Kinetics and chemomechanical properties of the F1-ATPase molecular motor

Mg²⁺ coordinating dynamics in Mg:ATP fueled motor proteins

The coordination of Mg(2+) with the triphosphate group of adenosine triphosphate (ATP) in motor proteins is investigated using data mining and molecular dynamics. The possible coordination structures available from crystal data for actin, myosin, RNA polymerase, DNA polymerase, DNA helicase, and F1-ATPase are verified and investigated further by molecular dynamics. Coordination states are evaluated using structural analysis and quantified by radial distribution functions, coordination numbers, and pair interaction energy calculations. The results reveal a diverse range of both transitory and stable coordination arrangements between Mg(2+) and ATP. The two most stable coordinating states occur when Mg(2+) coordinates two or three oxygens from the triphosphate group of ATP. Evidence for five-site coordination is also reported involving water in addition to the triphosphate group. The stable states correspond to a pair interaction energy of either ∼-2750 kJ/mol or -3500 kJ/mol. The role of water molecules in the hydration shell surrounding Mg(2+) is also reported.

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.

Conformational dynamics of ATP/Mg:ATP in motor proteins via data mining and molecular simulation

The conformational diversity of ATP/Mg:ATP in motor proteins was investigated using molecular dynamics and data mining. Adenosine triphosphate (ATP) conformations were found to be constrained mostly by inter cavity motifs in the motor proteins. It is demonstrated that ATP favors extended conformations in the tight pockets of motor proteins such as F(1)-ATPase and actin whereas compact structures are favored in motor proteins such as RNA polymerase and DNA helicase. The incorporation of Mg(2+) leads to increased flexibility of ATP molecules. The differences in the conformational dynamics of ATP/Mg:ATP in various motor proteins was quantified by the radius of gyration. The relationship between the simulation results and those obtained by data mining of motor proteins available in the protein data bank is analyzed. The data mining analysis of motor proteins supports the conformational diversity of the phosphate group of ATP obtained computationally.

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.

Monte Carlo simulation from proton slip to "coupled" proton flow in ATP synthase based on the bi-site mechanism

ATP synthase couples proton flow to ATP synthesis, but is leaky to protons at very low nucleotide concentration. Based on the bi-site mechanism, we simulated the proton conduction from proton slip to "coupled" proton flow in ATP synthase using the Monte Carlo method. Good agreement is obtained between the simulated and available experimental results. Our model provides deeper insight into the nucleotide dependence of ATP catalysis, and the kinetic cooperativity in three catalysis subunits. The results of simulation support the bi-site mechanism in ATP synthesis.

Dynamics of chemically powered nanodimer motors subject to an external force

The chemically powered self-propelled directed motions of nanodimer motors confined in a rectangular channel and subject to an applied external conservative force are investigated using hybrid molecular dynamics/multiparticle collision dynamics. The influence of factors, such as dimer sizes, chemical reaction type, and the nature of the interaction potentials between dimer monomers and solvent molecules, on the propulsion force and friction constant are examined. The stall force, for which the nanodimer has zero net velocity, and the thermodynamic efficiency of the motor are calculated. Both irreversible and reversible chemical reactions are considered. The simulation results are compared to theoretical predictions which are able to capture the major features of the self-propelled motion.

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.

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

The enzyme F(1)-ATPase is a rotary nanomotor in which the central gamma subunit rotates inside the cavity made of alpha(3)beta(3) subunits. The experiments showed that the rotation proceeds in steps of 120 degrees and each 120 degrees step consists of 80 degrees and 40 degrees substeps. Here the Author proposes a stochastic wave mechanics of the F(1)-ATPase motor and combines it with the structure-based kinetics of the F(1)-ATPase to form a chemomechanic coupled model. The model can reproduce quantitatively and explain the experimental observations about the F(1) motor. Using the model, several rate-limited situations about gamma subunit rotation are proposed, the effects of the friction and the load on the substeps are investigated and the chemomechanic coupled time during ATP hydrolysis cycle is determined.

How subunit coupling produces the gamma-subunit rotary motion in F1-ATPase

F(o)F(1)-ATP synthase manufactures the energy "currency," ATP, of living cells. The soluble F(1) portion, called F(1)-ATPase, can act as a rotary motor, with ATP binding, hydrolysis, and product release, inducing a torque on the gamma-subunit. A coarse-grained plastic network model is used to show at a residue level of detail how the conformational changes of the catalytic beta-subunits act on the gamma-subunit through repulsive van der Waals interactions to generate a torque that drives unidirectional rotation, as observed experimentally. The simulations suggest that the calculated 85 degrees substep rotation is driven primarily by ATP binding and that the subsequent 35 degrees substep rotation is produced by product release from one beta-subunit and a concomitant binding pocket expansion of another beta-subunit. The results of the simulation agree with single-molecule experiments [see, for example, Adachi K, et al. (2007) Cell 130:309-321] and support a tri-site rotary mechanism for F(1)-ATPase under physiological condition.

