In search of an optimal ring to couple microtubule
depolymerization to processive chromosome motions
Artem Efremov*†, Ekaterina L. Grishchuk*‡, J. Richard McIntosh*§, and Fazly I. Ataullakhanov†¶?
*Department of Molecular, Cellular, and Developmental Biology, University of Colorado, Boulder, CO 80309-0347;†National Research Center for
Hematology, Moscow 125167, Russia;‡Institute of General Pathology and Pathophysiology, Moscow 125315, Russia;¶Physics Department
Moscow State University, Moscow 119992, Russia; and?Center for Theoretical Problems of Physicochemical Pharmacology, Russian
Academy of Sciences, Moscow 119991, Russia
Contributed by J. Richard McIntosh, October 6, 2007 (sent for review August 31, 2007)
associated proteins that couple kinetochores to MT ends. A good
coupler should ensure a high stability of attachment, even when
force. The optimal coupler is also expected to be efficient in
converting the energy of MT depolymerization into chromosome
motility. As was shown years ago, a ‘‘sleeve’’-based, chromosome-
associated structure could, in principle, couple MT dynamics to
chromosome motion. A recently identified kinetochore complex
from yeast, the ‘‘Dam1’’ or ‘‘DASH’’ complex, may function as an
encircling coupler in vivo. Some features of the Dam1 ring differ
from those of the ‘‘sleeve,’’ but whether these differences are
significant has not been examined. Here, we analyze theoretically
the biomechanical properties of encircling couplers that have
properties of the Dam1/DASH complex, such as its large diameter
and inward-directed extensions. We demonstrate that, if the
coupler is modeled as a wide ring with links that bind the MT wall,
its optimal performance is achieved when the linkers are flexible
and their binding to tubulin dimers is strong. The diffusive move-
ment of such a coupler is limited, but MT depolymerization can
drive its motion via a ‘‘forced walk,’’ whose features differ signif-
icantly from those of the mechanisms based on biased diffusion.
Our analysis identifies key experimental parameters whose values
should determine whether the Dam1/DASH ring moves via diffu-
sion or a forced walk.
biased diffusion ? kinetochore ? mathematical model ? power stroke ?
rangements of 13 linear polymers of ??-tubulin dimers called
protofilaments (PFs) (1). Tubulin complexed with GTP elon-
gates PF ends, but this GTP is hydrolyzed soon after dimers have
added (2). The preferred conformation of the GDP dimers is
more bent, so the PFs tend to curve (3, 4). In a growing MT, this
tendency is counteracted by a relatively straight ‘‘GTP cap,’’ but
if the terminal layers are lost, the PFs are no longer restrained
(1, 2). As they splay out, the dimers dissociate from their
longitudinal neighbors, and the MT shortens (5, 6). Conse-
quently, the ends of depolymerizing MTs in vitro and in vivo
display curved PFs (3, 5–9). Representative images of kineto-
chore MTs are shown in Fig. 1A; the length of PF flares in
anaphase mammalian cells is 53 ? 7 nm, n ? 368 PFs (J.R.M.,
E.L.G., A.E., K. Zhudenkov, M. Morphew, et al., unpublished
It is well documented that MTs can move chromosomes or
microbeads that associate with their shortening ends (9–15).
Several hypotheses have explained these force-transducing at-
tachments with the help of a coupler that encircles the MT
(16–18). Movement in these models is driven ultimately by the
energy released during hydrolysis of tubulin-associated GTP
(?12 kBT; kB, Boltzmann constant). However, the design of a
coupler, e.g., its geometry and size and the nature of its
interaction with MTs, determines the fraction of this energy that
hromosome segregation during cell division depends on the
activities of microtubules (MTs), which are cylindrical ar-
can be converted into useful work. Below we analyze several
biomechanical designs, focusing on their efficiency and the
stability of their attachment to the MT end.
