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Abstract

We combine molecular simulations and mechanical modeling to explore the mechanism of energy conversion in the coupled rotary motors of F_oF₁-ATP synthase. A torsional viscoelastic model with frictional dissipation quantitatively reproduces the dynamics and energetics seen in atomistic molecular dynamics simulations of torque-driven γ-subunit rotation in the F₁-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_BT 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 F₁ rotary motors in ATP synthase, and explain the need for the finer stepping of the F₁ motor in the mammalian system, as seen in recent experiments. Compensating for incommensurate eightfold and threefold rotational symmetries in F_o and F₁, 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.