Force transduction by the microtubule-bound Dam1 ring
FForce transduction by the microtubule-bound Dam1 ring
Jonathan W ArmondDepartment of Physics and MOAC Doctoral Training Centre,University of Warwick, Coventry, CV4 7AL, UKMatthew S Turner ∗ Department of Physics,University of Warwick, Coventry, CV4 7AL, UK
Abstract
The coupling between the depolymerization of microtubules (MTs) and the motion of the Dam1 ringcomplex is now thought to play an important role in the generation of forces during mitosis. Our currentunderstanding of this motion is based on a number of detailed computational models. Although thesemodels realize possible mechanisms for force transduction, they can be extended by variation of any ofa large number of poorly measured parameters and there is no clear strategy for determining how theymight be distinguished experimentally. Here we seek to identify and analyze two distinct mechanismspresent in the computational models. In the first the splayed protofilaments at the end of the depolymer-izing MT physically prevent the Dam1 ring from falling off the end, in the other an attractive bindingsecures the ring to the microtubule. Based on this analysis, we discuss how to distinguish between com-peting models that seek to explain how the Dam1 ring stays on the MT. We propose novel experimentalapproaches that could resolve these models for the first time, either by changing the diffusion constantof the Dam1 ring (e.g., by tethering a long polymer to it) or by using a time varying load.
Key words: Brownian ratchet; burnt bridges; DASH; protofilaments; mitosis; motors ∗ Corresponding author. Address: Department of Physics, University of Warwick, Gibbet Hill Road, Coventry, CV4 7AL, UK,[email protected] a r X i v : . [ q - b i o . S C ] A p r orce transduction by the Dam1 ringorce transduction by the Dam1 ring
Key words: Brownian ratchet; burnt bridges; DASH; protofilaments; mitosis; motors ∗ Corresponding author. Address: Department of Physics, University of Warwick, Gibbet Hill Road, Coventry, CV4 7AL, UK,[email protected] a r X i v : . [ q - b i o . S C ] A p r orce transduction by the Dam1 ringorce transduction by the Dam1 ring Introduction
Mitosis is the mechanism of cell division in eukaryotic cells. In mitosis, chromosomes condense and arearranged at the center of the cell by the mitotic spindle. Microtubules (MTs) are protein fibers, composed of n parallel protofilaments (PFs, typically n =
13) forming a hollow cylinder. Each PF is built from stackedtubulin protein dimers. MTs emanate from centrosomes and attach to chromosome-bound kinetochores.Centrosomes are positioned at both poles of the cell forming a bipolar spindle. During anaphase chromo-somes are segregated and transported to the cell poles by the retraction of MTs, providing both daughtercells with a single copy of the cell’s chromosomes (1). To achieve segregation, depolymerizing kinetochore-attached microtubules (KMTs) must generate forces, e.g., to overcome chromosomal drag in the cytosol (2).There is evidence that mitotic MT force generation occurs in the absence of MT minus-end directed motorproteins (3) and when minus-end depolymerization is inhibited (4). Previously, a hypothetical sleeve hadbeen proposed to couple MT depolymerization to kinetochores (5, 6). A 10-protein complex, purified frombudding yeast (7), called Dam1 (or DASH) has been observed to form rings around MTs (8, 9). Dam1 ringshave been observed tracking depolymerizing MT plus ends in vitro (10) and an optical trap has been used tomeasure force-distance traces for Dam1-coated polystyrene beads attached to depolymerizing MTs (11–13).Intriguingly, Dam1 has been shown to be essential for chromosome segregation in budding yeast (14, 15)and important for avoiding mis-segregation problems in fission yeast (16).
Mechanisms of force transduction
Several models have been proposed to explain how the Dam1 ring can couple the kinetochore to a depoly-merizing MT so as to produce a force. Over two decades ago Hill proposed the first quantitative modeldescribing how a depolymerizing microtubule could be harnessed for the production of force (6). In thismodel a hypothetical sleeve surrounds the MT and provides the attachment to the kinetochore. An attractionbetween the sleeve and MT provides an energy barrier preventing detachment, but this sleeve may still beable to slide along the MT without paying the energy of detachment. More recent computational models aredetailed, mechanistic and micromechanical. One such model has taken into account the energy predicted tobe available due to the curling of PFs (17–19) and, following the discovery of the Dam1 ring, was extendedto reflect current structural knowledge and incorporate the hypothesis that Dam1 forms rigid transient linksto the MT (20). Another independent model postulated an electrostatic attraction maintaining the rings po-sition at the tip of the MT (21), combined with a powerstroke. All recent models include a combination ofthe following features: 1) the intrinsic diffusion of the Dam1 ring; 2) an effective powerstroke due to curlingPFs; 3) an attractive potential between Dam1 and the MT. However, whilst these models include many of therelevant physical features of the system and produce satisfactory simulations of a reliable force transductionsystem, the problem cannot be considered “solved” because many variants on these models are possible andthey have not been quantitatively compared to data. Furthermore, the lack of discriminatory experimentaldata precludes validation. In light of this we feel that much can be gained from rigorously analyzing thecontribution of the various features in order to determine their possible role.In what follows we describe two distinct “minimal” models, both of which describe a functional Dam1-mediated force transduction system. In the protofilament model the splaying PFs at the depolymerizingend physically prevent the ring from sliding off. In binding models an attraction between the ring andMT provides an energy barrier preventing detachment. The two models are not mutually exclusive – ahybrid model, incorporating both contributions, may also apply although one of the constituent mechanismswill typically dominate. While it is straightforward to modify our analysis to include such hybrid modelswe neglect them here for clarity, since our purpose is to differentiate the contributions. In common withprevious models we also neglect other molecular components, e.g., microtuble-associated proteins (MAPs)and kinases (22, 23), that certainly play important additional roles in vivo .Some previous studies have incorporated a powerstroke, arising from the motion of PFs, in driving themotion of the Dam1 ring (17, 20, 21). Indeed, it has been demonstrated that PFs can push a bead attached to orce transduction by the Dam1 ringorce transduction by the Dam1 ring
