# Bistability and oscillations in cooperative microtubule and kinetochore
dynamics in the mitotic spindle

Felix Schwietert and Jan Kierfeld  Physics Department, TU Dortmund University,
44221 Dortmund, Germany<EMAIL_ADDRESS>
###### Abstract

In the mitotic spindle microtubules attach to kinetochores via catch bonds
during metaphase, and microtubule depolymerization forces give rise to
stochastic chromosome oscillations. We investigate the cooperative stochastic
microtubule dynamics in spindle models consisting of ensembles of parallel
microtubules, which attach to a kinetochore via elastic linkers. We include
the dynamic instability of microtubules and forces on microtubules and
kinetochores from elastic linkers. A one-sided model, where an external force
acts on the kinetochore is solved analytically employing a mean-field approach
based on Fokker-Planck equations. The solution establishes a bistable force-
velocity relation of the microtubule ensemble in agreement with stochastic
simulations. We derive constraints on linker stiffness and microtubule number
for bistability. The bistable force-velocity relation of the one-sided spindle
model gives rise to oscillations in the two-sided model, which can explain
stochastic chromosome oscillations in metaphase (directional instability). We
derive constraints on linker stiffness and microtubule number for metaphase
chromosome oscillations. Including poleward microtubule flux into the model we
can provide an explanation for the experimentally observed suppression of
chromosome oscillations in cells with high poleward flux velocities.
Chromosome oscillations persist in the presence of polar ejection forces,
however, with a reduced amplitude and a phase shift between sister
kinetochores. Moreover, polar ejection forces are necessary to align the
chromosomes at the spindle equator and stabilize an alternating oscillation
pattern of the two kinetochores. Finally, we modify the model such that
microtubules can only exert tensile forces on the kinetochore resulting in a
tug-of-war between the two microtubule ensembles. Then, induced microtubule
catastrophes after reaching the kinetochore are necessary to stimulate
oscillations. The model can reproduce experimental results for kinetochore
oscillations in PtK1 cells quantitatively.

††: New J. Phys.

  * March 2020

Keywords: mitotic spindle, directional instability, microtubule dynamics,
kinetochore oscillations, bistability, stochastic simulation

##  1 Introduction

Proper separation of chromosomes during mitosis is essential for the
maintenance of life and achieved by the mitotic spindle, which is composed of
two microtubule (MT) asters anchored at the spindle poles. The spindle
contains three types of MTs classified according to their function [1]: astral
MTs interact with the cell membrane to position the spindle poles, interpolar
MTs interact with MTs from the opposite pole to maintain spindle length, and,
finally, kinetochore MTs link to the chromosomes via the kinetochores at the
centromere and can apply pulling forces via the linkage. The MT-kinetochore
bond is a catch bond [2], i.e., tightening under tension but the molecular
nature of the MT-kinetochore link is not exactly known and a complete
mechanistic understanding of the catch bond is missing [3, 4] but probably
involves Aurora B [5]; the Ndc80 complexes and Dam1 (in yeast) are believed to
play a major role in the MT-kinetochore link. One function of the spindle is
to align the chromosomes in the metaphase plate at the spindle equator. It has
been observed in several vertebrate cells that chromosomes do not rest during
metaphase but exhibit oscillations along the pole to pole axis known as
directional instability [6, 7, 8, 9, 10, 11, 12], whereas in Drosophila
embryos and Xenopus eggs a directional instability does not occur [13, 14]. If
present, these oscillations are stochastic and on the time scale of minutes,
i.e., on a much larger time scale than the dynamic instability of single MTs.
Both single kinetochores and the inter-kinetochore distance oscillate; inter-
kinetochore or breathing oscillations occur with twice the frequency of single
kinetochore oscillations [11].

A quantitative understanding of the underlying mechanics of the MT-
kinetochore-chromosome system is still lacking. In the past, several
theoretical models have been proposed that reproduce chromosome oscillations
[15, 16, 17, 18, 19, 20, 21]. (see table 1 and reviews [22, 23]) These models
have in common that they simplify to a quasi-one-dimensional geometry and
contain two ensembles of MTs growing from the two spindle poles that connect
to one chromosome that is represented by two kinetochores connected by a
spring (the cohesin bond). Kinetochores follow overdamped motion [16, 17, 18,
19, 20] or are assumed to reach force balance instantaneously under the
influence of MT depolymerization and polymerization forces (because the
friction force is small) [15, 21].

Several MT depolymerization and polymerization forces are included into the
models. The models neglect explicit spindle pole dynamics but possibly include
poleward MT flux [16, 20], which describes a constant flux of tubulin from the
plus-ends towards the spindle pole and is probably driven by plus-end directed
kinesin-5 motors pushing overlapping antiparallel interpolar MTs apart and
kinesin-13 proteins that depolymerize MTs at the centrosome [24]. The main
poleward forces on kinetochores are generated by depolymerization of MTs which
builds up and transmits a poleward force via the MT-kinetochore link. Only in
the model of Civelekoglu-Scholey et al. [16] the main poleward force is
generated by MT depolymerization motors at the spindle poles. In order to be
able to exert poleward pulling forces the MT-kinetochore bond needs to remain
intact during depolymerization and “slide” with the depolymerizing MT plus
end. The force that can be exerted depends on the nature of this bond and is
high if it is a catch bond that tightens under tension [2]. All models include
switching between polymerizing and depolymerizing MT states; in most models
this switching is caused by catastrophe and rescue events (dynamic instability
[25]), only Shtylla and Keener [17, 18] do not introduce explicit MT
catastrophes but catastrophe-like events are triggered by a chemical feedback
loop if MTs approach the kinetochore.

The two ensembles of MTs are engaged in a kind of tug-of-war and exert
antagonistic forces via the spring connecting kinetochores: poleward (P)
depolymerization forces of one ensemble generate an antipoleward (AP) force on
the other kinetochore. Experiments suggest that kinetochore MTs can only exert
P-directed pulling forces by depolymerization but are not able to directly
exert AP-directed pushing forces on the kinetochore during polymerization [7,
26]. During directional instability, the spindle switches between the left and
the right ensemble pulling actively in P-direction by depolymerization while
the respective other ensemble is passively following in AP-direction by
polymerization without actively pushing. Nevertheless, some models have
included AP-directed MT pushing forces [16, 17, 18, 20, 21]. Antagonistic AP-
directed forces on the kinetochores can also be generated by polar ejection
forces (PEFs); they originate from non-kinetochore MTs interacting with the
chromosome arms via collisions or chromokinesins belonging to the kinesin-4
and kinesin-10 families [27] and pushing them thereby towards the spindle
equator. The action of different P- and AP-directed forces can move
kinetochores back and forth and also tense and relax the inter-kinetochore
cohesin bond.

Models differ in their assumptions about the MT-kinetochore link and the
mechanism how MT dynamics is directed by mechanical forces to give rise to
kinetochore and inter-kinetochore distance oscillations. The model by Joglekar
and Hunt [15] uses the thermodynamic Hill sleeve model [28] for the MT-
kinetochore connection, which assumes equally spaced rigid linkers that
diffuse on the discrete MT lattice. Shtylla and Keener [17, 18] combine a
continuous version of the Hill sleeve model with a negative chemical feedback
between the force at the MT-kinetochore interface and the depolymerization
rate. In Hill sleeve models there is no effect of MT insertion and, thus,
force onto the MT dynamic instability, i.e., on catastrophe and rescue rates.
The Hill sleeve can transmit pulling forces onto the kinetochore up to a
critical force above which MTs pull out of the sleeve [15], and there is
evidence that the Hill sleeve exhibits catch-bond-like behavior [29]. More
recent studies show that the kinetochore is not rigid, as supposed in the Hill
sleeve model, but should be viewed as a flexible framework [30]. Moreover,
Ndc80 fibrils have been suggested as main force transmitter [31, 4, 32], which
motivated Keener and Shtylla to modify their Hill sleeve model by replacing
the rigid attachment sites with elastic linkers and allowing for a force
feedback onto MT depolymerization [33]. However, sleeve models remain
speculative as electron microscopy has not yet revealed appropriate structures
[34, 35]. Civelekoglu-Scholey et al. [16] proposed a model in which MTs and
kinetochores are linked by motor proteins (dyneins) walking towards the MT
minus end; these links have no catch-bond-like behavior. The links are assumed
to be able to transmit tension onto MTs that promotes MT rescue. In [21] no
explicit linkers are introduced but permanent MT-kinetochore links are assumed
that can transmit both pulling and pushing forces onto MTs. As the exact
nature of MT-kinetochore linking structures is not known, a model of the MT-
kinetochore linkage as a generic elastic structure seems reasonable, as in
recent models where the MTs are linked to the kinetochore via (visco-)elastic
springs [19, 20]. The MT-kinetochore bond can be modeled as a catch bond, and
the elastic linkers also transmit forces back onto the MT allowing for a force
feedback onto MT dynamics as it has been measured in [2].

In the model of Shtylla and Keener [17], MTs that are attached to the same
kinetochore share the force from the cohesin bond equally and exhibit
synchronous dynamics. The last assumption is contradictory to the experimental
observation that one kinetochore MT ensemble does not coherently
(de)polymerize but always consists of a mixture of both states [36, 37]. Klemm
et al. [21] take into account this observation by dividing each MT ensemble
into a growing and a shrinking sub-ensemble, but also make the strong
assumption of equal force sharing between the MTs within each sub-ensemble.
All other models allow for individual MT dynamics and for different forces
between MTs depending on the distances of MTs to the kinetochore.

The main mechanism for oscillations differs between models depending on the
main antagonistic AP-directed force that switches a depolymerizing P-directed
ensemble back into AP-directed polymerization. Switching can be triggered by
the AP-directed force that the other ensemble can exert via the cohesin spring
during depolymerization and by AP-directed PEFs if MT catastrophes are
suppressed or rescues promoted under tension. In the model by Joglekar and
Hunt [15] AP-directed PEFs are essential for switching. Civelekoglu-Scholey et
al. [16] proposed a model in which force is transmitted by motor proteins. By
variation of the model parameters they were able to reproduce a wide range of
chromosome behavior observed in different cell types. In this model, a
depolymerizing P-directed ensemble switches because tension in the cohesin
spring and PEFs promote rescue events. A modified model [19] uses viscoelastic
catch bonds and accounts for the observation that in PtK1 cells only
chromosomes in the center of the metaphase plate exhibit directional
instability [11]. They explain this dichotomy with different distributions of
PEFs at the center and the periphery of the metaphase plate. In the model by
Shtylla and Keener [17, 18] MT catastrophe-like events are only triggered by a
chemical feedback such that kinetochore oscillations become coupled to
oscillations of the chemical negative feedback system: AP-directed MT
polymerization exerts pushing forces onto the kinetochore but triggers
switching into a depolymerizing state, and MT depolymerization exerts
P-directed pulling forces and triggers switching back into a polymerizing
state.

Whereas in [15, 16, 19] AP-directed PEFs are present and in the model by
Joglekar and Hunt [15] also essential for realistic kinetochore oscillations,
Banigan et al. [20] presented a minimal model with simple elastic linkers and
neglecting PEFs. Referring to experiments with budding yeast kinetochores [2],
they modeled MT dynamics with force-dependent velocities, catastrophe and
rescue rates. In this model, kinetochore oscillations arise solely from the
collective behavior of attached MTs under force and the resulting interplay
between P-directed depolymerization forces and AP-directed polymerization
forces of the opposing MT ensembles. Force-dependent velocities, catastrophe
and rescue rates are essential to trigger switching of kinetochore motion and
oscillations in this model. MTs can exert pushing forces such that it is
unclear to what extent the oscillation mechanism remains functional if pushing
forces are absent as suggested experimentally. Also the recent model by Klemm
et al. [21], which aims to describe kinetochore dynamics in fission yeast,
does not rely on PEFs. It uses a permanent MT-kinetochore bond and
oscillations result from the interplay between MT depolymerization and
polymerization forces via force-dependence in MT dynamics; also in this model
MTs can exert pushing forces. Moreover, the model makes the strong assumption
of equal force sharing between all growing or shrinking MTs attached to a
kinetochore. The model also includes kinesin-8 motors that enhance the
catastrophe rate and have a centering effect on the chromosome positions.

Table 1: Overview of assumptions of models exhibiting stochastic chromosome oscillations. In the referred sections we discuss how poleward flux, PEFs and the absence of pushing forces affect kinetochore dynamics in the model used for this work. | linker | catch | equal | force-dep. | MT |  | pole-
---|---|---|---|---|---|---|---
Ref. (year) | model | bonds | force | MT | pushing | PEFs | ward
|  |  | sharing | rescue/cat. | forces |  | MT flux
Joglekar [15] (2002) | Hill sleeve |  | no | no | no | yes | no
Civelekoglu [16] (2006) | motor | no | no | yes | yes | yes | yes
Civelekoglu [19] (2013) | viscoelastic | yes | no | yes | no | yes | no
Shtylla [17, 18] (2010) | Hill sleeve |  | yes | no | yes | yes | no
Banigan [20] (2015) | elastic | yes | no | yes | yes | no | yes
Klemm [21] (2018) | permanent |  | yes | yes | yes | no | no
this work | elastic | yes | no | yes |  sec. 8 |  sec. 7 |  sec. 6

In all Refs. [15, 16, 17, 18, 19, 20, 21] a sufficient set of ingredients is
given for the respective model to exhibit oscillations including a specific
choice of parameter values. It is much harder to give necessary conditions and
parameter ranges for oscillations, which means to obtain quantitative bounds
on model parameters, than to give a sufficient set of conditions. This is the
aim of the present paper within a model that starts from the minimal model by
Banigan et al. and generalizes this model in several respects in later
sections, see table 1. In this way we discuss the complete inventory of
possible forces acting on the kinetochore and its influence on oscillations.

It is also difficult to trace the actual mechanism leading to oscillations. An
essential part in our quantitative analysis is a mean-field solution of the
one-sided minimal model of Banigan et al. [20], where a single kinetochore
under force is connected to one or several MTs that experience length-
dependent individual loads and feature force-dependent dynamics. Force-
velocity relations for a single kinetochore, which is connected to one or
several MTs have been investigated previously based on a sleeve-like MT-
kinetochore interface [17, 18, 33, 29]. Here, we can derive an analytical
solution of the one-sided minimal model from a novel mean-field approach. For
this purpose, we start from the Fokker-Planck equations for the length
distribution of the MT-kinetochore linkers. The only mean-field approximation
is to neglect stochastic velocity fluctuations of the attached kinetochore.
Our solution clearly shows that the force feedback of linkers onto the MT
depolymerization dynamics via catch (or permanent) bonds is essential for a
bistable force-velocity relation within the minimal model. Moreover, the
stationary state solution allows us to quantify the parameter range for a
bistability in the parameter plane of MT-kinetochore linker stiffness and MT
numbers.

By interpreting the force-velocity relation as phase space diagram for the
two-sided model as in [20], we show that bistability in the one-sided model is
a necessary condition for kinetochore oscillations in the two-sided model.
Beyond that, we are able (1) to quantify an oscillatory regime, in which
kinetochores exhibit directional instability, in the parameter plane of linker
stiffness and MT numbers predicting that linkers have to be sufficiently
stiff; (2) to describe kinetochore motion in this oscillatory regime,
calculate frequencies which agree with in vivo measurements [11] and to
explain frequency doubling of breathing compared to single kinetochore
oscillations; (3) to describe kinetochore motion in the non-oscillatory regime
as fluctuations around a fixed point; (4) to show that high poleward flux
velocities move the system out of the oscillatory regime and thereby explain
why directional instability has been observed in mitotic vertebrate cells but
not in Drosophila embryos and Xenopus eggs; (5) to show that polar ejection
forces reduce the amplitude of oscillations, induce a phase shift between
sister kinetochores and are necessary to align the chromosome at the spindle
equator; (6) to derive as necessary condition for oscillations that either MTs
must be able to apply pushing forces on the kinetochore or a catastrophe has
to be induced with increased catastrophe rate when a MT reaches the
kinetochore. All these results are validated by stochastic simulations; (7) to
provide a set of model parameters that reproduce experimental results for
kinetochore oscillations in PtK1 cells quantitatively.