Biased Diffusion of a Cylindrical Coupler. The 40-nm-long ‘‘sleeve,’’
proposed by Hill, had numerous binding sites for enclosed
tubulin dimers (Fig. 1B) (19, 20). Because protein–protein
interactions act over very short distances, the sleeve could be
only 0.1–0.2 nm wider than the MT, whose outside diameter is
25 nm. The interaction between a dimer and a binding site in the
sleeve was postulated to decrease the total MT-sleeve energy by
2.5 kBT, but the activation energy for the transitions between the
binding sites was 0.1–0.2 kBT. These assumptions are key for
MT-dependent motility in this model. As the sleeve diffuses on
Author contributions: F.I.A. designed research; A.E. performed research; E.L.G. and F.I.A.
analyzed data; and E.L.G. and J.R.M. wrote the paper.
The authors declare no conflict of interest.
§To whom correspondence should be addressed. E-mail: firstname.lastname@example.org.
This article contains supporting information online at www.pnas.org/cgi/content/full/
© 2007 by The National Academy of Sciences of the USA
+ - -
s r e k n i l h t i w g
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ring-like coupling. (A) Axial slices from electron tomograms of kinetochore
MTs from PtK1 (a and b), Schizosaccharomyces pombe (c), S. cerevisiae (d and
e). PFs have extensive flare (arrows) but different lengths. (B) In Hill’s model,
a sleeve has 65 low-affinity-binding sites along each PF (20). This mechanism
does not use the PF’s power stroke, because a MT tip is buried in a narrow
sleeve. (C) A PF power stroke pushes with maximal force on a ring, which is
?10-nm wider than the MT. This mechanism has no restrictions on PF length.
(D) In the electrostatic model, the ring moves at the ends of the curved PFs,
which therefore extend only ?1 dimer beyond the ring (based on ring diam-
eter and PF curvature). (E) A model that combines the power stroke, as in C,
B, but the bonds are now formed by the linkers. The resulting motions are
significantly different from those in other models.
Structure of kinetochore MT ends and models for force-bearing
www.pnas.org?cgi?doi?10.1073?pnas.0709524104 PNAS ?
November 27, 2007 ?
vol. 104 ?
no. 48 ?
system’s energy; with no opposing load, the steady state is
achieved when 90–95% of binding sites are occupied (20, 21). As
the MT shortens by loss of tubulin dimers through the narrow
channel of a sleeve, sleeve diffusion relocates the bonds to
preserve the overlap, thereby causing the sleeve to move with the
MT end. With increased opposing force, the amount of overlap
decreases; a ?15-pN tension leads to complete loss of attach-
100% efficiency, when all of the energy of GTP hydrolysis is
converted into useful work, would produce ?80 pN (20–22).
Although a coupler with Hill’s design has not yet been found,
this physically solid model has had a significant impact on
thinking in the mitosis field. Since the model was proposed,
however, it has been learned that the ends of shortening MTs,
is not compatible with the above mechanism. A long narrow
sleeve (or a shorter ring) can reside only over the cylindrical part
of the wall, away from the tip where the PFs flare. When the
sleeve is over the wall, all of its binding sites are occupied, so its
diffusion is not biased. However, a sleeve (or a ring) with
structure similar to Hill’s proposal might be biased to move with
a shortening MT by some other mechanism, as has been
proposed for directed motions of various microobjects (23–26).
Power Stroke Mechanism. Bending PFs have been proposed to
move a circular coupler by pushing on it directly (Fig. 1C) (12,
27). Indeed, the existence of a PF power stroke has been
experimentally confirmed with nonencircling couplers (13).
takes into account current knowledge of the PF bending pathway
suggests that the efficiency of a ring coupler can be very high, but
it depends strongly on ring diameter (22, 28). A narrow ring,
similar to Hill’s sleeve, constricts PF bending, and the resulting
force is ?10 pN (22). Maximal efficiency (?90%) is achieved by
a ring with diameter 35–40 nm, which is similar to the inner
diameter of a Dam1 ring (29, 30). Such a wide ring could bind
directly to the MT wall only when positioned asymmetrically, but
Dam1 rings appear to be coaxial with MTs. Furthermore, Dam1
rings assemble spontaneously around MTs, strongly suggesting
there is an interaction between the ring’s inner surface and the
MT wall (29, 30). We have hypothesized that such binding might
be accomplished by ‘‘linkage’’ between Dam1 heterodecamers
and tubulin dimers (22), an idea that is supported by recent
indications that the linking structure is flexible, but its exact
properties are unknown (30–32).
