letters to nature
Energy transduction in the
F1 motor of ATP synthase
Hongyun Wang & George Oster
Department of Molecular and Cellular Biology, and College of Natural Resources,
University of California, Berkeley, California 94720-3112, USA
.........................................................................................................................
ATP synthase is the universal enzyme that manufactures ATP
from ADP and phosphate by using the energy derived from a
transmembrane protonmotive gradient. It can also reverse itself
and hydrolyse ATP to pump protons against an electrochemical
gradient. ATP synthase carries out both its synthetic and hydrolytic cycles by a rotary mechanism1±4. This has been con®rmed in
the direction of hydrolysis5,6 after isolation of the soluble F1
portion of the protein and visualization of the actual rotation of
the central `shaft' of the enzyme with respect to the rest of the
molecule, making ATP synthase the world's smallest rotary
engine. Here we present a model for this engine that accounts
for its mechanochemical behaviour in both the hydrolysing and
synthesizing directions. We conclude that the F1 motor achieves
its high mechanical torque and almost 100% ef®ciency because it
converts the free energy of ATP binding into elastic strain, which
is then released by a coordinated kinetic and tightly coupled
conformational mechanism to create a rotary torque.
The general structure of ATP synthase is shown in Fig. 1. It
consists of two portions: a soluble component, F1, containing the
catalytic sites, and a membrane-spanning component, Fo, comprising the proton channel. The entire FoF1 structure is arranged as a
counter-rotating `rotor' and `stator' assembly. The stator portion
consists of the catalytic sites contained in the a3b3 hexamer,
together with subunits a, b2 and d. The rotor consists of 9±12
c-subunits arranged in a ring and connected to the g- and esubunits that form the central `shaft'. The mechanism of protonmotive force transduction has been discussed elsewhere7; we now
focus on the mechanochemistry of F1, which synthesizes ATP when
the g±e shaft is turned clockwise (viewed from the membrane) by
the protonmotive force, and which rotates counterclockwise to
pump protons when hydrolysing ATP.
Any mechanism proposed to explain the experimental results is
strongly constrained by thermodynamic, kinetic, structural and
mechanical considerations. In the hydrolysis direction, it must
reproduce the large torque and almost 100% mechanical
ef®ciency5,6; in the synthesis direction, ATP must be produced at
the measured rate when driven by torques generated by the protonmotive force in Fo. Kinetic constants and free energies must
correspond to measured values. A model must also be geometrically
compatible with the detailed structural information now available.
We constructed the model in three steps. First, kinematic motions
of the F1 subunits were analysed to determine the conformational
changes that drive rotation of the g-subunit. From this, we constructed a mathematical model describing the principal mechanical
and electrostatic interactions between the subunits. Finally, the
reaction sequences were incorporated to create a complete
mechanochemical model. The outline of this model can be
summarized as follows.
The reactivity of a catalytic site depends on its local conformation
and composition. The unique feature of F1 is that the reactivity of
each site depends on the states of the other two sites, and the
position of g. Each of the three catalytic sites, b1, b2 and b3, passes
through at least four chemical states:
E Ã
Empty
ATP
binding
Hydrolysis
! T Ã
ATP
bound
! D×P
Ã
ADP
and Pi
bound
Pi
release
! D Ã
ADP
bound
NATURE | VOL 396 | 19 NOVEMBER 1998 | www.nature.com
ADP
release
We denote by bi the chemical state (E, T, D×P, D) of the ith
b-subunit, so that a3b3 has a total of 43 ˆ 64 possible kinetic states.
The collection of these states can be visualized as the integer points
in a 4 3 4 3 4 cube (actually, a three-dimensional torus, because
when the state exits one face of the cube, it re-enters the opposite
face). The progression of the system through this chemical-state
space depends on the angular position of the g-subunit, which we
denote by v. The kinetics can be described by a set of equations (that
is, a Markov model) of the form
8
dbi =dt ˆ K…v; bi 2 1 ; bi‡1 † 3 bi
…2†
where i ˆ 1; 2; 3 and where K is the matrix of kinetic transition rates
for a catalytic site, which depends on the angular position of g and
the occupancy of the other sites. Thus the state of F1 is speci®ed by
its rotational position and its kinetic state: (v, b1, b2, b3).
The motion of g is described by balancing the torques on the
rotating g-subunit:
dv
z
dt
|{z}
ˆ
Viscous torque on g
t…v; b1 ; b2 ; b3 †
|‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚}
Torque between g and a3 b3
2
TL ‡
|{z}
Load torque
T B …t†
|‚{z‚}
…3†
Brownian torque
The viscous load on the rotating g-subunit is contained in the drage
coef®cient, z. TL represents any additional load torque (such as from
a laser trap). TB(t) is the brownian torque due to thermal ¯uctuations. The driving torques exerted on g by the b-subunits can be
expressed as the gradient of an elastic potential: t…v; b1 ; b2 ; b3 † ˆ
2 ]V…v; b1 ; b2 ; b3 †=]v. This potential is derived from the geometry
H+
a
c
Fo
ε
b
β
δ
γ
α
F1
Figure 1 Structure of ATP synthase. It has a transmembrane portion, Fo,
consisting of the stoichiometric subunits c12ab2, and a soluble portion, F1, which
consists of subunits a3b3ged. Functionally, the protein behaves as a `rotor' (c12ge)
and a `stator' (ab2da3b3), which are thought to counter-rotate during hydrolysis
and synthesis. The three catalytic sites are located at the ab interfaces, and the
proton channel is at the c±a interface. The dashed outline represents the
fragment of g that has not been resolved. In recent hydrolysis experiments5,6, the
! E
Empty
…1†
a3b3 subunit was attached to a coverslip and a ¯uorescently tagged actin ®lament
was attached to the g-subunit so that rotation of g could be observed; here, only
subunits a3b3g were present.
Nature © Macmillan Publishers Ltd 1998
279
letters to nature
and elastic properties of the mechanical model for F1. We constructed this mechanical model by the following procedure.
First, we analysed the kinematic motions of F1. To do this, we
imported the coordinates for a3b3g from the Protein Data Bank
into the molecular structure program Rasmol1,8. According to
Boyer's binding change mechanism, a 2p/3 rotation of g corresponds to a 2p/3 advance in the kinetic states of the three catalytic
sites and a 2p/3 advance of the conformational asymmetry of the
a3b3 hexamer. This allows us to derive the coordinates of F1 for
v ˆ 2p=3 from those at v ˆ 0. For angles between 0 and 2p/3, the
coordinates of F1 are calculated by interpolation in a cylindrical
coordinate system. From this we created stereo animations of the
complete motion of F1 through the hydrolysis cycle (see Supplementary Information). Although the interpolated positions obey all
steric constraints, they are probably not exact; nevertheless, they do
give an idea of the overall motion of the subunits.
Next, we placed a local coordinate system on each subunit so that
we could determine quantitatively the subunit's sequence of conformations. A study of the Protein Motions Database shows that
most conformational changes can be resolved into hinge and shear
motions9,10. The principal motion of the b-subunits consists of a
hinge motion that bends the top and bottom segments towards one
another by about 308 (Fig. 2), and two much smaller motions which
we ignore: a radial shear motion and rotation in the v direction. The
a-subunits undergo very little change in shape.
To see how the vertical bending motion is converted into a rotary
torque on the g-subunit, we took horizontal `serial sections'
through the a3b3g assembly to determine the cross-sectional
shapes of g and the central channel in a3b3 into which g ®ts.
Combining this information, we deduced that the curved and
asymmetric g-subunit is driven by the hinge motion of the
a
b-subunits in a fashion analogous to a hand on the crank of a car
jack (see Supplementary Information).
We now make an important assumption about how the free
energy from ATP hydrolysis is converted into the hinge-bending
motion of b: the energy to bend a b-subunit is conferred on the
catalytic site at the nucleotide-binding step in equation (1); that is,
the rapid sequential bonding of the nucleotide to the catalytic site is
converted 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 step
in equation (1). This strain energy tends to bend the b-subunit, and
the bending stress is converted by the protein's geometry into a
rotary torque on the g-subunit and to stresses on the other catalytic
sites (Fig. 2).
Viewed as an elastic body, the a3b3 hexamer is complicated, and
certainly not isotropic. However, we can construct a simpli®ed
mechanical equivalent that captures its principal motions11,12.
