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Folding-based molecular simulations revealmechanisms of the
rotary motor F1–ATPaseNobuyasu Koga* and Shoji Takada*†‡
*Graduate School of Science and Technology, Kobe University,
Rokkodai, Nada, Kobe 657-8501, Japan; and †Core Research for
Evolutional Scienceand Technology, Japan Science and Technology
Corporation, Rokkodai, Nada, Kobe 657-8501, Japan
Edited by Peter G. Wolynes, University of California at San
Diego, La Jolla, CA, and approved February 13, 2006 (received for
review November 5, 2005)
Biomolecular machines fulfill their function through large
confor-mational changes that typically occur on the millisecond
time scaleor longer. Conventional atomistic simulations can only
reach mi-croseconds at the moment. Here, extending the minimalist
modeldeveloped for protein folding, we propose the ‘‘switching
Go�model’’ and use it to simulate the rotary motion of
ATP-drivenmolecular motor F1–ATPase. The simulation recovers the
unidirec-tional 120° rotation of the �-subunit, the rotor. The
rotation wasinduced solely by steric repulsion from the �3�3
subunits, thestator, which undergoes conformation changes during
ATP hydro-lysis. In silico alanine mutagenesis further elucidated
which resi-dues play specific roles in the rotation. Finally,
regarding themechanochemical coupling scheme, we found that the
tri-sitemodel does not lead to successful rotation but that the
always-bi-site model produces �30° and �90° substeps, perfectly in
accordwith experiments. In the always-bi-site model, the number of
sitesoccupied by nucleotides is always two during the hydrolysis
cycle.This study opens up an avenue of simulating functional
dynamicsof huge biomolecules that occur on the millisecond time
scalesinvolving large-amplitude conformational change.
always-bi-site model � energy landscape � funnel � switching Go�
model
B iomolecular machines, such as the ribosome, transporter,and
molecular motors, fulfill their function through large-amplitude
conformational change. Structural information be-fore and after the
conformational change has been provided byx-ray crystallography and
other methods for many cases (1).However, these methods do not
directly observe the moleculardynamics that connects two-end
structures. These dynamicalaspects can be observed directly by
fluorescence and othertime-resolved spectroscopy; however, the
latter methods moni-tor local structure but do not give global
structural information.In this sense, molecular dynamics simulation
is potentiallypowerful because it can provide full time-dependent
structuralinformation about biomolecular machines (2–5). Yet,
functionalcycles of these systems typically take milliseconds or
longer,which is far beyond the current reach of molecular
simulationswith all-atom standard force-fields (6). Thus, a
complementaryapproach may be to coarse-grain the molecular
representation,thereby enabling the simulation orders of
magnitude-longer timescales.
Coarse-graining drops some details from the model,
whilehopefully keeping the essence. To achieve them, we need
someperspective on the physical process we are trying to simply.
Uponconformational change, some portions of molecules do
notsignificantly change their structures; here, the atomic motion
canbe well approximated by harmonic fluctuations. It has beenknown
that very simple C� models, such as the elastic networkmodel (7),
can reproduce low-frequency harmonic fluctuationfairly well.
Conversely, large-scale conformational change in-volves
rearrangement of nonlocal contacts between amino acids,which is
beyond the harmonic approximation (8, 9). For approx-imating these
rearrangements, we need to take into account forpartial disorder or
partial unfolding, which induces conforma-tional diversity (8). To
this end, we rely on the energy landscape
perspective of proteins, which has been established in
protein-folding studies. In this perspective, proteins evolved to
havefunnel-like energy landscapes, where the native structure is
atthe bottom of the funnel with the lowest effective energy
anddisordered states have higher energies and large
conformationalentropy (10, 11). Technically, a funnel-like energy
landscape caneasily be realized by Go� models, which have become
one of thestandard models for folding simulation studies (12–18).
Forrepresenting large-amplitude conformational dynamics, Go�models
are suitable because they account for both small f luc-tuations
around the native basin and large fluctuations thatinvolves local
unfolding. Here, we propose a previously unde-scribed computational
framework, which we call a ‘‘switching Go�model,’’ for simulating
large-amplitude motion of biomolecularmachines. We then apply it to
the rotary motion of F1–ATPase.
