Fluctuation theorem applied to F1-ATPase.

Abstract

In recent years, theories of nonequilibrium statistical mechanics such as the fluctuation theorem (FT) and the Jarzynski equality have been experimentally applied to micro and nanosized systems. However, so far, these theories are seldom applied to autonomous systems such as motor proteins. In particular, representing the property of entropy production in a small system driven out of equilibrium, FT seems suitable to be applied to them. Hence, for the first time, we employed FT in the single molecule experiments of the motor protein F1-adenosine triphosphatase (F1), in which the rotor γ subunit rotates in the stator α3β3 ring upon adenosine triphosphate hydrolysis. We found that FT provided the better estimation of the rotary torque of F1 than the conventional method.

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Fluctuation Theorem Applied to F1-ATPase
Kumiko Hayashi, Hiroshi Ueno, Ryota Iino, and Hiroyuki Noji*
The Institute of Scientific and Industrial Research, Osaka University, 8-1 Mihogaoka, Ibaraki 567-0047, Osaka, Japan
(Received 18 February 2010; published 28 May 2010)
In recent years, theories of nonequilibrium statistical mechanics such as the fluctuation theorem (FT)
and the Jarzynski equality have been experimentally applied to micro and nanosized systems. However, so
far, these theories are seldom applied to autonomous systems such as motor proteins. In particular,
representing the property of entropy production in a small system driven out of equilibrium, FT seems
suitable to be applied to them. Hence, for the first time, we employed FT in the single molecule
experiments of the motor protein F1-adenosine triphosphatase (F1), in which the rotor ? subunit rotates
in the stator ?3?3ring upon adenosine triphosphate hydrolysis. We found that FT provided the better
estimation of the rotary torque of F1than the conventional method.
DOI: 10.1103/PhysRevLett.104.218103 PACS numbers: 87.15.Ya, 05.40.?a
Experimental applications of the fluctuation theories of
nonequilibrium statistical mechanics [1–4] to micro and
nanosized systems have been conducted with technical
development in both the manipulation and observation
of objects on small scales [5–12]. Particularly, the fluctua-
tion theorem (FT), which represents the property of en-
tropy production in a small nonequilibrium system, has
been applied to various physical systems such as colloidal
particle systems, granular systems, and turbulent systems
[5–8].
The theories of nonequilibrium statistical mechanics
have recently been actively applied to biological systems
[8–12]. As to FT, the work fluctuation theorem was inves-
tigated in RNA hairpin systems [10]. Unlike RNA hairpins,
motor proteins autonomously move by using the free en-
ergy of chemical reactions. Can FT be verified for such an
autonomously nonequilibrium system? Which expression
ofFTisusefulinelucidatingthethermodynamicproperties
of motor proteins? To answer these questions, we extend
the applicability of FT to the single molecule experiment
of F1-adenosine triphosphatase (ATPase) (F1), in which the
? subunit rotates in the ?3?3ring with adenosine triphos-
phate (ATP) hydrolysis. We focus on its rotary torque
measurement, and propose FT as a new tool for nonequi-
librium force measurement of a wide range of biological
systems in vitro and in vivo.
Single molecule experiment.—F1is a rotary motor pro-
tein and a part of FoF1-ATPase/synthase [13–20]. The
minimum complex acting as a motor is the ?3?3? sub-
complex, and the ? subunit rotates in the ?3?3ring upon
ATP hydrolysis [13]. The three catalytic ? subunits hydro-
lyze ATP sequentially and cooperatively. Three ATP mole-
cules are hydrolyzed per turn, or in other words, the free
energy obtained from single ATP hydrolysis is used for a
120?rotation [14]. The conformations of the ? subunits
change as the elementary chemical steps such as the ATP
binding, the ATP hydrolysis (cleavage of the covalent
bond) and the product [adenosine diphosphate (ADP) and
inorganic phosphate] releases proceed [Fig. 1(a)], and the
coordinated push-pull motion of the C-terminal domains of
the ? subunits produces torque for the ? subunit to rotate
[15,16].
In our single molecule assay (see Methods), the rotation
of the ? subunit was observed as the rotation of a probe
attached to it [Fig. 1(b)] because the size of the ? subunit
(?2 nm) is too small for its rotation to be observed directly
under an optical microscope. The rotational angle, ?ðtÞ,
FIG. 1 (color).
and the red arrow represent the ? subunits and ? subunit of F1,
respectively. F1performs a 120?step rotation upon ATP hydro-
lysis comprising 80?and 40?substeps. (b) Schematic diagrams
of our experimental system (not to scale). The rotation of F1is
probed by a duplex of polystyrene beads (left) or an irregularly
shaped magnetic bead (right). The size of F1is about 10 nm and
that of a probe is in the range of 358–940 nm. (c) ATP-driven
rotations of F1probed by the magnetic beads at 1 mM ATP (red
line) and 100 nM ATP (blue line).