In this way, the complicated elastic behaviour of F1 can be
approximated by an assembly of rigid elements connected
by springs (Fig. 2b).
We treat the a3b3 hexamer as an elastic body with a pre-stress
arising from the free energy of assembly. First, we model the
intrinsic elasticity of the b-subunits as passive elastic elements at
the hinge points. Second, we represent the active stress created by
nucleotide binding as much stronger elastic elements that are
activated upon nucleotide binding. Third, by assigning reasonable
elastic constants to these elements (Box 1), we can compute elastic
potential energies based on the simpli®ed molecular structure
shown in Fig. 2, one potential curve for each of the 64 kinetic
S2
Box 1 Multisite reaction rates
Multisite reaction rates are constructed from the unisite rates by taking
into account the effects of coupling of each catalytic site to the other sites,
the two switches on the g-subunit, and the elastic energy of each site. The
elastic constants of the active and passive springs refer to Fig. 2.
Construction of the multisite reaction rate kM from the unisite reaction
rate kU:
βE
γ
α
S1
βT
γ
α
U
kM
forward …v; bi 2 1 ; bi‡1 † ˆ kforward 3 f…bi 2 1 ; bi‡1 † 3 g…v† 3 exp
l 3 DE…v†
kB T
U
kM
backward …v; bi 2 1 ; bi‡1 † ˆ kbackward 3 f…bi 2 1 ; bi‡1 † 3 g…v†
b
3 exp
Passive
spring
2 …1 2 l† 3 DE…v†
kB T
f…bi 2 1 ; bi‡1 † contains the effect of the occupancy of other sites. Binding of
Active
spring
ATP on other sites increases the reaction on this site. g(v) models the
effect of switches S1 and S2: rotation of g brings the switch regions on b
Bearing
surface
and g into proximity, which increases the reaction rate. DE(v) is the elastic
energy difference between the two chemical states of b. (see Supple-
Figure 2 Conformation changes in the b-subunit. a, Side-view cross-section
showing the conformational changes of the b-subunit in the empty (E) and ATPbound (T) states, and the corresponding rotation of the g-subunit as deduced
from an analysis of the PDB coordinates (see Supplementary Information for
mentary Information).
Elastic constants of active and passive springs:
kactive ˆ 10 pN nm 2 1 ; and kpassive ˆ 4 pN nm 2 1 :
procedures and animation moves). The two switch regions14, denoted by S1 and
The active and passive springs are pre-compressed by 4 nm. During
S2, control nucleotide binding and phosphate release, respectively, at each
hydrolysis, the additional displacement of these springs is 2 nm.
catalytic site. b, Mechanical model used to compute the elastic potential that
drive the rotation of the g-subunit (described in Supplementary Information). The
hinge-bending motion of b rotates the upper portion by ,308. This is converted
into a rotary torque on g by the asymmetric bend in g. The torque on g is
contributed by two b-subunits at a time, because each b has two free-energy
drops during each hydrolysis cycle (Fig. 3a).
280
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NATURE | VOL 396 | 19 NOVEMBER 1998 | www.nature.com
8
letters to nature
states. The derivatives of these elastic potentials yield the torques
that drive the rotation of g, as given in equation (3).
The exact procedure for constructing the potentials is given in the
Supplementary Information. Roughly, this consists of measuring
the bending motions of a b-subunit by the azimuthal angle, f,
between its upper and lower portions. During the course of its
conformational change, a b-subunit closes and opens its azimuthal
angle by about Df < 308. Trigonometry yields a relationship
between the bending angle, f, of a b-subunit and the rotation
angle of g : v ˆ f …f†. As the bending angle f closes, b pushes on the
off-axis bowed segment of g (Fig. 2). In this way, a rotary torque in
the v direction is generated by the geometric propagation of stress
from the catalytic site (the active spring) to g. As each of the
subunits cycles through its conformational range in sequence, a
continuous rotational torque is exerted on g.
a
The unisite (that is, at substoichiometric ATP concentration) rate
constants for the transitions given in equation (1) have been
measured13. But, at steady state the reactions at the three catalytic
sites are coordinated in two ways. First, the strain induced by
nucleotide binding at each site is communicated to the other two
sites. Second, there are at least two `switch points' on g that interact
with speci®c sites on the a3b3 hexamer14. Thus, the activation
energy barriers for the reaction at each site are in¯uenced by the
state 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 system
evolves through the 64 possible chemical states. Note that, as
measured externally, only part of the free energy of ATP is released
to the surroundings upon binding: the remainder is stored internally 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
8
a
E
B
C
8
F1+ATP
-10
S2
T
∆G/kBT
Revolutions
0
A
S1
DP
F1•ATP
6
4
F1•ADP•Pi
-20
2
D
F1•ADP
-30
E
F1
0
0
0
0.1
Reaction coordinate
0.2
10
0.4
20
0.6
0.2
0.3
Time (s)
30
40
50
1
0.8
0.4
b
40
Hydrolysis
direction
101
Rotation rate (Hz)
γ
E
20
∆G/kBT
1
0
PS1
PS2
E
Rotation rate (Hz)
b
T
S1
2
DP
3
-20
S2
100
6
Actin length 1 µm
4
2
0
0
20
40
Load torque (pN nm)
: 20 µM
: 2.0 µM
: 0.2 µM
: 60 nM
: 20 nM
4
D
5
-40
-1
0
1
Rotation angle (θ/π)
2
10-1
3
Figure 3 Energy pro®les. The free energy of F1 during its hydrolysis cycle
1
2
3
Actin length (µm)
4
Figure 4 Model predictions. a, Simulated rotational trajectories of the g-subunit
depends on the chemical state of each b-subunit, represented by bi ˆ …E; T; DP; D†,
relative to a3b3 at three different ATP concentrations and viscous loads. Trajectory
and the angular coordinate, v, of the g-subunit: DG(v,b1,b2,b3). a, Projection of the
A shows the rotation of g when the ambient ATP concentration is 2 mM. The
free energy onto the reaction coordinate of a single b-subunit. The free-energy
motor stochastically steps in increments of 2p/3, and the rate is limited by ADP
drop proceeds in 4 stages (see equation (1)). The free-energy drops are com-
release. In trajectories B and C, a drag coef®cient equivalent to an actin ®lament
puted from unisite reaction rates (the free-energy barriers are shown schemati-
1 mm long was imposed on the g subunit5,19. Trajectory B corresponds to an ATP
cally). The transition E ! T (nucleotide binding) is activated by switch S1; the
concentration of 2 mM. Here the viscous damping of the actin ®lament obscures
transition DP ! D (phosphate release) is activated by switch S2. b, Free-energy
the stepping behaviour. Trajectory C shows that the stepping behaviour
pro®le of the interaction between the g-subunit and a single b-subunit, showing
reappears, even with the drag of the actin ®lament, when ATP drops to 20 nM.
the 4 elastic periodic potentials along which the system is constrained to move.
In this situation, hydrolysis is limited by diffusion of ATP to a catalytic site. Note
Numbers trace the trajectory of a catalytic site through one cycle of hydrolysis,
that rare backward steps occur when nucleotide occasionally binds to the
which generates two power strokes. The primary power stroke (PS1) occurs upon
`wrong' site. The average velocities are as measured in ref. 19. b, The rotation
binding of nucleotide, which drives the bending of the b-subunit. During this
rate of g when subjected to viscous loads from a long actin ®lament. Data points
bending, the elastic stress turns g and compresses the passive elastic element.
are from ref. 19. Solid lines are computed from the model, and ®t the data at all
The secondary power stroke (PS2) is triggered by the release of phosphate,
loads and ATP concentrations. If an additional static load is imposed on a 1-mm
causing the elastic energy stored from the primary power stroke to be liberated
actin ®lament from, say, a laser trap, the predicted load±velocity relation is shown
during the ®rst power stroke of the next site in the sequence.
in the inset.
NATURE | VOL 396 | 19 NOVEMBER 1998 | www.nature.com
Nature © Macmillan Publishers Ltd 1998
281
letters to nature
bThr 304 (Escherichia coli sequence numbering); it activates binding of nucleotide successively to each catalytic site. The switch at
gArg 242 and bGlu 381 of the conserved DELSEED sequence in the
b-subunit (S2 in Figs 2, 3) activates release of phosphate at each site.
