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At saturating ATP, binding need no longer dominate the observed dwell, but saturation alone does not determine a dwell-time distribution. The standard interpretation of a distribution that rises from zero is that at least two sequential events remain rate limiting. Model a dwell as $T=T_1+T_2$, where the independent waiting times are distributed according to exponential distributions with rates $k_1,k_2$. Their convolution isThis hypoexponential distribution starts at zero, rises to a mode $\ln (k_2/k_1)/(k_2-k_1)$, then has an exponential tail. When the rates are equal, the continuous limit is the two-stage Erlang distribution,A single slow memoryless event instead gives $k{e}^{-kt}$, maximal at zero. Three 120-degree steps per revolution specify rotational stoichiometry, not three slow reactions within every dwell. The two-event interpretation is supported by the resolved catalytic-dwell chemical kinetics in the primary F1 experiment.^[ref]

The fit assumes sequential approximately irreversible events, constant rates during each dwell, negligible ATP-binding delay, negligible mechanical transit time, and detection of all relevant steps. Parallel pathways, backward transitions, inhibited states, probe drag and missed short dwells can change the histogram. The mathematical fit alone does not identify each event chemically or prove that two ATP molecules are hydrolyzed per step.

The central prediction of the ATP synthase ATP synthase rotary mechanism is relative motion of the central shaft and catalytic head, not simply a change in bulk ATP concentration. In the foundational single-molecule experiment, a visible marker coupled to the central gamma shaft reported its angular position relative to the immobilized catalytic head. Sustained directional rotations during ATP hydrolysis demonstrated rotary motion directly. The catalytic head contains three beta sites within an alpha-three beta-three assembly; their changing nucleotide affinities motivate the binding-change mechanism. ^[ref]

An angular trajectory gives more information than an average rotational speed. At low ATP, pauses before roughly 120-degree advances become longer as ATP falls, supporting an ATP-binding-limited wait. Reducing marker drag and improving temporal resolution reveals substeps and ATP-independent catalytic dwells. The early high-speed study resolved approximately 90-degree and 30-degree substeps, with more than one slow event during the catalytic dwell. Numerical angles vary with preparation and operational definition, so these observations should not be imposed as exact universal geometric increments.^[ref]

The dwell-time model in b tests serial chemical kinetics; the angle histogram tests preferred positions. Comparing probe loads checks whether viscous drag limits the observed speed. Measuring torque and angular displacement estimates mechanical work, while a separate hydrolysis readout tests chemical coupling. With one ATP per 120-degree step, the chemical budget per revolution is three ATP, but rotation alone does not count every chemical event or establish unity efficiency.

A probe can perturb the very process it reports, and inhibited pauses or unresolved reversals can bias averages. Isolated F1 observed during hydrolysis also does not demonstrate proton-driven ATP synthesis by intact membrane F0F1. Testing reverse synthesis requires evidence for chemical gene expression under the corresponding driving conditions; a rotation movie by itself is insufficient. These distinctions connect the single-molecule motion to the free-energy balance without overinterpreting it.