  F1F0 ATP synthase (ATPase) either facilitates the synthesis of ATP in the
mitochondrial membranes and bacterial inner membranes in a process driven by
the proton moving force (pmf), or uses the energy from ATP hydrolysis to pump
protons against the concentration gradient across the membrane. ATPase is
composed of two rotary motors, F0 and F1, which generate the opposing rotation
and compete for control of their shared central gamma-shaft. Here we present a
self-consistent physical model of the F1 motor as a simplified two-state
Brownian ratchet based on the asymmetry of torsional elastic energy of the
coiled-coil gamma-shaft. This stochastic model unifies the physical description
of linear and rotary motors and explains the stepped unidirectional rotation of
the $\gamma$-shaft, in agreement with the `binding-change' ideas of Boyer.
Substituting the model parameters, all independently known from recent
experiments, our model quantitatively reproduces the ATPase operation, e.g. the
`no-load' angular velocity is ca. 400~rad/s anticlockwise at 4 mM ATP, in close
agreement with experiment. Increasing the pmf torque exerted by F0 can slow,
stop and overcome the torque generated by F1, switching from ATP hydrolysis to
synthesis at a very low value of `stall torque'. We discuss the matters of the
motor efficiency, which is very low if calculated from the useful mechanical
work it produces - but is quite high when the `useful outcome' is measured in
the number of H+ pushed against the chemical gradient in the F1 ATP-driven
operation.