The interactions between the g- and the b-subunits at the switch
regions are electrostatic and are communicated to the catalytic
sites to produce entropic and steric effects that do not involve
appreciable changes in free energy.
From the con®gurational geometry and elastic constants, we can
compute the sequence of elastic potentials the g-subunit experiences from a b-subunit as it undergoes its cycle of nucleotide
binding, hydrolysis and product release (Fig. 3b). The elastic
energy curves of the other two b-subunits are identical to those
shown, but shifted by 2p/3 and -2p/3 respectively. The potential
experienced by g from the three driving b-subunits is the sum of
these three sets of potentials. In Fig. 3b, this is represented by the
three connected points (offset by 2p/3) on one set of potentials. At
one site, the cycle proceeds as indicated by the numbers 1±5,
starting where the empty site binds ATP. This drops the state
from point (1) on E to point (2) on T; this transition is triggered
by S1. The exact angle, v, at which ATP binds is stochastic; the
vertical arrow in Fig. 3b indicates a typical transition point; the
binding of nucleotide strains the catalytic site, and release of this
stress through the mechanical escapement shown in Fig. 2 rotates
the 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 secondary
power stroke. The energy drop from …4† ! …5† on the D curve in
Fig. 3b releases the elastic strain stored during the primary power
stroke. This takes place during the primary power stroke of the next
site. Thus, each 2p/3 rotation of g is driven by two bs: the binding of
nucleotide to one b and the release of elastic strain from the
previous b when phosphate is released.
All that remains to complete the model is to specify the chemical
rates, K(v, s), in equation (2), which control the transition between
the 64 different elastic potentials. The transition rates are determined by the free energy difference and the energy barrier between
adjacent states. As already discussed and summarized in Box 1, the
activation energy barrier of a catalytic site is affected by the chemical
state of the other two sites and the interaction with the two switch
regions on g (see Supplementary Information for a complete
description of the mathematical model and its numerical simulation). The results of these computations are summarized in Fig. 4.
Figure 4a shows typical trajectories for various ATP concentrations and viscous loads. At saturated ATP concentrations (2 mM)
and with no additional viscous load, the motor rates in steps of
2p/3, consuming one ATP per step (curve A). The rate-limiting step
under these conditions is the release of ADP15. At low ATP
concentrations, and with a large viscous load imposed by the
actin ®lament attached to the g-subunit5,6, the motor also behaves
as a stepper (curve C in Fig. 4a). In this situation, the motor is
limited by the diffusion of ATP to the catalytic sites16. At high ATP
concentrations and loads, the step-like progression of the motor is
obscured by the viscous drag (curve B in Fig. 4a). At low ATP
concentrations, the motor occasionally executes backward steps
when nucleotide binds to the `incorrect' site (curve C in Fig. 4a).
These curves reproduce quantitatively the measurements of Yasuda
et al.6. The mechanical ef®ciency of the motorÐcomputed as the
ratio of the average viscous dissipation per step to the free energy of
hydrolysisÐis very high, approaching 100% for large viscous loads.
This high ef®ciency is possible because the motor is not a heat
engine, but is driven almost completely by elastic strain energy.
Figure 4b shows the expected load±velocity relation, together
with 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
282
energy conferred by nucleotide binding in two stages. The primary
power stroke begins immediately upon nucleotide binding. This
power stroke performs two tasks: it drives the g-subunit in a
full 2p/3 rotation, and it also compresses the passive elastic element,
storing nearly half of the elastic energy of binding. This elastic
energy is released in a secondary power stroke that assists the next
hydrolysis site during its primary power stroke. This two-stage
release of stored energy smoothes the output torque and contributes
to the high mechanical ef®ciency of the motor.
The model has three other predictions (see Supplementary
Information): (1) it correctly predicts the measured rates of ATP
synthesis when driven in reverse by a torque corresponding to that
generated in Fo (refs 2, 17); (2) the proton ¯ux through Fo coupled
to the rotation of g is blocked by small concentrations of ADP but
then increases under normal synthesis conditions, in accordance
with results from `slip' experiments18; and (3) the average occupancy
of the catalytic sites as a function of ATP concentration agrees well
with experiments13.
In summary, the model contained in equations (1)±(3) can
quantitatively explain the principal features of the F1 motor. The
motor rotates in steps of 2p/3, generating up to 45 pN nm of torque
and consuming, on average, a single ATP per step, with a mechanical
ef®ciency approaching 100%. To perform in this way, the motor
cannot operate as a heat engine but converts the free energy of
nucleotide binding into elastic strain energy. This strain energy
drives the rotation of the g-subunit through a tightly coupled
mechanical mechanism, which is closely coordinated with the
kinetic transitions. The general features of energy transduction in
F1 may have implications for other protein motors driven by
nucleotide hydrolysis.
M
Received 8 June; accepted 11 September 1998.
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heart mitochondria. Nature 370, 621±628 (1994).
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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 proton
transport 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 mitochondrial
respiratory chain. J. Biol. Chem. 261, 14031±14038 (1986).
18. Groth, G. & Junge, W. Proton slip of the chloroplast ATPase: its nucleotide dependence, energetic
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Supplementary information is available on Nature's World Wide Web site (http://www.nature.com) or
as 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 of
the three con®gurations which simulated our interpolated videos; R. Nakamoto for advice on the switch 1
and 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 and
suggestions on the manuscript.
Correspondence and requests for materials should be addressed to G.O. (e-mail: [email protected].
edu).
Nature © Macmillan Publishers Ltd 1998
NATURE | VOL 396 | 19 NOVEMBER 1998 | www.nature.com
8
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
Additional results
The F1 model correctly predicts the ATP synthesis
rate when torque is applied to the g -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 interactions
between rotor and stator that are known to be important were missing in their
experiments. These interactions appear unnecessary in the hydrolysis direction, and so
we 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 must
be coordinated with the binding of reactants and the release of product. First, the
rotation of g must be retarded until ADP and phosphate are sequentially bound at the
catalytic site. Otherwise, rotation will frequently bypass the phosphate binding sector
resulting in a wasted rotation. Second, the catalytic site must trap ATP for release. That
is, since ATP is in equilibrium with ADP and Pi with Keq ~ 1, there must be a mechanism
to alter the catalytic site so that ATP is favored as rotation of g applies strain to the site
to open it up. The shapes of the potentials in Figure 3b in the text do not ensure this
sequence 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 to
synthesize ATP at the appropriate rate when a torque of ~ 45 pN-nm is applied to the
g-shaft, the amount necessary to release the tightly bound nucleotide from the catalytic
site 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 in
Figure 1 accomplished the task. The sequence of events is enumerated in the figure
caption. The shape of the E (Empty) potential retards rotation until ADP is bound. The
barrier 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, lest
the site release reactants rather than product. Interestingly, the modified potentials
shown in Figure 1 drive the motor in the hydrolysis direction somewhat more
efficiently than those in Figure 3b in the text. Figure 2a shows that ATP is produced at
the observed rate when this torque is applied to the g subunit.
At physiological conditions, the transmembrane protonmotive force in mitochondria is
220 ~ 230 mV (Stryer, 1995). The observed maximum ATP synthesis rate at saturated
oxygen supply is about 400 ATP/s/F1 (Boyer, 1993; Matsuno-Yagi, et al., 1986). During
ATP synthesis, the rotary torque applied to the g shaft of F1 is generated in F0 by a
protonmotive force. To simulate ATP synthesis rate as a function of the protonmotive
force, 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
1
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
2b, the correct ATP synthesis rate is achieved when the protonmotive force across Fo is
in the physiological range.
40
Synthesis
20
E
⑧
DG/kBT
0
⑥
⑨
T
④
⑦
②
⑤
-20
DP
③
D
①E
-40
-1
0
1
2
3
q/p
Figure 1. The free energy potentials required for nucleotide synthesis. In a typical cycle
starting 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), the
system drops to point (3) on the D curve when ADP binds to the site. Further
rotation drives the system up the D curve until, at point (4) phosphate is bound
and the system drops to the DP potential curve at point (5). Continued rotation
further strains the occupied catalytic site driving the system up the DP curve
until at some point (6) ATP is formed at the transition from (6) to (7). The system
is now trapped in the T state where ATP is bound tightly. Rotational torque from
Fo 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 the
empty, E, state. The ÔbumpsÕ on the elastic energy curves are necessary in order
to trap reactants as g turns to avoid futile rotations. These features can be
incorporated into the mechanical model as additional elastic elements. This does
not address their origin within the structure of F1 since there is no unique
2
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
relationship between the elastic potentials and a mechanical realization.
