We combine molecular simulations and mechanical modeling to explore the mechanism of energy conversion in the coupled rotary motors of F o F 1 -ATP synthase. A torsional viscoelastic model with frictional dissipation quantitatively reproduces the dynamics and energetics seen in atomistic molecular dynamics simulations of torquedriven γ-subunit rotation in the F 1 -ATPase rotary motor. The torsional elastic coefficients determined from the simulations agree with results from independent single-molecule experiments probing different segments of the γ-subunit, which resolves a long-lasting controversy. At steady rotational speeds of ∼1 kHz corresponding to experimental turnover, the calculated frictional dissipation of less than k B T per rotation is consistent with the high thermodynamic efficiency of the fully reversible motor. Without load, the maximum rotational speed during transitions between dwells is reached at ∼1 MHz. Energetic constraints dictate a unique pathway for the coupled rotations of the F o and F 1 rotary motors in ATP synthase, and explain the need for the finer stepping of the F 1 motor in the mammalian system, as seen in recent experiments. Compensating for incommensurate eightfold and threefold rotational symmetries in F o and F 1 , respectively, a significant fraction of the external mechanical work is transiently stored as elastic energy in the γ-subunit. The general framework developed here should be applicable to other molecular machines.bioenergetics | molecular motor | mechanochemical coupling F o F 1 -ATP synthase is essential for life. From bacteria to human, this protein synthesizes ATP from ADP and inorganic phosphate P i in its F 1 domain, powered by an electrochemical proton gradient that drives the rotation of its membrane-embedded F o domain (1-5). Its two rotary motors, F 1 and F o , are coupled through the γ-subunit forming their central shaft (2). ATP synthase is a fully reversible motor, in which the rotational direction switches according to different sources of energy (2, 6). In hydrolysis mode, the F 1 motor pumps protons against an electrochemical gradient across the membrane-embedded F o part, converting ATP to ADP and P i (7,8).F 1 has a symmetric ring structure composed of three αβ-subunits with the asymmetric γ-subunit sitting inside the ring (9, 10). Each αβ-subunit has a catalytic site located at the αβ-domain interface. The F 1 ring has a pseudothreefold symmetry with the three αβ-subunits taking three different conformations, E (empty), TP (ATP-bound), and DP (ADP bound) (9-11). The F o part is composed of a c ring and an a subunit (3, 12). Driven by protons passing through the interface of the c ring and the a subunit, the c ring rotates together with the γ-subunit (rotor) relative to the a subunit, which is connected to the F 1 ring through the peripheral stalk of the b subunit (stator) (12). Interestingly, in nature, one finds a large variation in the number of subunits in the c ring. In animal mitochondria, one finds c 8 rings, requiring a minimal number of e...