ATP synthase is the universal enzyme that manufactures ATPfrom ADP and phosphate by using the energy derived from atransmembrane protonmotive gradient. It can also reverse itselfand hydrolyse ATP to pump protons against an electrochemicalgradient. ATP synthase carries out both its synthetic and hydro-lytic cycles by a rotary mechanism1±4. This has been con®rmed inthe direction of hydrolysis5,6 after isolation of the soluble F1
portion of the protein and visualization of the actual rotation ofthe central `shaft' of the enzyme with respect to the rest of themolecule, making ATP synthase the world's smallest rotaryengine. Here we present a model for this engine that accountsfor its mechanochemical behaviour in both the hydrolysing andsynthesizing directions. We conclude that the F1 motor achievesits high mechanical torque and almost 100% ef®ciency because itconverts the free energy of ATP binding into elastic strain, whichis then released by a coordinated kinetic and tightly coupledconformational mechanism to create a rotary torque.
The general structure of ATP synthase is shown in Fig. 1. Itconsists of two portions: a soluble component, F1, containing thecatalytic sites, and a membrane-spanning component, Fo, compris-ing the proton channel. The entire FoF1 structure is arranged as acounter-rotating `rotor' and `stator' assembly. The stator portionconsists of the catalytic sites contained in the a3b3 hexamer,together with subunits a, b2 and d. The rotor consists of 9±12c-subunits arranged in a ring and connected to the g- and e-subunits that form the central `shaft'. The mechanism of proton-motive force transduction has been discussed elsewhere7; we nowfocus on the mechanochemistry of F1, which synthesizes ATP whenthe g±e shaft is turned clockwise (viewed from the membrane) bythe protonmotive force, and which rotates counterclockwise topump protons when hydrolysing ATP.
Any mechanism proposed to explain the experimental results isstrongly constrained by thermodynamic, kinetic, structural andmechanical considerations. In the hydrolysis direction, it mustreproduce the large torque and almost 100% mechanicalef®ciency5,6; in the synthesis direction, ATP must be produced atthe measured rate when driven by torques generated by the proton-motive force in Fo. Kinetic constants and free energies mustcorrespond to measured values. A model must also be geometricallycompatible with the detailed structural information now available.
We constructed the model in three steps. First, kinematic motionsof the F1 subunits were analysed to determine the conformationalchanges that drive rotation of the g-subunit. From this, we con-structed a mathematical model describing the principal mechanicaland electrostatic interactions between the subunits. Finally, thereaction sequences were incorporated to create a completemechanochemical model. The outline of this model can besummarized as follows.
The reactivity of a catalytic site depends on its local conformationand composition. The unique feature of F1 is that the reactivity ofeach site depends on the states of the other two sites, and theposition of g. Each of the three catalytic sites, b1, b2 and b3, passesthrough at least four chemical states:
EEmpty
Ãÿÿÿÿ!ATP
binding
TATP
bound
Ãÿÿÿÿ!HydrolysisD×P
ADPand Pibound
Ãÿÿÿÿ!Pi
releaseD
ADPbound
Ãÿÿÿÿ!ADP
releaseE
Empty�1�
We denote by bi the chemical state (E, T, D×P, D) of the ithb-subunit, so that a3b3 has a total of 43 � 64 possible kinetic states.The collection of these states can be visualized as the integer pointsin a 4 3 4 3 4 cube (actually, a three-dimensional torus, becausewhen the state exits one face of the cube, it re-enters the oppositeface). The progression of the system through this chemical-statespace depends on the angular position of the g-subunit, which wedenote by v. The kinetics can be described by a set of equations (thatis, a Markov model) of the form
dbi=dt � K�v; bi 2 1; bi�1� 3 bi �2�
where i � 1; 2; 3 and where K is the matrix of kinetic transition ratesfor a catalytic site, which depends on the angular position of g andthe occupancy of the other sites. Thus the state of F1 is speci®ed byits rotational position and its kinetic state: (v, b1, b2, b3).
The motion of g is described by balancing the torques on therotating g-subunit:
zdv
dt|{z}Viscous torque on g
� t�v; b1; b2; b3�|���������{z���������}Torque between g and a3b3
2 TL|{z}Load torque
� TB�t�|�{z�}Brownian torque
�3�
The viscous load on the rotating g-subunit is contained in the dragecoef®cient, z. TL represents any additional load torque (such as froma laser trap). TB(t) is the brownian torque due to thermal ¯uctua-tions. The driving torques exerted on g by the b-subunits can beexpressed as the gradient of an elastic potential: t�v; b1; b2; b3� �
2 ]V�v; b1; b2; b3�=]v. This potential is derived from the geometry
and elastic properties of the mechanical model for F1. We con-structed this mechanical model by the following procedure.
First, we analysed the kinematic motions of F1. To do this, weimported the coordinates for a3b3g from the Protein Data Bankinto the molecular structure program Rasmol1,8. According toBoyer's binding change mechanism, a 2p/3 rotation of g corre-sponds to a 2p/3 advance in the kinetic states of the three catalyticsites and a 2p/3 advance of the conformational asymmetry of thea3b3 hexamer. This allows us to derive the coordinates of F1 forv � 2p=3 from those at v � 0. For angles between 0 and 2p/3, thecoordinates of F1 are calculated by interpolation in a cylindricalcoordinate system. From this we created stereo animations of thecomplete motion of F1 through the hydrolysis cycle (see Supple-mentary Information). Although the interpolated positions obey allsteric constraints, they are probably not exact; nevertheless, they dogive an idea of the overall motion of the subunits.
Next, we placed a local coordinate system on each subunit so thatwe could determine quantitatively the subunit's sequence of con-formations. A study of the Protein Motions Database shows thatmost conformational changes can be resolved into hinge and shearmotions9,10. The principal motion of the b-subunits consists of ahinge motion that bends the top and bottom segments towards oneanother by about 308 (Fig. 2), and two much smaller motions whichwe ignore: a radial shear motion and rotation in the v direction. Thea-subunits undergo very little change in shape.
To see how the vertical bending motion is converted into a rotarytorque on the g-subunit, we took horizontal `serial sections'through the a3b3g assembly to determine the cross-sectionalshapes of g and the central channel in a3b3 into which g ®ts.Combining this information, we deduced that the curved andasymmetric g-subunit is driven by the hinge motion of the
b-subunits in a fashion analogous to a hand on the crank of a carjack (see Supplementary Information).
We now make an important assumption about how the freeenergy from ATP hydrolysis is converted into the hinge-bendingmotion of b: the energy to bend a b-subunit is conferred on thecatalytic site at the nucleotide-binding step in equation (1); that is,the rapid sequential bonding of the nucleotide to the catalytic site isconverted into an elastic strain energy of ,24 kBT (14 kcal mol-1)centred at the catalytic site. Binding may proceed in several stages:for example, weak binding, followed by a rapid thermal `zippering'of (mostly hydrogen) bondsÐprimarily to the phosphate groups.However, for simplicity we treat the binding as a single kinetic stepin equation (1). This strain energy tends to bend the b-subunit, andthe bending stress is converted by the protein's geometry into arotary torque on the g-subunit and to stresses on the other catalyticsites (Fig. 2).
Viewed as an elastic body, the a3b3 hexamer is complicated, andcertainly not isotropic. However, we can construct a simpli®edmechanical equivalent that captures its principal motions11,12.In this way, the complicated elastic behaviour of F1 can beapproximated by an assembly of rigid elements connectedby springs (Fig. 2b).
We treat the a3b3 hexamer as an elastic body with a pre-stressarising from the free energy of assembly. First, we model theintrinsic elasticity of the b-subunits as passive elastic elements atthe hinge points. Second, we represent the active stress created bynucleotide binding as much stronger elastic elements that areactivated upon nucleotide binding. Third, by assigning reasonableelastic constants to these elements (Box 1), we can compute elasticpotential energies based on the simpli®ed molecular structureshown in Fig. 2, one potential curve for each of the 64 kinetic
states. The derivatives of these elastic potentials yield the torquesthat drive the rotation of g, as given in equation (3).
The exact procedure for constructing the potentials is given in theSupplementary Information. Roughly, this consists of measuringthe bending motions of a b-subunit by the azimuthal angle, f,between its upper and lower portions. During the course of itsconformational change, a b-subunit closes and opens its azimuthalangle by about Df < 308. Trigonometry yields a relationshipbetween the bending angle, f, of a b-subunit and the rotationangle of g : v � f �f�. As the bending angle f closes, b pushes on theoff-axis bowed segment of g (Fig. 2). In this way, a rotary torque inthe v direction is generated by the geometric propagation of stressfrom the catalytic site (the active spring) to g. As each of thesubunits cycles through its conformational range in sequence, acontinuous rotational torque is exerted on g.
The unisite (that is, at substoichiometric ATP concentration) rateconstants for the transitions given in equation (1) have beenmeasured13. But, at steady state the reactions at the three catalyticsites are coordinated in two ways. First, the strain induced bynucleotide binding at each site is communicated to the other twosites. Second, there are at least two `switch points' on g that interactwith speci®c sites on the a3b3 hexamer14. Thus, the activationenergy barriers for the reaction at each site are in¯uenced by thestate of the other two sites and the angular position of the g-subunit.
