F1-ATPase Rotary Mechanism: Interpreting Results of Diverse Experimental Modes With an Elastic Coupling Theory

Abstract

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.

Keywords: ATP binding; F1-ATPase; concerted kinetics; cryo-electron microscopy; multi-state theory; rotary biomolecular motors; single-molecule imaging; stepping rotation of F1-ATPase.

Figure 1

F-ATPase structure showing the F1 and Fo components (left). F1-ATPase motor “transverse cross-section” view showing the central rotor in blue and $\alpha$ ring subunits in gold and $\beta$ subunits in red (center). Example of a trajectory showing stepping rotation of F1-ATPase (right), as imaged by a probe in single molecule experiment (top inset). A transition is shown (red points) between two dwells (blue dots) in the lower inset. Figures were reproduced, with permission from Weber (2010) and Volkán-Kacsó et al. (2019).

Scheme 1

Thermodynamic cycle for the nucleotide binding step.

Figure 2

Free energy curves for reactant and product states showing the definition of $\Delta G^0$ and $\lambda$ . These two parabolas are plotted vs. the reaction coordinate that describes the progress of the reaction. The transition state occurs at the intersection of the two parabolas, as also in Figure 1 in Marcus and Sutin (1985). For simplicity of presentation, some symbols and the labeling of work terms are omitted here.

Figure 3

Binding (forward) and release (back) rate and equilibrium rate constants vs. θ angle for Cy3-ATP in the presence (solid squares, circles, and triangles) and absence of Pi (open symbols) in solution. The experimental data of Adachi et al. corrected for missed events are compared with their theoretically predicted counterparts (solid lines). Dashed lines show the data without corrections. The figure is reproduced from Volkán-Kacsó and Marcus (2016).

Figure 4

Mean angular jump as a function of rotation angle (top) in fast rotation trajectories. Black dots indicate experimental points, the dashed line shows the predictions from a three-state model, and the continuous line was calculated assuming a fourth state. It is a three-occupancy state. The ATP* and ATP^h indicate that the ATP in the pocket is in different environments sue to conformational changes in their host subunits. The figure was reproduced from Volkán-Kacsó et al. (2019).

Figure 5

(A) Reported binding and release rate constants vs. controlled rotation angle for fluorescent ATP in the presence of Pi in solution. The reported uncorrected experimental data (squares) are compared with theoretical counterparts (solid lines) by calculating missed events and also correcting for an error due to replacing the time in the empty state T₀ by the total time T. Dashed lines show a fit to the experimental data. (B) F₁-ATPase structure at three different rotor angles with β subunits in green, α subunits in red and the γ subunit in black. (C) Cutaway of the three structures (PDB:1H8E) revealing the binding channel at the α − β interface and its narrowing as the rotor angle is changed. The figure was reproduced with permission from Volkán-Kacsó and Marcus (2017b).

Figure 6

Left: Histogram of angular positions in the dwells from ThF1 rotation trajectories. Right: Consolidated histogram of all three subunits after selecting for “high-quality” dwells. More recently, a correction method was developed for treating the asymmetry between subunits using a tilting algorithm. The figure was reproduced from Volkán-Kacsó et al. (2019).