Several models have been proposed to explain how the Dam1 ring can couple the kinetochore to a depoly-merizing MT so as to produce a force. Over two decades ago Hill proposed the first quantitative modeldescribing how a depolymerizing microtubule could be harnessed for the production of force (6). In thismodel a hypothetical sleeve surrounds the MT and provides the attachment to the kinetochore. An attractionbetween the sleeve and MT provides an energy barrier preventing detachment, but this sleeve may still beable to slide along the MT without paying the energy of detachment. More recent computational models aredetailed, mechanistic and micromechanical. One such model has taken into account the energy predicted tobe available due to the curling of PFs (17–19) and, following the discovery of the Dam1 ring, was extendedto reflect current structural knowledge and incorporate the hypothesis that Dam1 forms rigid transient linksto the MT (20). Another independent model postulated an electrostatic attraction maintaining the rings po-sition at the tip of the MT (21), combined with a powerstroke. All recent models include a combination ofthe following features: 1) the intrinsic diffusion of the Dam1 ring; 2) an effective powerstroke due to curlingPFs; 3) an attractive potential between Dam1 and the MT. However, whilst these models include many of therelevant physical features of the system and produce satisfactory simulations of a reliable force transductionsystem, the problem cannot be considered “solved” because many variants on these models are possible andthey have not been quantitatively compared to data. Furthermore, the lack of discriminatory experimentaldata precludes validation. In light of this we feel that much can be gained from rigorously analyzing thecontribution of the various features in order to determine their possible role.In what follows we describe two distinct “minimal” models, both of which describe a functional Dam1-mediated force transduction system. In the protofilament model the splaying PFs at the depolymerizingend physically prevent the ring from sliding off. In binding models an attraction between the ring andMT provides an energy barrier preventing detachment. The two models are not mutually exclusive – ahybrid model, incorporating both contributions, may also apply although one of the constituent mechanismswill typically dominate. While it is straightforward to modify our analysis to include such hybrid modelswe neglect them here for clarity, since our purpose is to differentiate the contributions. In common withprevious models we also neglect other molecular components, e.g., microtuble-associated proteins (MAPs)and kinases (22, 23), that certainly play important additional roles in vivo .Some previous studies have incorporated a powerstroke, arising from the motion of PFs, in driving themotion of the Dam1 ring (17, 20, 21). Indeed, it has been demonstrated that PFs can push a bead attached to orce transduction by the Dam1 ringorce transduction by the Dam1 ring K i n e t o c h o r e K i n e t o c h o r e V x
PF curls MicrotubuleTubulin dimersDam1 ring
V x∆G
A B -ε x=0 x=0 Linker complexes
Figure 1: Two general classes of models of Dam1 ring-microtubule coupling. In both cases force is generated by rectifyingBrownian motion, that is the ring diffuses to the right and the MT happens to unzip one segment or it is driven to the right bya powerstroke associated with unzipping. The dotted line denotes the point reached by MT unzipping ( x = A ) The ring issterically confined to the MT by PFs (protofilament model) ( B ) The ring is attractively bound to the MT surface with a free energyof binding ∆ G Dam1 (binding model). Below each model, the potential profile V in which the ring diffuses is shown as a functionof the distance x of the ring from the MT end, and a dotted line indicates the connection between profile and model. The load forceis the slope of V ( x ) for x >
0. In ( A ) there is a large (infinite) energy barrier preventing the Dam1 ring moving to x < B ) the ringmaintains only partial contact as it slides off the end of the MT ( − ε < x < x < − ε . See text for details. the side of a MT (24), with a force of about 5 pN per 1–2 PFs. However, it is also known that models in the“burnt bridges” (25) class require no powerstroke per se to generate motion (26). Rather, purely diffusiveBrownian motion can be rectified if “bridges” (here segments of MT) are lost (depolymerize) after theyhave been crossed. No instantaneous physical force is required, although the resultant rectified Brownianmotion does give rise to a force in the thermodynamic sense. Such models are, in turn, members of a largerclass of models known as “Brownian ratchets” (27). These models exhibit velocities that depend on appliedforce, and stall for sufficiently high forces, as also seen in more complex models (21). It is not clear a priori to what extent the powerstroke plays an important role. In the present work we also seek to answer thisquestion. Generalized model
We seek to analyze a general model that includes both a diffusive burnt bridge mechanism and a powerstroke,in order to determine their relative contribution. Here the powerstroke involves a depolymerization eventwhich “unzips” PFs and moves the position of the last unbroken section of MT; a new section takes on thisidentity when the previous one unzips (contains separated, splayed protofilaments). As a highly energeticpowerstroke this is assumed to occur even when the ring is very close to the MT end, with a rate thatgives rise to a depolymerization velocity v ps . The sequence of microscopic PF unzippering events givesrise to a well defined velocity for the last fully intact MT section, irrespective of the sequence in which theneighboring PFs unzip, and the precise MT helicity. Critically, we also assume that polymerized MT can belost with a second rate , giving a depolymerization velocity v bb , whenever the ring has diffused a distance δ from the end. This can be thought of as the depolymerization velocity of a bare MT because, in thiscase, there is no Dam1 ring anywhere on the MT. We make no prior assumptions as to which contributiondominates, rather we determine this by fitting the parameters v bb , v ps and δ to data for the variation of theDam1 velocity with load (12).Our model involves a clear distinction between two mechanisms (only) and represents the simplest pos-sible model capable of explaining this data. It can be biophysically motivated on the grounds that the Dam1 orce transduction by the Dam1 ringorce transduction by the Dam1 ring
We seek to analyze a general model that includes both a diffusive burnt bridge mechanism and a powerstroke,in order to determine their relative contribution. Here the powerstroke involves a depolymerization eventwhich “unzips” PFs and moves the position of the last unbroken section of MT; a new section takes on thisidentity when the previous one unzips (contains separated, splayed protofilaments). As a highly energeticpowerstroke this is assumed to occur even when the ring is very close to the MT end, with a rate thatgives rise to a depolymerization velocity v ps . The sequence of microscopic PF unzippering events givesrise to a well defined velocity for the last fully intact MT section, irrespective of the sequence in which theneighboring PFs unzip, and the precise MT helicity. Critically, we also assume that polymerized MT can belost with a second rate , giving a depolymerization velocity v bb , whenever the ring has diffused a distance δ from the end. This can be thought of as the depolymerization velocity of a bare MT because, in thiscase, there is no Dam1 ring anywhere on the MT. We make no prior assumptions as to which contributiondominates, rather we determine this by fitting the parameters v bb , v ps and δ to data for the variation of theDam1 velocity with load (12).Our model involves a clear distinction between two mechanisms (only) and represents the simplest pos-sible model capable of explaining this data. It can be biophysically motivated on the grounds that the Dam1 orce transduction by the Dam1 ringorce transduction by the Dam1 ring −
65 pN(24). This is because, with a powerstroke-only model, we would not expect the velocity of Dam1 ring tobe significantly slowed under a force as low as 2 pN; the data shows a significant slowing. We find that thelength scale δ controlling burnt-bridge reactions, a free fit parameter, is close to the axial length of a tubulindimer. We speculate that this may provide indication of “crack”-like splitting of the MT, as discussed below. Model