(a) Reaction scheme of F1. The green circles
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was calculated from the recorded images of the probe. In
Fig. 1(c), ?ðtÞ is plotted in the cases of 1 mM ATP and
100 nM ATP. While the rotation was continuous for 1 mM
ATP, it became stepwise for 100 nM ATP pausing every
120?. The ATP binding dwell (the time that a catalytic
site waits for ATP binding) is very short at high [ATP]
(<0:1 ms at 1 mM [17]) and the time constant of the dwells
for the ATP hydrolysis and that for the product releases
are also short (about 1 ms independent of [ATP] for the
wild-type F1[17]), and a rotation appears continuous when
the response time of a probe is longer than these time
scales.
In previous studies [14,17–19] assuming that a rotary
torque, N, acting on a probe attached to F1is constant, N
was estimated using the equation
N ¼ ?!;
(1)
where ! and ? are the mean angular velocity and frictional
drag coefficient of a rotary probe, respectively. From the
calculation of fluid mechanics, the functional forms of ?
for several probes are known as
?ac¼
?sb¼ 8??a3þ 6??ax2
?db¼ 16??a3þ 6??ax2
4??L3
3½lnðL=2rÞ?0:447?
for an actin filament;
for an a single bead;
for a duplex of beads;
1
1þ 6??ax2
2
(2)
where L and r are the length and radius of an actin
filament, a and xi(i ¼ 1, 2) are the radius and rotation
radius of each bead [Fig. 1(b)], and ? is the viscosity of a
medium (? ¼ 0:89 ? 10?9pNs=nm2at 25?C). For the
continuous rotation of the wild-type F1probed by the actin
filament (L ¼ 2 ?m) at 2 mM ATP, N was estimated to be
about 40 pNnm [18].
Although the estimation of the rotary torque of F1using
Eqs. (1) and (2) seems successful, a problem related to the
estimation of ? remains [20]. When a probe rotates near a
glass surface, the real value of ? is expected to be larger
than the value estimated by Eq. (2), which is derived from
the assumption that a rotation occurs in a bulk [14,19,20].
In fact, when the probe rotated near the glass surface
(?10 nm) [19], the torque has been estimated to be lower
than that measured when it rotated away from the glass
surface (?200 nm) [14,18]. Noting that this difficulty in
estimating ? is a common problem for force measurements
of biological motors such as bacterial flagella [21] and
RNA polymerase [22] which move near the surfaces of a
cell and a glass, we employ an expression of FT that can
estimate N without using the value of ? to overcome the
problem.
Fluctuation theorem.—For a continuous rotation of F1,
the time evolution of ?ðtÞ is assumed to be described by a
Langevin equation
?d?
dt¼ N þ ?ðtÞ;
where ? is a random force that represents the effect of
thermal noise, kBis the Boltzmann constant, and T is the
room temperature (T ¼ 25?C). We also assume that N is
constant as was suggested in previous studies [14,17–19].
On the basis of the above model, FT for the torque mea-
surement is expressed as
h?ðtÞ?ðt0Þi ¼ 2?kBT?ðt ? t0Þ; (3)
ln½Pð??Þ=Pð???Þ? ¼ N??=kBT;
where ?? ¼ ?ðt þ ?tÞ ? ?ðtÞ and Pð??Þ is the probability
distribution of ??.
Verification.—In the following, the torque determined
using Eqs. (1) and (2) and that determined using Eq. (4) are
(4)
denoted as N?and NFT, respectively. Their averaged values
are given in Table I (see supplementary material [23] for
those of each molecule). Below, we used irregular-shaped
magnetic beads (see Methods) and duplexes of spherical
beads as a probe, which were suitable to observe the
rotation of F1because of their anisotropic shapes.
For the continuous rotation of the particular F1probed
by the magnetic bead, Pð??Þ and ln½Pð??Þ=Pð???Þ? are
plotted for the cases ?t ¼ 2:5–10 ms in Fig. 2(a). The
slopes of the graphs in Fig. 2(a) (Right) are almost the
same for all cases ?t ¼ 2:5–10 ms, and the value of
the slopes corresponds to NFTaccording to Eq. (4). In
Table I (a), N?is compared with NFT. Although the mag-
netic beads have a broad distribution in size (200–700 nm)
and are not spherical, their approximate sizes were esti-
mated from the recorded images. We then estimated the
values of ?sbusing these approximate sizes assuming that
the beads were spherical. The fact NFT? 40 pNnm > N?
implies the effect of the glass surface on the estimation
of ?sb.
TABLE I.