An electrostatic mechanism has been also proposed for cou-
pling a 32-nm ring to MTs (33) in which PFs bend, as in ref. 22.
This ring is proposed to interact weakly with the MT wall (12 kBT
per ring) over a 3.5-nm gap between these structures. Ring and
tubulin are also supposed to have opposite charges, so the ring
is attracted strongly to the splayed PFs. In this model, ring
movement is promoted by the instantaneous removal of terminal
dimers; the ring is always at the PF ends, where it transduces
?12-pN force (Fig. 1D). MTs are described as nonhelical
(although MTs in vivo are helices), so the terminal dimers on all
PFs must dissociate synchronously. These restrictions appeared
to us ill-justified, so we sought a different approach.
A Combined Approach to Examine MT-Dependent Movements of
Various Ring Couplers. Tools are now available to build a versatile
model that addresses the mechanisms of Dam1 motility. For
example, thermal motions of the coupler (essential in a biased-
diffusion mechanism) can be analyzed with a Hill-like coupler,
which assumes an explicitly defined energy relationship with
tubulin. It is reasonable, though, to change the dimensions of the
coupler, as well as its number of tubulin-binding sites, to match
account the energetics of tubulin–tubulin interactions during PF
bending in a 13?3 MT (13 PFs arranged in a three-start helix).
Because the molecular details of Dam1–tubulin interactions are
not known, the model should explore the unknown parameters,
e.g., the number of subunits per ring, the number of bonds to the
MT wall, and the flexibility of linkers. Below, we define the
design of such a coupler.
Model Assumptions. The coupler is a ring with a fixed inner
diameter of 33 nm. As in the above models, the coupler is solid
and does not deform.
1. Because Dam1 complexes bind MT walls, we assume the ring
has extensions that bridge the gap between the ring subunits
and tubulin dimers in the wall of a straight MT. Each subunit
has ?4-nm-long linkers; for simplicity, we model them as rods
2. We assume a direct and specific interaction between tubulin
and the linker, characterized by negative potential energy d:
d??? ? ?kDAM ? e?
where kDAMand rDAMare parameters for the depth and width
of the potential well, respectively; ? is the distance between
the MT-proximal end of the linker and the closest binding site
on the MT wall; for simplicity, the sites on the MT wall are
positioned in the middle of tubulin’s outer surface. The
interaction sites used for model calculations should not be
confused with real contacts between Dam1 heterodecamers
and tubulin dimers. The interacting proteins usually form
various bonds (electrostatic, hydrophobic, and others) be-
tween multiple amino acid residues. Specific binding is char-
acterized by strong forces that act over a short range (34) and
are described appropriately by the above equation. The exact
parameters of the Dam1–tubulin interaction, kDAMand rDAM,
however, are not known. Cosedimentation of Dam1 com-
plexes with taxol-stabilized MTs estimated a dissociation
constant at 0.2 ?M (29), which suggests an interaction free
energy of ?15 kBT for each Dam1 subunit with the MT wall.
This value may overestimate Dam1-tubulin binding, e.g.,
because of the unknown strength of interactions between the
ring subunits. We will therefore analyze model solutions for
a range of Dam1-tubulin binding energies (parameter kDAM).
4. We assume that each tubulin dimer binds a single linker,
based on analogy with other MT-binding proteins, such as
kinesins (35). We did explore models based on two Dam1-
binding sites per tubulin dimer, but the results gave a poor
description of the Dam1 ring’s properties (below).
Mathematical Modeling and Data Analysis. A 13?3 MT was modeled
as an ensemble of subunits connected by fixed ‘‘interaction
original model (28) [Fig. 2A; see supporting information (SI)
Text and SI Table 1]. Here, however, a tubulin monomer is the
thermal fluctuations into account (SI Text). The position of the
ring coupler is defined by the three coordinates of its center
(xcyczc) and its Euler’s angles (?, ?, and ?) (Fig. 2B and SI Table
in SI Text. The model contains no additional assumptions that
would bias the ring’s diffusion. Numerical calculations were
carried out with the Metropolis method for Monte-Carlo sim-
ulation (36). Errors are standard deviations.
www.pnas.org?cgi?doi?10.1073?pnas.0709524104Efremov et al.