However, there are certain features of F1 that are candidate structures. Most
noticeably, the e subunitÑwhich is absent in the hydrolysis experiments (Noji, et
al., 1997; Yasuda, et al., 1998)Ñis in the right position to coordinate the
phosphate trap. The ADP and ATP traps also may be ascribed to e, and/or to
strain at the catalytic sites induced by additional elastic elements in b. Specific
assignment of these features awaits more detailed structural analysis of F1.
ATP synthesis rate [1/s]
600
400
200
0
-100
40
50
Torque from F0 [pN×nm]
3
60
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
ATP synthesis rate [1/s]
600
400
200
0
-100
100
200
300
Protonmotive force [mV]
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 is
applied 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 is
modeled by combining the F1 model in this study with the F0 model in our
previous 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 for
physiological protonmotive forces (shaded region). The protonmotive force is
given 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 nucleotide
concentrations 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 half
maximum synthesis rate (Matsuno-Yagi, et al., 1986).
The occupancy of the catalytic sites as a function
of ATP concentration
By inserting a tryptophan residue in position b331 of E. coli F1-ATPase, SeniorÕs group
designed an optical probe which directly monitors occupancy of the three catalytic sites
by nucleotides (Weber, et al., 1997). The relation between the ATP concentration and
4
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
# catalytic sites occupied per F1
the occupancy of the catalytic sites was recorded by repeating the experiment over a
wide range of ATP concentrations.
3
2
1
0 -7
10
-6
10
-5
-4
10
3
[ATP] = 10-7M
-3
# catalytic sites per F1
# catalytic sites per F1
3
2
1
0
-3
10
10
ATP concentration [M]
E
T
DP
[ATP] = 10 M
2
1
0
D
E
T
DP
D
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 of
the F1 model and the filled circles are the experimental data from Weber and
Senior (Weber, et al., 1997). The two bottom panels show the computed
-7
-3
occupancies of the catalytic sites for 10 M and 10 M (saturated) ATP
concentrations.
5
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
As another validity test for the F1 model, we compared the theoretical prediction of the
F1 model with their experimental results. We calculated the average number of catalytic
sites occupied per F1 as a function of the ATP concentration. In the top panel of Figure
3, the solid line represents the theoretical prediction and the filled circles represent the
experimental data from Weber and Senior (Weber, et al., 1997). The F1 model
quantitatively predicts the dependence of the catalytic site occupancy on the ATP
concentration. 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. In
the two bottom panels of Figure 3, the detailed occupancies of the catalytic sites are
computed for low and saturated ATP concentrations.
The proton flux through F0 coupled to the rotation
of g is blocked by small concentrations of ADP
The F-ATPase couples proton flow to ATP synthesis; in the absence of nucleotide ATP
synthesis cannot proceed. However, driven by the torque generated in Fo, the g shaft
can continue to rotate. This rotation of g allows an unproductive proton flow through
the membrane, usually called leakage or proton slip. Groth and Junge found that this
proton 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 the
productive and the unproductive proton flows) as a function of the ADP concentration
passes through a minimum. At low concentrations (less than 1 mM), ADP reduces the
proton leakage and so the total proton flow decreases with the ADP concentration
increases. At concentrations above 1 mM, ADP dramatically increases ATP synthesis
activityÑand the productive proton flux coupled to itÑso that the total proton flux
increases with the ADP concentration.
We found that the model reproduces this curious experimental observation: proton flux
through Fo coupled to the rotation of g is blocked by small concentrations of ADP, but
then increases at the normal synthesis conditions. Figure 4 shows the total proton flux
as a function of ADP concentration, predicted by the model. The total proton flux passes
through a minimum as the ADP concentration increases from 0.01 mM to 1 mM. Pi
concentration is 0.5 mM as in the experiments. This unexpected correlation with
experiment supports the assumptions of the model.
6
Proton flux [1/s]
T HE F 1
300
MOTOR:
SUPPLEMENTARY M ATERIAL
Pi 0.5 mM
200
100
0
10-8
10-6
10-4
ADP Concentration [M]
Figure 4. The total proton flux as a function of ADP concentration predicted by the
model. The total proton flux passes a minimum as the ADP concentration
increases 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).
7
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
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 extract
approximate conformational changes of each subunit. Figure 5a,b shows a side and top
view of F1 in ribbon display mode; Figure 5c shows a side view in spacefill display
mode, 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 binding
sites located at the a-b interfaces; the 3 catalytic sites are mostly on the b subunit, while
the 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 subunit
rotates with respect to the a3b3 hexamer as F1 hydrolyzes ATP. The bottom portion (in
Figure 5) of the a3b3 hexamer is symmetric and is thought to constitute the
hydrophobic ÔbearingÕ surface between a3b3 and g. This portion does not change
appreciably during the rotation of g. The g subunit and the top portion of the a3b3
hexamer are asymmetric and complementary, so that the asymmetry of the g subunit
fits the asymmetry of the hexamer, suggesting that the g subunit is rotated by changing
the asymmetry of the top a3b3 hexamer.
In order to analyze the motions of F1 we first place a cylindrical coordinate system on
the bottom portion of the a3b3 hexamer, since it undergoes little or no conformational
change during the rotation of g. Figure 6a shows a view from the bottom of the barrel
portion of F1 where the symmetry of the a3b3 hexamer is apparent. Three a-carbons
from three b-Glu26 residue groups (one from each b) are shown as black dots. These
three atoms form an equilateral triangle. We position the origin of our cylindrical
coordinate system at the center of this triangle and erect the z-axis perpendicular to the
plane of the triangle (pointing from F1 towards Fo as shown in Figure 6a). We number
the 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-axis
pointing upwards.
8
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
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 of
one a, one b and g. The aÕs are in yellow; the stationary lower portions of
the bÕs are in green and the moveable upper portions are in red. The two coils
of 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.
9
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
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, one
from each b. The coordinate system is determined by these three atoms (see
text). 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 rotates
one 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 chemical
states of the three catalytic sites and a 2p /3 advance of the conformational asymmetry
of the a3b3 hexamer. That is, after a 2p/3 rotation of the g subunit, b1 takes the
configuration 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 a3b3
hexamer propagates by 2p /3. Thus after the g subunit rotates by 2p/3, the new
coordinates of b1 are the original coordinates of b3 rotated by 2p /3 in the cylindrical
coordinate system. The new coordinates of b2 and b3 are calculated from the original
coordinates of b1 and b2 respectively. The new coordinates of b1 after a 4p /3 rotation
of 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 displacements
of g between 0 and 2p/3, we estimate the new coordinates of b1 as a linear interpolation
of the coordinates for q=0 and the coordinates for q=2p/3 in the cylindrical coordinate
system. The new coordinates of b2, b3, a1, a2 and a3 are calculated accordingly. In this
way, we can compute a sequence of configurations corresponding to different angular
displacements of g in F1.
10
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
Kinematics of the F1 motor
Once we knows how to compute the configuration of F1 for any angular displacement
of g, we can examine the kinematics of the F1 motor by comparing the configurations of
F1 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). For
clarity, we show only one a, one b and g, giving a cross sectional view of the a3b3
hexamer. As before, the top portion of the b subunits are colored red and the bottom
portion colored green.
From studies such as Figure 7, we can draw the following conclusions about the
conformational motions of F1:
·
During the rotation of g, there is very little conformational change in the bottom
portion 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 both
the middle ring and the bottom ring are centered on the z-axis of the coordinate
system 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 subunit
is 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 part
of g. This ring is also off the z-axis. The position of this top ring relative to the z-axis
determines the position of the top part of g relative to the z-axis. Since the bottom
part of g is held by two rings centered on the z-axis, the position of the top part of g
relative 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 offaxis top ring about the centerline, while the middle and bottom rings centered on the zaxis act as bearings for the lower portion of g. Thus the asymmetrically curved g subunit
is driven by the three b subunits in a fashion analogous to 3 arms cranking an
automobile jack. To visualize the motion of F1, we have used the interpolation program
to make movies of F1 as g rotates. These movies in MPEG and QuickTime formats can
be downloaded from the web site http://teddy.berkeley.edu:1024/ATP_synthase/.