In particular, we quantify lower bounds for linker stiffnesses that allow
oscillations, whose value depends on the behavior of MTs growing against the
kinetochore. If kinetochore MTs can exert pushing forces, we find oscillations
for linker stiffnesses
$>$16\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$; also
if MT catastrophes are induced upon reaching the kinetochore, we find
oscillations in a similar range of linker stiffnesses. These constraints
provide useful additional information on MT-kinetochore linkers whose
molecular nature is not completely unraveled up to now.

##  2 Mitotic spindle model

We use a one-dimensional model of the mitotic spindle (figure 1(a)), similar
to the model from [20]. The $x$-coordinate specifies positions along the one-
dimensional model, and we choose $x=0$ to be the spindle equator. The spindle
model contains a single chromosome represented by two kinetochores, which are
linked by a spring with stiffness $c_{\mathrm{k}}$ and rest length $d_{0}$.
Two centrosomes margin the spindle at $\pm x_{\mathrm{c}}$. From each
centrosome a constant number $M$ of MTs emerges with their plus ends directed
towards the spindle equator. Each MT exhibits dynamic instability [25] and
attaches to and detaches from the corresponding kinetochore stochastically.
Attached MTs are connected to the kinetochore by a linker, which we model as
Hookean polymeric spring with stiffness $c$ and zero rest length. This spring
exerts a force $F_{\mathrm{mk}}=-c(x_{\mathrm{m}}-X_{\mathrm{k}})$ on each MT,
and each MT exerts a counter force $-F_{\mathrm{mk}}$ on the kinetochore,
where $X_{\mathrm{k}}$ and $x_{\mathrm{m}}$ are kinetochore and MT position.

Figure 1:  One-dimensional model of the mitotic spindle. (a) Two-sided model:
$M$ MTs arise from each centrosome and can attach to / detach from the
corresponding kinetochore. (b) One-sided model: Left half of two-sided model.
The cohesin bond is replaced by the external force $F_{\mathrm{ext}}$. MTs are
not confined by a centrosome and permanently attached to the kinetochore. MT-
kinetochore distances $x_{i}=x_{\mathrm{m,}i}-X_{\mathrm{k}}$ are the only
relevant coordinates.

In the following we define all MT parameters for MTs in the left half of the
spindle model; for MTs in the right half position velocities $v$ and forces
$F$ have opposite signs. In the left half, tensile forces on the MT-
kinetochore link arise for $X_{\mathrm{k}}>x_{\mathrm{m}}$ and pull the MT in
the positive $x$-direction, $F_{\mathrm{mk}}>0$. In [2], the velocities of MT
growth $v_{\mathrm{m}+}$ and shrinkage $v_{\mathrm{m}-}$ as well as the rates
of catastrophe $\omega_{\mathrm{c}}$, rescue $\omega_{\mathrm{r}}$ and
detachment $\omega_{\mathrm{d}\pm}$ have been measured while MTs were attached
to reconstituted yeast kinetochores. They can all be described by an
exponential dependence on the force $F_{\mathrm{mk}}$ that acts on the MT plus
end:

$\displaystyle
v_{\mathrm{m}\pm}=v^{0}_{\pm}\exp\left(\frac{F_{\mathrm{mk}}}{F_{\pm}}\right),\qquad\omega_{i}=\omega^{0}_{i}\exp\left(\frac{F_{\mathrm{mk}}}{F_{i}}\right),$
(1)

(for $i=\mathrm{r},\mathrm{c},\mathrm{d}+,\mathrm{d}-$) with
$F_{+},~{}F_{\mathrm{r}},~{}F_{\mathrm{d+}}>0$ and
$F_{-},~{}F_{\mathrm{c}},~{}F_{\mathrm{d-}}<0$ for the characteristic forces,
because tension ($F_{\mathrm{mk}}>0$) enhances growth velocity, rescue and
detachment of a growing MT, while it suppresses shrinking velocity,
catastrophe and detachment of a shrinking MT (note that we use signed
velocities throughout the paper, i.e., $v_{\mathrm{m}-}<0$ and
$v_{\mathrm{m}+}>0$). Suppression of detachment of shrinking MTs is the catch
bond property of the MT-kinetochore link. The attachment rate is assumed to
follow a Boltzmann distribution,

$\displaystyle\omega_{\mathrm{a}}=\omega^{0}_{\mathrm{a}}\exp\left(\frac{c(X_{\mathrm{k}}-x_{\mathrm{m}})^{2}}{2k_{\mathrm{B}}T}\right),$
(2)

according to the MT-kinetochore linker spring energy.

The kinetochore motion is described by an overdamped dynamics,

$\displaystyle
v_{\mathrm{k}}\equiv\dot{X}_{\mathrm{k}}=\frac{1}{\gamma}\left(F_{\mathrm{kk}}+F_{\mathrm{km}}\right),$
(3)

with the friction coefficient $\gamma$, and the forces $F_{\mathrm{kk}}$ and
$F_{\mathrm{km}}=-\sum_{\rm att.~{}MTs}F_{\mathrm{mk}}$ originating from the
cohesin bond between kinetochores and the MT-kinetochore linkers of all
attached MTs, respectively.

We perform simulations of the model by integrating the deterministic equations
of motion for MTs ($\dot{x}_{\mathrm{m,i}}=v_{\mathrm{m}\pm,i}$ for
$i=1,...,M$) and kinetochores (eq. (3)) and include stochastic switching
events between growth and shrinking as well as for attachment and detachment
to the kinetochore for each MT. For integration we employ an Euler algorithm
with a fixed time step $\Delta t\leq 10^{-3}\,{\rm s}$ which is small enough
to ensure $\omega_{i}\Delta t\ll 1$ for all stochastic switching events (see
table 2). The algorithm is described in the supplementary material in more
detail. We use parameter values from experiments as listed in table 2.

Table 2: Model parameters.

  * Transition parameters | $\omega_{i}$ |  $\omega_{i}^{0}$ ($\mathrm{s}^{-1}$) |  $F_{i}$ ($\mathrm{p}\mathrm{N}$) |
---|---|---|---|---
catastrophe | $\omega_{\mathrm{c}}$ | $0.0019$ | $-2.3$ | [2]
rescue | $\omega_{\mathrm{r}}$ | $0.024$ |  -$6.4$ | [2]
detachment in growing state | $\omega_{\mathrm{d}+}$ | $0.000\,11$ |  -$3.8$ | [2]
detachment in shrinking state | $\omega_{\mathrm{d}-}$ | $0.035$ | $-4.0$ | [2]
attachment rate | $\omega_{\mathrm{a}}$ | $1.0$ | estimated
Velocity parameters | $v_{\mathrm{m}\pm}$ |  $v_{\pm}^{0}$ ($\mathrm{n}\mathrm{m}\mathrm{s}^{-1}$) |  $F_{\pm}$ ($\mathrm{p}\mathrm{N}$) |
growth | $v_{\mathrm{m}+}$ |  -00$5.2$ |  -$8.7$ | [2]
shrinking | $v_{\mathrm{m}-}$ | $-210.0$ | $-3.2$ | [2]
Other parameters | Symbol | Value |  |
cohesin bond stiffness | $c_{\mathrm{k}}$ | $20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$ | estimated
cohesin bond rest length | $d_{0}$ | $1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ | [7]
centrosome position | $x_{\mathrm{c}}$ | $6.8\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ | [10]
friction coefficient | $\gamma$ | $1\text{\,}\mathrm{pN}\text{\,}\mathrm{s}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$ | estimated
thermal energy | $k_{\mathrm{B}}T$ | $4\text{\,}\mathrm{pN}\text{\,}\mathrm{nm}$ | estimated

We start with the investigation of the minimal model, i.e. neglecting poleward
flux and PEFs and using the same simple spring model for the MT-kinetochore
linker as Banigan et al. where the MT plus ends are able to “overtake” the
kinetochore ($x_{\mathrm{m}}>X_{\mathrm{k}}$, again for MTs in the left half
of the spindle) and thereby exert pushing forces $F_{\mathrm{km}}>0$ on the
kinetochore (which could be interpreted as a compression of the MT-kinetochore
linker). Later, we will generalize the minimal model as described in the
introduction, see table 1. In a first step, we add poleward MT flux, which
describes a constant flux of tubulin from the plus-ends towards the spindle
pole [24], by shifting the MT velocities $v_{\mathrm{m}\pm}$. PEFs, which push
the kinetochore away from the pole [27], will be included in a second step as
external forces, which depend on the absolute positions of the kinetochores.
Finally, we will take account of the hypothesis that MTs are not able to apply
pushing forces on the kinetochore [7, 26] by modifying the model such that the
growth of a MT is stalled or that the MT undergoes a catastrophe when it
reaches the kinetochore.

At the centrosome, MTs are confined: It is reasonable to assume that they
undergo a forced rescue and detach from the kinetochore if they shrink to zero
length. If the mean distance of MTs from the spindle equator is sufficiently
small, $|\langle x_{\mathrm{m}}\rangle|\ll|x_{\mathrm{c}}|$, we can also
consider the MTs as unconfined ($|x_{\mathrm{c}}|\rightarrow\infty$). Then
both MT and kinetochore dynamics solely depend on their relative distances and
not on absolute positions, which simplifies the analysis.

##  3 Mean-field theory for bistability in the one-sided model

We first examine the one-sided model of Banigan et al. [20], which only
consists of the left half of the two-sided model with an external force
applied to the kinetochore (figure 1(b)). In simulations of this one-sided
spindle model, kinetochore movement exhibits bistable behavior as a function
of the applied force, i.e., within a certain force range there are two
metastable states for the same external force: In one state the MTs
predominantly grow and the kinetochore velocity is positive while in the other
state the kinetochore has a negative velocity as a consequence of mainly
shrinking MTs. It depends on the history which of these two states is assumed:
When the system enters the bistable area in consequence of a force change, the
kinetochore velocity will maintain its direction (following its current
metastable branch) until the force is sufficiently large that the system
leaves the bistable area again (the current metastable branch becomes
unstable). Later we will show that this hysteresis of the one-sided model can
explain stochastic chromosome oscillations in metaphase if two one-sided
models are coupled in the full two-sided model.

In the following, we present a Fokker-Planck mean-field approach that lets us
derive bistability analytically and obtain constraints for the occurrence of
bistability. We obtain a system of Fokker-Planck equations (FPEs) for the $M$
MT-kinetochore distances $x_{i}\equiv x_{\mathrm{m,}i}-X_{\mathrm{k}}$
($i=1,...,M$) and decouple the MT dynamics in a mean-field approximation,
which neglects kinetochore velocity fluctuations.

We make two assumptions. First we assume that all $M$ MTs are always attached
to the kinetochore. Because the MT-kinetochore links are catch bonds this
assumption is equivalent to assuming that these links are predominantly under
tension. We will check below by comparison with numerical simulations to what
extent this assumption can be justified. Secondly, we neglect that MTs are
confined by a centrosome. Then, as mentioned above, the only relevant
coordinates are the relative MT-kinetochore distances $x_{i}$, which measure
the extension of the $i$-th linker.

The MTs are coupled because they attach to the same kinetochore: each MT
experiences a force $F_{\mathrm{mk},i}=-cx_{i}$ from the elastic linker to the
kinetochore, which is under tension (compression) for $x_{i}<0$ ($x_{i}>0$);
the kinetochore is subject to the total counter force
$F_{\mathrm{km}}=c\sum_{i}x_{i}$. Therefore, the kinetochore velocity
$v_{\mathrm{k}}$ is a stochastic variable depending on all distances $x_{i}$,
on the one hand, but determines the velocities
$\dot{x}_{i}=v_{\text{m}\pm}(x_{i})-v_{\text{k}}$ of MTs relative to the
kinetochores, on the other hand. The equations can be decoupled to a good
approximation because the one-sided system assumes a steady state with an
approximately stationary kinetochore velocity $v_{\mathrm{k}}$ after a short
time (rather than, for example, a cooperative oscillation as for an MT
ensemble pushing against an elastic barrier [38]). In our mean-field
approximation we then assume a constant kinetochore velocity
$v_{\mathrm{k}}\equiv\langle v_{\mathrm{k}}\rangle$ and neglect all stochastic
fluctuations around this mean. This mean value is determined by the mean
linker extension $\langle x\rangle$ via

$v_{\mathrm{k}}=\frac{1}{\gamma}\left(F_{\mathrm{ext}}+cM\langle
x\rangle\right).$ (4)

Fluctuations around the mean value are caused by fluctuations of the force
$F_{\mathrm{km}}=c\sum_{i}x_{i}$ around its mean $\langle
F_{\mathrm{km}}\rangle=Mc\langle x\rangle$, which become small for large $M$
(following the central limit theorem).

If $v_{\mathrm{k}}$ is no longer a stochastic variable, the dynamics of the
MT-kinetochore extensions $x_{i}$ decouple. Then, the probability distribution
for the $M$ extensions $x_{i}$ factorizes into $M$ identical factors
$p_{\pm}(x_{i},t)$, where $p_{\pm}(x,t)$ are the probabilities to find one
particular MT in the growing ($+$) or shrinking ($-$) state with a MT-
kinetochore linker extensions $x$. We can derive two FPEs for the dynamical
evolution of $p_{\pm}(x,t)$,

$\displaystyle\partial_{t}p_{+}(x,t)$
$\displaystyle=-\omega_{\mathrm{c}}(x)p_{+}(x,t)+\omega_{\mathrm{r}}(x)p_{-}(x,t)-\partial_{x}\big{(}v_{+}(x)p_{+}(x,t)\big{)},$
(5) $\displaystyle\partial_{t}p_{-}(x,t)$
$\displaystyle=\phantom{-}\omega_{\mathrm{c}}(x)p_{+}(x,t)-\omega_{\mathrm{r}}(x)p_{-}(x,t)-\partial_{x}\big{(}v_{-}(x)p_{-}(x,t)\big{)},$
(6)

where $v_{\pm}(x)$ denotes the relative velocity of MT and kinetochore,

$\displaystyle v_{\pm}(x)\equiv
v_{\text{m}\pm}(x)-v_{\text{k}}=v_{\pm}^{0}\exp\left(-\frac{cx}{F_{\pm}}\right)-v_{\text{k}},$
(7)

and
$\omega_{\mathrm{c},r}(x)=\omega_{\mathrm{c},r}^{0}\exp\left(-{cx}/{F_{\mathrm{c},r}}\right)$.
The velocity $v_{\mathrm{k}}$ is no longer stochastic but self-consistently
determined by (4). We note that these FPEs are equivalent to single MT FPEs
with position-dependent velocities, catastrophe and rescue rates [39, 40, 41,
42].

