Mechanochemical coupling of kinesin studied with the neck linker swing model

0.18 MB·English

by  Yaogen Shu

Mechanochemical coupling of kinesin studied with the neck linker swing model
Yaogen Shu1,∗ and Hualin Shi1
1Institute of Theoretical Physics, The Chinese Academy of Sciences, P. O. Box 2735, Beijing 100080, China
(Dated: February 4, 2008)
6
0 Wehaveproposedthenecklinkerswingmodeltoinvestigatethemechanismofmechanochemical
0 couplingofkinesin. TheMichaelis-Menten-like curvefor velocityvsATPconcentration at different
2 loadshasbeenobtained,whichisinagreementwithexperiments. WehavepredictedthatMichaelis
constant doesn’t increase monotonically and an elastic instability will happen with increasing of
n
applied force.
a
J
PACSnumbers: 87.16.Nn
4
2
I. INTRODUCTION headbindsADP.Fordouble-headedbinding,theforward

] head is empty, whilst the rear head binds either ATP or
T
ADP·P.
O Kinesinisamolecularmotorthattransportsorganelles i
Kinesinworksinacyclicfashionforseveralintermedi-
. and membrane-bound vesicles along a microtubule in
o ate states. The Michaelis-Mentenrelationfor the rate of
various cells [1, 2]. It takes hundreds of 8 nm steps
i ATPhydrolysisisstillabasiclaw[4,9,10,23,24,25,26].
b (the size of tubulin heterodimers composed of α and β
The average stepping velocity, however, also has been
- subunits)[3, 4, 5] before detachment and the run length
q experimentally found obeying the Michaelis-Menten law
is longer than 1 µm[3, 4, 5, 6]. This is why kinesin is
[ for a range loads[27], which means the mechanochemical
called a processive motor.
1 couplingistight,i.e.,kinesinhydrolyzesonemolecularof
v Conventional kinesin (hereafter called ‘kinesin’) is a ATP for each 8 nm step[12]. With increasing of the ap-
0 dimer consisting of two identical ∼ 120kD chains, com- pliedforce,thesaturatingvelocitydecreasesasexpected,
4 monlyknownasheavychains. Eachheavychaincontains however, Michaelis constant surprisingly increases[27].
0 a N-terminalglobularheaddomain,a stalk regionwhich The role of force in the reaction kinetics can be used to
1 is responsibleforheavychaindimerization,atether that investigatethemechanismofmechanochemicalcoupling.
0
joinstheheadandstalk,andaC-terminalfan-shapedtail
6 In this paper, we present a neck linker swing model to
domainwhichusuallybindscargofortransportinginliv-
0 investigate the mechanism of mechanochemical coupling
ing cell or beadfor applying force in single molecule ma-
/ of kinesin. We will discuss the effect of applied force
o nipulatedexperiments[7,8]. Theheaddomainisahighly
on the reaction kinetics and the elastic instability of the
i
b conservedregionfor different members ofthe kinesin su- complex composed of neck linker and attached head.
- perfamily and contains two binding sites. One binds to
q
microtubule and the other to nucleotides. Two heads al-
:
v ternately hydrolyze one molecular of ATP for each 8 nm
i step[1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13]. Sometimes, the II. NECK LINKER SWING MODEL
X
8-nm step can be resolved into fast and slow substeps,
r
a eachcorrespondingtoadisplacementof∼4 nm[14]. The Whenaheadattachesmicrotubule,itstether becomes
tether is a ∼ 15-amino-acid segment. It becomes immo- a rigid rod that can be bent by an applied force, and we
bilizedandextendedtowardstheforwarddirectionwhen call this tether as a neck linker. Otherwise, we call the
its head binds microtubule and ATP, and reverts to a tether as a flexible chain if its head is detached[15].