F1–ATPase, the catalytic part of the energy conversion en-zyme
F0F1–ATP synthase, is known to act as a rotary motor uponATP
hydrolysis (19–24). Single-molecule experiments have vi-sualized
the rotation and elucidated some of the details; the�-subunit
rotates unidirectionally in discrete 120° steps with eachATP
hydrolysis (25), and the 120° step can be divided into 90°and 30°
substeps at lower ATP concentration (26). The dwelltime before the
90° substep depends on ATP concentration, andthe one before the 30°
substep includes two rates of an �1-mstime scale. These findings
suggest that the 90° substep involvesATP binding, and the 30°
substep may correspond to ATPhydrolysis and release of products
(ADP, phosphate, or both).These experiments have given much insight
into the rotationalmechanisms as described above; however, the
resolution of thosestudies is limited to �40 nm, which cannot
provide a completestructural view of the F1 dynamics.
The minimal catalytic complex of F1 contains seven
subunits�3�3�, whose x-ray structure (27) is depicted in Fig. 1a.
Thecentral stalk � is a rotor, whereas the alternately arranged �3
and�3 subunits form a ring acting as the stator of the rotary
motor.There are three catalytic nucleotide-binding sites, which are
atthe interfaces of �-subunits with the neighboring �-subunits.
Inthe crystal structure determined in 1994 (27), one catalytic
siteis filled with ATP analog and another with ADP, and the last
isempty. The structures of the ATP-bound subunits are denotedas �TP
and �TP, where �TP takes on the ‘‘closed form’’ in its Cterminus
(Fig. 1b Right Lower) (28) and the catalytic interface isloosely
packed. The ADP-bound �-subunit, �DP, also takes theclosed form,
and the interface to �DP is tightly packed. Thestructures of
nucleotide-free subunits are denoted as �E and �E,where �E has
‘‘open form’’ in its C terminus (Fig. 1b Left Lower)(28). From the
static structure, we can deduce that the catalysis-induced bending
motion of � between closed and open formsproduce torque on �
leading to rotation (27, 29). Given the staticx-ray structures at
high resolution and the dynamic single
Conflict of interest statement: No conflicts declared.
This paper was submitted directly (Track II) to the PNAS
office.
‡To whom correspondence should be sent at the * address. E-mail:
[email protected].
© 2006 by The National Academy of Sciences of the USA
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2006 � vol. 103 � no. 14 � 5367–5372
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molecule observations at low resolution, currently missing is
thetime-dependent structural insight that connects the two.
In this work, by using our proposed switching Go� model,
werealize structure-based molecular simulations of
F1–ATPaserotation upon hydrolysis, providing us with
time-dependentstructural insights on F1 rotation. We address which
parts ofthe molecule play what roles in rotation in a
site-specificmanner. Furthermore, combining all available
experimentaldata with simulation results, we identify the
mechanochemicalcoupling mechanism, which has long been under debate
in thefield. This work opens an avenue of simulating
large-scalemotion involved in dynamical function of large
biomolecularcomplexes by folding-based model.
ResultsSwitching Go� Model. There are many versions of Go�
models, whichconcisely realize perfect funnel energy landscape (13)
of proteins(13–18). Among them, we use the version of Clementi et
al. (18),where amino acids are represented by single spheres
centered atC� atoms, which are sequentially connected by virtual
bonds, andinteractions are specified so that structures closer to
the nativestructure are more stable (30). See Materials and Methods
fordetails.
The switching Go� model is based on the idea that the shapeof
the funnel depends on chemical composition of the system andthus is
altered upon ligand binding (31). We illustrate this notionfor the
conformational transition in �-subunit of F1–ATPasefrom the
nucleotide-free state ‘‘E’’ to the ATP-bound state ‘‘TP’’upon
binding to ATP (Fig. 1b). In this case, we consider twofunnels; one
for E state, at the bottom of which is the �Estructure (Fig. 1b,
cyan funnel), and another one for the TP state,at the bottom of
which is the �TP structure (Fig. 1b, greenfunnel). Go� potentials
of the �-subunit for the states E and TPare denoted as V�(R��E�)
and V�(R��TP�), respectively, where
R� collectively represents the coordinates of amino acids in
the�-subunit, and E and TP indicate the nucleotide state.
