Discrete- versus continuous-state descriptions of the F1-ATPase molecular motor
aa r X i v : . [ q - b i o . S C ] A p r Discrete- versus continuous-state descriptions of the F -ATPase molecular motor E. Gerritsma and P. Gaspard
Center for Nonlinear Phenomena and Complex Systems,Universit´e Libre de Bruxelles, Code Postal 231, Campus Plaine, B-1050 Brussels, Belgium
A discrete-state model of the F -ATPase molecular motor is developed which describes not onlythe dependences of the rotation and ATP consumption rates on the chemical concentrations ofATP, ADP, and inorganic phosphate, but also on mechanical control parameters such as the frictioncoefficient and the external torque. The dependence on these mechanical parameters is given tothe discrete-state model by fitting its transition rates to the continuous-angle model of P. Gaspardand E. Gerritsma [J. Theor. Biol. (2007) 672-686]. This discrete-state model describes thebehavior of the F motor in the regime of tight coupling between mechanical motion and chemicalreaction. In this way, kinetic and thermodynamic properties of the F motor are obtained such asthe Michaelis-Menten dependence of the rotation and ATP consumption rates on ATP concentrationand its extension in the presence of ADP and P i , their dependences on friction and external torque,as well as the chemical and mechanical thermodynamic efficiencies. Keywords: molecular motor, F -ATPase, mechanochemical coupling, stochastic process, nonequi-librium thermodynamics. I. INTRODUCTION F o F -ATPase is a ubiquitous protein producing adenosine triphosphate (ATP) in mitochondria [1, 2]. In vivo , theF o part of this protein is embedded in the inner membrane of mitochondria and is rotating as a turbine when a protoncurrent flows through it, across the membrane. This turbine drives the rotation of a shaft inside the hydrophylicF part. This latter is composed of three α - and three β -subunits spatialy alternated as a hexamer ( αβ ) andforming a barrel for the rotation of the shaft made of a γ -subunit [3, 4]. Upon rotation, the central γ -shaft inducesconformational changes in the hexamer, leading to the synthesis of ATP in catalytic sites located in each β -subunit.In their experimental work, Kinosita and coworkers have succeeded to build a nanomotor by separating the F partand attaching an actin filament or a colloidal bead to its γ -shaft. In vitro , ATP hydrolysis drives the rotation of thisnanomotor, transforming chemical free energy from ATP into the mechanical motion of the γ -shaft [5, 6]. This motionproceeds in steps of 120 ◦ , revealing the three-fold symmetry of F -ATPase [3, 4, 5]. Furthermore, the experimentshave shown that each of these steps is subdivided into two substeps [7, 8]. The first substep of about 90 ◦ is attributedto ATP binding to an empty catalytic β -subunit of F , followed by conformational changes of this subunit and ofthe whole hexamer as a consequence of its binding to the other subunits. This deformation of the protein generatesa torque on the γ shaft, hence its rotation by about 90 ◦ . The secondary substep of about 30 ◦ follows the hydrolysisof ATP and results into the release of the products adenosine diphosphate (ADP) and inorganic phosphate (P i ),completing the 120 ◦ step [6, 9].In our previous work [12], we carried out a theoretical study of the stochastic chemomechanics of the F -ATPasemolecular motor on the basis of the experimental observations reported in Ref. [7]. The stochasticity of the motion isthe consequence of the nanometric size of the F motor making it sensitive to the thermal and molecular fluctuationsdue to the atomic structure of the protein and surrounding medium [10, 11]. In our paper [12], the rotation angle wastaken as a continuous random variable and the stochastic process ruled by six coupled Fokker-Planck equations forthe biased diffusion of the angle and the random jumps between the six chemical states considered in our model. Thiscontinuous-angle model allows us to reproduce accurately the experimental observations of Ref. [7] and, in particular,the random trajectories of the γ -shaft with the steps and also the substeps [12]. Taking the angle as a continuousvariable provides a realistic description of the random motion of the motor.However, coarser descriptions are often considered in which the angle (or the position in the case of linear motors)performs discrete jumps instead of varying continuously. The discretization is considered not only because it isalways required in order to simulate the random process of diffusion for the angle [13], but also because the angle isobserved to jump by finite amounts corresponding to the steps and substeps observed in the experiments [7, 8]. Thisobservation suggests an alternative description with discrete states associated with the steps and substeps. As longas the steps and substeps can be identified with the chemical states of the rotary motor, this fully discrete descriptioncan be set up on the basis of chemical kinetics. This description could in principle be deduced by coarse grainingthe continuous-angle description, but is most often taken as a model with parameters to be fitted to experimentalobservations. In such coarse-grained descriptions in terms of discrete states, the stochastic process is no longer aprocess with both diffusion and jumps, but becomes a purely jump process ruled by a so-called master equation[10, 11]. Such discrete-state descriptions provide simpler models of molecular motors, relying on similar but differentassumptions compared to the finer descriptions using continuous random variables for angle or position beside thediscrete chemical states. In this context, the question arises of understanding the relationship between the two levelsof description of the molecular motor. Since both discrete- and continuous-state models of molecular motors havebeen proposed in the literature, it is becoming important to develop methods to relate both levels of description,which is the purpose of the present paper.The comparison between the two levels of description is interesting not only for the methodology, but also because itcan reveal important properties of the molecular motor such as the nature of the coupling between their chemistry andmechanics, i.e., the question whether the mechanochemical coupling is tight or loose. As discussed in the following,this property can be expressed in terms of the nonequilibrium thermodynamics of the molecular motor.The paper is organized as follows. In section II, we present in detail the discrete-state description of the F molecularmotor. In section III, this description is compared with our previous model with a continuous angle variable. In sectionIV, we discuss the properties of the F motor in the light of the comparison between the discrete- and continuous-statedescriptions and, in particular, the question of the mechanochemical coupling and its thermodynamic efficiencies. Theconclusions are drawn in section V. II. DISCRETE-STATE DESCRIPTIONA. Chemistry of the F motor In the discrete-state description, the discrete values of the angle of the γ -shaft correspond to the chemical states sothat the mechanical motion of the motor is directly controlled by its chemistry.The motor is powered by the hydrolysis of ATP into ADP and P i :ATP ⇋ ADP + P i (1)This reaction is driven by the difference of chemical potential ∆ µ between the three species (ATP, ADP, and P i ):∆ µ = µ ATP − µ ADP − µ P i (2)where the chemical potential µ X is equal to the corresponding Gibbs free energy per molecule defined as µ X = µ + k B T ln [X] c (3)with X = ATP, ADP, or P i , the absolute temperature T , Boltzmann’s constant k B = 1 .
