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Modelling chromosome dynamics in
mitosis: a historical perspective on
models of metaphase and anaphase in
eukaryotic cells
Gul Civelekoglu-Scholey1 and Daniela Cimini2
Review
Cite this article: Civelekoglu-Scholey G,
Cimini D. 2014 Modelling chromosome
dynamics in mitosis: a historical perspective on
models of metaphase and anaphase in
eukaryotic cells. Interface Focus 4: 20130073.
http://dx.doi.org/10.1098/rsfs.2013.0073
One contribution of 10 to a Theme Issue
‘Computational cell biology: from the past to
the future’.
Subject Areas:
computational biology
Keywords:
mitosis, microtubule, kinetochore,
chromosome, mitotic spindle, mitotic
apparatus
Authors for correspondence:
Gul Civelekoglu-Scholey
e-mail: [email protected]
Daniela Cimini
e-mail: [email protected]
1
Department of Molecular and Cellular Biology, University of California, Davis, CA 95616, USA
Department of Biological Sciences and Virginia Bioinformatics Institute, Virginia Tech, Blacksburg,
VA 24061, USA
2
Mitosis is the process by which the genome is segregated to form two identical
daughter cells during cell division. The process of cell division is essential to
the maintenance of every form of life. However, a detailed quantitative understanding of mitosis has been difficult owing to the complexity of the process.
Indeed, it has been long recognized that, because of the complexity of the
molecules involved, their dynamics and their properties, the mitotic events
that mediate the segregation of the genome into daughter nuclei cannot be
fully understood without the contribution of mathematical/quantitative modelling. Here, we provide an overview of mitosis and describe the dynamic and
mechanical properties of the mitotic apparatus. We then discuss several quantitative models that emerged in the past decades and made an impact on our
understanding of specific aspects of mitosis, including the motility of the
chromosomes within the mitotic spindle during metaphase and anaphase,
the maintenance of spindle length during metaphase and the switch to spindle
elongation that occurs during anaphase.
1. Introduction
Mitotic cell division is an absolute requirement for survival of both uni- and
multi-cellular organisms. Indeed, cell division errors are typically associated
with lethality in unicellular organisms; and development of multi-cellular
organisms requires a number of tightly regulated and highly ordered cell divisions. Moreover, cell division is essential for morphogenesis and tissue
homeostasis. Cell division is the final stage of a cell life cycle (the cell cycle)
and is the stage during which the genetic material is partitioned equally
between two daughter cells. In preparation for mitosis, the DNA is replicated
during a previous stage (S-phase) of the cell cycle, but the two DNA molecules
produced during replication of each chromosome are held together by specialized chromosome-associated proteins that form the cohesin complexes. Thus,
each mitotic chromosome consists of two paired sister chromatids. Segregation
of sister chromatids into two daughter cells relies on dynamic interactions
between the microtubules of the mitotic spindle and the chromosomes. These
interactions and the forces produced lead to a series of highly coordinated
and dynamic events that have fascinated scientists for centuries [1]. Mitosis
has also been a subject of interest to scientists outside of the life sciences, including physicists, who have long been interested in the mechanical properties of
the mitotic apparatus components and the forces they produce, and mathematicians, who have long been interested in developing mathematical models of
mitosis. Here, we first provide an overview of the mitotic process, and then
we describe some of the key models of mitosis that have been developed
over the years. These models will be described in chronological order to provide
a historical perspective of how, thanks to the increased knowledge of mitosis at
the cellular and molecular level, these mathematical models have become more
and more complex and comprehensive.
& 2014 The Author(s) Published by the Royal Society. All rights reserved.
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2. An overview of mitosis
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prometaphase
metaphase
merotelic attachment
anaphase
telophase/
cytokinesis
kinetochore
centrosome
chromosome
microtubule/
microtubule bundle
Figure 1. Diagrammatic representation of mitosis. The five stages of mitosis are
depicted from top to bottom. For simplicity, only two chromosomes are depicted
here. During the first stage of mitosis (prophase), the chromosomes undergo condensation, whereas the duplicated centrosomes move around the nucleus, whose
envelope is still intact. Prometaphase begins when the nuclear envelope breaks
down, thus allowing interaction between the kinetochores and the microtubules of
the mitotic spindle. During prometaphase, chromosomes establish kinetochore–
microtubule attachment and progressively move to the spindle equator. When all
the chromosomes become aligned at the spindle equator (forming the metaphase
plate), the cell is said to be in metaphase. During anaphase, the sister chromatids
separate from each other and move to opposite poles of the mitotic spindle. Finally,
during telophase, the nuclear envelope begins to reassemble around the decondensing chromosomes. During cytokinesis, which temporally overlaps with the late
stages of mitosis, an actin–myosin contractile ring cleaves the cytoplasm to
complete cell division. (See text for further details.) (Online version in colour.)
