The coupled chemomechanics of the F(1)-ATPase molecular motor

Modeling work-speed-accuracy trade-offs in a stochastic rotary machine

Molecular machines are stochastic systems that catalyze the energetic processes keeping living cells alive and structured. Inspired by the examples of F_{1}-ATP synthase and the bacterial flagellum, we present a minimal model of an externally driven stochastic rotary machine. We explore the trade-offs of work, driving speed, and driving accuracy when changing driving strength, speed, and the underlying system dynamics. We find an upper bound on accuracy and work for a particular speed. Our results favor slow driving when tasked with minimizing the work-accuracy ratio and maximizing the rate of successful cycles. Finally, in the parameter regime mapping to the dynamics of F_{1}-ATP synthase, we find a significant decay of driving accuracy at physiological rotation rates, raising questions about how ATP synthase achieves reasonable or even remarkable efficiency in vivo.

Optimal control of rotary motors

Single-molecule experiments have found near-perfect thermodynamic efficiency in the rotary motor F_{1}-ATP synthase. To help elucidate the principles underlying nonequilibrium energetic efficiency in such stochastic machines, we investigate driving protocols that minimize dissipation near equilibrium in a simple model rotary mechanochemical motor, as determined by a generalized friction coefficient. Our simple model has a periodic friction coefficient that peaks near system energy barriers. This implies a minimum-dissipation protocol that proceeds rapidly when the system is overwhelmingly in a single macrostate but slows significantly near energy barriers, thereby harnessing thermal fluctuations to kick the system over energy barriers with minimal work input. This model also manifests a phenomenon not seen in otherwise similar nonperiodic systems: Sufficiently fast protocols can effectively lap the system. While this leads to a trade-off between accuracy of driving and energetic cost, we find that our designed protocols outperform naive protocols.

Torque-coupled thermodynamic model for F_{o}F_{1}-ATPase

F_{o}F_{1}-ATPase is a motor protein complex that utilizes transmembrane ion flow to drive the synthesis of adenosine triphosphate (ATP) from adenosine diphosphate (ADP) and phosphate (Pi). While many theoretical models have been proposed to account for its rotary activity, most of them focus on the F_{o} or F_{1} portions separately rather than the complex as a whole. Here, we propose a simple but new torque-coupled thermodynamic model of F_{o}F_{1}-ATPase. Solving this model at steady state, we find that the monotonic variation of each portion's efficiency becomes much more robust over a wide range of parameters when the F_{o} and F_{1} portions are coupled together, as compared to cases when they are considered separately. Furthermore, the coupled model predicts the dependence of each portion's kinetic behavior on the parameters of the other. Specifically, the power and efficiency of the F_{1} portion are quite sensitive to the proton gradient across the membrane, while those of the F_{o} portion as well as the related Michaelis constants for proton concentrations respond insensitively to concentration changes in the reactants of ATP synthesis. The physiological proton gradient across the membrane in the F_{o} portion is also shown to be optimal for the Michaelis constants of ADP and phosphate in the F_{1} portion during ATP synthesis. Together, our coupled model is able to predict key dynamic and thermodynamic features of the F_{o}F_{1}-ATPase in vivo semiquantitatively, and suggests that such coupling approach could be further applied to other biophysical systems.

Light Effect on Water Viscosity: Implication for ATP Biosynthesis

Previous work assumed that ATP synthase, the smallest known rotary motor in nature, operates at 100% efficiency. Calculations which arrive to this result assume that the water viscosity inside mitochondria is constant and corresponds to that of bulk water. In our opinion this assumption is not satisfactory for two reasons: (1) There is evidence that the water in mitochondria prevails to 100% as interfacial water. (2) Laboratory experiments which explore the properties of interfacial water suggest viscosities which exceed those of bulk water, specifically at hydrophilic interfaces. Here, we wish to suggest a physicochemical mechanism which assumes intramitochondrial water viscosity gradients and consistently explains two cellular responses: The decrease and increase in ATP synthesis in response to reactive oxygen species and non-destructive levels of near-infrared (NIR) laser light, respectively. The mechanism is derived from the results of a new experimental method, which combines the technique of nanoindentation with the modulation of interfacial water layers by laser irradiation. Results, including the elucidation of the principle of light-induced ATP production, are expected to have broad implications in all fields of medicine.