38 10 − J/K, and the referenceconcentration c = 1 mole per liter at which the chemical potential of species X takes its standard value µ .The standard Gibbs free energy of ATP hydrolysis takes the value ∆ G = − ∆ µ = − . − . −
50 pN nm at the temperature of 23 ◦ C, the external pressure of 1 atm, and pH 7 [14]. We notice that ATP hydrolysisprovides a significant amount of free energy of − ∆ G = 12 . k B T above the thermal energy k B T = 4 . i satisfy [ATP][ADP][P i ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) eq = exp ∆ G k B T ≃ . − M − (4)showing that ATP tends to hydrolyze into its products. In Eq. (4) and the following, we no longer write the referenceconcentration c , assuming that the concentrations are counted in mole per liter (M).The kinetic scheme of our model is based on the phenomenological observations of 120 ◦ rotation of the γ -shaft perconsumed ATP molecule. In accordance with Ref. [7], the first substep, the 90 ◦ rotation of the γ -shaft, is induced bythe binding of ATP to an empty catalytic site. The second substep, the 30 ◦ rotation of the γ -shaft, is induced by therelease of ADP and P i . The process can be summarized by the following chemical schemeATP + [ ∅ , γ ( θ )] | {z } state 1 W +1 ⇋ W − [ATP ‡ , γ ( θ + 90 ◦ )] | {z } state 2 W +2 ⇋ W − [ ∅ , γ ( θ + 120 ◦ )] | {z } state 1 +ADP + P i (5)From left to right: In state 1, ATP can bind to an empty ( ∅ ) β -catalytic site of F with the γ -shaft at angular position θ . State 1 is thus defined by [ ∅ , γ ( θ )]. Binding of ATP induces the 90 ◦ rotation of the γ -shaft, which we represent by γ ( θ + 90 ◦ ) and fills this catalytic site. ATP ‡ stands for any transition state of ATP between the initial triphosphatemolecule to the products of hydrolysis ADP and P i before the evacuation of the β -catalytic site. State 2 is thusrepresented by [ATP ‡ , γ ( θ + 90 ◦ )]. If F proceeds to hydrolysis, the products ADP and P i are released together, whichinduces the secondary 30 ◦ rotation and empties a β -subunit.Notice that Eq. (5) does not necessarily represent the same β -subunit during different real catalytic states. Followingour reference data [7], ADP and P i are released together, but from a different β -subunit then the ATP binding one.In this case, Eq. (5) represents different β -subunits; one that binds ATP and one that releases ADP and P i fromanother ATP molecule previously bound to an other β -catalytic subunit. Furthermore, if the concentration [ATP]is below the nanomolar, F can perform uni-site catalysis [15]. In this other case, Eq. (5) would represent a single β -subunit in its two main catalytic states. The relevance of Eq. (5) to both cases is a consequence of our approachwhich consists in looking only at the contributions of ATP hydrolysis to the rotation of γ . Very recent experimentalobservations have shown that ADP and P i may be released from different β -subunits [6]. If both product moleculesare released at different times, a more complete model could be considered by adding a third state to the kineticscheme. Here, the scheme is simplified by lumping together the two states subsequent to hydrolysis in a regime wherethe time delay between the releases of ADP and P i is short enough, as also considered elsewhere [12].Since F is a hexamer composed of three β -subunits, the reactions (5) appear three times for the angles θ + 120 ◦ n with n = 0 , ,
2, so that the motor has a total of six chemical states. However, by the three-fold symmetry of theF motor, these six states can be regroup three by three since the process repeats itself similarly every 120 ◦ in each β -subunit. We can therefore identify the three states of type 1 together in state 1 and the same for state 2, reducingthe motor dynamics to a process with only two discrete states.According to the mass-action law of chemical kinetics, the reaction rates W ρ in Eq. (5) depend on the molecularconcentrations in the solution surrounding the motor as follows W +1 = k +1 [ATP] (6) W − = k − (7) W +2 = k +2 (8) W − = k − [ADP][P i ] (9)where the quantities k ρ ( ρ = ± , ±
2) are the constants of the forward and backward reactions of binding and unbindingof ATP or ADP with P i and, [ATP], [ADP], [P i ] represent the concentrations of each species. k +1 is the constant ofATP binding often denoted k on and k − the ATP unbinding constant k off , while k +2 is the constant of ATP catalysisdenoted k cat . The two-state model includes the possibility of ADP and P i binding and thus ATP synthesis, withthe corresponding kinetic constant k − . Since the rates W ρ are measured in s − , the constant k +1 has the units ofM − s − , k − and k +2 the units of s − , and k − the units of M − s − .An important aspect of the experiments with beads or actin filaments attached to the γ -shaft is that the behavior ofthe molecular motor also depends on the friction ζ of the attached objects moving in the viscous medium surroundingF , as well as on the external torque τ which is applied in some experiments [8, 16]. As a consequence, the reactionconstants (6)-(9) depend on both the friction ζ and the external torque τ . Indeed, a discrete-state description in whichthe motion is directly coupled to the chemical reactions leaves no place to the modeling of mechanical aspects such asfriction and external torque. On the other, a continuous-angle description based on a stochastic Newtonian equationof Langevin type would provide a way to include the mechanical forces of friction and external torque by using the lawsof Newtonian mechanics. This was achieved in our previous paper [12], allowing us to study the dependence on frictionand external torque. In order to take into account these mechanical aspects in the present discrete-state model, wehave to incorporate the effect of friction ζ and external torque τ into the reaction constants k ρ ( ζ, τ ) ( ρ = ± , ±
2) andto find these dependences by comparison with the continuous-angle model [12] which includes the mechanical aspects.This will be achieved in section III where the discrete-state model will be completed in this way. We notice that adiscrete-state model with six chemical states has been studied in Ref. [17] in the particular case where the externaltorque is vanishing and for a fixed value of the friction coefficient. Here, our purpose is to investigate situationswhere the molecular motor is submitted to an external torque to understand how the coupling between chemistry andmechanics can be formulated in terms of the reaction rates (6)-(9) of the discrete-state model.
B. Thermodynamics of the F motor In order for the description to be consistent with thermodynamics, let us summarize here the conditions that thechemical and mechanical properties of the motor must satisfy.The internal energy of the motor changes with the rotation angle θ and the numbers N X of molecules (X=ATP,ADP, P i ) entering the catalytic sites according to the Gibbs relation dE = T dS + τ dθ + X X µ X dN X (10)where τ denotes the external torque, µ X the chemical potential (3), T the temperature, and S the thermodynamicentropy. The changes in molecular numbers due to the overall reaction (1) satisfy dN ATP = − dN ADP = − dN P i (11)where dN X is the number of molecules of species X entering the motor. Consequently, the Gibbs relation (10) becomes dE = T dS + τ dθ + ∆ µ dN ATP (12)in terms of the chemical potential (2) of the overall reaction. In an open system such as a motor, energy and entropyvary a priori because of exchanges with the surrounding medium or possibly due to internal variations: dE = d e E + d i E (13) dS = d e S + d i S (14)The laws of thermodynamics only rule the internal variations:first law: d i E = 0 (15)second law: d i S ≥ H dE = 0 and H dS = 0. For the isothermal process of ananomotor in the heat bath given by the surrounding medium, the entropy irreversibility produced over a motor cycleis thus given by I d i S = − I d e S = 1 T I ( τ dθ + ∆ µ dN ATP ) ≥ V ≡ π (cid:28) dθdt (cid:29) (18)and the mean rate of ATP consumption R ≡ (cid:28) dN ATP dt (cid:29) (19)the entropy production is given by d i Sdt = 2 πτT V + ∆ µT R ≥ πτ /T and ∆ µ/T , and the corresponding fluxes orcurrents, V and R [18, 19, 20]. All these quantities vanish at the thermodynamic equilibrium. Mechanical or chemicalenergies are provided from the exterior if either the torque τ or the reaction free energy ∆ µ are non vanishing.In general, molecular