Sándor Volkán-Kacsó

Evidence has been mounting that in the rotational cycle of F1-ATPase there is a concerted ATP binding and ADP release that yields a million-fold acceleration in the rate of the product ADP release. We developed a theory of reaction kinetics to investigate the relationship between the concerted nucleotide exchange and previous single-molecule forced rotation data from Adachi, K. Nat. Commun. 2012, 3, 1022. We extracted from these data angle-dependent rate constants for nucleotide binding and release. The rate constants were then used in a unified kinetic scheme, also consistent with other single-molecule and ensemble experiments, to obtain analytical equations for nucleotide occupancy change events from nano- to millimolar ATP concentrations. A theory-experiment comparison revealed novel evidence about the concerted mechanism: it is determined by correlated conformational changes in the F1-ATPase ring, and its kinetic signature is a unified angle-dependent function of the nucleotide binding and release rate constants, which is independent of ATP concentration.

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 F₁-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 F₁-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 F₁-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 F₁ domain.

Significance: In single-molecule imaging of the F 1 - ATPase rotation by nanoprobes, there are jumps in the angular position. Their distribution is used in a method for detecting hidden states in the transitions during rotation steps. Complementing stalling experiments, this method reveals, now in free rotation without magnetic tweezers, that the motor can be in either of 2 states at a rotation angle. A comparison between the experimental trajectories and a multistate theory reveals that an 80° substep of the coupled ATP binding and ADP release involves an intermediate state reminiscent of a 3-occupancy structure. Its lifetime ( ∼ 10 μs) is comparable with the frame time of the imaging, so, by detecting this short-lived state, the method provides an increased effective time resolution.

We summarize the different types of single molecule experiments on the F₁ component of F_OF₁-ATP Synthase and what has been learned from them. We also describe results from our recent studies on interpreting the experiments using a chemical-mechanical theory for these biological motors.

The theory of elastic group transfer for the binding and release rate constants for nucleotides in F₁-ATPase as a function of the rotor angle is further extended in several respects. (i) A method is described for predicting the experimentally observed lifetime distribution of long binding events in the controlled rotation experiments by taking into account the hydrolysis and synthesis reactions occurring during these events. (ii) A method is also given for treating the long binding events in the experiments and obtaining the rate constants for the hydrolysis and synthesis reactions occurring during these events. (iii) The theory in the previous paper is given in a symmetric form, an extension that simplifies the application of the theory to experiments. It also includes a theory-based correction of the reported “on” and “off” rates by calculating the missed events. A near symmetry of the data about the angle of −40° and a “turnover” in the binding rate data vs. rotor angle for angles greater than ∼∼ 40° is also discussed.

The recent experimental, theoretical and computational advances in the field of F₁-ATPase single-molecule microscopy are briefly surveyed. The role of theory is revealed in the statistical analysis, interpretation and prediction of single-molecule experimental trajectories, and in linking them with atomistic simulations. In particular, a theoretical model of elastically coupled molecular group transfer is reviewed and a detailed method for its application in stalling and controlled rotation experiments is provided. It is shown how the model can predict, using previous experiments, the rates of ligand binding/release processes (steps) and their exponential dependence on rotor angle in these experiments. The concept of Brønsted slopes is reviewed in the context of the single-molecule experiments, and the rate versus rotor angle relations are explained using the elastic model. These experimental data are treated in terms of the effect of thermodynamic driving forces on the rates assuming that the rotor shaft is elastically coupled to stator ring subunits in which the steps occur. In the application of the group transfer model on an extended angular range processes leading up to the transfer are discussed. Implications for large-scale atomistic simulation are suggested for the treatment of torque-generating steps.

Significance: The investigation of nucleotide binding and release dynamics vs. rotor shaft rotation in the F ₁ -ATPase enzyme is necessary to reveal biological function. We elucidate the mechanism of the exponential-like change of binding and release rate (and thus the equilibrium) constants when probed against the rotor angle at the single-molecule level. We extend our group transfer theory proposed for the stalling experiments to treat controlled rotation experiments. The model correctly predicts the controlled rotation data on fluorescent ATP without any adjustable parameters. The theory provides a framework able to treat the binding and release of various nucleotides. In the process we also learn about the properties of the fluorescent nucleotide Cy3-ATP.

A theoretical model of elastically coupled reactions is proposed for single molecule imaging and rotor manipulation experiments on F₁-ATPase. Stalling experiments are considered in which rates of individual ligand binding, ligand release, and chemical reaction steps have an exponential dependence on rotor angle. These data are treated in terms of the effect of thermodynamic driving forces on reaction rates, and lead to equations relating rate constants and free energies to the stalling angle. These relations, in turn, are modeled using a formalism originally developed to treat electron and other transfer reactions. During stalling the free energy profile of the enzymatic steps is altered by a work term due to elastic structural twisting. Using biochemical and single molecule data, the dependence of the rate constant and equilibrium constant on the stall angle, as well as the Børnsted slope are predicted and compared with experiment. Reasonable agreement is found with stalling experiments for ATP and GTP binding. The model can be applied to other torque-generating steps of reversible ligand binding, such as ADP and P_i release, when sufficient data become available.