Figure 3a shows the free-energy levels of a b-subunit as the systemevolves through the 64 possible chemical states. Note that, asmeasured externally, only part of the free energy of ATP is releasedto the surroundings upon binding: the remainder is stored intern-ally as elastic deformation and released when phosphate is released.The switch denoted by S1 in Figs 2 and 3 is located at gGln 269 and
bThr 304 (Escherichia coli sequence numbering); it activates bind-ing of nucleotide successively to each catalytic site. The switch atgArg 242 and bGlu 381 of the conserved DELSEED sequence in theb-subunit (S2 in Figs 2, 3) activates release of phosphate at each site.The interactions between the g- and the b-subunits at the switchregions are electrostatic and are communicated to the catalyticsites to produce entropic and steric effects that do not involveappreciable changes in free energy.
From the con®gurational geometry and elastic constants, we cancompute the sequence of elastic potentials the g-subunit experi-ences from a b-subunit as it undergoes its cycle of nucleotidebinding, hydrolysis and product release (Fig. 3b). The elasticenergy curves of the other two b-subunits are identical to thoseshown, but shifted by 2p/3 and -2p/3 respectively. The potentialexperienced by g from the three driving b-subunits is the sum ofthese three sets of potentials. In Fig. 3b, this is represented by thethree connected points (offset by 2p/3) on one set of potentials. Atone site, the cycle proceeds as indicated by the numbers 1±5,starting where the empty site binds ATP. This drops the statefrom point (1) on E to point (2) on T; this transition is triggeredby S1. The exact angle, v, at which ATP binds is stochastic; thevertical arrow in Fig. 3b indicates a typical transition point; thebinding of nucleotide strains the catalytic site, and release of thisstress through the mechanical escapement shown in Fig. 2 rotatesthe g shaft from �2� ! �3�. This is the primary power stroke.Approximately 1808 opposite S1 on g is the second switch S2,which triggers the release of phosphate (the transition from point(3) on DP to point (4) on D in Fig. 3b), initiating the secondarypower stroke. The energy drop from �4� ! �5� on the D curve inFig. 3b releases the elastic strain stored during the primary powerstroke. This takes place during the primary power stroke of the nextsite. Thus, each 2p/3 rotation of g is driven by two bs: the binding ofnucleotide to one b and the release of elastic strain from theprevious b when phosphate is released.
All that remains to complete the model is to specify the chemicalrates, K(v, s), in equation (2), which control the transition betweenthe 64 different elastic potentials. The transition rates are deter-mined by the free energy difference and the energy barrier betweenadjacent states. As already discussed and summarized in Box 1, theactivation energy barrier of a catalytic site is affected by the chemicalstate of the other two sites and the interaction with the two switchregions on g (see Supplementary Information for a completedescription of the mathematical model and its numerical simula-tion). The results of these computations are summarized in Fig. 4.
Figure 4a shows typical trajectories for various ATP concentra-tions and viscous loads. At saturated ATP concentrations (2 mM)and with no additional viscous load, the motor rates in steps of2p/3, consuming one ATP per step (curve A). The rate-limiting stepunder these conditions is the release of ADP15. At low ATPconcentrations, and with a large viscous load imposed by theactin ®lament attached to the g-subunit5,6, the motor also behavesas a stepper (curve C in Fig. 4a). In this situation, the motor islimited by the diffusion of ATP to the catalytic sites16. At high ATPconcentrations and loads, the step-like progression of the motor isobscured by the viscous drag (curve B in Fig. 4a). At low ATPconcentrations, the motor occasionally executes backward stepswhen nucleotide binds to the `incorrect' site (curve C in Fig. 4a).These curves reproduce quantitatively the measurements of Yasudaet al.6. The mechanical ef®ciency of the motorÐcomputed as theratio of the average viscous dissipation per step to the free energy ofhydrolysisÐis very high, approaching 100% for large viscous loads.This high ef®ciency is possible because the motor is not a heatengine, but is driven almost completely by elastic strain energy.
Figure 4b shows the expected load±velocity relation, togetherwith the velocity as a function of the length of an attached actin®lament5,6. The motor can generate torques of more than 40 pN nm.The mechanical escapement allows the motor to release the strain
energy conferred by nucleotide binding in two stages. The primarypower stroke begins immediately upon nucleotide binding. Thispower stroke performs two tasks: it drives the g-subunit in afull 2p/3 rotation, and it also compresses the passive elastic element,storing nearly half of the elastic energy of binding. This elasticenergy is released in a secondary power stroke that assists the nexthydrolysis site during its primary power stroke. This two-stagerelease of stored energy smoothes the output torque and contributesto the high mechanical ef®ciency of the motor.
The model has three other predictions (see SupplementaryInformation): (1) it correctly predicts the measured rates of ATPsynthesis when driven in reverse by a torque corresponding to thatgenerated in Fo (refs 2, 17); (2) the proton ¯ux through Fo coupledto the rotation of g is blocked by small concentrations of ADP butthen increases under normal synthesis conditions, in accordancewith results from `slip' experiments18; and (3) the average occupancyof the catalytic sites as a function of ATP concentration agrees wellwith experiments13.
In summary, the model contained in equations (1)±(3) canquantitatively explain the principal features of the F1 motor. Themotor rotates in steps of 2p/3, generating up to 45 pN nm of torqueand consuming, on average, a single ATP per step, with a mechanicalef®ciency approaching 100%. To perform in this way, the motorcannot operate as a heat engine but converts the free energy ofnucleotide binding into elastic strain energy. This strain energydrives the rotation of the g-subunit through a tightly coupledmechanical mechanism, which is closely coordinated with thekinetic transitions. The general features of energy transduction inF1 may have implications for other protein motors driven bynucleotide hydrolysis. M
Received 8 June; accepted 11 September 1998.
1. Abrahams, J., Leslie, A., Lutter, R. & Walker, J. Structure at 2.8 AÊ resolution of F1-ATPase from bovine
heart mitochondria. Nature 370, 621±628 (1994).2. Boyer, P. The binding change mechanism for ATP synthaseÐsome probabilities and possibilities.
Biochim. Biophys. Acta 1140, 215±250 (1993).
3. Fillingame, R. H. Coupling H+ transport and ATP synthesis in F1Fo-ATP synthases: glimpses of
interacting parts in a dynamic molecular machine. J. Exp. Biol. 200, 217±224 (1997).
4. Cross, R. & Duncan, T. Subunit rotation in FoF1-ATP synthases as a means of coupling protontransport through Fo to the binding changes in F1. J. Bioen. Biomembr. 28, 403±408 (1996).
5. Noji, H., Yasuda, R., Yoshida, M. & Kinosita, K. Direct observation of the rotation of F1-ATPase.
Nature 386, 299±302 (1997).
6. Yasuda, R., Noji, H., Kinosita, K., Motojima, F. & Yoshida, M. Rotation of the g subunit in F1-ATPase:
Evidence that ATP synthase is a rotary motor enzyme. J. Bioen. Biomembr. 29, 207±209 (1997).7. Elston, T., Wang, H. & Oster, G. Energy transduction in ATP synthase. Nature 391, 510±514 (1998).
8. Shirakihara, Y. et al. The crystal structure of the nucleotide-free a3 b3 subcomplex of F1-ATPase from
the thermophilic Bacillus PS3 is a symmetric trimer. Structure 5, 825±836 (1997).
9. Gerstein, M., Lesk, A. & Chothia, C. Structural mechanisms for domain movements in proteins.
Biochemistry 33, 6739±6749 (1994).10. Gerstein, M. in Protein Motions (ed. Subbiah, S.) 81±90 (Chapman and Hall, New York, 1996).
11. Alberts, B. The cell as a collection of protein machines: preparing the next generation of molecular
biologists. Cell 92, 291±294 (1998).
12. Rojnuckarin, A., Kim, S. & Subramaniam, S. Brownian dynamics simulations of protein folding:
access to milliseconds time scale and beyond. Proc. Natl Acad. Sci. USA 95, 4288±4292 (1998).13. Senior, A. Catalytic sites of Escherichia coli F1-ATPase. J. Bioen. Biomembr. 24, 479±483 (1992).
14. Al-Shawi, M. & Nakamoto, R. Mechanism of energy coupling in the FoF1-ATP synthase: the
uncoupling mutation, gM23K, disrupts the use of binding energy to drive catalysis. Biochemistry
36, 12954±12960 (1997).15. Hasler, K., Engelbrecht, S. & Junge, W. Three-stepped rotation of subunits g and e in single molecules
of F-ATPase as revealed by polarized, confocal ¯uorometry. FEBS Lett. 426, 301±304 (1998).
16. Yasuda, R., Noji, H., Kinosita, K. & Yoshida, M. F1 ATPase is a stepper motor. Biophys. J. 74, A1
(1998).
17. Matsuno-Yagi, A. & Hate®, Y. Kinetic modalities of ATP synthesis: regulation by the mitochondrialrespiratory chain. J. Biol. Chem. 261, 14031±14038 (1986).
18. Groth, G. & Junge, W. Proton slip of the chloroplast ATPase: its nucleotide dependence, energetic
threshold, and relation to an alternating site mechanism of catalysis. Biochemistry 32, 8103±8111
(1993).
19. Yasuda, R., Noji, H., Kinosita, K. & Yoshida, M. F1-ATPase is a highly ef®cient molecular motor thatrotates with discrete 1208 steps. Cell 93, 1117±1124 (1998).
Supplementary information is available on Nature's World Wide Web site (http://www.nature.com) oras paper copy from the London editorial of®ce of Nature.
Acknowledgements. We thank J. Walker for discussions on the motions of the b-subunit and for videos ofthe three con®gurations which simulated our interpolated videos; R. Nakamoto for advice on the switch 1and 2 interactions; M. Yoshida, K. Kinosita and their co-workers for inspiring the construction of the F1
model by their ingenious experiments; and M. Grabe and K. Kinosita for insightful comments andsuggestions on the manuscript.