motors can be driven out of equilibrium by either the external torque τ or the molecularconcentrations of ATP, ADP and P i entering ∆ µ , possibly in combination. In this regard, the thermodynamic forces2 πτ /T and ∆ µ/T are independent control parameters. However, in a regime where the chemistry and the mechanicsof the molecular motor are tightly coupled, three ATP molecules are consumed per revolution which is expressed bythe conditions that the ATP consumption rate is three times the velocity, i.e.,tight coupling: V = 13 R (21)In this case, the entropy production (20) becomestight coupling: 1 k B d i Sdt = AR ≥ A ≡ π τk B T | {z } mechanics + ∆ µk B T | {z } chemistry (23)which is here written in dimensionless form. In the regime of tight coupling, the mechanical and chemical thermo-dynamic forces are thus no longer independent control parameters but are replaced by the unique chemomechanicalaffinity (23). This affinity vanishes at equilibrium. By using Eq. (3), we obtain the equilibrium condition for theconcentrations: [ATP][ADP][P i ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) eq = exp 1 k B T (cid:18) ∆ G − π τ (cid:19) ≃ . − M − exp (cid:18) − π τk B T (cid:19) (24)in the presence of the external torque τ . This result shows that the chemical equilibrium is displaced in the presenceof an external torque. In vivo , such an external torque comes from the F o part of ATPase and the transmembranepH difference in mitochondria. This displacement of equilibrium finds its origin in the mechanochemical couplingachieved in the F rotary motor but happens only if an external torque is enforced on the γ -shaft.In the discrete-state description, the discrete values of the angle of the γ -shaft correspond to the chemical states sothat the mechanical motion is tightly coupled to the chemistry of F , a feature which is characteristic of such discretemodels. Therefore, the relations (21), (22), and (23) apply to such tight-coupling models. C. Master equation description
The chemistry of F is a stochastic process because the arrival of each substrate molecule (ATP or ADP withP i ) is a random event in time. Accordingly, the time evolution of the molecular motor is described in terms of theprobabilities P σ ( t ) to find it in one or the other of the two states σ = 1 , dP σ ( t ) dt = X ρ,σ ′ [ W ρ ( σ ′ | σ ) P σ ′ ( t ) − W − ρ ( σ | σ ′ ) P σ ( t )] (25)with a sum over the two reactions ρ = 1 , σ ′ = 1 , σ ′ ρ −→ σ or after thereverse transition σ − ρ −→ σ ′ . The quantity W ρ ( σ ′ | σ ) is the transition rate per unit time from the state σ ′ to the state σ due to the reaction ρ , which can be identified with the rate of the corresponding reaction between the two chemicalstates σ and σ ′ . The master equation conserves the total probability P σ P σ ( t ) = 1 for all times t . Using the specificvalues of the four reaction rates of Eq. (5) resumed in Eqs. (6)-(9), we write the time evolution of the states 1 and 2with respectively empty and occupied catalytic site as dP dt = ( W − + W +2 ) P − ( W +1 + W − ) P (26) dP dt = ( W +1 + W − ) P − ( W − + W +2 ) P (27)with the normalisation condition P + P = 1 always satisfied.The mean consumption rates of the different species are given by ddt h N ATP i = W +1 P − W − P (28) ddt h N ADP i = ddt h N P i i = W − P − W +2 P (29)Since the γ -shaft rotates by a substep of 90 ◦ during the first reaction ρ = +1 and by a substep of 30 ◦ during thesecond reaction ρ = +2, the mean angle of the γ -shaft evolves in time according to ddt h θ i = π W +1 P − W − P ) + π W +2 P − W − P ) (30)in radian per second. Stationary state.
The master equation admits a time-independent stationary solution such that ( d/dt ) P (st) σ = 0.The stationary solution for each two states is given by P (st)1 = k − + k +2 k +1 [ATP] + k − + k +2 + k − [ADP][P i ] (31) P (st)2 = k +1 [ATP] + k − [ADP][P i ] k +1 [ATP] + k − + k +2 + k − [ADP][P i ] (32)where we used the normalization condition P (st)1 + P (st)2 = 1. We notice that this state is stationary in the statisticalsense because an individual motor continues to fluctuate in time. The probabilities (31)-(32) define the statisticaldistribution of its fluctuations between the states 1 and 2 after sampling over a long enough lapse of time and overmany random realizations.In the stationary state, the mean angular velocity of the γ -shaft is thus given by V = 12 π ddt h θ i st = 13 k +1 k +2 [ATP] − k − k − [ADP][P i ] k +1 [ATP] + k − + k +2 + k − [ADP][P i ] (33)and the mean rates of molecular consumption by R = ddt h N ATP i st = − ddt h N ADP i st = − ddt h N P i i st = 3 V (34)The condition (21) of tight chemomechanical coupling is thus satisfied. Equilibrium state.
At thermodynamical equilibrium, all the nonequilibrium constraints vanish. In this case, thestationary solution represents the equilibrium probability P (eq) σ and obeys the detailed balance conditions [21] W ρ ( σ ′ | σ ) P (eq) σ ′ = W − ρ ( σ | σ ′ ) P (eq) σ (35)for all reactions ρ = 1 , σ, σ ′ = 1 ,
2. As a consequence, the mean velocity (33) and the rate(34) vanish which occurs under the condition: [ATP][ADP][P i ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) eq = k − k − k +1 k +2 (36)Comparing with the condition obtained from the thermodynamics of the motor, we get the equilibrium constant ofthe reaction: K eq ≡ k − k − k +1 k +2 = exp 1 k B T (cid:18) ∆ G − π τ (cid:19) ≃ . − M − exp (cid:18) − π τk B T (cid:19) (37)which depends on the external torque τ acting on the molecular motor. We see that considerations of equilibriumthermodynamics give a constraint allowing us to fix one reaction constant if we know the three other constants.However, equilibrium thermodynamics does not provide enough relations in order to determine the four reactionconstants. We have therefore to use data from the kinetics of the motor, i.e., when the motor is out of equilibrium.This is the purpose of the following subsection. D. Determination of the reaction constants
The motor is in a nonequilibrium stationary state if the chemomechanical affinity (23) is non-vanishing, i.e., if theequilibrium condition (36) is not satisfied. In this case, the mean velocity (33) does not vanish and can be comparedwith experimental data [7] or with the continuous-angle model [12] which is complete with respect to the mechanics.In this way, each one of the reaction constants k ρ can be determined by appropriate limits at fixed values of ζ and τ ,as explained here below.First of all, the mean velocity (33) can be rewritten in the following form: V = V max ([ATP] − K eq [ADP][P i ])[ATP] + K M + K P [ADP][P i ] (38)in terms of the constants V max ≡ k +2 (39) K M ≡ k − + k +2 k +1 (40) K P ≡ k − k +1 (41)together with the equilibrium constant K eq defined by Eq. (37). The knowledge of these four constants is equivalentto knowing the four reaction constants k ± and k ± . Each one of the three constants (39), (40), and (41) can bedetermined in a specific regime of functioning of the molecular motor.The constants (39) and (40) can in particular be determined in the absence of the products of hydrolysis, [ADP][P i ] =0, in which case the velocity follows a typical Michaelis-Menten kinetics: V = V max [ATP][ATP] + K M (42)Here, we see that Eq. (40) is the Michaelis-Menten constant defined as the ATP concentration at which the velocity V equals V max / ≪ K M where the rotationis limited by the slow arrival of ATP molecules and the saturation regime [ATP] ≫ K M where the velocity reachesits maximum value fixed by the finite rate of release of ADP and P i . Accordingly, the constant (39) is the maximumangular velocity of the motor, which is reached in a regime where the ATP concentration is large enough with respectto the Michaelis-Menten constant: V max = lim [ATP] →∞ V (43)the limit meaning that [ATP] ≫ K M . The Michaelis-Menten constant itself can be determined at low ATP concen-trations according to K M = lim [ATP] → [ATP] (cid:18) V max V − (cid:19) (44)this other limit meaning that [ATP] ≪ K M .The third constant (41) characterizes the decrease of the velocity as the concentrations of the products ADP and P i are increased. It turns out as we shall see in the following that this constant is larger by several orders of magnitudethan the equilibrium constant: K P ≫ K eq (45)Accordingly, the term involving the equilibrium constant in the numerator of Eq. (38) can be neglected. Hence, theconstant K P can be determined in the regime where[ATP] + K M K P ≪ [ADP] , [P i ] ≪ [ATP] K eq (46)by taking the difference of the inverses of the velocities at two different concentrations of ADP, keeping fixed the otherconcentrations: K P = V max [ATP][P i ] ∆(1 /V )∆[ADP] (47)with the notation ∆ X = X − X .Another consequence of the inequality (45) is obtained by using the definition (37) of the equilibrium constant: k − = K eq K P k +2 ≪ k +2 (48)whereupon the Michaelis-Menten constant (40) is essentially independent of k − in the present case: K M = k +2 k +1 (49)and thus directly determines the constant k +1 of ATP binding once the constant k +2 of product release is obtainedthanks to the maximum velocity (39). Subsequently, the constant k − of the binding of ADP and P i is determinedfrom the constant K P as explained here above. Finally, the constant k − of ATP unbinding is determined with theequilibrium constant (37). In summary, we find successively: k +2 = 3 V max (50) k +1 = k +2 K M (51) k − = k +1 K P (52) k − = k +1 k +2 k − K eq (53) III. COMPARISON WITH THE CONTINUOUS-STATE DESCRIPTION