Rigid Linkers Decrease the Number of Dam1–MT Connections, Thereby
Reducing the Affinity of This Coupler for the MT Wall. First, we
considered geometrical aspects of how a rigid 33-nm ring with
linkers would form connections with the wall of a 13?3 MT.
There is a mismatch between the symmetry of such a MT and the
Dam1 ring, which is thought to have ?16 subunits (31, 37). We
have examined the minimum energy configurations for rings
with 13–25 subunits. When linkers are modeled as stiff 4-nm
rods, three or fewer linkers can bind dimers in the MT wall (Fig.
2C). Additional bonds are sterically prohibited by the incom-
patibilities of a helical MT and a rigid planar ring. A small
number of connections is disadvantageous, because it impairs
the stability of ring–MT attachment. If the energy of a single
Dam1-tubulin bond is ?2 kBT, similar to the sleeve–MT bond
(20), the paraxial force that would detach the 13 subunit ring
from the MT end is only 4 pN. Stronger binding increases the
stability of attachment, but it is still far from the maximum
possible, because of the limited number of ring–MT bonds. In
rings with more subunits, the number of possible ring–MT
connections is even smaller, so this conclusion is independent of
the number of ring subunits.
Rigid Dam1–Tubulin Linkers also Limit the Efficiency of Energy Trans-
duction. As a MT begins to depolymerize, bending PFs push on
the coupler. The total paraxial force on the ring can be esti-
force is applied to the linkers. As the PFs bend, all linkers are
affected, regardless of whether they have formed a bond with the
MT wall. The geometrical differences between the noninteract-
ing ring (22) and the coupler considered here are not important.
Indeed, a 33-nm ring with any number of nondeformable
4-nm-long linkers is mechanically equivalent to a narrow ring.
Therefore, the force experienced by a ring with rigid linkers is
virtually identical to that calculated for a rigid ring with a
diameter ?26 nm. For example, when the free energy of a
dimer–linker bond is ?2 kBT, the maximal force generated is
only 12 pN; for stronger binding, it is even smaller.
Based on these considerations, we conclude that couplers with
stiff linkers are far from optimal. To examine the effect of
flexible linkers, we modeled them as linear rods that could
stretch and compress along their length (parameter kspring) and
bend away from the ring’s plane (parameter kflex) (SI Text).
Model calculations show that 10-fold variations in a linker’s
longitudinal stiffness have little effect on model solutions (not
shown), so this parameter was fixed at kspring? 0.13 N/m for all
our calculations. For a linker 4 ? 1 ? 1 nm, the corresponding
and MTs (38). The flexural rigidity kflexwas varied from 5–2,000
kBT/Rad. As described above, very rigid linkers (?2,000 kBT/
Rad) are equivalent to a narrow rigid ring, which is a poor
transducer (22). When linkers are very soft (? 5 kBT/Rad), their
impact on the mechanics of the system is negligible; the ring
behaves like the 33-nm ring that does not interact with the MT
wall (22). Thus, this parameter was examined over a range of
20–200 kBT/ Rad. A 10-fold change in linker-bending stiffness in
this range does not appreciably affect the ring’s properties, e.g.,
its diffusion, so all calculations were carried out for kflex? 20
The Preferred Configuration of the Dam1 Ring Is Tilted Relative to the
MT Axis. When linkers are flexible, they can reach more binding
sites on the MT surface, so a 13-subunit ring can establish
connections with 13 dimers in the MT wall. However, rings with
more subunits can still bind only 13 sites, because the number of
bonds is limited by the number of PFs in the MT. To establish
more bonds, the linkers in a 33-nm Dam1 ring with 15–25
subunits would have to stretch ?1.5-fold, an implausible sce-
nario. Because the free energy of the ring–MT system is deter-
mined by the number of protein–protein bonds established, not
the number of available linkers, rings with more subunits should
behave much like a ‘‘minimal’’ ring with only 13 subunits per
y c n
e r f e v i t a l e r
tilting angle θ
theory; one bond per dimer
0.27 ± 0.07, N=586
theory; two bonds per dimer
0.11 ± 0.07, N=282
0.25 ± 0.15, N=28
tilting angle (Rad)
n i s θ
# of ring
lateral bonds (blue dots) with adjacent PFs (green), and three are longitudinal junctions, one within the dimer and two with dimers in the same PF (red dots).
