Dynamics of the bacterial flagellar motor with multiple stators

The bacterial flagellar motor drives the rotation of flagellar filaments and enables many species of bacteria to swim. Torque is generated by interaction of stator units, anchored to the peptidoglycan cell wall, with the rotor. Recent experiments [Yuan, J. & Berg, H. C. (2008) PNAS 105, 1182-1185] show that near zero load the speed of the motor is independent of the number of stators. Here, we introduce a mathematical model of the motor dynamics that explains this behavior based on a general assumption that the stepping rate of a stator depends on the torque exerted by the stator on the rotor. We find that the motor dynamics can be characterized by two time scales: the moving-time interval for the mechanical rotation of the rotor and the waiting-time interval determined by the chemical transitions of the stators. We show that these two time scales depend differently on the load, and that their crossover provides the microscopic explanation for the existence of two regimes in the torque-speed curves observed experimentally. We also analyze the speed fluctuation for a single motor using our model. We show that the motion is smoothed by having more stator units. However, the mechanism for such fluctuation reduction is different depending on the load. We predict that the speed fluctuation is determined by the number of steps per revolution only at low load and is controlled by external noise for high load. Our model can be generalized to study other molecular motor systems with multiple power-generating units.

💡 Research Summary

The paper presents a quantitative model of the bacterial flagellar motor that accounts for the experimentally observed independence of rotation speed from the number of stator units at near‑zero load. The authors assume that the stepping rate of each stator depends inversely on the torque it exerts on the rotor: higher torque reduces the probability of a forward step, while lower torque increases it. This torque‑dependent chemical transition rate is incorporated into a hybrid mechanical‑chemical description. The rotor’s angular position and velocity are treated as continuous variables, whereas each stator’s state (stepping or waiting) is discrete. Two characteristic time scales emerge from the model: (1) the moving‑time interval τ_move, the duration required for the rotor to rotate under the current torque, and (2) the waiting‑time interval τ_wait, the average time a stator spends in the chemical waiting state before the next step.

Under low‑load conditions τ_move is much shorter than τ_wait, so the rotor rotates almost continuously. In this regime the stepping rate of each stator quickly reaches a torque‑saturated value, and adding more stators does not increase the average stepping frequency; consequently the motor speed remains constant regardless of stator number, reproducing the Yuan‑Berg observation. As the external load increases, τ_move becomes comparable to or longer than τ_wait. The rotor now experiences intermittent pauses, and the overall torque generated depends on how many stators are simultaneously ready to step. Adding stators reduces the effective waiting time, raising the average torque and altering the speed‑load relationship. The crossover between the τ_move‑dominated and τ_wait‑dominated regimes provides a microscopic explanation for the two distinct regions (high‑speed/low‑torque and low‑speed/high‑torque) seen in torque‑speed curves.

The authors also analyze speed fluctuations using the coefficient of variation (CV). They find that increasing the number of stators smooths the rotation because more steps per revolution reduce statistical noise. However, the origin of fluctuation reduction differs with load. At low load, the smoothing is purely a consequence of the larger number of small steps (step‑count effect). At high load, external thermal and mechanical noise dominate, and the benefit of additional stators is limited. This dual mechanism predicts that the CV will be determined by step number only in the low‑load regime, while in the high‑load regime it will be governed mainly by external noise.

Finally, the authors argue that their framework is readily extensible to other molecular motors that contain multiple power‑generating units, such as ATP synthase or myosin ensembles. By adjusting the torque‑dependent stepping function, the model can capture a wide range of efficiency, torque‑speed, and fluctuation characteristics across different biological nanomachines. In summary, the paper offers a coherent, physics‑based explanation for the load‑dependent behavior of the flagellar motor, reconciles previously puzzling experimental data, and provides a versatile platform for studying multi‑unit molecular motors.