In this section, we compare the aforementioned discrete-state description with a continuous-state description wepreviously reported on [12]. The continuous-state description considers the rotation angle as a continuous randomvariable instead of supposing the angle to jump by a finite amount at each reactive event. Accordingly, the continuous-state description allows us to incorporate mechanical aspects in the modeling which should otherwise be assumed ina discrete-state model, as explained here above. Therefore, the continuous-state description provides the dependenceof the rotational velocity on both the friction ζ and the external torque τ and is thus more complete in this regard.Consequently, the dependence of the reaction constants of the discrete-state model on friction and external torquecan be determined by comparison with the results of the simulation of our continuous-state model [12], which is thepurpose of the present section. A. Fokker-Planck equation description
In the continuous-state model [12], the system is found at a given time t in one of the six chemical states σ = 1 , , ..., γ -shaft at an angle 0 ≤ θ < π . There are six chemical states because the three β -subunits can be eitherempty or occupied by a molecule of ATP or by the products ADP and P i of hydrolysis.Consequently, the system is described by six probability densities p σ ( θ, t ) normalized according to P σ =1 R π p σ ( θ, t ) dθ = 1. The time evolution of the probability densities is ruled by a set of six Fokker-Planckequations coupled together by the terms describing the random jumps between the chemical states σ due to the twochemical reactions (ATP binding and release of the products ADP and P i ) with their corresponding reversed reactions[12]: ∂ t p σ ( θ, t ) + ∂ θ J σ ( θ, t ) = X ρ =1 , X σ ′ ( = σ ) [ p σ ′ ( θ, t ) w ρ,σ ′ → σ ( θ ) − p σ ( θ, t ) w − ρ,σ → σ ′ ( θ )] (54)where the probability current densities are given by J σ ( θ, t ) = − D ∂ θ p σ ( θ, t ) + 1 ζ [ − ∂ θ U σ ( θ ) + τ ] p σ ( θ, t ) (55)The diffusion coefficient D can be expressed in terms of the friction coefficient ζ according to Einstein’s relation D = k B T /ζ . The friction coefficient ζ can be evaluated for a bead attached to the γ -shaft, a bead duplex, or acylindrical filament, as presented elsewhere [7, 12, 14, 23]. In the case of a bead of radius r attached off-axis with itsdistance at a distance x = r sin α from the rotation axis, the friction coefficient is given by ζ = 2 πηr (cid:0) α (cid:1) (56)with the water viscosity η = 10 − pN s nm − and α = π/ σ , the γ -shaft is submitted to the external torque τ and the internal torque − ∂ θ U σ due to the free-energy potential U σ ( θ ) of the motor with its γ -shaft at the angle θ . Applying an externaltorque to the motor has the effect of tilting the potentials into U σ ( θ ) − τ θ which eases the rotation or makes it harder,depending on the sign of τ . B. Coarse graining into a discrete-state model
The correspondence with the discrete-state description in terms of the master equation (25) can in principle beestablished by coarse graining the continuous angle into discrete states. These discrete states correspond to the angularintervals θ σ < θ < θ σ + 2 π/ γ -shaft spends most of its time while in the chemical state σ . Accordingly,the probabilities ruled by the master equation (25) are related to the probability densities of the continuous-statedescription (54) according to P σ ( t ) = Z θ σ +2 π/ θ σ p σ ( θ, t ) dθ (57)where the angular integral is carried out over the aforementioned intervals. In this way, a fully discrete descriptioncould be inferred from the continuous-state description. In general, this reduction from one description to the other bythe aforementioned coarse graining leads to non-Markovian equations. In the case where there is a net separation oftime scales between the dwell times and the transit times between the discrete states, the non-Markovian effects maybe negligible and a description in terms of a Markovian equation such as the master equation (25) may be obtained.This is the situation we here consider.In this case, the transition rates W ρ ( σ ′ | σ ) of the master equation (25) can be deduced from the solution of theFokker-Planck equations (54). We emphasize that the transition rates W ρ ( σ ′ | σ ) of the master equation (25) do nottake the same values as the rates w ρ,σ ′ → σ ( θ ) appearing in the Fokker-Planck equations (54). Indeed, the latter onesdepend on the angle although the former ones do not. A method of deduction of the former from the latter with Eq.(57) would show that the transitions rates of the discrete-state description are given by intergrals of the continuous-angle rates in Eq. (54) combined with the probability densities of the quasi-stationary dwell states corresponding tothe substeps. In spite of their conceptual and numerical differences, the transition rates appearing in both descriptionsare in correspondence as far as the chemical reaction they describe are concerned, as shown in Table I. TABLE I: Comparison of the transition rates of the discrete model ruled by the master equation (25) with those of the continuousmodel ruled by the Fokker-Planck equations (54). The transition rates are in correspondence by the chemical reaction theydescribe: ATP binding for ρ = +1; ATP unbinding for ρ = −
1; the release of ADP and P i for ρ = +2; the binding of ADP andP i for ρ = −
2. We notice that the rates of the continuous model depend on the angle θ of the γ -shaft. This dependence is ofArrhenius type in terms of the inverse temperature β = ( k B T ) − and the free-energy potentials U ( θ ) and ˜ U ( θ ) (for respectivelythe empty and occupied catalytic sites) and U ‡ ( θ ) and ˜ U ‡ ( θ ) (for respectively the transition states of ATP binding or unbindingand of the release or binding of ADP and P i ). See Ref. [12] for the forms of these potentials and the numerical values of theparameters of our continuous model. W ρ ( σ ′ | σ ) w ρ,σ ′ → σ ( θ ) k +1 [ATP] k [ATP] exp ˘ − β ˆ U ‡ ( θ ) − U ( θ ) − G ◦ ATP ˜¯ k − k exp n − β h U ‡ ( θ ) − ˜ U ( θ ) io k +2 ˜ k exp n − β h ˜ U ‡ ( θ ) − ˜ U ( θ + π ) io k − [ADP][P i ] ˜ k [ADP][P i ] exp n − β h ˜ U ‡ ( θ ) − U ( θ ) − G ◦ ADP − G ◦ P i io We notice that the rates w ρ,σ ′ → σ ( θ ) appearing in the Fokker-Planck equations (54) only concern the chemicalreactions and, therefore, do not depend on the friction coefficient ζ and the external torque τ which only enter in thecurrent densities (55) appearing in the left-hand side of the Fokker-Planck equations (54). In contrast, the reactionconstants k ρ shown in Table I necessarily depend on the friction coefficient ζ and the external torque τ , which otherwisewould not appear in the discrete model given by the master equation (25). Accordingly, we now need to determinethe dependences k ρ ( ζ, τ ) of the discrete-state reaction constants on both friction and external torque, which is thepurpose of the next subsection.0 TABLE II: Values of the coefficients of the Taylor expansions (59)-(60) of the functions a ρ ( τ ) and b ρ ( τ ) giving the reactionconstants k +1 ( ζ, τ ), k +2 ( ζ, τ ), and k − ( ζ, τ ), according to Eq. (58).coefficient k +1 ( ζ, τ ) k +2 ( ζ, τ ) k − ( ζ, τ ) units a (0) ρ − . − . − .
