Contact-Dependent Ion Gating Explains Directional Asymmetry in the Bacterial Flagellar Motor

Abstract

The bacterial flagellar motor (BFM) is a rotary molecular machine driven by the ion electrochemical potential across the cell membrane. Recent cryo-EM structures reveal a cogwheel-like architecture in which multiple stators engage a large rotor. A longstanding puzzle is the directional asymmetry of its torque-speed relation: concave in counterclockwise (CCW) rotation but nearly linear in clockwise (CW) rotation. Here, we develop a stochastic mechanochemical model that explicitly incorporates rotor-stator coupling and detailed ion translocation kinetics. By integrating physiological torque-speed data with recent measurements of rotor-stator relative motion, we show that under physiological conditions the motor operates in a tight engagement regime, rendering the torque-speed relation largely insensitive to the specific form of mechanical interactions. This finding rules out differences in rotor-stator mechanics as the origin of CW-CCW asymmetry. Guided by cryo-EM structures, we propose a contact-dependent gating mechanism in which the MotA-FliG interaction modulates the ion release rate of the MotB subunit proximal to the FliG ring. Molecular dynamics simulations indicate tighter MotA-FliG contact in the CW motor, implying a reduced ion release rate compared to CCW. Our model demonstrates that differential gating strength accounts for the observed asymmetry: stronger gating in CCW shortens torque-free waiting phases, enhances torque generation, and produces a concave torque-speed curve, whereas weaker gating in CW yields lower torque and a linear relation. This structure-based framework quantitatively links molecular asymmetry to motor function and identifies specific interfaces for targeted perturbation and mutational studies.

Figures (6)

• Figure 1: Model of two coupled nano-rings. (A) Rotor-stator coupling in CW (left) and CCW (right) rotations. In CCW rotation the stator (MotAB) engages the rotor's outer periphery, whereas in CW rotation it engages the rotor's inner periphery. (B) Periodic rotor-stator interaction potential $V_r$ with depth $h_r$. (C) Periodic stator free-energy landscape $V_s$. The stator generates positive torque $\tau_+=h_s/\Delta\theta_s$, and jumps to the next cycle with rate $k(\theta_s)$. Here we only show the "no-gating" ion kinetics in the CW mode.
• Figure 2: The stator-rotor slippage and the stall torque. (A) Trajectories of the relative angle $\varphi=\theta_r-\alpha\theta_s$ in the deterministic-slipping (blue) and stochastic-slipping (red) regimes. (B) Engagement fraction $\gamma=\left \langle \omega_{r} \right \rangle / \alpha\left \langle \omega_{s} \right \rangle$ as a function of rotor load $\xi_r$ and rotor-stator coupling depth $h_r$ (with $h_s=12k_BT$). Red dashed line: $h_r\approx 15.4 k_BT$ inferred from experimenthosu2025torque. (C) Stall torque versus $h_{r}$ ($\xi_r\approx 3.3$ pN nm s for 1$\mu$m bead assay). Increasing $h_{r}$ suppresses slipping and drives the stall toward the upper bound $\tau_{+}/\alpha$. The torque reduction follows an exponential decay (dashed line, Eq. \ref{['slipping']}). At $h_r=12k_BT$, $\gamma \sim 0.55$ while the torque drop is $\sim 3\%$. (D) Stall torque versus IMF. Assuming $\tau_+$ and $k$ scale linearly with IMF, $\tau_{\text{stall}}$ rises linearly at low IMF (dashed line) but bends downward upon entering the determinist-slipping regime. Experimental data (triangles, from Lo2013) remain linear, implying tight engagement and $h_r\geq 12k_BT$.
• Figure 3: Torque-speed curves in the strong coupling regime. (A) The torque–speed curve is essentially unchanged over a broad range of rotor–stator interaction potentials $V_{r}$. (B) Varying the chemical transition rate $k$ shifts the speed but leaves the stall torque and the linear shape of the curve unchanged. (C) Stator mechanochemical dynamics. Top: In the absence of gating, after moving down the potential gradient generating torque, the stator dwells in a torque-free waiting phase at the bottom of the potential well (blue double arrow) before jumping to the next cycle (the green arrow). Results from panels A&B are from this "no-gating" model. Bottom: With gating, a transition (jumping) is triggered earlier along the downhill branch (red gate) bypassing the torque-free waiting phase near the minimum, which increases the torque-generating fraction of the cycle. $t_w$: waiting time; $t_m$: moving time. (D) Torque-speed curves with gating (red line) or no gating (black line). Gating suppresses the waiting phase and flattens the high-load slope, which produces a concave torque–speed curve.
• Figure 4: Mechanochemical dynamics of a stator unit during one torque-generating cycle ($\Delta \theta_s=2\pi/5$). (A) Schematic of the two-ion pathways through the two MotB subunits. The branching of the two pathways (upper and lower) is determined by early release of ion at MotB$_2$ (proximal to FliG, light blue) with rate $k_g$. At $\theta_s=0$ (state I), ion from the periplasm binds to MotB$_1$ (distal from FliG, dark blue) when the MotA-MotB charging channel (half circle markers) is aligned (state I). If the ion on MotB$_2$ is released through the gating mechanism (red arrow), the cycle follows the upper branch (state II); otherwise it follows the lower branch (state II'). MotB$_1$ then drives a $\pi/5$ power stroke of the MotA ring (blue arrows; states III and III'). After the power stroke, the ion is released through the MotA-MotB release channel (dashed purple line, state IV and IV'). At $\theta_s=\pi/5$, an ion binds to MotB$_2$ and drives the next $\pi/5$ power stroke (state VI). If MotB$_2$ has not undergone early release, the system remains in the lower branch and waits until ion release occurs through a baseline pathway (rate $k$, state V'), generating a torque-free waiting phase. After a period ($2\pi/5$), the stator disengages from the current FliG and engages to the next FliG (not shown). (B) Stator potential and ion kinetics for the distal MotB$_1$. Ion binds at $\theta_s=0$ and releases during $\pi/5<\theta_s<2\pi/5$ with rate $k_a$ (purple bar). (C) State potential and ion kinetics for the proximal MotB$_2$. Ion could be released either through the gating channel with rate $k_g$ (red bar) at $\theta_s=0$ or near $\theta_s=\pi/5$ at the base-line rate $k$ (green bar).
• Figure 5: Dynamics of instantaneous torque and the torque-speed relations for CCW and CW motors. The only difference for CCW and CW motors is: $k_{g,\text{CW}}=0$ and $k_{g,\text{CCW}}=k_g$. Other parameters used in simulations are given in SI. (A) Instantaneous torque dynamics in CCW motors at high load ($\xi_r=0.15$ pN nm s). Gating keeps the stator predominantly in the positive-torque region. Ions on MotB$_1$ (black line) and MotB$_2$ (red line) driving the stator alternatively (switching between $\tau_+$ and 0). (B) Instantaneous torque dynamics in CW motors at high load. Near the equilibrium position (purple arrows, the waiting phase), MotB$_1$ alternates between $\tau_+$ and 0 (zoomed inset), and MotB$_2$ switches between at 0 and $\tau_-$, yielding zero net torque on average (state V' in Fig. 4). (C) Torque–speed relations with the ion translocation kinetics given in Fig. 4. The torque-speed curve is linear for CW motors with $k_g=0$. Increasing the gating rate $k_g$ enhances concavity of the torque–speed relation. (D) Torque-speed curves for CCW motors with different IMF agree with experimental data from Lo2013. Here, we assume $\tau_+$ scales linearly with IMF as in Fig. 2D, and other parameters (e.g., $k_g$ and $k$) are given in the SI.