The Dam1 ring complex is reported to be capable of axial movement with respect to the MT (10). Therefore,we treat the Dam1 ring as a particle undergoing one-dimensional Brownian motion in a potential V ( x ) (shown for two different models in Fig. 1). The fully intact MT extends away from the depolymerizing endfor x > x = A ). The following Fokker-Plankequation (28) determines the probability density φ ( x , t ) for the ring’s position relative to the (moving) end, ∂ φ∂ t = D ∂∂ x (cid:18) ∂ φ∂ x + k B T ∂ V ∂ x φ (cid:19) , (1)where D is the diffusion constant of the ring. This approach is appropriate providing the depolymerizationvelocity v of the MT is not too fast (bounds given later in this section), otherwise we must instead treat this asa full moving boundary problem. Since the microtubule depolymerization is here quasistatically slow withrespect to the diffusive relaxation of the ring, we can neglect the drag force on the ring, except as discussedin Supporting Material.In the following we assume the Dam1 ring is sufficiently stable that it can only dissociate by slipping offthe tip. For simplicity we restrict our analysis to continuous depolymerization processes only and discountthe possibility of rescue and polymerization. Although it would be straightforward to include such processeswe believe that they would distract from the central results of this paper.A force − ∂ V / ∂ x appears in Eq. 1. This is the magnitude of the applied force f on the Dam1 ring whileon the MT ( x >
0) since the ring must do work to move against this force. Hence, from Eq. 1 it can be shownthat, for constant (or slowly varying) f the probability distribution φ ( x ) is of Boltzmann form φ ( x ) = fk B T exp (cid:18) − f xk B T (cid:19) , (2)where the ring typically explores a characteristic diffusion length λ = k B T / f from the MT end and positivevalues of f here indicate loads pulling in the negative x direction (towards the MT end). We assume that thedepolymerization is quasistatically slow . This is appropriate provided the time for the MT to depolymerizethe distance λ is much larger than the relaxation time for a ring to diffuse this distance. This in turnrequires λ / v ( f ) (cid:29) λ / D . In this case the distribution of the ring position is always close to the equilibriumprobability distribution that it would have on an MT that was not depolymerizing. This sets an upper boundon the depolymerization velocity, or equivalently a lower bound on the load force, beyond which our theoryis at best semiquantitative; solving D / v ( f ) = k B T / f with Eq. 4, we estimate these values to be 500 nm/s and0.04 pN respectively. Under these conditions the average ring velocity is equivalent to the depolymerizationvelocity of the MT. orce transduction by the Dam1 ringorce transduction by the Dam1 ring
0) since the ring must do work to move against this force. Hence, from Eq. 1 it can be shownthat, for constant (or slowly varying) f the probability distribution φ ( x ) is of Boltzmann form φ ( x ) = fk B T exp (cid:18) − f xk B T (cid:19) , (2)where the ring typically explores a characteristic diffusion length λ = k B T / f from the MT end and positivevalues of f here indicate loads pulling in the negative x direction (towards the MT end). We assume that thedepolymerization is quasistatically slow . This is appropriate provided the time for the MT to depolymerizethe distance λ is much larger than the relaxation time for a ring to diffuse this distance. This in turnrequires λ / v ( f ) (cid:29) λ / D . In this case the distribution of the ring position is always close to the equilibriumprobability distribution that it would have on an MT that was not depolymerizing. This sets an upper boundon the depolymerization velocity, or equivalently a lower bound on the load force, beyond which our theoryis at best semiquantitative; solving D / v ( f ) = k B T / f with Eq. 4, we estimate these values to be 500 nm/s and0.04 pN respectively. Under these conditions the average ring velocity is equivalent to the depolymerizationvelocity of the MT. orce transduction by the Dam1 ringorce transduction by the Dam1 ring Fθ x0 0δ θ Figure 2: Schematic energy landscape underlying PF unzipping. The proposed free energy F landscape of a tubulin dimer at theend of the MT is shown (right) as a function of the distance of the Dam1 ring from the MT end, x , and a reaction coordinate forthe unzippering, the angle θ moved by the tubulin dimer (see diagram at left). The diagram is shown for illustrative purposes onlyand is not quantified in this work. Here θ = θ increases and the dimer moves out, ultimately forming the base of a splayed PF. The unzippering is an activated process with anenergy barrier (the height of the ridge on the right) that is different for a powerstroke ( x < δ ) and a burnt bridges reaction ( x > δ ),leading to velocities v ps and v bb respectively. The energy landscape must have at least these basic features in order to give rise tothe two depolymerization rates consistent with the data. Force dependent depolymerization velocity
The powerstroke and burnt-bridge reactions can be thought of as arising from transitions over an energybarrier of the form shown in Fig. 2, where the free energy F of PF curling is shown as varying with protofil-ament angle θ and the distance of the Dam1 ring from the MT tip x . The figure shows only a putativeschematic of the free energy of PF curling reaction, and should not be confused with the potential V ( x ) inwhich the Dam1 ring diffuses.PFs may produce a power stroke that pushes the ring with force f pf , estimated from experimental evi-dence to be 30 −
65 pN (24). This is the slope down the descending valley, diagonally right to left, in Fig. 2.Provided that the load force f (cid:28) f pf the powerstroke will give rise to a depolymerization velocity v ps that isthe rate at which the last intact dimer on the MT crosses the highest part of the ridge-like energy barrier inFig. 2 ( x < δ ). Since the estimate for f pf is so much larger than any force considered here, it is reasonable tomake the limited assumption that v ps is constant for all experimentally measurable load forces of a few pNor less.In addition the MT can also depolymerize when the Dam1 ring is further than a critical distance δ fromthe end of the MT. In this case the burnt-bridge reaction gives rise to a depolymerization velocity v bb thatis the rate at which the last intact dimer on the MT crosses the lower part of the ridge-like energy barrier inFig. 2 ( x > δ ). That the rate of MT unzippering is retarded when the Dam1 ring is near the MT end is aresult of the fact that the velocity decreases as the load force is increased and the ring is more often closerto the MT end. Although it is not necessary to interpret our model in terms of the Dam1 ring physicallyoccluding the unzippering of the tubulin dimers, this interpretation may not be unreasonable, particularly inview of the fact that we find δ to be comparable with the axial length of the last intact ring of tubulin dimers.The resultant velocity due to both mechanisms is the sum of the probability that the ring is close to theMT end x < δ , multiplied by the powerstroke velocity, and the probability that it is far x > δ , multiplied bythe burnt-bridge velocity, v = v ps (cid:18) − (cid:90) ∞ δ φ ( x ) dx (cid:19) + v bb (cid:90) ∞ δ φ ( x ) dx = (cid:0) v bb − v ps (cid:1) (cid:90) ∞ δ φ ( x ) dx + v ps (3) orce transduction by the Dam1 ringorce transduction by the Dam1 ring