By telophase, the chromosomes have reached the spindle
pole regions, and start to decondense. Meanwhile, the nuclear
envelope starts to reassemble around the decondensing
chromosomes to form two daughter interphase nuclei.
Cytokinesis normally starts during the later stages of mitosis
Interface Focus 4: 20130073
The ultimate goal of mitosis is the equal partitioning of the
replicated genome into two daughter cells (figure 1). In
eukaryotic cells, mitosis is typically divided into five distinct
stages: prophase, prometaphase, metaphase, anaphase and
telophase (figure 1). Cytokinesis, which follows mitosis and
partly overlaps with it, leads to partitioning of the cytoplasm,
thus completing cell division [2]. This review, however, will
exclusively focus on mitosis, or nuclear division.
In mitotic prophase, the chromatin condenses, so that individual chromosomes become discernible. Prophase is also marked
by a change in microtubule dynamics, which leads to the
disassembly of interphase microtubules (long and not very
dynamic), and the assembly of mitotic microtubules (short and
highly dynamic). The centrosomes, which are replicated during
S-phase and serve as microtubule-organizing centres in most
animal cells, move apart during prophase and an aster starts
forming around each duplicated centrosome [3] with microtubule
plus ends extending outwards. This process initiates the assembly
of a dynamic macromolecular machine: the mitotic spindle.
The breakdown of the nuclear envelope marks the onset
of prometaphase, and a bipolar spindle forms when microtubule plus ends extending from opposite asters overlap and
form bundles enabled by the concerted action of multiple microtubule-based motors and microtubule-associated proteins
(MAPs). During this phase, the dynamic microtubule plus
ends switch between polymerization and depolymerization
phases within the nuclear region, where they may encounter a
kinetochore, a specialized protein structure that mediates
chromosome–microtubule attachment. Kinetochores can
initially establish lateral interactions with microtubules [4,5],
but these attachments are subsequently converted into end-on
attachments, and this is believed to increase microtubule stability [6]. In most model organisms, individual kinetochores
bind multiple microtubules, which will form a microtubule
bundle, also referred to as kinetochore-fibre or k-fibre. Typically,
chromosomes first establish monotelic attachment (one sister
kinetochore attached and one unattached). These monotelic
chromosomes can either become amphitelically attached
(sister kinetochores bound to microtubules from opposite
poles) and then move to the spindle equator or they can move
to the spindle equator, thanks to lateral kinetochore–microtubule interactions, and then become amphitelically attached
[4,5]. During prometaphase, chromosomes progressively
become amphitelically attached and aligned at the spindle
equator in a process called chromosome congression.
After all chromosomes have aligned at the spindle equator,
the cell is said to be in metaphase, and the chromosomes lined
up at the spindle equator are said to form the metaphase plate.
In many cell types, chromosomes at the metaphase plate exhibit continuous oscillations back and forth about the spindle
equator [7].
Anaphase onset is marked by the abrupt and synchronous
separation of the sister chromatids, which is due to the sudden
degradation of the cohesin complexes between the sister chromatids [8]. During anaphase, the two sister chromatids, now
daughter chromosomes, move to opposite spindle poles as
their respective k-fibres shorten. This process is also referred to
as anaphase A, to distinguish it from anaphase B, in which the
spindle elongates, thus moving the two groups of segregating
chromosomes further apart. In most organisms, there is temporal overlap between anaphase A and B.
