### abstract ###
The bacterial flagellar motor is a highly efficient rotary machine used by many bacteria to propel themselves.
It has recently been shown that at low speeds its rotation proceeds in steps.
Here we propose a simple physical model, based on the storage of energy in protein springs, that accounts for this stepping behavior as a random walk in a tilted corrugated potential that combines torque and contact forces.
We argue that the absolute angular position of the rotor is crucial for understanding step properties and show this hypothesis to be consistent with the available data, in particular the observation that backward steps are smaller on average than forward steps.
We also predict a sublinear speed versus torque relationship for fixed load at low torque, and a peak in rotor diffusion as a function of torque.
Our model provides a comprehensive framework for understanding and analyzing stepping behavior in the bacterial flagellar motor and proposes novel, testable predictions.
More broadly, the storage of energy in protein springs by the flagellar motor may provide useful general insights into the design of highly efficient molecular machines.
### introduction ###
Bacteria swim by virtue of tiny rotary motors that drive rotation of helical flagella.
These motors are powered by a transmembrane proton flux which is converted into torque.
However, little is known about the detailed mechanisms of energy conversion, or torque generation.
Recently, a new result has provided direct insight into motor operation CITATION : at low speeds, the bacterial flagellar motor proceeds by steps.
This stepping is stochastic in nature, as manifested by the occurrence of occasional backward steps even for motors locked in one rotation direction.
What is the origin of motor steps and how can these steps be reconciled with the near perfect efficiency of the motor observed at low speeds CITATION ? We argue that steps, including backward steps, are an inevitable consequence of the physical structure of the motor a stator driving a bumpy rotor through a viscous medium.
In response to chemotactic signals, flagellar motors switch from counterclockwise to clockwise rotation causing cells to tumble or change directions.
In Escherichia coli, the basic mechanism of torque generation appears to be the same for both directions of motor rotation CITATION.
Torque is generated by the passage of FORMULA ions through the cytoplasmic membrane.
As shown schematically in Fig.
1A, torque is applied to the rotor, including the flagellum, by the stator, which is comprised of independent torque-generating units anchored to the peptidoglycan cell wall.
The exact number of torque-generating units can vary from motor to motor, with the maximum estimated to be at least 11 CITATION.
The rotor includes 26 circularly arrayed FliG proteins that contact the MotA/B complexes.
The torque-speed relation of the motor has been measured under a range of conditions CITATION CITATION.
The maximum torque in the high load, low speed regime tracks the electrochemical potential difference or proton motive force across the membrane, and the motor operates with nearly perfect efficiency CITATION.
Whereas torque and efficiency fall off at high speeds, proton flux and motor rotation are always strongly coupled with FORMULA protons passing through the membrane per MotA/B unit per rotation CITATION .
Recent experiments, where rotation was measured by attaching a polystyrene bead to a flagellar stump driven by a counterclockwise-locked FORMULA chimaeric motor at low speeds, revealed that the motor proceeds by steps CITATION.
The steps have average size FORMULA, which corresponds to 26 steps per rotation, exactly the number of copies of FliG around the rotor.
Occasional backward steps are observed and, interestingly, these are smaller on average than forward steps.
These observations, as well as the stepping mechanism itself, have so far remained unexplained.
It has been suggested that stepping is caused by the stochastic passage of ions.
However, as pointed out in CITATION, the energy provided by passage of a single ion can only move the rotor attached to a FORMULA polystyrene bead by FORMULA, much less than the typical observed step size.
Here we propose a simple physical model to explain stepping: the stator applies nearly constant torque to the rotor, but, at the same time, contact forces on the rotor produce a potential and therefore an additional torque with approximately the 26-fold periodicity of FliG.
Flagellar rotation is viewed as a circular random walk in a bumpy potential biased to favor rotation in a particular direction by the torque exerted by the stator elements.
Our model naturally accounts for the existence of backward steps, as well as the discrepancy between forward and backward step sizes, and also predicts that step statistics depend on the absolute position of the rotor around the circle.
Our predictions are found to be consistent with the available data, including angular diffusion of the motor CITATION, and suggest how steps could be used to study the physical structure of the motor.
A novel testable prediction is that the torque-speed relation will become sublinear at very low torques.
Our model for stepping relies on two main assumptions: constant or nearly constant torque between stator and rotor and an approximately 26-fold periodic contact potential.
All torque-generating units apply torque simultaneously and additively.
Following the model of Meister et al. CITATION, we assume that each MotA/B complex acts as a set of protein springs that reversibly store the energy available from FORMULA translocations.
The protein springs are attached to fixed sites of the rotor circumference.
When an FORMULA passes through the membrane, it causes a spring to detach from its attachment site, stretch, and reattach to the next site.
At stall, all springs are maximally stretched, such that the PMF matches the energy necessary to stretch a spring to its next site.
At low speeds, the rotor moves and springs relax, but these are quickly restretched by FORMULA passage, so that the system remains in quasi-equilibrium with the torque set by the PMF.
Spring stretching may vary slightly among units, but since there are several motor units, we assume that the instantaneous torque self-averages and is nearly constant in time.
Under this scenario, steps cannot be explained at the level of a single pair of MotA/B and FliG subunits, but must arise at the global level of the rotor-stator interaction.
There are contact forces between the stator and the rotor.
These forces may be caused by contact between the MotA/B stator units and FliG proteins, but also possibly by contact with FliF, FlgH or FlgI proteins each of which forms a circle of 26 copies.
There may be other periodicities to the contact forces as well, arising from the filament and the hook, which are 11-fold periodic, from FlgK and FlgL, FlgB, FlgC and FlgF and FliE.
We assume that a 26-fold periodicity is dominant, in agreement with experimental observations.
We therefore collect all contact forces in a potential FORMULA which we suppose to be approximately 26-fold periodic .
Since the motor operates at the molecular scale, its rotation is intrinsically stochastic as it is subject to random thermal fluctuations.
Another potential source of noise is fluctuations of the torque applied by the individual MotA/B stators, due to the discrete nature of the proton flux.
However, in presence of multiple independent stator units, this noise averages out and can be neglected.
Under the combined influence of the applied torque, the contact potential, and thermal fluctuations the rotor performs a circular and continuous random walk in a tilted, approximately periodic potential, which we model by the following Langevin equation:FORMULAwhere FORMULA is the drag coefficient, FORMULA the total torque exerted via protein springs by the stators, and where the potential FORMULA includes the torque, and the approximately 26-fold periodic contact potential:FORMULAThe term FORMULA represents Gaussian white noise and accounts for thermal fluctuations: FORMULA, where FORMULA is the rotor diffusion coefficient, related to the temperature and the drag coefficient via Einstein's relation: FORMULA.
In experiments, a load is attached to a flagellar stump and this load is largely responsible for the drag.
For simplicity, we assume that linkage between motor and load is instantaneous, as the relaxation is rapid compared to the typical stepping time .