65 pN (24). This is the slope down the descending valley, diagonally right to left, in Fig. 2.Provided that the load force f (cid:28) f pf the powerstroke will give rise to a depolymerization velocity v ps that isthe rate at which the last intact dimer on the MT crosses the highest part of the ridge-like energy barrier inFig. 2 ( x < δ ). Since the estimate for f pf is so much larger than any force considered here, it is reasonable tomake the limited assumption that v ps is constant for all experimentally measurable load forces of a few pNor less.In addition the MT can also depolymerize when the Dam1 ring is further than a critical distance δ fromthe end of the MT. In this case the burnt-bridge reaction gives rise to a depolymerization velocity v bb thatis the rate at which the last intact dimer on the MT crosses the lower part of the ridge-like energy barrier inFig. 2 ( x > δ ). That the rate of MT unzippering is retarded when the Dam1 ring is near the MT end is aresult of the fact that the velocity decreases as the load force is increased and the ring is more often closerto the MT end. Although it is not necessary to interpret our model in terms of the Dam1 ring physicallyoccluding the unzippering of the tubulin dimers, this interpretation may not be unreasonable, particularly inview of the fact that we find δ to be comparable with the axial length of the last intact ring of tubulin dimers.The resultant velocity due to both mechanisms is the sum of the probability that the ring is close to theMT end x < δ , multiplied by the powerstroke velocity, and the probability that it is far x > δ , multiplied bythe burnt-bridge velocity, v = v ps (cid:18) − (cid:90) ∞ δ φ ( x ) dx (cid:19) + v bb (cid:90) ∞ δ φ ( x ) dx = (cid:0) v bb − v ps (cid:1) (cid:90) ∞ δ φ ( x ) dx + v ps (3) orce transduction by the Dam1 ringorce transduction by the Dam1 ring f (pN)0200400600 R i ng v e l o c i t y , v ( n m / s ) v ps = 55 nm/s Figure 3: The variation of velocity of the Dam1 ring with applied load. The velocity falls as the force increases because the motionmust increasingly rely on the energetic powerstroke. Note that, although the graph appears to suggest an absence of a stalling force,a significantly higher forces the assumption of constant v ps would fail and the ring would stall. The curve is produced from the bestfit of δ and v ps in Eq. 4 and data from (12). The velocity follows from Eqs. 3 and 2 v = ( v bb − v ps ) exp (cid:18) − f δ k B T (cid:19) + v ps , (4)The variation of this velocity with load is shown in Fig. 3 for v bb =
580 nm/s (29), and the values v ps =
55 nm/s and δ =
14 nm that correspond to the best fit to data (12). Since a “burnt-bridges”-onlymodel fails to fit the data sufficiently (i.e. v ps >
0) it suggests that a powerstroke plays a role in forcedDam1 motion. It should be noted that, although in this model v → v ps as f → ∞ , we do not suggest thisis a physical feature of the system. Rather it is the consequence of the assumption that protofilaments areperfectly rigid and the powerstroke reaction is asymptotically strong. Our model would need modificationfor forces approaching f pf . As discussed later, PFs are estimated to require tens of pN to bend. Two models for Dam1 ring retention
We now proceed to calculate the mean time the Dam1 ring will remain on a MT and transduce force ∗ .This time is controlled by different physics in the protofilament and the binding models, see Figs. 1 and 4.However, in both cases, the velocity of the ring is governed by the model described above, see Eq. 4. Runtime: Binding model
The binding model involves a ring diffusing on a MT according to Eq. 1, leading to a depolymerizationvelocity as given in Eq. 4. However, in order to detach from the MT end the ring must overcome a linearpotential imposed by the Dam1-MT binding energy ∆ G Dam1 as it slides off the end of the ring. In this respectit is similar to Hill’s model (6). Previous models invoked a ∆ G Dam1 that also determined the roughness ofthe energy landscape through “linkers” (20) whose existence is supported by binding studies (33). Herewe don’t make this assumption, rather ∆ G Dam1 could be due to less specific interactions without significantenergy barriers between neighboring sites (34) but, importantly, can vary independently of the diffusion ∗ Recently it has been discovered that the Dam1 oligomers track the tip of depolymerizing MTs without forming a ring (30, 31).It seems unlikely that a protofilament model could operate without a full ring, however, it is not known to what extent, if atall, small oligomers contribute to force production. Furthermore, it has been shown that 16-20 Dam1 complexes are present at thekinetochore during metaphase (32), enough to form the ring. We await the result of experiments where tension is applied to putativeDam1 oligomers. orce transduction by the Dam1 ringorce transduction by the Dam1 ring
The binding model involves a ring diffusing on a MT according to Eq. 1, leading to a depolymerizationvelocity as given in Eq. 4. However, in order to detach from the MT end the ring must overcome a linearpotential imposed by the Dam1-MT binding energy ∆ G Dam1 as it slides off the end of the ring. In this respectit is similar to Hill’s model (6). Previous models invoked a ∆ G Dam1 that also determined the roughness ofthe energy landscape through “linkers” (20) whose existence is supported by binding studies (33). Herewe don’t make this assumption, rather ∆ G Dam1 could be due to less specific interactions without significantenergy barriers between neighboring sites (34) but, importantly, can vary independently of the diffusion ∗ Recently it has been discovered that the Dam1 oligomers track the tip of depolymerizing MTs without forming a ring (30, 31).It seems unlikely that a protofilament model could operate without a full ring, however, it is not known to what extent, if atall, small oligomers contribute to force production. Furthermore, it has been shown that 16-20 Dam1 complexes are present at thekinetochore during metaphase (32), enough to form the ring. We await the result of experiments where tension is applied to putativeDam1 oligomers. orce transduction by the Dam1 ringorce transduction by the Dam1 ring Figure 4: Various sketches of a ring on a microtubule. ( A ) In this configuration the ring is further than δ from the tip of the MT, sothe MT depolymerizes with velocity v bb . Unzipped protofilaments are shown dotted as they do not affect depolymerization. ( B ) Insome other configuration, the ring is closer to the tip than δ , so the MT depolymerizes with velocity v ps . In ( C-F ) the detachmentmechanisms are shown. This is either insensitive to PFs (
C-D ; binding model) or sensitive to PFs (
E-F ; protofilament model). In( C ) and ( E ) the ring has not yet escaped. In ( D ) and ( F ) the ring has escaped from the MT. constant D . This, in turn, is fixed by the smoothness of the underlying energy landscape experienced by thering as it diffuses along the MT (distinct from the energy landscape experienced by an unzippering PF shownin Fig. 2). This model assumes that the splayed PFs play no role, either because they are transient (rapidlybreaking) or otherwise interact negligibly with the ring as it slides off the end of the MT. Although clearlyan extreme approximation it forms the natural opposite limit to the protofilament model discussed in thenext section. Under a load force the ring is in the well of a “tick”-shaped potential with two linear domains(inset Fig. 1 B ). To move to the left (towards negative x ) it must partially unbind from the MT, to move tothe right (positive x ) it must do work against the applied force. The potential gradients experienced by theDam1 ring determine the load force f (while on the MT, x >
