Hopping and stalling of processive molecular motors

ARTICLE IN PRESS Journal of Theoretical Biology 261 (2009) 43–49

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Hopping and stalling of processive molecular motors Yasuhiro Imafuku , Neil Thomas 1, Katsuhisa Tawada Department of Biology, Graduate School of Sciences, Kyushu University, Fukuoka 812-8581, Japan

a r t i c l e in fo

abstract

Article history: Received 13 March 2009 Received in revised form 7 July 2009 Accepted 8 July 2009 Available online 21 July 2009

When a two-headed molecular motor such as kinesin is attached to its track by just a single head in the presence of an applied load, thermally activated head detachment followed by rapid re-attachment at another binding site can cause the motor to ‘hop’ backwards. Such hopping, on its own, would produce a linear force–velocity relation. However, for kinesin, we must incorporate hopping into the motor’s alternating-head scheme, where we expect it to be most important for the state prior to neck-linker docking. We show that hopping can account for the backward steps, run length and stalling of conventional kinesin. In particular, although hopping does not hydrolyse ATP, we find that the hopping rate obeys the same Michaelis–Menten relation as the ATP hydrolysis rate. Hopping can also account for the reduced processivity observed in kinesins with mutations in their tubulin-binding loop. Indeed, it may provide a general mechanism for the breakdown of perfect processivity in two-headed molecular motors. & 2009 Elsevier Ltd. All rights reserved.

Keywords: Brownian motion Monte Carlo Kinesin

1. Introduction Processive molecular motors play a central role in intracellular transport along the cytoskeleton. Important examples are kinesin and myosin V, both of which have two heads that are believed to step alternately along microtubules or actin filaments, respectively (Hackney, 1994, 1995; Duke and Leibler, 1996; Hancock and Howard, 1999; Mehta et al., 1999; Rief et al., 2000). This alternating-head picture can account for the processive motion observed in single-molecule experiments on these motors (Svoboda et al., 1993; Schnitzer and Block, 1997; Visscher et al., 1999; Schnitzer et al., 2000; Rief et al., 2000; Mehta, 2001; Spudich and Rock, 2002; Veigel et al., 2002; Kaseda et al., 2003). However, the motors are not in fact perfectly processive: at low loads, they typically take a 100 or more steps along their tracks before dissociating (Funatsu et al., 1995; Schnitzer et al., 2000; Sakamoto et al., 2003; Clemen et al., 2005); both motors also occasionally step backwards, particularly under a high load (Rief et al., 2000; Nishiyama et al., 2002; Carter and Cross, 2005). Furthermore, both kinesin and myosin V exhibit ‘stalling’ at high loads, whereby the motors come to a halt and then dissociate from their tracks (Schnitzer et al., 2000; Rief et al., 2000). We present here a simple model that can account for this non-processive behaviour.  Corresponding author. Tel.:/./fax: +81 92 642 2634.

E-mail addresses: [email protected] (Y. Imafuku), [email protected] (N. Thomas), [email protected] (K. Tawada). 1 Present address: Physics Department, Birmingham University, Birmingham B15 2TT, UK. 0022-5193/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2009.07.011

Kaseda et al. (2002) performed a key experiment that sheds light on the breakdown of processivity in kinesin. They observed lower processivity in a range of kinesins with different mutations in the tubulin-binding loop. The force–velocity relations for these mutant kinesins cannot be explained using the alternating-head model alone (Thomas et al., 2002). However, the experimental results suggest that tubulin binding is involved in the breakdown of processivity. We propose here that the non-processive motion arises from detachment of kinesin when it is bound to tubulin by just a single head. At low loads, the rapid Brownian motion of the detached head allows it to re-attach to a nearby binding site. Both head detachment and rapid re-attachment are thermally activated processes. Hence, they effectively cause the motor to ‘hop’ from one binding site to another. High loads slow down the processive forward stepping of the motor, and when it detaches from tubulin it is pulled a long way backwards by the applied force. Hopping can therefore produce the stalling behaviour that is observed in real processive motors. Section 2 of this paper presents the basic physical model for hopping by a molecular motor that is attached to its track by a single head. Although thermally activated hopping is a common phenomenon in solid-state physics, we believe that this may be the first time that hopping has been considered for a biological molecular motor. We demonstrate in Section 3 that hopping can account for the backward steps (Nishiyama et al., 2002; Carter and Cross, 2005), run length (Schnitzer et al., 2000; Rosenfeld et al., 2003) and stalling force (Visscher et al., 1999) in conventional kinesin. Section 4 then shows that hopping can also account for the reduced processivity of the mutant kinesins studied by Kaseda et al. (2002). Moreover, hopping may occur in other two-headed

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motors, such as myosin V (Rief et al., 2000; Clemen et al., 2005) and myosin VI (Rock et al., 2001). The hopping model presented here may therefore provide a general insight into the breakdown of processivity in molecular motors.