We start simulations from the �E structure using the
Go�potential V�(R��E�) appropriate for the nucleotide-free state
E�(blue funnel in Fig. 1b). Upon binding ATP, we then switch
theenergy to another Go� potential V�(R��TP�), appropriate for
theATP-bound state TP� (green funnel in Fig. 1b). Fig. 1c
plotsdegree of conformational change as a function of time.
Theamount of conformational change is defined by a set of aminoacid
pairs that are in the native contact at the final structure,
�TP,and that are not in the native contact at the initial
structure, �E(for the definition of the native contact, see
Materials andMethods). Within the set, the fraction that is formed
is used asthe measure of conformational change. We performed
molec-ular dynamics simulations (32, 33) at a temperature T1
andswitched the Go� potential at the 10,000th time step,
observingthe structural relaxation of the � from �E to �TP in 200
steps (Fig.1c). Simulation of relatively large-scale conformational
changeswas achieved within a very short time by switching Go�
model.
Modeling the �3�3� Complex of F1–ATPase. Next, we move on to
themodel of the F1 complex. The model of the �3�3� complex ofF1 has
several important features. (i) A Go� potential V� funnelsthe
simulated �-subunit into its structure of the 1994 complexso that
the � only f luctuates around its native structure. (ii) Forthe
�3�3 ring, we used the switching Go�
potentialV�3�3(R��1,��2,��3�X��1,��2,��3), where the ring structure
is fun-neled to the conformation characterized by nucleotide
statesof the three ��-subunits X��1,��2,��3. Each of the three
��-subunits may have nucleotide states of either TP, DP � Pi, DP,or
E states (here, we omit the possibility that Pi alone is bound,but
the ADP is released based on structural observation).Thus,
X��1��2��3 varies among 4
3 � 64 types of nucleotidestates. Here, there is some room of
uncertainty in mapping thenucleotide states to the reference
structures, which will bedescribed thoroughly later. For example,
when X��1,��2,��3 isthe nucleotide state {DP, E, TP}��1,��2,��3,
the stable confor-mations of three ��-subunits are ��DP
1994, ��E1994, and ��TP
1994.[The superscript 1994 means that structures are taken from
thex-ray structure determined in 1994 (27).] To induce
confor-mational changes of the �3�3 ring, this Go� potential
isswitched, e.g., for the simulation of conformational changeupon
ATP binding at the catalytic site of the ��2, we switch theGo�
potential from V�3�3(R��1,��2,��3�{DP, E, TP}��1,��2,��3) to
Fig. 2. F1–ATPase rotary simulation by all-at-once catalysis
approximation.The vertical axis is the � rotation angle. Each line
represents the individualtrajectory, where the energy is switched
from V�3�3(R��1,��2,��3�{DP, E,TP}��1,��2,��3) to
V�3�3(R��1,��2,��3�{E, TP, DP}��1,��2,��3) at the 30,000th step.
(a)Time courses of � rotation upon single ATP hydrolysis at the
temperature T1.The � angles averaged over the time steps
20,000–30,000 and 50,000–60,000are �15.5° and 105.8°, respectively.
(b) The � angle vs. degree of conforma-tional change of the �3�3
ring (horizontal axis). For the latter, 0 means thestructure of
{��DP
1994, ��E1994, ��TP
1994}, and 1 corresponds to that of {��E1994, ��TP
1994,��DP
1994}.
Fig. 1. Structure of F1–ATPase and illustration of switching Go�
model. (a)Structure of F1–ATPase by Abrahams et al. (27), viewed
from the outermembrane side. The central stalk � is the rotor. The
��-subunits of theATP-bound (yellow), the ADP-bound (gray), and the
nucleotide-free parts aredenoted as (�TP, �TP), (�DP, �DP), and
(�E, �E). (b) Concept of switching Go�model. In the simulation of
ATP binding, we start the simulation withV�(R��E�), in which the
simulated structure of the �-subunit is funneled to the�E
structure, and switch it at a certain time to V�(R��TP�), where the
structureof simulated � is funneled to the �TP structure. (c) Time
courses of simulationsof a single �-subunit upon ATP binding at
temperature T1. The switch fromV�(R��E�) to V�(R��TP�) is conducted
at the 10,000th step. The value of thevertical axis close to 0.0
(1.0) means the structure is close to �E (�TP).
