-ATPase STATOR REGULATION STUDIED

WITH A RESONANCE MODEL

Yao-Gen Shu and Zhong-Can Ou-Yang

Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China

Keywords:

Molecular motor, F

-ATPase, Stator regulation, Binding change mechanism, Resonance model.

Abstract:

The F

-ATPase activity was regulated through external links on the exposed stator. The regulation tendency

of synthesis was the same with that of hydrolysis. A resonance model has been proposed to illustrate these

regulation phenomena. The novel model not only has deepened our understanding of the “binding change

mechanism”, but also was very useful to develop the rotary motor into a biosensor.

1 INTRODUCTION

-ATPase is the ubiquitous rotary motor that uses

the transmembrane electrochemical potential to syn-

thesize ATP in bacteria, chloroplasts and mitochon-

dria. The holoenzyme is a complex of two rotary

motors, F

and F

, which are mechanically coupled

by a common central stalk (“rotor”), c

-ε-γ. The

membrane embedded F

unit converts the proton mo-

tive force(p.m.f) into mechanical rotation of the “ro-

tor”, thereby causing cyclic conformational change of

crown (“stator”) in F

and driving ATP synthe-

sis. A striking characteristic of this motor is its re-

versibility. It may rotate in the reverse direction for

ATP hydrolysis and utilize the excess energy to pump

protons across the membrane. In F

motor, the eccen-

tric rotation of γ subunit and the cyclic conformational

change of α

crown, are mechanically coupled at

near 100% efﬁciency(Boyer,1997; Leslie and Walker,

2000; Weber and Senior, 2003; Kinosita et al., 2004;

Junge, 2004; Wilkens, 2005; Nakamoto et al., 2008;

Ballmoos et al., 2008; Yasuda et al., 1998; Shu et al.,

2010).

In the proof-of-principle demonstrations, the

holoenzyme activity was regulated by changing of the

transmembrane gradient of proton(P¨anke and Rum-

berg, 1996; Turina et al., 2003; Diez et al., 2004;

Steigmiller et al., 2008). Moreover, the F

activ-

ity was studied in detail through an actin ﬁlament

or a bead on the “rotor”(Yasuda et al., 1998; Itoh

et al., 2004). On the other hand, several pioneering

investigations(Boyer, 1993; Abrahams et al., 1994;

Noji et al., 1997; Nishizaka et al., 2004) have re-

vealed that the cyclic conformational change of α

crown “stator” is tightly coupled with the rotation of

γ “rotor”. A fascinating question arises: can we reg-

ulate the holoenzyme activity through a convenient

method? Yue Group’s recent experiments(Cheng

et al., 2010) about virus detection have demonstrated

that the holoenzyme activity can be regulated by the

external links on the exposed α (or β) subunits. The

external links not only inhibit the enzyme but also ac-

tivate it, and the regulation tendency of synthesis is

the same with that of hydrolysis. However, what is

the implied mechanism?

It is impossible for the two movements to transfer

energy at near 100% efﬁciency(Yasuda et al., 1998)

unless they work in a resonance fashion through

an elastic element. Actually, Boyer’s “binding

change mechanism”(Boyer, 1993), Walker’s struc-

ture(Abrahams et al., 1994) and Oster’s hinge bend-

ing model(Wang and Oster, 1998) have partly implied

that F

-ATPase works in a fashion of driven oscilla-

tion(Kargovsky et al., 2009; W¨achter et al., 2011).

The driving force is supplied by the eccentric rotation

of γ subunit for synthase or by the binding of ATP

for hydrolase, whereas the passive oscillation is the

cyclic conformational change of α

crown, whose

frequency is the driving frequency. If we regard α

or β subunit as a harmonic oscillator, α and β sub-

units will vibrate at its natural frequency. Once the

driving frequency matches the natural frequency, res-

onance occurs. At resonancefrequency, the amplitude

of steady state response is the biggest in proportion to

the amount of driving force, which is the most conve-

nient for binding of substrates and releasing of prod-

ucts. The driving force is essentially just pumping en-

ergy back into the system to compensate for the loss

132

Shu Y. and Ou-Yang Z..

FoF1-ATPase STATOR REGULATION STUDIED WITH A RESONANCE MODEL.