The nonlinear chemo-mechanic coupled dynamics of the F 1 -ATPase molecular motor

The ATP synthase consists of two opposing rotary motors, F0 and F1, coupled to each other. When the F1 motor is not coupled to the F0 motor, it can work in the direction hydrolyzing ATP, as a nanomotor called F1-ATPase. It has been reported that the stiffness of the protein varies nonlinearly with increasing load. The nonlinearity has an important effect on the rotating rate of the F1-ATPase. Here, considering the nonlinearity of the γ shaft stiffness for the F1-ATPase, a nonlinear chemo-mechanical coupled dynamic model of F1 motor is proposed. Nonlinear vibration frequencies of the γ shaft and their changes along with the system parameters are investigated. The nonlinear stochastic response of the elastic γ shaft to thermal excitation is analyzed. The results show that the stiffness nonlinearity of the γ shaft causes an increase of the vibration frequency for the F1 motor, which increases the motor's rotation rate. When the concentration of ATP is relatively high and the load torque is small, the effects of the stiffness nonlinearity on the rotating rates of the F1 motor are obvious and should be considered. These results are useful for improving calculation of the rotating rate for the F1 motor and provide insight about the stochastic wave mechanics of F1-ATPase.

CHEMOMECHANICAL COUPLING AND STOCHASTIC THERMODYNAMICS OF THE F1-ATPase MOLECULAR MOTOR WITH AN APPLIED EXTERNAL TORQUE

The effects of external torque on the F 1-ATPase rotary molecular motor are studied from the viewpoint of recent advances in stochastic thermodynamics. This motor is modeled in terms of discrete-state and continuous-state stochastic processes. The dependence of the discrete-state description on external torque and friction is obtained by fitting its transition rates to a continuous-angle model based on Newtonian mechanics with Langevin fluctuating forces and reproducing experimental data on this motor. In this approach, the continuous-angle model is coarse-grained into discrete states separated by both mechanical and chemical transitions. The resulting discrete-state model allows us to identify the regime of tight chemomechanical coupling of the F 1 motor and to infer that its chemical and mechanical efficiencies may reach values close to the thermodynamically allowed maxima near the stalling torque. We also show that, under physiological conditions, the F 1 motor is functioning in a highly-nonlinear-response regime, providing a rotation rate a million times faster than would be possible in the linear-response regime of nonequilibrium thermodynamics. Furthermore, the counting statistics of fluctuations can be obtained in the tight-coupling regime thanks to the discrete-state stochastic process and we demonstrate that the so-called fluctuation theorem provides a useful method for measuring the thermodynamic forces driving the motor out of equilibrium.

On chemomechanical coupling of the F(1)-ATPase molecular motor

F(1)-ATPase catalyzes ATP hydrolysis to drive the central gamma-shaft rotating inside a hexameric cylinder composed of alternating alpha and beta subunits. Experiments showed that the rotation of gamma-shaft proceeds in steps of 120 degrees and each 120 degrees -rotation is composed of an 80 degrees substep and a 40 degrees substep. Here, based on the previously proposed models, an improved physical model for chemomechanical coupling of F(1)-ATPase is presented, with which the two-substep rotation is well explained. One substep is driven by the power stroke upon ATP binding, while the other one resulted from the passage of gamma-shaft from previous to next adjacent beta subunits via free diffusion. Using the model, the dynamics and kinetics of F(1)-ATPase, such as the rotating time of each substep, the dwell time at each pause and the rotation rate, are analytically studied. The theoretical results obtained with only three adjustable parameters reproduce the available experimental data well.