(b) Summary of NFT(pN nm) (magnetic bead). Note that n
represents the number of molecules investigated.
(a) Torque estimated using the different methods.
(a)probe
N?(pN nm)
28 ? 5:9a,e
23 ? 3:8a,e
31 ? 3:3a,e
19 ? 1:8b,e
NFT(pN nm)
35 ? 2:8a,c,e
38 ? 2:5a,c,e
31 ? 2:6a,c,e
26 ? 1:2b,c,e
magnetic bead, n ¼ 6
duplex (340 nm), n ¼ 6
duplex (470 nm), n ¼ 4
duplex (179 nm), n ¼ 3
(b) F1(wild-type) F1(wild-type) F1(?E190D)
V1
n ¼ 6n ¼ 2
38 ? 2:7a,d,f
n ¼ 3
37 ? 2:4a,d,f
n ¼ 5
33 ? 2:2a,d,f
35 ? 2:8a,c,e
aThe recording rate was 2000 fps.
bThe recording rate was 1000 fps.
cNFTwas estimated in the case ?t ¼ 10 ms.
dNFTwas estimated in the case ?t ¼ 2:5 ms.
eContinuous rotation.
fStepping rotation.
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To compare NFTand N?more precisely, we used the
duplexes of spherical beads [14,17]. When ?dbwas used
for a duplex, we set x1¼ 0 and x2¼ 2a assuming that the
duplex typically attached to F1perpendicularly to the rota-
tional direction because it is difficult to know precisely
how a probe attaches to F1using an optical microscope
[17]. In the case of the duplex of 340 nm polystyrene beads
(2a ¼ 340 nm), which are similar in size to the mag-
netic beads, we again found that NFT? 40 pNnm > N?
[Table I (a)].
We also investigated a larger probe (the duplex of
470 nm polystyrene beads), the most part of which is
considered to rotate far from a glass surface, and a smaller
probe (the duplex of 179 nm polystyrene beads), whose
response time is considered to be close to the time constant
of the ATP hydrolysis dwell of thewild-type F1[17]. In the
former case, as expected, it was found NFT¼ N?within
statistical error bar [Table I (a)]. Both NFTand N?were
slightly smaller than 40 pNnm because the rotary torque
may show the load dependence resulted from the intrinsic
behavior of F1although we cannot exclude the possibility
that the torque was lower for this large probe because it
may scratch the glass surface during its rotation. In the
latter case, both NFTand N?were much smaller than
40 pNnm although NFT> N?[Table I (a)] because the
existence of the dwells much affected on the fluctuation
of the probe, and invalidated the use of Eqs. (1) and (3).
Dependence on the recording rate and ?t.—In our
single molecule assays, the average rate of a continuous
rotation for the wild-type F1depends on the size of a probe
and is in the range of 3–40 Hz (see [23] for the rotation rate
of each probe). Since we needed to accurately measure the
forward and backward fluctuation in ?ðtÞ to calculate the
left-hand side of Eq. (4), the images of the rotating probes
were recorded with a high-speed camera [24] at various
recording rates [500–30000 frames per second (fps)] each
of which was much higher than the rotation rate. In
Fig. 2(b) using the same molecule, NFTis plotted as a
function of ?t for the different recording rates. It is seen
that for large ?t, NFTconverges to almost the same value
for all recording rates. Note that below 500 fps, the nega-
tive values of ??, which are needed for the calculation of
the left-hand side of Eq. (4), were hardly measured, and
reliable estimation of Pð??Þ=Pð???Þ was difficult. For
FIG. 2 (color).
cases ?t ¼ 2:5 ms (red line), 5.0 ms (yellow line), 7.5 ms (green
line), and 10 ms (blue line) for the particular F1probed by the
magnetic bead (left). ln½Pð??Þ=Pð???Þ? as a function of
??=kBT (right). The slope was 38 pNnm in the case ?t ¼
10 ms. The recording rate was 2000 fps. (b) NFTas a function
of ?t for the recording rates, 500 fps (red line), 1000 fps (blue
line), 2000 fps (green line), 3000 fps (orange line), 6000 fps
(pink line), 10000 fps (aqua line), and 30000 fps (violet line).
Here, we used the same molecule probed by the same magnetic
bead at 1 mM ATP.
(a) Probability distribution, Pð??Þ, for theFIG. 3 (color).
We analyzed ?ðtÞ encircled in red (about 100–300 steps). The
steps were determined by eye. (b) Probability distributions,
Pð??Þ, for the cases ?t ¼ 0:5 ms (red line), 1.0 ms (yellow
line), 1.5 ms (green line), and 2.0 ms (blue line), and 2.5 ms
(aqua line) for the particular F1probed by the magnetic bead
(left). ln½Pð??Þ=Pð???Þ? as a function of ??=kBT (right). The
slope was 40 pNnm in the case ?t ¼ 2:5 ms. The recording rate
was 2000 fps.