11
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
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 close
contact between g and the annulus in a3b3. The bottom two contact circles
remain approximately concentric, while the top contact circle rotates off-center
about the vertical axis (dashed line). The corresponding animation movie is
F1_abg.mpeg.
12
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
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 with
specular 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 with
specular 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 with
specular highlights.
F1_abg.mpeg
Cross-section view of one a, one b and g in cartoon display
mode.
F1_abg_fill.mpeg
Cross-section view of one a, one b and g in spacefill display
mode.
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 display
mode.
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 motor
As shown in Figure 7, the g subunit is constrained by two rings at the middle and the
bottom levels. When the top ring moves around the z-axis, the g subunit must rotate to
accommodate all three rings. That is, the motion of the top ring around the z-axis drives
the rotation of the g subunit. The motion of the top ring is driven, in turn, by hinge
motions of each b subunit.
The structure of an empty b subunit suggests the ÔopenÕ configuration is the relaxed
state of the empty b subunits (Shirakihara, et al., 1997). The model assumes that the
13
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
hinge motion of a b subunit from ÔopenÕ to ÔclosedÕ is driven by the binding of
nucleotide to the catalytic site. We call this introduction of binding free energy at the
catalytic site the primary power stroke. It can be modeled as an elastic element which is
switched on (stretched and engaged) upon ATP binding at the catalytic site. We shall call
this elastic element the Ôactive springÕ.
We further assume that the hinge motion of a b subunit from ÔclosedÕ back to ÔopenÕ is
driven by the ÔrecoilÕ elasticity resulting from the compression of the primary power
stroke. That is, the secondary power stroke can be modeled by a spring that is
compressed during the hinge motion from ÔopenÕ to ÔclosedÕ of the primary power
stroke. This spring represents the intrinsic elasticity of the b subunits, which is
established during the initial assembly of the a3b3 hexamer. For this reason, we call this
the Ôpassive springÕ. Thus at the completion of the primary power stroke the active
spring is switched off, and the b subunit can recoil back to the ÔopenÕ configuration
driven by the passive spring. This mechanical model captures the essential energy
transfers that take place within F1 during the hydrolysis cycle. The timing and sequence
of the energy transfers are controlled by the chemistry of the hydrolysis cycle, which
we now discuss.
Kinetics
During ATP synthesis and hydrolysis, each b subunit passes through at least four
chemical states (Equation 1 in the text):
Hydrolysis
ATP binding
P release
ADP release
i
E ¬¾¾¾¾® T ¬¾¾¾
¾® D × P ¬¾¾¾
® D ¬¾¾¾
¾® E
Empty
ATP
bound
ADP & Pi
bound
ADP
bound
Empty
(1)
b 1, b 2, b 3)
We denote by b i the chemical state of the ith b subunit, and denote by S = (b
the chemical state of the a3b3 system. For each b subunit, the possible chemical states
are
{ 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
(3)
These 4 = 64 possible kinetic states can be visualized as the integer points in a 4x4x4
cube. Actually the kinetic states form a 3-dimensional torus, since when the system exits
one face of the cube, it reenters the opposite face.
The reaction process at each b catalytic site can be described by the Markov process:
14
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
E ¬
¾®
T
b
b
D ¬
¾® D × P
(4)
It is important to stress that the progression of the system through the 64-state cube is
stochastic. That is, we do not impose a deterministic sequence of chemical states; we
shall specify only the transition rates between states, and let the system evolve
stochastically as the dynamics dictates. Thus the formulation given here differs
fundamentally from other kinetic schemes proposed for ATP synthase. However,
under some circumstances (e.g. ambient concentrations), there may be a most probable
path. 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 b i-1,
bi and bi+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 the
bÕs are in chemical states bi-1, bi and bi+1, respectively. Here q is a parameter. By fixing
the g subunit at an angular position, we are able to isolate the kinetics and study its
governing equation. In the subsequent sections, we will combine the kinetics and the
dynamics to build a complete model for F1 motor.
The governing equation for the Markov process on bi is
ér(q; t, b i -1 , E, b i +1 ) ù é - S E
ú ê
ê
d êr(q; t, b i -1 , T, b i +1 ) ú ê k E ® T
=
dt êr(q; t, b i -1 , DP, b i +1 )ú ê 0
ú ê
ê
êër(q; t, b i -1 , D, b i +1 ) úû ë k E ® D
0
k T®E
-ST
k DP ® T
- S DP
k DP ® D
k T ® DP
0
k D ® E ù ér(q; t, b i -1 , E, b i +1 ) ù
0 ú êêr(q; t, b i -1 , T, b i +1 ) úú
ú×
k D ® DP ú êr(q; t, b i -1 , DP, b i +1 )ú
ê
ú
- S D úû êër(q; t, b i -1 , D, b i +1 ) úû
(5)
where
S E = k E®T + k E®D
S T = k T ® E + k T ® DP
S DP = k DP ® T + k DP ® D
(6)
S D = k D ® E + k D ® DP
All transition rates depend on q, the angular position of g relative to b, and the chemical
states of the other two bÕs:
k E ® T = k E ® T (q, b i -1 , b i +1 ),
k T ® DP = k T ® DP (q, b i -1 , b i +1 ),
L
In the matrix-vector form, equation (5) is
15
(7)
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
d
ri (q; t, b i -1, b i +1 ) = K(q, b i -1 , b i +1 ) × ri (q; t, b i -1, b i +1 )
dt
(8)
ér(q; t, b i -1 , E, b i +1 ) ù
ú
ê
r(q; t, b i -1 , T, b i +1 ) ú
ê
ri (q; t, b i -1 , b i +1 ) =
êr(q; t, b i -1 , DP, b i +1 )ú
ú
ê
êër(q; t, b i -1 , D, b i +1 ) úû
(9)
where
In Equation 2 in the text, we write equation (8) symbolically as:
db i
= K(q, b i -1 , b i +1 ) × b i
dt
,
i = 1, 2, 3
(10)
There are 16 choices for the combination of (bi-1, bi+1). So there are 16 different
probability density vector, ri , each of which is a vector with 4 components. These riÕs
form a vector with 64 elements.
ér(q; t, E, E, E ) ù
ú
ê
êr(q; t, E, E, T ) ú
êr(q; t, E, E, DP ) ú
ú
ê
r(q; t, E, E, D) ú
ê
r(q; t ) =
êr(q; t, E, T, E ) ú
ú
ê
M
ú
ê
êr(q; t, D, D, DP )ú
ú
ê
êër(q; t, D, D, D) úû
ü
ï
ï
ï
ï
ï
ý 64 components
ï
ï
ï
ï
ï
þ
(11)
Combining equation (5) for all combinations of (bi-1, bi+1), we have the governing
equation for the probability vector, r(q, t), of the a3b3 system:
d
r(q; t ) = K(q) × r(q; t )
dt
(12)
where the transition matrix K(q) is a 64x64 sparse matrix with only 6x64=384 non-zero
elements. The elements of the transition matrix of the system, K(q), are related to the
elements of the transition matrix of the ith b, K(q, bi-1, bi+1), as:
16
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
k ( E , E , E )® ( E , T , E ) (q) = k E ® T (q, E, E ),
2p
k ( E , E , E ) ® ( E , E , T ) (q ) = k E ® T æ q , E, E ö ,
è
ø
3
2p
k ( E , E , E ) ® ( T , E , E ) (q ) = k E ® T æ q +
, E, E ö ,
è
ø
3
(13)
L
Hence we only need to consider the transition matrix of one b for 16 different chemical
state combinations of the other two bÕs.
Coupling of the reaction on b to the rotation of g
To simulate the chemical reactions taking place at each catalytic site, we must construct
the transition matrix of one b for 16 different chemical state combinations at the other
two bÕs. Since each transition matrix has 8 non-zero elements (see equation (5)), there
are 8 ´ 16 independent transition rates. However, we can build all of these starting
from 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 under
consideration (q dependent).