382 - a (1) ρ . − . − .
29 10 − (pN nm) − a (2) ρ . − . − . − (pN nm) − b (0) ρ − . − . − .
338 - b (1) ρ − . − − . − . − (pN nm) − b (2) ρ . − . − − . − (pN nm) − C. Dependence of the reaction constants on friction and external torque
The original and key feature of our work is the inclusion in the discrete-model reaction constants k ρ ( ρ = ± , ± k ρ on both friction and external troque by fitting them tothe simulations of our continuous-angle model [12]. Since this latter has been fitted to experimental data, the presentfitting procedure is comparable to a fitting to the experimental data of Ref. [7]. We use the method presented insubsection II D. In this way, we evaluate successively the reaction constants thanks to Eqs. (50)-(53) for differentvalues of the external torque τ and the friction ζ .In this regard, an essential observation is that the reaction constants become independent of the friction coefficient ζ at low friction and decrease as the inverse of the friction at high friction: k ρ ∝ ζ − . The motor is functioningin a reaction-limited regime at low friction and in a friction-limited regime at high friction [12]. At high friction,the substeps are no longer visible since the rotation is slow down by the friction, in which case the Fokker-Planckequations (54) suggest indeed that the rate constants should scale as ζ − .The crossover between the low- and high-friction regimes can be well described by giving the following analyticalform to the reaction constants: k ρ ( ζ, τ ) = 1e a ρ ( τ ) + e b ρ ( τ ) ζ (58)with ρ = ± , ±
2. The coefficients of the function in the denominator are taken as exponentials in order to guaranteethe positivity of the reaction constants, the friction coefficient ζ being always non-negative. The coefficients e a ρ ( τ ) can be determined in the low-friction regime and the coeffcients e b ρ ( τ ) in the high-friction regime.The dependence on the external torque τ appears in the functions a ρ ( τ ) and b ρ ( τ ), which are taken as expansionsin powers of τ limited to the second order: a ρ ( τ ) = a (0) ρ + a (1) ρ τ + a (2) ρ τ + O ( τ ) (59) b ρ ( τ ) = b (0) ρ + b (1) ρ τ + b (2) ρ τ + O ( τ ) (60)with ρ = 1 ,
2. The coefficients of these expansions are fitted in intervals of values of the external torque whichare typically | τ | <
20 pN nm. The values of the coefficients of Eqs. (59)-(60) for the reaction constants k +1 ( ζ, τ ), k +2 ( ζ, τ ), and k − ( ζ, τ ) are given in Table II.The last constant for ATP unbinding is finally obtained by using Eq. (53) as k − ( ζ, τ ) = k +1 ( ζ, τ ) k +2 ( ζ, τ ) k − ( ζ, τ ) exp 1 k B T (cid:18) ∆ G − π τ (cid:19) (61)with ∆ G = −
50 pN nm.
IV. PROPERTIES OF THE F MOTOR
The comparison between the discrete and continuous descriptions sheds a new light on the properties of the F motor. On the one hand, the continuous-angle model reproduces the experimental observations of Ref. [7] and its1simulation can test the assumptions of the discrete-state model, in particular, the assumption of tight coupling. Onthe other hand, the discrete-state model can be treated analytically thanks to its simplicity. In this way, the fittingof the discrete-state model to the continuous one allows us to investigate more closely the properties of the F motorin the regime of validity of the discrete-state model. A. Tight versus loose chemomechanical coupling
In order to determine the regime of tight coupling between the chemistry and the mechanics of the F motor, boththe angular velocity V and the ATP consumption rate R have been simulated with the continuous-angle model (54) fordifferent values of the external torque τ and chemical potential difference ∆ µ , which are the corresponding affinities.We depict in Fig. 1 the curves in the plane ( τ, ∆ µ ) where either the velocity vanishes, V = 0, or the ATP consumptionrate vanishes, R = 0. The value of the external torque where the velocity vanishes is the so-called stalling torque.Since the variations of the chemical potential difference ∆ µ can only be one dimensional in such a representation, wetake a combined variation of the concentrations of the difference species as explained in the caption of Fig. 1. Thetwo curves V = 0 and R = 0 intersect at the origin ( τ = 0 , ∆ µ = 0) which is the thermodynamic equilibrium point.We notice that the curve V = 0 is above the curve R = 0 in the plane of the chemical potential difference ∆ µ versusthe torque as it should according to the second law of thermodynamics (20). V = 0 R = 0tight couplingtorque (pN nm) Dm k B T ln10 V > 0 V < 0 R < 0 R > 0 FIG. 1: Chemical potential difference ∆ µ in units of k B T ln 10 versus the external torque τ for the situations where the rotationrate V (circles) and the ATP consumption rate R (squares) vanish in the continuous model (54) and compared with thestraight line ∆ µ = − πτ / A = 0. The concentrations are fixed accordingto [ATP] = 4 . . a − M and [ADP][P i ] = 10 − . a − M, in terms of the quantity a = ∆ µ/ ( k B T ln 10). The bead attachedto the γ -shaft has the diameter d = 2 r = 80 nm and the temperature is of 23 degrees Celsius. The torque where V = 0 iscalled the stall torque. The determination of the curves V = 0 and R = 0 is difficult close to the thermodynamic equilibriumpoint ( τ = 0 , ∆ µ = 0) because both the rotation rate V and the ATP consumption rate R are very small in this region, whichexplains the absence of dots close to the origin. In the tight-coupling regime, the condition (21) should hold, which implies that the mechanical and chemicalaffinities τ and ∆ µ are no longer independent but should combined into the unique chemomechanical affinity (23). Inthis regime, the vanishing of the rates, V = 0 and R = 0, should thus occur on the curve where the chemomechanicalaffinity (23) vanishes, i.e., along the straight line ∆ µ = − π τ (62)This is observed in Fig. 1 for values of the external torque extending from zero down to about τ ≃ −
30 pN nmand for chemical potential difference from zero up to ∆ µ ≃ k B T , which delimits the zone where the tight-couplingassumption is satisfied. Outside this zone for higher values of ∆ µ or lower values of τ , the F motor is no longerfunctioning in the tight-coupling regime and the chemomechanical coupling becomes loose [24].2Since the coincidence of the curves V = 0 and R = 0 along the straight line (62) is a feature of the discrete-statemodel (27), we may expect it describes the F motor in the aforementioned zone of tight coupling. B. Rotation rate versus ATP concentration
Random trajectories of the discrete model can be simulated thanks to Gillespie’s numerical algorithm [25, 26].Examples of random trajectories are depicted in Fig. 2 for different values of ATP concentration, illustrating theMichaelis-Menten kinetics described by Eq. (42). At low concentrations [ATP] ≪ K M ≃ µ M, the motor isessentially waiting for the arrival of new ATP molecules with its γ -shaft at some angle 120 ◦ n ( n integer). Instead,at high concentrations [ATP] ≫ K M , the rotation is limited by the release of the hydrolysis products ADP and P i from its catalytic sites with its γ -shaft at some angle 90 ◦ + 120 ◦ n ( n integer), as clearly seen in Fig. 2. In thisrespect, the random trajectories of the discrete model reproduce the jumps by 90 ◦ and 30 ◦ corresponding to thesubsteps experimentally observed in Ref. [7]. We notice, however, that the discrete model cannot reproduce thesmall-amplitude fluctuations around the dwell angles as the continuous-angle model does [12].The crossover between the regime at low ATP concentration and the saturation regime of the Michaelis-Mentenkinetics is seen in Fig. 3 where we directly compare Eq. (42) of the discrete model with experimental data from Ref.[7]. At low ATP concentrations, the rotation rate is proportional to the ATP concentration while the rotation ratereaches its maximum value of about 130 rev/s in the saturation regime.In order to appreciate the nonequilibrium thermodynamics of the molecular motor, it is interesting to depict therotation rate as a function of the affinity (23) instead of the ATP concentration. Indeed, the former is a substituteof the latter, as shown by expressing the concentrations in terms of the chemical potentials by Eq. (3) and using thedefinitions of the chemomechanical affinity (23) and of the equilibrium constant (37) to get[ATP] = K eq [ADP][P i ] e A (63)In this way, we recover the equilibrium relation (36) between the concentrations with Eq. (37) in the thermodynamicequilibrium state A = 0 corresponding to given concentrations of ADP and P i . Substituting Eq. (63) into Eq. (38),we obtain the following expression for the rotation rate: V = V max (cid:0) e A − (cid:1) e A − V max L (64)with the constant L ≡ V max K eq [ADP][P i ]( K eq + K P )[ADP][P i ] + K M (65)This coefficient controls the linear response of the molecular motor because V ≃ ( LA for A ≪ V max for A ≫ A in a highly nonlinear way,in contrast to what is often supposed. The nonlinear dependence is very important as observed in Fig. 4. The linearregime extends around the thermodynamic equilibrium point at ∆ µ = 0 where the function V ( A ) is essentially flatbecause the linear-response coefficient takes the very small value L ≃ − s − . Since the affinity is about A ≃ . ≃ − M, [ADP] ≃ − M, and [P i ] ≃ − M [14], the rotation ratewould take the extremely low value V ≃ LA/ ≃ . A allows the rotation rate to reach the maximumvalue V max ≃
130 rev/s under physiological conditions.