Correspondence and requests for materials should be addressed to G.O. (e-mail: [email protected]).
The F1 model correctly predicts the ATP synthesisrate when torque is applied to the gggg-shaft
The hydrolysis-driven motor investigated by the Japanese group (Noji, et al., 1997;Yasuda, et al., 1998) did not contain the e subunit. Therefore, some of the interactionsbetween rotor and stator that are known to be important were missing in theirexperiments. These interactions appear unnecessary in the hydrolysis direction, and sowe modeled the elastic potentials as simply as possible to reproduce their results.However, in order to perform correctly in the synthesis direction, the rotation of g mustbe coordinated with the binding of reactants and the release of product. First, therotation of g must be retarded until ADP and phosphate are sequentially bound at thecatalytic site. Otherwise, rotation will frequently bypass the phosphate binding sectorresulting in a wasted rotation. Second, the catalytic site must trap ATP for release. Thatis, since ATP is in equilibrium with ADP and Pi with Keq ~ 1, there must be a mechanismto alter the catalytic site so that ATP is favored as rotation of g applies strain to the siteto open it up. The shapes of the potentials in Figure 3b in the text do not ensure thissequence of events, so they will not work properly in the synthesis direction. Therefore,we investigated what alterations in the potentials would allow the F1 model tosynthesize ATP at the appropriate rate when a torque of ~ 45 pN-nm is applied to theg-shaft, the amount necessary to release the tightly bound nucleotide from the catalyticsite of a b subunit (Elston, et al., 1998).
We found that adding two ÔbumpsÕ to the DP (ADP¥Pi) and T (ATP) curves as shown inFigure 1 accomplished the task. The sequence of events is enumerated in the figurecaption. The shape of the E (Empty) potential retards rotation until ADP is bound. Thebarrier on D (ADP) potential for trapping phosphate could be supplied by the e-subunit,which is absent in the hydrolysis experiments. The shapes of the DP (ADP¥Pi) and T(ATP) potentials trap nucleotide in the ATP form until it can dissociate into solution, lestthe site release reactants rather than product. Interestingly, the modified potentialsshown in Figure 1 drive the motor in the hydrolysis direction somewhat moreefficiently than those in Figure 3b in the text. Figure 2a shows that ATP is produced atthe observed rate when this torque is applied to the g subunit.
At physiological conditions, the transmembrane protonmotive force in mitochondria is220 ~ 230 mV (Stryer, 1995). The observed maximum ATP synthesis rate at saturatedoxygen supply is about 400 ATP/s/F1 (Boyer, 1993; Matsuno-Yagi, et al., 1986). DuringATP synthesis, the rotary torque applied to the g shaft of F1 is generated in F0 by aprotonmotive force. To simulate ATP synthesis rate as a function of the protonmotiveforce, we combined the F1 model in this study with the F0 model in the previous study(Elston, et al., 1998) by constraining the F1 and F0 to rotate at the same rate. In Figure
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
2
2b, the correct ATP synthesis rate is achieved when the protonmotive force across Fo isin the physiological range.
-1 0 1 2 3-40
-20
0
20
40
DG/kBT
q/p
①
②
E
E③
④
⑤
⑥
⑦
D
Synthesis
⑧
DP
⑨ T
Figure 1. The free energy potentials required for nucleotide synthesis. In a typical cyclestarting in the empty state, E, at (1), torque from Fo rotates g to the left. This
strains the empty catalytic site, raising its elastic energy. At some point (2), thesystem drops to point (3) on the D curve when ADP binds to the site. Furtherrotation drives the system up the D curve until, at point (4) phosphate is boundand the system drops to the DP potential curve at point (5). Continued rotationfurther strains the occupied catalytic site driving the system up the DP curveuntil at some point (6) ATP is formed at the transition from (6) to (7). The systemis now trapped in the T state where ATP is bound tightly. Rotational torque fromFo must now pry open the active site by straining the system from (7) to (8)
whereupon the nucleotide is released and the system drops to point (9) in theempty, E, state. The ÔbumpsÕ on the elastic energy curves are necessary in orderto trap reactants as g turns to avoid futile rotations. These features can be
incorporated into the mechanical model as additional elastic elements. This doesnot address their origin within the structure of F1 since there is no unique
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
3
relationship between the elastic potentials and a mechanical realization.However, there are certain features of F1 that are candidate structures. Mostnoticeably, the e subunitÑwhich is absent in the hydrolysis experiments (Noji, et
al., 1997; Yasuda, et al., 1998)Ñis in the right position to coordinate thephosphate trap. The ADP and ATP traps also may be ascribed to e, and/or tostrain at the catalytic sites induced by additional elastic elements in b. Specificassignment of these features awaits more detailed structural analysis of F1.
40 50 60-100
0
200
400
600
Torque from F0 [pN×nm]
ATP
synt
hesis
rate
[1/s
]
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
4
100 200 300-100
0
200
400
600
Protonmotive force [mV]
ATP
synt
hesis
rate
[1/s
]
Figure 2. (a) The ATP synthesis rate as a function of the torque applied to the g shaft.
The model predicts the correct synthesis rate when a torque of ~ 45 pN-nm isapplied to F1 (shaded region).
(b) The ATP synthesis rate as a function of the protonmotive force (pmf)imposed across the membrane-spanning Fo subunit. The F1F0 complex ismodeled by combining the F1 model in this study with the F0 model in ourprevious study (Elston, et al., 1998). When the F1 and F0 models are constrained
to rotate at the same rate, the model predicts the correct synthesis rate forphysiological protonmotive forces (shaded region). The protonmotive force isgiven by pmf = Dy + RT/F DpH, where Dy is the transmembrane potential, R is
the gas constant and F is FaradayÕs constant. In both (a) and (b), the nucleotideconcentrations are [ATP] = 0.2 mM, [ADP] = 0.1 mM and [Pi] = 2 mM. In the
experiments of Matsuno-Yagi, this set of concentrations corresponds to the halfmaximum synthesis rate (Matsuno-Yagi, et al., 1986).
The occupancy of the catalytic sites as a functionof ATP concentration
By inserting a tryptophan residue in position b331 of E. coli F1-ATPase, SeniorÕs groupdesigned an optical probe which directly monitors occupancy of the three catalytic sitesby nucleotides (Weber, et al., 1997). The relation between the ATP concentration and
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
5
the occupancy of the catalytic sites was recorded by repeating the experiment over awide range of ATP concentrations.
E T DP D 0
1
2
3
[ATP] = 10-7M
# ca
talyt
ic sit
es p
er F
1
E T DP D 0
1
2
3
[ATP] = 10-3M
10-7 10-6 10-5 10-4 10-3 0
1
2
3
ATP concentration [M]
# ca
talyt
ic sit
es o
ccup
ied
per F
1
# ca
talyt
ic sit
es p
er F
1
Figure 3. Top panel shows the average number of catalytic sites occupied per F1 as a
function of the ATP concentration. The solid line is the theoretical prediction ofthe F1 model and the filled circles are the experimental data from Weber and
Senior (Weber, et al., 1997). The two bottom panels show the computedoccupancies of the catalytic sites for 10-7 M and 10-3 M (saturated) ATP
concentrations.
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
6
As another validity test for the F1 model, we compared the theoretical prediction of theF1 model with their experimental results. We calculated the average number of catalyticsites occupied per F1 as a function of the ATP concentration. In the top panel of Figure3, the solid line represents the theoretical prediction and the filled circles represent theexperimental data from Weber and Senior (Weber, et al., 1997). The F1 modelquantitatively predicts the dependence of the catalytic site occupancy on the ATPconcentration. Here a catalytic site in the empty state, E, is labeled as unoccupied,otherwise it is labeled as occupied. So an occupied site could be in T, DP or D states. Inthe two bottom panels of Figure 3, the detailed occupancies of the catalytic sites arecomputed for low and saturated ATP concentrations.
The proton flux through F0 coupled to the rotationof gggg is blocked by small concentrations of ADP
The F-ATPase couples proton flow to ATP synthesis; in the absence of nucleotide ATPsynthesis cannot proceed. However, driven by the torque generated in Fo, the g shaftcan continue to rotate. This rotation of g allows an unproductive proton flow throughthe membrane, usually called leakage or proton slip. Groth and Junge found that thisproton slip can be blocked by small concentrations of ADP (Groth, et al., 1993).Moreover, in the presence of 0.5 mM Pi, the total proton flow (the sum of theproductive and the unproductive proton flows) as a function of the ADP concentrationpasses through a minimum. At low concentrations (less than 1 mM), ADP reduces theproton leakage and so the total proton flow decreases with the ADP concentrationincreases. At concentrations above 1 mM, ADP dramatically increases ATP synthesisactivityÑand the productive proton flux coupled to itÑso that the total proton fluxincreases with the ADP concentration.
We found that the model reproduces this curious experimental observation: proton fluxthrough Fo coupled to the rotation of g is blocked by small concentrations of ADP, butthen increases at the normal synthesis conditions. Figure 4 shows the total proton fluxas a function of ADP concentration, predicted by the model. The total proton flux passesthrough a minimum as the ADP concentration increases from 0.01 mM to 1 mM. Piconcentration is 0.5 mM as in the experiments. This unexpected correlation withexperiment supports the assumptions of the model.