The word ÔmultisiteÕ suggests that factor #1 is most important; however, the fast
multisite reactions are only observed when more than one catalytic site is occupied and
the g subunit is present (Kaibara, et al., 1996). This implies that the g subunit plays an
important role in the multisite reactions, and so factors #2 and #3 must be taken into
consideration.
We can express these factors mathematically as:
æ l × DE(q) ö
U
kM
÷
forward (q, b i -1 , b i +1 ) = k forward × f (b i -1 , b i +1 ) × g(q) × expç
è k BT ø
k
M
backward
(q, bi -1, bi +1 ) = k
U
backward
(14)
æ -(1 - l ) × DE(q) ö
× f (b i -1 , b i +1 ) × g(q) × expç
÷
k BT
è
ø
M
U
where k represents the multisite transition rates between two chemical states, k
represents the unisite rates. The function f(bi-1, bi+1) represents the effect of the other
two sites, g(q) simulates the effect of the two switches on g, and DE(q) is the elastic
energy difference between the two chemical states of the b under consideration. To
17
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
keep the notation simple, we shall drop the superscript M for multisite rates from now
on.
Table 2 summarizes the mathematical formulation of the multisite reaction rates along
with the unisite rates and other parameter values.
PARAMETERS
VALUES
k
Unisite reaction
rates of E. Coli.
(Senior, 1992)
k
k
k
U
5 -1
E®T
-1
= 1.1 ´ 10 s M , k
U
-1 -1
T®DP
= 1.2 ´ 10 s ,
DP®D
= 1.2 ´ 10 s ,
U
k
-3 -1
U
k
-3 -1
D®E
= 1.6 ´ 10 s ,
k
U
-5 -1
= 2.5 ´ 10 s
T®E
U
-2 -1
DP®T
= 4.3 ´ 10 s
D®DP
= 4.8 ´ 10 s M
U
-4 -1
U
2 -1
E®D
= 1.8 ´ 10 s M
-1
0
DGE®T = DG E®T + log([ATP]/M)×kBT
0
U
U
DG E®T = log(k E®T / k T®E) = 22.2 kBT
U
T®DP
DGT®DP = log(k
Free energy drops
of unisite reaction
of E. Coli.
/k
U
DP®T)
= 1.0 kBT
0
DGDP®D = DG DP®D - log([Pi]/M)×kBT
0
U
U
DG DP®D = log(k DP®D / k D®DP) = 0.9 kBT
0
DGD®E = DG D®E - log([ADP]/M)×kBT
0
U
U
DG D®E = log(k D®E / k E®D) = -11.6 kBT
18
-1
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
PARAMETERS
VALUES
æ l × DE(q) ö
U
kM
÷
forward (q, b i -1 , b i +1 ) = k forward × f (b i -1 , b i +1 ) × g(q) × expç
è k BT ø
æ -(1 - l ) × DE(q) ö
U
kM
÷
backward (q, b i -1 , b i +1 ) = k backward × f (b i -1 , b i +1 ) × g(q) × expç
k BT
è
ø
Construction of
the multisite
M
reaction rate, k ,
from the unisite
U
rate, k .
f(bi-1, bi+1) simulates the effect of the occupancy of the other two
sites (i.e. binding of ATP on other sites increases the reaction on
this 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 chemical
states of the b in consideration (q dependent).
See Table 3 for details of the multisite reaction rates.
Elasticity constants
of the active and
passive springs
kActive = 10 pN/nm, kPassive = 4 pN/nm
The active and passive springs are pre-stretched (precompressed) by 4 nm. During the reaction, the additional
displacement of these springs is 2 nm.
Table 2. Unisite reaction rates, elasticity constants and the mathematical formulation of
the 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 accommodated
easily into this framework and simulated. Next we discuss the multisite reaction rates in
detail.
Transitions between E and T
k E ® T (q, b i -1 , b i +1 ) = k UE ® T × fE , T (b i -1 , b i +1 ) × g E , T (q),
k T ® E (q, b i -1 , b i +1 ) = k TU® E × fE , T (b i -1 , b i +1 ) × g E , T (q)
æ ( E - E T ( p)) - ( E E - E E (0)) ö
× expç T
÷
k BT
è
ø
19
(15)
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
b i -1 = b i +1 = E (unisite mode)
ì1
fE , T (b i -1 , b i +1 ) = í
(multisite mode)
î150 otherwise
(16)
0 < q < p/2
ì1
ï
ATP binding promoted by switch #1
ïï
g E , T (q) = í0.05 - p / 2 < q < 0 and one of (b i-1 , b i +1 ) is E
ï
allows ATP binding on the wrong site
ï
ïî0
otherwise
(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 states
of the other two sites is modeled as either putting the system into the unisite mode or
the multisite mode.
Equation (17) contains both the effect of stress at the catalytic site and the effect of
switch #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 the
system 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 the
region [-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 the
q-range [-p/2, 0] leads to a backward step. Switch #1 on the g subunit promotes ATP
binding in the q-range [0, p/2], but does not completely inhibit the ATP binding in the
q-range [-p/2, 0]. This allows the occasional backward steps.
Transitions between T and D×× P
æ -EB ö
k T ® DP (q, b i -1 , b i +1 ) = k TU® DP × fT , DP (b i -1 , b i +1 ) × g T , DP (q) × expç DP ÷
è k BT ø
æ - E BT ö
U
k DP ® T (q, b i -1 , b i +1 ) = k DP
×
f
b
,
b
×
g
q
×
exp
(
)
(
)
ç
÷
T , DP
i -1
i +1
T , DP
®T
è k BT ø
(18)
b i -1 = b i +1 = E (unisite mode)
ì1
fT , DP (b i -1 , b i +1 ) = í 5
(multisite mode)
î10 otherwise
(19)
g T , DP (q) = 1
(20)
20
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
The total elastic energy of a b consists of three parts:
E = EA + EP + EB
A
(21)
P
E and E are the elastic energies associated with the ÔactiveÕ and ÔpassiveÕ springs,
B
respectively, and E 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 is
always engaged. Thus the total elastic energies of the b for states E, T, D×P and D are
given respectively by
EE
ET
E DP
ED
=
= EA
= EA
=
EP
+ EP
+ EP
EP
+
+
+
+
E BE
E BT
E BDP
E BD
(22)
B
DP
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 elastic
B
components (other than the two major springs) of the b. Correspondingly E T is that
part of the elastic energy of the b in state T.
In equation (18), E
We use superscript ÔBÕ to denote the additional elastic energy associated with the
ÔbumpsÕ in the elastic energy curves. These bumps are not necessary when F1 is
hydrolyzing ATP and functioning as a motor. However, they are essential when F1 is
turned in the reverse direction by Fo to synthesize ATP. The function of these bumps in
ATP synthesis is to prevent ÔfutileÕ rotations. When the g subunit is turned one full
revolution 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.
21
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
3. After the binding of Pi, ATP does not form and b either stays in state D×P or loses its
PI 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 catalytic
site 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 synthesis
reaction to the rotation of g. More specifically, as g rotates, we guide the b from state E
into 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 the
catalytic site).
B
B
B
DP
The important roles of the additional elastic energies (E E, E T, E
and E
B
D)
in the
ATP synthesis are shown in Figure 1.
Transitions between D×× P and D
U
k DP ® D (q, b i -1 , b i +1 ) = k DP
® D × fDP , D (b i -1 , b i +1 ) × g DP , D (q)
æ ( E - E DP ( p)) ö
× expç DP
÷
k BT
è
ø
k D ® DP (q, b i -1 , b i +1 ) = k DU ® DP × fDP , D (b i -1 , b i +1 ) × g DP , D (q)
(24)
æ ( E - E D (0)) ö
× expç D
÷
k BT
è
ø
ì1
fDP , D (b i -1 , b i +1 ) = í 6
î10
b i -1 = b i +1 = E (unisite mode)
otherwise
(multisite mode)
ìï1 p < q < 4 p
q
g DP , D ( ) = í
3
îï0 otherwise
(switch # 2)
(25)
(26)
In equation (26), Pi release (Pi binding in the ATP synthesis direction) is regulated by
switch #2 on the g subunit.
