Rotational dynamics of ATP synthase: mechanical constraints and energy dissipative channels

Abstract The proton motive force (PMF) across the inner mitochondrial membrane delivers approximately 0.2 eV of energy per proton, powering the F o F 1 -ATP synthase molecular motor. Here, we provide a detailed accounting of how this energy is utilized: Approximately 75–83 % is transduced into the chemical free energy of ATP synthesis, while the remaining 17–25 % is dissipated through internal friction, viscous drag, proton leakage, electroviscous effects, elastic deformations, and information-theoretic costs. Each dissipation channel is quantitatively evaluated, revealing that internal friction in the F 1 motor is the dominant loss mechanism. In this work, we did not account for the possible energy contribution due to the intrinsic electrostatic potential of the enzyme itself. In addition to this energy bookkeeping, we also examine the quantum mechanical constraints on the F o unit’s rotation. We find that, as can be expected, the energy spacing between quantized rotational states is several orders of magnitude smaller than thermal energies at physiological temperature, and that the tunneling probability through rotational barriers is practically zero. Furthermore, the biological rotation speed (∼100–650 revolutions per second (rps)) is between one and three orders of magnitude below the quantum limit implied by quantization of angular momentum of the c-ring (which would have been ca . 13,000 to 62,000 rps (depending on the size of the c-ring (17–8 subunits, respectively)) in the first rotational energy level of the c-ring). Nevertheless, experimental estimates of the rotation rates in isolated c-rings suggest rates in the vicinity of 43,000 rps, right within our theoretical quantum estimates. However, ATP synthase as a whole operates firmly within the classical regime, despite its nanoscale dimensions, which highlights its evolutionary optimization for robust and efficient energy conversion at the quantum–classical boundary. This is the result of the rotatory coupling between the F o and the much slower F 1 unit. ATP synthase’s purely classical behavior showcases a remarkable evolutionary optimization of one of life’s most essential rotary motor engineered so as to thrive far from the quantum limit, securing its function against the uncertainties of the quantum world. As Schrödinger stated in What is Life? (1944): “ The submicroscopic world is full of fluctuations. But in large aggregates of atoms, the law of large numbers ensures that these fluctuations become negligible ” a prediction directly confirmed here in the context of the rotational stability of the F o unit.