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
7
10-8 10-6 10-4
300
200
100
0
ADP Concentration [M]
Prot
on fl
ux [1
/s]
Pi 0.5 mM
Figure 4. The total proton flux as a function of ADP concentration predicted by themodel. The total proton flux passes a minimum as the ADP concentrationincreases from 0.01 mM to 1 mM (PI = 0.5 mM). This is in accord with the
experimental results in Figure 2a of (Groth, et al., 1993).
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
8
Construction of the F1 motor model
Kinematics and dynamics
Extracting conformational changes from WalkerÕs structure
The molecular structure (PDB) file of F1-ATPase by WalkerÕs lab (Abrahams, et al., 1994)contains the coordinates of three aÕs, three bÕs and part of g. This file was used to extractapproximate conformational changes of each subunit. Figure 5a,b shows a side and topview of F1 in ribbon display mode; Figure 5c shows a side view in spacefill displaymode, and Figure 5d shows a side view of one a, one b and g in spacefill display mode.The a3b3 hexamer forms an annulus around the g subunit. There are six ATP bindingsites located at the a-b interfaces; the 3 catalytic sites are mostly on the b subunit, whilethe noncatalytic sites are mostly on the a subunit.
In the experiments of Noji, et. al (Noji, et al., 1997), (Yasuda, et al., 1997) the g subunitrotates with respect to the a3b3 hexamer as F1 hydrolyzes ATP. The bottom portion (inFigure 5) of the a3b3 hexamer is symmetric and is thought to constitute thehydrophobic ÔbearingÕ surface between a3b3 and g. This portion does not changeappreciably during the rotation of g. The g subunit and the top portion of the a3b3hexamer are asymmetric and complementary, so that the asymmetry of the g subunitfits the asymmetry of the hexamer, suggesting that the g subunit is rotated by changingthe asymmetry of the top a3b3 hexamer.
In order to analyze the motions of F1 we first place a cylindrical coordinate system onthe bottom portion of the a3b3 hexamer, since it undergoes little or no conformationalchange during the rotation of g. Figure 6a shows a view from the bottom of the barrelportion of F1 where the symmetry of the a3b3 hexamer is apparent. Three a-carbonsfrom three b-Glu26 residue groups (one from each b) are shown as black dots. Thesethree atoms form an equilateral triangle. We position the origin of our cylindricalcoordinate system at the center of this triangle and erect the z-axis perpendicular to theplane of the triangle (pointing from F1 towards Fo as shown in Figure 6a). We numberthe bÕs in the positive rotation direction around the z-axis (clockwise in Figure 6a).Figure 6b is a perspective view of the bottom barrel portion of F1 with the z-axispointing upwards.
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
9
Figure 5 Geometry of F1. The top row shows a ribbon view from the side (a) and top(b). (c) is a solid (spacefill) view from the side. (d) is a solid view from side ofone a, one b and g. The aÕs are in yellow; the stationary lower portions ofthe bÕs are in green and the moveable upper portions are in red. The two coilsof g are shown in blue and gray. The corresponding animation movies are (a)F1_side.mpeg, (b) F1_top.mpeg, (c) F1_side_fill.mpeg, (d) F1_abg_fill.mpeg.
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
10
Figure 6 Bottom view (a) and perspective view (b) of the bottom barrel portion of F1.The three black dots show three a-carbons from three b-Glu26 residues, onefrom each b. The coordinate system is determined by these three atoms (seetext). The color scheme is the same as in Figure 5: aÕs in yellow, bÕs in green.
Next we determine the coordinates of each b and a subunit after the g subunit rotatesone third of a revolution. According to BoyerÕs binding change mechanism (Boyer,1993), a 2p /3 rotation of the g subunit corresponds to a 2p /3 advance of the chemicalstates of the three catalytic sites and a 2p /3 advance of the conformational asymmetryof the a3b3 hexamer. That is, after a 2p/3 rotation of the g subunit, b1 takes theconfiguration of b3 which is 2p /3 behind b1, b2 takes the configuration of b1 and b3
takes the configuration of b2 so that the conformational asymmetry of the a3b3hexamer propagates by 2p /3. Thus after the g subunit rotates by 2p/3, the newcoordinates of b1 are the original coordinates of b3 rotated by 2p /3 in the cylindricalcoordinate system. The new coordinates of b2 and b3 are calculated from the originalcoordinates of b1 and b2 respectively. The new coordinates of b1 after a 4p /3 rotationof g are calculated by rotating the original coordinates of b2 which is 4p /3 behind b1.After a 2p rotation of g, b1 assumes its original coordinates. For angular displacementsof g between 0 and 2p/3, we estimate the new coordinates of b1 as a linear interpolationof the coordinates for q=0 and the coordinates for q=2p/3 in the cylindrical coordinatesystem. The new coordinates of b2, b3, a1, a2 and a3 are calculated accordingly. In thisway, we can compute a sequence of configurations corresponding to different angulardisplacements of g in F1.
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
11
Kinematics of the F1 motor
Once we knows how to compute the configuration of F1 for any angular displacementof g, we can examine the kinematics of the F1 motor by comparing the configurations ofF1 for different angular displacements of g. Figure 7 shows four configurations of one a,one b and g corresponding to angular displacements of g of q = (0, p/4, p/2, p). Forclarity, we show only one a, one b and g, giving a cross sectional view of the a3b3hexamer. As before, the top portion of the b subunits are colored red and the bottomportion colored green.
From studies such as Figure 7, we can draw the following conclusions about theconformational motions of F1:
· During the rotation of g, there is very little conformational change in the bottomportion of the a3b3 hexamer.
· The bottom portion of the a3b3 hexamer forms a symmetric bearing. In Figure 7,the bearing section contained between the middle and bottom rings. Note that boththe middle ring and the bottom ring are centered on the z-axis of the coordinatesystem defined in Figure 6, and they remain unchanged during the rotation of g.
· The bottom part of the g subunit fits into the middle and bottom rings. The g subunitis asymmetric, and the top part of the g is off the z-axis.
· The top ring is formed by the tips of the a3b3 hexamer which surrounds the top partof g. This ring is also off the z-axis. The position of this top ring relative to the z-axisdetermines the position of the top part of g relative to the z-axis. Since the bottompart of g is held by two rings centered on the z-axis, the position of the top part of grelative to the z-axis determines the angular displacement of g.
From these observations we conclude that the g subunit is driven by rotation of the off-axis top ring about the centerline, while the middle and bottom rings centered on the z-axis act as bearings for the lower portion of g. Thus the asymmetrically curved g subunitis driven by the three b subunits in a fashion analogous to 3 arms cranking anautomobile jack. To visualize the motion of F1, we have used the interpolation programto make movies of F1 as g rotates. These movies in MPEG and QuickTime formats canbe downloaded from the web site http://teddy.berkeley.edu:1024/ATP_synthase/.
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
12
Figure 7 Configurations of one a, one b and g for several angular displacements of g:(a) q = 0, (b) q = p/4, (c) q = p/2, (d) q = p. The circles show the regions of closecontact between g and the annulus in a3b3. The bottom two contact circlesremain approximately concentric, while the top contact circle rotates off-centerabout the vertical axis (dashed line). The corresponding animation movie isF1_abg.mpeg.
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
13
F1_3d.mpeg Perspective view of a3b3 and g in cartoon display mode.
F1_3d_fill_spc.mpeg Perspective view of a3b3 and g in spacefill display mode withspecular highlights.
F1_top.mpeg Top view of a3b3 and g in cartoon display mode.
F1_top_fill_spc.mpeg Top view of a3b3 and g in spacefill display mode withspecular highlights.
F1_side.mpeg Side view of a3b3 and g in cartoon display mode.
F1_side_fill_spc.mpeg Side view of a3b3 and g in spacefill display mode withspecular highlights.
F1_abg.mpeg Cross-section view of one a, one b and g in cartoon displaymode.
F1_abg_fill.mpeg Cross-section view of one a, one b and g in spacefill displaymode.
F1_beta.mpeg Cross-section view of b in cartoon display mode.
F1_beta_fill.mpeg Cross-section view of b in spacefill display mode.
F1_beta_site.mpeg A zoom-in view of b near the catalytic site in cartoon displaymode.
F1_beta_top.mpeg Top view of b in cartoon display mode.
F1_alpha.mpeg Cross-section view of a in cartoon display mode.
F1_Model.mpeg Animation based on our mechanical model.
Table 1. Index of movies showing the motions of F1.
Dynamics of the F1 motorAs shown in Figure 7, the g subunit is constrained by two rings at the middle and thebottom levels. When the top ring moves around the z-axis, the g subunit must rotate toaccommodate all three rings. That is, the motion of the top ring around the z-axis drivesthe rotation of the g subunit. The motion of the top ring is driven, in turn, by hingemotions of each b subunit.
The structure of an empty b subunit suggests the ÔopenÕ configuration is the relaxedstate of the empty b subunits (Shirakihara, et al., 1997). The model assumes that the
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
14
hinge motion of a b subunit from ÔopenÕ to ÔclosedÕ is driven by the binding ofnucleotide to the catalytic site. We call this introduction of binding free energy at thecatalytic site the primary power stroke. It can be modeled as an elastic element which isswitched on (stretched and engaged) upon ATP binding at the catalytic site. We shall callthis elastic element the Ôactive springÕ.