2
prophase
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Forces generated in the mitotic spindle are required for proper positioning of the sister chromatids within the spindle,
for mitotic progression and for accurate segregation of the
genome during mitosis (reviewed in [9]). Moreover, forces
generated in the mitotic spindle control spindle organization
and length (reviewed in [10]). Microtubules in the spindle
can generate polymerization ratcheting forces that push the
chromosome arms and/or the spindle poles, and depolymerization of microtubules at their plus or minus ends can pull
the kinetochores poleward with or without the help of microtubule depolymerases. In addition, multiple minus- and
plus-end-directed microtubule-based motor proteins exert
forces on the kinetochores, chromosome arms or the centrosomes by moving towards the plus or minus end of their
microtubule tracks within the spindle. Although the microtubule-based mitotic spindle maintains an overall stable
structure and organization, the microtubules that form the spindle scaffold display rapid dynamics, as assayed by fluorescence
recovery after photobleaching [11]. The rapid fluorescence
recovery, with a half-life ranging from a few seconds to a few
minutes depending on the organism, is attributed mainly to
dynamic instability of microtubule plus ends, and to a lesser
extent to poleward transport of the microtubule lattice and
depolymerization of centrosome-associated microtubule
minus ends in a process called microtubule poleward flux [12].
Following its assembly at the onset of prometaphase, the
mitotic spindle goes through multiple phases of elongation
and steady-state length maintenance until its disassembly
during telophase. The elongation rate and the steady-state
length of the spindle during these phases are determined
through a balance of forces generated by the antagonistic
and/or complementary action of microtubule dynamics at
plus ends, depolymerization at minus ends, and microtubule-based motor proteins associated with spindle
microtubules, chromosomes, kinetochores and spindle poles
(the mechanic and dynamic control of spindle length is
reviewed in [10]).
The k-fibres of amphitelically attached sister chromatids
pull the sister kinetochores in opposite directions and thereby
exert tension and stretch the cohesin bonds between the sisters.
These forces mediate the congression of the sister chromatids to
the spindle equator during prometaphase [13] in a process
4. The chronology of mathematical modelling
of mitosis
As the structure, organization and dynamics of the protein
machine called the mitotic apparatus was being visualized
and the inventory of the molecules involved was being
gathered, scholars of mitosis set out to gain a quantitative understanding of the dynamic events that lead to chromosome
segregation, in terms of the interacting molecular parts and
physical and chemical principles. Significant efforts have been
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3. The mechanics and dynamics of mitotic
spindle, microtubules and chromosomes
during mitosis
often accompanied by the oscillation of the sister chromatids
along the pole–pole axis. The oscillations of the sister chromatids between the spindle poles persist throughout metaphase in
many vertebrate culture cells, and is termed chromosome
directional instability [7]. The tension between the sisters
may also play a role in cell cycle progression by silencing the
mitotic checkpoint, as suggested by micromanipulation studies
in which mechanical stretch of monotelically attached chromosomes resulted in precocious anaphase (reviewed in [9]). In
addition, classic laser microsurgery experiments showed that
disrupting kinetochore–microtubule attachment on one sister
chromatid of an amphitelic chromosome resulted in the rapid
movement of both sister chromatids (whole chromosome)
towards the pole associated with the intact k-fibre [14], indicating that the balance of forces between the k-fibres of
amphitelically attached sister chromatids determines their
position and their motility rate within the spindle.
The maximum force that the spindle components can exert
on a chromosome during anaphase was measured and found
to be approximately 700 pN in classic micromanipulation experiments [15], corresponding to a 10–15 pN force (typical stall
force of a few motor proteins) per individual microtubule in a
k-fibre [16]. Other mechanical properties of the mitotic spindle,
including the elastic modulus of chromosomes, the viscosity of
the cytoplasm and the magnitude of polar ejection forces (forces
directed away from the spindle poles and acting on the chromosome arms [17]), were also estimated based on the analysis of the
three-dimensional motion of chromosome arms over time [18].
More recently, the distribution of polar ejection forces within the
spindle was also determined by analysing the changes in velocity and range of motion of chromosomes subsequent to
arm-severing by ultrafast laser microsurgery [19].
In an attempt to identify the magnitude of tension exerted
on pairs of sister kinetochores oscillating between the spindle
poles, where one sister moves poleward (leading) and the
other moves away from its pole (trailing), recent studies
used probes to mark the pole-proximal and the centromeric
ends of pairs of sister kinetochores undergoing directional
instability in metaphase spindles of cultured vertebrate cells
[20]. The authors found that the poleward-moving sister
kinetochore is compressed, whereas its trailing sister is
stretched, and attributed this finding to the presence of an
elevated friction at the site of active force generation (i.e. at
the poleward-moving sister kinetochore).
Finally, studies using metaphase spindles formed in
Xenopus laevis egg extracts tested the possible contribution of
an elastic matrix to forces generated within the spindle by
using microneedles to skew the spindle during metaphase
[21]. The findings of such studies suggested that a spindle
matrix does not contribute significantly to the motility of the
spindle parts, but instead intrinsic forces dominate movements.
