Integration of F1 ATPase kinetic data and thermodynamics reveals a hidden tri-site intermediate

F 1 -ATPase is the catalytic motor of ATP synthase and plays a central role in cellular energy production. It is a reversible rotary motor that converts between mechanical rotation and the chemical energy of ATP with near 100% efficiency. Single-molecule experiments have revealed that this function relies on the rotation-angle dependence of nucleotide binding, release, and ATP hydrolysis/synthesis. However, a quantitative model that directly incorporates experimentally measured angle-dependent kinetic data and satisfies thermodynamic consistency has remained elusive. Here, we establish a data-driven mathematical model grounded in experimentally measured rate constants and constrained by thermodynamic consistency. A minimal model based on these rates fails to reproduce experimental behavior under thermodynamic constraints. By introducing a tri-site intermediate as a transient state, we develop an extended model that quantitatively reproduces both ATP-hydrolysis–driven rotation and ATP synthesis under externally applied torque. Our analysis reveals that the tri-site intermediate is short-lived during ATP hydrolysis but becomes prominent during ATP synthesis, where it appears prior to the rate-limiting ATP release and increases under high synthesis conditions. These results provide a quantitative framework for chemo-mechanical coupling in F 1 -ATPase and establish a generalizable approach for describing energy conversion in molecular motors.