We further assume that the hinge motion of a b subunit from ÔclosedÕ back to ÔopenÕ isdriven by the ÔrecoilÕ elasticity resulting from the compression of the primary powerstroke. That is, the secondary power stroke can be modeled by a spring that iscompressed during the hinge motion from ÔopenÕ to ÔclosedÕ of the primary powerstroke. This spring represents the intrinsic elasticity of the b subunits, which isestablished during the initial assembly of the a3b3 hexamer. For this reason, we call thisthe Ôpassive springÕ. Thus at the completion of the primary power stroke the activespring is switched off, and the b subunit can recoil back to the ÔopenÕ configurationdriven by the passive spring. This mechanical model captures the essential energytransfers that take place within F1 during the hydrolysis cycle. The timing and sequenceof the energy transfers are controlled by the chemistry of the hydrolysis cycle, whichwe now discuss.
KineticsDuring ATP synthesis and hydrolysis, each b subunit passes through at least fourchemical states (Equation 1 in the text):
E T D P D EEmpty
ATP binding
ATPbound
ADP & P
release
ADP
ADP release
Emptyi
¬ ®¾ ¾ ¾¾ ¬ ®¾ ¾ ¾¾ × ¬ ®¾ ¾¾ ¬ ®¾ ¾ ¾¾Hydrolysis
bound
P
bound
i (1)
We denote by bbbbi the chemical state of the ith b subunit, and denote by S = (bbbb1, bbbb2, bbbb3)the chemical state of the a3b3 system. For each b subunit, the possible chemical statesare
{ E, T, D××××P, D }. (2)
The space of the possible kinetic states for the a3b3 system is:
{ E, T, D××××P, D } ´ { E, T, D××××P, D } ´ { E, T, D××××P, D }. (3)
These 43 = 64 possible kinetic states can be visualized as the integer points in a 4x4x4cube. Actually the kinetic states form a 3-dimensional torus, since when the system exitsone face of the cube, it reenters the opposite face.
The reaction process at each b catalytic site can be described by the Markov process:
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
15
E T¬ ®¾
¬ ®¾ ×b b
D D P
(4)
It is important to stress that the progression of the system through the 64-state cube isstochastic. That is, we do not impose a deterministic sequence of chemical states; weshall specify only the transition rates between states, and let the system evolvestochastically as the dynamics dictates. Thus the formulation given here differsfundamentally from other kinetic schemes proposed for ATP synthase. However,under some circumstances (e.g. ambient concentrations), there may be a most probablepath. We formulate the stochastic dynamics of the three catalytic sites as follows.
Let r(q; t, bi-1, bi, bi+1) be the probability at time t that the bÕs are in chemical states bbbbi-1,bbbbi and bbbbi+1, respectively, while g is fixed at an angular position q. Note that r(q; t, bi-1, bi,bi+1) is NOT the probability density that g is at the angular position q at time t, and thebÕs are in chemical states bbbbi-1, bbbbi and bbbbi+1, respectively. Here q is a parameter. By fixingthe g subunit at an angular position, we are able to isolate the kinetics and study itsgoverning equation. In the subsequent sections, we will combine the kinetics and thedynamics to build a complete model for F1 motor.
The governing equation for the Markov process on bi is
ddt
t E
t T
t DP
t D
k k
k k
k
i i
i i
i i
i i
E T E D E
E T T DP T
T
r q b br q b br q b br q b b
; , , ,
; , , ,
; , , ,
; , , ,
- +
- +
- +
- +
® ®
® ®
®
( )( )( )( )
é
ë
êêêêê
ù
û
úúúúú
=
--
1 1
1 1
1 1
1 1
0
0
0
SS
DPDP DP D DP
E D DP D D
i i
i i
i i
i i
k
k k
t E
t T
t DP
t D
--
é
ë
êêêê
ù
û
úúúú
×
( )( )( )( )
é
ë
êêêêê
ù
û
úú
®
® ®
- +
- +
- +
- +
SS0
1 1
1 1
1 1
1 1
r q b br q b br q b br q b b
; , , ,
; , , ,
; , , ,
; , , ,
úúúú
(5)
where
SSSS
E E T E D
T T E T DP
DP DP T DP D
D D E D DP
k k
k k
k k
k k
= += +
= += +
® ®
® ®
® ®
® ®
(6)
All transition rates depend on q, the angular position of g relative to b, and the chemicalstates of the other two bÕs:
k k
k k
E T E T i i
T DP T DP i i
® ® - +
® ® - +
= ( )= ( )
q b b
q b b
, , ,
, , ,
1 1
1 1
L
(7)
In the matrix-vector form, equation (5) is
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
16
ddt
t ti i i i i i i ir q b b q b b r q b b; , ( , , ) ; ,, ,- + - + - +( ) = × ( )1 1 1 1 1 1K (8)
where
r q b b
r q b br q b br q b br q b b
i i i
i i
i i
i i
i i
t
t E
t T
t DP
t D
; , ,
; , , ,
; , , ,
; , , ,
; , , ,
- +
- +
- +
- +
- +
( ) =
( )( )( )( )
é
ë
êêêêê
ù
û
úúúúú
1 1
1 1
1 1
1 1
1 1
(9)
In Equation 2 in the text, we write equation (8) symbolically as:
ddt
iii i i
b q b b b= × =- +K( , , ) , , ,1 1 1 2 3 (10)
There are 16 choices for the combination of (bi-1, bi+1). So there are 16 differentprobability density vector, ri , each of which is a vector with 4 components. These riÕsform a vector with 64 elements.
rr q
r qr qr qr qr q
r qr q
;
; , , ,
; , , ,
; , , ,
; , , ,
; , , ,
; , , ,
; , , ,
t
t E E E
t E E T
t E E DP
t E E D
t E T E
t D D DP
t D D D
( ) =
( )( )( )( )( )
( )( )
é
ë
êêêêêêêêêêê
ù
û
úúúúúúúúúú
M
úú
ü
ý
ïïïïï
þ
ïïïïï
64 components (11)
Combining equation (5) for all combinations of (bi-1, bi+1), we have the governingequation for the probability vector, r(q, t), of the a3b3 system:
ddt
t trr rrq q q; ( ) ;( ) = × ( )K (12)
where the transition matrix K(q) is a 64x64 sparse matrix with only 6x64=384 non-zeroelements. The elements of the transition matrix of the system, K(q), are related to theelements of the transition matrix of the ith b, K(q, bi-1, bi+1), as:
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
17
k k E E
k k E E
k k E E
E E E E T E E T
E E E E E T E T
E E E T E E E T
, , , ,
, , , ,
, , , ,
, , ,
, , ,
, , ,
( )® ( ) ®
( )® ( ) ®
( )® ( ) ®
( ) = ( )
( ) = -æè
öø
( ) = +æè
öø
q q
q q p
q q p
23
23
L
(13)
Hence we only need to consider the transition matrix of one b for 16 different chemicalstate combinations of the other two bÕs.
Coupling of the reaction on bbbb to the rotation of ggggTo simulate the chemical reactions taking place at each catalytic site, we must constructthe transition matrix of one b for 16 different chemical state combinations at the othertwo bÕs. Since each transition matrix has 8 non-zero elements (see equation (5)), thereare 8 ´ 16 independent transition rates. However, we can build all of these startingfrom the uni-site reaction rates listed in Table 2 (Senior, 1992).
The multisite reaction rates are affected by several factors:
1. The chemical states of the other two sites, bi-1 and bi+1,
2. The positions of switches S1 and S2 on g relative to b (q dependence),
3. The elastic energy difference between the two chemical states of the b underconsideration (q dependent).
The word ÔmultisiteÕ suggests that factor #1 is most important; however, the fastmultisite reactions are only observed when more than one catalytic site is occupied andthe g subunit is present (Kaibara, et al., 1996). This implies that the g subunit plays animportant role in the multisite reactions, and so factors #2 and #3 must be taken intoconsideration.
We can express these factors mathematically as:
k k f g
k k f g
forwardM
i i forwardU
i i
backwardM
i i backwardU
i i
q b b b b q l q
q b b b b q l q
, , ,
, , ,( )
- + - +
- + - +
( ) = × ( ) × ( ) × × ( )æèç
öø÷
( ) = × ( ) × ( ) × - - × ( )æèç
ö
1 1 1 1
1 1 1 1
1
expE
k T
expE
k T
B
B
D
Døø÷
(14)
where kM represents the multisite transition rates between two chemical states, kU
represents the unisite rates. The function f(bi-1, bi+1) represents the effect of the othertwo sites, g(q) simulates the effect of the two switches on g, and DE(q) is the elasticenergy difference between the two chemical states of the b under consideration. To
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
18
keep the notation simple, we shall drop the superscript M for multisite rates from nowon.
Table 2 summarizes the mathematical formulation of the multisite reaction rates alongwith the unisite rates and other parameter values.
PARAMETERS VALUES
Unisite reactionrates of E. Coli.(Senior, 1992)
kUE®T = 1.1 ´ 105 s-1 M-1, kU
T®E = 2.5 ´ 10-5 s-1
kUT®DP = 1.2 ´ 10-1 s-1, kU
DP®T = 4.3 ´ 10-2 s-1
kUDP®D = 1.2 ´ 10-3 s-1, kU
D®DP = 4.8 ´ 10-4 s-1 M-1
kUD®E = 1.6 ´ 10-3 s-1, kU
E®D = 1.8 ´ 102 s-1 M-1
Free energy dropsof unisite reactionof E. Coli.