more mobile conformation when phosphate is released An idealized scheme of the relations among nu-
after ATP hydrolysis[15]. If the tether is replaced by a cleotides, microtubule and the tether’s conformation is
randomsequenceofaminoacids[16]orifitiscross-linked shown in Figure 1a[15, 28, 29, 30]. For attached head,
to nucleotides site[17], the kinesin will lose the capabil- if its catalytic cleft is empty, the neck linker tends to be
ity of stepping. Therefore, the conformational change of perpendicular to microtubule. ATP binding will change
tether seems to be necessary for kinesin to step[15]. the catalyticcleft’s conformation. The allostericinterac-
The motility of kinesin can be explained by a “hand tion between two binding sites in the attached head will
over hand” model[18, 19, 20, 21]. The two heads trigger neck linker to swing to the forward direction[15].
alternately repeat single- and double-headed binding The neck linker swing model consists of two chemi-
with microtubule. A simplified binding mode has been cal transitions (k1 and k3) and two mechanical substeps
proposed[22]. For single-headed binding, the attached (k2 and k4) as shown in Figure 1b. This model is con-
head either binds ATP or is empty, whilst the detached sistent with the widely accepted model for the kinetic
mechanism of kinesin[31]. We just keep their main pro-
cesses and rearrange them into steps in our model. This
approximation is reasonable for the case that ADP con-
∗Electronicaddress: [email protected] centration is low.
2
a power stroke recovery stroke the free head and bead will be thrown forward. A
flexble chain
power stroke occurs. We assume the motion of bead,
l
D f stalk by which optical tweezer can applies force on motor, is
bead
neck linker overdamped, the average velocity is
T 1
Microtubule hx˙i= (hfci+hfbi−f) (2)
ζ
b
D D
where ζ is the viscous coefficient, hfci and hfbi are av-
f f f
erage forces generated from ATP binding and releasing
(1) (4) P D k4 of the bending energy respectively, that is, hfci+hfbi=
n n+1 (∆E +E )/(1L+ℓ). The power stroke rate then is
1 bend 2
D
f k1 k3 f k1 k (f)= hx˙i = 1/ζ ∆E1+Ebend −f (3)
(2) T k2 (3) T D 2 L/2+ℓ L/2+ℓ(cid:18) L/2+ℓ (cid:19)
∆E is the energy that power stroke outputs, and orig-
1
FIG. 1: (a): An idealized scheme of the relations among nu- inates from the catalytic cleft’s conformational change
cleotides, microtubule and the tether’s conformation(not to
induced by ATP binding[15].
scale). (b): Neck linker swing model. Two chemical transi-
tionsand twomechanicalsubstepsarecoupled alternately to
eachother. T,DandPrepresentsATP,ADPandPi respec-
C. chemical transition 2: detached head’s ADP
tively.
releasing and attached head’s ATP hydrolysis
In this model, the mechanochemical cycle of kinesin With starting of the power stroke, the random mo-
includes four steps: ATP binding; power stroke; ADP tion of the free head will be biased to the forward direc-
releasing + ATP hydrolysis and recovery stroke. tion and the flexible chain will be stretched due to neck
linker’s throwing and the interaction from the nearest
β-tubulin monomer, which greatly increases the proba-
bility that the free head reaches the next binding site on
the track. It first bind weakly, eventually, releases its
A. chemical transition 1: ATP binding ADP and attaches strongly to the track. The stretched
chain will shrink and transform into a rigid rod, which
Whenaheadstronglyattachestothemicrotubuleand helps the rear head to split the bound ATP, and one of
waits for ATP binding, the neck linker will be bent by products, phosphate will be released[15, 22, 30]. This
the applied force f to the backward direction with dis- reaction weakens the binding of the rear head with the
placement ℓ as shown in Figure 1a. The bending energy microtubule and leads it to detach from the track with
E ≈ 1κℓ2 and the force f = dE /dℓ ≈ κℓ in the ADP[15,22,30]. Weassumetherateoftheprocessfrom
bend 2 bend
case of slight bending, where κ is the bending rigidity. theendofpowerstroketothedetachingofrearhead,k ,
3
Theattachedheadwillfinallybearthebendingenergy is independent of applied force.
andthe energybarrierfor ATP binding will increase. So
ATP binding rate can be written as:
D. mechanical substep 2: recovery stroke
k (f)=k e−Ebend/kBT[ATP] (1)
1 b
The tether linked the forward head now becomes the
wherek isATP bindingconstantwithoutappliedforce,
b neck linker[15, 22, 30], and will swing to the mechanical
k isBoltzmannconstant,andT isabsolutetemperature.