2. Single-headed hopping Fig. 1 illustrates hopping by a molecular motor. For simplicity, we consider a motor that consists of just a single head A joined to a spring S of stiffness l0. The spring tethers the motor to a bead C that is subjected to a constant force f. The head of the motor can also bind to a filament or microtubule at discrete attachment sites separated by a distance u0. In Fig. 1a, the motor is bound to attachment site n, whilst in Fig. 1b it has acquired sufficient thermal energy to detach. Both the detached motor and the bead execute Brownian motion, and at the same time the bead drifts in response to the load f. If the detached motor comes into contact with another binding site then it may re-attach at that site, as shown in Fig. 1c. Detachment followed by rapid re-attachment therefore allows the motor to ‘hop’ from one binding site to another. The constant load f applied to the bead in Fig. 1 produces two separate contributions to the average distance moved by hopping. First of all, when the motor detaches, its spring recoils. The motor therefore quickly moves backwards a distance f/l0 relative to the bead, as indicated in Fig. 1b. Secondly, the bead itself moves backwards in response to the load at a drift velocity of vdrift ¼ mf ¼ f =6pZr;

ð2:1Þ

where we have used Stokes’ law to determine the drift mobility m for a spherical bead of radius r in a medium of viscosity Z. If the motor in Fig. 1b is detached for an average time t, then Eq. (2.1) implies that the bead drifts an average distance mft. Hence, the average hopping distance d is given by

d ¼ f =l0 þ mf t:

ð2:2Þ

To calculate the force–velocity relation for the hopping process, we write the hopping velocity as v ¼ k0 pd;

ð2:3Þ

where k0 is the detachment rate constant, and p is the probability that the motor is attached. (For simplicity, we take k0 here to be a constant. However, in some cases k0 may depend on the load; for instance, Nishizaka et al. (1995) observed an increase in the detachment rate of rigorized myosin II cross-bridges at high loads.) At steady state, the attachment and detachment rates must balance, in which case k0 p ¼ ð1  pÞ=t:

ð2:4Þ

Hence the attachment probability in Eq. (2.3) is p ¼ 1=ð1 þ k0 tÞ:

ð2:5Þ

It therefore follows from Eqs. (2.2), (2.3) and (2.5) that the hopping velocity is v ¼ ðk0 f =l0 Þ½ð1 þ l0 mtÞ=ð1 þ k0 tÞ:

ð2:6Þ

Note that the above derivation assumes that the bead is in equilibrium before the motor detaches. Since the bead comes into equilibrium with a time constant of 1/l0m, and the average attachment time is 1/k0, we require that l0mbk0. It is useful to consider two limiting cases for the hopping velocity in Eq. (2.6). In the first case, when tb1/k0, the displacement dEmft and the attachment probability pE1/k0t, so v  vdrift ¼ mf :

ð2:7Þ

Fig. 1. Single-headed hopping of a molecular motor. A spring of stiffness l0 tethers the motor to a bead subjected to a constant force f. The head of the motor can bind to a filament or microtubule at discrete attachment sites separated by the distance u0. (a) The motor is bound to attachment site n. (b) The motor has detached. The detached motor and the bead execute Brownian motion, and at the same time the bead drifts in response to the load. (c) If the detached motor comes into contact with another binding site, then it may re-attach as shown. (d) Force–velocity relations for the single-headed hopping model for a bead of diameter of 0.5 mm when l0 ¼ 0.95 pN nm1 and the detachment rate constant k0 ¼ 10 s1. Points show the results of Monte Carlo simulations using different values for the attachment rate constant k1 as indicated. Also shown are the theoretical force–velocity relations for the viscous-drag regime (heavy line) and the purehopping regime (light line) according to Eqs. (2.7) and (2.8).

