The catalytic site must have a mechanism to distinguish between the ATP hydrolysis and synthesis cycles. During the hydrolysis cycle, one open catalytic site binds ATP molecule, closes, and then hydrolyzes ATP. Phosphate and ADP are released on the way back to the open conformation. However, during the ATP synthesis cycle, an open catalytic site binds phosphate and ADP, while closing, synthesizes ATP, then releases ATP while resuming the open conformation. The cycle is generally written as shown in . Although the ATP hydrolysis process and ATP synthesis process appear to be the reciprocal of each other, the β-subunit closing and β-subunit opening in the hydrolysis direction are not reciprocal of each other. The opening of the β-subunit associated with product release in hydrolysis is not simply the reverse of the closing of β associated with ATP binding. (Although the opening of β associated with ATP release in synthesis may be the reversal of the closing of β associated with ATP binding in hydrolysis, at near equilibrium, it must be.) Thus, at the angstrom level, where the γ-phosphate bond is either made or broken, the collective motions of the catalytic site atoms must be cyclic; this is illustrated in . Thus, we can conclude that the positions of the catalytic residues must traverse a periodic geometrical orbit in opposite directions during hydrolysis or synthesis. (Of course, because of thermal fluctuations, these orbits reflect the average positions of the catalytic atoms.) Although this may be obvious, it is not obvious that this cyclic orbit is reflected in the observable nanometer level motions of the β-subunits as they undergo their cyclic hinge bending motions. To see this, we next discuss the constraints structure imposes on the motions of the β-subunits. Kinematics of coupling between the catalytic site and the γ-subunit: The tight-coupling approximation. In hydrolysis mode, the eccentric shape of the γ shaft converts the hinge-bending motions of the β-subunits into a rotary motion analogous to turning the crank on an automobile jack (, ). Conversely, in synthesis mode, the rotation of the γ shaft drives the β-subunits through their hinge-bending cycle. Several authors have found evidence that the mechanical coupling between the β and γ subunits is tightened by a circumferential “track” of hydrophobic and positively charged residues around the γ shaft that guides the tips of the β-subunits (, ). This track is shown in . unwraps the γ shaft and plots these residues on γ in the θ-z plane, where θ is the rotational angle of the γ shaft and z is the vertical coordinate along the rotation axis. Providing that the tip of each β-subunit remains in contact with the γ shaft, each rotation of the γ shaft drives every β tip along a cyclic circumferential path. Here we will assume that the β-tips follow this path so that the rotation of the γ shaft is tightly coupled to the bending of the β-subunits.