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V�3�3(R��1,��2,��3�{DP, TP, TP}��1,��2,��3). (iii) For the
interac-tion between the � and the �3�3 ring V�3�3��, we used a
simple,short-ranged steric repulsion between amino acids. Very
im-portantly, the switching Go� potential was imposed only for
theintra-�3�3 ring interactions, but not the � rotation, and thus
therotation is not biased by design. (iv) The N termini of the
three�-subunits and the C termini of � are constrained to
theirinitial positions by springs Vspring, which mimics the
His-tagused to fix F1 in the single molecule experiments (19). The
totalpotential is the sum of four terms: V � V� � V�3�3 � V�3�3��
�Vspring. See Materials and Methods and Table 1, which ispublished
as supporting information on the PNAS web site, fordetails.
F1 Simulation of All-at-Once Catalysis. The first set of F1
simulationsstarted with the 1994 structure and the
potentialV�3�3(R��1,��2,��3�{DP, E, TP}��1,��2,��3) for which the
lowestenergy state corresponds to the 1994 structure, {��DP
1994, ��E1994,
��TP1994} (27). After the 30,000th time steps, the potential
is
switched to V�3�3(R��1,��2,��3�{E, TP, DP}��1,��2,��3), in which
thesimulated structures are funneled to the structure, {��E
1994,��TP
1994, ��DP1994}. The switching corresponds to a single cycle of
the
catalysis consisting of ADP release from the catalytic site of
the��1 (termed site 1 hereafter), ATP binding at site 2, and
ATPhydrolysis at site 3, which are conducted all at once. (Here,
the
phosphate, Pi, release may occur before or after reaching
thenucleotide state of the 1994 structure.) Repeating the
simula-tions 20 times, we monitored the rotary angle of � as a
functionof time. In all trajectories, the � angle weakly fluctuated
around�15.5° before the switching, and after the switching, the
�rotated �120° in a counterclockwise direction, consistent withthe
observations of the single-molecule experiment (19, 25) (Fig.2a;
see also Movie 1, which is published as supporting informa-tion on
the PNAS web site). Fig. 2b plots the � angle and thedegree of
structural relaxation in the �3�3 ring, indicating thatstructural
relaxation in the �3�3 ring is followed by the � rotation.The �
rotation was induced solely by steric repulsion upon
theconformational change in the �3�3 ring. Although factors
notincluded in this simulation, such as hydrophobic and
side-chaininteractions, may perturb the results quantitatively,
structuralcomplementarities are of primary importance for the
rotation.
Critical Residues for Producing Torque. To decipher which parts
ofthe � and �3�3 are essential for transferring the torque,
weperformed a series of simulations analogous to
Ala-scanningmutagenesis (34). In silico mutagenesis removed the
stericinteractions of five consecutive residues in �, from �(i � 2)
to�(i � 2), and for each i we repeated 40 simulations of
all-at-oncecatalysis just as described in the previous subsection.
Thenumber of trajectories that successfully completed the
120°rotation is shown at two different temperatures (T1 and T2
�5T1) in Fig. 3 as a function of central residue number i. At
thelower temperature T1, we found three critical regions
requiredfor the � rotation: �Ala-1–Asp-5, �Ile-13–Met-25, and
��la-231–Ala-235. We highlighted these three portions in white
andthe portions of the �-subunits that contact them in yellow in
Fig.4. Both �Ile-13–Met-25, and ��la-231–Ala-235 form contactswith
the portion of � that shows the major structural change (theyellow
portion in the C-terminal of the �-subunit colored by cyanin Fig. 4
a and c). This portion of � is not present in the highlyconserved
DELSEED region but is localized nearby. This resultis consistent
with the experimental inference that the DEL-SEED region does not
play a critical role in torque transfer (35).The portions around
the �Ile-13–Met-25 and �Ala-231–Ala-235regions are likely the sites
of action of the � rotation. The otherimportant site, �Ala-1–Asp-5,
keeps contacts with �- and �-sub-units at their middle portion, the
switch II region that does notshow significant structural change
(Fig. 4 b and c). Thus,�Ala-1–Asp-5 probably works as a fulcrum.
