Minimal model for collective kinetochore–microtubule dynamics

Significance

Coordinated metaphase chromosome motions are driven by microtubule (MT) dynamics. MTs stochastically switch between growing and shrinking states with rates that depend on forces and biochemical factors acting at the kinetochore–MT interface. Single-MT behavior is known from in vitro experiments, but it is unclear how many MTs cooperate to control chromosome dynamics. We construct and experimentally test a minimal model for collective MT dynamics. The force dependence of the MTs leads to bistable and hysteretic dynamics. This produces chromosome oscillations and error-correcting behavior, as observed in vivo. Our model provides a mechanistic, predictive framework in which we can incorporate further biological complexity.

Abstract

Chromosome segregation during cell division depends on interactions of kinetochores with dynamic microtubules (MTs). In many eukaryotes, each kinetochore binds multiple MTs, but the collective behavior of these coupled MTs is not well understood. We present a minimal model for collective kinetochore–MT dynamics, based on in vitro measurements of individual MTs and their dependence on force and kinetochore phosphorylation by Aurora B kinase. For a system of multiple MTs connected to the same kinetochore, the force–velocity relation has a bistable regime with two possible steady-state velocities: rapid shortening or slow growth. Bistability, combined with the difference between the growing and shrinking speeds, leads to center-of-mass and breathing oscillations in bioriented sister kinetochore pairs. Kinetochore phosphorylation shifts the bistable region to higher tensions, so that only the rapidly shortening state is stable at low tension. Thus, phosphorylation leads to error correction for kinetochores that are not under tension. We challenged the model with new experiments, using chemically induced dimerization to enhance Aurora B activity at metaphase kinetochores. The model suggests that the experimentally observed disordering of the metaphase plate occurs because phosphorylation increases kinetochore speeds by biasing MTs to shrink. Our minimal model qualitatively captures certain characteristic features of kinetochore dynamics, illustrates how biochemical signals such as phosphorylation may regulate the dynamics, and provides a theoretical framework for understanding other factors that control the dynamics in vivo.

Microtubule (MT) dynamics are critical for cell division. Plus ends of spindle MTs interact with kinetochores, protein complexes that assemble at the centromere of each chromosome, and these dynamic MTs exert forces to move chromosomes. Individual MTs are “dynamically unstable,” spontaneously switching between a polymerizing state and a depolymerizing state (1) with growth, shortening, and switching rates that are regulated by the forces exerted at the MT tips (2–6). For many eukaryotes, however, multiple MTs are connected to each kinetochore, giving rise to collective MT behavior that is not well understood and can be entirely different from the behavior of individual MTs. Here, we develop a model of collective MT dynamics based on the measured force-dependent dynamics of individual MTs.

Accurate chromosome segregation depends on correctly biorienting the kinetochore pairs by attaching sister kinetochores to opposite spindle poles. Properly attached kinetochores undergo center-of-mass (CM) and breathing oscillations that are regulated by collective MT dynamics (7–12). Incorrect attachments—such as syntelic attachment of both kinetochores to the same pole—must be corrected (13–17). Tension may cue this process because bioriented kinetochore pairs are under tension while syntelically attached kinetochores are not (7, 9, 15, 17, 18). Error correction is also mediated by Aurora B kinase phosphorylating MT-binding kinetochore proteins (13–17, 19–21). A consistent theory of metaphase kinetochore–MT dynamics should capture CM and breathing oscillations for correctly attached pairs and elucidate the contributions of tension and phosphorylation to syntelic error correction.

Several models suggest that chromosome oscillations result from competition between poleward MT-based pulling and antipoleward “polar ejection” forces (22–24). Another model proposes that oscillations occur via a general mechanobiochemical feedback (25). Models of force-dependent MTs interacting with the same object also exhibit cooperative behavior (5, 26–29). However, these models do not explain error correction dynamics. Thus, the underlying physical mechanisms coordinating metaphase chromosome motions are unclear.

We address these issues by developing a minimal model for collective MT dynamics based on in vitro measurements of single MTs interacting dynamically with kinetochore proteins (4, 6, 20, 21). In the model, MT polymerization and rescue are promoted by tension and inhibited by compression, whereas depolymerization and catastrophe are enhanced by compression and reduced by tension. With just these features, we find a robust and versatile mechanism by which force-dependent MTs coupled to the same kinetochore may drive metaphase chromosome motions. The force–velocity relation for a MT bundle is fundamentally different from that of a single dynamically unstable MT, exhibiting bistable behavior. Bistability gives rise to kinetochore oscillations and is shifted by phosphorylation to produce error correction. The model qualitatively predicts kinetochore motions in our experiments in which Aurora B is hyperactivated in bioriented kinetochore pairs. Thus, we find that many characteristics of metaphase kinetochore dynamics emerge simply from the force coupling of many MTs to the same kinetochore, and chemical signals such as phosphorylation can regulate this physical mechanism.