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(i.e. late anaphase/telophase), when a cleavage furrow containing an actin–myosin contractile ring forms and pinches
the cell membrane of the dividing cell in a region corresponding to the spindle equator. This process divides the cytoplasm
into two, thus completing cell division.
Despite model system-dependent variations in spindle
size, total number of chromosomes, duration of different
mitotic phases, dynamics of metaphase chromosomes, extent
of chromosome segregation during anaphase A and B and
deployment of force generators, the mechanisms underlying
mitotic chromosome movements are general. Nevertheless,
these mechanisms are used and combined in various ways
to adapt to a cell-specific geometry, and to the structure and
properties of its mitotic constituents.
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Among many puzzling phenomena, one that remains to be
solved is how kinetochores are able to maintain attachment
to shortening microtubule tips during metaphase and anaphase. Indeed, one of the earliest theoretical models (the
Hill sleeve model; figure 2a) of a mitotic event concerned this
process [25]. In this classic study, based on thermodynamic
and kinetic arguments, the author quantitatively shows how
a shortening microtubule tip can remain attached to and maintain a steady insertion depth inside a kinetochore region
referred to as the ‘sleeve’. It is assumed that the tubulin subunits in the microtubule lattice interact with a finite number
of sites along the length of this sleeve with moderate affinity.
However, because the translocation of the microtubule inside
the sleeve requires detachment of existing bonds, driven by
the thermal fluctuations of the binding sites inside the sleeve,
as well as re-binding, the potential barrier/resistance increases
for increased number of interactions between the sleeve and the
lattice. This effectively slows down the relative movement of
the microtubule inside the sleeve and a steady-state penetration length of the lattice inside the sleeve is maintained,
even for a microtubule depolymerizing at approximately
1 mm min21.
The Hill sleeve idea was developed further in a forcebalance approach [26] (figure 3a), where the authors
accounted for the metaphase oscillations (directional instability) of sister kinetochores capable of forming attachments
with multiple microtubules through Hill sleeves arranged
in parallel at each kinetochore, and anchored to the kinetochore via Hookean springs. In this study, tension forces
between the sister kinetochores arising from the stretched
cohesin bonds and the polar ejection forces were taken into
account. However, poleward flux of the microtubules was
not accounted for. As in reference [25], here too the poleward
movement of a sleeve with respect to its microtubule requires
the detachment and re-binding of all bonds, setting a large
energy barrier for the movement at high sleeve insertion
depths. This mechanism effectively prevents the poleward
movement of a sleeve (and hence its kinetochore) when the
microtubule is in the polymerization state, which naturally
leads to deeper insertion of the microtubule into the sleeve,
while favouring the poleward movement of a sleeve (and
hence its kinetochore) when the microtubule is in the depolymerization state, which reduces the insertion depth. Thus, in
this force-balance Hill sleeve model, the primary factor
driving the poleward or anti-poleward movement of the
metaphase kinetochore is microtubule dynamics [26].
An alternative to the Hill sleeve model for microtubule–
kinetochore interactions was put forward based on the presence
and requirement of microtubule-based motors at the kinetochore for chromosome motility in many organisms. Theoretical
investigation of this model in a force-balance approach [27]
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(b)
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4.1. Kinetochore– microtubule attachments and
chromosome motility
(a)
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placed in developing mathematical models of various mitotic
subprocesses, including spindle assembly, metaphase chromosome oscillations and anaphase chromosome segregation.
Here, we do not discuss models of mitotic spindle assembly
and establishment of kinetochore attachment [22–24]. Instead,
we primarily focus on the quantitative models of metaphase
chromosome dynamics and briefly review the models of metaphase spindle length control and anaphase spindle elongation.
(c)
key
inner–outer plates and
fibrous corona
kinesin13’s
centrosome
dynein
CENP-E
microtubule
Ndc80
Hill sleeve
Figure 2. Kinetochore– microtubule interactions considered in quantitative
models. (a) Hill sleeve model: kinetochore – microtubule bonds are arranged
in series and require nearly synchronous detachment and re-attachment of all
bonds for relative movement of the microtubule and the kinetochore.