In this viscous-drag regime, the motor is detached for most of the time. Hence, the bead velocity is determined by Eq. (2.1). The opposite extreme occurs when t51/l0m. In that case, dEf/l0 and pE1, so we find v  vhop ¼ k0 f =l0 :

ð2:8Þ

In this pure-hopping regime, the motor is attached for most of the time, but it hops backwards a distance f/l0 at a rate of k0 times per

ARTICLE IN PRESS Y. Imafuku et al. / Journal of Theoretical Biology 261 (2009) 43–49

45

second. Since the bead does not drift significantly during the extremely short detachment time t, the hopping distance d is determined purely by contraction of the spring, Hence, the hopping velocity vhop does not depend on t or m. The pure-hopping and viscous-drag regimes represent extreme cases where either t51/l0m or tb1/k0, respectively. An intermediate hopping regime occurs when 1/l0m5t51/k0, in which case v  k0 mf t:

ð2:9Þ

In this case, the displacement dEmft, whilst the attachment probability pE1. As in the pure-hopping regime, the motor is attached for most of the time and it hops backwards at a rate of k0 times per second. However, the longer detachment time t means that the average hopping distance d is determined by drift of the bead rather than by contraction of the spring. During the final phase of the hopping process in Fig. 1c, the motor re-attaches after the load has pulled the bead backwards a distance mft. If the bead position is x, then re-attachment of the motor at site m entails a spring extension of x-mu0 and requires an elastic energy of l0(x-mu0)2/2. Since this energy arises from thermal fluctuations, we may write the rate constant k01,m(x) for re-attachment at site m as k01;m ðxÞ ¼ k1 exp½l0 ðx  mu0 Þ2 =2kT;

ð2:10Þ

where k1 is a constant. A similar relation applies to the attachment rate constant of myosin II cross-bridges in muscle (Thomas and Thornhill, 1998). The motor can re-attach at site m by stretching or compressing the spring. However, the symmetrical Gaussian position dependence of k01,m(x) in Eq. (2.10) means that the average spring extension immediately after re-attachment is zero. Once the motor has re-attached, the load pulls the bead backwards a distance f/l0 as it stretches the spring back to its original length in Fig. 1a. The motor and the bead therefore move in slightly different ways during hopping. After detachment, the motor first moves backwards a distance f/l0 when its spring recoils. The detached motor then drifts backwards a distance mft with the bead and reattaches with (on average) zero spring extension. In contrast, the bead first drifts a distance mft backwards, and the load then pulls it a further distance f/l0 against the spring after the motor reattaches. Hopping therefore displaces both the motor and the bead by the same distance d in Eq. (2.2). The detachment time t in Eq. (2.6) plays a key role in the hopping. However, in general, one cannot calculate t analytically. We have therefore performed Monte Carlo simulations of the hopping model, incorporating drift and Brownian motion of the bead together with random attachment and detachment of the motor (Thomas and Thornhill, 1998). Fig. 1d shows the force–velocity relations obtained for a wide range of values of k1 when k0 ¼ 10 s1. As expected from Eq. (2.6), the force–velocity relations are approximately linear. Also shown are the theoretical force–velocity relations for the viscous-drag regime (heavy line) and the pure-hopping regime (light line) according to Eqs. (2.7) and (2.8).

3. Hopping of kinesin Fig. 2 incorporates hopping into an alternating-head scheme for the processive motion of kinesin along a microtubule (Hackney, 1994, 1995; Duke and Leibler, 1996; Hancock and Howard, 1999; Rice et al., 1999; Thomas et al., 2002). The kinesin dimer is attached to an optically trapped bead and moves through four states at tubulin attachment site n before proceeding to site n+1 a distance u0 ( ¼ 8 nm) along the microtubule. In state 1, head

Fig. 2. Four-state cycle for the stepping of a kinesin homodimer. Hopping is shown here for state 2, which precedes neck-linker docking (Thomas et al., 2002).

A is nucleotide free and is attached to site n, whilst head B is detached and still binds ADP. The neck linkers of both heads at this stage are undocked (Rice et al., 1999). Head A then binds ATP to form state 2, and that allows its neck linker to dock, producing state 20 . Head B then swings forward and attaches to site n+1 to form the two-headed attachment complex shown as state 3. Following this, head A detaches and releases inorganic phosphate (Pi), which allows its neck linker to undock (Rice et al., 1999), whilst head B remains attached to site n+1 and releases ADP. The cycle then repeats itself with the roles of heads A and B interchanged. The kinesin motor therefore hydrolyses one molecule of ATP for each 8 nm step as it advances towards the ‘plus’ end of the microtubule. Hopping may occur for any of the states where the kinesin dimer in Fig. 2 is attached to the microtubule by just a single head. However, we believe that hopping in state 2 is likely to be most important. This is because an applied force inhibits neck-linker docking (Thomas et al., 2002). Hence, when the kinesin motor pulls against a load, it is trapped for much of its cycle in state 2 with its neck linker undocked. Trapping the motor in state 2 is therefore likely to increase the importance of hopping for this state, as indicated in Fig. 2. Hopping in state 2 can account for the occasional backward steps observed by Nishiyama et al. (2002) in conventional kinesin. If we define the directionality ratio r as the ratio of forward- to