0) and the resultant force f ε (while detachingfrom the MT over the small distance − ε < x < x , is therefore given by − ∂ V ∂ x = − f x ≥ f ε = ∆ G Dam1 ε − f − ε ≤ x ≤ ε is the unbinding region. If the ring is in the region x < − ε then it is lost, and if lost we assume itnever returns, hence we have V → − ∞ for x < − ε .Symmetry from electron microscopy (10) and copy number (32) experiments suggest 16 complexes arerequired to form the Dam1 ring, however, the total bond energy may not be additive and this should thereforebe regarded as an extreme upper bound on the total binding energy.The detachment of the ring can be cast as a classical Kramers escape problem (35). To solve Eq. 1 withEq. 5 we followed the method in (36, 37). In this way we obtain the lifetime of the metastable state directlyfrom the Laplace transformed version of Eq. 1, with initial condition φ ( x , ) = δ ( x ) , where δ ( x ) is the Diracdelta function, although the precise form of this initial condition is unimportant. The mean time the ringremains on the MT is the runtime ττ = ( k B T ) D f ε e f εε kBT − f − e f εε kBT + f ε ε k B T − f ε , (6)where ε is a small distance. orce transduction by the Dam1 ringorce transduction by the Dam1 ring
0) and the resultant force f ε (while detachingfrom the MT over the small distance − ε < x < x , is therefore given by − ∂ V ∂ x = − f x ≥ f ε = ∆ G Dam1 ε − f − ε ≤ x ≤ ε is the unbinding region. If the ring is in the region x < − ε then it is lost, and if lost we assume itnever returns, hence we have V → − ∞ for x < − ε .Symmetry from electron microscopy (10) and copy number (32) experiments suggest 16 complexes arerequired to form the Dam1 ring, however, the total bond energy may not be additive and this should thereforebe regarded as an extreme upper bound on the total binding energy.The detachment of the ring can be cast as a classical Kramers escape problem (35). To solve Eq. 1 withEq. 5 we followed the method in (36, 37). In this way we obtain the lifetime of the metastable state directlyfrom the Laplace transformed version of Eq. 1, with initial condition φ ( x , ) = δ ( x ) , where δ ( x ) is the Diracdelta function, although the precise form of this initial condition is unimportant. The mean time the ringremains on the MT is the runtime ττ = ( k B T ) D f ε e f εε kBT − f − e f εε kBT + f ε ε k B T − f ε , (6)where ε is a small distance. orce transduction by the Dam1 ringorce transduction by the Dam1 ring Runtime: Protofilament model
The protofilament model involves a ring diffusing on a MT according to Eq. 1, leading to a depolymerizationvelocity Eq. 4, as before. However, in order to detach from the MT end the ring has to wait until allprotofilaments have broken (depolymerized), leaving a sufficiently “blunt” end to the MT for the ring tosimply slide off, see Fig. 1 A . We no longer require the Dam1 ring to overcome a Dam1-MT binding energy.Electron microscopy reveals that short, separated PFs splay outwards at the depolymerizing MT end (18, 19)and it is quite plausible that these block the escape of the ring; the elastic energy required to straighten acurled PF (38) follows from measurements of their rigidity (39, 40) and is of the order of tens of k B T persubunit, i.e. very large.The “frayed” PFs near the end of the MT are curved and laterally separate. The unzipping (depoly-merization) of the MT lattice (see Fig. 1A) is most accurately described as a process which transfers lengthfrom the polymerized MT into separated PFs. The unzipping is thought to be driven by the stored elasticenergy in the αβ -tubulin units in the lattice (41). When not constrained by lateral bonds, PFs relax into acurved state. We model unzipping as a Poisson process with rate k unzip . Each unzipping event extends everyPF curl by some microscopic, or subunit, length b , leading to a depolymerization velocity v = bk unzip . Thismicroscopic length might be the tubulin dimer repeat distance b , if the MT splits between a particular pair ofPFs, or otherwise smaller than this. When the ring is within a small length δ unzipping is inhibited. To morecarefully analyze this process note that the time between unzipping events t unzip is an exponential randomvariable, with probability density function p unzip ( t ) = k unzip exp ( − k unzip t ) , and mean (cid:104) t unzip (cid:105) = k unzip = bv . (7)The distribution of the ring position in this model follows Eq. 2. Detachment occurs when all PF curllengths reach zero † . Since v is a function of the applied force f , according to Eq. 4, (cid:104) t unzip (cid:105) increases underload. From Eq. 2, we have that the characteristic distance of the ring from the tip is λ = k B T / f . Thecharacteristic time for the ring to diffuse this length and escape is λ / D which is much less than (cid:104) t unzip (cid:105) fortypical parameters whenever f > .
15 pN (see
Supporting Material ). Thus it is not unreasonable to assumethat the ring might disengage from the MT extremely rapidly as soon as all curled PFs reach zero length.We assume that tubulin subunits on the frayed PFs break independently according to a Poisson processwith rate k break . The depolymerization of PFs then follows from the loss of all PF material beyond the break,as in previous computational models (42). A PF curl reaches zero length if the axial bond nearest to theunzipping point breaks, see Fig. 1 A . Since this occurs with a rate k break the waiting time t pf,i for PF curl i to break off completely is an exponential random variable. The wait time for all n PF curls breaking is theorder statistic t pf = max i t pf,i . The distribution function for this time is P pf ( t ) = ( − exp ( − k break t )) n and themean wait time (see section 4.6 from (43) or (44)) is (cid:104) t pf (cid:105) = n ∑ i = (cid:104) t pf,i (cid:105) n − i + = H n k break , (8)where H n = ∑ ni = i − is the harmonic number, roughly log n for n (cid:29) (cid:104)·(cid:105) denotes the ensemble average. The Dam1 ring will therefore no longer be secured to theMT end and will detach after a time t pf provided that no unzipping events having taken place during thetime t pf . If the MT has unzipped then the PFs extend (from their base), effectively “restarting” the waitingprocess.Fundamentally we are interested in the mean runtime τ , this being the time taken for the curled PFs to alldepolymerize completely even while the MT is simultaneously undergoing stochastic unzipping events. τ † Extensions of our model to the case of loosely-fitting rings is straightforward, involving attachment whenever the PF curlsexceed some finite length L . Our results are qualitatively insensitive to this modification, provided the ring rarely detaches at lowforce. Furthermore, for such rings the molecular length b becomes irrelevant as the characteristic timescale is L / v . orce transduction by the Dam1 ringorce transduction by the Dam1 ring
Supporting Material ). Thus it is not unreasonable to assumethat the ring might disengage from the MT extremely rapidly as soon as all curled PFs reach zero length.We assume that tubulin subunits on the frayed PFs break independently according to a Poisson processwith rate k break . The depolymerization of PFs then follows from the loss of all PF material beyond the break,as in previous computational models (42). A PF curl reaches zero length if the axial bond nearest to theunzipping point breaks, see Fig. 1 A . Since this occurs with a rate k break the waiting time t pf,i for PF curl i to break off completely is an exponential random variable. The wait time for all n PF curls breaking is theorder statistic t pf = max i t pf,i . The distribution function for this time is P pf ( t ) = ( − exp ( − k break t )) n and themean wait time (see section 4.6 from (43) or (44)) is (cid:104) t pf (cid:105) = n ∑ i = (cid:104) t pf,i (cid:105) n − i + = H n k break , (8)where H n = ∑ ni = i − is the harmonic number, roughly log n for n (cid:29) (cid:104)·(cid:105) denotes the ensemble average. The Dam1 ring will therefore no longer be secured to theMT end and will detach after a time t pf provided that no unzipping events having taken place during thetime t pf . If the MT has unzipped then the PFs extend (from their base), effectively “restarting” the waitingprocess.Fundamentally we are interested in the mean runtime τ , this being the time taken for the curled PFs to alldepolymerize completely even while the MT is simultaneously undergoing stochastic unzipping events. τ † Extensions of our model to the case of loosely-fitting rings is straightforward, involving attachment whenever the PF curlsexceed some finite length L . Our results are qualitatively insensitive to this modification, provided the ring rarely detaches at lowforce. Furthermore, for such rings the molecular length b becomes irrelevant as the characteristic timescale is L / v . orce transduction by the Dam1 ringorce transduction by the Dam1 ring N that occur before the PFs all successfully breakand the Dam1 ring can disengage. N is geometrically distributed with mean (cid:104) N (cid:105) = / P detach with P detach theprobability that the curled PFs depolymerize completely before the next unzippering event. Thus τ = (cid:104) t unzip (cid:105) P detach . (9)The ring detaches if the PFs break before an unzipping occurs, i.e. with probability that t pf < t unzip , P detach = (cid:90) ∞ dt p unzip ( t ) (cid:90) t p pf ( t (cid:48) ) dt (cid:48) , (10)where p pf = dP pf / dt is the probability density function for t pf . Evaluating the integral with respect to t (cid:48) P detach = (cid:90) ∞ P pf ( t ) p unzip ( t ) dt = (cid:90) ∞ (cid:16) − e − k break t (cid:17) n k unzip e − k unzip t dt , (11)Binomially expanding the integrand, integrating term-by-term and substituting back into Eq. 9 we obtain τ = k unzip (cid:32) n ∑ j = (cid:18) nj (cid:19) k unzip ( − ) j jk break + k unzip (cid:33) − . (12)where (cid:0) nj (cid:1) = n ! / j ! ( n − j ) !. Time varying applied forces
We now consider an oscillating applied force of the form f ( t ) = f sin ω t + f (13)Provided the period is sufficiently long ω − (cid:29) λ / D our quasi-static approximation for the ring positionshould give an accurate estimate for its probability density φ ( x , t ) .The depolymerization velocity will be retarded according to Eq. 4, relating v to f ( t ) . Protofilament model under oscillating force
The probability that the MT does not unzip in a time t after the time at which the last unzipping occurred ˜ t is ¯ P unzip ( t ; ˜ t ) = − P unzip ( t ; ˜ t ) = exp (cid:18) − (cid:90) ˜ t + t ˜ t v ( t (cid:48) ) b dt (cid:48) (cid:19) , (14)Since Eq. 14 depends explicitly on ˜ t , we perform an average over ˜ t , appropriately weighted, to give thecomplementary distribution of times between unzipping events¯ P unzip ( t ) = (cid:90) π / ω ¯ P unzip ( t ; ˜ t ) v ( ˜ t ) N b d ˜ t = (cid:90) π / ω exp (cid:18) − (cid:90) ˜ t + t ˜ t v ( t (cid:48) ) b dt (cid:48) (cid:19) v ( ˜ t ) N b d ˜ t (15)involving a normalization constant N = (cid:82) π / ω v ( ˜ t ) b d ˜ t . orce transduction by the Dam1 ringorce transduction by the Dam1 ring
The probability that the MT does not unzip in a time t after the time at which the last unzipping occurred ˜ t is ¯ P unzip ( t ; ˜ t ) = − P unzip ( t ; ˜ t ) = exp (cid:18) − (cid:90) ˜ t + t ˜ t v ( t (cid:48) ) b dt (cid:48) (cid:19) , (14)Since Eq. 14 depends explicitly on ˜ t , we perform an average over ˜ t , appropriately weighted, to give thecomplementary distribution of times between unzipping events¯ P unzip ( t ) = (cid:90) π / ω ¯ P unzip ( t ; ˜ t ) v ( ˜ t ) N b d ˜ t = (cid:90) π / ω exp (cid:18) − (cid:90) ˜ t + t ˜ t v ( t (cid:48) ) b dt (cid:48) (cid:19) v ( ˜ t ) N b d ˜ t (15)involving a normalization constant N = (cid:82) π / ω v ( ˜ t ) b d ˜ t . orce transduction by the Dam1 ringorce transduction by the Dam1 ring P detach = (cid:90) ∞ dt p unzip ( t ) (cid:90) t p pf ( t (cid:48) ) dt (cid:48) = (cid:90) ∞ dt (cid:48) p pf ( t (cid:48) ) (cid:90) ∞ t (cid:48) p unzip ( t ) dt = (cid:90) ∞ ¯ P unzip ( t ) p pf ( t ) dt . (16)The probability density of t pf is p pf ( t ) = ddt P pf ( t ) = ddt (cid:16) − e − k break t (cid:17) n = nk break e − k break t (cid:16) − e − k break t (cid:17) n − . (17)Finally the runtime is τ = (cid:2) P detach (cid:104) k unzip ( t ) (cid:105) (cid:3) − = P detach ω π (cid:90) π / ω bv ( t ) dt , (18)where 1 / P detach is the mean number of steps before detachment. Binding model under oscillating force
The generalization of Eq. 6 to the case of time-varying force (Eq. 13) is straightforward, τ = ω π (cid:90) π / ω ( k B T ) D f ε (cid:32) e f ε ε / k B T − f − e f ε ε / k B T + f ε ε / k B T − f ε (cid:33) dt , (19)where f and f ε are now time-dependent potential gradients, according to Eq. 13 with Eq. 5. Results and Discussion
We identified the following parameters using data reported in the experimental literature; v bb =
580 nm/s(29), D = . ± . µ m s − (10); for the protofilament model we assume b = n =
13 to betypical.Table 1 in Franck et al. (12) (see Table S1 in the Supporting Material) lists velocities at f = . . δ = ± . v ps = ± . δ is intriguingly close to a tubulin axial repeat length (8 nm or 1.5 times this, dueto helicity). A simple picture might be of a sleeve that suppresses depolymerization while it sits over thenext intact tubulin dimers in the PFs that are about to split. This supports the idea that the MT depolymerizesby first splitting in a linear fashion, perhaps along its seam, with the other PF pairs splitting apart somewhatbehind this leading crack-like defect. Indeed, materials do typically split along linear cracks, where theelastic stresses are concentrated (45). In particular, splitting between random PFs would yield step sizes that,due to helicity, could be a small fraction of the tubulin size. It would be hard to physically motivate a range δ that is more than ten times the incremental depolymerization step size. Why would such a depolymerizationprocess, involving little or no motion of PFs more than 1 nm from the last fully polymerized section of MT,be highly sensitive to the presence of a Dam1 ring more than 10 nm distant? We therefore consider our orce transduction by the Dam1 ringorce transduction by the Dam1 ring
13 to betypical.Table 1 in Franck et al. (12) (see Table S1 in the Supporting Material) lists velocities at f = . . δ = ± . v ps = ± . δ is intriguingly close to a tubulin axial repeat length (8 nm or 1.5 times this, dueto helicity). A simple picture might be of a sleeve that suppresses depolymerization while it sits over thenext intact tubulin dimers in the PFs that are about to split. This supports the idea that the MT depolymerizesby first splitting in a linear fashion, perhaps along its seam, with the other PF pairs splitting apart somewhatbehind this leading crack-like defect. Indeed, materials do typically split along linear cracks, where theelastic stresses are concentrated (45). In particular, splitting between random PFs would yield step sizes that,due to helicity, could be a small fraction of the tubulin size. It would be hard to physically motivate a range δ that is more than ten times the incremental depolymerization step size. Why would such a depolymerizationprocess, involving little or no motion of PFs more than 1 nm from the last fully polymerized section of MT,be highly sensitive to the presence of a Dam1 ring more than 10 nm distant? We therefore consider our orce transduction by the Dam1 ringorce transduction by the Dam1 ring f (pN)020406080 R un t i m e , t ( s ) ProtofilamentBinding
Figure 5: Runtime of the protofilament and binding models. The runtime τ of each model is calculated using the parameters fittedas described in the Results section. Although, it may seem that distinguishing the models by varying force is possible due to thedifference between their predicted behaviour, as shown here, the difference is close to experimental error ( ± .
