Discrete- versus continuous-state descriptions of the F1-ATPase molecular motor

A discrete-state model of the F1-ATPase molecular motor is developed which describes not only the dependences of the rotation and ATP consumption rates on the chemical concentrations of ATP, ADP, and inorganic phosphate, but also on mechanical control parameters such as the friction coefficient and the external torque. The dependence on these mechanical parameters is given to the discrete-state model by fitting its transition rates to the continuous-angle model of P. Gaspard and E. Gerritsma [J. Theor. Biol. 247 (2007) 672-686]. This discrete-state model describes the behavior of the F1 motor in the regime of tight coupling between mechanical motion and chemical reaction. In this way, kinetic and thermodynamic properties of the F1 motor are obtained such as the Michaelis-Menten dependence of the rotation and ATP consumption rates on ATP concentration and its extension in the presence of ADP and Pi, their dependences on friction and external torque, as well as the chemical and mechanical thermodynamic efficiencies.

💡 Research Summary

The paper presents a discrete‑state kinetic model for the rotary molecular motor F1‑ATPase and demonstrates how this simplified description can faithfully reproduce the behavior captured by a more detailed continuous‑angle model. The authors begin by reviewing the continuous model introduced by Gaspard and Gerritsma, which treats the motor’s rotation angle as a continuous variable governed by a Fokker‑Planck‑type equation. In that framework, the transition rates depend explicitly on the mechanical load (external torque) and the viscous drag (friction coefficient), as well as on the chemical potentials of ATP, ADP, and inorganic phosphate (Pi). While accurate, the continuous model requires solving partial differential equations and is not readily amenable to direct comparison with experimental kinetic data.

To overcome these limitations, the authors construct a three‑state discrete model representing the essential steps of the catalytic cycle: ATP binding, hydrolysis (including the power stroke), and ADP·Pi release. The dynamics are described by a master equation with four transition rates (forward and reverse for each step). Crucially, the authors fit these rates to the results of the continuous model, thereby embedding the dependence on friction (γ) and external torque (τ) into simple exponential functions of the form k = k₀ exp