Interestingly, anytrajectories that did not achieve a 120° rotation
of � in thesemutagenesis simulations still showed a � rotation up
to 30° from
Fig. 3. Simulations analogous to Ala-scanning mutagenesis (34).
The num-ber of completely rotated trajectories, out of 40 trials,
is plotted against thecentral residue i of �, around which five
consecutive residues are ‘‘mutated.’’All-at-once catalysis
simulations were conducted at two temperatures, T1 (red)and T2 � 5
� T1 (blue).
Fig. 4. Residues that transfer the torque. According to the
results represented in Fig. 3, the three critical parts in � are
plotted in white: �Ala-1–Asp-5,�Ile-13–Met-25, and
�Ala-231–Ala-235. a–c are different views of the identical
structure taken from a simulated snapshot after 30° rotation. The �
subunits ingreen, cyan, and magenta are in the nucleotide states
Tp, E, and DP, respectively, before all-at-once catalysis switch.
Shown are views from �TP (a), �E (b), and�E (c). The subunit in
pink (only in b) is the �-subunit in DP state. In b, only �1–44 is
represented. The residues in � that are within 11 Å from the three
criticalparts of � are in yellow.
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the initial 1994 structure (27). (We show an example of
thesimulations in Fig. 7, which is published as supporting
informa-tion on the PNAS web site.) This result implies that the
initial 30°rotation is less sensitive to steric interactions. At
higher tem-peratures, the results were less clear, although a
similar tendencywas found.
Mechanochemical Coupling: Tri-Site Model vs. Always-Bi-Site
Model.Next, we address how the catalytic reactions are related
toconformational changes in �3�3 and � rotation substeps.
Exper-iments showed that the 90° substeps are driven by ATP
binding(26), and two, and only two, �-subunits are occupied by
nucle-otides before the ATP binding (36). We first assumed that
the1994 structure and its corresponding nucleotide state {DP,
E,TP}��1,��2,��3 are on the pathway. Starting from this state,
thenext step suggested by the above experiments is ATP binding
atsite 2 of the 1994 structure (27), which is expected to make a
90°substep (Fig. 5a). This scenario is consistent with the
modelgenerally called the tri-site model (37). To test this
hypothesis,we performed simulations starting from
V�3�3(R��1,��2,��3�{DP,E, TP}��1,��2,��3), of which the lowest
energy state correspondsto the 1994 structure, {��DP
1994, ��E1994, ��TP
1994}, and at the30,000th step, we switched it to
V�3�3(R��1,��2,��3�{DP, TP,TP}��1,��2,��3), which funneled the
simulated structures to{��DP
1994, ��TP1994, ��TP
1994}. However, the simulations exhibited afatal collision
between the �3�3 and � and did not show thesubstep or any
unidirectional rotation (Fig. 5b). This resultsuggests that, for
avoiding the steric clash, ��1 must open itsconformation before or
concomitant with the closure of theC-terminal domain of ��2 upon
ATP binding. All conceivablecatalytic schemes starting from the
1994 structure with {DP, E,TP}��1,��2,��3 were simulated. Those
failed to reproduce resultsconsistent with experiments. As Abrahams
et al. (27) suggested,the nucleotide state of {DP, E,
TP}��1,��2,��3 with the 1994
structure may correspond to the MgADP-inhibition state and isnot
on the pathway.
Here, three additional implications from experiments
furtherinform the mechanistic hypothesis. (i) A FRET
experimentsuggests that the 1994 structure corresponds to the one
formedafter a 90° substep (38). (ii) A structural change in �
uponphosphate release has been reported (39). These two
observa-tions and the above simulation results led us to redefine
thenucleotide state of the 1994 structure as {DP � Pi,
E,TP}��1,��2,��3, which is on-pathway, and to assume that
the�-subunit changes its structure upon phosphate release
fromclosed to ‘‘half-closed’’ form (28), the latter taken from
thestructure reported in 2001 (40). Furthermore, (iii) one
scenariothat is consistent with the experiments of Nishizaka et al.
