Efficiently driving F$_1$ molecular motor in experiment by suppressing nonequilibrium variation

Abstract

F$_1$-ATPase (F$_1$) is central to cellular energy transduction. Forcibly rotated by another motor F$_\mathrm{o}$, F$_1$ catalyzes ATP synthesis by converting mechanical work into chemical free energy stored in the molecule ATP. The details of how F$_\mathrm{o}$ drives F$_1$ are not fully understood; however, evaluating efficient ways to rotate F$_1$ could provide fruitful insights into this driving since there is a selective pressure to improve efficiency. Here, we show that rotating F$_1$ with an angle clamp is significantly more efficient than a constant torque. Our experiments, combined with theory and simulation, indicate that the angle clamp significantly suppresses the nonequilibrium variation that contributes to the futile dissipation of input work.

Figures (4)

• Figure 1: Single-molecule control of F$_1$ as a model system of ATP synthase. (a) We externally rotate F$_1$ derived from thermophilic Bacillus PS3 Noji_1997 by applying torque on the probe attached to its $\gamma$-shaft. We observe the rotation by video microscopy at 4 kHz. Inset: F$_\mathrm{o}$F$_1$-ATP synthase. (b) We realize constant-torque and angle-clamp driving modes by varying the amplitudes and phases of the alternating-current voltages induced on the four electrodes (A--D) Toyabe_2010. (c) The free-energy profile results from the interaction between the $\gamma$-shaft and stator of F$_1$ and the chemical free-energy change $\Delta\mu$ of ATP hydrolysis. We rotate the probe attached to the $\gamma$-shaft in the ATP-synthetic direction by a constant torque (left) or by a trapping torque with the trap center rotated at a constant rate (right).
• Figure 2: Rotational trajectories and angular distributions. (a) Rotational trajectories, (b) angular distributions, and (c) rotational rates averaged over a 0.05s time window, for forced ATP-synthetic rotations under constant torque (blue) and angle clamp (red) and free ATP-hydrolytic rotations (gray). [ATP] = [ADP] = 0.4µ M.
• Figure 3: (a) Average work to rotate the $\gamma$-shaft 120$^\circ$, for [ATP]=[ADP]=0.4µ M (circles), 2µ M (squares), and 10µ M (triangles) by constant torque (blue filled symbols) and angle clamp (red open symbols). Solid curves show fits of simulations [SI Sec. S2 Note1]. The negative control $-\mathrm{F}_1$ corresponds to a freely rotating Brownian dimeric probe pinned on a glass surface without F$_1$. 159 trajectories (32 molecules) and 278 trajectories (34 molecules) are measured for the angle-clamp and constant-torque modes, respectively. $W_\mathrm{trq}$ is averaged across bins with 1-Hz width. (b) The Stokes efficiency $\eta_\mathrm{Stokes}$ [Eq. \ref{['eq:eta:Stokes']}] for different [ATP]=[ADP], calculated using the mean of $W_\mathrm{d}$ for rotational rates between 4Hz and 6Hz. (c) $\eta_\mathrm{Stokes}$ in angle clamp was separately evaluated (using linear fit slope of work vs. global mean velocity, SI Sec. S1.3 Note1) for molecules with large or small $k$ values (threshold is $70\,\,k_\mathrm{B}T/\mathrm{rad^2}$, and different [ATP]=[ADP] concentrations are mixed). Error bars indicate standard errors.
• Figure 4: The mechanism for reduced work under angle clamp. Schematic of the mechanism (top) and representative examples of torque balance (bottom) under constant torque (a) and angle clamp (b), for average rotation rate $\approx5Hz$. The net torque is obtained from the local mean velocity as $N(\theta)=\Gamma\nu(\theta)$. Subtracting the external torque $N_\mathrm{ext}(\theta)$ gives the composite torque by F$_1$ and diffusional flow: $N_\mathrm{F1}(\theta)+N_\mathrm{diff}(\theta)=N(\theta) - N_\mathrm{ext}(\theta)$. (c) Dissipation estimates $q$ [Eq. \ref{['eq:Wdis:nu:2']}] (bars) and $W_\mathrm{d}=W-\Delta\mu$ with $W$ calculated by Eq. \ref{['eq:W:angle']} (circles) per 120$^\circ$ rotation, for indicated average rotation rate and $\mathrm{[ATP]=[ADP]}=10µ M$. $q$ and its decompositions are obtained by integrating the terms in Eq. \ref{['eq:Wdis:nu:2']} over 120$^\circ$ rotation.