The Dam1 linker binds to a tubulin monomer (yellow dot). Axis z coincides with the MT axis and points to its plus end; x and y axes lie in a perpendicular plane,
and x points to PF no. 13. (B) Angular variables of ring orientation. (C) MT wall-ring configurations as calculated in the model. Purple lines show ring backbones.
(D) Shown is a 13-fold ring, but similar models can be drawn for a range of ring symmetries, because the number of bonds is defined by the MT surface, which
has a lower radial symmetry than the ring. (E) Preferred ring orientations. A ring that is perfectly perpendicular to the MT axis would correspond to a central
dot, where ? ? 0. The ring is sensitive to the MT seam, as seen from the nonsymmetric distribution of data points relative to the orange line. (F) Experimental
have resulted from specimen distortions that frequently accompany the negative staining procedure.
Ring with linkers on a 13?3 MT. (A) Interaction points between dimers (light, ? tubulins; dark, ? tubulins). Each dimer interacts at seven points; four are
Efremov et al. PNAS ?
November 27, 2007 ?
vol. 104 ?
no. 48 ?
linkers. We have therefore carried out all calculations with rings
of 13 subunits, each of which has a flexible linker.
When such a ring encircles a MT, it orients to maximize the
number of Dam1–tubulin bonds, while minimizing their total
bending deformations, thus minimizing the total system’s en-
ergy. Calculations show there are several configurations with
similarly low energies, but in all of them, the ring is tilted relative
to the MT axis (Fig. 2D). Fig. 2E shows projections of the unit
vector normal to the ring (red arrow on Fig. 2B) onto the plane
perpendicular to the MT axis (dark circle on Fig. 2B), as viewed
from the MT-plus end. Each dot corresponds to a configuration
characterized by the ring’s tilting angle ? (radii of the diagram’s
circles show sin? with 0.05 increments; scale on the left) relative
to positions of the different PFs (the numbered radial spokes).
The dots are distributed nonuniformly because of the MT’s
helicity and asymmetry, which results from its seam. Most
frequently, the ring is oriented so its axis forms a 15–16° angle
with PF no. 5.
We have compared the predicted distribution of ?s with the
orientations of Dam1 rings, as visualized by electron microscopy
(Fig. 2F). If there is only one Dam1-binding site per dimer
(assumption 4), the degrees of tilting of the theoretical ring
have one bond per monomer, the predicted average tilting angle
is ?2.4-fold smaller than that observed. In general, more binding
sites per tubulin dimer should lead to a less pronounced tilting,
whereas rings that do not have specific binding sites on MT
surfaces should have no tilt.
A Weakly Bound Ring Will Follow a Shortening MT End via Biased
Diffusion. After eliminating unimportant parameters and unfa-
vorable designs, we analyzed the diffusion of the resulting
coupler on a MT. When ring–MT interaction is very weak (kDAM
? 2 kBT), ring diffusion is fast and depends little on bond
strength (Fig. 3A). For stronger binding (2–5 kBT), the depen-
dence is exponential, and a 1-kBT change reduces diffusion
10-fold. This result is almost insensitive to the exact value of
linker flexibility within the above-determined range. Experi-
ments suggest that Dam1 complexes diffuse on an MT surface
the MT-encircling rings, this implies relatively weak Dam1-
tubulin binding, which matches the model coupler with kDAM
As a MT shortens, the random walk of such a ring becomes
biased (Fig. 3B; SI Movie 1). When the ring approaches the MT
end, its motion in this direction is interrupted, because the flared
PFs present a significant barrier; the energy required to
straighten them, so the ring could slide over that area, is ?12 kBT
per tubulin dimer. This energy is so high the ring’s thermal
energy is not sufficient to pass this barrier. Although technically
the PFs push on the weakly bound ring when it comes in contact,
this aspect of their interactions is mechanically insignificant,
because of the low resistance to ring sliding.