(36) isthat the site occupancy of nucleotides is always 2 (41).
Finally,we suggest simultaneous reactions of ATP binding and
ADPrelease. The catalysis scheme thus starts from {DP � Pi,
E,TP}��1,��2,��3 with the 1994 structure. Then, the phosphate
isreleased at site 1, which is followed by the simultaneous
reactionsof ATP binding at site 2 and the ADP release at site 1.
Finally,ATP hydrolysis occurs at site 3 (Fig. 6a).
Simulations consistent with this mechanism were conducted
Fig. 5. Simulation of a step in the tri-site model. (a) Cartoon
diagrams. Thediagrams correspond to the part of the tri-site model.
(Upper) The transitionof the nucleotide states in the catalytic
scheme. (Lower) The reference struc-tures used in the simulation.
Each circle represents an ��-subunit, and char-acters in it
indicate the nucleotide state of the ��-subunit at the top and
thereference structure at the bottom. Red arrow indicates the angle
of � obtainedby simulations. (b) Time courses of � angle based on
the catalytic model. Eachline represents individual trajectory. ATP
binding was conducted at the30,000th step.
Fig. 6. Simulation with the always-bi-site model. (a) Cartoon
diagrams. Seethe legend of Fig. 5 for description of Upper and
Lower. (b) Time courses of �angle based on the catalytic model.
Each line represents individual trajectory.Simulations contain four
stages shown in a, and each stage corresponds to30,000 time steps.
(c) Histogram of the � angle from simulations shown in b.
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(Fig. 6 a and b). We first simulated with V�3�3(R��1,��2,��3�{DP
�Pi, E, TP}��1,��2,��3), where site 1 is funneled to ��DP of
the1994 structure; the reference structure is {��DP
1994, ��E1994,
��P1994}. At the 30,000th step, this potential is switched
to
V�3�3(R��1,��2,��3�{DP, E, TP}��1,��2,��3), where site 1 has
thefunnel centered at the ��DP�
2001, the half-closed � structure in the2001 complex (40); the
reference structure is {��DP�
2001, ��E1994,
��TP1994}. Next, we switched the potential to
V�3�3(R��1,��2,��3�{E,
TP, TP}��1,��2,��3) at the 60,000th step, and
V�3�3(R��1,��2,��3�{E,TP, DP}��1,��2,��3) at the 90,000th step. The
former potentialfunnels the simulated structures to {��E
1994, ��TP1994, ��TP
1994} andthe latter to {��E
1994, ��TP1994, ��DP
1994}. The resulting trajectoriesand histogram are shown in Fig.
6 b and c (see Movie 2, whichis published as supporting information
on the PNAS web site)where we found the �30° substep upon the
phosphate releaseand the �80° substep upon the coupled reactions of
the ATPbinding and ADP release, which are perfectly consistent
withexperiments. Interestingly, the �-subunit rotated �10° uponATP
hydrolysis. This finding might be related to the discrepancybetween
90° � 30° substeps with ATP as a substrate and 80° �40° substeps
with slowly hydrolysable substrate ATP[�-S] (42).
Discussion and ConclusionThe results from the experiment and
simulation led us to thefollowing rotational mechanisms. Starting
from the 1994 struc-ture and the nucleotide state {DP � Pi, E,
TP}��1,��2,��3,remodeling of the site 1 �-subunits from closed to
half-closedform upon phosphate release makes space between the �3�3
andthe �, leading to 30° rotation of �. Next, ATP-binding at site
2induces the �-subunits closing motion, which pushes the �-sub-unit
primarily through contacts near �Ile-13–Met-25, and
��la-231–Ala-235 leading to � rotation. This motion further
induces� opening at site 1, thus accelerating the ADP release. The
latter� opening motion at site 1 is crucial for making space for �
tocomplete �80° rotation (see Movie 2). The ATP-binding, the
80°rotation, and ADP release are so tightly coupled that they are
inthe single kinetic process. Hydrolysis might induce � rotation
of�10°. We note that current simulations do not rule out
theso-called bi-site model (43) at very low ATP concentration.
