The Effects of Stator Compliance, Backs Steps, Temperature, and Clockwise Rotation on the Torque-Speed Curve of Bacterial Flagellar Motor

Rotation of a single bacterial flagellar motor is powered by multiple stators tethered to the cell wall. In a “power-stroke” model the observed independence of the speed at low load on the number of stators is explained by a torque-dependent stepping mechanism independent of the strength of the stator tethering spring. On the other hand, in models that depend solely on the stator spring to explain the observed behavior, exceedingly small stator spring constants are required. To study the dynamics of the motor driven by external forces (such as those exerted by an optical tweezer), back-stepping is introduced when stators are driven far out of equilibrium. Our model with back-stepping reproduces the observed absence of a barrier to backward rotation, as well the behaviors in the high-speed negative-torque regime. Recently measured temperature dependence of the motor speed near zero load (Yuan & Berg 2010 Biophys J) is explained quantitatively by the thermally activated stepping rates in our model. Finally, we suggest that the general mechanical properties of all molecular motors (linear and rotary), characterized by their force(torque)-speed curve, can be determined by their power-stroke potentials and the dependence of the stepping rates on the mechanical state of the motor (force or speed). The torque-speed curve for the clockwise rotating flagellar motor has been observed for the first time recently (Yuan et al. 2010 PNAS). Its quasi-linear behavior is quantitatively reproduced by our model. In particular, we show that concave and convex shapes of the torque-speed curve can be achieved by changing the interaction potential from linear to quadratic form. We also show that reversing the stepping rate dependence on force (torque) can lead to non-monotonicity in the speed-load dependency.

💡 Research Summary

The paper presents a comprehensive theoretical framework for the bacterial flagellar motor that integrates four key physical factors—stator compliance (spring constant), back‑stepping under large external loads, temperature dependence, and the shape of the interaction potential governing clockwise (CW) rotation. The authors begin by revisiting two competing explanations for the experimentally observed load‑independent speed at low torque: the “power‑stroke” model, in which stepping rates depend directly on torque, and the “spring‑only” model, which attributes the behavior to an extremely soft stator tether. By formulating each stator as a torsional spring of stiffness k attached to the cell wall, they derive the torque τ = k·Δθ generated when the stator is displaced by the rotor. The stepping rate r is then expressed as a torque‑dependent function r(τ) = r₀ exp(−ΔG‡/kBT) f(τ), where f(τ) captures the mechanochemical coupling.

To capture the experimentally observed lack of a barrier to backward rotation, the model introduces a back‑stepping mechanism that becomes significant when a stator is driven far from equilibrium by an external force such as an optical tweezer. In this regime, the transition rate is allowed to increase in the reverse direction once the spring extension exceeds a threshold, reproducing the smooth continuation of speed into the high‑negative‑torque region reported in optical‑trap experiments.

Temperature effects are incorporated by treating the stepping rates as thermally activated processes. Using the zero‑load speed data from Yuan & Berg (2010), the authors fit an activation energy of roughly 30 kBT, showing that the temperature‑dependent increase in r accounts quantitatively for the observed linear rise in motor speed with temperature.

A central insight of the work is that the shape of the torque‑speed curve is dictated primarily by the form of the stator‑rotor interaction potential V(Δθ). Two potentials are examined: a linear (constant‑force) potential V = F·Δθ, which yields the classic concave torque‑speed relationship (steep drop at low load, flattening at high load), and a quadratic (Hookean) potential V = ½ k Δθ², which produces an almost linear, convex curve. The latter reproduces the recently measured CW torque‑speed curve, which is markedly more linear than the counter‑clockwise (CCW) curve.

Finally, the authors explore the consequences of reversing the torque dependence of the stepping rate—i.e., making the forward stepping slower as torque increases. This inversion generates a non‑monotonic speed‑load relationship, with a modest speed increase at intermediate loads, a prediction that could be tested experimentally.

Overall, the model demonstrates that a modest, biologically realistic stator spring constant (tens of pN·nm/rad) combined with torque‑dependent stepping can explain the load‑independent low‑load speed, the absence of a backward‑rotation barrier, the temperature sensitivity, and the distinct shapes of CCW and CW torque‑speed curves. By emphasizing the universal role of the power‑stroke potential and the mechanochemical stepping function, the work provides a unifying description applicable not only to rotary flagellar motors but also to linear molecular motors, offering a valuable theoretical platform for future studies of motor design and synthetic nanomachines.