B equilibrium position. A recovery stroke occurs. Similar
This relationship holds whether [ATP](ATP concentra-
with the power stroke, the rate of this step
tion) is high or low. During attached head is waiting for
ATP binding, the free head can’t reach any binding site 1/ζ ∆E −E
2 bend
onthetrackbecausetheend-to-enddistanceofthe flexi- k4(f)= −f (4)
L/2−ℓ L/2−ℓ
blechainisnotlongenough. Thesesitesareseparatedby (cid:18) (cid:19)
a fixed distance (L=8 nm) along the rigid microtubule. where∆E originatesfromthecatalyticcleft’sconforma-
2
tionalchangeinducedbyADPreleasing,whichisconfor-
mationally the recovery process of ATP binding.
B. mechanical substep 1: power stroke Thecycleisnowreadytorepeat,the differenceisthat
rolesofthesetwopartnerheadshaveexchanged. Theki-
Once the attached head binds an ATP molecule, neck nesindimerhashydrolyzedoneATPandmovedforward
linker will move to the new equilibrium position, and 8 nm with two substeps[15, 28, 29, 30]. Although there
3
aremorethantwochemicaltransitionsasabove,allother
TABLE I: The fitted parameters for experimental vmax and
chemical processes are rapid rate transitions (random-
ness shows that there are only two to three rate-limiting KM versus f[27] with kBTr =4.1pN·nm
transitions[27,32,33]). Welumpallotherchemicaltran- fstall κ kb k3 ζ
sitions into k . (pN) (pN·nm−1) (µM−1·s−1) (s−1) (pN·s/nm)
3
6.50 1.88 1.33 120 1.45×10−3
III. FITTING AND RESULTS
ourmodelis independent ofATP concentration. We can
From the time series shown in Figure1b, the average rewrite Eq.(6) and Eq.(7) as
time to complete a single enzymatic cycle, hτi, can be
computed conveniently[33, 34]. The average velocity of ln KML =−lnk + 1/(2kBT)f2 (9)
b
kinesin moving along microtubule is vmax κ
(cid:18) (cid:19)
L 1 1 (1+2f¯)2 (1−2f¯)2
v = L =L 4 1 −1 = vmax[ATP] (5) vmax = k3 + k0 (cid:20)1−f¯∆(1+f¯) + 1−f¯∆(1−f¯)(cid:21)(10)
hτi k K +[ATP]
i! M
Xi=1 andusethemtofitthemeasureddataofvmaxandKMat
different loads[27], where f¯ = f/[f (1+f /(κL))],
with ∆ stall stall
and f¯=f/(κL). The fitted values of f , κ, k , k and
stall b 3
1 1 1 −1 ζ arelistedintableI,andthetwofittedcurves,vmax ∼f
vmax = L k + k + k (6) and KM ∼f, are shown in Figure 2(a).
(cid:18) 2 3 4(cid:19) The fitted stall force f = 6.5 pN is consistent with
stall
eEbend/kBT vmax measured value[27]. The total energy outputted in one
K = (7)
M
kb L cycle, 2∆E ≈ 74 pN·nm, is less than ∆µ, which demon-
strated again that the fitted values are reasonable. The
wherevmaxandKMaresaturatingvelocityandMichaelis efficiency of mechanochemical coupling of kinesin, η, is
constant respectively. Obviously, the average velocity
about93%. However,the efficiencyofmotor,f L/∆µ,
stall
obeys the Michaelis-Menten law as observed[27].
is about 65%, which is in agreement with experiments.