DGE®T = DG0E®T + log([ATP]/M)×kBT
DG0E®T = log(kU
E®T / kUT®E) = 22.2 kBT
DGT®DP = log(kUT®DP / kU
DP®T) = 1.0 kBT
DGDP®D = DG0DP®D - log([Pi]/M)×kBT
DG0DP®D = log(kU
DP®D / kUD®DP) = 0.9 kBT
DGD®E = DG0D®E - log([ADP]/M)×kBT
DG0D®E = log(kU
D®E / kUE®D) = -11.6 kBT
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
19
PARAMETERS VALUES
Construction ofthe multisitereaction rate, kM,from the unisiterate, kU.
k k f g
k k f g
forwardM
i i forwardU
i i
backwardM
i i backwardU
i i
q b b b b q l q
q b b b b q l q
, , ,
, , ,( )
- + - +
- + - +
( ) = × ( ) × ( ) × × ( )æèç
öø÷
( ) = × ( ) × ( ) × - - × ( )æèç
ö
1 1 1 1
1 1 1 1
1
expE
k T
expE
k T
B
B
D
Døø÷
f(bi-1, bi+1) simulates the effect of the occupancy of the other twosites (i.e. binding of ATP on other sites increases the reaction onthis site).
g(q) simulates the effect of the two switches S1 and S2 on g (i.e.rotation of g brings the switches close to (or away from) this site,which increases (or decreases) the reaction on this site).
DE(q) is the elastic energy difference between the two chemicalstates of the b in consideration (q dependent).
See Table 3 for details of the multisite reaction rates.
Elasticity constantsof the active andpassive springs
kActive = 10 pN/nm, kPassive = 4 pN/nm
The active and passive springs are pre-stretched (pre-compressed) by 4 nm. During the reaction, the additionaldisplacement of these springs is 2 nm.
Table 2. Unisite reaction rates, elasticity constants and the mathematical formulation ofthe multisite reaction rates.
It should be pointed out that the formulation in equation (14) is a very general one.Once more experimental data are available, more refined models can be accommodatedeasily into this framework and simulated. Next we discuss the multisite reaction rates indetail.
Transitions between E and T
k k f g
k k f g
E E E E
E T i i E TU
E T i i E T
T E i i T EU
E T i i E T
T T E E
® - + ® - +
® - + ® - +
( ) = × ( ) × ( )( ) = × ( ) × ( )
×- ( )( ) - - ( )(
q b b b b q
q b b b b q
p
, , , ,
, , ,
, ,
, ,
1 1 1 1
1 1 1 1
0exp
))æèç
öø÷kBT
(15)
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
20
fE
E T i ii i
, ,b bb b
- +- +( ) =
= =ìíî
1 11 11
150
(unisite mode)
otherwise (multisite mode)(16)
g
otherwise
E T i,
/
/ ,q
q p
p q b b( ) =
< <
- < < ( )
ì
í
ïïï
î
ïïï
+
1 0 2
2 0
0
1
ATP binding promoted by switch #1
0.05 and one of is Eallows ATP binding on the wrong site
i-1(17)
where EE and ET are the total elastic energy of the b in state E and T, respectively.
In equation (16), fE, T(bi-1, bi+1) takes only two values. The effect of the chemical statesof the other two sites is modeled as either putting the system into the unisite mode orthe multisite mode.
Equation (17) contains both the effect of stress at the catalytic site and the effect ofswitch #1. The configuration of the catalytic site is related to the angular position, q, of g.The site is ÔopenÕ near q = 0 (Figure 7a) and is ÔclosedÕ near q = p (Figure 7d). Since thesystem is periodic in the q-direction, q = -p represents the same angular position as q =p. In equation (17), ATP binding is allowed only when the angular position, q, is in theregion [-p/2, p/2] where the catalytic site is ÔopenÕ. The ATP binding in the q-range [0,p/2] contributes to the forward rotation of the motor while the ATP binding in theq-range [-p/2, 0] leads to a backward step. Switch #1 on the g subunit promotes ATPbinding in the q-range [0, p/2], but does not completely inhibit the ATP binding in theq-range [-p/2, 0]. This allows the occasional backward steps.
Transitions between T and D××××P
k k f gE
T
k k f gE
T DP i i T DPU
T DP i i T DPDPB
DP T i i DP TU
T DP i i T DPTB
® - + ® - +
® - + ® - +
( ) = × ( ) × ( ) × -æèç
öø÷
( ) = × ( ) × ( ) × -
q b b b b q
q b b b b q
, , ,
, , ,
, ,
, ,
1 1 1 1
1 1 1 1
expk
expk
B
BBT
æèç
öø÷
(18)
fE
T DP i ii i
, ,b bb b
- +- +( ) =
= =ìíî
1 11 1
5
1
10
(unisite mode)
otherwise (multisite mode)(19)
gT DP, q( ) = 1 (20)
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
21
The total elastic energy of a b consists of three parts:
E E E EA P B= + + (21)
EA and EP are the elastic energies associated with the ÔactiveÕ and ÔpassiveÕ springs,respectively, and EB is the elastic energy associated with the ÔbumpsÕ required during
synthesis (see below).
The active spring is switched on only in states T and D×P while the passive spring isalways engaged. Thus the total elastic energies of the b for states E, T, D×P and D are
given respectively by
E E E
E E E E
E E E E
E E E
EP
EB
TA P
TB
DPA P
DPB
DP
DB
= += + += + += +
(22)
In equation (18), EBDP is the part of the elastic energy of the b in state D×P that is caused
by the additional interaction between the b and the g, and by the additional elasticcomponents (other than the two major springs) of the b. Correspondingly EB
T is thatpart of the elastic energy of the b in state T.
We use superscript ÔBÕ to denote the additional elastic energy associated with theÔbumpsÕ in the elastic energy curves. These bumps are not necessary when F1 ishydrolyzing ATP and functioning as a motor. However, they are essential when F1 isturned in the reverse direction by Fo to synthesize ATP. The function of these bumps inATP synthesis is to prevent ÔfutileÕ rotations. When the g subunit is turned one fullrevolution in the reverse direction, on average, each b should pass once through the
cycle
E D D P T E® ® × ® ® (23)
and produce one ATP. Failure of a b to produce an ATP is a futile rotation. Futile
rotations can come about in several ways:
1. ADP does not bind on the catalytic site and b stays in state E.
2. After binding of ADP, no Pi binds on the catalytic site and b either stays in state D or
loses its ADP and reverts back to state E.
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
22
3. After the binding of Pi, ATP does not form and b either stays in state D×P or loses itsPI and/or ADP, reverting back to states D or E.
4. After the formation of ATP from ADP and Pi, ATP is not released from the catalyticsite and b either stays in state T or reverts back to states D×P, D or E.
To prevent these futile rotations, we need to coordinate sub-steps in the ATP synthesisreaction to the rotation of g. More specifically, as g rotates, we guide the b from state Einto state D (ADP binding), then into state D×P (Pi binding), then into state T (forming
ATP from ADP and Pi), and finally back into state E again (releasing ATP from thecatalytic site).
The important roles of the additional elastic energies (EBE, EB
T, EBDP and EB
D) in the
ATP synthesis are shown in Figure 1.
Transitions between D××××P and D
k k f g
E E
T
k k f g
DP D i i DP DU
DP D i i DP D
DP DP
D DP i i D DPU
DP D i i DP D
® - + ® - +
® - + ® - +
( ) = × ( ) × ( )
×- ( )( )æ
èçöø÷
( ) = × ( ) × ( )
×
q b b b b q
p
q b b b b q
, , ,
, , ,
, ,
, ,
1 1 1 1
1 1 1 1
expkB
expexpkB
E E
TD D- ( )( )æ
èçöø÷
0
(24)
fE
DP D i ii i
, ,b bb b
- +- +( ) =
= =ìíî
1 11 1
6
1
10
(unisite mode)
otherwise (multisite mode)(25)
gDP D,(q p q p
( ) = < <ìíï
îï1
43
switch # 2)
0 otherwise(26)
In equation (26), Pi release (Pi binding in the ATP synthesis direction) is regulated byswitch #2 on the g subunit.
Transitions between D and E
k k f gET
k k f gE
D E i i D EU
D E i i D EEB
E D i i E DU
D E i i D EDB
® - + ® - +
® - + ® - +
( ) = × ( ) × ( ) × -æèç
öø÷
( ) = × ( ) × ( ) × -
q b b b b q
q b b b b q
, , ,
, , ,
, ,
, ,
1 1 1 1
1 1 1 1
expk
expk
B
BTT
æèç
öø÷
(27)
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
23
fE
D E i ii i
, ,b bb b
- +- +( ) =
= =´
ìíî
1 11 1
4
1
3 10
(unisite mode)
otherwise (multisite mode)(28)
g siteDP D, qp q p
( ) = - < < æèç
öø÷
ìíï
îï
123
23
binding regulated by
configuration0 otherwise
(29)
In equation (29), the ADP release (ADP binding in the ATP synthesis direction) isregulated by the configuration of the catalytic site.
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
24
PARAMETERS VALUES
kE®T and kT®E
k k f g
k k f g
E E E E
E T i i E TU
E T i i E T
T E i i T EU
E T i i E T
T T E E
® - + ® - +
® - + ® - +
( ) = × ( ) × ( )( ) = × ( ) × ( )
×- ( )( ) - - ( )(
q b b b b q
q b b b b q
p
, , , ,
, , ,
, ,
, ,
1 1 1 1
1 1 1 1
0exp
))æèç
öø÷kBT
fE
E T i ii i
, ,b bb b
- +- +( ) =
= =ìíî
1 11 11
150
(unisite mode)
otherwise (multisite mode)
g
otherwise
E T i,
/
/ ,q
q p
p q b b( ) =
< <
- < < ( )
ì
í
ïïï
î
ïïï
+
1 0 2
2 0
0
1
ATP binding promoted by switch #1
0.05 and one of is Eallows ATP binding on the wrong site
i-1
ET is the total elastic energy of the b in state T.