C. Rotation rate versus friction
In Fig. 5, we show the effect of friction on the motor’s velocity in the absence of ADP or P i . At low friction, thevelocity saturates exhibing the reaction-limited regime. At high friction, we see the rapid decrease of the velocitywith increasing friction in the friction-limited regime [12]. In this latter regime, the velocity decreases as the inverseof the friction coefficient V ∝ ζ − in consistency with the analytical form (58) given to the reaction constants. A3 r o t a ti on (r e v ) time (s) d = 40 nm [ATP] = 2 mM[ATP] = 20 m M[ATP] = 2 m M FIG. 2: Simulation of random trajectories of the discrete-state model for [ADP][P i ]=0, a temperature of 23 degrees Celsius, abead of diameter d = 2 r = 40 nm, and a zero external torque. -8 -7 -6 -5 -4 -3 -2 exp. (40 nm bead)theory r o t a ti on r a t e (r e v / s ) [ATP] (M) FIG. 3: Mean rotation rate of the γ -shaft of F in revolutions per second, versus the ATP concentration [ATP] in mole per literfor [ADP][P i ]=0. In accordance with the experimental setup [7], the diameter of the bead is d = 2 r = 40 nm, the temperatureis of 23 degrees Celsius, and the external torque is zero. The circles are the experimental data from Ref. [7]. The solid line isthe result of numerical simulation of the two-state model. -200204060801001201400 4 8 12 16 20 24[ADP][P i ] = 10 -6 M [ADP][P i ] = 10 -7 M [ADP][P i ] = 10 -8 M r o t a ti on r a t e (r e v / s ) d = 40 nm Dm k B T FIG. 4: Mean rotation rate versus the chemical potential difference ∆ µ in units of the thermal energy k B T . The thermodynamicequilibrium corresponds to ∆ µ = 0. The ATP concentration is given in terms of the chemical potential difference by [ATP] =[ADP][P i ] exp[(∆ µ − ∆ µ ) / ( k B T )] ≃ . − M − [ADP][P i ] exp[∆ µ/ ( k B T )] since ∆ µ = − ∆ G = 50 pN nm. The results ofthe discrete model (solid lines) are compared with the continuous model (dots) for three different values of [ADP][P i ]. Thediameter of the bead is d = 2 r = 40 nm, the temperature 23 degrees Celsius, and the external torque zero. [ATP] = 2 mM (theory)[ATP] = 2 mM (beads)[ATP] = 2 mM (filaments)[ATP] = 2 m M (theory)[ATP] = 2 m M (beads)[ATP] = 2 m M (filaments) r o t a ti on r a t e (r e v / s ) viscous friction (pN nm s) FIG. 5: The mean rotation rate in revolutions per second versus the viscous friction coeffcient ζ for a bead, a bead duplex, ora filament attached to the γ -shaft. The circles and triangles are the experimental data in Fig. 2 of Ref. [7] at [ATP] = 2 mM(squares and downward triangles) and [ATP] = 2 µ M (circles and upward triangles). The squares and circles correspond to thesingle beads and bead duplexes, the triangles to the actin filaments. The solid lines are the results of the present model with[ADP][P i ] = 0, the temperature of 23 degrees Celsius, and zero external torque. -8 -7 -6 -5 -4 -3 -2 -1 [ATP] = 10 -5 M[ATP] = 10 -6 M[ATP] = 10 -7 M r o t a ti on r a t e (r e v / s ) [ADP] (M) d = 40 nm [ P i ] = 10 -3 M FIG. 6: Mean rotation rate of a bead of radius r = 20 nm attached to the γ -shaft in revolutions per second versus [ADP](in mole per liter) and varying ATP concentrations. The concentration of P i is 10 − M and the temperature of 23 degreesCelsius, and the external torque vanishing. The simulations of the continuous-angle model (squares, diamonds, and circles) arecompared with the analytical expression (38) for the discrete model of Table II (lines). similar behavior is observed at [ATP] = 2 mM in the saturation regime of the Michaelis-Menten kinetics and at ATPconcentrations lower than the Michaelis-Menten constant for [ATP] = 2 µ M < K M = 17 µ M. The agreement is asgood as for the continuous-angle model [12].