In our model, ∆E and ∆E are independent of ATP
1 2 It is the motion of a silica bead that is measured in
concentration. Wedefine theefficiencyofmechanochem-
experiment[27]. The radius of the bead, R, is 0.25 µm.
icalcoupling ofkinesinasη =(∆E +∆E )/∆µandas-
1 2 The viscousity of water at room temperature, η˜ , is
w
sume it is a constant,where ∆µ is the free energyexcess about 0.9×10−9pN·s/nm2. The Stokes drag coefficient
inonemoleculeATPhydrolysis. Itmustbenotedthatη
of the bead can be estimated by 6πRη˜ , and is about
w
defined here is differentfromthe efficiency of motor. We 4.24×10−3pN·s/nm. Thefittedviscouscoefficientofbead
assume kinesin has adjusted ∆E and ∆E to achieve
1 2 in our model, thus, is comparable with the Stokes drag
optimal kinetic velocity in evolution. From Eq.(3) and
coefficient for a sphere in 0.5 µm diameter at room tem-
Eq.(4), we yield ∆E = ∆E ≡ ∆E (< 1∆µ), which
1 2 2 perature.
is reasonable with the fact that the catalytic cleft’s con-
The widely accepted model for the kinetic mechanism
formational change induced by ADP releasing is confor-
of kinesin[31] has proposed that the ATP binding rate
mationally the recovery process of that induced by ATP is about 2 µM−1·s−1 and the rate from free head’s ADP
binding. For simplicity, we introduce the rate of power
releasing to attached head’s detaching can be estimated
(or recovery)stroke without load, k0 =4∆E/(ζL2). about 70 s−1. In our model, ATP dissociating rate k−1
AccordingtoEqs.(3)and(4), the powerstrokeratek
2 isn’ttakenintoaccount,thisiswhyk isslightlylessthan
b
decreases with increasing of applied force f, whilst the
the proposed value. k also approaches the estimated
3
recoverystrokeratek increasesasshowninFigure2(d).
4 value.
At a critical applied force,k becomes zero. We call this
2
critical force as stall force. It can be derived as
IV. DISCUSSION
∆E+E (f )
f = bend stall (8)
stall L/2+ℓ(f )
stall A. KM no longer increases monotonically with load
Ifappliedforceis largerthanthe stallforce,the forcein-
duced by conformationchangecan’tovercomeit to push Although the recovery stroke will go faster and faster
the kinesin forward along the microtubule. with increasing of applied force, the power stroke will
For the case of slight bending of neck linker, Eq.(8) spend more and more time and finally stalls at the stall
can be rewritten as f ≈ 2∆E/[L(1+f /(κL))] < force, which leads to the concave down of saturating
stall stall
∆µ/L ∼ 10 pN with ∆µ ≈ 80 pN·nm under physiolog- velocity. With low applied force, Michaelis constant
ical conditions[35], which is in agreement with the ex- will nearly increase exponentially because eEbend/kBT in-
perimental data[27]. It must be noted that stall force in creases faster than vmax decreases. With high force,
4
with increasing of applied force, and its force-extension
(a) (c)
-1⋅V, (nms)max s• VKmMax µK, (M)M plied force, (pN) bending instability extension reCla.tiovnelcoacnitbyleimvfieitrttsienudgsbaAypTpaPliseicmdopnfolcyrecneeltarasattticisoamntuordaetlinaggaainn.d
p
A
Load (pN) Extension, (nm) With these reasonable fitted parameters, we can use
(b) (d)
Eq.(5)directlytocomputethedependenceofthevelocity
-1⋅city, (nms) s 2 mM ATP → -1Rate, (s) 0.1 mM ATP kkk432 roAenslalstohiaoodnwanistiisnnatFguoirgoaudtrienagg2r(aebne)dm,eltinmhteitttionhgeeoxArpeTetriPicmacleonvntecaleolncdtitrayat-tafi[oo2rn7cs]e..
