A Minimal Chemo-mechanical Markov Model for Rotary Catalysis of F ₁ -ATPase

F ₁ -ATPase, the catalytic domain of ATP synthase, is pivotal for mechano-chemical energy conversion in mitochondria. Aiming at a minimal yet quantitative and thermodynamically consistent model for F ₁ -ATPase rotary catalysis mechanism, here we developed a chemo-mechanical Markov model that involves the relevant conformational and chemical degrees of freedom and reproduces all relevant experimental data. We systematically evaluated over 14,000 model variants, combining exhaustive Bayesian search in the large parameter space of transition rates with cross-validation. Unexpectedly, we find that a fully functional minimal model requires four distinct β -subunit conformations. Further, our model reconciles the decade-long bi-site vs . tri-site controversy, clarifying that both pathways contribute depending on ATP concentration. Additionally, our model suggests a Brownian-ratchet-like mechanism that explains the observation that one ATP hydrolysis event can trigger larger than 120 ^° rotations, thereby explaining seemingly over 100% efficiency. Beyond this prototypic example of a complex biomolecular machine, our approach should enable one to study many other enzymatic mechanisms that implement close coupling between conformational motions, substrate binding, and chemical reactions.