Rocking ratchet based on F1-ATPase in the absence of ATP

Bartussek, Hänggi and Kissner studied a rocking ratchet system, in which a Brownian particle is subject to an asymmetric periodic potential together with an oscillating force, and found that the direction of the macroscopic current can be reversed by changing the parameter values characterizing the model [Europhys. Lett., 28 (1994) 459]. In this letter, we apply their ratchet theory to a rotary motor-protein, F1-ATPase. In this work, we construct a model of a rocking ratchet in which F1-ATPase rotates not as a result of ATP hydrolysis but through the influence of an oscillating force. We then study the motion of F1-ATPase on the basis of molecular dynamics simulations of this coarse-grained protein model. Although in the absence of ATP, F1-ATPase exhibits directionless Brownian motion when there exists no oscillating force, we observe directional motion when we do apply an oscillating force. Furthermore, we observe that the direction of rotation is reversed when we change the oscillation frequency. Introduction. – Recently, theories of nonequilibrium statistical mechanics have been applied to many biological systems. Among such studies, the fluctuation-dissipation theorem has been investigated with the purpose of understanding the motion of myosins in an actin network [1], the fluctuation theorem has been used to quantify the fluctuations of cell motion [2], and the Jarzynski equality and the Crooks theorem have been applied to RNA hairpins in order to measure their free energies [3]. Recent technical developments have made it possible to investigate small systems, including biological systems, and this has provided another way to test theories of non-equilibrium statistical mechanics in addition to the investigation of theoretical models of stochastic processes. Ratchet theories have been proposed to describe certain types of non-equilibrium statistical behavior [4–6], and they predict that together, spatial asymmetry and nonequilibrium effect, can cause directional motion in stochastic systems [5]. Among the various ratchet models, that consisting of a Brownian particle in an asymmetric periodic potential together with an oscillating force is called a “rocking ratchet.” For such ratchets, it has been shown that the direction of the current can be reversed by changing the parameter values characterizing the model [6]. Although the application of such current reversal to nanosized systems is mentioned in Ref. [6], as far as we know, there is yet no such experiment. In this letter, we numerically investigate the rocking ratchet theory in application to a nano-sized bio-molecule to which many theories of non-equilibrium statistical mechanics have been applied in recent years. We study a rotary motor-protein, F1-ATPase, in which a γ-subunit rotates in the center of α3β3-subunits [7, 8]. In this letter, we construct a model of a rocking ratchet in which F1-ATPase rotates through the influence of an oscillating force, and particularly study the motion of F1ATPase in the absence of ATP. Because in this case no nucleotide is attached to the β-subunits, such a state is called a nucleotide-free state. In the absence of ATP, it was observed that the γ-subunit exhibits directionless Brownian motion with rotational steps of ±120 [9]. This indicates that the rotational potential experienced by a γ-subunit is periodic with a period of 120, reflecting the three-fold symmetry of α3β3-subunits. Then, if this periodic potential, which reflects the interaction between a γ-subunit and α3β3-subunits, is asymmetric, we expect that a γ-subunit will display a directional rotation when we apply an oscillating force. In order to test this conjecture, we performed molecular dynamics simulations of a protein model, in which a protein molecule is regarded as consisting of α-carbon atoms, employing a coarse-grained representation of the amino-acid residues [10]. We find that the γ-subunit ro-