elo ← • 5 µM ATP k1 It is clear that there are three distinctive regimes of ap-
V
plied force: (1) If the load is low, chemical transition 2,
the process from free head’s ADP releasing to attached
Load (pN) Load (pN)
head’s ATP hydrolysis,k , is the rate-limiting transition
3
at saturating [ATP] and v ≈ Lk ≈ 900 nm/s, which is
FIG.2: (a): Fittingto experimentalvmax and KM vsf[27]. 3
consistent with what is known about the biochemistry
The fitted parameters are listed in table I. (b): Dependence
of kinesin[37, 38], while ATP binding is the rate-limiting
of velocity on load at saturated andlimiting ATPconcentra-
tions. The curves are directly from Eq.(5) analytically with transition at low [ATP] and v ≈ Lk1 ≈ 50 nm/s with
parametersshownintableI,andtheexperimentaldatacome 5 µM. (2) If load is high, kinesin will stall at the same
from Ref[27]. (c): Applied force vs extension. The relation force as discussed in section III whether ATP concen-
f =κℓwithκshownintableIisingood agreementwiththe tration is high or low. (3) If applied force is moderate,
experimentaldata[22]at lowforce. (d): Dependenceofrates the velocity-force curves display different shapes at low
on applied force at 0.1 mM ATP. and saturating ATP concentration respectively. At very
low ATP concentration, the velocity decreases exponen-
tially as k and looks like linear with load because ATP
1
power stroke becomes the rate-limiting step for the sin- binding is the rate-limiting step. At saturating [ATP],
gle enzymatic cycle, so, KM will fall fast as vmax shown the velocity is concave down with load as discussed in
in Figure2(a). Michaelis constant no longer increases section IVA.
monotonicallywith load. The big errorbar at highforce
implies that this prediction is possible.
D. two mechanical substeps
The two mechanical substeps in this model are con-
tributed by the two heads respectively. The recovery
B. neck linker’s bending stroke is always a rapid rate step which corresponds to
the observedfastsubstep[14]. Ifamoderateloadsuchas
Actually, the neck linker’s bending has been uncon- 1.5 pNacts onthe beadandATP concentrationis main-
sciously revealed in experiment[22]. With an applied tained at 0.1 mM, the two chemical transitions have the
force moving toward the end of microtubule, a simply same rate as shown in Figure2(d). The time spent in
elastic model can fit all extension of kinesin-microtubule ATP binding equals that spent in chemical transition 2.
complex for double-headed attaching. But the force- Theoretically, 4.8 nm slow substep and 3.2 nm fast sub-
extensionrelationinFigure2(c)can’tbefittedbyasame step can be detected directly by single molecular manip-
simply elastic model for all extension for single-headed ulated techniques suchas opticaltweezers. The different
attaching. With our neck linker swing model, we can load-dependence of these substeps’ rate may be revealed
image there are three distinctive regions of force. If the in the future experiment.
force is low, the “Extension” is mainly contributed by
neck linker bending. We compare the relation f = κℓ
to the experimental force-extension data[22] in Figure V. CONCLUSION
2(c). It is very surprising that our model is in agree-
ment with experiment very well. With the increasing Weproposedthenecklinkerswingmodelwhichdivides
of applied force, a phenomenon of elastic instability[36] the single enzymatic cycle into two chemical transitions
will inevitably happen to the complex composed of neck and two mechanical substeps. Each chemical transition
linker and attached head. This is why the extension will induce the conformational change in the catalytic
rapidly increases while the “applied force” falls fast as cleftandgenerateacorrespondedmechanicalstroke. The
themeasureddatainFigure2(c). Afterreachingthenew model can be used to explain the observed substeps[14].
mechanical equilibrium, the complex will further extend The different load-dependence of these two strokes’ rate
5
may be revealed in the future experiment. We have in- experiment at low force[22]. With these reasonable fit-
vestigated the mechanism of mechanochemical coupling ted parameters, we can directly use Eq.(5) to describe
of kinesin by the influence of applied force on the bend- therelationbetweentheaveragevelocityandloadatdif-
ing of neck linker. When attached head is waiting for ferent ATP concentrations, which is in good agreement
ATPbinding,thenecklinkerisbentbytheappliedforce. to experimental data[27] as shown in Figure 2(b). In
Theattachedheadbearsthenecklinker’sbendingenergy addition, we have predicted Michaelis constant doesn’t
andtheenergybarrierforATPbindingincreases. Thisis increasemonotonicallyandanelasticinstabilitywillhap-
whyMichaelisconstantincreaseswithappliedforce. Our pentothecomplexcomposedofnecklinkerandattached
theoreticalanalysisofaveragevelocityofmotorinEq.(5) head with increasing of applied force.
alsoobeysMichaelis-Mentenlawandhasbeenusedtofit
the observed saturating velocity and Michaelis constant
at different loads[27]. The fitted values of chemical re-
action rates are in agreement with those in the widely
Acknowledgements
accepted model[31], and the fitted viscous coefficient of
bead is alsocomparablewith the Stokes drag coefficient.
