of the contact potential
from rotation data and see how and whether it varies in time and among motors, potentially yielding new insight into the dynamics of motor assembly and reorganization. To this end a more sophisticated approach to learning the potential may be required, e.g.
employing maximum likelihood techniques. Lastly, one still needs to understand the mechanism of torque generation, including the role played by the discreteness of ion translocation, the chemical nature of protein springs and their attachment sites, as well as the energy conversion process.