We will now obtain the force-velocity relation of the whole MT ensemble by
first solving the FPEs (5) and (6) in the stationary state
$\partial_{t}p_{\pm}(x,t)=0$ and then calculating the mean linker extension
$\langle x\rangle$ for given kinetochore velocity $v_{\mathrm{k}}$ using the
stationary distribution $p_{\pm}(x)$. The external force that is necessary to
move the kinetochore with velocity $v_{\mathrm{k}}$ then follows from (4),

$F_{\mathrm{ext}}=\gamma v_{\text{k}}-cM\langle x\rangle(v_{\mathrm{k}}).$ (8)

The MT-kinetochore distance $x$ is limited to a maximal or a minimal value
$x_{\mathrm{max}}$ or $x_{\mathrm{min}}$ for a given kinetochore velocity
$v_{\mathrm{k}}>0$ or $<0$, respectively, see table 3. These limiting values
are reached if the relative MT-kinetochore velocities vanish after the linker
extension $x$ has adjusted: First we consider $v_{k}<0$ and a shrinking MT. If
we start with a compressed linker ($x>0$), the MT starts to shrink fast, the
compression is reduced and the linker may get under tension ($x<0$) because
the relative velocity is negative, $\dot{x}=v_{-}(x)<0$. The MT-kinetochore
distance $x$ continues to decrease until $\dot{x}=v_{-}(x_{\mathrm{min}})=0$
in (7), where the shrinking velocity of the MTs is the same as the prescribed
kinetochore velocity ($v_{\mathrm{m},-}=v_{\mathrm{k}}$). Further shrinking to
$x<x_{\mathrm{min}}$ is not possible but distances $x>x_{\mathrm{min}}$ can
always be reached if MTs are rescued. If $v_{k}<0$ and the MT grows, on the
other hand, there is no upper bound on $x$, as the relative velocity
$\dot{x}=v_{+}(x)$ is always positive; $x$ starts to grow into the compressive
regime $x>0$ and continues to grow without upper bound (for very large
compressive linker extensions, MT growth is suppressed, but the kinetochore
still moves such that $v_{+}(x)\approx-v_{\mathrm{k}}>0$). Analogously, if
$v_{k}>0$ and MTs grow, $x$ grows until $\dot{x}=v_{+}(x_{\mathrm{max}})=0$,
and smaller distances can be reached by catastrophe but there is no lower
bound on $x$ for shrinking MTs. Linker extensions $x_{\mathrm{max}}$
($x_{\mathrm{min}}$) are reached as stationary states if catastrophes
(rescues) are suppressed (for example, because of large forces), such that MTs
can grow (shrink) for sufficiently long times. If the external force
$F_{\mathrm{ext}}$ is prescribed rather than a kinetochore velocity, all MTs
reach a stationary state with the same velocity $\tilde{v}_{\pm}$ given by
(8), $F_{\mathrm{ext}}=\gamma\tilde{v}_{\pm}-cMx_{\mathrm{max,min}}$. In this
stationary state, both MT-tips and kinetochore move with the same velocity

$\displaystyle\tilde{v}_{\pm}\equiv\frac{MF_{\pm}}{\gamma}\,W\left(\frac{\gamma
v^{0}_{\pm}}{MF_{\pm}}\exp\left(\frac{F_{\mathrm{ext}}}{MF_{\pm}}\right)\right),$
(9)

where $W()$ denotes the Lambert-W function (defined by $x=W(x)e^{W(x)}$).

Table 3:  Maximal or a minimal value $x_{\mathrm{max}}$ or $x_{\mathrm{min}}$
of the stationary linker extension distribution $p(x)$ from conditions
$v_{-}(x_{\mathrm{min}})=0$ and $v_{+}(x_{\mathrm{max}})=0$. The distance
$x_{\mathrm{min}}$ ($x_{\mathrm{max}}$) is a function of the prescribed
kinetochore velocity $v_{\mathrm{k}}$ and has to be specified separately
depending on the direction of $v_{\mathrm{k}}$; $x_{\mathrm{min}}$
($x_{\mathrm{max}}$) is approached if the MTs shrink (grow) for a sufficiently
long time.

  * | MT shrinks | MT grows
---|---|---
$v_{\mathrm{k}}>0$ | $v_{-}(x)<-v_{\mathrm{k}}\;{\rm always}$ | $v_{+}(x)>0\;\text{for}\;x<x_{\mathrm{max}}$
| $x_{\mathrm{min}}=-\infty$ | $x_{\mathrm{max}}=({F_{+}}/{c})\ln\left({v^{0}_{+}}/{v_{\mathrm{k}}}\right)$
$v_{\mathrm{k}}<0$ | $v_{-}(x)<0\;\text{for}\;x>x_{\mathrm{min}}$ | $v_{+}(x)>v_{\mathrm{k}}\;{\rm always}$
| $x_{\mathrm{min}}=({F_{-}}/{c})\ln\left({v^{0}_{-}}/{v_{\mathrm{k}}}\right)$ | $x_{\mathrm{max}}=\infty$
$v_{\mathrm{k}}=0$ | $v_{-}(x)<0\;{\rm always}$ | $v_{+}(x)>0\;{\rm always}$
| $x_{\mathrm{min}}=-\infty$ | $x_{\mathrm{max}}=\infty$

In the complete absence of stochastic switching between growth and shrinking
by catastrophes and rescues, the MT ensemble reaches stationary states with
peaked distributions $p_{+}(x)\propto\delta(x_{\mathrm{max}}-x)$ and
$p_{-}(x)\propto\delta(x-x_{\mathrm{min}})$. Stochastic switching shifts and
broadens these peaks, and the FPEs (5) and (6) lead to a distribution
$p_{\pm}(x,t)$ of linker extensions $x$ in the growing and shrinking states
with statistical weight $p_{\pm}(x,t)>0$ in the whole interval
$x_{\mathrm{min}}\leq x\leq x_{\mathrm{max}}$. At the boundaries
$x_{\mathrm{min}}$ and $x_{\mathrm{max}}$ of this interval, the probability
current density

$\displaystyle j(x,t)\equiv v_{+}(x,t)p_{+}(x,t)+v_{-}(x,t)p_{-}(x,t)$ (10)

has to vanish. Furthermore, in any stationary state
($\partial_{t}p_{\pm}(x,t)=0$) the current density is homogeneous, as can be
seen by summing up the FPEs (5) and (6):

$\displaystyle
0=\partial_{x}(v_{+}(x)p_{+}(x)+v_{-}(x)p_{-}(x))=\partial_{x}j(x).$ (11)

Together with $j=0$ at the boundaries this implies that $j=0$ everywhere in a
steady state. The resulting relation $v_{+}(x)p_{+}(x)=-v_{-}(x)p_{-}(x)$ can
be used to reduce the stationary FPEs to a single ordinary differential
equation with the solution [41]

$\displaystyle
p_{\pm}(x)=\frac{\pm\mathcal{N}}{v_{\pm}(x)}\exp\left(-\int\left(\frac{\omega_{\text{c}}(x)}{v_{+}(x)}+\frac{\omega_{\text{r}}(x)}{v_{-}(x)}\right)\mathrm{d}x\right)$
(12)

for the stationary distribution of linker extensions $x$ in the growing and
shrinking states. The normalization constant $\mathcal{N}$ must be chosen so
that the overall probability density $p(x)\equiv p_{+}(x)+p_{-}(x)$ satisfies
$\int_{x_{\mathrm{min}}}^{x_{\mathrm{max}}}p(x)\mathrm{d}x=1$. Obviously,
$p_{\pm}(x)=0$ for $x>x_{\mathrm{max}}$ and $x<x_{\mathrm{min}}$. The
stationary probability densities $p_{\pm}(x)$ from (12) can then be used to
calculate the mean distance $\langle x\rangle$ as a function of the
kinetochore velocity $v_{k}$, which enters through (7) for $v_{\pm}(x)$. The
integral in the exponent in (12) as well as the normalization can be evaluated
numerically to obtain an explicit $\langle x\rangle(v_{\mathrm{k}})$-relation,
which is shown in figure 2(a).

Figure 2:  Mean-field results compared to stochastic simulations of the one-
sided model. (a) The master curve $\langle x\rangle(v_{\mathrm{k}})$ from the
mean-field approach (red line) agrees with simulation results for different
MT-numbers $M=5,20,50,200$. The dashed lines mark $x_{\rm
min,max}(v_{\mathrm{k}})$ from table 3. We run simulations with constant
external forces and average over 80 simulations for each force. Initially, the
MT-kinetochore distance is either $x_{\mathrm{min}}$ or $x_{\mathrm{max}}$
while all MTs grow or shrink with velocity $\tilde{v}_{\pm}$, respectively.
The system then enters a (meta-)stable state, in which we measure the mean
kinetochore velocity and MT-kinetochore distances. The marker size depicts the
time the system rests in this state on average, which is a measure for its
stability (maximum marker size corresponds to
$t_{\mathrm{rest}}\geq$1000\text{\,}\mathrm{s}$$). As predicted, the mean-
field approach turns out to be correct in the limit of many MTs, and in this
limit the $\langle x\rangle(v_{\mathrm{k}})$-relation is independent of the
MT-number $M$. (b) Resulting force-velocity relations for different MT-numbers
$M=5,20,50,200$. The dashed lines show the large velocity limit
$v_{\mathrm{k}}\approx\tilde{v}_{\pm}$ given by (9). We used a linker
stiffness of $c=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$ both in (a) and (b).

At this point it should be noticed that in the mean-field theory the $\langle
x\rangle(v_{\mathrm{k}})$-relation is independent of the MT number $M$.
Therefore, we call it master curve henceforth. In figure 2(a) we compare the
mean-field theory result to stochastic simulations and find that the mean-
field approach becomes exact in the limit of large $M$, where fluctuations in
the kinetochore velocity around its mean in (4) can be neglected.

The master curve is a central result and will be the basis for all further
discussion. Together with the force-balance (8) on the kinetochore, the master
curve will give the force-velocity relation for the MT-kinetochore system. A
positive slope of the master curve, as it can occur for small
$v_{\mathrm{k}}\approx 0$ (see figure 2(a)), gives rise to an instability of
the MT-kinetochore system: Then, a positive kinetochore velocity fluctuation
$\delta v_{\mathrm{k}}>0$ leads to a MT-kinetochore linker compression
$\delta\langle x\rangle>0$. According to the force-balance (8), a compression
$\delta\langle x\rangle>0$ puts additional forward-force on the kinetochore
leading to a positive feedback and further increase $\delta v_{\mathrm{k}}>0$
of the kinetochore velocity. This results in an instability, which will
prevent the system to assume mean linker extensions $\langle x\rangle$ in this
unstable regime. This is confirmed by stochastic simulation results in figure
2(a), which show that the unstable states are only assumed transiently for
very short times. Therefore, occurrence of a positive slope in the master
curve in figure 2(a) is the essential feature that will give rise to
bistability in the one-sided model and, finally, to oscillations in the full
two-sided model.

Now we want to trace the origin of this instability for small
$v_{\mathrm{k}}\approx 0$. If the MTs are growing (shrinking) for a long time,
all linker extensions assume the stationary values $x\approx
x_{\mathrm{max}}(v_{\mathrm{k}})$ ($x\approx
x_{\mathrm{min}}(v_{\mathrm{k}})$) from table 3, where the MT-velocity adjusts
to the kinetochore velocity, $v_{\mathrm{k}}\approx v_{\text{m}\pm}(x)$. If
the kinetochore velocity increases in these states by a fluctuation (i.e.,
$\delta v_{\mathrm{k}}>0$), the MT-kinetochore linkers are stretched (i.e.,
$\delta x<0$), which slows the kinetochore down again resulting in a stable
response. This is reflected in the negative slopes of both $x_{\rm
max}(v_{\mathrm{k}})$ (for $v_{\mathrm{k}}>0$) and $x_{\rm
min}(v_{\mathrm{k}})$ (for $v_{\mathrm{k}}<0$). Because of constant stochastic
switching between catastrophes and rescues the mean linker extension exhibits
fluctuations about $x_{\rm max}$ and $x_{\rm min}$, but we expect also the
master curve $\langle x\rangle(v_{\mathrm{k}})$ to have a negative slope for a
wide range of velocities $v_{\mathrm{k}}$. Figure 2(a) shows that this is
actually the case for kinetochore velocities $v_{\mathrm{k}}$ around the
force-free growth or shrinking velocities $v^{0}_{\pm}$ of the MTs, i.e., if
the imposed kinetochore velocity $v_{\mathrm{k}}$ roughly “matches” the force-
free growing or shrinking MT velocity. Then a small mismatch can be
accommodated by small linker extensions $x$, which do not dramatically
increase fluctuations by triggering catastrophe or rescue events.

The situation changes for small negative or small positive values of the
kinetochore velocity around $v_{\mathrm{k}}\approx 0$. For
$v_{\mathrm{k}}\lesssim 0$, MT-kinetochore linkers develop logarithmically
growing large negative extensions $x_{\rm min}$ (see table 3) corresponding to
a slow kinetochore trailing fast shrinking MTs that strongly stretch the
linker. Likewise, for $v_{\mathrm{k}}\gtrsim 0$, MT-kinetochore linkers
develop logarithmically growing large positive extensions $x_{\rm max}$
corresponding to a slow kinetochore trailing fast growing MTs that strongly
compress the linker. Around $v_{\mathrm{k}}\approx 0$, the system has to
switch from large negative $x$ to large positive $x$ because the resulting
tensile force $F_{\mathrm{mk}}=-cx$ on the shrinking MT will destabilize the
shrinking state and give rise to MT rescue at least for $x<-F_{r}/c$.

Therefore, also the mean value $\langle x\rangle$ switches from negative to
positive values resulting in a positive slope of the master curve if the
stationary distributions $p_{-}(x)$ and $p_{+}(x)$ remain sufficiently peaked
around the linker extensions $x_{\mathrm{min}}$ and $x_{\mathrm{max}}$, also
in the presence of fluctuations by catastrophes and rescues. In the
supplementary material, we show that the stationary distributions assume a
power-law behavior $p_{+}(x)\propto(x_{\mathrm{max}}-x)^{\alpha_{+}}$
[$p_{-}(x)\propto(x-x_{\mathrm{min}})^{\alpha_{-}}$] around $x_{\mathrm{max}}$
[$x_{\mathrm{min}}$] for $v_{\mathrm{k}}>0$ [$v_{\mathrm{k}}<0$] with
exponents $\alpha_{\pm}$ that depend on the MT-kinetochore stiffness $c$ as
$\alpha_{\pm}+1\propto 1/c$ in the presence of fluctuations. It follows that
distributions are peaked (i.e., have a large kurtosis) and bistability emerges
if the MT-kinetochore linker stiffness $c$ is sufficiently large such that
deviations of the MT velocity from the kinetochore velocity become suppressed
by strong spring forces. This is one of our main results. We also find that
$\alpha_{\pm}+1\propto(|v_{\mathrm{k}}/v_{\pm}^{0}|)^{-1-|F_{\pm}/F_{\mathrm{c,r}}|}$
such that the distributions become also peaked around $x_{\mathrm{min,max}}$
in the limit of large velocities $|v_{\mathrm{k}}|$. Then the velocity
approaches $v_{\mathrm{k}}\approx\tilde{v}_{\pm}(F_{\mathrm{ext}})$ for a
prescribed external force such that $\tilde{v}_{\pm}$ from (9) represents the
large velocity and large force limit of the force-velocity relation of the
kinetochore (see figure 2(b)).

In the unstable regime around $v_{\mathrm{k}}\approx 0$, the linker length
distribution $p(x)$ is typically broad without pronounced peaks and has a
minimal kurtosis (as a function of $v_{\mathrm{k}}$) in the presence of
catastrophe and rescue fluctuations. In this regime the system assumes a state
with a heterogeneous stationary distribution of growing and shrinking MTs,
i.e., the total probabilities to grow or shrink become comparable, $\int
p_{+}(x)\mathrm{d}x\sim\int p_{-}(x)\mathrm{d}x$. If the kinetochore velocity
is increased, $\delta v_{\mathrm{k}}>0$, the system does not react by $\delta
x<0$, i.e., by increasing the average tension in the linkers in order to pull
MTs forward, but by switching MTs from the shrinking to the growing state (on
average), which then even allows to relax the average linker tension.

