The ATP synthase (F-ATPase) is a highly complex rotary machine that synthesizes ATP, powered by a proton electrochemical gradient. Why did evolution select such an elaborate mechanism over arguably simpler alternating-access processes that can be reversed to perform ATP synthesis? We studied a systematic enumeration of alternative mechanisms, using numerical and theoretical means. When the alternative models are optimized subject to fundamental thermodynamic constraints, they fail to match the kinetic ability of the rotary mechanism over a wide range of conditions, particularly under low-energy conditions. We used a physically interpretable, closed-form solution for the steady-state rate for an arbitrary chemical cycle, which clarifies kinetic effects of complex free-energy landscapes. Our analysis also yields insights into the debated "kinetic equivalence" of ATP synthesis driven by transmembrane pH and potential difference. Overall, our study suggests that the complexity of the F-ATPase may have resulted from positive selection for its kinetic advantage.ATP synthase | kinetic mechanism | free-energy landscape | nonequilibrium steady state | evolution T he F-ATPase performs molecular-level free-energy (FE) transduction to phosphorylate ADP and yield ATP, the primary energy carrier that drives a vast range of cellular processes. It spurred Mitchell's chemiosmotic hypothesis (1) and Boyer's now-validated proposal for the binding-change mechanism (2) in which a proton electrochemical gradient is transduced to rotationbased mechanical energy and then back to chemical FE as ADP is phosphorylated. The F-ATPase's two-domain uniaxial rotary structure is conserved across all three domains of life (3-6), and details of its function have been the subject of a multitude of studies (e.g., refs. 7-30).Here, we address a relatively narrow question with potentially significant evolutionary implications: Why is ATP synthesized by a rotary mechanism instead of a potentially much simpler alternating-access mechanism (Fig. 1)? Using thermodynamic and kinetic constraints, we address the question of whether evolution tends to arrive at optimal molecular processes (31), building on established concepts of optimality-derived evolutionary convergence (32, 33). Presumably, performance advantages in the central task of ATP synthesis would be under significant evolutionary pressure. Previous modeling studies of the F-ATPase have addressed structural and mechanistic questions about the rotary mechanism (e.g., refs. 11-20), but not the metaissue of the mechanism itself compared with alternatives.To assess whether the rotary mechanism possesses any intrinsic performance advantage, we constructed a series of kinetic models abstracted from known mechanisms (Fig. 1). Beyond the rotarybased model, we considered a series of alternating-access analogs (Fig. 2), building on the demonstrated capacity for ATP-hydrolyzing transporters to be driven in reverse to synthesize ATP (34, 35). The discrete-state models do not include structural details, bu...