Mathematical Model

Our many-MT model is composed of a minimal set of mechanical and biochemical processes. The aim is to test whether the simple rules governing individual MT dynamics are sufficient to generate the complex behaviors observed during metaphase.

In the model (Fig. 1A), each kinetochore is associated with N dynamically unstable MTs. Each MT is in either a growing state in which it stochastically polymerizes at force-dependent rate $k_{+}\left(F\right)$, or a shrinking state, in which it stochastically depolymerizes at force-dependent rate $k_{-}\left(F\right)$. Each MT can stochastically switch from growing to shrinking at force-dependent catastrophe rate $k_c\left(F\right)$, and from shrinking to growing at force-dependent rescue rate $k_r\left(F\right)$ (Fig. 1B).

Fig. 1.

Minimal model for kinetochore–MT dynamics and the single kinetochore force–velocity relation. (A) N MTs (red) are attached to each kinetochore (green) by springs. MTs of varying lengths, ${\ell }_i$, may be in shrinking (flared MTs) or growing (pointed MTs) states. Kinetochores at $x_{\mathrm{L}}$ and $x_{\mathrm{R}}$ are connected by a chromatin spring. (B) A growing ($+$) MT with n tubulin subunits can add a subunit at rate $k_{+}\left(F\right)$ or become a shrinking ($-$) MT at rate $k_c\left(F\right)$. A shrinking MT can lose a subunit at rate $k_{-}\left(F\right)$ or switch to the growing state at rate $k_r\left(F\right)$. (C) The velocity, v, of a kinetochore under an external force, $F_{\text{ext}}$ (Insets) has three regimes. For large compressive forces (Left), the kinetochore moves backward as its MTs stably shrink. For large tensions (Right), the kinetochore moves forward as its MTs stably grow. For small forces, $F_{-}<F_{\text{ext}}<F_{+}$, both collectively growing and shrinking MT states are stable. The kinetochore exhibits hysteresis depending on its loading history.

Following the experimental observations of refs. 2 and 6, we assume that tension exponentially enhances polymerization and rescue while exponentially suppressing depolymerization and catastrophe, and compression increases depolymerization and catastrophe while decreasing polymerization and rescue.

Forces are transmitted from the kinetochore to MTs through springs with constant ${\kappa }_m$ attaching the kinetochore to the MTs. Attached MTs push or pull the kinetochore in the x direction via these springs, which model a soft kinetochore–MT interface. The main qualitative result is unchanged if the MTs do not support compression. A detached MT is compressed by the kinetochore if long enough, but otherwise experiences no force. Qualitative results are unchanged up to $N{\kappa }_m\approx$ 30 pN/nm (Fig. S1).

Fig. S1.

The bistable force–velocity relation is a robust feature of the minimal model. The single-kinetochore model exhibits bistability when MTs are attached to the kinetochore by springs with spring constant ${\kappa }_m=$ 0.4 pN/min (10 times the stiffness of the MT–kinetochore springs used to obtain the results in the main text).

MTs randomly detach from the kinetochore at force-independent rate $k_{d,+}$ while growing and $k_{d,-}$ while shortening. Incorporating force dependence (4, 6) does not alter our main results (Supporting Information). MT tips a distance $\delta <{\ell }_a$ from the kinetochore attach at force-independent rate $k_a$.

MTs are pulled away from their kinetochores at poleward flux velocity $v_p$ (30–32) to maintain a positive average tension in the two–coupled-kinetochore system.

To understand collective MT dynamics we consider a single kinetochore and its N MTs, with the kinetochore subjected to an external force, $F_{\text{ext}}$ (Insets to Fig. 1C). To study chromosome pair oscillations, we connect two kinetochores with a spring with constant ${\kappa }_k<N{\kappa }_m$ (Fig. 1A). We study these scenarios using Brownian dynamics (Materials and Methods). All parameter values are listed in Table 1 and Table S1.

Table 1.

Model parameters

Parameter description	Symbol	Value (ref.)
Rate constants
Zero force polymerization rate	$k_{+}^0$	0.7 s−1 (6)
Zero force depolymerization rate	$k_{-}^0$	20 s−1 (6)
Zero force rescue rate	$k_r^0$	0.02 s−1 (6)
Zero force catastrophe rate	$k_c^0$	0.003 s−1 (6)
Zero force rescue rate ($+\text{Rap}$)	$k_r^0$	0.005 s−1 (20)
Zero force catastrophe rate ($+\text{Rap}$)	$k_c^0$	0.005 s−1 (21)
Detachment rate in growing state	$k_{d,+}$	10−4 s−1 (4, 6)
Detachment rate in shrinking state	$k_{d,-}$	8 × 10−4 s−1 (4, 6)
Attachment rate	$k_a$	0.02 s−1 (6, 18, 19)
Poleward flux velocity	$v_p$	0.15 μm/min (30–32)
Force dependence of rates
Force sensitivity of polymerization	$c_{+}$	0.18 pN−1 (6)
Force sensitivity of depolymerization	$c_{-}$	−0.24 pN−1 (6)
Force sensitivity of rescue	$c_r$	0.32 pN−1 (6)
Force sensitivity of catastrophe	$c_c$	−0.72 pN−1 (6)
Mechanical properties
MT–kinetochore spring stiffness	${\kappa }_m$	0.04 pN/nm (5, 12, 22, 54)
Interkinetochore spring stiffness	${\kappa }_k$	0.04 pN/nm (22, 55, 56)
Kinetochore drag	ζ	4 × 10−6 kg/s (57–59)