Weakly Bound Rings Are Energy-Efficient, but Have a Low Stability of
Attachment. We examined the force-transduction by a weakly
bound ring (kDAM? 1–6 kBT) by analyzing its motions under a
load that was distributed evenly among its subunits. In vivo, a
Dam1 ring is probably attached to a chromosome via other
proteins that transmit the load in roughly this fashion (Fig. 3C
Inset). Unlike Hill’s sleeve, this coupler slows with increasing
is 60–70 pN. Because the maximal force that is energetically
possible from a disassembling MT is ?80 pN (22), the weakly
bound ring achieves up to 90% efficiency in energy transduction.
What happens when this ring stops moving toward the MT
minus end? The ring blocks separation of lateral tubulin–tubulin
direction is blocked by flared PFs. Such a ring is stalled on the
MT, but it will not detach so long as there are some splayed PFs
upstream from the ring. However, as the longitudinal bonds
between tubulin dimers in the flared PFs dissociate, the PFs
upstream from the ring will shorten and no longer protect the
ring from detachment. If the opposing force exceeds what is
required to break all ring–MT bonds, the ring will disconnect
from the MT end. Strikingly, the detachment force for a weakly
bound ring is ?15 pN, significantly less than its stalling force in
the presence of flared PFs (Fig. 3D). This feature of a weakly
bound ring seems highly disadvantageous for use as a coupler in
Strongly Bound Ring Couplers Move via a Forced Walk Mechanism. If
the energy of the linker–tubulin binding is large (kDAM? 9–15
kBT), the dynamic and coupling properties of the ring change
dramatically. Such rings would show negligible diffusion
(?10?2? ?m2per sec), but bending PFs can exert enough force
to displace them in a robust and deterministic manner. This
contrasts with the saltatory motions of a weakly bound coupler
(Fig. 4A), so we call such motility a ‘‘forced walk.’’ Ring
movements will be unidirectional, although they occur by
various paths, even when calculations are repeated for rings
with the same binding energy and initial conditions (Fig. 4A).
The paths differ because of the variable pauses, which fre-
quently interrupt the ring’s motion and PFs splitting. The
pausing is caused by the stochastic behavior of individual PFs
and the random transitions of the ring between successive
preferred positions on the MT lattice (SI Movie 2). As a PF
bends and pushes on the associated linker, the linker remains
at its binding site but bends away from ring’s plane, accumu-
lating energy in linker strain. The linker’s transition to the
next-closest binding site occurs abruptly with an 8-nm step (SI
Movie 3). The center of the ring moves less regularly, because
the ring can form up to 13 linker-mediated bonds with the MT
(Fig. 4B). Interestingly, when the same ring is placed on a
nonhelical MT (13?0), the ring does not move under the
depolymerization force. MT helicity is significant, because it
e i c i f f e
o i s u f f i d
(µm2) s /
KDAM , binding energy (k BT)
e c n
a t s i d
e l e v a r t
e c r o f
opposing load (pN)
y t i c o l e v
) n i m
KDAM, binding energy (kBT)
050100 150 200
surface is fast only for kDAM? 6 kBT. (B) Thermal motions of a weakly bound
ring (kDAM 3 kBT) are biased by the shortening MT end (shown is the z
coordinate of the uppermost lateral bond between two adjacent PFs). (C)
Force-velocity curves for weak and strong binding. (D) Detachment force
grows for stronger linker-tubulin binding, but the stalling force gets smaller.
Role of the ring-MT-binding energy. (A) Ring diffusion on the MT
www.pnas.org?cgi?doi?10.1073?pnas.0709524104Efremov et al.
allows linkers to step asynchronously, thereby minimizing the
energy barriers for ring transitions.
The number of bonds that can form between the MT and the
ring is independent of linker-binding strength. The detachment
force, however, increases with growing linker affinity (Fig. 3D).