(a) ATP-driven rotation of F1at 100 nM ATP.
FIG. 4 (color).
5 molecules) and V1(blue line, 5 molecules). The rotations
were probed by magnetic beads and recorded at 2000 fps.
NFT as a function of ?t for F1 (red line,
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small ?t, our numerical simulations suggested that the
behavior of NFTmight reflect the slight distortion of the
rotary potential of F1(see [23]).
Application I: Stepping rotations of wild-type F1at low
[ATP] and mutant F1(?E190D).—For the wild-type F1,
the rotation became stepwise at 100 nM ATP. In this case,
?? was calculated using the 120?steps encircled in red
[Fig. 3(a)], assuming that the time evolution of ?ðtÞ during
120?steps is also described by Eq. (3). In Fig. 3(b), Pð??Þ
and ln½Pð??Þ=Pð???Þ? are plotted for the cases ?t ¼
0:5–2:5 ms. These values of ?t are smaller than those
used at 1 mM ATP because Pð??Þ for ?t larger than
2.5 ms were hard to be measured precisely for the stepping
rotation due to a small number of samples.
Next, the mutant F1ð?E190DÞ was investigated.
Because the glutamic acid at 190 of the ? subunit (E190)
acts as a general base in the catalysis of ATP hydrolysis,
when this residue is mutated into aspartic acid, ATP hy-
drolysis at the catalytic site is greatly decelerated and
F1ð?E190DÞ shows 120?stepping even at 1 mM ATP
[25]. The fact seen in Table I (b) that the torque was almost
the same for the wild type and mutant within statistical
error bar implies that the torque caused by the global
structural change of the ? subunit was not affected greatly
by the local mutation while the time constant of ATP
hydrolysis was greatly affected. (Note that the torque of
the mutant has not been reported previously.)
Application II: V1-ATPase.—We applied FT (4) to an-
other rotary motor V1, which is a part of VoV1-ATPase
[26].InV1,theDsubunitrotatesin theA3B3ring,and120?
steps are observed at low [ATP] (in our case ½ATP? ¼
10 ?M). Using Eqs. (1) and (2), the rotary torque of V1
has been reported to be smaller than that of F1[26]. In
Fig. 4, it is seen that NFTof V1was smaller than that of F1.
This tendency was similar to that observed in the previous
study [26].
Summary.—Our results are summarized in Table I (b).
Investigating the rotations of F1probed by the several
kinds of beads and comparing our results with those in
Ref. [18], we checked the validity of Eqs. (3) and (4) and
found that NFTmeasured by using FT (4) was more accu-
ratethanN?calculatedfromthefrictiondragcoefficientby
using Eqs. (1) and (2) when the rotations occurred near the
glass surface. Further, FT (4) may be applied to, e.g., bac-
terial flagella, linear motor proteins such as kinesins and
myosins, and the transportations of vesicles and organelles
in a cell. Because it is not easy to estimate friction drag
coefficients in a cell, a nondestructive force measurement
method such as FT (4) will be in demand in the future.
Methods.—The
?3?3?
C193SÞ3?ðHis10Þ3?S107C=I210C] of F1-ATPase from
thermophilic Bacillus PS3 (called the wild-type F1), the
slow ATPcleavage
?3?3?
C193SÞ3?ðHis10=E190DÞ3?S107C=I210C]
mutant F1), and the V1complex from Thermus thermophi-
lus were expressed in E. coli, purified and biotinylated
[18,26]. As to the rotation assay, we used streptavidin-
subcomplex[?ðHis6=
subcomplex[?ðHis6=
(calledthe
coated magnetic beads (200–700 nm, Seradyn) which are
useful when F1 is operated by magnetic tweezers or
streptavidin-coated polystyrene beads. The magnetic beads
were pulverized by using the ultrasonic bath sonicator to
make them smaller. The rotary motion of F1(or V1) was
observed immobilizing the ?3?3ring on a Ni-NTA glass
surface (see Material and Methods in [23] for detail).
We thank members of our laboratory for discussions and
technical advice. We thank Dr. H. Takagi for discussions
on Eq. (4); and Dr. K. Yokoyama, Dr. H. Imamura, Dr. M.
Nakano, and Y. Apa Nishikawa for discussions and provid-
ing samples, related to V1. This work was supported by a
JSPS Grant-in-aid (to K.H.), and Grants-in-aids for Scien-
tific Research from MEXT (18074005 and 18201025 to
H.N. and 21107517 (to R.I.), and a grant (to H.N.) from
the Post-Silicon Alliance, ISIR at Osaka University.
*hnoji@sanken.osaka-u.ac.jp
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