Using the force-balance (8) on the kinetochore, the master curve is converted
to a force-velocity relation for the MT-kinetochore system; the results are
shown in figure 2(b). The bistability in the master curve directly translates
to a bistability in the force-velocity relation of the MT ensemble, and we
obtain a regime with three branches of possible velocities for the same
external force. The upper and the lower branches agree with our simulation
results and previous simulation results in [20], and our mean-field results
become exact in the limit of large $M$, see figure 2(b). These branches
correspond to the two stable parts of the master curve with negative slope,
that are found for kinetochore velocities $v_{\mathrm{k}}$ roughly matching
the force-free growth or shrinking velocities $v^{0}_{\pm}$ of the MTs. The
mid branch corresponds to the part of the master curve with positive slope,
where the system is unstable. Also figure 2(b) demonstrates that this
instability is confirmed by stochastic simulations results.

Finally, we note that a simpler theoretical approach, where it is assumed that
all linkers assume identical extensions $x_{i}\approx x$ and all attached MTs
are in the same state (growing or shrinking), is exact for a single MT ($M=1$)
by definition but not sufficient to obtain a bistable force-velocity relation
for MT ensembles ($M>1$) (see supplementary material). The same assumption of
identical MT positions has already been used to study an ensemble of MTs that
are connected to the same kinetochore via Hill sleeve like linkers [17, 29].
The model of Klemm et al. [21] divides each MT ensemble into a growing and a
shrinking sub-ensemble, and assumes equal load sharing only between MTs within
each sub-ensemble. We can show that, together with a force-sensitive rescue
force, this is sufficient to obtain a bistable force-velocity relation in a
corresponding one-sided model.

##  4 Bistability gives rise to oscillations in the two-sided model

As already worked out by Banigan et al. [20], the bistability in the force-
velocity relation of the one-sided MT ensemble can be considered to be the
cause for stochastic oscillations in the two-sided model. Each ensemble can be
either on the lower branch of the force-velocity relation, where it mainly
depolymerizes and exerts a P-directed pulling force ($v_{\mathrm{k}}<0$) or on
the upper branch where it mainly polymerizes and exerts an AP-directed pushing
force ($v_{\mathrm{k}}>0$). The external force in the one-sided model is a
substitute for the spring force
$F_{\mathrm{kk}}=c_{\text{k}}\left(X_{\mathrm{k,r}}-X_{\mathrm{k,l}}-d_{0}\right)$
of the cohesin bond in the full model with a stiffness $c_{\mathrm{k}}$ and
rest length $d_{0}$, see table 2. Since the cohesin force is a linear function
of the inter-kinetochore distance, the force-velocity relation can be treated
as distance-velocity (phase space) diagram for the two kinetochores (see
figure 3(a)), where both kinetochores move as points on the force-velocity
relation. The cohesin bond always affects the two kinetochores in the same way
because action equals reaction: if the cohesin spring is stretched, both
kinetochores are pulled away from their pole (AP), if it is compressed, both
kinetochores are pushed polewards (P). Thus, the kinetochores always have the
same position on the $F_{\mathrm{kk}}$-axis in the
$F_{\mathrm{kk}}$-$v_{\mathrm{k}}$ diagram in figure 3(a), if
$F_{\mathrm{kk}}$ on the horizontal axis is defined as the force on the
kinetochore in AP-direction (i.e., $F_{\mathrm{kk,l}}\equiv F_{\mathrm{kk}}$
and $F_{\mathrm{kk,r}}\equiv-F_{\mathrm{kk}}$ for the left/right kinetochore).
Likewise, we define $v_{\mathrm{k}}$ on the vertical axis as the velocity in
AP-direction (i.e., $v_{\mathrm{k,l}}\equiv\dot{X}_{\mathrm{k,l}}$ and
$v_{\mathrm{k,r}}\equiv-\dot{X}_{\mathrm{k,r}}$ for the left/right
kinetochore). The upper/lower stable branch of the force-velocity relation is
denoted by $v^{\pm}_{\mathrm{k}}(F_{\mathrm{kk}})$. Typically, a kinetochore
on the upper (lower) branch has $v^{+}_{\mathrm{k}}>0$
($v^{-}_{\mathrm{k}}<0$) and, thus moves in AP-(P-)direction. Using
$F_{\mathrm{kk}}=c_{\text{k}}\left(X_{\mathrm{k,r}}-X_{\mathrm{k,l}}-d_{0}\right)$
for the spring force, we find
$\dot{F}_{\mathrm{kk}}=-c_{\text{k}}\left(v_{\mathrm{k,r}}+v_{\mathrm{k,l}}\right)$,
i.e., kinetochores move with the sum of their AP-velocities along the force-
velocity curve in the $F_{\mathrm{kk}}$-$v_{\mathrm{k}}$ diagram.

Figure 3:  Bistability gives rise to oscillations in the two-sided model.
(a,b) Different states of sister kinetochore motion can be deduced from the
bistability of the force-velocity relation: either both kinetochores are in
the upper branch (0) or one is in the upper and the other one in the lower
branch (2, $2^{\prime}$). In the first case, both kinetochores move away from
their pole (AP) towards each other. Thus, the spring force $F_{\text{kk}}$
decreases until it reaches $F_{\mathrm{min}}$. Since the upper branch is not
stable anymore below $F_{\mathrm{min}}$, either the left (1) or the right
($1^{\prime}$) kinetochore switches to the lower branch and changes direction
to poleward movement (P). The system is then in state 2 or $2^{\prime}$, where
both kinetochores move into the same direction: the leading kinetochore P, the
trailing kinetochore AP. As P- is much faster than AP-movement (MT shrinking
is much faster than growth), the inter-kinetochore distance and the spring
force are increasing. Above $F_{\mathrm{max}}$ only AP-movement is stable,
which is why the leading kinetochore changes direction (3, $3^{\prime}$) and
the system switches to state 0 again. (c) Solution of the equations of motion
(19) for $c=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$ and $M=25$ with an imposed periodic order of states
($0-2-0-2^{\prime}-0-...$). The initial condition is
$F_{\text{kk}}=F_{\mathrm{max}}$ (both kinetochores at the right end of the
upper branch). For an animated version see video S1 in the supplementary
material.

Oscillations arise from the two kinetochores moving through the hysteresis
loop of the bistable force-velocity relation as described in figure 3(a).
Three states are possible (see figure 3(b)). In state $0$, both kinetochores
move in AP-direction (i.e., in opposite directions) relaxing the
$F_{\mathrm{kk}}$-force from the cohesin bond, i.e., on the upper branch and
to the left in the $v_{\mathrm{k}}$-$F_{\mathrm{kk}}$-diagram with velocity
$\dot{F}_{\mathrm{kk}}=-2c_{\text{k}}v^{+}_{\mathrm{k}}<0$. After reaching the
lower critical force $F_{\mathrm{min}}$ of the hysteresis loop, one of the two
kinetochores reverses its direction and switches to the lower branch resulting
into states $2$ or $2^{\prime}$ where one kinetochore continues in AP-
direction with $v^{+}_{\mathrm{k}}>0$ while the other is moving in P-direction
with $v^{-}_{\mathrm{k}}<0$ (i.e., both move in the same direction). In the
$v_{\mathrm{k}}$-$F_{\mathrm{kk}}$-diagram, this results in a motion to the
right with velocity
$\dot{F}_{\mathrm{kk}}=c_{\text{k}}(-v^{-}_{\mathrm{k}}-v^{+}_{\mathrm{k}})>0$
because MTs typically shrink much faster than they grow ($-v^{0}_{-}\gg
v^{0}_{+}$, see table 2). Moving on opposite P- and AP-branches increases the
kinetochore distance and builds up $F_{\mathrm{kk}}$-force in the cohesin
bond. After reaching the upper critical force $F_{\mathrm{max}}$ of the
hysteresis loop, it is always the kinetochore on the lower branch moving in
P-direction which switches back and state $0$ is reached again. This behavior
is in agreement with experimental results [11]. The system oscillates by
alternating between state $0$ and one of the states $2$ or $2^{\prime}$ (which
is selected randomly with equal probability).

For each of the states 0, 2 and $2^{\prime}$ depicted in figure 3(ab) the two
branches $v^{\pm}_{\mathrm{k}}=v^{\pm}_{\mathrm{k}}[F_{\mathrm{kk}}]$ provide
deterministic equations of motion for the kinetochore positions. Inserting
$F_{\mathrm{kk}}=c_{\text{k}}\left(X_{\mathrm{k,r}}-X_{\mathrm{k,l}}-d_{0}\right)$
we obtain both kinetochore velocities as functions of the kinetochore
positions and find

$\displaystyle\eqalign{\text{state
0:}&\dot{X}_{\mathrm{k,l}}=\phantom{-}v^{+}_{\mathrm{k}}\big{[}c_{\text{k}}\left(X_{\mathrm{k,r}}-X_{\mathrm{k,l}}-d_{0}\right)\big{]}>0,\\\
&\dot{X}_{\mathrm{k,r}}=-v^{+}_{\mathrm{k}}\big{[}c_{\text{k}}\left(X_{\mathrm{k,r}}-X_{\mathrm{k,l}}-d_{0}\right)\big{]}<0,\\\
\text{state
2:}&\dot{X}_{\mathrm{k,l}}=\phantom{-}v^{-}_{\mathrm{k}}\big{[}c_{\text{k}}\left(X_{\mathrm{k,r}}-X_{\mathrm{k,l}}-d_{0}\right)\big{]}<0,\\\
&\dot{X}_{\mathrm{k,r}}=-v^{+}_{\mathrm{k}}\big{[}c_{\text{k}}\left(X_{\mathrm{k,r}}-X_{\mathrm{k,l}}-d_{0}\right)\big{]}<0,\\\
\text{state
$2^{\prime}$:}&\dot{X}_{\mathrm{k,l}}=\phantom{-}v^{+}_{\mathrm{k}}\big{[}c_{\text{k}}\left(X_{\mathrm{k,r}}-X_{\mathrm{k,l}}-d_{0}\right)\big{]}>0,\\\
&\dot{X}_{\mathrm{k,r}}=-v^{-}_{\mathrm{k}}\big{[}c_{\text{k}}\left(X_{\mathrm{k,r}}-X_{\mathrm{k,l}}-d_{0}\right)\big{]}>0.}$
(19)

Solving these equations gives idealized deterministic trajectories of the
sister kinetochores, when we also assume that the left and the right
kinetochore pass the lower branch alternately such that the order of states is
a periodic sequence $0-2-0-2^{\prime}-0-...$ as shown in the example in figure
3(c). Then single kinetochores oscillate with half the frequency of inter-
kinetochore (breathing) oscillations, just as observed in PtK1 cells [11].
Moreover, we can obtain numerical values of the frequencies directly from the
trajectories. For a MT-kinetochore linker stiffness
$c=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$ and
20–25 MTs per kinetochore, which is a realistic number for mammalian cells
[43], we get periods of 206–$258\text{\,}\mathrm{s}$ and
103–$129\text{\,}\mathrm{s}$ for kinetochore and breathing oscillations,
respectively. These values coincide with experimental results of
$239\text{\,}\mathrm{s}$ and $121\text{\,}\mathrm{s}$ measured in PtK1 cells
[11].

The calculated trajectories are idealized since they neglect stochastic
fluctuations that occur in simulations of the two-sided model and have two
main effects on the kinetochore dynamics which already arise in simulations
that comply with the assumptions behind the mean-field theory (no confinement
($x_{\mathrm{c}}\to\infty$) and permanent bonds ($\omega_{\mathrm{d}}=0$)):
Firstly, the sister kinetochores do not pass the lower branch alternately but
in random order. Therefore, we observe phases where one kinetochore moves in
AP-direction for several periods, while the other one changes its direction
periodically but moves polewards on average (figure 4(a)). Since this does not
influence the trajectory of the inter-kinetochore distance, breathing
oscillations still occur in a more or less regular manner, which allows us to
measure their frequencies by Fourier analysis. We will show below that
additional polar ejection forces suppress this random behavior and force the
kinetochores to pass the lower branch alternately. As a second effect of the
stochastic character of the simulation, kinetochores do not change the branch
instantaneously after crossing the critical forces $F_{\mathrm{max}}$ or
$F_{\mathrm{min}}$. Instead, they tend to maintain their primary state for a
while (figure 4(b)) and follow the metastable states that we also observe in
the one-sided model (figure 2(b)). Hence, the frequencies we measure in the
simulations are smaller than those we calculate from the Fokker-Planck mean-
field approach (figure 4(c)). The latter effect vanishes in the limit of many
MTs (large $M$): the switching points approach the theoretical values
$F_{\mathrm{max}}$ and $F_{\mathrm{min}}$, and the simulated breathing
frequencies converge to our mean-field predictions.

Figure 4:  Oscillations in stochastic simulations of the unconfined model
compared to mean-field results. (a) Kinetochore trajectories and breathing
oscillations in the two-sided model without confinement
($x_{\mathrm{c}}\to\infty$) and detachment ($\omega_{\mathrm{d}}=0$). The
kinetochores behave as described in figure 3 with a random order of states
$2/2^{\prime}$. The breathing oscillations are regular enough to assign a
frequency by Fourier analysis, see (d). With less MTs oscillations are more
fluctuative. (b) Kinetochore velocity against cohesin force in simulations of
the unconfined two-sided model without detachment (green). For many MTs the
velocity follows very precisely the predicted hysteresis from the mean-field
approach (red). For animated versions see videos S2 ($M=25$) and S3 ($M=500$)
in the supplementary material. (c) Double-logarithmic plot of frequencies of
breathing oscillations as a function of MT number $M$: calculated from the
mean-field approach according to figure 3 (red) and measured in simulations of
the unconfined (green diamonds) as well as the confined model with detachable
catch bonds (blue circles) and with permanent attachment (orange triangles).
Confinement becomes relevant for large MT numbers. In the presence of
detachable catch bonds only $75\text{\,}\mathrm{\char 37\relax}$ of the MTs
are attached on average, which corresponds to a simple shift of the curve to
lower MT numbers. (d) Trajectories from (a) in Fourier space. While
$\tilde{X}_{\mathrm{k,r}}$ has its maximum at $f=0$ due to the random order of
states in figure 3, $\Delta\tilde{X}_{\mathrm{k}}$ has a distinct peak that
becomes sharper for large $M$ indicating regular breathing oscillations. For
all simulations the MT-kinetochore linker stiffness was
$c=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$.

So far we have demonstrated that the mean field theory correctly describes
kinetochore dynamics in simulations of the unconfined model where we suppress
detachment in order to prevent unattached MTs from shrinking towards infinity.
As shown in figure 5(ab), kinetochore oscillations also survive in simulations
of the confined model independently of whether the MTs are able to detach from
the kinetochore, i.e., to rupture the catch bond. However, confinement by the
centrosome influences the kinetochore dynamics in the limit of large $M$:
since more MTs exert a higher force on the kinetochore, it is possible that
one of the two sisters gets stuck at the centrosome for a while (see figure
5(ab)). Hence, the frequencies measured in the confined two-sided model
deviate from the frequencies in the unconfined case above $M\approx 200$.

Figure 5:  Dynamics in the confined model with detachable MTs. (a) Kinetochore
positions $X_{\mathrm{k}}$ and inter-kinetochore distance $\Delta
X_{\mathrm{k}}$ over time in simulations with a total number of $M=25$ and
$M=100$ MTs per spindle pole. Oscillations as described in figure 3 are
recognizable. With 100 MTs one kinetochore can get stuck to the centrosome for
a while. (b) Distribution of kinetochore positions. The kinetochores are not
aligned to the spindle equator and for $M=100$ they are most likely to be
found near the centrosomes. (c) Number of attached MTs $M^{\mathrm{att}}$ over
time. MTs are more likely to be attached when the correspondent kinetochore is
near the centrosome since the free MTs can reattach to the kinetochore faster
in that case. (d) Distribution of $M^{\mathrm{att}}$. On average
$75\text{\,}\mathrm{\char 37\relax}$ of the MTs are attached independently of
the total MT number $M$.