Tension applied to the coupler provides a load that slows ring
movement (Figs. 3C and 4C). The dwell periods become longer,
and when ring–MT binding is very strong, the ring stalls even
under a small load (Fig. 3D). When kDAM?13 kBT, the ring will
maintain attachment, even when stalled at a blunt MT end,
thereby increasing the fidelity of its MT-dependent motion.
Thus, strong ring–MT binding improves the coupler’s stability of
attachment but reduces the load it can carry.
Specifying the Ring’s Design. We have developed a versatile
molecular-mechanical model of ring–MT coupling by combining
highlights from previous theoretical approaches; assumptions
bending PFs and the ring. Different aspects of these interactions
were analyzed to narrow the range of important model param-
eters. This approach has identified two possible mechanisms for
from the mechanism proposed by Hill and Kirschner (19, 20).
The two modes of movement and other ring characteristics have
been explained in terms of specific features of the coupler’s
MTs (29, 30) is important for maximizing force transduction
(22). The exact number of subunits in the ring is virtually
inconsequential (when it is ?13, the number of PFs in a MT), but
the number of bonds that can be formed between the ring and
the MT wall is critical. The latter is greatly influenced by the
flexibility of the binding structures. We have chosen linkers
whose flexibility is just enough to maximize the number of bonds
yet not limit the force transduction processes. The main model
conclusions, however, are unchanged with a 10-fold change of
this parameter. Recent structural studies indicate that the core
of the ring, not just the linkers, may be flexible (31, 32). The
mechanical properties of a rigid ring with flexible linkers should
be highly similar to those of a flexible ring with rigid linkers, but
this issue will require further work.
Strength of Ring–MT Binding. The parameter that is most crucial
for MT-dependent ring movement is the strength of linker
binding to the MT wall. We have analyzed couplers with a 1- to
15-kBT range of this energy, which encompasses the strengths of
typical protein–protein interactions as well as the various esti-
mates of Dam1–tubulin-binding energy. Some data support the
idea that Dam1–MT interactions are weak. The measured
diffusion of the Dam1 complexes that were interpreted as rings
(37) suggests that Dam1–tubulin-binding energy is ?4 kBT (Fig.
3A). The forces that detach Dam1-coated microbeads from
shortening MTs in vitro are ?3 pN (14), implying even weaker
binding (?2 kBT, Fig. 3D). Without knowledge of the pathway
for Dam1 oligomerization and MT binding, the Dam1–MT
dissociation constant determined biochemically (29) does not
define binding energy unambiguously and can be used only as an
upper estimate (15 kBT). However, the MT affinity of a mutant
Dam1 complex with reduced ability to form rings is the same as
that of wild-type Dam1 (29, 32), suggesting that a large fraction
of the observed interaction corresponds to Dam1–MT binding.
We therefore think this energy is toward the upper end of the
Biased-Diffusion Mode. When the binding is relatively weak (? 6
will be biased by MT shortening. The latter motion, however, is
produced by a mechanism that is different from Hill’s model, in
which the sleeve moves directionally only when MT is not fully
inserted. Ring motions are affected simply by the presence of the
flared PFs at the shortening MT end. Such a coupler, in
principle, can move under a large opposing load, but for loads
?12 pN, its retention by the MT end depends entirely on the
presence of flared PFs. If the PFs should happen to shorten, the
ring would swiftly be removed from the MT end, even by a small
load. If the Dam1-tubulin binding is 1–3 kBT, thermal motions
would detach such rings from a ‘‘blunt’’ MT end almost every
time they approach it, even without a load. Most of the plus ends
of MT growing in vitro are blunt, although some have long PF
extensions (5, 6). These, however, are curved only gently com-
pared with the PFs on depolymerizing MTs (39) and are unlikely
to prevent detachment of 33-nm rings from growing MT tips, as
suggested in ref. 14. Thus, the MT attachment of the weakly
bound ring should be highly vulnerable to variations in the rate
and direction of MT length change. In vivo, moving chromo-
somes frequently pause and even reverse their direction. There-
fore, for this coupling to be biologically successful, there must be
some additional mechanism that ensures its attachment. For
example, the stability of Dam1 ring attachment might be aug-
mented by other kinetochore proteins and/or proteins that bind
MT plus ends (40). Such interactions are possible, but their
effects on ring motility are speculative.
an attractive solution to the above problem. Such rings will
diffuse by undetectable amounts under normal experimental
conditions. They can, however, be forced to walk by the action
of PF power strokes. The persistent association of such a ring
of its design and does not require additional assumptions. Rings
that bind very strongly (?15 kBT) would maintain their associ-
ation with MT ends of any geometry, even under a large
opposing force (Fig. 3D). Such rings, however, move very slowly
with the shrinking MT end, and they will stall frequently (Fig.
