![](bg5.png)
resonance mechanism, however, has satisfactorily il-
lustrated these phenomena. Each external link serves
as a harmonic oscillator. The more oscillators are
linked in series, the more resonance frequencies may
be selected. If the system does in a resonance fash-
ion, the motor works in a most efficient way. Here,
we consider that motor will equiprobably match ev-
ery resonance frequency. The average resonance fre-
quency is statistically equal to the ensemble experi-
mental activity. Actually, the lowest resonance fre-
quency will be closer and closer to zero with adding
of oscillator or the increasing of inertial mass. How-
ever, it is impossible for F
1
motor to stop in virtue
of the driving of F
o
or binding of ATP. This lowest
frequency that is close to zero has to be cancelled in
statistics and the average resonance frequency then
will jump. This is one of reasons that the value of
No.5 is higher than that of No.4 in Fig.2. The criti-
cal lowest resonance frequency is waiting to be deter-
mined by further experiments. Our novel resonance
model together with systematic kinetics study(Shu
and Lai, 2008) has predicted that the relative activity
of hydrolysis is higher(lower) than that of synthesis if
the latter is less(higher) than 1, which has been exper-
imentally demonstrated.
The frequencyspectrum indicated in Fig.4 is wait-
ing to be tested by a mass of single molecule experi-
ments. If there is enough single molecule experimen-
tal data, we can firstly confirm whether the distribu-
tion of activity of single motor is the same with that of
our model. Secondly, we can affirm whether the mo-
tor equiprobably matches every resonance frequency
or not. Finally, we can estimate the standard devia-
tion, σ
i
, for each Gaussian distribution. Here, for sim-
plest, we consider that motor equiprobably matches
every resonance frequency. The fitted elastic coeffi-
cient can be used to estimate the elasticity of linking
protein. It seems reasonable that k
4
is much higher
than others in view of the stiff capsid of H9 virus.
The fitted rate of P
i
releasing is also logical because
it is higher than
hω
0
i/2π.
It is clear that the mechanical movement of F
1
comprises the eccentric rotation of γ subunit and the
conformational change of the α
3
β
3
crown, and they
are tightly coupled with each other. However, pre-
vious theoretical study only focus on the rotation of
γ subunit. In fact, the conformational change of the
α
3
β
3
crown is very important for the entire motor,
and plays an active role in hydrolysis. On the other
hand, the role that α subunit plays in the conforma-
tional change of the α
3
β
3
crown has not been studied
although the structure of α subunit resembles that of β
subunit with the exception of catalytic site. This work
is try to investigate what α subunit functions in the
rotary motor and has expanded the “binding change
mechanism”.
In contrast with Langevin equation that ignores
the inertial force due to overdamping, our resonance
model does neglect the damping force because the en-
tire motor doesn’t output any mechanical work and
dissipates hardly heat in medium. Additionally, the
scale of conformational change of the α
3
β
3
crown is
very small, thereby the average velocity of oscilla-
tion is so low that λ
i
h ˙x
i
i ≪ m
i
h ¨x
i
i, which is possi-
ble although the values have not been estimated. It
should be noted that the elastic coefficient is indepen-
dent of the additional links. Although the elastic co-
efficient can in fact depend on the additional links due
to electrostatic interaction, in the simplest version of
the model, we assume that all elastic coefficients are
links-independent.
ACKNOWLEDGEMENTS
This work is partially supported by the NSFC grant
No.91027045 and No.11075015, and the National
Basic Research Program of China (973 Program) un-
der the grant No. 2007CB935903.
REFERENCES
Abrahams, J. P., Leslie, A. G. W., Lutter, R., and Walker,
J. E. (1994). Structure at 2.8˚a resolution of F
1
-
ATPase from bovine heart mitochondria. Nature,
370:621–628.
Ballmoos, C. V., Cook, G. M., and Dimroth, P. (2008).
Unique rotary ATP synthase and its biological diver-
sity. Annu. Rev. Biophys., 37:43–64.
Boyer, P. D. (1993). The binding change mechanism
for ATP synthase-some probabilities and possibilities.
Biochim. Biophys. Acta., 1140:215–250.
Boyer, P. D. (1997). The ATP synthase-a splendid molecu-
lar machine. Annu. Rev. Biochem., 66:717–749.
Cheng, J., Zhang, X. A., Shu, Y. G., and Yue, J. C. (2010).
F
o
F
1
-ATPase activity regulated by external links on β
subunits. Biochem. Biophys. Res. Commun., 391:182–
186.
Diez, M., Zimmermann, B., B¨orsch, M., K¨onig, M.,
Schweinberger, E., Steigmiller, S., Reuter, R.,
Felekyan, S., Kudryavtsev, V., Seidel, C. A. M., and
Gr¨aber, P. (2004). Proton-powered subunit rotation
in single membrane-bound F
o
F
1
-ATP synthase. Nat.
Struct. Mol. Biol., 11:135–141.
Itoh, H., Takahashi, A., Adachi, K., Noji, H., Yasuda,
R., Yoshida, M., and Jr, K. K. (2004). Mechani-
cally driven ATP synthesis by F
1
-ATPase. Nature,
427:465–468.
BIODEVICES 2012 - International Conference on Biomedical Electronics and Devices
136