(b) Kinetochore motors model: a tug of war between the plus- and
minus-end-directed motors, CENP-E and dynein, located at the kinetochore
powers the movement of the kinetochore along the microtubule lattice,
whereas members of the kinesin-13 family depolymerases located at the
kinetochore and at the centrosome shorten the lattice at both ends, leading
to ‘pacman’ and ‘poleward flux’, respectively. (c) Non-motor kinetochore–
microtubule coupler model: flexible non-motor proteins, Ndc80, located at
the outer kinetochore plate bind to and detach from the microtubule independently from one another and serve as dynamic couplers between the
kinetochore and the spindle microtubules. (Online version in colour.)
(figure 3a) was based on the presence of two antagonistic kinetochore motors, dynein and CENP-E, and the two members of the
kinesin-13 family microtubule depolymerases located at the
kinetochore and at the spindle poles, incorporating the poleward
flux of kinetochore–microtubules (figure 2b). This model successfully accounted for metaphase and anaphase chromosome
behaviour in a wide range of organisms through adjustments
in model parameters. Notably, this model reproduced qualitatively different metaphase chromosome dynamics, such
as directional instability and steady positioning, and a wide
range of anaphase chromosome-to-pole velocities. It also
quantitatively showed that the transition from metaphase positioning to anaphase poleward motility can be achieved merely
through the degradation of the cohesin bonds between the
sister chromatids and the removal of the polar ejection forces
[27]. This model formed the basis of several other models that
were proposed later [28–30].
Two models, derived from the Civelekoglu-Scholey et al.
[27] force-balance model, were developed by the Tournier
group to investigate chromosome behaviour in fission yeast
[28,29]. Their first force-balance model addressed the
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(a)
force-balance model
5
= µ vchr
F chr
drag
=
F chr
drag
S(F
KT-Fcohesin-FPE-Fpoly)
FPE
FKT
Fcohesin
(b)
Fpoly
Fsliding
sliding and switch from depolymerization to elongation model
metaphase: sliding and depolymerization
steady-state spindle length
anaphase B: sliding and no depolymerization
spindle elongation
slide and cluster model
metaphase: sliding and rapid turnover
steady-state spindle length
anaphase B: sliding and slow turnover
spindle elongation
Figure 3. The force-balance approach and models for spindle length control. (a) Force-balance models assume the sum of the forces generated by various spindle parts to
be balanced by the viscous drag forces on the moving module (e.g. the chromosome). Viscous drag forces are proportional to the rate of movement and the friction
coefficient of the module. Forces exerted on the module are determined based on simple diagrams through consideration of various spindle parts interacting with the
module. (b) Two competing models for spindle length changes from steady state (metaphase) to elongation (anaphase B). Top panel: the sliding and switch from minusend depolymerization to spindle elongation model. In this model, the sliding of antiparallel overlapping microtubules by bipolar kinesin-5 motors (in red) is balanced by
depolymerization of microtubule minus ends near the poles during metaphase and the cessation of microtubule minus-end depolymerization engages the sliding motors
to push the poles apart and to elongate the spindle in anaphase. Bottom panel: the slide and cluster model. In this model, microtubule dynamics is restricted to plus
ends, microtubules are nucleated near the spindle equator, and their sliding rate (resulting from the combined action of dynein and kinesin-5 motors) and the half-life of
the microtubules determine the steady-state spindle length. Slowing down of microtubule plus-end dynamics (e.g. slower turnover by reduced catastrophe at microtubule
plus ends) can lead to spindle elongation. In this model, microtubule minus ends do not depolymerize. (Online version in colour.)
correction of merotelic attachments during anaphase in fission yeast and did not account for amphitelic kinetochore
pairs [28]. This model was derived from the CivelekogluScholey et al. and Brust-Mascher et al. models [27,31] with
additional simplifying assumptions. A second quantitative
model developed by the Tournier group accounted for the
dynamics of amphitelic kinetochore pairs [29]. Similar to
their previous study [28], this second study by the Tournier
group also relied upon a macroscopic approach [29].