EE is the total elastic energy of the b in state E.
kT®DP and kDP®T
k k f gE
T
k k f gE
T DP i i T DPU
T DP i i T DPDPB
DP T i i DP TU
T DP i i T DPTB
® - + ® - +
® - + ® - +
( ) = × ( ) × ( ) × -æèç
öø÷
( ) = × ( ) × ( ) × -
q b b b b q
q b b b b q
, , ,
, , ,
, ,
, ,
1 1 1 1
1 1 1 1
expk
expk
B
BBT
æèç
öø÷
fE
T DP i ii i
, ,b bb b
- +- +( ) =
= =ìíî
1 11 1
5
1
10
(unisite mode)
otherwise (multisite mode)
gT DP, q( ) = 1
EBDP is the part of the elastic energy of the b in state D×P that
is caused by the additional interaction between the b and theg, and by the additional elastic components (other than thetwo major springs) of the b.
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
25
PARAMETERS VALUES
kDP®D and kD®DP
k k f g
E E
T
k k f g
DP D i i DP DU
DP D i i DP D
DP DP
D DP i i D DPU
DP D i i DP D
® - + ® - +
® - + ® - +
( ) = × ( ) × ( )
×- ( )( )æ
èçöø÷
( ) = × ( ) × ( )
×
q b b b b q
p
q b b b b q
, , ,
, , ,
, ,
, ,
1 1 1 1
1 1 1 1
expkB
expexpkB
E E
TD D- ( )( )æ
èçöø÷
0
fE
DP D i ii i
, ,b bb b
- +- +( ) =
= =ìíî
1 11 1
6
1
10
(unisite mode)
otherwise (multisite mode)
gDP D,(q p q p
( ) = < <ìíï
îï1
43
switch # 2)
0 otherwise
kD®E and kE®D
k k f gET
k k f gE
D E i i D EU
D E i i D EEB
E D i i E DU
D E i i D EDB
® - + ® - +
® - + ® - +
( ) = × ( ) × ( ) × -æèç
öø÷
( ) = × ( ) × ( ) × -
q b b b b q
q b b b b q
, , ,
, , ,
, ,
, ,
1 1 1 1
1 1 1 1
expk
expk
B
BTT
æèç
öø÷
fE
D E i ii i
, ,b bb b
- +- +( ) =
= =´
ìíî
1 11 1
4
1
3 10
(unisite mode)
otherwise (multisite mode)
g siteDP D, qp q p
( ) = - < < æèç
öø÷
ìíï
îï
123
23
binding regulated by
configuration0 otherwise
EBD is the part of the elastic energy of the b in state D that is
caused by the additional interaction between the b and theg, and by the additional elastic components (other than thetwo major springs) of the b.
Table 3. Multisite reaction rates
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
26
Calculating the elastic potentialsFigure 8 shows how the bending angle, f, of the b subunit is coupled to the rotationalangle, q, of the g subunit. In the left panel, the plane shown is perpendicular to therotation axis of g. The dashed circle represents the cross-section of g at the hydrophobicÔbearingÕ section where the g is held tightly at two levels (middle and bottom levels inFigure 7). The solid filled circle represents the cross section of the g at the ÔdrivingÕ levelwhere g contacts the bending b subunits. The three small filled circles represent thethree hinge points on the three bÕs. The solid lines between the hinge points and the gare the projection of the top parts of the bÕs onto the plane perpendicular to the g axis.L(q) is the length of the top part of each b projected onto the plane perpendicular to theg axis; it is a function of the rotation angle, q, of g . In the right panel, the plane shown isparallel to the g axis. The top and bottom sectors of b are shown schematically as linesegments. The dashed line represents the relaxed position of the top sector of an emptyb. The azimuthal angle between the top and bottom sectors, f, measures the bendingangle of b. L(f) is the length of the top part of b projected onto the plane perpendicularto the g axis; it is a function of the bending angle, f. By equating L(q) and L(f), weobtain a relation between the bending angle, f, and the rotational angle, q. If we assumethe active and passive springs connecting the top and bottom parts of each b are linearin the bending angle, f, we can calculate the potential caused by these two springs as afunction of the rotational angle, q.
Df
L(f)
b
SIDE VIEWTOP VIEW
L(q)q
g at bearing level
g at driving level
f
Figure 8 Coupling between the rotation angle, q, of the g subunit and the bendingangle, f, of the b subunit. In the left panel, the plane shown is perpendicular tothe g axis. In the right panel, the plane shown is parallel to the g axis. See text for
details.
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
27
We can calculate the potentials in a simpler way which is somewhat more intuitive.Because the change of the bending angle is small (Df ~ 30û), the length, L, of the top partof b projected onto the plane perpendicular to the g axis is approximately a linearfunction of the bending angle, f. Therefore, the active spring and the passive springconnecting the top and bottom parts of each b are also linear in the length, L, of the toppart of the b projected onto the plane perpendicular to the g axis. Of course, the linearrelation between f and L is an approximate one, as is the assumption that the springsare linear in f. In this sense, it is no more reasonable to assume the springs are linear interms of f than to assume the springs are linear in terms of L. Thus we shall view theactive and passive springs on each b as linear springs in the projected length L.
L(q)qr2
r1
Figure 9 Mathematical calculation of the projected length, L, as a function of therotation angle, q.
As shown in Figure 9, the projected length, L, of the top part of b as a function of q isgiven by
L r r r r rq q( ) = + - ( ) -12
22
1 2 12 cos (30)
where r1 = 1 nm and r2 = 4 nm.
The elastic energy due to the passive spring as a function of q is
E k L L L k LP Pecompress
Pecompressq q( ) = × ( ) - ( )( ) +[ ] - × [ ]1
20
12
2 2D DPr Pr (31)
where the passive spring is pre-compressed by DLPrecompress = 4 nm.
The elastic energy due to the active spring as a function of q is
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
28
E k L L L k LA Aestress
Aestressq p q( ) = × ( ) - ( )( ) +[ ] - × [ ]1
212
2 2D DPr Pr (32)
where the active spring is prestressed by DLPrestress = 4 nm.
Viewing the passive and active springs in the direction of the projected length Lsimplifies the calculation of the potentials, and makes the bending motion of each bdirectly comparable to the linear motion of each piston in a three-cylinder engine.
The active spring is switched on only in states T and D×P while the passive spring isalways engaged. The elastic energy in state D is due solely to the passive spring.However, the energy in state E requires an additional component because threeexperimental observations would be violated were we to model the elastic energy ofthe empty state by only the passive spring:
1. Rotation of g driven by a proton flux is smaller in the absence of nucleotide than
under normal synthesis conditions (Groth, et al., 1993).
2. Under hydrolysis conditions at low nucleotide concentrations rotation of g proceeds
stepwise, with 3 steps per revolution (Yasuda, et al., 1998).
3. The model should operate in reverse to synthesize ATP from ADP and phosphate.
To meet these requirements, we must modify the elastic potential in the empty stateover what is provided by the passive spring. To do this we write the total free energiesof each b for states E, T, D×P and D as
E E E
E E E E
E E E E
E E E
EP
EB
TA P
TB
DPA P
DPB
DP
DB
q q qq q q q
q q q qq q q
( ) = ( ) + ( )( ) = ( ) + ( ) + ( )
( ) = ( ) + ( ) + ( )( ) = ( ) + ( )
(33)
In this equation, the superscripts A and P refer to the elastic potentials of the active andpassive springs, respectively, as computed above. The superscript (B) represents thepart of the elastic energy of the b caused by an additional interaction between the b andg, and/or a. The additional elastic component in empty state, EB
E, makes emptyba pairs stiff; that is it creates a deep potential barrier against rotation of the g subunitwhen three catalytic sites are empty. It also prevents the g subunit from diffusing over alarge angle when the catalytic site is empty, thus enforcing the observed steppingbehavior.
In the hydrolysis direction, EBT, EB
DP and EBD are unnecessary. but in the synthesis
direction all four additional terms (EBE, EB
D, EBDP and EB
T) are required for (i) trappingADP, (ii) trapping phosphate, (iii) forming ATP and (iv) releasing ATP (see Figure 1).
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
29
For example, in the synthesis direction, EBE raises the free energy on the E curve such
that the transition E ®®®® D is promoted and the backward transition D ®®®® E is prohibitedwhen the system is driven up the E curve by a mechanical torque.
Mathematical equations and numerical solutions
Langevin equation formulationSince inertia is negligible, equating the viscous drag on the rotating g to the other forcesacting on it yields the Langevin equation (Doering, 1990; Risken, 1989),
z q qq
g g a b
ddt
dV
dT t
Viscoustorque
TorqueLoadtorque
Browniantorque
L B
on between
and 3
{ 1 24 34{ 123
= - ( ) - + Á;
( )S
3
(34)
where z is the rotational drag coefficient, q is the angular coordinate, V(q; S) is the sumof the elastic energies of the three bÕs, TL is the load torque, and ÁB is the Browniantorque due to thermal fluctuations. S = (b1, b2, b3) represents the kinetic state of thethree catalytic sites. S can take any of the 43 = 64 possible kinetic states in the space
{ E, T, D××××P, D } ´ { E, T, D××××P, D } ´ { E, T, D××××P, D }
The probability of state S evolves according to kinetic equation (12), which we rewritesymbolically as
ddt
S S= ×K( )q (35)
To compute the torque generated by the F1 motor, equation (34) must be solvedsimultaneously with the Markov process governing the kinetic transitions on the threecatalytic sites (equation (35)).