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ð3:1Þ

where R is the steady-state ATP hydrolysis rate, k0 is the detachment rate constant, and p2 is the probability that the motor is in state 2. The other terms on the right-hand side of Eq. (3.1) are defined by Thomas et al. (2002). We have also assumed that the concentrations of ADP and Pi are essentially zero. The points in Fig. 3a show the directionality ratio determined experimentally by Nishiyama et al. (2002) for kinesin at various loads when [ATP] ¼ 1 mM (triangles) and [ATP] ¼ 10 mM (squares). The curve is a fit of Eq. (3.1) to the experimental data. The single fitting parameter here is the detachment rate constant k0 ¼ 6.771.0 s1. The other parameters used for the kinesin model are given in the appendix. The experimental data in Fig. 3a agree very well with the load dependence expected from the hopping theory. The curvature of the theoretical fit arises from the quadratic load dependence in the exponential term in Eq. (3.1), which is a characteristic feature of the neck-linker docking model of Thomas et al. (2002). Furthermore, the experimental directionality ratio is independent of [ATP], in agreement with Eq. (3.1). In contrast, we find that the directionality ratio for hopping in state 1 is load dependent but is proportional to [ATP], whilst the ratio for hopping in state 20 (and, in principle, state 3) is independent of both load and [ATP]. Hence, only hopping in state 2 produces both the correct load dependence and the insensitivity of the directionality ratio to [ATP]. The rate of backward hopping k0p2 in Eq. (3.1) must therefore exhibit the same Michaelis–Menten behaviour as the ATP hydrolysis rate R (Visscher et al., 1999; Fisher and Kolomeisky, 2001; Thomas et al., 2002). Both Nishiyama et al. (2002) and Carter and Cross (2005) interpreted the ATP dependence of the backward-stepping rate in kinesin as evidence that backward steps hydrolyse ATP. However, in our model that is not the case: backward steps due to hopping do not hydrolyse ATP. The ATP dependence of the hopping rate arises instead from the probability p2 that the motor is in state 2 in Fig. 2. It is straightforward to show from the kinetics of the kinesin cycle (Fisher and Kolomeisky, 2001; Thomas et al., 2002) that (in the absence of ADP and/or Pi) the ATP hydrolysis rate R and the occupation probabilities for all of the states (except the ATP-binding state 1) obey the same Michaelis–Menten relation. Hence, it is incorrect to conclude from their ATP dependence that backward steps involve ATP hydrolysis: the only correct conclusion that can be drawn is that backward steps do not occur in the ATP-binding state. Only hopping in state 2 produces the correct load dependence of the hopping rate. Indeed, Carter and Cross (2005) observed a load dependence similar to that in Fig. 3a. Fitting the hopping model to their data produces a detachment rate constant k0 ¼ 5.470.7 s1. This result agrees within experimental error with the value obtained above using the data of Nishiyama et al. (2002). The hopping model can therefore account for the backward steps in kinesin observed by both groups. Nishiyama et al. (2002) also measured the average dwell time before backward steps. We have used an analytical method equivalent to that of Shaevitz et al. (2005) to calculate the dwell times analytically for the four-state kinesin model, using the hopping rate constant k0 that was fitted to the directionality ratio in Fig. 3a. The predicted dwell times (solid curve) in Fig. 3b agree very well with the experimental data of Nishiyama et al. (2002) for [ATP] ¼ 1 mM (triangles). The agreement is not quite so good for [ATP] ¼ 10 mM, where the predicted dwell times (dashed curve)

10

1

0.1

0.01 0

2

4

6

8

10

Load / pN 600 500 Dwell time / ms

k220 k20 3 k34 exp½ðf  f0 Þ2 =2lkT   k0 k320 k20 2 þ k34 ðk20 2 þ k20 3 Þ

400 300 200 100 0

0

2

4

6

8

Load / pN 2000 1600 Run Length / nm

r ¼ R=k0 p2 ¼

Directionality Ratio

backward-stepping rates, then the four-state kinesin model with hopping in state 2 predicts that