DOI: 10.5220/0003753401320137

In Proceedings of the International Conference on Biomedical Electronics and Devices (BIODEVICES-2012), pages 132-137

ISBN: 978-989-8425-91-1

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

Figure 1: The schematic illustration of the activity regu-

lation assay, showing the experimental geometry (not to

scale). The other two same links are not painted. All the

external links, α antibody, streptavidin, H9 antibody and

H9 virus, are regarded as harmonic oscillators as shown in

Fig.3.

of energy due to ATP synthesis or protons pumping

as well as damping, which is the most efﬁcient way

for motor to work. The rotary speed of motor, which

corresponds to the measurable holoenzyme synthesis

activity, is equal to the resonance frequency of “sta-

tor”.

2 EXPERIMENTAL RESULTS

A series of harmonic oscillators have been designed

to link to the exposed α subunits. The schematic il-

lustration of this activity regulation assay is shown in

Fig.1. F

-ATPase is embedded in a chromatophore,

in which the transmembrane p.m.f is stored. The ﬁrst

external harmonic oscillator is the α antibody, the sec-

ond one is the streptavidin, the third one is H9 avian

inﬂuenza virus (hereafter called H9 virus) antibody

and the fourth one is the H9 virus.

The holoenzyme activity with different links has

been measured as follows: Firstly, each α subunit was

bound by an α antibody, the holoenzyme was inhib-

ited, and the relative synthesis and hydrolysis activity

decreased to 0.71 and 0.84 respectively. Secondly,

the streptavidin was linked to α antibody through a

small biotin, the enzyme was inhibited continuously,

and the relative activities of synthesis and hydroly-

sis declined to 0.60 and 0.73 respectively. Thirdly,

H9 antibody was linked to streptavidin through biotin,

the holoenzyme was inhibited again, and the relative

activities of synthesis and hydrolysis dropped to 0.51

and 0.61 respectively. However,when H9 virus bound

to its antibody ﬁnally, the modiﬁed enzyme was acti-

vated, and its relative activities jumped to 0.90 and

Relative synthesis /hydrolysis activity

❍: hydrolysis

❑: synthesis

●, ■: Theoretical data

Figure 2: Different activities of enzyme with different links

on α subunit shown in Fig.3. The statistical value (mean ±

s.e.m.) of each system was computed by 50∼60 samples.

The experiment has been repeated independently more than

ﬁve times. The native F

-ATPase activity was taken as a

control and others were expressed in proportion to the con-

trol. The solid dots are calculated by Eq.(4).

0.93 respectively. It is very amazing that the activity

of the modiﬁed holoenzyme, in which a series of ex-

ternal proteins and virus are linking on “stator”, can

resume to a nativelevel. These results are correspond-

ingly indicated in Fig.2 No. 2, 3, 4 and 5. The activ-

ity of No. 5 is about two times that of No. 4, which

means that the No. 4 complex is a sensitive biosensor

of virus detecting.

To identify whether H9 virus affects F

-ATPase

activity directly, we measured the activity of native

buffer incubated with H9 virus. The result shown in

Fig.2 No.6 has proved that the H9 virus doesn’t have

any direct effect on the holoenzyme activity no matter

the enzyme functions as a synthase or hydrolase.

hydrolysis activity can be expectably regu-

lated by varying the load on the “rotor”(Yasuda et al.,

1998). With increasing of the length of ﬁlament at-

tached on the “rotor”, the damping force will increase,

thereby inhibiting the F

hydrolysis activity, which

has been approvingly interpreted by traditional mod-

els(Sun et al., 2004; Xing et al., 2005; Jonathan et al.,

2006). Our experiment has revealed that the holoen-

zyme activity also can be regulated by the external

links on the “stator”. The regulation tendency of syn-

thesis is the same with that of hydrolysis. However, it

is very difﬁculty for us to understand this regulation

mechanism using traditional model. Here, we intro-

duce the resonance mechanism to statistically illus-

trate these phenomena.

FoF1-ATPase STATOR REGULATION STUDIED WITH A RESONANCE MODEL

133

Figure 3: Resonance model. m

, m

and m

repre-

sent the inertial mass(Dalton molecular weight ) of α sub-

unit, α antibody, streptavidin, H9 antibody and H9 virus re-

spectively. k

is the elastic coefﬁcient of corresponded har-

monic oscillator.

