Unidirectional rotary nanomotors powered by an electrochemical potential gradient

We examine the dynamics of biological nanomotors within a simple model of a rotor having three ion-binding sites. It is shown that in the presence of an external dc electric field in the plane of the rotor, the loading of the ion from the positive side of a membrane (rotor charging) provides a torque leading to the motor rotation. We derive equations for the proton populations of the sites and solve these equations numerically jointly with the Langevin-type equation for the rotor angle. Using parameters for biological systems, we demonstrate that the sequential loading and unloading of the sites lead to the unidirectional rotation of the motor. The previously unexplained phenomenon of fast direction-switching in the rotation of a bacterial flagellar motor can also be understood within our model.

💡 Research Summary

The paper presents a minimal yet quantitative model for the operation of biological rotary nanomotors, focusing on the bacterial flagellar motor (BFM). The authors consider a circular rotor of radius ~2 nm that carries three equally spaced proton‑binding sites (A, B, C). Each site can be either empty or occupied by a single proton, thereby acquiring a charge +e when occupied. The rotor is embedded in a membrane that maintains an electrochemical potential gradient Δμ = μ_out − μ_in across its two sides (the “positive” side being the extracellular side). In addition, a static dc electric field E is applied in the plane of the rotor.

The central idea is that when a site becomes protonated on the positive side, the resulting charge experiences a force F = q E, which, because the charge is located at a distance R from the rotation axis, generates a torque τ = q R E sin(θ − θ_i). The sum of the torques from all three sites provides a net mechanical drive that rotates the rotor. As the rotor turns, the geometric relationship between the sites and the field changes, causing the torque to vary continuously with the rotor angle θ.

Proton transfer dynamics are described by a set of master‑equation‑type rate laws. The forward (charging) rate for site i is
k_i^in = k_0 exp