If we enable detachment in our simulations we find that the number of attached
MTs correlates with the kinetochore position (see figure 5(c)) since due to
the exponential distribution of free MTs and the distance dependent attachment
rate (2) detached MTs are more likely to reattach to the kinetochore the
closer it is to the centrosome. Moreover, on average, about
$75\text{\,}\mathrm{\char 37\relax}$ of the MTs are attached independently of
the total MT number (see figure 5(cd)). Therefore, the catch bond nature of
the link leads to an effective behavior similar to a system without detachment
but with less MTs, which explains the difference in frequencies between the
confined models with and without detachment in figure 4(c). We conclude that
detachment does not play a major role for the occurrence of kinetochore
oscillations in cells with many MTs as despite detachment there are always
enough MTs attached to justify our mean-field approximation. Hence, (periodic)
changes in the number of attached MTs as they can be seen in figure 5(c) are
rather a passive consequence than an active source of kinetochore
oscillations. This argumentation may not be tenable, if just a few MTs are
attached to a kinetochore, so that even detachment of a single MT effects the
total force acting on the kinetochore significantly. Then, detachment can be
the primary cause of directional instability as worked out by Gay et al. [44],
who modeled the mitotic spindle of fission yeast.

Taking into account the results of the last paragraph, we will mainly
investigate the unconfined model with permanently attached MTs in the
following sections. This procedure is reasonable as we do not lose any
qualitative key features of kinetochore dynamics on the one hand, and, on the
other hand, gain a much better comparability of our mean field theory with the
appropriate stochastic simulations.

We finally note that in all cases we examined (confined / unconfined system,
permanent / detachable bonds) the kinetochore oscillations become more
fluctuative if less MTs are attached. This leads to the conclusion that
kinetochore oscillations are a result of the collective dynamics of an
ensemble of MTs that exhibit a force-dependent dynamic instability
individually. Such a behavior can not be described correctly based on the
simple assumption that all linkers have the same extension, i.e., that MTs
share the load equally and all attached MTs are in the same state (growing or
shrinking), (see supplementary material). Therefore, the model of Shtylla and
Keener [17] which does assume equal load sharing and synchronous MT dynamics
requires a chemical feedback as an additional mechanism in order to obtain
kinetochore oscillations. The model of Klemm et al. [21] divides each MT
ensemble into a growing and a shrinking sub-ensemble, and assumes equal load
sharing only between MTs within each sub-ensemble. Together with a force-
sensitive rescue force, this is sufficient to obtain oscillations.

##  5 Constraints on linker stiffness and MT number for bistability and
oscillations

###  5.1 Constraints for bistability in the one-sided model

We already argued above in Sec. 3 that bistability (and thus oscillations) can
only emerge if the MT-kinetochore linker is sufficiently stiff. To analyze the
influence of the linker stiffness $c$ and the MT number $M$ on bistability
quantitatively, the transformation from the master curve to the force-velocity
relation is visualized in figure 6(a) as search for the intersections of the
master curve with linear functions

$\displaystyle\langle x\rangle=\frac{1}{cM}(\gamma
v_{\mathrm{k}}-F_{\mathrm{ext}}).$ (20)

In the limit of large $M$ these linear functions have zero slope. Bistable
force-velocity relations with three intersection points are only possible if
the master curve has positive slope for intermediate $v_{\mathrm{k}}$
resulting in a maximum and minimum. The extrema of the master curve vanish,
however, in a saddle-node bifurcation if the linker stiffness drops below
$c_{\mathrm{bist}}=$7.737\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$, which is, therefore, a lower bound for the occurrence of
bistability. In the case of finite MT numbers $M$, bistable force-velocity
relations can only be found if the slope in the inflection point of the master
curve exceeds $\gamma/cM$ (the slope of the linear function (20)). This allows
us to quantify a bistable regime in the parameter plane of linker stiffness
$c$ and MT number $M$ as shown in figure 6(b).

Figure 6:  Constraints for bistability in the one-sided model. (a) Master
curves for different linker stiffnesses $c$ and linear functions according to
(20). In the limit of large $M$ the linear functions have zero slope and
bistability occurs if the master curve has two extrema, which is the case for
$c>c_{\mathrm{bist}}$. For finite $M$ bistable solutions are possible if the
linear functions have a smaller slope than the inflection point of the master
curve. (b) Resulting bistable regime in the parameter plane of linker
stiffness $c$ and MT number $M$.

###  5.2 Constraints for oscillations in the two-sided model

We showed in Sec. 4 that bistability of the one-sided model is a necessary
condition for oscillations in the two-sided model. Now we show that
bistability in the one-sided model is, however, not sufficient for
oscillations in the full model. If the force-velocity relation is interpreted
as phase space diagram for the two kinetochores, kinetochores only switch
branches in the $v_{\mathrm{k}}$-$F_{\mathrm{kk}}$-diagram if their velocity
changes its sign at the turning points $F_{\mathrm{min}}$ and
$F_{\mathrm{max}}$. If this is not the case and one of the two branches
crosses $v_{\mathrm{k}}=0$ (e.g. the right branch for
$c=$10\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$ in
figure 6(a), which transforms to the upper branch of the force-velocity
relation), the intersection point is a stable fixed point in the phase space
diagram (see figure 7(a)). At this fixed point kinetochore motion will relax
to zero velocity and just exhibit fluctuations around an equilibrium distance
instead of oscillations.

Figure 7:  Kinetochore dynamics in the non-oscillatory regime. (a) Schematic
explanation of kinetochore motion in the non-oscillatory regime based on the
force-velocity relation. Where the upper branch crosses zero velocity, inter-
kinetochore distance has a fixed point, around which it fluctuates. With
higher linker stiffnesses $c$ the fixed point comes closer to the left turning
point $F_{\mathrm{min}}$. When $c$ is just slightly smaller than
$c_{\mathrm{osc}}$, fluctuations can be large enough for the kinetochore
distance to leave the upper stable branch. Then, one of the two sister
kinetochores passes once through the lower branch. (b,c) This behavior can be
observed in simulations. While at
$c=$10\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$
kinetochores just fluctuate around the fixed point, at
$c=$14\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$ the
kinetochores occasionally pass through the hysteresis loop. Simulations were
performed with an unconfined system and 100 MTs on each side.

As a sufficient condition for oscillations we have to require – besides
bistability – a strictly positive velocity in the upper and a strictly
negative velocity in the lower branch in the
$v_{\mathrm{k}}$-$F_{\mathrm{kk}}$-diagram. Based on this condition we
quantify an oscillatory regime in the parameter plane of linker stiffness $c$
and MT number $M$ in figure 8(a). In the limit of many MTs the sufficient
condition for oscillations can be formulated in terms of the master curve: the
maximum of the master curve has to be located at a positive and the minimum at
a negative velocity. This is the case for
$c>c_{\mathrm{osc}}=$15.91\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$, which is, therefore, a lower bound for the occurrence of
oscillations. This constraint on the linker stiffness for metaphase chromosome
oscillations provides additional information on MT-kinetochore linkers whose
molecular nature is not known up to now.

Figure 8:  Constraints for oscillations in the two-sided model. (a)
Oscillatory regime in the parameter plane of linker stiffness $c$ and MT
number $M$. (b) Mean inter-kinetochore distance according to (22) (red) and
measured in simulations (blue) with $M=100$. Below
$c_{\mathrm{osc}}=$15.91\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$ (dashed line) both results match, whereas in the oscillatory regime
mean inter-kinetochore distance diverges from the fixed point, and its
standard deviation increases notably.

Because of stochastic fluctuations, the transition between oscillatory and
non-oscillatory regime is not sharp in our simulations. In the non-oscillatory
regime kinetochores fluctuate around a fixed point of inter-kinetochore
distance, where the upper branch crosses $v_{\mathrm{k}}=0$. However, these
fluctuations can be large enough for the inter-kinetochore distance to shrink
and leave the upper branch on the left side, especially for stiffnesses $c$
slightly below $c_{\mathrm{osc}}$. If that happens, one kinetochore passes
once through the lower branch of the force-velocity relation just as in an
oscillation. The difference to genuine oscillations is that these are randomly
occurring single events (resulting in a Poisson process). Randomly occurring
oscillations are visualized in figure 7 for $c<c_{\mathrm{osc}}$ and
$c\lesssim c_{\mathrm{osc}}$. Moreover, the force-velocity relations as well
as the kinetochore trajectories measured in corresponding simulations are
shown.

In the non-oscillatory regime, the fixed point should determine the mean
inter-kinetochore distance $\langle\Delta X_{\mathrm{k}}\rangle=\langle
X_{\text{k,r}}-X_{\text{k,l}}\rangle$. Solving the FPEs for $v_{\text{k}}=0$,
we compute the (external) force $F_{0}$ that has to be applied to one
kinetochore to stall its motion:

$\displaystyle F_{0}=\gamma v_{\text{k}}-cM\langle x\rangle=-cM\langle
x\rangle(v_{\text{k}}=0).$ (21)

In the two-sided model this force is applied to the kinetochores by the
cohesin bond at the fixed point. With $F_{\text{kk}}=c_{\text{k}}(\Delta
X_{\mathrm{k}}-d_{0})$ we compute the corresponding mean inter-kinetochore
distance:

$\displaystyle\langle\Delta
X_{\mathrm{k}}\rangle=\frac{F_{0}}{c_{\text{k}}}+d_{0}=-\frac{cM}{c_{\text{k}}}\langle
x\rangle(v_{\text{k}}=0)+d_{0}.$ (22)

Figure 8(b) shows that simulations agree with this result in the non-
oscillatory regime. At $c_{\mathrm{osc}}$ the transition to the oscillatory
regime can be recognized, where the mean inter-kinetochore distance deviates
from the fixed point (22). Moreover, the variance of $\Delta X_{\mathrm{k}}$
increases significantly at $c_{\mathrm{osc}}$ due to the transition to the
oscillatory regime.

In order to provide an overview and to make orientation easier for the reader,
we summarize in figure 9 where the stochastic simulations from the last three
sections and the master curves in figure 6(a) are located in the parameter
plane of linker stiffness $c$ and MT number $M$, and which regime they are
part of.

Figure 9:  Locations in $c$-$M$-parameter plane of the master curves from
figure 6(a) and the simulations from figures 2, 4, 5, 7 and 8.

##  6 Poleward microtubule flux suppresses oscillations

An effect we have not included so far is poleward MT flux, which was observed
in several metazoan cells (table 4). It describes the constant flux of tubulin
from the plus-ends towards the spindle pole and is probably driven by plus-end
directed kinesin-5 motors pushing overlapping antiparallel MTs apart as well
as kinesin-13 proteins that are located at the centrosome and depolymerize the
MTs at their minus-ends [24]. During metaphase, spindle and MT length can be
maintained by simultaneous polymerization at the plus-ends [45], which results
in a behavior similar to treadmilling of MTs [46].

Table 4:  Metaphase poleward flux velocities $v_{\mathrm{f}}$ and occurrence
of directional instability. For a more detailed review of poleward flux
measurements see [45]

  * Cell type |  $v_{\mathrm{f}}($\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$$) | Directional instability
---|---|---
LLC-PK1 (porcine) |  $8.3$ [8] | yes [8]
PtK1 (rat kangaroo) |  $7.7$ [47] | yes [11]
PtK2 (rat kangaroo) |  $10$ [8] | yes [12]
Newt lung |  $9.0$ [48] | yes [6]
U2OS (human) |  $8.8$ [9] | yes [9]
Drosophila embryo |  $32$ [49] | no [13]
Xenopus egg |  $37$ [50] | no [14]

Poleward flux can be easily included in our model by subtracting a constant
flux velocity $v_{\mathrm{f}}$ from the MT velocity. Then, the relative MT-
kinetochore velocities (7) become

$\displaystyle
v_{\pm}(x)=v_{\pm}^{0}\exp\left(-\frac{cx}{F_{\pm}}\right)-v_{\mathrm{f}}-v_{\mathrm{k}}.$
(23)

Hence, the flux velocity can be treated as an offset to the constant
kinetochore velocity in the solution of the stationary FPEs. The final effect
is a shift of both the master curves and the force-velocity relations by
$v_{\mathrm{f}}$ towards smaller kinetochore velocities $v_{\mathrm{k}}$ as
shown in figure 10(a). If the shift is so large that the left turning point
$F_{\mathrm{min}}$ of the force-velocity hysteresis is located at a negative
velocity, poleward flux suppresses directional instability because a fixed
point emerges, and we expect similar behavior as for intermediate linker
stiffnesses in the previous section (see figure 7). In the limit of many MTs,
the maximum flux velocity that still allows directional instability is given
by the velocity in the maximum of the master curve, which provides the
boundary of the oscillatory regime in the parameter plane of linker stiffness
$c$ and poleward flux velocity $v_{\mathrm{f}}$ (figure 10(b)). Phase space
diagrams (figure 10(c)) and kinetochore trajectories (figure 10(d)) from
simulations with appropriate flux velocities confirm our arguments exhibiting
similar behavior as for intermediate linker stiffnesses in figure 7. For small
flux velocities the boundary of the oscillatory regime in figure 10(b)
approaches our above result
$c_{\mathrm{osc}}=$15.91\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$. For increasing flux velocities the oscillatory regime shrinks, and
its boundary has a maximum at
$c\approx$50\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$ with
$v_{\mathrm{f}}\approx$3.11\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$$.
We conclude that kinetochore oscillations can be suppressed by moderate flux
velocities independently of the linker stiffness.

Figure 10:  Poleward flux suppresses oscillations. (a) Due to (23), the force-
velocity relation is shifted by the amount of the flux velocity
$v_{\mathrm{f}}$ towards smaller kinetochore velocities. If the flux is slower
than the kinetochore velocity $v_{\mathrm{min}}$ in the left turning point
$F_{\mathrm{min}}$, the kinetochores still oscillate. For larger flux
velocities, a fixed point arises on the upper branch and the kinetochores
behave as described in figure 7. (b) Oscillatory regime in the parameter plane
of $c$ and $v_{\mathrm{f}}$ in the limit of many MTs. Fast poleward flux
suppresses kinetochore oscillations for arbitrary linker stiffnesses $c$.
(b,c) Phase space diagrams and MT trajectories from simulations of the
unconfined two-sided model with
$c=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$ and
$M=100$. While at
$v_{\mathrm{f}}=$2\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$$ the system
is still in the oscillatory regime, where hysteresis is recognizable in phase
space, at $v_{\mathrm{f}}=$4\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$$
kinetochores show fluctuative motion as described in figure 7.

Our theory also agrees with and explains simulation results in [20], where,
for large flux velocities, suppression of kinetochore oscillations were
observed but at the same time maintenance of bistability. Moreover, our
results explain the experimentally observed correlation between flux velocity
and directional instability. Kinetochore oscillations have been observed in
the mitotic vertebrate cells listed in table 4 (LLC-PK1, PtK1/2, newt lung,
U2OS) which have poleward flux velocities not exceeding
$10\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$, whereas in the mitosis of
a Drosophila embryo as well as in meiosis of a Xenopus egg, where flux
velocities are three to four times higher, chromosomes do not exhibit
directional instability.