D. Rotation in the presence of ATP hydrolysis products
In the presence of ADP and P i in the environment of the motor, Eq. (38) shows that the rotation rate decreases,as expected since these products tend to counteract ATP hydrolysis that is powering the motor. This phenomenon isknown as ADP inhibition [7, 14]. There are two possible causes of the decrease of rotation rate if the concentrationsof ADP and P i are positive: (1) the term in the numerator of Eq. (38) where the equilibrium constant K eq multiplies[ADP][P i ]; (2) the term in the denominator of (38) where the constant K P multiplies [ADP][P i ]. As explained in5 -10-50510 -40 -20 0 20 40[ATP] = 4.9 10 M -7 V (continuous) R /3 (continuous) V (discrete) r a t e s ( s - ) torque (pN nm) (a) -20-10010203040 -40 -20 0 20 40[ATP] = 4.9 10 M -6 V (continuous) R /3 (continuous) V (discrete) r a t e s ( s - ) torque (pN nm) (b) FIG. 7: Mean rotation rate V of the γ -shaft of F in revolutions per second versus the external torque for (a) [ATP] = 4 . − Mand (b) [ATP] = 4 . − M (circles for the continuous-state model and solid line for the discrete-state model). The otherconcentrations are fixed to [ADP] = 10 − M and [P i ] = 10 − M. The squares show the reaction rate R divided by three in orderto display the regime of tight coupling where V = R/
3. The diameter of the bead is d = 2 r = 160 nm and the temperature of23 degrees Celsius. subsection II D and can be checked with Table II, the inequality (45) holds by several orders of magnitude so that themain cause of the decrease of the rotation rate is the term in the denominator of Eq. (38) due to the reaction ρ = − i to the catalytic sites of the F motor.As observed in Fig. 6 which compares the continuous and discrete models, this effect manifests itself above millimolarconcentrations of ADP if inorganic phosphate is in millimolar concentration. We notice that the decrease of therotation rate goes as the inverse of the concentrations of ADP and P i , V ∝ ([ADP][P i ]) − as described by Eq. (38)given the fact that the inequality (45) holds. E. Dependence on the external torque
Figure 7 depicts the dependence of the rotation and ATP consumption rates on the external torque τ for both thecontinuous-angle and discrete-state models, showing their agreement in the range of validity of tight coupling. Thisrange of validity is observed in Fig. 1 to extend down to τ ≃ −
30 pN nm. The reason is that the mechanical motionbecomes decoupled from the chemical reactions if the external torque is too much tilting the internal free-energypotential surfaces of the motor. For vanishing or moderate values of the external torque, the potential surfaces havebarriers which stop mechanical motion and allow for jumps between potential surfaces corresponding to differentchemical states of the molecular motor, coupling in this way the mechanical motion to the chemical reactions. Beyond6some threshold for the external torque, these barriers disappear causing the decoupling between mechanics andchemistry [12]. This decoupling may happen at both negative and positive thresholds for the external torque, as seenin Fig. 7 for the continuous-angle model. The tight-coupling condition V = R/ | τ | <
30 pN nm for the continuous-angle model, although it always holds by assumption for the discrete-state model.Therefore, the range of validity of the discrete model is restricted to the interval | τ | <
30 pN nm of values of theexternal torque, as already discussed about Fig. 1.Figure 7 also shows the remarkable feature of the discrete model to reproduce the phenomenon of stalling torquethat the mean rotation rate can be stopped at a negative critical value of the external torque opposing the rotationalmotion. Indeed, the value of the external torque where both the rotation and ATP consumption rates vanish canbe obtained in the discrete model by Eq. (64), which shows that the chemomechanical affinity should vanish at thestalling torque. Whereupon, we recover the condition (62) giving the stalling torque as τ stall = − π ∆ µ = − π (cid:18) ∆ µ + k B T ln [ATP][ADP][P i ] (cid:19) (67)in the tight-coupling regime. For [ADP][P i ] = 10 − M , the chemical potential difference takes the values ∆ µ =56 . µ = 65 . . − M and [ATP] = 4 . − M. Hence, Eq. (67)gives respectively the stalling torques τ stall = − . τ stall = − . F. Chemical and mechanical efficiencies
Under a negative external torque τ <
0, the F motor can synthesize ATP, in which case the ATP consumption rateas well as the rotation rate are negative, R <
V <
0. In this regime of ATP synthesis, the chemical efficiency isdefined as the ratio of the free energy stored in the synthesized ATP over the mechanical power due to the externaltorque: η c ≡ − ∆ µ R πτ V (68)such that 0 ≤ η c ≤ η m ≡ − πτ V ∆ µ R (69)The mechanical efficiency satisfies 0 ≤ η m ≤ V >
R > R = 3 V , these conditions coincide and the chemicaland mechanical efficiencies (68) and (69) becometight coupling: η c = 1 η m = − µ πτ (70)in agreement with Eq. (A27) derived in the Appendix from the assumption of linear response. In the tight-couplingregime, the chemical and mechanical efficiencies can reach the maximal unit value at the stalling torque where Eq.(67) holds. This remarkable result is observed in Fig. 8 depicting the chemical and mechanical efficiencies versus theexternal torque under conditions corresponding to the chemical potential difference ∆ µ = 56 . τ stall = − . < τ <
0, the external torque is opposed to the mean rotation rate but themotor consumes ATP, so that the mechanical efficiency (69) is postive. In this interval, the coupling is tight so thatthe mechanical efficiency computed with the continuous-angle model (squares) agrees with the prediction (70) of tightcoupling which holds for the discrete model. Indeed, the mechanical efficiency reaches the unit value at the stallingtorque marked by the vertical line.On the other hand, the motor synthesizes ATP under the action of the external torque for τ < τ stall = − . e ff i c i e n c i e s torque (pN nm) [ATP] = 4.9 10 M -7 [ADP] = 10 M -4 [P i ] = 10 M -3 h c h m d = 160 nm FIG. 8: Chemical efficiency (68) and mechanical efficiency (69) versus the external torque τ in the continuous-state model(respectively circles and squares joined by a solid line) and compared with the prediction (70) of tight coupling (dashedlines). The vertical solid line indicates the stalling torque at τ = − . . − M,[ADP] = 10 − M, and [P i ] = 10 − M. The diameter of the bead is d = 2 r = 160 nm and the temperature of 23 degrees Celsius.These conditions are identical as in Fig. 7a. The predictions of tight coupling (dashed lines) are respectively η c = τ stall /τ for τ < τ stall , and η m = τ /τ stall for τ stall < τ <
0, with τ stall = − . V. CONCLUSIONS
In the present paper, we have studied one of the simplest possible stochastic processes describing the stochasticchemomechanics of the F -ATPase molecular motor, as observed in Ref. [7]. This description considers the twodiscrete states corresponding to the steps and substeps observed in the rotary motion of the γ -shaft of F [7]. Thistwo-state discrete model is based on the master equation ruling the time evolution of the probabilities of the twodiscrete states. The model is analytically solvable, which provides us with an interesting insight in our understandingof the behavior of the F motor. The discrete model is set up by fitting its rate coefficients to simulations with ourpreviously continuous-angle model [12]. Since this latter was itself fitted to the observations of Ref. [7], both thediscrete and continuous models describe the same experimental system.The comparison between the discrete-state and continuous-angle descriptions reveals important properties of the F motor. Indeed, the discrete-state model presupposes that the mechanical motion and the chemical reactions poweringthe motor are tightly coupled, although the continuous-angle model does not. Accordingly, the comparison betweenboth models clearly reveal the regime of tight coupling. Thanks to the analytic tractability of the discrete-state model,the consequences of the transition between the tight- and loose- coupling regimes can be understood for the kineticand thermodynamic properties of the F motor.An important difference between the continuous and discrete models is that mechanical properties such as thefriction of the object attached to the γ -shaft in the liquid surrounding the motor or the external torque acting on theshaft explicitly appear in the Fokker-Planck equations defining the continuous model, but does not appear in the masterequation of the discrete model. Indeed, this latter is defined by its transition rates and its rate constants although theformer also contains a biased diffusive part involving both the friction coefficient ζ and the external torque τ . In orderto describe these mechanical properties in the discrete-state model, we thus have to give a dependence on the frictioncoefficient ζ and the external torque τ in the rate constants, which are also the the constants of the reactions of the F motor with ATP, ADP, and P i . Accordingly, the discrete-state model is essentially a model of the chemical kineticsof the F motor with reaction constants k ρ ( ζ, τ ) depending on mechanical parameters. This dependence is fitted tonumerical simulations of the continuous-angle model. This fitting is performed in the tight-coupling regime wherethe discrete-state model is supposed to correspond to the continuous-angle model. In this way, the two-state model iscompleted by including the dependences on both the chemical and mechanical control parameters. The random timeevolution of the discrete two-state model can be simulated thanks to Gillespie’s numerical algorithm [25, 26].In the discrete-state model, the tight coupling between mechanics and chemistry implies that the motor is driven outof equilibrium by the unique chemomechanical affinity (23) combing the external torque with the chemical potentialdifference of ATP hydrolysis. Thermodynamic considerations shows that the chemical equilibrium is displaced by theexternal torque acting on the motor if this latter strictly functions in the tight-coupling regime.Thanks to the solvability of the two-state model, the stationary solutions of the master equation can be exactly8deduced, allowing us to show analytically that the mean rotation rate obeys a Michaelis-Menten kinetics with respectto the ATP concentration. Moreover, the analytical formula (38) is obtained for the rotation rate in the presence ofboth ATP and the products of ATP hydrolysis, i.e., ADP and P i .The highly nonlinear dependence of the mean rotation rate of the γ -shaft (64) on the chemomechanical affinity(23) shows that the F motor is not functioning in the linear-response regime defined by Onsager’s linear-responsecoefficents, but instead typically runs in a nonlinear-response regime which is more the feature of far-from-equilibriumsystems than of close-to-equilibrium systems. This remarkable property allows the motor to rotate under physiologicalconditions at about 130 rev/s although it would be a million times slower if the motor was functioning in a linear-response regime.Furthermore, the crossover between the reaction-limited and friction-limited regimes is well described by the two-state model as the friction coefficient ζ is increased. Although, the mean rotation rate is nealry independent of thefriction coefficient in the reaction-limited regime at low friction, it decreases as the inverse of the friction coefficientat high friction because the reaction rates (58) fitted to the continuous-angle model appropriately take this featureinto account.We have also investigated the behavior of the F motor in a surrounding filled with ADP and inorganic phosphate.As shown by Eq. (38), the mean rotation rate decreases as the concentrations of ADP and P i exceed a crossovervalue. The reason is that the release of the products of ATP hydrolysis in the motor is counteracted by the reversereaction of binding of the products to the catalytic sites, which has the effect of slowing down the motor.The dependence of the rotation and ATP consumption rates on the external torque is also of great interest becauseit reveals the interval of values of the external torque where the tight-coupling condition holds. It is in this regime thatthe discrete-state model provides a good description of the motor. This comparison shows that the stalling torquewhere the rotation and ATP consumption rates vanish is given in terms of the chemical potential difference of ATPhydrolysis and thus depend on the concentrations of the involved species according to Eq. (67).Moreover, the nonequilibrium thermodynamics of the F motor has been developed, allowing us to study thechemical and mechanical efficiencies defined in Ref. [27]. In the tight-coupling regime, these efficiencies can reachtheir maximal unit value near the stalling torque. The coupling between mechanics and chemistry is so tight thatthe expected deviations between the stalling torque and the values of the torque where the efficiencies reach theirmaximum value turn out to be very small. The coincidence of these different values of the external torque is theconsequence of the high degree of tight coupling achieved in the F motor, but would not necessarily hold for a motorfunctioning in a loose-coupling regime.In conclusion, the two-state model and its comparison with the continuous-angle model of our previous paper [12]is providing a powerful method to study the kinetic and thermodynamic properties of the F motor and, especially,the coupling between its mechanics and chemistry. Acknowledgments.