1200 800 400 0 0

2

4 Load / pN

6

Fig. 3. (a) Load dependence of the directionality ratio for kinesin. Points represent the experimental data of Nishiyama et al. (2002) when [ATP] ¼ 1 mM (triangles) and [ATP] ¼ 10 mM (squares). The curve is a fit of Eq. (3.1) from the hopping model to the experimental data with a single fitting parameter k0 ¼ 6.771.0 s1. The other parameters used for the kinesin model are given in the appendix. (b) Load dependence of the dwell time for kinesin backward steps. Points represent the experimental data of Nishiyama et al. (2002) when [ATP] ¼ 1 mM (triangles) and [ATP] ¼ 10 mM (squares). The curves show the load dependence of the dwell time predicted by the hopping model when [ATP] ¼ 1 mM (solid line) and [ATP] ¼ 10 mM (dashed line) using the same parameters as in (a). (c) Load dependence of the kinesin run length. Points represent the experimental data of Schnitzer et al. (2000) when [ATP] ¼ 2 mM (triangles) and [ATP] ¼ 5 mM (squares). The curve is a fit of Eq. (3.2) to the experimental data with a single fitting parameter k0 ¼ 3.7370.30 s1. The other parameters used for the kinesin model are given in the appendix.

are shorter than the experimental dwell times (squares) at high loads. However, there are large uncertainties in the experimental data and also some uncertainty in the rate constants and other parameters for the kinesin cycle itself. Indeed, Nishiyama et al. (2002) found that the Michaelis constant Km ¼ 3673 mM for their kinesin preparation at zero load, which is much lower than the

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value determined by Schnitzer et al. (2000), who found that Km ¼ 8574 mM at a load of 1 pN. This must reflect either a genuine difference between the two kinesin preparations or systematic experimental errors. The parameters used here for the different kinesin preparations are given in the appendix. Hopping entails head detachment followed by rapid reattachment. However, as discussed by Schnitzer et al. (2000), detachment on its own may be the critical factor that determines a molecular motor’s run length. If detachment from state 2 always terminated the motor’s processive run (instead of producing a short backward hop), then we would expect the average run length L to be given by L ¼ u0 R=k0 p2 :

ð3:2Þ

Fig. 3c shows the results of Schnitzer et al. (2000) for the load dependence of the kinesin run length together with a theoretical curve obtained by fitting Eq. (3.2) to the data at high [ATP]. This fit produces a detachment rate constant k0 ¼ 3.870.3 s1. As one would expect, this value for k0 is lower than that required for hopping in Fig. 3a, since small backward hops were not included in the experimental measurements of the run length. Schnitzer et al. (2000) found that the run length decreased slightly at low [ATP], which implies that detachment may sometimes occur in state 1 of the kinesin cycle in Fig. 2. However, in a more recent study of an engineered kinesin, Rosenfeld et al. (2003) found the run length to be independent of [ATP]. Hence, detachment from state 2 seems to be the dominant factor that determines both the rate of backward hopping and the run length in kinesin.

47

Fig. 4a shows the force–velocity relations for wild-type and mutant kinesins studied by Kaseda et al. (2002). The solid curves represent fits of Eq. (4.2) to the experimental data points for the wild-type homodimer (denoted by WT/WT) and two mutant homodimers (denoted by L8/L8 and L11/L11) using k320 and k20 3 as fitting parameters, as described in the appendix. Dashed lines are the force–velocity relations for the heterodimers (WT/L8 and WT/L11) predicted by the hopping model using the parameters determined for the homodimers. Fig. 4a also shows the experimental data for the WT/L12 mutant heterodimer. There are no force–velocity data for the non-processive L12/L12 homodimer (Kaseda et al., 2002). Hence, as described in the appendix, we have fitted the hopping model directly to the WT/L12 heterodimer data in Fig. 4a (dashed-dotted curve). It is clear from Fig. 4a that the hopping model can account for the force–velocity relations of the wild-type and mutant kinesins. In particular, the model predicts that a motor stalls when v ¼ 0 in Eq. (4.1). The stalling force fs must therefore satisfy the implicit equation fs ¼ u0 Rðfs Þ=mhop p2 ðfs Þ;

ð4:3Þ

where both the ATPase rate R and the probability p2 are functions of load f. The right-hand side of this equation is similar to the expressions for the directionality r in Eq. (3.1) and the run length

4. Hopping and stalling of mutant kinesins Kaseda et al. (2002) observed lower processivity in several kinesins with mutations in the tubulin-binding loop. Such mutations may promote hopping by weakening the binding of the mutant heads to tubulin. The hopping model for kinesin therefore provides a basis for analysing these mutant kinesins. In the presence of hopping in state 2, the average velocity of a kinesin homodimer may be written as

v ¼ u0 R  m0 fp2 k320 =k20 3 : where m0 is a constant.

