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A model-based method is used to extract a short-lived state in the rotation kinetics of the F1-ATPase of a bacterial species, Paracoccus denitrificans (PdF1). Imaged as a single molecule, PdF1 takes large 120ø steps during it rotation. The apparent lack of further substeps in the trajectories not only renders the rotation of PdF1 unlike that of other F-ATPases, but also hinders the establishment of its mechano-chemical kinetic scheme. We addressed these challenges using the angular velocity extracted from the single-molecule trajectories and compare it with its theoretically calculated counterpart. The theory-experiment comparison indicate the presence of a 20μs lifetime state, 40o after ATP binding. We identify a kinetic cycle in which this state is a three-nucleotide occupancy state prior to ADP release from another site. A similar state was also reported in our earlier study of the Thermophilic bacillus F1-ATPase (lifetime ∼10μs), suggesting thereby a common mechanism for removing a nucleotide release bottleneck in the rotary mechanism.

In this chapter, we review single-molecule observations of rotary motors, focusing on the general theme that their mechanical motion proceeds in substeps with each substep described by an angle-dependent rate constant. In the molecular machine F1-ATPase, the stepping rotation is described for individual steps by forward and back reaction rate constants, some of which depend strongly on the rotation angle. The rotation of a central shaft is typically monitored by an optical probe. We review our recent work on the theory for the angle-dependent rate constants built to treat a variety of single-molecule and ensemble experiments on the F1-ATPase, and relating the free energy of activation of a step to the standard free energy of reaction for that step. This theory, an elastic molecular transfer theory, provides a framework for a multistate model and includes the probe used in single-molecule imaging and magnetic manipulation experiments. Several examples of its application are the following: (a) treatment of the angle-dependent rate constants in stalling experiments, (b) use of the model to enhance the time resolution of the single-molecule imaging apparatus and to detect short-lived states with a microsecond lifetime, states hidden by the fluctuations of the imaging probe, (c) treatment of out-of-equilibrium “controlled rotation” experiments, (d) use of the model to predict, without adjustable parameters, the angle-dependent rate constants of nucleotide binding and release, using data from other experiments, and (e) insights obtained from correlation of kinetic and cryo-EM structural data. It is also noted that in the case where the release of ADP would be a bottleneck process, the binding of ATP to another site acts to accelerate the release by 5–6 orders of magnitude. The relation of the present set of studies to previous and current theoretical work in the field is described. An overall goal is to gain mechanistic insight into the biological function in relation to structure.

During brief 120° transitions between long catalytic dwells, single F _1 -ATPase molecules exhibit angular jumps that vary with rotation angles. Using the angular jump profile enables the detection of fast states in the mechano-chemical scheme of the enzyme, states that are difficult to capture from single-molecule trajectories due to the fluctuations of the imaging nanoprobe. In a previous work, a short-lived, three occupancy state was postulated from a multi-state, probabilistic theory to explain the mean angular jump profile. An assumption in the theory was that the ‘mixing’ of chemical states is negligible during jumps. In a mixing event, two subsequent angular positions recorded by the imaging apparatus belong to two different chemical states of the motor enzyme due to fast reactions within a recording frame. In this paper, we provide an enhanced method for the detection of fast states. On one hand, we show using Langevin simulations that state mixing leads to faster mean angular jump, shifting up the profile. Consequently, the improved method provides a correction to the angular position and lifetime of the postulated three-occupancy metastable state. On the other hand, we show that when F _1 -ATPase is subject to torques opposing rotation in hydrolysis direction, the torques shift down the dwell angles without affecting the angle-dependent reaction rates. The torques improve detection capability for the fast state by increasing dwell times which is made evident by the flattening of the mean angular jump profile within 40°–60° from the catalytic dwell. In the three-occupancy state release of ADP occurs in concert with the binding of ATP to a different site in the F _1 -ATPase. Similarly, in the full ATP synthase when torques are created by the proton gradient in the F _O region, the release of the product ATP is presumably accelerated by the binding of ADP to a different site in the F _1 domain.