Indeed, all kinetochore components are represented by a
homogeneous viscoelastic ‘unit’ that can attach/detach
from microtubule plus ends, assumed to be homogeneously
distributed within the spindle [29]. However, because the
positions and dynamic states of microtubule plus ends are
not accounted for, neither the effect of kinetochore tension
on microtubule plus-end dynamics nor the effect of microtubule plus-end dynamics on the attachment/detachment from
the kinetochore are considered [29]. In particular, the kinetochore–microtubule binding and detachment in this model
occur at fixed, average rates (implemented stochastically),
implying that a kinetochore is equally likely to encounter a
‘free’ microtubule plus end from prometaphase through anaphase (‘free’ microtubule plus ends are assumed to be as
abundant in all phases), and does so regardless of the
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F chr
drag
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F chr
drag = sum of all forces exerted on the chromosome
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and Hunt model [26], in this new model, bonds with a polymerizing microtubule are generally ‘weak’, and those with
a depolymerizing microtubule are generally ‘strong’, in
agreement with recent in vitro biophysical studies [37].
Following the identification of ring-like protein complexes
at the kinetochore–microtubule interface in budding yeast
[44,45], and the slender fibrils connecting the kinetochore to
the flaring out tips of depolymerizing microtubules in vertebrate cells [46], new quantitative models addressed the
ability of these ‘couplers’ to mediate stable attachment
of the kinetochores to dynamic microtubule tips [46,47]. By
investigating the biomechanical properties of the coupling
interaction between a ring-like structure and a microtubule tip
in a theoretical framework, Efremov et al. [47] found that the
optimal coupling occurs when the ring is bound to the microtubule via flexible linkers that bind strongly. They further
showed that by such a coupling mechanism the kinetochore
ring can faithfully follow a depolymerizing microtubule tip by
harnessing the energy released from the power stroke as the protofilaments in the lattice curl outwards, and maintain attachment
with straight (polymerizing) microtubule tips under load [47].
Finally, the authors showed that the slender fibrils observed in
electron tomographic images in vertebrate cell kinetochores
[46] can form efficient couplers between the kinetochore and
the microtubules and can sustain load, even in the absence of
rings around the microtubule tips, if the binding is strong [47].
In fact, this study showed that direct kinetochore–microtubule
coupling via fibrils, modelled as elastic elements, works more
efficiently than kinetochore–microtubule coupling via rings
and elastic links along depolymerizing microtubules, because
the microtubule depolymerization rate becomes limited by the
ring’s sliding rate in the latter case. These theoretical models of
kinetochore–microtubule couplers, however, have not yet been
extended to account for the binding of a kinetochore to multiple
microtubules, or to account for sister kinetochore dynamics in
metaphase or anaphase spindles, in the presence of polar
ejection forces, cohesin spring tension and with microtubules
undergoing poleward flux.
Another notable theoretical approach in understanding
kinetochore motility emerged from studies aimed at understanding the behaviour of chromosomes in the budding yeast
Saccharomyces cerevisiae [48–50]. In the budding yeast, where
spindle microtubules do not undergo poleward flux, and
only a single microtubule binds to each kinetochore, the maintenance of each kinetochore’s attachment to the spindle pole
relies upon the dynamic instability of a single microtubule
plus end. The computational models in these studies addressed
both the mechanical (e.g. tension dependent) and the molecular (e.g. kinesin-5) regulation of the plus-end dynamics of a
kinetochore-bound microtubule, and successfully accounted
for the behaviour of yeast kinetochores in metaphase and
made valuable experimentally testable predictions [49,50].
Several other theoretical models in the literature proposed
to account for the metaphase and anaphase behaviour of
chromosomes, but used speculative approaches with unidentified interactions and/or feedback loops, or made gross
simplifying assumptions with the aim of providing a simple
physical description of these events. Unfortunately, these
strategies compromised the purpose of theoretical modelling,
which is to provide a quantitative understanding of the behaviour resulting from interacting molecules, based on the
principles of physics and chemistry. Some of the mathematical
and computational models of kinetochore attachment to
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kinetochore position within the spindle (‘free’ microtubule
plus ends are uniformly distributed along the pole –pole
axis of the spindle) [29]. Furthermore, because individual
microtubules are not accounted for, a microtubule is assumed
to instantly switch to polymerization or depolymerization
and follow the direction of the kinetochore upon attachment
[29]. Thus, some key questions, such as (i) how are the
growth/shrinkage state and rate of different microtubules
attached to each kinetochore coordinated and (ii) how are
the sister kinetochores coordinated to give rise to the
observed sister kinetochore movements, are not and cannot
be addressed in this simplified framework.
Another force-balance model [32], developed independently of the model by Civelekoglu-Scholey et al. [27], used a
simplified approach to describe and account for movements
and oscillations of monotelic chromosomes. In this model,
the chromokinesin motors anchored on the chromosome
arms generated anti-poleward forces by walking towards the
plus ends of the spindle microtubules they encounter, whereas
the opposing poleward force, dependent on kinetochore
attachment to microtubules, was assumed to be constant.