3 RESONANCE MODEL

The “stator” of F

motor is a crown with alternat-

ing α and β subunits. Each catalytic site located in

β subunit is at different stage of the enzyme kinetic

cycle: one binds ATP, another binds ADP, and the

third keeps empty. All three sites coordinate for the

next 120

◦

step. The different stage of catalytic site is

tightly coupled to the different conformation of β and

α subunit(Nishizaka et al., 2004; Diez et al., 2004;

Itoh et al., 2004). We deﬁne the cyclic conformational

change of α

crown as an oscillation. The oscil-

lations of three α subunits are identical with the ex-

ception of 120

◦

phase. Hence, the activity regulation

assay may be simply cartooned in Fig.3 with one α

subunit. Each link, including α subunit, functions as a

spring with its inertial mass, and is simply regarded as

a harmonic oscillator. Five harmonic oscillators have

constructed ﬁve systems. Each harmonic oscillator is

a damped, driven harmonic oscillator. The dynamic

equation of the ith system is

X+ KX+ Λ

X = F+ Γ (1)

where

M = I













, X =













K =







+ k

−k

0 .. . .. .

−k

+ k

−k

. .. . −k

i−1

+ k

−k

0 .. . .. . −k







and

Λ = I













,F =







(t)







,Γ =







(t)







I is an identity matrix. M

X is the inertial force and

is usually ignored in linear motors motility study due

to overdamping. i = 0,1,2, 3,4 represents α subunit,

α subunit antibody, streptavidin, H9 antibody and H9

virus respectively, the inertial mass, m

, is in propor-

tion to its Dalton molecular weight, that is, m

∝ 55,

170, 60, 170 and ∞ (KD) for i =0, 1, 2, 3, and 4 re-

spectively, x

is the displacement of the ith oscillator.

KX is the restoring force. Λ

X is the damping force.

F is the driving force ( f

(t) = 0 for i ≥ 1). Γ is the

random force characterized by a Gaussian distribu-

tion with zero mean (hΓi = 0). The general solution

of Eq.(1) is a sum of a transient solution that depends

on initial conditions, and a steady state one that de-

pends only on driving frequency,driving force, restor-

ing force and damping force. Of course, we only dis-

cuss the latter.

motor consists of two rotary motors acting in

opposition: F

motor generates a mechanical torque

using the hydrolysis energy of ATP, while F

motor

generates a rotary torque in the opposite direction us-

ing a transmembrane p.m.f. When the motor func-

tions as a synthase, the driving force of α subunit,

+ (t), is generated by the eccentric rotation of γ sub-

unit that is impelled by the transmembraneproton ﬂux

of chromatophore, whereas the incurred counterforce,

(t), comes of the releasing of ATP in β subunit, the

net driving force then f

(t) = f

+ (t) − f

(t). When

the motor functions as a hydrolase, the driving force

of α subunit, f

(t), is generated by the binding of

ATP in β subunit, whereas the incurred counterforce,

− (t), comes of the transmembrane protons pump-

ing, the net driving force then f

(t) = f

(t)− f

− (t).

At steady state, F

-ATPase converts directly chemi-

cal energy at nearly 100% efﬁciency(Oster and Wang,

2000; Kinosita et al., 2000) between p.m.f and energy

currency, ATP, which means that motor doesn’t out-

put any mechanical work and dissipates hardly heat

in medium, hence, the net driving force f

(t) ≈ 0.

Resonance is the tendency of a system to oscil-

late with the maximum amplitude at a certain fre-

BIODEVICES 2012 - International Conference on Biomedical Electronics and Devices

134

–

)

–

)

–

)

–

)

<ω



>=0.73

<ω



>=1.0

(<ω

<ω

0/1

(<ω

<ω



>=0.61

<ω

(<ω

<ω



>=0.51

<ω

(<ω

<ω



>=0.90

<ω

(<ω

Figure 4: Right: Distribution of time average resonance fre-

quency. The ensemble statistic average value is calculated

by Eq.(3).Left: Distribution of elastic coefﬁcient. All of

them are assumed Gaussian distribution. The ﬁtted aver-

age values k

= 0.35, 0.13, 0.06 and 8 for i = 1,2,3,4

respectively. The standard deviation, σ

, is not sensitive

to the average value. Here, σ

=0.1, 0.04, 0.02 and 0.3

for i = 1,2,3,4 respectively, and σ

hω

i=0.3 for P

(hω

i).