Transitions between D and E
æ -EB ö
k D ® E (q, b i -1 , b i +1 ) = k DU ® E × fD, E (b i -1 , b i +1 ) × g D, E (q) × expç E ÷
è k BT ø
k E ® D (q, b i -1 , b i +1 ) = k
U
E®D
æ - E BD ö
× fD, E (b i -1 , b i +1 ) × g D, E (q) × expç
÷
è k BT ø
22
(27)
T HE F 1
MOTOR:
ì1
fD, E (b i -1 , b i +1 ) = í
4
î3 ´ 10
SUPPLEMENTARY M ATERIAL
b i -1 = b i +1 = E (unisite mode)
otherwise
(multisite mode)
ì
2p
2p
ï1 <q<
g DP , D (q) = í
3
3
ïî0 otherwise
æ binding regulated byö
ç
÷
è site configuration ø
In equation (29), the ADP release (ADP binding in the ATP synthesis direction) is
regulated by the configuration of the catalytic site.
23
(28)
(29)
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
PARAMETERS
VALUES
k E ® T (q, b i -1 , b i +1 ) = k UE ® T × fE , T (b i -1 , b i +1 ) × g E , T (q),
k T ® E (q, b i -1 , b i +1 ) = k TU® E × fE , T (b i -1 , b i +1 ) × g E , T (q)
æ ( E - E T ( p)) - ( E E - E E (0)) ö
× expç T
÷
k BT
è
ø
kE®T and kT®E
b i -1 = b i +1 = E (unisite mode)
ì1
fE , T (b i -1 , b i +1 ) = í
(multisite mode)
î150 otherwise
0 < q < p/2
ì1
ï
ATP binding promoted by switch #1
ïï
g E , T (q) = í0.05 - p / 2 < q < 0 and one of (b i-1 , b i +1 ) is E
ï
allows ATP binding on the wrong site
ï
ïî0
otherwise
ET is the total elastic energy of the b in state T.
EE is the total elastic energy of the b in state E.
æ -EB ö
k T ® DP (q, b i -1 , b i +1 ) = k TU® DP × fT , DP (b i -1 , b i +1 ) × g T , DP (q) × expç DP ÷
è k BT ø
kT®DP and kDP®T
æ - E BT ö
U
k DP ® T (q, b i -1 , b i +1 ) = k DP
×
f
b
,
b
×
g
q
×
exp
(
)
(
)
ç
÷
T , DP
i -1
i +1
T , DP
®T
è k BT ø
b i -1 = b i +1 = E (unisite mode)
ì1
fT , DP (b i -1 , b i +1 ) = í 5
(multisite mode)
î10 otherwise
g T , DP (q) = 1
B
E DP 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 elastic components (other than the
two major springs) of the b.
24
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
PARAMETERS
VALUES
U
k DP ® D (q, b i -1 , b i +1 ) = k DP
® D × fDP , D (b i -1 , b i +1 ) × g DP , D (q)
æ ( E - E DP ( p)) ö
× expç DP
÷
k BT
è
ø
kDP®D and kD®DP
k D ® DP (q, b i -1 , b i +1 ) = k DU ® DP × fDP , D (b i -1 , b i +1 ) × g DP , D (q)
æ ( E - E D (0)) ö
× expç D
÷
k BT
è
ø
ì1
fDP , D (b i -1 , b i +1 ) = í 6
î10
b i -1 = b i +1 = E (unisite mode)
otherwise
(multisite mode)
ìï1 p < q < 4 p
g DP , D (q) = í
3
ïî0 otherwise
(switch # 2)
æ -EB ö
k D ® E (q, b i -1 , b i +1 ) = k DU ® E × fD, E (b i -1 , b i +1 ) × g D, E (q) × expç E ÷
è k BT ø
kD®E and kE®D
æ -EB ö
k E ® D (q, b i -1 , b i +1 ) = k UE ® D × fD, E (b i -1 , b i +1 ) × g D, E (q) × expç D ÷
è k BT ø
ì1
fD, E (b i -1 , b i +1 ) = í
4
î3 ´ 10
b i -1 = b i +1 = E (unisite mode)
otherwise
(multisite mode)
ì
2p
2p
ï1 <q<
g DP , D (q) = í
3
3
ïî0 otherwise
B
æ binding regulated byö
ç
÷
è site configuration ø
E D is the part of the elastic energy of the b in state D that is
caused by the additional interaction between the b and the
g, and by the additional elastic components (other than the
two major springs) of the b.
Table 3. Multisite reaction rates
25
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
Calculating the elastic potentials
Figure 8 shows how the bending angle, f, of the b subunit is coupled to the rotational
angle, q, of the g subunit. In the left panel, the plane shown is perpendicular to the
rotation axis of g. The dashed circle represents the cross-section of g at the hydrophobic
ÔbearingÕ section where the g is held tightly at two levels (middle and bottom levels in
Figure 7). The solid filled circle represents the cross section of the g at the ÔdrivingÕ level
where g contacts the bending b subunits. The three small filled circles represent the
three hinge points on the three bÕs. The solid lines between the hinge points and the g
are 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 the
g axis; it is a function of the rotation angle, q, of g . In the right panel, the plane shown is
parallel to the g axis. The top and bottom sectors of b are shown schematically as line
segments. The dashed line represents the relaxed position of the top sector of an empty
b. The azimuthal angle between the top and bottom sectors, f, measures the bending
angle of b. L(f) is the length of the top part of b projected onto the plane perpendicular
to the g axis; it is a function of the bending angle, f. By equating L(q) and L(f), we
obtain a relation between the bending angle, f, and the rotational angle, q. If we assume
the active and passive springs connecting the top and bottom parts of each b are linear
in the bending angle, f, we can calculate the potential caused by these two springs as a
function of the rotational angle, q.
TOP VIEW
SIDE VIEW
g at bearing level
Df
g at driving level
L(q)
L(f)
q
f
b
Figure 8 Coupling between the rotation angle, q, of the g subunit and the bending
angle, f, of the b subunit. In the left panel, the plane shown is perpendicular to
the g axis. In the right panel, the plane shown is parallel to the g axis. See text for
details.
26
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
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 part
of b projected onto the plane perpendicular to the g axis is approximately a linear
function of the bending angle, f. Therefore, the active spring and the passive spring
connecting the top and bottom parts of each b are also linear in the length, L, of the top
part of the b projected onto the plane perpendicular to the g axis. Of course, the linear
relation between f and L is an approximate one, as is the assumption that the springs
are linear in f. In this sense, it is no more reasonable to assume the springs are linear in
terms of f than to assume the springs are linear in terms of L. Thus we shall view the
active and passive springs on each b as linear springs in the projected length L.
r1
r2
L(q)
q
Figure 9 Mathematical calculation of the projected length, L, as a function of the
rotation angle, q.
As shown in Figure 9, the projected length, L, of the top part of b as a function of q is
given by
L(q) = r12 + r2 2 - 2 r1 r2 cos(q) - r1
(30)
where r1 = 1 nm and r2 = 4 nm.
The elastic energy due to the passive spring as a function of q is
E P (q ) =
1 P
k × ( L(q) - L(0)) + DL Pr ecompress
2
[
2
]
-
1 P
k × DL Pr ecompress
2
[
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
27
2
]
(31)
T HE F 1
E A (q ) =
MOTOR:
SUPPLEMENTARY M ATERIAL
1 A
k × ( L( p) - L(q)) + DL Pr estress
2
[
2
]
-
1 A
2
k × [ DL Pr estress ]
2
(32)
where the active spring is prestressed by DLPrestress = 4 nm.
Viewing the passive and active springs in the direction of the projected length L
simplifies the calculation of the potentials, and makes the bending motion of each b
directly 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 is
always 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 three
experimental observations would be violated were we to model the elastic energy of
the 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 state
over what is provided by the passive spring. To do this we write the total free energies
of each b for states E, T, D×P and D as
E E (q )
E T (q )
E DP (q)
E D (q )
=
E P (q )
= E A (q ) + E P (q )
= E A (q ) + E P (q )
=
E P (q )
+
+
+
+
E BE (q)
E BT (q)
E BDP (q)
E BD (q)
(33)
In this equation, the superscripts A and P refer to the elastic potentials of the active and
passive springs, respectively, as computed above. The superscript (B) represents the
part of the elastic energy of the b caused by an additional interaction between the b and
B
g, and/or a. The additional elastic component in empty state, E E, makes empty
ba pairs stiff; that is it creates a deep potential barrier against rotation of the g subunit
when three catalytic sites are empty. It also prevents the g subunit from diffusing over a
large angle when the catalytic site is empty, thus enforcing the observed stepping
behavior.