The stall force is independent of ATP concentrationand We acknowledge useful discussions with Ou-Yang
its fitted value is consistent with the observed data in Zhong-can and Ming Li. This work was supported by
experiment[27]. Thefittedbendingrigidityofnecklinker Special Fund for Theoretical Physics of Postdoctor and
can be used to explain the relation of force-extension in National Science Foundation of China.
[1] T.Kreis and R.D.Vale , Guidebook to the Cytoskeletal [21] W.R.Schief et al.,PNAS,101, 1183 (2004)
and Motor Proteins, (Oxford Univ. Press, Oxford, ed. [22] K.Kawaguchi and S.Ishiwata, Science, 291, 667 (2001)
2, 1999) pp.398-402 [23] S.P.Gilbert et al., Nature,373, 671 (1995)
[2] N.Hirokawa, Science, 279, 519 (1998) [24] S.P.Gilbert,M.L.Moyer,K.A.Johnson,Biochemistry,37,
[3] S.M.Block, L.S.B.Goldstein and B.J.Schnapp, Nature, 792 (1998)
348, 348 (1990) [25] M.L.Moyer,S.P.Gilbert,K.A.Johnson,Biochemistry,37,
[4] K.Svobodaet al., Nature365, 721 (1993) 800 (1998)
[5] J.Howard, Annu.Rev.Physiol. 58, 703 (1996) [26] Y.G.Shu and H.L.Shi, PRE, 69, 021912 (2004)
[6] K.Kawaguchi and S.Ishiwata, Biochem. Biophys. Res. [27] K.Visscher, M.J.Schnitzer and S.M.Block, Nature 400,
Commun. 272, 895 (2000) 184 (1999)
[7] K.Hirose, L.A.Amos, Cell. Mol. Life Sci. 56, 184(1999) [28] R.D.Vale and R.A.Milligan, Science 288, 88 (2000)
[8] N.Hirokawa et al.,Cell 56, 867(1989) [29] H.Sosa et al.,Nat. Struct.Biol., 8, 540 (2001)
[9] J.Howard,A.J.HudspethandR.D.Vale,Nature342,154 [30] P.Nelson, Biological Physics: Energy, Information, Life,
(1989) (W.H. Freeman and Co., 2004),pp.441-444.
[10] D.D.Hackney,Nature377, 448 (1995) [31] R.A.Cross, TRENDS in Biochemical Sciences, 29, 301
[11] W.Huaet al., Nature388, 390 (1997) (2004)
[12] M.J.Schnitzer and S.M.Block, Nature 388, 386 (1997) [32] S.M.Block et al.,PNAS 100, 2351 (2003)
[13] D.L.Coy, M.Wagenbach and J.Howard, J. Biol. Chem. [33] M.J.Schnitzer andS.M.Block, Cold SpringHarborSym-
274, 3667 (1999) posia on QuantitativeBiology, 60, 793 (1995)
[14] M.Nishiyama et al.,Nat. Cell Biol., 3, 425 (2001) [34] SunneyXie, Single Mol. 2, 229 (2001)
[15] S.Riceet al.,Nature402, 778 (1999) [35] L.Stryer, Biochemistry, Fourth Edition (New York:
[16] R.B.Case et al., Curr. Biol., 10, 157 (2000) Freeman,1995),pp.443-462.
[17] M.Tomishige and R.D.Vale, J. Cell. Biol. 151, 1081 [36] L.D.Landau and E.M.Lifshitz, Theory of Elasticity, (Pe-
(2000) framon Press, 1986), pp.70-84.
[18] K.Kasedal, H.Higuchi and K.Hirosel, Nat. Cell Biol., 5, [37] D.D.Hackney,PNAS,85, 6314 (1988)
1079 (2003) [38] S.P.Gilbert and K.A.Johnson, Biochemistry, 33, 1951
[19] C.L.Asbury,A.N.FehrandS.M.Block,Science,302,2130 (1994)
(2003)
[20] A.Yildiz et al.,Science, 303, 676 (2004)