Reconstruction of the Bacterial Flagellar Motor's Energy Landscape, Viscous Load, and Torque Generation Across Diffusion Regimes

Abstract

The bacterial flagellar motor (BFM) converts transmembrane ion flux into directed mechanical rotation, driving bacterial motility. Despite extensive study, the frictional forces and energetics governing its torque generation remain poorly understood. Here, we combine single-molecule rotation measurements with stochastic thermodynamics to quantitatively estimate its effective torque, viscous drag and activation energy barriers. We present three complementary methods based on solutions to the Smoluchowski equation for overdamped diffusion in a tilted periodic potential, which use as input the steady-state angular velocity and rotational diffusion data from individual \textit{E. coli} motors spanning different dynamical regimes. Crucially, these three methods require neither active external torque control, nor prior knowledge of the system's viscous drag or the motor's torque output. The first method assumes as input a model-dependent sinusoidal potential (single Fourier mode), albeit with unknown periodicity, yielding closed-form results in the low- and high-tilt limits, whereas the last two methods use a full Fourier-based reconstruction to output the now model-independent potential landscape. These approaches yield consistent estimates of the potential periodicity ($\approx$ 26-fold symmetry), energy barrier height ($\approx 2\!-\!4 k_{\rm B}T$), and internal friction coefficient ($\approx 0.1$ pN nm s rad$^{-2}$). Our results reveal that the BFM's torque-velocity relationship deviates significantly from the linear approximation near the critical tilt, where angular diffusion is maximized. More broadly, our framework provides a coherent strategy for reconstructing nanoscale energy landscapes from single-molecule data and is generalizable to other stepping molecular motors operating in cyclic conditions.