##  7 Polar ejection forces provide an alternating oscillation pattern and
chromosome alignment at the spindle equator

So far, we have not included polar ejection forces (PEFs). They originate from
non-kinetochore MTs interacting with the chromosome arms and pushing them
thereby towards the spindle equator, either through collisions with the
chromosome arms or via chromokinesins [27], and provide additional pushing
forces on kinetochores. Therefore, they can be included into the model by
adding forces $F_{\mathrm{PEF,r}}(X_{\mathrm{k,r}})$ and
$F_{\mathrm{PEF,l}}(X_{\mathrm{k,l}})$ acting on kinetochores, which depend on
the absolute position of the kinetochores [19]. Due to the exponential length
distribution of free MTs as well as the spherical geometry of the MT asters,
the density of non-kinetochore MTs decreases monotonically with the distance
from the spindle pole. Therefore, we assume that PEFs reach their maximum at
the centrosome and vanish at the spindle equator (located at $x=0$), where
opposite PEFs compensate each other. This assumption is supported by the
monotonic PEF distribution that has been measured in vivo by Ke et al. [51].
Here, we will only discuss linearized PEFs

$\displaystyle F_{\mathrm{PEF,l}}(X_{\mathrm{k,l}})=-kX_{\mathrm{k,l}},\qquad
F_{\mathrm{PEF,r}}(X_{\mathrm{k,r}})$ $\displaystyle=kX_{\mathrm{k,r}},$ (24)

where the spring constant $k$ defines the strength of the forces, and the
signs are chosen so that a positive force acts in AP-direction. We show in
figure S3 in the supplementary material that other force distributions do not
differ qualitatively in their influence on the kinetochore dynamics.

To determine kinetochore trajectories of the two-sided model in the presence
of PEFs, we can start from the same force-velocity relations as for the basic
one-sided model. In the presence of PEFs, the total forces $F_{\mathrm{k,l}}$
and $F_{\mathrm{k,r}}$ that act on the left and the right kinetochore in AP-
direction depend on the absolute kinetochore positions $X_{\mathrm{k,l}}$ and
$X_{\mathrm{k,r}}$:

$\displaystyle F_{\mathrm{k,l}}$ $\displaystyle=F_{\mathrm{kk}}(\Delta
X_{\mathrm{k}})+F_{\mathrm{PEF,l}}(X_{\mathrm{k,l}}),$ (25) $\displaystyle
F_{\mathrm{k,r}}$ $\displaystyle=F_{\mathrm{kk}}(\Delta
X_{\mathrm{k}})+F_{\mathrm{PEF,r}}(X_{\mathrm{k,r}}).$ (26)

We can investigate the motion of kinetochores in the full two-sided model
again by using a phase space diagram; in the presence of PEFs we use a
$v_{\mathrm{k}}$-$F_{\mathrm{k}}$-diagram with the total force
$F_{\mathrm{k}}$ in AP-direction on the horizontal axis and the velocity
$v_{\mathrm{k}}$ in AP-direction on the vertical axis. Because the total
forces contain the external PEFs they are no longer related by action and
reaction and, thus, the two kinetochores no longer have the same position on
the $F_{\mathrm{k}}$-axis, but they still remain close to each other on the
$F_{\mathrm{k}}$-axis as long as the cohesin bond is strong enough.

A kinetochore on the upper/lower branch moves in AP-/P-direction with
$v^{\pm}_{\mathrm{k}}(F_{\mathrm{k}})$ if $v^{+}_{\mathrm{k}}>0$
($v^{-}_{\mathrm{k}}<0$). A kinetochore on the upper AP-directed branch will
relax its AP-directed PEFs, while a kinetochore on the lower P-directed branch
will build up AP-directed PEFs. After a time of equilibration the kinetochores
behave as described in figure 11. When one kinetochore changes its direction
from P to AP (switches to the upper branch) the sister kinetochore, which was
on the upper branch before, becomes the leading kinetochore (here, “leading”
refers to the position in the force velocity phase space). Therefore, the
kinetochores do not reach the left turning point $F_{\mathrm{min}}$ at the
same time so that it is always the leading kinetochore that switches to the
lower branch. Since in general the absolute P-velocity is much larger than the
AP-velocity ($-v_{-}$ for the lower branch is much larger than $+v_{+}$ for
the upper branch), the AP-directed PEF contribution to the total force
increases faster on the lower branch than on the upper one. As a result, the
P-moving kinetochore overtakes its sister on the $F_{\mathrm{k}}$-axis before
switching back to the upper branch such that the leading kinetochore
automatically becomes the trailing kinetochore in the next oscillation period
(again, “leading” and “trailing” in terms of phase space positions). This
periodic change of kinetochore positions in the force-velocity diagram leads
to both regular breathing and regular single kinetochore oscillations, as the
kinetochores alternately pass the lower branch. Solving appropriate equations
of motions similar to (19) for each of the states depicted in figure 11(ab),
we determine the deterministic trajectories in figure 11(c) confirming this
regular alternating oscillation pattern.

Figure 11:  Kinetochore motion in the presence of PEFs. (a,b) At the beginning
of state 1 the left kinetochore (green) has just switched from P- to AP-
movement, so that both kinetochores are on the upper branch. Both kinetochores
move in AP-direction, which means that both the cohesin force and the PEFs
decrease and both kinetochores move left in the force-velocity diagram. Due to
different PEFs, the right kinetochore (red) reaches the left turning point
$F_{\mathrm{min}}$ first and switches to the lower branch, which marks the
start of state 2. This state is dominated by the fast P-movement of the right
kinetochore, which causes a steep increase of both $F_{\mathrm{kk}}$ and
$F_{\mathrm{PEF,r}}$. Therefore, the right kinetochore moves to the right in
the force-velocity diagram. Meanwhile, the left sister still moves in AP-
direction and $F_{\mathrm{k,l}}$ increases slightly as the increase of
$F_{\mathrm{kk}}$ is larger than the decrease of $F_{\mathrm{PEF,l}}$. Since
$\dot{F}_{\mathrm{k,r}}>\dot{F}_{\mathrm{k,l}}$, the right kinetochore
overtakes its sister on the $F_{\mathrm{k}}$-axis before it reaches the right
turning point and switches to the upper branch. The then following states
$1^{\prime}$ and $2^{\prime}$ are the exact opposite to 1 and 2 with swapped
kinetochores. (c) Solution of the corresponding equations of motion for
$c=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$,
$k=$10\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$ and
$M=25$. For an animated version see video S4 in the supplementary material.

The alternating oscillation pattern robustly survives in stochastic
simulations in the presence of moderate PEFs
($k\sim$10\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$)
as we demonstrate in figure 12(a) by means of the kinetochore trajectories in
real space. In figure 12(b), emergence of regular oscillations is illustrated
in Fourier space: Whereas for rather small values of $k$ single kinetochore
oscillations are still irregular resulting in a nearly monotonic decreasing
Fourier transform, for
$k=$10\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$
single kinetochore motion has a distinct peak in the Fourier space indicating
a regular shape of oscillations in real space. Moreover, frequency doubling of
breathing compared to single kinetochore oscillations can directly be
recognized by comparing the corresponding Fourier transforms. As a consequence
of regular oscillations, the kinetochores stay near the spindle equator and
can not get stuck to one of the centrosomes as in the basic model, see
histograms of kinetochore positions in figure 12(c). We conclude that PEFs are
necessary to assure proper chromosome alignment in the metaphase plate at the
spindle equator. This is consistent with an experiment by Levesque and Compton
[52], who observed mitosis of vertebrate cells after suppressing the activity
of chromokinesins and, thus PEFs. This resulted in $17.5\text{\,}\mathrm{\char
37\relax}$ of the cells in at least one chromosome not aligning at the
equator, but locating near a spindle pole.

Figure 12:  Kinetochore dynamics under the influence of PEFs. (a) Kinetochore
trajectories with different PEF constants $k$ from simulations with $M=100$,
$c=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$ and
without confinement at the spindle poles. The PEFs force the kinetochores to
oscillate regularly and to stay near the spindle equator. For
$k=$10\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$
kinetochores oscillate as described in figure 11. Since with strong PEFs
kinetochores tend to switch to the lower branch simultaneously when reaching
$F_{\mathrm{min}}$ in the phase space at the same time, for
$k=$1000\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$
oscillations are in antiphase due to symmetric initial conditions before the
system equilibrates at $t\approx$1500\text{\,}\mathrm{s}$$. After
equilibration, periods of antiphase oscillations reappear over and over again
due to fluctuations. Stronger PEFs cause a more fluctuative kinetochore
motion. Especially for moderate MT numbers, this can lead to suppression of
kinetochore oscillations. For animated versions of phase space trajectories
see videos S5 ($k=$10\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$) and S6
($k=$1000\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$)
in the supplementary material. (b) Single (right) kinetochore and breathing
oscillations in Fourier space. For weak PEFs
($k=$1\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$)
single kinetochore oscillations are still irregular and
$\tilde{X}_{\mathrm{k,r}}$ has its maximum at $f=0$. If
$k=$10\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$,
$\tilde{X}_{\mathrm{k,r}}$ has a distinct peak at half the breathing
frequency, indicating regular oscillations as described in figure 11 and
frequency doubling of breathing compared to single kinetochore oscillations.
With sufficiently strong PEFs
($k\gtrsim$100\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$) frequency doubling is lost as a consequence of antiphase
oscillations and the peaks of $\tilde{X}_{\mathrm{k,r}}$ and
$\Delta\tilde{X}_{\mathrm{k}}$ coincide with each other. (c) Histograms of
kinetochore positions and inter-kinetochore distances for the realistic case
of $M=25$. Chromosomes are aligned at the spindle equator despite missing
confinement at the centrosome. The range of kinetochore positions is narrower
and the distances smaller if PEFs are stronger.

Moreover, PEFs reduce the amplitude and increase the frequency of
oscillations. The amplitude decreases for increasing PEF strength $k$ as the
kinetochores have to cover a smaller distance between the turning points at
$F_{\mathrm{min}}$ and $F_{\mathrm{max}}$. The increase of the frequency is
linear in $k$, which can be deduced from the linear increase of
$|\dot{F}_{\mathrm{k}}|$:

$\displaystyle|\dot{F}_{\mathrm{k,l}}|$
$\displaystyle=\left|c_{\text{k}}\left(v_{\mathrm{k,r}}+v_{\mathrm{k,l}}\right)+kv_{\mathrm{k,l}}\right|,$
(27) $\displaystyle|\dot{F}_{\mathrm{k,r}}|$
$\displaystyle=\left|c_{\text{k}}\left(v_{\mathrm{k,r}}+v_{\mathrm{k,l}}\right)+kv_{\mathrm{k,r}}\right|$
(28)

(defining $v_{\mathrm{k,l}}\equiv\dot{X}_{\mathrm{k,l}}$ and
$v_{\mathrm{k,r}}\equiv-\dot{X}_{\mathrm{k,r}}$ as the velocities in AP-
direction as before).

Since PEFs do not have any influence on the underlying master curves and
force-velocity relations, they do not affect the kinetochore velocities
$v_{\mathrm{k}}$ and never completely suppress kinetochore oscillations in the
deterministic Fokker-Planck model, but only reduce their amplitude and
increase their frequency. For strong PEFs, however, this gives rise to
kinetochore motion with a fluctuative character, see figure 12 (see also video
S6 in the supplementary material). The same observation was made in the model
of Civelekoglu-Scholey et al. [19]. Additionally, we detect sister kinetochore
oscillations being in antiphase if PEFs are strong enough
($k\gtrsim$100\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$), see figure 12(a). This follows from the phase space velocities
$\dot{F}_{\mathrm{k}}$ being dominated by the strong PEFs compared to inter-
kinetochore tension: Imagine, both kinetochores are in the upper branch of the
phase space and reach the turning point $F_{\mathrm{min}}$ at nearly the same
time. When now one of the two kinetochores switches to the lower branch and
starts moving polewards, its sister does not change its direction in phase
space as in state $2/2^{\prime}$ in figure 11(a) but continues moving left
since the decrease of PEFs due to its poleward motion can not be compensated
by the increasing AP-directed cohesin tension if $k\gg c_{\mathrm{k}}$. As a
consequence, the kinetochore will switch to the lower branch just after its
sister and both kinetochores pass the lower branch simultaneously, i.e. move
apart from each other, finally resulting in antiphase oscillations. While the
antiphase behavior vanishes after a certain time of equilibration in the
deterministic model, in stochastic simulations periods of antiphase
oscillations can be observed over and over again regardless of whether the
system has been equilibrated before. A characteristic of antiphase
oscillations is the loss of frequency doubling which also appears in the
Fourier space where the peaks of single kinetochore and breathing motion
coincide with each other if PEFs are strong, see figure 12(b). Since antiphase
kinetochore oscillations have not been observed experimentally, we conclude
that in vivo PEFs are weak compared to the inter-kinetochore tension but
strong enough to assure chromosome alignment at the spindle equator. Compared
to experimental results [6, 7, 10, 11, 12, 19], in our model,
$k=$10\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$
seems a reasonable choice as it assures regular oscillations with frequency
doubling, keeps the inter-kinetochore distance within a suitable range of
$1.2\pm 0.7\text{\,}\mathrm{\SIUnitSymbolMicro m}$, and aligns kinetochores in
a realistic maximum distance of $3\text{\,}\mathrm{\SIUnitSymbolMicro m}$ from
the spindle equator with a standard deviation of
$0.88\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in the lifelike case of $M=25$.

##  8 Catastrophe promotion at the kinetochore is required to stimulate
directional instability if microtubules can not exert pushing forces

So far, we assumed that MTs are also able to exert pushing forces on the
kinetochore. During oscillations we find, on average, slightly less (48%) MT-
kinetochore links under tension, while a substantial part of linkers also
exerts pushing forces. Two experimental results suggest, however, that MTs do
not directly exert pushing forces on the kinetochore: In [7], it was shown
that the link between chromosomes is always under tension; the experiments in
[26] demonstrated that, after removal of the cohesin bond, AP-moving
kinetochores immediately stop indicating that kinetochore MTs can not exert
pushing forces, while P-moving kinetochores continue moving due to MT pulling
forces.

In view of these experimental results and in order to answer the question
whether MT pushing forces are essential for bistability and oscillations, we
analyze variants of our basic model, where MT growth is confined at the
kinetochore, i.e., where the relative coordinate
$x=x_{\mathrm{m}}-X_{\mathrm{k}}$ is limited to $x\leq 0$ such that MTs can
only exert tensile forces on the kinetochore. This implies that the
kinetochore undergoes a catastrophe if it reaches the kinetochore, i.e., if
the relative coordinate reaches $x=0$ from below in the one-sided model.
Different choices for the corresponding catastrophe rate
$\omega_{\mathrm{c}}^{\mathrm{kin}}$ at $x=0$ are possible: (i) A reflecting
boundary, i.e., $\omega_{\mathrm{c}}^{\mathrm{kin}}=\infty$, where a
catastrophe is immediately triggered if the MT plus-end reaches the
kinetochore. (ii) A “waiting” boundary condition, where the relative velocity
$v_{+}=v_{\mathrm{m}+}-v_{\mathrm{k}}=0$ stalls if the MT reaches $x=0$ (in
the simulation, we set the MT velocity to $v_{\mathrm{m}+}=v_{\mathrm{k}}$).
In contrast to the reflecting boundary condition, the catastrophe rate
$\omega_{\mathrm{c}}^{\mathrm{kin}}$ at the kinetochore is finite such that
the MT waits at the kinetochore until it undergoes a catastrophe for a mean
waiting time $1/\omega_{\mathrm{c}}^{\mathrm{kin}}$, as similarly observed in
metaphase of PtK1 cells [36]. Because $x=0$ also results in
$F_{\mathrm{mk}}=0$, the force-free catastrophe rate seems a natural choice,
$\omega_{\mathrm{c}}^{\mathrm{kin}}=\omega^{0}_{\mathrm{c}}$ [see (1)], which
should be realized in the absence of any additional catastrophe regulating
proteins at the centromere. (iii) If catastrophes are promoted by regulating
proteins, but not immediately as for (i), we obtain intermediate cases of
waiting boundary conditions with
$\omega^{0}_{\mathrm{c}}<\omega_{\mathrm{c}}^{\mathrm{kin}}<\infty$. In
mammalian cells, such regulating mechanisms could be provided by the kinesin
MCAK, which is localized at the centromere during metaphase [53] and has been
reported to increase the catastrophe rate of MTs roughly 7-fold [54].
Therefore, waiting boundary conditions with an increased catastrophe rate
appear to be the most realistic scenario. We introduce a numerical catastrophe
enhancement factor $n\geq 1$ characterizing the increased catastrophe rate,
$\omega_{\mathrm{c}}^{\mathrm{kin}}=n\omega^{0}_{\mathrm{c}}$. Within this
general scenario reflecting boundary conditions (i) are recovered for
$n=\infty$ and (ii) waiting boundary conditions with the zero force
catastrophe rate for $n=1$. We will discuss the general case (iii) in the
following.