It is surprising that steric repulsion between the � and the�3�3
ring alone reproduced not only 120° rotation, but also 30°and 90°
substeps. Here, the shape complementarities betweenthe � and �3�3
play crucial roles in functional dynamics of F1.It may be viewed as
the ‘‘dynamic version of lock-and-key’’mechanism; the shape of the
keyhole, the �3�3 ring, changesin the catalytic cycle, which
rotates the key, the �-subunit, bystructural complementarities. It
may be one reason for F1 toexhibit more deterministic motion (25)
than that of linearmolecular motors such as myosin (44).
As in the F1–ATPase rotary motion, biomolecular functionalmotion
takes milliseconds, a time scale that is orders of magni-tude
longer that the current reach of atomistic molecular dy-namics
simulations (3, 4). A folding-based simulation modelprovides a
powerful tool of deducing functional dynamics oflarge biomolecular
complexes from static structural information,once structures in
multiple states are experimentally supplied.Here, we reported an
initial application of this approach toF1–ATPase.
There are many ways to further improve the current model ofF1.
First of all, a remarkable feature of F1–ATPase is
itsreversibility. By enforcing the rotation of �, F1–ATPase
cansynthesize ATP from ADP and Pi (45). For simulating it, we
needto take into consideration the feedback mechanism from the�3�3
ring structure to the chemical reaction of nucleotide.Second, the
current switching Go� model introduced the ‘‘verticalexcitation’’
for jumping from the initial funnel to the final funnel(see Fig.
1b). Physically, this way of simulation corresponds
to‘‘photoexcitation,’’ whereas the real process studied is of
course
thermally activated. The present approach may be viewed as
arough approximation of the real process for addressing
struc-ture–function relationship. For pursuing more physical
aspects,however, the present approach needs to be modified.
Recently,methods in this direction were investigated (8, 9, 46).
Third, wedid not explicitly include nucleotide molecules, such as
ATP,here. Explicit inclusion of ligand molecules is desired.
Improve-ment toward these directions is needed.
Materials and MethodsMolecular Simulation with Go� Models.
Clementi’s version of the C�Go� model has the effective energy
function
V�R�X � �bonds
Kr�r i � rX,i2 � �angles
K��� i � �X,i2
� �dihedral
K��1�1 � cos�� i � �X,i�
� K��3�1 � cos 3�� i � �X,i�
� �i�j�3
nativecontact
� 5� rX,ijr ij �12
� 6� rX,ijr ij �10�
� �i�j�3
nonnativecontact
�D1r ij �12
. [1]
Here, R collectively represents the Cartesian coordinates of
C�atoms. X signifies the native structure of the simulated
protein.In this equation, the first term keeps the chain
connectivity; ri isthe length of the ith virtual bond between the
ith and i�1-thamino acids. The second and the third terms represent
the localtorsional interactions; �i, and i stand for the virtual
bond anglebetween the i�1-th and ith bonds and the virtual dihedral
anglearound the i-h bond, respectively. The fourth and fifth terms
arenonlocal interactions. The former includes interactions
between‘‘native contact’’ pairs of amino acids that are spatially
close inthe native structure (see below for details), and the last
term isa nonspecific repulsion between amino acid pairs. The rij is
thedistance between the C� atoms of the ith and jth amino
acids.Structural information on X is used for setting the
parameters;all of the constant parameters with the subscript X
indicate thevalues of the corresponding variables at the structure
X. Forexample, because the variable rij is the distance between the
ithand jth residues, the parameter rXij has the value of rij in the
Xstructure. Native contact in the fifth term is defined as below.
Ifone of the nonhydrogen atoms in the ith amino acid is within
adistance of 6.5 Å from any nonhydrogen atom in the jth aminoacid,
we define the pair of the ith and jth amino acids as beingnative
contact (amino acid pairs that are not in the native contactare
called nonnative pairs). We note that all of the terms exceptthe
last one are set up so that each term has the lowest energywhen the
conformation R coincides with the structure X: theseterms realize
the funnel-like energy landscape. Parameters Kr �100.0, K� � 20.0,
K�
(1) � 1.0, K�(3) � 0.5, � 0.36, and D1 � 4.0
are used throughout the present work. In the switching Go�
modelproposed here, the reference structure X is switched
uponchange in the nucleotide state.
