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

TL;DR

This work addresses how the bacterial flagellar motor transduces electrochemical energy into rotation by reconstructing the LP ring–rod energy landscape from passive single-molecule data. It develops and applies three complementary methods based on the Smoluchowski equation for overdamped diffusion in a tilted periodic potential to extract the energy barrier $E_{\rm a}$, periodicity $L$, and drag $\gamma$ from observables $\omega_{\rm ss}$, $D_{\rm ss}$, and $P_{\rm ss}(\phi)$. Across 95 motors, the authors find a ~26-fold periodicity, barrier heights of roughly $2$–$4\,k_BT$, and a dominant internal drag around $0.1$ pN·nm·s·rad$^{-2}$, with torque–velocity behavior deviating from linearity near the critical tilt. The framework provides a general, model-independent route to nanoscale energy landscape reconstruction for cyclic molecular motors and can inform explanations of motor efficiency, regulation, and evolution in biological systems.

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.

Figures (8)

• Figure 1: a) Side view of the BFM's protein structure (by David Goodsell 10.2210/rcsb_pdb/mom_2024_12 under CC-BY-4.0 license). b) Composed top view displaying key folding periodicity of the BFM's components and a schematic representation of the torques involved ($f_{\rm dr}$, $f_{\rm op}$, $f$) in its motion. c) Schematic representation of the BFM bead assay and the relevant viscous loads present, $\gamma_{\textsc{b}}$ and $\gamma_{\textsc{r}}$, for the bead and internal ring, respectively. d) Example of the angular position time series obtained from single-molecule experiments tracking the center of mass of the attached bead. e) Ensemble Empirical Mode Decomposition (EEMD) analysis of the stepping size for a set of 95 individual traces showing a mean stepping very close to 26 steps per revolution. f) Normalized probability distribution of the angular position folded over the main periodicity found by the EEMD analysis (C.I. denotes confidence interval).
• Figure 2: Illustrative theoretical results for Method 1 and application to experimental data to reconstruct: $E_{\rm a}$ (methods 1 to 3), $\gamma$ (methods 1, 2), effective torque (methods 1, 3), and $V_{\rm eq}$ (methods 2, 3). (a) Theoretical parametric plot $D_{\rm ss}$ vs. $\omega_{\rm ss}$ along with the method 1 approximations, Eqs. (\ref{['Eq:Dss/vss/Kramers']}), (\ref{['Eq:Dss/vss/Pertur']}), (\ref{['Eq:1/Dcr']}), used to determine $D_{\rm ss}$ for a model potential. (b) Relative error between $f$ and the Method 1 torque estimates, Eqs. (\ref{['Eq:f/Kramerexp-2']}), (\ref{['Eq:f/Pertur-2']}), (\ref{['Eq:f/Padde']}), as a function of $\epsilon$ for a model potential. (c) Parametric plot $D_{\rm ss}$ vs. $\omega_{\rm ss}$ for the experimental data and best fit model (rainbow curve parametrized by $\epsilon$) using the interpolation Eq. (\ref{['Eq:1/Dcr']}). The shaded gray area shows the 95% confidence interval; the black line is the exact numerical result for the same parameters used in the fit. (d) Corresponding torque values obtained from the experimental $\omega_{\rm ss}$ values ($f_0$, corrected models, $f_{\rm opt}$, Eqs. (4) (9) best fit and asymptotic behavior included). (e) Resulting energy landscapes from individual traces using (left) method-2 and (right) method-3, color bar represents the corresponding experimental $\omega_{\rm ss}$ values. (f) Distribution of $\gamma_0$ values calculated from $f_0$ and method-2 barriers. In all panels, the model parameters and best fit parameters are listed in their respective legends.
• Figure 3: Example trace showing showing: (a) (solid) accumulated angle versus time plot, (dashed) longest uninterrupted portion used to calculate $\omega_{\rm ss}$, and inset $\{x(t),y(t)\}$ coordinates every 60 points, (b) MSD analysis for increasing $\Delta t$ used to determine $D_{\rm ss}$.
• Figure 4: Set of traces showing: MSD analysis for increasing $\Delta t$ used to determine $D_{\rm ss}$.
• Figure 5: EEMD analysis of the trace shown in Fig. \ref{['Fig:PreAnalysis_ss']}a. (a) The IMFs ordered in decreasing frequency, (b) the corresponding spectra as function of the steps per rotation, and (c) angular histogram of the original signal $H(\phi)\rvert_{2\pi}$ (dashed line) and the filtered signal $H(\phi)^\dagger\rvert_{2\pi}$ (solid line).