This research is financially supported by the “Communaut´e fran¸caise de Belgique” (contract“Actions de Recherche Concert´ees” No. 04/09-312), by the “Fonds pour la Formation `a la Recherche dans l’Industrieet l’Agriculture” (F. R. I. A. Belgium), and the National Fund for Scientific Research (F. N. R. S. Belgium, contractF. R. F. C. No. 2.4577.04).
APPENDIX A: THERMODYNAMIC RELATIONS BETWEEN AFFINITIES AND CURRENTS
In this appendix, the nonequilibrium thermodynamics of the F motor is presented in the linear regime very closeto the equilibrium.
1. The general case
The molecular motor can be driven out of equilibrium by the two independent affinities which are the mechanicaland the chemical affinities proportional respectively to the torque τ and the chemical potential difference ∆ µ . Theseaffinities are the corresponding fluxes or currents can be defined as A m ≡ π τk B T ↔ J m ≡ V (A1) A c ≡ ∆ µk B T ↔ J c ≡ R (A2)9in which case the thermodynamic entropy production (20) takes the following form:1 k B d i Sdt = A m J m + A c J c ≥ J m = J m ( A m , A c ) (A4) J c = J c ( A m , A c ) (A5)which vanish at the thermodynamic equilibrium where the affinities vanish, A m = A c = 0. Close to equilibrium, thecurrents can be expanded in powers of the affinities, which defines the linear response coefficients L ij as J m = L A m + L A c + O (2) (A6) J c = L A m + L A c + O (2) (A7)The microreversibility implies the Onsager reciprocity relation: L = L (A8)and the non-negativity of the entropy production (A3) the inequality L L ≥ L with L , L ≥ A m , A c ), the curve where the velocity vanishes behaves around the origin as J m = 3 V = 0 : A c = − L L A m + O ( A ) (A10)while the curves where the ATP consumption rate vanishes is given by J c = R = 0 : A c = − L L A m + O ( A ) (A11)Near the origin, the curve (A10) has therefore a more negative slope than the curve (A11) as the consequence of theinequality (A9) resulting from the second law of thermodynamics [27].
2. The case of tight coupling
In the case of a tight coupling between the mechanics and the chemistry of the molecular motor, these two curveshave the same slope around the thermodynamic equilibrium point. Indeed, the tight-coupling condition (21) readstight coupling: J m = J c ≡ J (A12)so that the entropy production becomestight coupling: 1 k B d i Sdt = AJ ≥ A ≡ A m + A c (A14)Therefore, the mechanical and chemical affinities are no longer independent in the tight-coupling regime where thechemomechanical affinity (A14) becomes the unique nonequilibrium driving force. Accordingly, the unique current(A12) is a function of this unique affinity: J = J ( A ) = LA + O ( A ) (A15)0and all the linear response coefficients become related to each other by L ≡ L = L = L = L (A16)In this case, the inequality (A9) reaches the equality: L L = L (A17)which is also characteristic of tight coupling.
3. The efficiencies
The chemical and mechanical efficiencies (68)-(69) can be expressed as follows in terms of the affinities and thecurrents: η c = − A c J c A m J m = 1 η m (A18)In the regime of linear response where the currents are the linear functions (A6)-(A7) of the affinities, the efficienciescan be written in the form η c = − − εα α + 1 α + 1 − ε = 1 η m (A19)in terms of the coefficient α ≡ L L A m A c (A20)and the constant ε ≡ − L L L (A21)such that 0 ≤ ε ≤ ≤ η max ≡ − √ ε √ ε ≤ A m , A c ) of the affinities with respect to the curves where the rotationand ATP consumption rates vanish since [27] η m = η max if α = − √ ε (A23) J m = 3 V = 0 if α = − ε (A24) J c = R = 0 if α = − η c = η max if α = − − √ ε (A26)In the tight-coupling limit, the coefficient (A21) vanishes because of Eq. (A17), ε = 0, so that the chemical andmechanical efficiencies take the value η c = 1 η m = − α = − A c A m (A27)according to Eqs. (A19) and (A16). In this case, the four conditions (A23)-(A26) coincide in α = − η max = 1. Therefore, tightcoupling favors the optimization of the efficiencies. [1] B. Alberts, D. Bray, A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter, Essential Cell Biology (Garland Publishing,New York, 1998). [2] H. Wang and G. Oster, Nature , 279 (1998).[3] J. P. Abrahams, A. G. W. Leslie, R. Lutter, and J. E. Walker, Nature , 621 (1994).[4] R. Ian Menz, J. E. Walker, and A. G. W. Leslie, Cell , 331 (2001).[5] H. Noji, R. Yasuda, M. Yoshida, and K. Kinosita Jr., Nature , 299 (1997).[6] K. Adachi, K. Oiwa, T. Nishizaka, S. Furuike, H. Noji, H. Itoh, M. Yoshida, and K. Kinosita Jr., Cell , 309 (2007).[7] R. Yasuda, H. Noji, M. Yoshida, K. Kinosita Jr., and H. Itoh, Nature , 898 (2001).[8] H. Itoh, A. Takahashi, K. Adachi, H. Noji, R. Yasuda, M. Yoshida, and K. Kinosita Jr., Nature , 465 (2004).[9] K. Shimabukuro, R. Yasuda, E. Muneyuki, K. Y. Hara, K. Kinosita Jr., and M. Yoshida, Proc. Natl. Acad. Sci. U. S. A. , 14731 (2003).[10] G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems (Wiley, New York, 1977).[11] C. W. Gardiner,
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