15 s) and both modelspresent similar functional form. Only two data points with sufficient statistics were available to perform this fitting (12) making itdifficult to draw any conclusions from this approach. The fit provides values for ∆ G Dam1 for the binding model and k break for theprotofilament model. estimate of the characteristic range δ for the burnt-bridges reaction (within which the Dam1 ring occludesunzipping) to be quite reasonable.Combining the available data for velocity and detachment frequency we find, on average, τ = . . f = . . τ we choose ε = ∆ G Dam1 = ± . k B T . Independently,we fit k break for the protofilament model and find k break = . ± .
63 s − . Fitting these parameters to just twodata points does not provide strong evidence for these particular values. However, uncertainty in the exactparameter values should not detract from the main value of this work; to provide a model that explains theDam1 force sensitivity and to distinguish between binding and protofilament models. Comparison of the fitwith data is shown in Fig. 5. Variation of intrinsic depolymerization velocity
The protofilament model exhibits the most sensitivity to the intrinsic (bare) MT depolymerization velocity v bb , as is shown in Fig. 6 A . For the protofilament model, τ is strongly dependent on (cid:104) t unzip (cid:105) and consequently v bb . The binding model, on the other hand, is only weakly dependent on v (see Supporting Material ), and onthis range of v bb we can assume that depolymerization is quasistatically slow with respect to ring diffusion.The result can be understood physically by realising that as v bb increases, the rate of PF unzipping k unzip alsoincreases, while k break remains constant making it less likely that the PFs will break off sufficiently quicklyto release the ring.An experimental test that might be able to distinguish which model operates could be achieved, e.g., byaddition of a depolymerization inducing agent, such as Ca + or XMCAK1. Changing of diffusion coefficient
The diffusion constant D of the ring is determined by the ring’s dimensions and the roughness of the bindingenergy landscape along the MT, rather than the magnitude of the binding energy itself. A more roughlandscape reduces the mobility of the ring. Fig. 6 B shows the effect of the diffusion constant on the runtimefor both models. Only the binding model is sensitive to change in D , having reduced runtime with fasterdiffusion. This is because the increased mobility of the ring increases the chance it is able to scale the orce transduction by the Dam1 ringorce transduction by the Dam1 ring
The diffusion constant D of the ring is determined by the ring’s dimensions and the roughness of the bindingenergy landscape along the MT, rather than the magnitude of the binding energy itself. A more roughlandscape reduces the mobility of the ring. Fig. 6 B shows the effect of the diffusion constant on the runtimefor both models. Only the binding model is sensitive to change in D , having reduced runtime with fasterdiffusion. This is because the increased mobility of the ring increases the chance it is able to scale the orce transduction by the Dam1 ringorce transduction by the Dam1 ring
200 400 600 800 1000Bare depolymerisation velocity, v bb (nm/s)050100150 R un t i m e , t ( s ) ProtofilamentBinding v bb = n m / s A - - D ( m m /s)020406080100120 R un t i m e , t ( s ) BindingProtofilament D = . m m / s B w / p (Hz)303540 R un t i m e , t ( s ) ProtofilamentBinding C Figure 6: Model discrimination. The panels show variation of runtime τ with ( A ) bare MT depolymerization velocity v bb , ( B )diffusion coefficient D , and ( C ) frequency of applied force ω / π , for both models under load f = .
45 pN, chosen because bothmodels predict the same nominal τ and v at this load (see Fig. 5). ( A ) The runtime τ increases exponentially with v bb for theprotofilament model, whilst the binding model is insensitive. This is because the protofilament model directly depends on v , butthe binding model does not. ( B ) Restricted diffusion suppresses detachment for the binding model because τ is inversely relatedto D , due to the reduced impetus to escape the potential barrier. The protofilament model, on the other hand, is not affected by D since t unzip is independent of D . Distinguishing between models will be easiest by experimental reduction of D , for example byattachment of a long polymer. ( C ) The binding model is sensitive only to the amplitudes f (here 0.1 pN) and f (here 0.43 pN), notthe frequency ω . The rate of detachment for the protofilament model instead strongly depends on the frequency: roughly speakingthe ring is lost more quickly when the high force part of the cycle persists for long enough for the PFs to completely depolymerizein this time, i.e. when the period is long. potential barrier constraining it to the MT.Although it may be possible to alter D biochemically, for example by phosphorylation (30), it is difficultto do so independently of ∆ G Dam1 . Decreasing D may be better accomplished by attaching a long inertpolymer to the complex to increase viscous drag. Effect of time-varying loading force
The runtime in the binding model is sensitive only to the instantaneous force provided 2 π / ω (cid:29) λ / D , seethe low frequency portion of Fig. 6 C . If f = π / ω max is on the order of 1 kHz. The runtime in theprotofilament model is sensitive to the time over which changes in v persist. If the force is oscillating witha long period then the rate of detachment will be greater in the high force part of the cycle than if the periodis short. This is because the Dam1 ring takes some time to detach if it needs to first wait for the PF curlsto break, see Fig. 6 C . Sigmoidal increase of τ would be a signature of a system that depends on a secondtime (1/ k break ), like the protofilament model; insensitivity of τ to frequency would imply a binding-stylecoupling. Conclusion
Our results indicate a power stroke does contribute to the effective force generated during depolymerizationbut only becomes dominant at over 2 pN load. We show how a faster depolymerization mechanism mustoperate at lower loads and argue that the Dam1 ring suppresses depolymerization when it is close to the MTend.We have shown that either of two rather different Dam1-MT coupling mechanisms might be operatingunder piconewton loads. Both models have comparable performance under load; their differences onlybecome apparent under novel experimental conditions. Structural studies cannot resolve the question ofwhich model operates in vivo . We suggest several methods for using runtime statistics to determine whichclass of model best describes the coupling of the Dam1 ring to depolymerizing MTs. Note that throughoutthis study we have assumed that depolymerization is sufficiently slow compared to ring diffusion that wecan consider the distribution of the ring’s position to be quasi-equilibrated. Over the range of parameterswe have considered this assumption is valid to within 1% of the predicted velocity. The characteristic orce transduction by the Dam1 ringorce transduction by the Dam1 ring
Our results indicate a power stroke does contribute to the effective force generated during depolymerizationbut only becomes dominant at over 2 pN load. We show how a faster depolymerization mechanism mustoperate at lower loads and argue that the Dam1 ring suppresses depolymerization when it is close to the MTend.We have shown that either of two rather different Dam1-MT coupling mechanisms might be operatingunder piconewton loads. Both models have comparable performance under load; their differences onlybecome apparent under novel experimental conditions. Structural studies cannot resolve the question ofwhich model operates in vivo . We suggest several methods for using runtime statistics to determine whichclass of model best describes the coupling of the Dam1 ring to depolymerizing MTs. Note that throughoutthis study we have assumed that depolymerization is sufficiently slow compared to ring diffusion that wecan consider the distribution of the ring’s position to be quasi-equilibrated. Over the range of parameterswe have considered this assumption is valid to within 1% of the predicted velocity. The characteristic orce transduction by the Dam1 ringorce transduction by the Dam1 ring in vivo factors such as microtubule-associatedproteins (MAPs) or kinases. However, some of these factors operate to increase or reduce the depolymer-ization rate of the microtubule, a parameter included in the model. We therefore expect the general resultsto remain largely applicable. Furthermore, we have assumed Dam1 to be present as a ring. Recent work(30, 31) has raised the possibility that Dam1 may operate as short oligomers or single complexes. If wecan assume these oligomers interact with PFs in a comparable fashion as a ring would, our model wouldbe indistinguishable for rings or oligomers. If not, our model may be of use to determine whether ring oroligomer is present based on e.g., differing diffusion constant.