Neither the dynamic instability of microtubule plus ends nor
the poleward flux of microtubules was accounted for in this
model. Nevertheless, the authors showed that the heterogeneity of the microtubule plus-end distribution within the
spindle, together with a force-sensitive dissociation property
of the chromokinesin motors are sufficient to account for the
oscillatory behaviour of monotelic chromosomes.
The original Civelekoglu-Scholey force-balance model [27]
was more recently adapted and extended to understand the
dichotomy of chromosome behaviour seen in metaphase
PtK1 cells [30], where peripheral sister kinetochores remain
steady at the metaphase plate, whereas sister kinetochores in
the middle of the metaphase plate oscillate back and forth
between the poles [30,33–35]. In this recent model, nonmotor proteins, modelled as elastic bonds, provide the link
between the kinetochore and the microtubules (figure 2c).
This was based on recent studies that identified the KMN
network of proteins, specifically the Ndc80 complex, as the
‘coupler’ between the kinetochore and the microtubule
(reviewed in references [36–40]). As for the kinetochorebound motors in the original Civelekoglu-Scholey model
[27], but in contrast with the Hill sleeve bonds proposed by
Joglekar & Hunt [26], these elastic bonds bind to and detach
from the microtubule lattice independently from one another
(non-synchronously) [30]. While the primary factor driving
the poleward kinetochore movement is poleward flux by
pulling on and stretching the microtubule-bound protein
complexes at the kinetochore, the depolymerization rate of
kinetochore–microtubule plus ends is governed by the
dynamics of these bonds, which in turn is regulated by tension
forces [30,41]. Like in the initial Civelekoglu-Scholey model
[26], the elastic bonds have a high association constant for the
microtubule, but, in this new model, the detachment rates for
these bonds are assumed to be force sensitive and to display
different kinetics depending on whether the microtubule tips
are polymerizing or depolymerizing [30]. Specifically, it is
assumed that bonds detach easily under low and high tension,
but the bond life is increased under moderate tension from a
depolymerizing microtubule tip, a ‘catch bond’ interaction,
and the detachment rate from a polymerizing microtubule tip
increases linearly with increasing tension, a ‘slip bond’ interaction [30,42,43]. Consequently, in contrast with the Joglekar
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4.2. Metaphase spindle length and anaphase
spindle elongation
5. Concluding remarks
Despite the discovery of the key components that drive
chromosome segregation in the mitotic spindle, and the characterization of their chemical and biophysical properties, many
questions, particularly related to the bewildering dynamic of
the mitotic apparatus, remain to be answered. For instance, (i)
how does a kinetochore bind to multiple microtubule plus
ends that are dynamic? (ii) How do kinetochores maintain faithful attachments with dynamic microtubule plus ends for
accurate segregation, while allowing the correction of misattachments? and (iii) How can a spindle constituted of highly
dynamic microtubules maintain a constant length? Realistically,
the answers to these questions must take into account the chemical and physical properties of the molecular components of the
mitotic apparatus and its environment, as well as the interactions
between such components, and therefore inevitably require
mathematical/computational modelling.
Models that in the past made an impact on our understanding of mitotic events were often based on cartoon depictions
providing snap shots of a dynamic process, and quantitatively
7
Interface Focus 4: 20130073
During metaphase, which lasts for tens of seconds to hours
depending on the organism, the spindle poles are maintained
at constant spacing, giving rise to a steady spindle length.
During this period, however, the microtubules that make
up the spindle scaffold undergo two types of highly dynamic
processes: dynamic instability of microtubule plus ends and
microtubule poleward flux resulting from the microtubules’
poleward translocation by the combined action of multiple
motor proteins, coupled to their depolymerization at their
minus ends. Quantitative understanding of how such
dynamic microtubules can maintain the spindle at a constant
length, for extended periods of time in some species, has been
a popular subject of theoretical investigation.