B
B
B
In the hydrolysis direction, E T, E DP and E D are unnecessary. but in the synthesis
B
B
B
B
direction all four additional terms (E E, E D, E DP and E T) are required for (i) trapping
ADP, (ii) trapping phosphate, (iii) forming ATP and (iv) releasing ATP (see Figure 1).
28
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
B
For example, in the synthesis direction, E E raises the free energy on the E curve such
that the transition E ® D is promoted and the backward transition D ® E is prohibited
when the system is driven up the E curve by a mechanical torque.
Mathematical equations and numerical solutions
Langevin equation formulation
Since inertia is negligible, equating the viscous drag on the rotating g to the other forces
acting on it yields the Langevin equation (Doering, 1990; Risken, 1989),
dq
z
dt
{
Viscous
torque on g
dV(q; S)
= - T
(t )
B3
{L + Á
12
dq43
142
Load
Brownian
Torque between
g and a 3b 3
torque
(34)
torque
where z is the rotational drag coefficient, q is the angular coordinate, V(q; S) is the sum
of the elastic energies of the three bÕs, TL is the load torque, and ÁB is the Brownian
torque due to thermal fluctuations. S = (b1, b2, b3) represents the kinetic state of the
3
three catalytic sites. S can take any of the 4 = 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 rewrite
symbolically as
d
S = K(q) × S
dt
(35)
To compute the torque generated by the F1 motor, equation (34) must be solved
simultaneously with the Markov process governing the kinetic transitions on the three
catalytic sites (equation (35)).
Fokker-Planck equation formulation
In the Fokker-Planck formulation corresponding to equations (34) and (35), the F1
motor 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 at
time t, and the bÕs are in chemical states bi-1, bi and bi+1, respectively. Define the
probability density vector as:
29
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
ér(q, t, E, E, E ) ù
ú
ê
êr(q, t, E, E, T ) ú
êr(q, t, E, E, DP ) ú
ú
ê
êr(q, t, E, E, D) ú
êr(q, t, E, T, E ) ú
r(q, t ) = ê
ú
êr(q, t, E, T, T ) ú
êr(q, t, E, T, DP ) ú
ú
ê
M
ú
ê
êr(q, t, D, D, DP )ú
ú
ê
êër(q, t, D, D, D) úû
ü
ï
ï
ï
ï
ï
ïï
ý 64 components
ï
ï
ï
ï
ï
ï
ïþ
(36)
This probability density vector evolves according to the convective diffusion equations
¶ r(q, t ) 1 ¶ ìæ dV(q)
¶ 2r(q, t )
ü
ö
=
+
+
+ K(q) × r(q, t )
T
r
q
,
t
D
(
)
íç
ý
L÷
ø
¶t
z ¶q îè dq
¶q2
þ
(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
ù
éV1(q)
ú
ê
V2 (q)
ú
ê
ú
ê
O
V(q) = ê
ú
V63 (q)
ú
ê
ú
ê
V
q
64 ( )û
ë
1444444
424444444
3
64 columns
ü
ï
ïï
ý 64 rows
ï
ï
ïþ
(38)
where
V1(q) = E E (q + 2 p / 3) + E E (q) + E E (q - 2 p / 3)
V2 (q) = E E (q + 2 p / 3) + E E (q) + E T (q - 2 p / 3)
V3 (q) = E E (q + 2 p / 3) + E E (q) + E DP (q - 2 p / 3)
V4 (q) = E E (q + 2 p / 3) + E E (q) + E D (q - 2 p / 3)
V5 (q) = E E (q + 2 p / 3) + E T (q) + E E (q - 2 p / 3)
V6 (q) = E E (q + 2 p / 3) + E T (q) + E T (q - 2 p / 3)
M
V63 (q) = E D (q + 2 p / 3) + E D (q) + E DP (q - 2 p / 3)
V64 (q) = E D (q + 2 p / 3) + E D (q) + E D (q - 2 p / 3)
30
(39)
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
Numerical solution of Fokker-Planck equations
The dynamic and kinetic behaviors of the F1 motor are governed by the Fokker-Planck
equations (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 our
numerical methods for general Fokker-Planck equations involves the analysis of the
equations; 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 two
coupled equations.
Consider the equations:
¶r1 1 ¶
¶ 2r
=
f1¢ (q)r1 ) + D 21 - k12 (q) × r1 + k 21 (q) × r2
(
424444
3
¶t z ¶q
¶q 1444
1444
424444
3 Probability flow in
Pr obability flow
in the q-direction
the reaction direction
¶r2 1 ¶
¶ 2 r2
=
+
f
q
r
D
¢
(
)
( 2 2 ) ¶q2 -1k444
21 (q) × r2 + k 12 (q) × r1
424444
3
¶t
z ¶q
1444424444
3 Probability flow in
Pr obability flow
in the q-direction
(40)
the reaction direction
Motion of the system
S1
q
K21
k12
S2
q
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
31
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
state 2, and k21(q) is the transition rate from state 2 to state 1. Equations (40) describe
the conservation of probability at any location q in each state. Figure 10 shows all inflows 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 the
q-direction into intervals of size Dq, and call the center point of the n-th interval qn . Let
rm, n(t) be the probability that the system is in the n-th interval at time t, and is in state
m. Figure 11 shows the discrete probability in-flows and out-flows in both the q and
kinetic directions. The conservation of probability in the n-th interval in state m leads to
differential difference equation of the form:
dr1, n
dt
(
)
= F1, n -1/ 2 × r1, n -1 - B1, n -1/ 2 + F1, n +1/ 2 × r1, n
+ B1, n +1/ 2 × r1, n +1 - k12, n × r1, n + k 21, n × r2, n
dr2, n
dt
(
(41)
)
= F2, n -1/ 2 × r2, n -1 - B2, n -1/ 2 + F2, n +1/ 2 × r2, n
+ B2, n +1/ 2 × r2, n +1 - k 21, n × r2, n + k12, n × r1, n
where the kinetic transition rates in the n-th interval, k12, n and k21, n, are given by
( )
k 21, n = k 21(q n )
k12, n = k12 q n
(42)
Fm, n+1/2 is the forward jump rate from n-th interval to the (n+1)-th interval and B m, 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 FokkerPlanck equations:
·
The numerical method should maintain the principle of detailed balance. The ratio of
the jump rates between two intervals is determined by the potential difference of
the two intervals:
Fm, n +1/ 2
Bm, n +1/ 2
·
( )
(
)
æ f q - f m q n +1 ö
= expç m n
÷
k BT
ø
è
(43)
For a uniform probability distribution, the net flux across qn+1/2 in state m (the
boundary between the n-th and (n+1)-th intervals) is proportional to the derivative
of potential fm. When the derivative of potential fm is a constant, independent of n,
this yields
32
T HE F 1
(F
m, n +1/ 2
MOTOR:
SUPPLEMENTARY M ATERIAL
)
- Bm, n +1/ 2 × Dq =
( )
(
f m q n - f m q n +1
)
(44)
z × Dq
Solving jump rates Fm, n+1/2 and Bm, n+1/2 from equations (43) and (44), we obtain
Fm, n +1/ 2 =
D
×
( Dq ) 2
-
( )
(
f m q n - f m q n +1
æ f
expç - m
è
k BT
q n - f m q n +1 ö
÷ -1
k BT
ø
( )
( )
(
(
)
f m q n - f m q n +1
Bm, n +1/ 2 =
D
×
( Dq)2
æf
expç m
è
)
(45)
)
k BT
q n - f m q n +1 ö
÷ -1
k BT
ø
( )
(
)
F1, n+1/2
S1
qn-1
qn
k12
k21
B1, n+1/2
q
F2, n+1/2
S2
qn-1
qn+1
qn
qn+1
B2, n+1/2
q
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 in
the q-direction, equations (41) are solved using a Crank-Nicolson scheme.
33
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
References
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Boyer, P. 1993. The binding change mechanism for ATP synthase--some probabilities
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34
T HE F 1
MOTOR:
SUPPLEMENTARY M ATERIAL
Stryer, L. 1995. Biochemistry. W. H. Freeman, New York.
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35