≡ m

hω

quency. At this resonance frequency, even small pe-

riodic driving force can produce large amplitude vi-

bration because the system stores vibrational energy.

When damping is small, the resonance frequency is

approximately equal to the natural frequency of the

system(Ogata, 2004). So, if we only consider the

time average resonance frequency of the ith system

at steady state for small damping, Eq.(1) may be sim-

pliﬁed into

Xi = −KhXi (2)

where h·i means time average value. Eq.(2) can be

used to determine the distribution of the time average

resonance frequency of ith system, P

(hω

i), if that of

native holoenzyme, P

(hω

i), has been given. Con-

sidering the diversity of motors and external links, we

assume hω

i and all k

(i ≥ 1) are Gaussian distribu-

tion as inserted in right top and shown in the left row

of Fig.4. The right row of Fig.4 is the corresponding

(hω

i). The system that comprises n harmonic oscil-

lators in series possesses n natural frequencies, that is,

motor has n frequencies to match for resonance. The

frequency spectrum indicated in Fig.4 can be directly

tested by a mass of single molecule experiments. The

ensemble averageresonance frequency of α

crown

is equal to the average holoenzyme activity. For sim-

plest, we consider that motor equiprobably matches

every resonance frequency, the ensemble statistic time

average resonance frequency of ith system can be cal-

culated statistically by

hω

i =

...

Z Z

(hω

i)hω

idhω

idk

...dk

(3)

Systematic kinetics study of F

-ATPase(Shu

and Lai, 2008) has derived the formula of saturated

rotary speed. The rates of ATP synthesis and hydrol-

ysis have been given by

= 3k

−

i,T

, v

−

i,D

−

i,P

−

i,D

+ k

−

i,P

respectively, where k

−

is the releasing rate of corre-

sponding product, and T, D and P represent ATP, ADP

and phosphate respectively. Because phosphate is too

small to be blocked in the catalytic cleft, we consider

that the releasing rate of phosphate, k

−

i,P

, doesn’t rely

on the conformation of β subunit and is constant, k

−

The releasing of ATP(ADP) is tightly coupled to the

conformational change of β subunit because the size

of ATP(ADP) can be compared with the size of cat-

alytic cleft, that is, k

−

i,T

−

i,D

) =

hω

i/2π. The relative

synthesis and hydrolysis activities are deﬁned as

≡

hω

, ρ

≡

= ρ

1+ b

+ b

(4)

respectively, where b ≡ k

−

hω

i/2π). Eq.(4) pre-

dicts that ρ

> ρ

if ρ

< 1 as indicated in Fig.2, oth-

erwise, ρ

< ρ

which has also been demonstrated by

our recent experiment with links on β subunit. The

ﬁtted b = 1.58, a reasonable value.

4 DISCUSSIONS

Different from previous activity regulation as-

says(Yasuda et al., 1998; P¨anke and Rumberg, 1996;

Turina et al., 2003; Diez et al., 2004; Steigmiller et al.,

2008; Itoh et al., 2004), our regulation based on the

stator links. The external links not only inhibit the en-

zyme but also resume it, and the regulation tendency

of synthesis is the same with that of hydrolysis. It is

very difﬁculty for power stroke model(Wang and Os-

ter, 1998) to interpret this regulation mechanism. The

FoF1-ATPase STATOR REGULATION STUDIED WITH A RESONANCE MODEL

135

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

motor to stop in virtue

of the driving of F

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

, 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

is much higher

than others in view of the stiff capsid of H9 virus.

The ﬁtted rate of P

releasing is also logical because

it is higher than

hω

i/2π.

It is clear that the mechanical movement of F

comprises the eccentric rotation of γ subunit and the

conformational change of the α

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

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

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

crown is

very small, thereby the average velocity of oscilla-

tion is so low that λ

h ˙x

i ≪ m

h ¨x

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.

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