4D), so very strong binding may not be optimal. Microbeads
coated by Dam1 heterodecamers maintain attachment to MT
tips, even as the MT elongates (14), a behavior more consistent
n ( e c n
a t s i d d
e l e v a r t
h t i w g
n i n
e t r o
h s T
n ( e c n
a t s i d d
e l e v a r t
linker # 1
linker # 9
KDAM= 13 kBT
n i r e
) n i m
KDAM, binding energy (kBT)
n ( e c n
a t s i d d
e l e v a r t
the stochasticity and asynchrony in PF splitting and walking of different
linkers. (B) The linkers step asynchronously (shown are the z coordinates of
their MT-associated ends) and in 8-nm steps (arrows point to some), whereas
kBT). (D) The rate of coupler’s movement with the depolymerizing MT end is
a blueprint for the strength of its affinity to the MT wall and thus for the
mechanism of its motion.
Forced walk of the ring. (A) With increasing ring-tubulin affinity, the
Efremov et al.PNAS ?
November 27, 2007 ?
vol. 104 ?
no. 48 ?
with weak Dam1–MT binding that permits ring diffusion. How-
ever, these movements were seen in the absence of soluble
bead-free Dam1, and it remains to be determined whether such
coupling is ring-based.
Our model defines an inverse relationship between the effi-
ciency of force transduction and the stability of coupler attach-
ment, leaving a relatively narrow window of ‘‘successful’’
is optimal by the postulated standards (Fig. 3D, shaded area). A
compromise between reduced efficiency of force transduction
and increased strength of attachment appears important for the
coupler in an organism like Saccharomyces cerevisiae, where a
kinetochore is stably attached to only one MT (41, 42). We
therefore favor a tight-binding mechanism with kDAM? 10–14
kBT and consider it better suites for budding yeast than the one
based on biased diffusion. However, a choice between different
coupler designs is impossible on strictly theoretical grounds. For
example, although the forces that oppose kinetochore motions
may be large in some organisms (43), in budding yeast, their size
has not been measured.
Experimental Approaches to Study the Mechanism of Dam1 Motility.
If the Dam1 ring is similar to our theoretical ring, our results
suggest that its mechanism of motility in vitro can be determined
by characterizing ring motion at a depolymerizing MT end. If the
ring moves by biased diffusion, the MT end serves only to ratchet
the ring’s random walk. Thus, the coupler’s speed will be
as in any other model in which the ring slides freely on the MT
rate of MT disassembly. A reduced rate of PF splitting has been
verified experimentally for the force-transducing couplers under
opposing tension (13, 44). Examination of the Dam1 ring’s
motility in the absence of a load, however, is lacking. Additional
information will be available from an analysis of mutant Dam1
complexes with reduced MT binding (29). If strong ring–MT
binding is facilitated by linkers, complexes with reduced binding
linker should slow MT depolymerization less than wild type. If,
the sliding (32, 37), the above mutant should retard MT disas-
sembly more than wild type. Experimental testing of these
predictions should help clarify the mechanism of Dam1 ring
We thank M. Molodtsov, N. Goudimchuk, and other members of the
McIntosh and Ataullakhanov laboratories for help and assistance and
A. I. Vorobjev for support. We are also grateful to D. Drubin, G. Oster,
K. Bloom, J. Scholey, and D. Odde for discussions. This work was
supported by National Institutes of Health Grant GM33787 (to J.R.M.)
and U.S. Civilian Research and Development Foundation Grant
CGP2006B#2863 (to F.I.A. and J.R.M.).
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