Fokker-Planck equation formulation
In the Fokker-Planck formulation corresponding to equations (34) and (35), the F1motor is described by a vector function consisting of 64 probability density functions.
Let r(q, t, bi-1, bi, bi+1) be the probability density that g is at the angular position q attime t, and the bÕs are in chemical states bbbbi-1, bbbbi and bbbbi+1, respectively. Define theprobability density vector as:
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
30
rr q
r qr qr qr qr qr qr q
r qr q
,
, , , ,
, , , ,
, , , ,
, , , ,
, , , ,
, , , ,
, , , ,
, , , ,
, , , ,
t
t E E E
t E E T
t E E DP
t E E D
t E T E
t E T T
t E T DP
t D D DP
t D D D
( ) =
( )( )( )( )( )( )( )
( )( )
é
M
ëë
êêêêêêêêêêêêêê
ù
û
úúúúúúúúúúúúúú
ü
ý
ïïïïïïï
þ
ïïïïïïï
64 components (36)
This probability density vector evolves according to the convective diffusion equations
¶ q¶ z
¶¶q
qq
q¶ q
¶qq q
rrrr
rrrr
,,
,,
t
td
dT t D
ttL
( ) = ( ) +æèç
öø÷ ( )ì
íî
üýþ
+ ( ) + ( ) × ( )1 2
2
VK (37)
where D = kBT / z is the rotational diffusion coefficient. The 64x64 transition matrix K(q)is defined in equation (12). The potential matrix V(q) is
6464 q q p q q p( ) = +( ) + ( ) + -( )E E ED D D2 3 2 3/ /
(39)
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
31
Numerical solution of Fokker-Planck equations
The dynamic and kinetic behaviors of the F1 motor are governed by the Fokker-Planckequations (37). In this subsection, we discuss the numerical methods for solving them.
Equations (37) is a system with 64 coupled equations. The full construction of ournumerical methods for general Fokker-Planck equations involves the analysis of theequations; details of the numerical analysis will be presented in a separate publication.Here we describe the numerical method for a model Fokker-Planck system with twocoupled equations.
Consider the equations:
¶r¶ z
¶¶q
f q r ¶ r¶q
q r q r
¶r¶ z
¶¶q
f q r ¶ r¶q
q
q
11 1
21
2 12 1 21 2
22 2
22
2
1
1
tD k k
tD
obability
obability
= ¢( )( ) + - ( ) × + ( ) ×
= ¢ ( )( ) +
Pr
Pr
flowin the -direction
Probability flow inthe reaction direction
flowin the -direction
1 24444 344441 24444 34444
1 224444 344441 24444 34444- ( ) × + ( ) ×k k21 2 12 1q r q r
Probability flow inthe reaction direction
(40)
S2
q
q
S1
k12
Motion of the system
K21
Figure 10 Probability flows in the q-direction and the kinetic reaction direction.
The system governed by equations (40) has two kinetic states, denoted 1 and 2. rm (q, t)is the probability density that the system is at location q at time t, and is in state m = 1,2.fm (q) is the potential of the system in state m. k12(q) is the transition rate from state 1 to
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
32
state 2, and k21(q) is the transition rate from state 2 to state 1. Equations (40) describethe conservation of probability at any location q in each state. Figure 10 shows all in-flows and out-flows at a given location q in states 1 and 2.
The numerical discretization is constructed as follows. First we divide the domain in theq-direction into intervals of size Dq, and call the center point of the n-th interval qn . Letrm, n(t) be the probability that the system is in the n-th interval at time t, and is in statem. Figure 11 shows the discrete probability in-flows and out-flows in both the q andkinetic directions. The conservation of probability in the n-th interval in state m leads todifferential difference equation of the form:
d
dtF B F
B k k
d
dtF B F
nn n n n n
n n n n n n
nn n n n
rr r
r r r
rr
1,
2,
= × - +( ) ×
+ × - × + ×
= × - +
- - - +
+ +
- - -
1 1 2 1 1 1 1 2 1 1 2 1
1 1 2 1 1 12 1 21 2
2 1 2 2 1 2 1 2 2
, / , , / , / ,
, / , , , , ,
, / , , / , ++
+ +
( ) ×
+ × - × + ×
1 2 2
2 1 2 2 1 21 2 12 1
/ ,
, / , , , , ,
r
r r r
n
n n n n n nB k k
(41)
where the kinetic transition rates in the n-th interval, k12, n and k21, n, are given by
k k
k k
n n
n n
12 12
21 21
,
,
= ( )= ( )
q
q(42)
Fm, n+1/2 is the forward jump rate from n-th interval to the (n+1)-th interval and Bm, n+1/2
is the jump rate from (n+1)-th interval back to the (n+1)-th interval. The jump rates Fm,
n+1/2 and Bm, n+1/2 are constructed to preserve two important properties of the Fokker-Planck equations:
· The numerical method should maintain the principle of detailed balance. The ratio ofthe jump rates between two intervals is determined by the potential difference ofthe two intervals:
F
B k Tm n
m n
m n m n
B
, /
, /
exp+
+
+=( ) - ( )æ
èç
ö
ø÷
1 2
1 2
1f q f q(43)
· For a uniform probability distribution, the net flux across qn+1/2 in state m (theboundary between the n-th and (n+1)-th intervals) is proportional to the derivativeof potential fm. When the derivative of potential fm is a constant, independent of n,this yields
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
33
F Bm n m nm n m n
, / , /+ ++-( ) × =
( ) - ( )×1 2 1 2
1DDDD
qf q f q
z q(44)
Solving jump rates Fm, n+1/2 and Bm, n+1/2 from equations (43) and (44), we obtain
FD k T
k T
BD k T
k T
m n
m n m n
m n m n
m n
m n m n
m n m n
B
B
B
B
, /
, /
exp
exp
+
+
+
+
+
+
=( )
×-
( ) - ( )
-( ) - ( )æ
èç
ö
ø÷ -
=( )
×
( ) - ( )
( ) - ( )æ
1 2
1
1
1 2
1
1
2
2
1DD
DD
q
f q f q
f q f q
q
f q f q
f q f q
èèç
ö
ø÷ - 1
(45)
qn+1
qn+1 qn
qn
S2
q
q
S1
k21
qn-1
qn-1
F2, n+1/2
F1, n+1/2
B2, n+1/2
B1, n+1/2 k12
Figure 11 Conservation of probability in the n-th interval in states 1 and 2.
Once we know the transition rates in the kinetic direction and the discrete jump rates inthe q-direction, equations (41) are solved using a Crank-Nicolson scheme.
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
34
References
Abrahams, J., A. Leslie, R. Lutter, and J. Walker. 1994. Structure at 2.8� resolution of F1-
ATPase from bovine heart mitochondria. Nature 370:621-628.
Boyer, P. 1993. The binding change mechanism for ATP synthase--some probabilitiesand possibilities. Biochim. Biophys. Acta 1140:215-250.
Doering, C. 1990. Modeling complex systems: Stochastic processes, stochasticdifferential equations, and Fokker-Planck equations. In 1990 Lectures in ComplexSystems. L. Nadel, and D. Stein, editors. Addison-Wesley, Redwood City, CA. 3-51.
Elston, T., H. Wang, and G. Oster. 1998. Energy transduction in ATP synthase. Nature
391:510-514.
Groth, G., and W. Junge. 1993. Proton slip of the chloroplast ATPase: its nucleotidedependence, energetic threshold, and relation to an alternating site mechanismof catalysis. Biochemistry 32:8103-8111.
Kaibara, C., T. Matsui, T. Hisabori, and M. Yoshida. 1996. Structural asymmetry of F1-
ATPase cuased by the g subunit generates a high affinity nucleotide binding site.
J. Biol. Chem. 271:2433-2438.
Matsuno-Yagi, A., and Y. Hatefi. 1986. Kinetic Modalities of ATP Synthesis: Regulationby the Mitochondrial Respiratory Chain. J. Biol. Chem. 261:14031-14038.
Noji, H., R. Yasuda, M. Yoshida, and K. Kinosita. 1997. Direct observation of therotation of F1-ATPase. Nature 386:299-302.
Risken, H. 1989. The Fokker-Planck Equation. Springer-Verlag, New York.
Senior, A. 1992. Catalytic Sites of Escherichia coli F1-ATPase. Journal of Bioenergetics and
Biomembranes 24:479-483.
Shirakihara, Y., A. Leslie, J. Abrahams, J. Walker, T. Ueda, Y. Sekimoto, M. Kambara, K.Saika, Y. Kagawa, and M. Yoshida. 1997. The Crystal Structure of the Nucleotide-Free a3 b3 Subcomplex of F1-ATPase from the Thermophilic Bacillus
PS3 is a Symmetric Trimer. Structure 5:825-836.
THE F1 MOTOR: SUPPLEMENTARY MATERIAL
35
Stryer, L. 1995. Biochemistry. W. H. Freeman, New York.
Weber, J., and A. E. Senior. 1997. Catalytic mechanism of F1-ATPase. Biochim. Biophys.
Acta 1319:19-58.
Yasuda, R., H. Noji, K. Kinosita, F. Motojima, and M. Yoshida. 1997. Rotation of the gSubunit in F1-ATPase; Evidence That ATP Synthase Is a Rotary Motor
Enzyme. Journal of Bioenergetics and Biomembranes 29:207-209.
Yasuda, R., H. Noji, K. Kinosita, and M. Yoshida. 1998. F1-ATPase is a highly efficient
molecular motor that rotates with discrete 120¡ steps. Cell 93:1117-1124.
