Modeling of chromosome motility during mitosis
Melissa K Gardner and David J Odde
Chromosome motility is a highly regulated and complex
process that ultimately achieves proper segregation of the
replicated genome. Recent modeling studies provide a
computational framework for investigating how microtubule
assembly dynamics, motor protein activity and mitotic spindle
mechanical properties are integrated to drive chromosome
motility. Among other things, these studies show that
metaphase chromosome oscillations can be explained by a
range of assumptions, and that non-oscillatory states can be
achieved with modest changes to the model parameters. In
addition, recent microscopy studies provide new insight into
the nature of the coupling between force on the kinetochore
and kinetochore–microtubule assembly/disassembly.
Together, these studies facilitate advancement toward a
unified model that quantitatively predicts chromosome motility.
Department of Biomedical Engineering, University of Minnesota,
7-132 Hasselmo Hall, 312 Church Street S.E., Minneapolis,
Minnesota 55455, USA
Corresponding author: Odde, David J (email@example.com)
Current Opinion in Cell Biology 2006, 18:639–647
This review comes from a themed issue on
Cell division, growth and death
Edited by Bill Earnshaw and Yuri Lazebnik
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# 2006 Elsevier Ltd. All rights reserved.
During mitosis, dynamic microtubules mediate the
proper segregation of the replicated genome into each
of two daughter cells. These dynamic microtubules are
essential components of the mitotic spindle, which is
additionally composed of spindle poles, kinetochores
and the replicated chromosomes themselves (reviewed
Microtubules have an inherent polarity, and are generally
attached at their minus-end to the poles of the mitotic
spindle [4,5]. Microtubule plus-ends are often associated
with a kinetochore, a complex protein-based structure
that serves as the essential mechanical linkage between
dynamic microtubules and chromatin (reviewed in [6,7]).
The structure and molecular composition of the kineto-
chore is now emerging, with the finding that multiple
distinct protein complexes cooperate to achieve and
maintain chromosome–microtubule coupling and regu-
late kinetochore-attached microtubule (kMT) plus-end
assembly (reviewed in [7,8]).
Microtubules are most dynamic at their plus-ends, with
extended periods of polymerization (growth) and depo-
lymerization (shortening) [9,10]. Changes between these
two states are stochastic, characterized by ‘catastrophe’
events (a switch from the growing to the shortening state)
process termed dynamic instability . During meta-
phase, the dynamic instability of kMTs can contribute to
the oscillations of chromosomes, a behavior called ‘direc-
tional instability’ . Although microtubule plus-ends
are dynamic in a wide range of mitotic spindles, the
magnitude of observed chromosome oscillations due to
directional instability varies between organisms [13–18]
(reviewed in ). In general, the dynamic instability
behavior of microtubules is thought to be responsible for
kinetochore attachment during prophase and prometa-
phase and for the alignment of kinetochores during
metaphase, such that sister chromatids are ultimately
segregated into each of two nascent daughter cells during
anaphase (reviewed in [20–22]). The congression of sister
spindle poles is a characteristic hallmark of metaphase,
after which correction of segregation errors is relatively
limited [23–25]. Therefore, prometaphase, metaphase
and anaphase represent key phases in the accurate seg-
regation of chromosomes.
The inherent complexity of the mitotic process, or even a
single phase of mitosis such as metaphase, has made it
challenging to infer the underlying mechanisms of chro-
mosome motility directly from experimental observation.
To manage this complexity, mathematical and computa-
tional models have recently been developed to integrate
further investigation of mitotic chromosome motility, in
particular its control at the kinetochore. Here we review
mitosis. Key common elements of these theoretical mod-
els for kinetochore motility include the critical role of
kMT dynamic instability, and the importance of forces
exerted at the kinetochore in either directly or indirectly
regulating kMT dynamic instability. Other elements of
spindle dynamics are considered in some of these models,
depending on the specific model organism. These ele-
ments include the following: microtubule poleward flux
(experimental characterization in [14,26–28]); force gen-
eration at the kinetochore via microtubule-associated
molecularmotors(reviewed in );microtubule
Current Opinion in Cell Biology 2006, 18:639–647
depolymerases at the kinetochore (experimental charac-
terization in [30–33]); polar ejection forces (experimental
characterization in [16,34,35]); the mechanical properties
of the kinetochore; spatial gradients in the parameters of
kMT dynamic instability (experimental characterization
in [36–39]); and kMT attachment to and detachment
from kinetochores (i.e. turnover). Modelsforeach specific
organism highlight common features as well as specific
differences between particular organisms.