In our basic model, where MTs can exert pushing forces on kinetochores, the
pushing phases where $x>0$ can also be interpreted as a an effective waiting
phase at the kinetochore with a catastrophe rate, which is effectively
increased by the pushing forces. Therefore, the behavior of our basic model
resembles a model with waiting boundary conditions with an increased
catastrophe rate $n>1$ at the kinetochore. MT pushing forces are not essential
for bistability and oscillations and have a similar effect as an increased
catastrophe rate at the kinetochore as our detailed analysis will show.

In the Fokker-Planck solution for the one-sided model, all confining boundary
conditions limit the maximum MT-kinetochore distance $x_{\mathrm{max}}$ to
zero, where it is positive in the basic model. When $x_{\mathrm{max}}$ is
negative in the basic model (for $v_{\mathrm{k}}>v^{0}_{+}$, see table 3),
confining boundary conditions do not modify the basic model, since the MTs are
not able to reach the fast kinetochore. For negative kinetochore velocities
$v_{\mathrm{k}}<v^{0}_{-}$, the minimum distance $x_{\mathrm{min}}$ becomes
positive while $x_{\mathrm{max}}$ is zero. Then, all confining boundary
conditions fix the MT tips to the kinetochore position as they do not shrink
fast enough to move away from the poleward-moving kinetochore after a
catastrophe resulting in $\langle x\rangle=0$ and $F_{\mathrm{ext}}=\gamma
v_{\mathrm{k}}$. All in all, confinement leads to the following maximal and
minimal values for the MT-kinetochore distance $x$ modifying table 3:

$\displaystyle
x^{\mathrm{conf}}_{\mathrm{max}}=\cases{0,&$v_{\mathrm{k}}<v^{0}_{+}$\\\
x_{\mathrm{max}},&$v_{\mathrm{k}}\geq v^{0}_{+}$,}\qquad
x^{\mathrm{conf}}_{\mathrm{min}}=\cases{0,&$v_{\mathrm{k}}<v^{0}_{-}$\\\
x_{\mathrm{min}},&$v_{\mathrm{k}}\geq v^{0}_{-}$.}$ (29)

We calculate the master curves $\langle x\rangle(v_{\mathrm{k}})$ for all
three types of confining boundary conditions (see figure 13(a)). Because
$x^{\mathrm{conf}}_{\mathrm{max}}\leq 0$ for any confining boundary condition,
also $\langle x\rangle<0$, i.e., the complete master curves lie in the regime
of tensile MT-kinetochore linker forces reflecting the fact that pushing
forces are strictly suppressed. Therefore, the MT-kinetochore catch bond is on
average under tension establishing a more firm MT-kinetochore connection
during the stochastic chromosome oscillations in metaphase. Oscillations then
become a tug-of-war, in which both sets of MTs only exert pulling forces onto
each other.

Figure 13:  Microtubule confinement at the kinetochore. (a) Master curves of a
system with a waiting boundary condition for various
$\omega_{\mathrm{c}}^{\mathrm{kin}}=n\,\omega_{\mathrm{c}}^{0}$ and
$c=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$. (b)
Regimes in the parameter plane of $c$ and $\omega_{\mathrm{c}}^{\mathrm{kin}}$
in the limit of many MTs. Outside the blue region, the master curve is
bistable. In the orange region, the left branch of the master curve and,
therefore, the lower branch of the $v_{\mathrm{k}}$-$F_{\mathrm{kk}}$-diagram
cross $v_{\mathrm{k}}=0$, which leads to a fixed point suppressing
oscillations (see text), whereas in the red region oscillations are possible.
In stochastic simulations, kinetochores already oscillate at much smaller
$\omega_{\mathrm{c}}^{\mathrm{kin}}$ than predicted by the master curves.
Additionally, a new kind of fixed point, which is depicted in (c), emerges in
the shaded region. (c,d) Phase space diagrams and kinetochore trajectories
from simulations of the unconfined two-sided model with
$c=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$ and
$M=100$. The blue dots mark the new kind of fixed point, where the leading
kinetochore in the lower branch moves with the same velocity as the trailing
kinetochore in the upper branch. Then the inter-kinetochore distance remains
constant, while the center of mass moves with a constant velocity as in (d)
for $\omega_{\mathrm{c}}^{\mathrm{kin}}=20\,\omega_{\mathrm{c}}^{0}$ at
$t\approx$25\,000\text{\,}\mathrm{s}$$. In the presence of PEFs, these fixed
points are absent and the shaded region in (b) does not apply.

With a waiting boundary condition at the kinetochore, the probability
densities $p_{\pm}(x,t)$ have to be supplemented with the probability $Q(t)$
to find a MT at the kinetochore ($x=0$). Besides the FPEs (5) and (6) for the
probability densities, we also have to solve the equation for the time
evolution of $Q(t)$:

$\displaystyle\partial_{t}Q(t)=v_{+}(0)p_{+}(0,t)-\omega_{\mathrm{c}}^{\mathrm{kin}}Q(t).$
(30)

The analogous model for a free MT that grows against a rigid wall has already
been solved in [55, 41]. In the stationary state, (30) leads to
$Q=p_{+}(0){v_{+}(0)}/{\omega_{\mathrm{c}}^{\mathrm{kin}}}$. For the
probability densities $p_{\pm}(x)$ we get the same solution as for the basic
model without confinement, except for the normalization constant. The overall
probability density can then be written as $p(x)=p_{+}(x)+p_{-}(x)+Q\delta(x)$
and has to satisfy
$\int_{x^{\mathrm{conf}}_{\mathrm{min}}}^{x^{\mathrm{conf}}_{\mathrm{max}}}p(x)\mathrm{d}x=1$.

From the overall probability density $p(x)$ we obtain the master curves, which
we show in figure 13(a) for $n=1,5,20,50,200,\infty$ and a linker stiffness of
$c=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$.
Again we can analyze the master curves for extrema to obtain constraints on
linker stiffness $c$ and catastrophe enhancement factor
$n=\omega_{\mathrm{c}}^{\mathrm{kin}}/\omega^{0}_{\mathrm{c}}$ for the
occurrence of bistability and oscillations. The results of this analysis are
shown in figure 13(b) as colored regions. It turns out that extrema in the
master curve and, thus, bistability occur if the linker stiffness is
sufficiently high $c>c_{\mathrm{bist}}$. For the zero force catastrophe rate
$n=1$ we find a high threshold value
$c_{\mathrm{bist}}=$178\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$, in the limit of a reflecting boundary $n=\infty$ a very low
threshold
$c_{\mathrm{bist}}=$1.218\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$.

We remind that a sufficient condition for oscillations is the absence of a
stable fixed point, where one of the two branches in the
$v_{\mathrm{k}}$-$F_{\mathrm{kk}}$-diagram crosses $v_{\mathrm{k}}=0$. In
contrast to the basic model, the maxima of the master curve are now located at
a positive velocity for $n>1$. Therefore, oscillations are suppressed by a
fixed point $v^{-}_{\mathrm{k}}=0$ on the lower branch in the
$v_{\mathrm{k}}$-$F_{\mathrm{kk}}$-diagram, which occurs if the velocity is
positive in the minimum of the master curve. In general, oscillations occur if
the linker stiffness is sufficiently high $c>c_{\mathrm{osc}}$. Again we find
a high threshold value
$c_{\mathrm{osc}}=$280\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$ for $n=1$ and a low threshold
$c_{\mathrm{osc}}=$1.237\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$ for a reflecting boundary condition ($n=\infty$).

For $n<10$ the threshold values remain high. Moreover, at such high linker
stiffnesses and for for small $n$, the simulations of the two-sided model do
not show the expected behavior. For $n=1$ and high linker stiffnesses in the
oscillatory regime the kinetochore trajectories do not exhibit regular
oscillations. Naively, one could argue that kinetochore oscillations are
suppressed due to the lack of a pushing force and can be restored by
additional PEFs. However, this is not the case, since, as stated above, PEFs
do not affect the master curve that determines the regime of kinetochore
motion. One reason for the absence of oscillations is that, for the zero force
catastrophe rate ($n=1$) the waiting time
$1/\omega_{\mathrm{c}}^{\mathrm{kin}}\sim 500{\rm s}$ (see table 2) at the
kinetochore is large compared to the typical oscillation periods, which are in
the range of $100-200{\rm s}$.

Figure 13(b) also shows that oscillations require increased catastrophe rates
with $n\gtrsim 20$ over a wide range of linker stiffnesses from
$c=$10\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$ to
$c=$200\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$.
For $n>1$, at the boundary between bistable and oscillatory regime in figure
13(b), a fixed point $v^{-}_{\mathrm{k}}=0$ on the lower branch of the
$v_{\mathrm{k}}$-$F_{\mathrm{kk}}$ phase space diagrams appears, which can
suppress oscillations. This fixed point is, however, less relevant because the
kinetochores will only occasionally pass the lower branch simultaneously,
which is necessary to reach this fixed point. Furthermore, this fixed point is
located near the right turning point $F_{\mathrm{max}}$ so that the
kinetochores can easily leave the fixed point by a stochastic fluctuation (as
in figure 7). For these two reasons, in stochastic simulations, oscillations
already occur for $n\gtrsim 5$, that is at a much lower $n$ than the
deterministically predicted $n\gtrsim 20$, but not for $n=1$, i.e., in the
absence of a catastrophe promoting mechanism.

The fixed point analysis of the $v_{\mathrm{k}}$-$F_{\mathrm{kk}}$ phase space
diagrams reveals that also a new type of fixed point corresponding to a non-
oscillatory motion emerges for $n\lesssim 100$ in the shaded regions in figure
13(b). In this new type of fixed point, the leading P-moving kinetochore in
the lower branch of the master curve has the same velocity as the trailing AP-
moving kinetochore in the upper branch (see figure 13(c)) so that
$\dot{F}_{\mathrm{kk}}=-c_{\text{k}}\left(v_{\mathrm{k,r}}+v_{\mathrm{k,l}}\right)=0$,
and the inter-kinetochore distance remains constant, while the center of mass
moves with a constant velocity (see figure 13(d)). In the presence of PEFs,
however, this new type of fixed point does not survive because for the P-
moving kinetochore the AP-directed PEFs increase, whereas they decrease for an
AP-moving kinetochore. Then the upper blue dot in figure 13(c) moves to the
left, while the lower blue point moves to the right such that this new type of
fixed point is unstable in the presence of PEFs. Therefore, in the entire
shaded region in figure 13(b) PEFs are essential to re-establish oscillations.

We conclude that both the linker stiffness
$c>$10\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$ and
the catastrophe rate $\omega_{\mathrm{c}}^{\mathrm{kin}}$ at the kinetochore
($n\gtrsim 20$ or $n\gtrsim 5$ in the presence of stochastic fluctuations)
have to be sufficiently large to obtain bistability and oscillations. Because
additional catastrophe promoting proteins are necessary to increase the
catastrophe rate at the kinetochore, the lowest values of $n$, which still
enable oscillations, might be advantageous in the cellular system. We note
that poleward flux can influence existence and positions of fixed points: An
intermediate flow velocity can eliminate a fixed point on the lower branch by
moving it into the unstable area of the phase space diagram. If flux is
sufficiently large it can establish additional fixed points on the upper
branch of the phase space diagrams, which suppress oscillations as in the
basic model.

Moreover, the linker stiffness has to be sufficiently high to give linker
extensions compatible with experimental results. An important part of the MT-
kinetochore linkage is Ndc80, which is a rod-like fibril of total length
around $60\,{\rm nm}$ [56, 57] consisting of two coiled-coil regions with a
flexible hinge that can adopt bending angles up to $120^{\circ}$ with a broad
distribution [57]. This bending corresponds to linker length changes of
$|x|\sim$50\text{\,}\mathrm{nm}$$. Moreover, fluorescent labeling showed total
intra-kinetochore stretches around $100\text{\,}\mathrm{nm}$ [58] or
$50\text{\,}\mathrm{nm}$ [12]. Therefore, we regard linker extensions
$x\lesssim$100\text{\,}\mathrm{nm}$$ as realistic values. For large $n\gg 20$
only a small linker stiffness is necessary to enable oscillations. At the
small threshold stiffness, the average linker length $|\langle x\rangle|$ is
typically $1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in this regime. Increasing
the linker stiffness leads to a decreasing linker length $|\langle x\rangle|$.
We conclude that, for $n\gg 20$, experimental observations of linker
extensions $|x|\lesssim$100\text{\,}\mathrm{nm}$$ put a stronger constraint on
linker stiffness than the experimental observations of oscillations. Linker
stiffnesses significantly above
$5\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$ and,
thus, far above $c_{\mathrm{osc}}$ are necessary to obtain a realistic linker
length.

For $n\sim 10-20$, which is compatible with the experimental result $n\sim 7$
for the catastrophe promoter MCAK [54], and a linker stiffness
$c=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$, the
increased catastrophe rate at the kinetochore leads to a realistic behavior
with linker extensions $x\sim$100\text{\,}\mathrm{nm}$$, which are also
compatible with the experimental results [56, 57, 58, 12] (see figure 13(a)).
This parameter regime is within the shaded regions in figure 13(b) and PEFs
are necessary to establish oscillations. The linker extension is independent
of PEFs.

For an increased catastrophe rate around $n\sim 10-20$ and a linker stiffness
$c=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$, the
more realistic model with waiting boundary conditions at the kinetochore
exhibits a similar behavior as our basic model because pushing phases where
$x>0$ in the basic model have a similar duration as waiting times at the
kinetochore in the more realistic model.

##  9 Model parameters can be adjusted to reproduce kinetochore oscillations
in PtK1 cells

So far, we took the experimentally measured parameters for MT transitions and
velocities from table 2 for granted in order to analyze the effects of
poleward flux, PEFs and confinement at the kinetochore by means of our mean-
field theory. These values stem from experiments with yeast kinetochores [2],
which can only bind one MT [59], whereas the mean-field theory is only correct
if the kinetochores are attached to multiple MTs as in metazoan cells.
Moreover, in budding yeast, the Ndc80 fibrils are connected to MTs via ring-
like Dam1 complexes, which do not appear in metazoan cells [60]. In this
section, we demonstrate that by adjusting the parameters of MT dynamics our
model can reproduce experimental data of metazoan spindles using the example
of PtK1 cells.

