Theory for rates, equilibrium constants, and Brønsted slopes in F ₁ -ATPase single molecule imaging experiments

Abstract

A theoretical model of elastically coupled reactions is proposed for single molecule imaging and rotor manipulation experiments on F 1 -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. S ingle molecule imaging directly demonstrated a stepping rotation in F 1 -ATPase (1) that was resolved into ∼ 40°and ∼ 80°s ubsteps (initially reported as ∼ 30°and ∼ 90°; cf. refs. 2 and 3), and much information has been extracted by elaborate techniques at the single molecule level (3-6). Complementing experimental tools such as X-ray spectroscopy (7) and ensemble biochemical methods (8), single molecule experiments reveal key details of the coupling between enzymatic processes and rotation (9-12). A detailed picture of highly coordinated substeps has emerged in which the binding of solution ATP to an empty subunit and release of hydrolyzed ADP from the clockwise neighbor (viewed from the rotor side) occur in concert during the ∼ 80°rotation step (13). As depicted in Fig. 1 and Table 1, the subsequent ∼ 40°r otation is coordinated with the hydrolysis of ATP in the third subunit and the release of P i from the subunit that just released ADP (13).Recent stalling (6,13,14) and controlled rotation (15) experiments provide additional insight into the dynamics of the coupling between the rotation of the central shaft and the reaction steps in the stator ring subunits. These experiments yielded the rate constants of various steps, binding and release of ligands, and hydrolysis/synthesis reaction, as a function of the stalled rotor angle θ. The rates show an exponential dependence on θ over a wide range, such as a range of 80°for ATP binding and 40°for hydrolysis. Free energy profiles for the initial and final dwell angles at a specific 40°or 80°step are given in Fig. 2. For intermediate stalling angles, the profile is intermediate between these two limits.In stalling experiments (14) the freely rotating shaft of the ATPase is stalled by magnetic tweezers upon reaching a dwell angle. Rotation experiments have resolved two dwells: the binding dwell (before...

Biophysical comparison of ATP synthesis mechanisms shows a kinetic advantage for the rotary process

“…Previous modeling studies of the F-ATPase have addressed structural and mechanistic questions about the rotary mechanism (e.g., refs. [11][12][13][14][15][16][17][18][19][20], but not the metaissue of the mechanism itself compared with alternatives.…”

What can be learned about the enzyme ATPase from single-molecule studies of its subunit F1?

“…The elastic behavior is due to the twisting and bending of structural elements in the ATPase, such as the rotor and the β lever arm, described by a spring constant κ (Volkán-Kacsó & Marcus, 2015). The linearity of the plot of the logarithm of the equilibrium constant versus θ supports the assumption that the same κ can be used in the two terms in Eq.…”

“…The theory will be described (Volkán-Kacsó & Marcus, 2015, 2017a) by providing (1) an equation for the (Braig et al 2000) viewed in the plane of the α 3 β 3 ring (right). The left subfigure was reproduced with journal permission from Weber (Weber, 2010).…”

“…We introduced into the expressions for the free energy of the reactants and of the products an elastic coupling (Sielaff et al 2008) between the rotor and the stator (Volkán-Kacsó & Marcus, 2015) involved in the step. In particular, to the two-state model originally used to derive Eq.…”

“…Boyer, 1993 and references therein). To this end we describe a chemical-mechanical theory (Volkán-Kacsó & Marcus, 2015) that we have formulated to treat several types of singlemolecule experiments on this biomolecular motor (Adachi et al 2012;Watanabe et al 2012). The theory draws heavily on a theory of electron and other transfers in solution (Marcus, 1968(Marcus, , 1993Marcus & Sutin, 1985).…”