Acknowledgements
The authors thank George Rowlands and Jonathan Millar for useful discussions.
References
1. Alberts, B., A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter. 2002. Molecular biology of thecell. Garland Science, New York, fourth edition.2. Inou´e, S., and E. Salmon. 1995. Force generation by microtubule assembly/disassembly in mitosis andrelated movements. Mol. Biol. Cell 6:1619–1640.3. Coue, M., V. Lombillo, and J. McIntosh. 1991. Microtubule depolymerization promotes particle andchromosome movement in vitro. J. Cell Biol. 112:1165–1175.4. Ganem, N. J., K. Upton, and D. A. Compton. 2005. Efficient Mitosis in Human Cells Lacking PolewardMicrotubule Flux. Curr. Biol. 15:1827–1832.5. Koshland, D. E., T. J. Mitchison, and M. W. Kirschner. 1988. Polewards chromosome movement drivenby microtubule depolymerization in vitro. Nature 331:499–504.6. Hill, T. L. 1985. Theoretical Problems Related to the Attachment of Microtubules to Kinetochores.Proc. Natl. Acad. Sci. U. S. A. 82:4404–4408.7. Cheeseman, I. M., C. Brew, M. Wolyniak, A. Desai, S. Anderson, N. Muster, J. R. Yates, T. C. Huffaker,D. G. Drubin, and G. Barnes. 2001. Implication of a novel multiprotein Dam1p complex in outerkinetochore function. J. Cell Biol. 155:1137–1146.8. Miranda, J. L., P. D. Wulf, P. K. Sorger, and S. C. Harrison. 2005. The yeast DASH complex formsclosed rings on microtubules. Nat. Struct. Mol. Biol. 12:138–143.9. Westermann, S., A. Avila-Sakar, H.-W. Wang, H. Niederstrasser, J. Wong, D. G. Drubin, E. Nogales,and G. Barnes. 2005. Formation of a Dynamic Kinetochore-Microtubule Interface through Assemblyof the Dam1 Ring Complex. Mol. Cell 17:277–290.10. Westermann, S., H.-W. Wang, A. Avila-Sakar, D. G. Drubin, E. Nogales, and G. Barnes. 2006. TheDam1 kinetochore ring complex moves processively on depolymerizing microtubule ends. Nature440:565–569. orce transduction by the Dam1 ringorce transduction by the Dam1 ring
1. Alberts, B., A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter. 2002. Molecular biology of thecell. Garland Science, New York, fourth edition.2. Inou´e, S., and E. Salmon. 1995. Force generation by microtubule assembly/disassembly in mitosis andrelated movements. Mol. Biol. Cell 6:1619–1640.3. Coue, M., V. Lombillo, and J. McIntosh. 1991. Microtubule depolymerization promotes particle andchromosome movement in vitro. J. Cell Biol. 112:1165–1175.4. Ganem, N. J., K. Upton, and D. A. Compton. 2005. Efficient Mitosis in Human Cells Lacking PolewardMicrotubule Flux. Curr. Biol. 15:1827–1832.5. Koshland, D. E., T. J. Mitchison, and M. W. Kirschner. 1988. Polewards chromosome movement drivenby microtubule depolymerization in vitro. Nature 331:499–504.6. Hill, T. L. 1985. Theoretical Problems Related to the Attachment of Microtubules to Kinetochores.Proc. Natl. Acad. Sci. U. S. A. 82:4404–4408.7. Cheeseman, I. M., C. Brew, M. Wolyniak, A. Desai, S. Anderson, N. Muster, J. R. Yates, T. C. Huffaker,D. G. Drubin, and G. Barnes. 2001. Implication of a novel multiprotein Dam1p complex in outerkinetochore function. J. Cell Biol. 155:1137–1146.8. Miranda, J. L., P. D. Wulf, P. K. Sorger, and S. C. Harrison. 2005. The yeast DASH complex formsclosed rings on microtubules. Nat. Struct. Mol. Biol. 12:138–143.9. Westermann, S., A. Avila-Sakar, H.-W. Wang, H. Niederstrasser, J. Wong, D. G. Drubin, E. Nogales,and G. Barnes. 2005. Formation of a Dynamic Kinetochore-Microtubule Interface through Assemblyof the Dam1 Ring Complex. Mol. Cell 17:277–290.10. Westermann, S., H.-W. Wang, A. Avila-Sakar, D. G. Drubin, E. Nogales, and G. Barnes. 2006. TheDam1 kinetochore ring complex moves processively on depolymerizing microtubule ends. Nature440:565–569. orce transduction by the Dam1 ringorce transduction by the Dam1 ring orce transduction by the Dam1 ringorce transduction by the Dam1 ring
1. Alberts, B., A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter. 2002. Molecular biology of thecell. Garland Science, New York, fourth edition.2. Inou´e, S., and E. Salmon. 1995. Force generation by microtubule assembly/disassembly in mitosis andrelated movements. Mol. Biol. Cell 6:1619–1640.3. Coue, M., V. Lombillo, and J. McIntosh. 1991. Microtubule depolymerization promotes particle andchromosome movement in vitro. J. Cell Biol. 112:1165–1175.4. Ganem, N. J., K. Upton, and D. A. Compton. 2005. Efficient Mitosis in Human Cells Lacking PolewardMicrotubule Flux. Curr. Biol. 15:1827–1832.5. Koshland, D. E., T. J. Mitchison, and M. W. Kirschner. 1988. Polewards chromosome movement drivenby microtubule depolymerization in vitro. Nature 331:499–504.6. Hill, T. L. 1985. Theoretical Problems Related to the Attachment of Microtubules to Kinetochores.Proc. Natl. Acad. Sci. U. S. A. 82:4404–4408.7. Cheeseman, I. M., C. Brew, M. Wolyniak, A. Desai, S. Anderson, N. Muster, J. R. Yates, T. C. Huffaker,D. G. Drubin, and G. Barnes. 2001. Implication of a novel multiprotein Dam1p complex in outerkinetochore function. J. Cell Biol. 155:1137–1146.8. Miranda, J. L., P. D. Wulf, P. K. Sorger, and S. C. Harrison. 2005. The yeast DASH complex formsclosed rings on microtubules. Nat. Struct. Mol. Biol. 12:138–143.9. Westermann, S., A. Avila-Sakar, H.-W. Wang, H. Niederstrasser, J. Wong, D. G. Drubin, E. Nogales,and G. Barnes. 2005. Formation of a Dynamic Kinetochore-Microtubule Interface through Assemblyof the Dam1 Ring Complex. Mol. Cell 17:277–290.10. Westermann, S., H.-W. Wang, A. Avila-Sakar, D. G. Drubin, E. Nogales, and G. Barnes. 2006. TheDam1 kinetochore ring complex moves processively on depolymerizing microtubule ends. Nature440:565–569. orce transduction by the Dam1 ringorce transduction by the Dam1 ring orce transduction by the Dam1 ringorce transduction by the Dam1 ring orce transduction by the Dam1 ringorce transduction by the Dam1 ring