In particular, how multiple motors with opposite
polarity working on the same or different overlapping microtubules and which combination of motors, with which type
of properties, can achieve this goal has been addressed in
an important computational model in which details of microtubule plus-end dynamics, motor kinetics and force generation,
elastic properties of the microtubules and viscosity of the
spindle environment were all accounted for [54]. The authors
in this work surprisingly concluded that only motors referred
to as ‘hetero-complexes’, i.e. bipolar motors with a minusend- and a plus-end-directed head at opposite ends, were
able to maintain a stable centrosome separation [54]. This
early computational model was later developed further and
has been successfully modified, extended and applied to
account for spindle length maintenance or spindle bipolarity
in many organisms [55–59].
Steady spindle length maintenance during metaphase
and the subsequent linear spindle elongation was later
addressed in a force-balance framework where the plus-end
dynamics of microtubules was accounted for [31] (figure 3b,
top panel). The underlying qualitative model based on experimental observations suggested that, during metaphase,
plus-end-directed bipolar motors on anti-parallel microtubule
overlaps slide them apart, whereas the minus ends that
reach the poles depolymerize, and at the onset of anaphase
B (spindle elongation) the depolymerization at the poles
ceases, engaging the bipolar motors to push the poles apart
to elongate the spindle [60]. Because the molecules involved
in this process, their dynamic properties, their numbers and
biophysical properties were previously known or identified
in the study, this mathematical/computational model provided a quantitative test for the ‘switch from poleward flux
to spindle elongation’ model. It showed that the elongation
rate was determined solely by the sliding rate of the motors
and the extent of the suppression of depolymerization at the
poles, and predicted that the elongation rate was robust to substantial changes in the numbers of motors, microtubules and
the microtubule plus-end dynamics, but a net microtubule
plus-end polymerization, resulting from dynamic instability,
was necessary for complete elongation [31]. Some of these predictions were later tested experimentally [61]. Similar to this
model, a recent experimental and computational study in
HeLa cells showed that the length of the central spindle that
forms during late anaphase is determined by the sliding of
anti-parallel overlapping interchromosomal microtubules,
whose length and position depend on depolymerization at
their minus end by Kif2a, which in turn is controlled by an
Aurora B gradient [62].
An alternative to the model ‘switch from poleward flux to
spindle elongation’ was later proposed to account for the
steady spindle length observed in metaphase spindles
formed in Xenopus laevis egg extracts, and it was referred to
as the ‘slide and cluster’ model [63] (figure 3b, bottom
panel). In this force-balance model, microtubules are
nucleated near the spindle equator, where they are organized
to form anti-parallel overlaps. Initially, plus-end-directed
bipolar motors (sliding motors) slide these newly nucleated microtubules apart polewards, and minus-end-directed
motors (clustering motors) anchored at the microtubules’ minus ends also pull them polewards along other
microtubules, helping the bipolar motors [63]. As the microtubules move poleward, first, the minus-end-directed motors on
newly nucleated anti-parallel overlapping microtubules near
the equator start pulling them inwards, antagonizing the
bipolar motors, then the microtubule plus-end dynamics parameters favour full depolymerization of the microtubules over
time, effectively setting a maximal microtubule length and life
time. Thus, in the ‘slide and cluster’ model [63], microtubules
near the equator slide outwards towards the poles, but their
minus ends do not depolymerize when they reach the poles.
In addition, microtubules near the poles are pulled towards
the equator by the minus-end-directed motors on overlapping
microtubules whose minus ends are closer to the equator.
Within the framework of this model, a steady spindle length
can be achieved and maintained, and it simply depends on
the sliding rate and the relative ratio of the sliding to clustering
motors and the plus-end dynamics of the microtubules [63].
Finally, this model posits that a change in the plus-end
dynamics of the microtubules (e.g. a reduction in the catastrophe frequency of microtubule plus ends) can tip the
balance and can, in principle, lead to the elongation of the
spindle to a different steady-state length. However, these
possibilities have not yet been tested.
rsfs.royalsocietypublishing.org
microtubules and/or metaphase and anaphase kinetochore/
chromosome behaviour discussed here, and some others not
considered here, have been previously reviewed [51–53].
Downloaded from http://rsfs.royalsocietypublishing.org/ on June 15, 2017
parameters and experimental testing of the model predictions.
Commitment to pursue such interdisciplinary studies that integrate experimental and computational approaches will be
essential for the achievement of a full, comprehensive quantitative understanding of mitosis, a key process for the
perpetuation of life.
Acknowledgements. We thank all the members of the Cimini laboratory
for useful discussion and helpful comments. We apologize to all
the colleagues whose work could not be cited because of space
limitations.
Funding statement. Work in the Cimini laboratory is supported by NSF
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