The energy function for the F1 complex consists of four terms:V
� V� � V�3�3 � V�3�3�� � Vspring. V� is a Go� model for �subunit,
in which the bottom structure X corresponds to the �structure in
the 1994 complex. V�3�3(R��1,��2,��3�X��1,��2,��3) is theswitching
Go� model for the �3�3 ring, where the referencestructure X depends
on the nucleotide states. For example, whenX � {DP, E, TP}, the
stable conformations of three ��-subunits
Koga and Takada PNAS � April 4, 2006 � vol. 103 � no. 14 �
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are ��DP1994, ��E
1994, and ��TP1994 (see Crystal Structures Used in
Materials and Methods). V�3�3�� is the short-range steric
repul-sion defined as
V�3�3�� � �i��3�3, j��
�D2�rij12, [2]
where D2 � 6.0 Å. Vspring is introduced to keep the F1 complexon
a plane and is defined as
Vspring � �i��19,�29,�39,�272
� k�Li � Li,02 �Li � Li,0 ,0 �Li � Li,0 ,[3]
where k � 0.18, Li is the distance of the ith residue from its
initialposition, and Li,0 � 1.5 Å (i � �19, �29, �39), and Li,0 �
2.0 Åfor i � �272. See Supporting Text, which is published as
sup-porting information on the PNAS web site, for some
moredetails.
Molecular dynamics simulations were performed by integrat-ing
Newton equation with ‘‘velocity Verlet’’ algorithm (32), andthe
velocity rescaling proposed by Berendsen et al. (33) was usedto
realize constant temperature simulation. Temperatures usedare T1 �
4 � 10�3 and 5T1. The mass for all amino acids wasidentical.
Crystal Structures Used. We used the reference structures of
the�3�3� complex that correspond to the bottoms of the funnels
inGo� potentials. All of the reference structures were taken
fromthe two crystal structures determined in 1994 and 2001 by
Walker and coworkers, which we refer as the 1994 (27) and
the2001 (40) structures, respectively. As for the � stalk, we used
theconformation in the 1994 structure as the reference
structurethroughout the work. Conversely, the reference structure
of the�3�3 ring varied depending on the states of three
catalyticbinding sites. Because the catalytic nucleotide binding
sites arelocated at the interfaces between the �- and �-subunits,
wetreated the �- and �-subunits that sandwich the catalytic
sitesgrouped together, which are colored by analogous hue in Fig.
1a.From the 1994 and 2001 structures, we took four
differentreference structures of grouped ��-subunits. We denoted
eachof them as ��A
y : ��DP1994, ��E
1994, ��TP1994, or ��DP�
2001. The superscripty represents the parent structure, from
which the structure of ��is taken; y is either 1994 or 2001. In the
1994 structure, A is TP,DP, or E. ��TP
1994 has ATP-analog in the catalytic site; thestructure of the
�-subunit is in the closed form in C-terminalregion, and the
interface is loosely packed. ��DP
1994 includes ADPin the catalytic site; the �-subunit structure
is similar to that inTP1994, whereas the interface is tightly
packed. ��E
1994 does nothave nucleotides at the catalytic site, and the
�-subunit takes theopen form in C-terminal region. In the 2001
structure, we pickedup one ��-structure as the reference structure,
which the�-subunit takes as a half-closed form (28): its C-terminal
isbetween the closed form in ��TP
1994 and the open form in ��E1994.
We call it ��DP�2001.
We thank Jeffery Saven and Peter Wolynes for many helpful
suggestions.N.K. is supported by Japan Society for the Promotion of
ScienceResearch Fellowship for Young Scientists. This work was
supported inpart by the ‘‘Water and Biomolecules’’ program from the
Ministry ofEducation, Culture, Sports, Science, and Technology
(Japan).
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5372 � www.pnas.org�cgi�doi�10.1073�pnas.0509642103 Koga and
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