Nonequilibrium Energetics of a Single ${\mathbf{F}}_1$-ATPase Molecule

Molecular motors drive mechanical motions utilizing the free energy liberated from chemical reactions such as ATP hydrolysis. Although it is essential to know the efficiency of this free energy transduction, it has been a challenge due to the system’s microscopic scale. Here, we evaluate the single-molecule energetics of a rotary molecular motor, ${\mathrm{F}}_1$-ATPase, by applying a recently derived nonequilibrium equality together with an electrorotation method. We show that the sum of the heat flow through the probe’s rotational degree of freedom and the work against an external load is almost equal to the free energy change per a single ATP hydrolysis under various conditions. This implies that ${\mathrm{F}}_1$-ATPase works at an efficiency of nearly 100% in a thermally fluctuating environment.

Figure 1

(a) Schematic of ${\mathrm{F}}_1$-ATPase molecule. The central $\gamma$ shaft rotates unidirectionally when ATP is hydrolyzed to ADP and phosphate. $\delta$ and $\epsilon$ subunits are omitted for simplicity. (b) Typical trajectory at low ATP concentration ($0.4\mu \mathrm{M}$ ATP, $0.4\mu \mathrm{M}$ ADP, 1 mM ${\mathrm{P}}_{\mathrm{i}}$). An ATP binding reaction triggers a 120° step.

Figure 2

(a) Experimental setup. A dimeric polystyrene particle is attached to the $\gamma$ shaft of ${\mathrm{F}}_1$-ATPase fixed on the top glass surface. Four electrodes are coated on the bottom glass surface. Distance between electrodes is $50\mu \mathrm{m}$. (b) Electrorotation method. A rotating electric field at a frequency of 10 MHz was generated at the center of four electrodes by applying sinusoidal voltages with a phase shift of $\pi /2$. Dielectric objects in this rotating electric field have a dielectric moment rotating at 10 MHz. The phase delay between the electric field and the dielectric moment resulted in the torque on the objects. Temperature rise due to the electric field was at most $0.3°\mathrm{C}$, which was smaller than the temperature variance of the environment. Not to scale.

Figure 3

Typical examples of fluctuations $\tilde{C}(f)$ (lines) and responses $2T{\tilde{R}}^{\prime }(f)$ (circles) of the rotational velocities. The shaded area corresponds to the half of the integral in (1). The rotational trajectory is shown in the inset. rots: rotations. (a) Rotational Brownian particle in the absence of ATP, ADP, and ${\mathrm{P}}_i$. (b)–(f) Particles driven by ${\mathrm{F}}_1$-ATPase at indicated concentrations of substrates. (f) Under an external load of $17.7\mathrm{pN}\mathrm{nm}/\mathrm{rad}$.

Figure 4

Amount of heat dissipation per a step $Q_{\mathrm{rot}}$, the work on an external load per a step $W$, and the free energy difference of an ATP hydrolysis $\Delta \mu$. $Q_{\mathrm{rot}}=Q_s+Q_v$ is separately plotted: the contribution from the steady motion $Q_s$ and that from nonequilibrium fluctuations $Q_v$. The horizontal thick lines indicate $\Delta \mu$ calculated on the basis of $\Delta \mu °$ reported in the literature [25, 26, 27, 28, 29]. Since $\Delta \mu °$ are different in the literature, the error of the estimation of $\Delta \mu$ is indicated as the thickness of the line: $72.4–78.4\mathrm{pN}\mathrm{nm}$ (i–iv) and $109–115\mathrm{pN}\mathrm{nm}$ (v). In (i–iv), the ratio $[\mathrm{ATP}]/[\mathrm{ADP}][{\mathrm{P}}_i]$ and accordingly $\Delta \mu$ are held to be constant except for a minor difference due to changes in the ionic strength, concentration of free ${\mathrm{Mg}}^{2+}$, etc. Error bars indicate the standard deviations. Number of samples are (i) 15, (ii) 8, (iii) 7, (iV) 5, and (V) 7, respectively.