ð4:2Þ

WT/WT WT/L8 L8/L8 WT/L11

200

WT/L12

0

ð4:1Þ

The first term on the right-hand side of this equation is the normal stepping velocity of a processive motor (Thomas et al., 2001). The motor moves forwards a distance u0 for each complete cycle in Fig. 2 and hydrolyses one molecule of ATP. Hence the stepping velocity is proportional to the steady-state ATPase rate R, which can be calculated from the rate constants for the kinesin model (Fisher and Kolomeisky, 2001; Thomas et al., 2002). The second term on the right-hand side of Eq. (4.1) represents the hopping velocity. It is negative because hopping opposes the forward stepping of the motor. As required by Eq. (2.6), the hopping velocity is proportional to the load f, whilst Eq. (2.3) also requires it to be proportional to the probability p2 of finding the motor in state 2. We have subsumed the other contributions to the hopping velocity into the hopping mobility mhop, which may be determined by fitting Eq. (4.1) to experimental data. In the intermediate hopping regime, Eq. (2.9) implies that mhop is proportional to k0/k1, where k0 and k1 are the detachment and attachment rate constants for kinesin hopping in state 2. Since the transition from states 20 to 3 for the kinesin cycle in Fig. 2 also involves head attachment, we may expect that k0/k1 should be proportional to k320 /k20 3. We may therefore rewrite Eq. (4.1) as

400

L11/L11

0

2

4 Load / pN

6

8

WT/WT

80

WT/L8

WT/WT

ATPase Rate / s-1

v ¼ u0 R  mhop fp2 :

Velocity / nm s-1

600

60 40

L8/L8

WT/L8

WT/L11

L8/L8

L11/L11

WT/L11

WT/L12 L12/L12

L11/L11 WT/L12

20

L12/L12

0 0

2

4 Load / pN

6

8

Fig. 4. Hopping model applied to kinesin binding-loop mutants. (a) Force–velocity relations for mutant kinesin homodimers (WT/WT, L8/L8, L11/L11 and L12/L12) and heterodimers (WT/L8, WT/L11 and WT/L12). Points represent the experimental data of Kaseda et al. (2002). Solid lines are fits to the homodimer data using the hopping model for kinesin with the parameters given in the appendix. Dashed lines are the force–velocity relations for the heterodimers predicted by the hopping model using the parameters determined for the homodimers. (b) ATPase rate for the mutant kinesins. The curves represent the load-dependent ATPase rate predicted by the hopping model, whilst the points represent the experimental measurements (normalized to wild-type kinesin) of Kaseda et al. (2002) at essentially zero load.

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L in Eq. (3.2). Hence the hopping model predicts that the stalling force should be independent of [ATP]. The measurements on mutant kinesins by Kaseda et al. (2002) were all performed with [ATP] ¼ 1 mM. However, Visscher et al. (1999) observed only a slight increase in the stalling force in wild-type kinesin on increasing ATP concentration from 5 mM to 2 mM. Fig. 4a shows that wild-type kinesin and the L8, L11 and L12 mutants have progressively lower stalling forces. This mainly arises from an increase in the hopping rate for head A in state 2 in Fig. 2, leading to an increased hopping mobility mhop in Eq. (4.3). Hopping therefore disrupts the processive stepping of the motor but it does not affect the ATPase rate R. However, the weaker binding of the mutant heads may also increase the rate of detachment of head B in state 3 (represented here by the rate constant k320 ). Since the motor is still bound by the other head, detachment of head B in state 3 does not produce hopping. Instead, it promotes the backward transition to state 20 , which slows down the kinesin cycle. Fig. 4a reflects this slowing down through the reduced velocity of the mutants at zero load. Since the hopping velocity in Eq. (4.1) vanishes at zero load, the decrease in the unloaded velocity cannot be attributed to hopping. Fig. 4b illustrates the reduction in the ATPase rate for the mutant kinesins. The curves represent the load-dependent ATPase rate predicted by the hopping model using the same parameters as in Fig. 4a, whilst the points represent the experimental measurements (normalized to wild-type kinesin) of Kaseda et al. (2002) using a microtubule-gliding assay. The reduced ATPase rate of the mutants is clearly apparent in the figure. The theoretical curves show a similar trend to the experimental data, except for the WT/L12 heterodimer, whose ATPase rate is unexpectedly high (Kaseda et al., 2002). Fig. 4b demonstrates that even mutants with low processivity such as L11/L11 are predicted to maintain most of their ATPase activity at loads up to 4 pN. Measurements of the load dependence of the ATPase rate for the mutant kinesins could therefore provide an additional test of the hopping model. A final point concerns thermodynamic consistency. The rate constants for the four-state kinesin model in Fig. 2 must satisfy the thermodynamic relation (Thomas et al., 2001) k12 k220 ðf Þk20 3k34 =k43 ðf Þk320 k20 2k21 ¼ exp½ðDG þ u0 f Þ=kT;

