PHYSICAL REVIEW E 93, 022408 (2016) Torque generation through the random movement of an asymmetric rotor: A potential rotational mechanism of the γ subunit of F 1 -ATPase Y. C. Chou and Yi-Feng Hsiao Department of Physics, National Tsing-Hua University, Hsinchu, Taiwan, Republic of China Gwo-Jen Hwang Department of Electronic Engineering, St. John’s University, Tamsui District, New Taipei City, Taiwan, Republic of China Kiwing To Institute of Physics, Academia Sinca, Taipei, Taiwan, Republic of China (Received 16 September 2015; revised manuscript received 5 January 2016; published 18 February 2016) The rotation of the γ subunit of F 1 -ATPase is stochastic, processive, unidirectional, reversible through an external torque, and stepwise with a slow rotation. We propose a mechanism that can explain these properties of the rotary molecular motor, and that can determine the direction of rotation. The asymmetric structures of the γ subunit, both at the tip of the shaft (C and N termini) and at the part (ε subunit) protruding from the α 3 β 3 subunits, are critical. The torque required for stochastic rotation is generated from the impulsive reactive force due to the random collisions between the γ subunit and the quasihexagonal α 3 β 3 subunits. The rotation is the result of the random motion of the confined asymmetric γ subunit. The steps originate from the chemical reactions of the γ subunit and physical interaction between the γ subunit and the flexible protrusions of the α 3 β 3 subunits. An external torque as well as a configurational modification in the γ subunit (the central rotor) can reverse the rota- tional direction. We demonstrate the applicability of the mechanism to a macroscopic simulation system, which has the essential ingredients of the F 1 -ATPase structure, by reproducing the dynamic properties of the rotation. DOI: 10.1103/PhysRevE.93.022408 I. INTRODUCTION F 0 F 1 -ATPase is a rotary molecular motor that provides the energy for numerous cellular activities in the living cells through ATP hydrolysis [1–4]. In addition, when the direction of rotation is reversed, F 0 F 1 -ATPase can catalyze ATP syn- thesis through the consumption of the energy obtained from protons that flow through the membrane with electrochemical potential difference [5,6]. F 0 F 1 -ATPase consists of two parts: F 0 and F 1 . The F 0 part is embedded in the membrane, and F 1 is the water-soluble portion of F 0 F 1 -ATPase. The terms F 0 and F 1 are coupled molecular motors. Energy is released from the ATP hydrolysis as the γ subunit (the rotor) of F 1 -ATPase rotates within the hexagonal compartment surrounded by the α 3 β 3 subunits (the stator). The γ subunit rotation occurs counterclockwise viewed from the F 0 side during ATP hydrolysis. After exhaustive studies in recent years, researchers recognize that the γ subunit rotates with steps which are coupled with chemical reactions such as binding or releasing of ADP, phosphate ion, and ATP, and synthesis or hydrolysis of ATP [4–9]. However, only the chemical energy and reaction are considered as the sources of the torque in the previously proposed mechanisms for the rotation [10–16]. No mechanical details of the generation of the torque required for the rotation are provided. Therefore, it is difficult to determine the direction of rotation. A. Torque generation mechanism This study proposes a torque-generating mechanism, and we demonstrate the applicability to a macroscopic simulation system. The interplay of two critical factors, which have been neglected in the previous studies on the mechanism of the γ -subunit rotation, may play a role in the generation of torque required for the rotation. The first factor is that the γ -subunit structure is asymmetric [2,3,15]. Thus the line of force of the reactive force due to the random collision between the γ subunit and the surrounding compartment may not traverse the center of mass (CM) of the γ subunit. Therefore, an impulsive torque upon CM is generated by the collision. Consequently, the γ subunit rotates with respect to its CM. The other missing factor for the generation of torque is the random motion of the γ subunit [17,18]. For any nanometer scale molecules, the extent of the random motion would be the same order of magnitude of their size. Therefore, the effects of the random motion on the rotation cannot be overlooked. The trapped γ subunit randomly collides with its surroundings, and the reactive force generates torque required for the rotation around the central axis of the hexagonal compartment surrounded by the α 3 β 3 subunits. The randomly generated torque is the source of the stochastic nature of the rotation. In addition, the asymmetric structure of the rotor (the γ subunit) provides the lever arm of the reactive force with respect to the rotational axis, and determines the direction of the torque and the rotational direction. The asymmetric structure, confinement, and random motion of the rotor are all that is needed for the generation of torque for the rotary motion of the rotor. II. EXPERIMENTAL METHODS A. Simulation device The axlelike asymmetric coiled structure of the γ sub- unit penetrates deeply into the central cavity of the α 3 β 3 pseudohexagon [2,3,19–22]. The remaining part of the γ 2470-0045/2016/93(2)/022408(8) 022408-1 ©2016 American Physical Society