Our model exhibits a large difference of P versus AP-velocity ($\sim 100$ vs.
$\sim$4\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$$, see figure 8) which
is the origin of frequency doubling and also appears in PtK1 cells but not in
this extent ($\sim 19$ vs.
$\sim$16\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$$) [11]. As a
consequence, in our model both kinetochores move towards each other in AP-
direction (state 0 in figure 3) most of the time, whereas in the experiment,
mostly one kinetochore moves in P- while the trailing sister is moving in AP-
direction (state $2/2^{\prime}$ in figure 3). In a first step we will respect
these results by adjusting the master curve (or force velocity relation) in a
way that the two stable branches fit the experimentally measured velocities.
This objective will be achieved by modifying the force-free MT velocities
$v_{\pm}^{0}$ (shifting the upper / lower branch up- or downwards) and the
corresponding characteristic forces $F_{\pm}$ (altering the slope of the upper
/ lower branch). Moreover, as a last parameter of MT dynamics, we will change
the rescue rate $\omega_{\mathrm{r}}^{0}$ in order to adjust the MT-
kinetochore distance to a realistic value. In a second step we will fit the
measured frequencies and amplitudes by varying the parameters that do not
affect the master curves ($c_{\mathrm{k}}$, $k$).

Using the model with confinement at the kinetochore, we assume a ten times
increased catastrophe rate
$\omega_{\mathrm{c}}^{\mathrm{kin}}=10\omega_{\mathrm{c}}^{0}$ according to
experimental results [54]. We set the linker stiffness to
$c=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$ and
keep it unchanged henceforth since this value results in strongly bistable
master curves and the manifold consequences that a further modification of $c$
has on kinetochore dynamics are hard to handle. The flux velocity is
$v_{\mathrm{f}}=$8\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$$ (see table
4). The force-free MT growth velocity $v_{+}^{0}$ has to be greater than
$v_{\mathrm{f}}$ for two reasons: Firstly, detached MTs would not have a
chance to reach the kinetochore again, otherwise. Secondly, this choice
prevents a fixed point at the upper branch, as the left turning point in phase
space (maximum of the master curve) is located at $v_{+}^{0}-v_{\mathrm{f}}$,
when the MTs are confined at the kinetochore. We increase the force-free
growth velocity roughly four-fold to
$v_{+}^{0}=$20\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$$, so that the
minimum AP-velocity
$v_{+}^{0}-v_{\mathrm{f}}=$12\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$$
in the left turning point $F_{\mathrm{min}}$ lies below the observed mean
velocity of $\sim$16\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$$. In order
to adjust the maximum AP-velocity, we reduce the characteristic force in MT
growth to $F_{+}=$5\text{\,}\mathrm{p}\mathrm{N}$$, which leads to a steeper
upper branch in the phase space diagram. The force-free shrinking velocity
$v_{-}^{0}$ should be smaller than the observed P-velocity since the lower,
P-directed branch always lies above it. Analogously to the upper branch and
$F_{+}$, also the slope of the lower branch can be adjusted by varying the
characteristic force $F_{-}$: An increase of $F_{-}$, i.e. a decrease of its
absolute value, steepens the lower branch and thereby slows down the poleward
motion. It turns out that it is a good choice to keep the values for
$v_{-}^{0}$ and $F_{-}$ from table 2 unchanged. Finally, we reduce the rescue
rate $\omega_{\mathrm{r}}^{0}$, which lets MTs shrink to smaller lengths
$x_{\mathrm{m}}$ (the minimum of the master curve is shifted downwards) and
increases the MT-kinetochore distance $|x|=|X_{\mathrm{k}}-x_{\mathrm{m}}|$ to
a realistic value.

Since we enable detachment in this section, we set $M=35$ as it results in a
mean number of $\sim 20$ attached MTs. Finally, we adjust the strength of PEFs
$k$ and the cohesin bond stiffness $c_{\mathrm{k}}$ to the following
conditions: Firstly, the PEFs have to be strong enough to assure proper
chromosome alignment at the equator as well as a regular oscillation pattern,
but should not dominate compared to the inter-kinetochore tension in order to
prevent antiphase oscillations. Secondly, $k$ and $c_{\mathrm{k}}$ affect the
amplitude and the frequency of kinetochore oscillations which should resemble
experimental results in the same manner: An increase of both $k$ and
$c_{\mathrm{k}}$ decreases the amplitude and increases the frequency. We find
that $k=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$
and
$c_{\mathrm{k}}=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$ fulfill both conditions. In table 5, we list all parameters that we
have changed compared to table 2.

Table 5:  Parameters to reproduce of kinetochore oscillations in PtK1 cells.
Parameters not listed here have been unchanged compared to table 2.

  * Description | Symbol | Value
---|---|---
zero force rescue rate | $\omega_{\mathrm{r}}^{0}$ | $0.012\text{\,}{\mathrm{s}}^{-1}$
zero force MT growth velocity | $v_{+}^{0}$ | $20\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$
characteristic force of MT growth | $F_{+}$ | $5\text{\,}\mathrm{p}\mathrm{N}$
catastrophe rate at the kinetochore | $\omega_{\mathrm{c}}^{\mathrm{kin}}$ | $0.019\text{\,}{\mathrm{s}}^{-1}$
MT flux velocity | $v_{\mathrm{f}}$ | $8\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$
PEF coefficient | $k$ | $20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$
cohesin bond stiffness | $c_{\mathrm{k}}$ | $20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$
MT-kinetochore linker stiffness | $c$ | $20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$
number of MTs | $M$ | 35

The resulting kinetochore dynamics is shown in figure 14. The simulated
kinetochore trajectories in figure 14(a) are very similar to the experimental
results in [11, 19] as they exhibit frequency doubling of breathing compared
to single kinetochore oscillations and move predominantly in phase, i.e. there
is a leading P- and a trailing AP-kinetochore (state $2/2^{\prime}$ in figure
3). The motion of the inter-kinetochore distance is rather fluctuative,
resulting in a broad Fourier transform, in which the maximum at the breathing
frequency is hardly recognizable, see figure 14(d). This is the only
significant difference to the real kinetochore motion. The distributions of
kinetochore positions as well as inter-kinetochore and MT-kinetochore
distances (figure 14(e-g)) are in good agreement with experimental results
[19].

Figure 14:  Reproduction of kinetochore oscillations in PtK1 cells. (a)
Kinetochore positions and inter-kinetochore distance over time. Although the
breathing oscillations are rather fluctuative, frequency doubling is
recognizable. (b) Number of attached MTs over time. (c) Kinetochore motion in
phase space (green) compared to the mean-field force-velocity relation (red,
calculated with the mean number of attached MTs). For an animated version see
video S7 in the supplementary material. (d) Position of the right kinetochore
and inter-kinetochore distance in Fourier space. Fluctuative breathing
oscillations lead to a Fourier transform with broad maxima, which are almost
only recognizable in the smoothed curve (dark blue). (e-h) Distributions of
kinetochore positions $X_{\mathrm{k}}$, inter-kinetochore distance $\Delta
X_{\mathrm{k}}$, MT-kinetochore distance $|x|$, and the number of attached MTs
$M^{\mathrm{att}}$.

In table 6, we list several characteristic quantities of kinetochore
oscillations that have also been determined experimentally for PtK1 cells.
Comparison with our model results shows quantitative agreement. In particular,
the large discrepancy in the P- and AP-velocities is eliminated.

Table 6:  Characteristic quantities of model kinetochore oscillations compared
to experimental results in PtK1 cells.

  * Description | Model | Experiment |
---|---|---|---
mean P velocity | $21.5\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$ | $19.0\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$ | [11]
mean AP velocity | $15.7\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$ | $15.7\text{\,}\mathrm{nm}\text{\,}{\mathrm{s}}^{-1}$ | [11]
single kinetochore frequency | $4.27\text{\,}\mathrm{mHz}$ | 4.14–$4.23\text{\,}\mathrm{mHz}$ | [11]
breathing frequency | $\sim$8.6\text{\,}\mathrm{mHz}$$ | $8.25\text{\,}\mathrm{mHz}$ | [11]
mean inter-kinetochore distance | $1.83\pm 0.42\text{\,}\mathrm{\SIUnitSymbolMicro m}$ | $1.90\pm 0.44\text{\,}\mathrm{\SIUnitSymbolMicro m}$ | [19]
mean MT-kinetochore distance | $0.081\pm 0.042\text{\,}\mathrm{\SIUnitSymbolMicro m}$ | $0.11\pm 0.04\text{\,}\mathrm{\SIUnitSymbolMicro m}$ | [19]
standard deviation of kinetochore position | $0.76\text{\,}\mathrm{\SIUnitSymbolMicro m}$ | 0.5–$1.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ | [19]
mean number of attached MTs | 21.4 | 20–25 | [43]

##  10 Discussion

We provided an analytical mean-field solution of the one-sided spindle model
introduced by Banigan et al. [20], which becomes exact in the limit of large
MT numbers. The mean-field solution is based on the calculation of the mean
linker extension $\langle x\rangle$ as a function of a constant kinetochore
velocity $v_{\mathrm{k}}$ (the master curve). Together with the equation of
motion of the kinetochore we obtained the force-velocity relation of the one-
sided model from the master curve. Our solution clearly shows that the force
feedback of linkers onto the MT depolymerization dynamics is essential for a
bistable force-velocity relation within the minimal model. The shape of the
distribution $p_{\pm}(x)$ of linker lengths (12) is governed by this force
feedback, and we traced the bistability to the peakedness (kurtosis) of this
distribution.

Bistability of the force-velocity relation in the one-sided model is a
necessary (but not sufficient) condition for oscillations in the two-sided
model. Interpreting the bistable force-velocity relation as phase space
diagram, we mathematically described kinetochore oscillations as an emergent
result of collective dynamics of coupled MTs that exhibit dynamic instability
individually. Our theory becomes exact in the limit of large MT numbers $M$.
This interpretation of oscillations is underpinned by the experimental
observations that kinetochore oscillations in budding yeast [61, 62, 63],
where each kinetochore is attached to one MT [59], as well as in fission yeast
[64, 21], where two to four MTs interact with the same kinetochore [65], have
a considerably more fluctuative character than the regular oscillations in
vertebrate cells [6, 7, 8, 9, 10, 11, 12] with $\sim 20$ MTs per kinetochore
[43, 66]. Moreover, we were able to deduce idealized kinetochore oscillations,
whose periods conform with experimental results [11]. For a MT-kinetochore
linker stiffness
$c=$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$ and
20–25 MTs per kinetochore, we get periods of 206–$258\text{\,}\mathrm{s}$ and
103–$129\text{\,}\mathrm{s}$ for kinetochore and breathing oscillations,
respectively. Our approach reproduced the frequency doubling of breathing
compared to single kinetochore oscillations, observed in the experiment [11].
Both in the model and in the experiment this doubling originates from the
different velocities of AP- and P-moving kinetochores, which ensure that a
P-to-AP switch ($3/3^{\prime}$ in figure 3) always follows an AP-to-P switch
of the same kinetochore ($1/1^{\prime}$ in figure 3). In the model the
velocity difference is, however, much larger. As a consequence, in our model
with 20–25 MTs an AP-to-P switch follows 96–$119\text{\,}\mathrm{s}$ after a
P-to-AP switch of the sister kinetochore, which is $93\text{\,}\mathrm{\char
37\relax}$ of a breathing period, whereas in PtK2 cells a mean interval of
merely $6\text{\,}\mathrm{s}$ has been measured [12]. In other words, in our
model, most of the time both kinetochores move towards each other in AP-
direction (state 0 in figure 3), whereas in the experiment, mostly one
kinetochore moves in P- while the trailing sister is moving in AP-direction
(state $2/2^{\prime}$ in figure 3). In our model, different AP- and
P-velocities are based on the fact that the MT shrinkage is much faster than
growth. The model parameters for MT dynamics were taken from experimental
measurements with yeast kinetochores [2], which, however, are distinct from
metazoan kinetochores in two main points: firstly, they can only attach to one
MT [59]; secondly, the Ndc80 fibrils are connected to MTs via ring-like Dam1
complexes, which do not appear in metazoan cells [60]. We show in section 9
that this discrepancy can be eliminated by adjusting some MT parameters and,
moreover, the model can reproduce kinetochore oscillations in PtK1 cells
quantitatively.

In experiments with HeLa cells Jaqaman et al. [67] observed an increase of
oscillation amplitudes and periods when they weakened the cohesin bond. In our
model, a smaller cohesin stiffness $c_{\mathrm{k}}$ has the same two effects
as the inter-kinetochore distance has to be larger to reach the turning points
$F_{\mathrm{min}}$ and $F_{\mathrm{max}}$ of the hysteresis loop, and the
phase space velocity
$\dot{F}_{\mathrm{kk}}=c_{\text{k}}\left(v_{\mathrm{k,r}}+v_{\mathrm{k,l}}\right)$
and, therefore, the frequencies are proportional to $c_{\mathrm{k}}$.

Our analytical approach also allowed us to go beyond the results of [20] and
quantify constraints on the linker stiffness $c$ and the MT number for
occurrence of bistability in the one-sided model and for the occurrence of
oscillations in the full model. We found that bistability requires linker
stiffnesses above
$c_{\mathrm{bist}}\simeq$8\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$. Bistability is, however, not sufficient for oscillations. Our
phase space interpretation showed that bistability only leads to directional
instability if the two branches of the force-velocity relation are also
separated by the zero velocity line. This condition quantifies the oscillatory
regime in the parameter plane of $c$ and $M$. We predict that oscillations
should only be observable if the MT-kinetochore linker stiffness is above
$c_{\mathrm{osc}}\simeq$16\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro
m}}^{-1}$$. Our model can thus provide additional information on the MT-
kinetochore linkers whose molecular nature is unknown up to now. Several Ndc80
fibrils, which cooperatively bind to the MT, are an important part of the MT-
kinetochore link and the stiffness of this Ndc80 link has been determined
recently using optical trap measurements [68]. These experiments found
stiffnesses above
$\sim$20\text{\,}\mathrm{pN}\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{-1}$$,
which are compatible with our bounds. Moreover, they found a stiffening of the
link under force, which could be included in our model in future work.

The derivation of the lower bound for the stiffness for the occurrence of
oscillations is based on the occurrence of a new zero AP-velocity fixed point
in the force-velocity diagram of the kinetochores, which suppresses
oscillations upon decreasing the stiffness. Also the influence of poleward
flux to the system could be analyzed by a fixed point analysis of the force-
velocity diagram. Since poleward MT flux shifts the force-velocity towards
smaller AP-velocities of the kinetochore, the upper branch may cross zero
velocity establishing again a zero velocity fixed point suppressing
oscillations. This explains why high flux velocities suppress directional
instability and rationalizes the correlation between kinetochore oscillations
and poleward flux observed in several cells (table 4). It has been observed in
newt lung cells that oscillations are occasionally ($11\text{\,}\mathrm{\char
37\relax}$ of time) interrupted by phases in which the kinetochores pause
their motion [6] analogously to resting in the fixed point in our model. This
indicates that the spindle of newt lung cells operates near the boundary
between the oscillatory and the non-oscillatory regime.

Also experimental results in [69, 70, 71, 72] on the effects of
phosphorylation of Hec1, which is part of mammalian Ndc80 complex, onto
kinetochore dynamics can be rationalized by our force-velocity diagram of the
kinetochores. Dephosphorylation leads to hyper-stable MT-kinetochore
attachments, increases the inter-kinetochore distance, damps or completely
suppresses oscillations, and lets the kinetochores more often be found in a
“paused state”. The increase of the inter-kinetochore distance can be
explained with the hyper-stable MT-kinetochore attachments: in the oscillatory
regime, the bistable area of the force-velocity relation increases if more MTs
are attached to the kinetochore (figure 2(b)); in the non-oscillatory regime,
the mean distance $\langle\Delta X_{\mathrm{k}}\rangle$ is a linear function