ð4:4Þ

where DG is the free-energy decrease due to the hydrolysis of one molecule of ATP. Only the rate constants k220 (f) and k43(f) here depend on the load f (Thomas et al., 2001), and it is straightforward to verify that Eq. (4.4) is obeyed. The four-state kinesin model is therefore consistent with thermodynamics. However, strictly speaking, we should not use k320 and k20 3 alone as fitting parameters in Fig. 4a, since that changes the value of the ratio on the left-hand side of Eq. (4.4), whilst the right-hand side is fixed. Since the transition from states 3 to 4 in Fig. 2 also involves head detachment, one can restore thermodynamic consistency by varying k34 and k43 such that the ratio k20 3k34/k43k320 remains constant. We find that this constraint produces fits similar to those in Fig. 4a for the wild-type, L8 and L11 homodimers (data not shown). However, we are unable to fit the results for the WT/ L12 heterodimer in this way, because the rapid head detachment requires a very high value for k34. This problem may arise because the rapid detachment of head A in the L12 mutant is not the ratelimiting factor in the transition from states 3 to 4, which also involves release of ADP by head B. Treating ADP release as a separate transition in the cycle would ensure its strict thermodynamic consistency. However, there are not enough experimental data to determine the two additional rate constants that this more detailed analysis requires.

5. Concluding remarks The essence of the hopping model in Fig. 1 is that head detachment followed by rapid re-attachment allows the motor to hop from one binding site to another. When the head detaches, its spring recoils and the bead drifts backwards under the applied load. Brownian motion of the detached head then allows it to reattach to another binding site. On average, the motor hops backwards a distance d that is proportional to the load f. The Monte Carlo simulations of hopping in Fig. 1d therefore produce a linear force–velocity relation, as expected from Eq. (2.6). In this respect, one might regard hopping as an example of ‘protein friction’ (Tawada and Sekimoto, 1991). We find that the detachment time t plays a key role in the hopping. In the pure-hopping regime, t is very short and drift of the bead can be neglected. The opposite is true in the viscous-drag regime, where t is long and drift of the bead dominates. There is also an intermediate regime, where the head makes occasional long hops during which the bead drifts backwards. Hopping may occur in the two-headed kinesin motor when it is bound to tubulin by just a single head. We expect this to occur mainly when the motor in Fig. 2 is trapped by an applied load in state 2 of the alternating-head scheme prior to neck-linker docking. Hopping can account for the backward steps (Nishiyama et al., 2002; Carter and Cross, 2005), run length (Schnitzer et al., 2000; Rosenfeld et al., 2003) and stalling (Visscher et al., 1999) that are observed in conventional kinesin. Moreover, although hopping itself does not hydrolyse ATP, we find that the hopping rate (in the absence of ADP and Pi) obeys the same Michaelis–Menten relation as the ATP hydrolysis rate. Hopping can also account for the reduced processivity of the mutant kinesins studied by Kaseda et al. (2002). They observed lower processivity in a range of kinesins with different mutations in the tubulin-binding loop. Weaker tubulin binding promotes hopping, which is represented by a negative linear term in the force–velocity relation in Eq. (4.1). The weaker binding also promotes detachment of the leading head when both heads are attached in state 3. This slows down the kinesin cycle and reduces the rate of ATP hydrolysis. The hopping model is not specific to kinesin. We may expect hopping to occur in other two-headed molecular motors, such as myosin V (Rief et al., 2000; Clemen et al., 2005) and myosin VI (Rock et al., 2001). The perfect processivity of the alternating-head model should perhaps be regarded as an idealization. Hopping provides a simple mechanism for this perfect processivity to break down.

Acknowledgement We are grateful to the Japan Society for the Promotion of Science, the British Council, and the Ministry of Education, Culture, Sports, Science & Technology in Japan for support for one of us (NT) to work at Kyushu University.