Given the multiple phenomena that mediate mitosis, it is
not always obvious how they behave as an integrated
system. To manage the inherent complexity and to pre-
dict emergent properties, a number of recent studies have
used computational modeling. Similar to experimental
work, modeling of chromosome motility is now beginning
to develop a set of common methods, tools, terminologies
and standards. In the work reviewed here, the modeled
‘system’ has been defined as the mitotic spindle, with
special attention paid to the kinetochore–microtubule
interface that largely governs chromosome motility dur-
ing mitosis (other aspects of mitosis modeling are
reviewed in [40,41]). In each simulation, the character-
istics of the spindle components are defined on the basis
of experimental observations (e.g. plus-end directed
motors will tend to pull kinetochores in the direction
of the kMT plus-end). Beyond the general behaviors of
model components, simulation requires quantification
(e.g. how quickly and/or with how much force plus-end
directed motors pull kinetochores towards the kMT plus-
end) and these quantities are defined as model para-
meters (e.g. stall forces and unloaded velocities), with
associated parameter values. Finally, in order to account
for cell-to-cell variation as well as for variation in behavior
within a single cell, the models reviewed here incorpo-
rated stochastic ranges of behavior. These ‘Monte Carlo’
simulations are accomplished through the use of compu-
ter-generated random numbers and calculated probabil-
ities (see Box 1). In addition to these modeling studies,
recent kMT structural data obtained using electron
microscopy and in vitro force measurements obtained
from single depolymerizing microtubules impinge on
the model assumptions and provide key tests of model
A kinetochore motility model for multiple
microtubule attachment: PtK1 cells
In 1985, Terrell Hill proposed a model to describe micro-
tubule–kinetochore interactions, providing a starting
point for the current modeling efforts . Specifically,
this model describes the interaction between a depoly-
merizing microtubule and a set of binding sites on the
kinetochore. Here, the kinetochore is described as a
(Figure 1a). Thermal fluctuations allow sliding of the
kinetochore sleeve relative to the kMT. Deeper insertion
of the kMT plus-end into the sleeve increases the
number of favorable wall contacts that the microtubule
makes with the kinetochore, and so reduces the free
energy of the kinetochore–microtubule interaction (i.e.
deeper insertion is energetically favorable; Figure 1a,
violet arrow). Conversely, removal of the kMT plus-
end from the kinetochore sleeve is energetically unfavor-
able and hence requires an energy input, which is pro-
vided by tension between the kinetochore and the kMT
(Figure 1a, blue arrow). Since energy is force multiplied
by distance, the force required to detach the kinetochore
from the kMT is equal to the energy of wall binding per
unit length of microtubule inserted into the sleeve. Hill
estimated the detachment force to be ?16 pN (= ?65
sites ? 10 pN-nm/site/40 nm) for meiotic grasshopper
spermatocytes. Note that in this model the kinetochore
remains attached indefinitely to a kMT when the tension
is below the detachment force, but that above this force
the kMT loses attachment rapidly (within ?1 s or less).
This model provides a theoretical basis for how a ?40 nm
kinetochore outer plate could in principle be sufficient to
provide persistent kMT–kinetochore attachments over a
wide range of tension values even while the kMT poly-
merizes and depolymerizes.
More recently, Joglekar and Hunt developed a detailed
mechanistic model to describe kinetochore motility dur-
ing mitosis, based on experimental data collected in
potoroo kidney epithelial (PtK1) cells [43??]. This model
is based theoretically on the Hill sleeve model, but is
Cell division, growth and death
Box 1 Stochastic simulation via the ‘Monte Carlo’ method
The ‘Monte Carlo’ method is useful for simulating both the mean and
standard deviation of an observed experimental observation, thus
adding experimentally observed ‘noise’ to otherwise deterministic
computer simulations. For example, simulation of microtubule
dynamic instability requires that the microtubule plus-end undergoes
both growth and shortening excursions, with abrupt and approxi-
mately random switching between these two states. The frequency
of switching from growth to shortening is characterized by the
catastrophe frequency (kc, units s?1), while the frequency of
switching from shortening to growing is characterized by the rescue
frequency (kr, units s?1). Therefore, in simulation, the probability of a
rescue event for a shortening microtubule is calculated via the
prðrescueÞ ¼ 1 ? e?ðkrÞðtÞ
where t is the simulated time step in seconds. This establishes the
probability of a rescue event occurring for a shortening microtubule at
any given time step, based on the rescue frequency. Now that a
probability is calculated, a computer-generated random number
between 0 and 1 can be compared to the probability of the rescue
event at every time step. If the random number is less than the
probability, the rescue event will occur. For example, if the probability
is calculated to be 0.06 and the random number is 0.008, a rescue
event would occur during the simulated time step, and the micro-
tubule would now begin to grow. The same method is used to
calculate the probability of catastrophe and thus simulate random
catastrophe events for growing microtubules. In general, the occur-
rence of any random event can be simulated with this approach.
Further details are given in Gillespie (1977) .
Current Opinion in Cell Biology 2006, 18:639–647 www.sciencedirect.com
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the shape and stability of microtubule plus-ends. This modeling is useful
to better understand how the physical configuration of microtubule plus-
ends could be related to their behavior in vivo.
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Modeling of chromosome motility during mitosis Gardner and Odde647
Current Opinion in Cell Biology 2006, 18:639–647