Fluctuation Theorem Applied to ${\mathbf{F}}_1$-ATPase

In recent years, theories of nonequilibrium statistical mechanics such as the fluctuation theorem (FT) and the Jarzynski equality have been experimentally applied to micro and nanosized systems. However, so far, these theories are seldom applied to autonomous systems such as motor proteins. In particular, representing the property of entropy production in a small system driven out of equilibrium, FT seems suitable to be applied to them. Hence, for the first time, we employed FT in the single molecule experiments of the motor protein ${\mathrm{F}}_1$-adenosine triphosphatase (${\mathrm{F}}_1$), in which the rotor $\gamma$ subunit rotates in the stator ${\alpha }_3{\beta }_3$ ring upon adenosine triphosphate hydrolysis. We found that FT provided the better estimation of the rotary torque of ${\mathrm{F}}_1$ than the conventional method.

Figure 1

(a) Reaction scheme of ${\mathrm{F}}_1$. The green circles and the red arrow represent the $\beta$ subunits and $\gamma$ subunit of $F_1$, respectively. ${\mathrm{F}}_1$ performs a 120° step rotation upon ATP hydrolysis comprising 80° and 40° substeps. (b) Schematic diagrams of our experimental system (not to scale). The rotation of ${\mathrm{F}}_1$ is probed by a duplex of polystyrene beads (left) or an irregularly shaped magnetic bead (right). The size of ${\mathrm{F}}_1$ is about 10 nm and that of a probe is in the range of 358–940 nm. (c) ATP-driven rotations of ${\mathrm{F}}_1$ probed by the magnetic beads at 1 mM ATP (red line) and 100 nM ATP (blue line).

Figure 2

(a) Probability distribution, $P(\Delta \theta )$, for the cases $\Delta t=2.5\mathrm{ms}$ (red line), 5.0 ms (yellow line), 7.5 ms (green line), and 10 ms (blue line) for the particular ${\mathrm{F}}_1$ probed by the magnetic bead (left). $\ln [P(\Delta \theta )/P(-\Delta \theta )]$ as a function of $\Delta \theta /k_{B}T$ (right). The slope was $38\mathrm{pN}\mathrm{nm}$ in the case $\Delta t=10\mathrm{ms}$. The recording rate was 2000 fps. (b) $N_{\mathrm{FT}}$ as a function of $\Delta t$ for the recording rates, 500 fps (red line), 1000 fps (blue line), 2000 fps (green line), 3000 fps (orange line), 6000 fps (pink line), 10 000 fps (aqua line), and 30 000 fps (violet line). Here, we used the same molecule probed by the same magnetic bead at 1 mM ATP.

Figure 3

(a) ATP-driven rotation of ${\mathrm{F}}_1$ at 100 nM ATP. We analyzed $\theta (t)$ encircled in red (about 100–300 steps). The steps were determined by eye. (b) Probability distributions, $P(\Delta \theta )$, for the cases $\Delta t=0.5\mathrm{ms}$ (red line), 1.0 ms (yellow line), 1.5 ms (green line), and 2.0 ms (blue line), and 2.5 ms (aqua line) for the particular ${\mathrm{F}}_1$ probed by the magnetic bead (left). $\ln [P(\Delta \theta )/P(-\Delta \theta )]$ as a function of $\Delta \theta /k_{B}T$ (right). The slope was $40\mathrm{pN}\mathrm{nm}$ in the case $\Delta t=2.5\mathrm{ms}$. The recording rate was 2000 fps.

Figure 4

$N_{\mathrm{FT}}$ as a function of $\Delta t$ for ${\mathrm{F}}_1$ (red line, 5 molecules) and ${\mathrm{V}}_1$ (blue line, 5 molecules). The rotations were probed by magnetic beads and recorded at 2000 fps.