Appendix A. Parameters for kinesin homodimers Conventional kinesin in Fig. 3. To model the kinesin preparations of Carter and Cross (2005) and Schnitzer et al. (2000), we used the following parameters in the four-state kinesin model (Thomas et al., 2002): f0 ¼ 1 pN, l ¼ 1.72 pN nm1, K12 ¼ 2 mM1 s1 (Moyer et al., 1998), k21 ¼ 827 s1, k220 ¼ 630 s1, k20 2 ¼ 10 s1, k20 3 ¼ 166 s1, k320 ¼ 1.74 s1 and k34 ¼ 217 s1. We set k43 ¼ 0, since [Pi] and [ADP] were essentially zero. Fitting Eq. (3.1) by nonlinear regression to the data of Carter and Cross (2005) using MATHEMATICA (Wolfram Research, Champain, IL, USA), we found that

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the detachment rate constant was k0 ¼ 5.470.7 s1. A separate fit of Eq. (3.2) to the run-length data of Schnitzer et al. (2000) at high [ATP] in Fig. 3c yielded k0 ¼ 3.7570.30 s1. To model the kinesin preparation of Nishiyama et al. (2002), we used the same parameters as above, except that we set K12 ¼ 3.8 mM1 s1, k21 ¼ 79.3 s1 and k20 3 ¼ k34 ¼ 305 s1 in order to obtain their Michaelis–Menten parameters of Vmax ¼ 960 nm s1 and Km ¼ 36 mM1. Fitting Eq. (3.1) to the data of Nishiyama et al. (2002) in Fig. 3a yielded k0 ¼ 6.771.0 s1. Wild-type and mutant kinesins in Fig. 4. For wild-type and mutant kinesin homodimers, we fixed the following parameters: f0 ¼ 1 pN, l ¼ 1.72 pN nm1, K12 ¼ 2 mM1 s1, k21 ¼ 827 s1, k20 2 ¼ 630 s1, k20 2 ¼ 10 s1, k34 ¼ 217 s1 and k43 ¼ 0. We then used MATHEMATICA to fit Eq. (4.2) by non-linear regression to the data of Kaseda et al. (2002) in Fig. 4a when m0 ¼ 750 nm s1 pN1. The values of the fitted parameters for wild-type kinesin were k20 3 ¼ 166714 s1 and k320 ¼ 1.771.2 s1. The fitted parameters for the L8 homodimer were k20 3 ¼ 96718 s1 and k320 ¼ 1876 s1, whilst for the L11 homodimer we found k20 3 ¼ 69713 s1 and k320 ¼ 205752 s1. For the L12 heterodimer, we set k20 3 ¼ 69 s1, and the single fitted parameter was k320 ¼ 837773 s1. References Carter, N.J., Cross, R.A., 2005. Mechanics of the kinesin step. Nature 435, 308–312, doi:10.1038/nature03528. ¨ Clemen, A.E.-M., Vilfan, M., Jaud, J., Zhang, J., Barmann, M., Rief, M., 2005. Force-dependent stepping kinetics of myosin-V. Biophys. J. 88, 4402–4410, doi:10.1529/biophysj.104.053504. Duke, T., Leibler, S., 1996. Motor protein mechanics: a stochastic model with minimal mechanochemical coupling. Biophys. J. 71, 1235–1247, doi:10.1016/ S0006-3495(96)79323-2. Fisher, M.E., Kolomeisky, A.B., 2001. Simple mechanochemistry describes the dynamics of kinesin molecules. Proc. Natl. Acad. Sci. USA 98, 7748–7753, doi:10.1073/pnas.141080498. Funatsu, T., Harada, Y., Tokunaga, M., Saito, K., Yanagida, T., 1995. Imaging of single fluorescent molecules and individual ATP turnovers by single myosin molecules in aqueous solution. Nature 374, 555–559, doi:10.1038/374555a0. Hackney, D.D., 1994. Evidence for alternating-head catalysis by kinesin during microtubule-stimulated ATP hydrolysis. Proc. Natl. Acad. Sci. USA 91, 6865–6869. Hackney, D.D., 1995. Highly processive microtubule-stimulated ATP hydrolysis by dimeric kinesin head domains. Nature 377, 448–450, doi:10.1038/377448a0. Hancock, W.O., Howard, J., 1999. Kinesin’s processivity results from mechanical and chemical coordination between the ATP hydrolysis cycles of the two motor domains. Proc. Natl. Acad. Sci. USA 96, 13147–13152. Kaseda, K., Higuchi, H., Hirose, K., 2002. Coordination of kinesin’s two heads studied with mutant heterodimers. Proc. Natl. Acad. Sci. USA 99, 16058–16063, doi:10.1073/pnas.252409199. Kaseda, K., Higuchi, H., Hirose, K., 2003. Alternate fast and slow stepping of a heterodimeric kinesin molecule. Nat. Cell Biol. 5, 1079–1082, doi:10.1038/ncb1067.

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