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

We report a theoretical study of the F(1)-ATPase molecular rotary motor experimentally studied by R. Yasuda, H. Noji, M. Yoshida, K. Kinosita Jr., H. Itoh [Nature 410 (2001) 898]. The motor is modeled as a stochastic process for the angle of its shaft and the chemical state of its catalytic sites. The stochastic process is ruled by six coupled Fokker-Planck equations for the biased diffusion of the angle and the random jumps between the chemical states. The model reproduces the experimental observations that the motor proceeds by substeps and the rotation rate saturates at high concentrations of adenosine triphosphate or at low values of the friction coefficient. Moreover, predictions are made about the dependence of the rotation rate on temperature, and about the behavior of the F(1) motor under the effect of an external torque, especially, in the regime of synthesis of adenosine triphosphate.

Dynamic and coordinating domain motions in the active subunits of the F1-ATPase molecular motor

F1-ATPase is a rotary molecular motor crucial for various cellular functions. In F1-ATPase, the rotation of the gammadeltaepsilon subunits against the hexameric alpha(3)beta(3) subunits is highly coordinative, driven by ATP hydrolysis and structural changes at three beta subunits. However, the dynamical and coordinating structural transitions in the beta subunits are not fully understood at the molecular level. Here we examine structural transitions and domain motions in the active subunits of F1-ATPase via dynamical domain analysis of the alpha(3)beta(3)gammadeltaepsilon complex. The domain movement and hinge axes and bending residues have been identified and determined for various conformational changes of the beta-subunits. P-loop and the ATP-binding pocket are for the first time found to play essential mechanical functions additional to the catalytic roles. The cooperative conformational changes pertaining to the rotary mechanism of F1-ATPase appears to be more complex than Boyer's 'bi-site' activity. These findings provide unique molecular insights into dynamic and cooperative domain motions in F1-ATPase.

Cooperativity in the motor activities of the ATP-fueled molecular motors

Kinesin, myosin and F1-ATPase are multi-domain molecular motors with multiple catalytic subunits. The motor mechanochemics are achieved via the conversion of ATP hydrolysis energy into forces and motions. We find that the catalysis of these molecular motors do not follow the simple Michaelis-Menten mechanism. The motor activities, such as the hydrolysis or processive rates, of kinesin, myosin and F1-ATPase have a complex ATP-dependent cooperativity. To understand this complexity in kinetics and mechanochemics, we develop a conformation correlation theory of cooperativity for the ATP-fueled motor proteins. The quantitative analysis and simulations indicate that cooperativity is induced by the conformational coupling of binding states of different subunits and prevails in the motor activities.

Biomolecular motors: the F1-ATPase paradigm

The realization that many essential functions of living cells are performed by nanoscale motors consisting of protein complexes has given rise to an intense effort to understand their mechanisms. Considerable progress has been made in the past two years by a combination of biophysical techniques and theoretical analysis. Single-molecule studies have played a spectacular role for a variety of motors including kinesin, myosin, and polymerases. The understanding of F(1)-ATPase, the smallest biomolecular rotary motor, has made particular progress by the interplay of experimental and theoretical studies; the latter have provided information not available from experiment.

Protein structural transitions and their functional role

Living cells are a collection of molecular machines which carry out many of the functions essential for the cell's existence, differentiation and reproduction. Most, though not all, of these machines are made up of proteins. Because of their complexity, an understanding of how they work requires a synergistic combination of experimental and theoretical studies. In this paper we outline our studies of two such protein machines. One is GroEL, the chaperone from Escherichia coli, which aids in protein folding; the other is F(1)-ATPase, a motor protein which synthesizes and hydrolyses ATP.

Complex cooperativity of ATP hydrolysis in the F(1)-ATPase molecular motor

F(1)-ATPase catalyses ATP hydrolysis and converts the cellular chemical energy into mechanical rotation. The hydrolysis reaction in F(1)-ATPase does not follow the widely believed Michaelis-Menten mechanism. Instead, the hydrolysis mechanism behaves in an ATP-dependent manner. We develop a model for enzyme kinetics and hydrolysis cooperativity of F(1)-ATPase which involves the binding-state changes to the coupling catalytic reactions. The quantitative analysis and modeling suggest the existence of complex cooperative hydrolysis between three different catalysis sites of F(1)-ATPase. This complexity may be taken into account to resolve the arguments on the binding change mechanism in F(1)-ATPase.