Table S1.

Additional model parameters

Parameter description	Symbol	Value (ref.)
Tubulin subunit size	σ	10 nm
MT–kinetochore interaction range	$l_a$	200 nm (61)
Number of kinetochore MTs	N	30 (37, 60)
Interkinetochore spring rest length	$\mathrm{\Delta }{x}_0$	1,000 nm
Spindle length	L	2,000σ

Results

Relation Between Kinetochore and MT Velocities.

To understand MT collective dynamics, we calculated the steady-state velocity of a single kinetochore with many attached MTs. In steady state, the kinetochore velocity is the average velocity of the MTs attached to the kinetochore (Supporting Information).

We first consider the case in which MT rate constants are independent of the force applied at the MT tip so that the dynamics of the individual MTs are decoupled. Hill (33) calculated the mean velocity of independent MTs to be the following:

$$v=\left(k_{+}{k}_r-k_{-}{k}_c\right)/\left(k_r+k_c\right).$$ [1]

Deviations from this average are small (decreasing as $1/\sqrt{N}$). Thus, the behavior of a kinetochore with many attached MTs is entirely different from that of a kinetochore with a single MT even when individual MT dynamics are decoupled. Instead of dynamic instability, the kinetochore with many independent MTs has a stable steady-state velocity given by Eq. 1.

In reality, MT rate constants depend on the forces applied to the MT tips (2–6). Generalizing Eq. 1 (Supporting Information), we find the average collective velocity of force-dependent MTs:

$$v=\left(\langle k_{+}\rangle \langle k_r\rangle -\langle k_{-}\rangle \langle k_c\rangle \right)/\left(\langle k_r\rangle +\langle k_c\rangle \right),$$ [2]

where $\langle \cdot \rangle$ denotes an average over the steady-state distribution of MTs about the kinetochore. Because the rates depend exponentially on force, the force-dependent velocity (Eq. 2) can be very different from the force-independent velocity (Eq. 1).

Coupled Dynamically Unstable Microtubules Are Collectively Bistable.

In living cells, kinetochores and their $N\approx 30$ MTs experience forces due to their connections to other kinetochores and spindle components (7–9, 12, 34). To understand collective MT behavior under these conditions, we computed the steady-state velocity of the model kinetochore with $N=30$ (Supporting Information) attached MTs under an external force, $F_{\text{ext}}$ (Fig. 1C).

Large tensions ($F_{\text{ext}}>F_{+}$) favor polymerization and rescue, so from Eq. 2 we expect stable collective MT growth ($v>0$); we observe this at the far right in Fig. 1C. Large compressive forces ($F_{\text{ext}}<F_{-}$) promote catastrophe and depolymerization, so the MTs should collectively shrink ($v<0$), as we observe at the far left in Fig. 1C.

To understand why growth at high tension and shrinking at high compression are stable, consider a collectively growing MT state in which one MT undergoes catastrophe and shrinks. As the kinetochore moves forward and the shrinking MT retracts, tension on the MT increases. However, the pulling force of one shrinking MT cannot overcome the forces exerted by the many growing MTs. Moreover, the small pulling force distributed over many growing MTs is unlikely to induce catastrophe. Instead, tension on the shrinking MT induces its rescue, and the kinetochore continues to move forward.

When the magnitude of $F_{\text{ext}}$ is sufficiently small, the kinetochore has two possible steady-state velocities, with two associated MT distributions (Fig. S2). One velocity is positive (antipoleward) and slow; the other is negative (poleward) and fast (Fig. 1C). For $F_{-}<F_{\text{ext}}<F_{+}$, MT bundles exhibit hysteresis: v is determined by the initial state. If most of the MTs are initially shrinking (growing), then $v<0$ ($v>0$) in the final steady state. The speeds differ because depolymerization is faster and more force-sensitive than polymerization (Table 1). Bistability arises because tension promotes rescue while compression promotes catastrophe. Thus, collective rescue and collective catastrophe require different forces, leading to hysteresis as in the bar magnet model in ref. 35.

Fig. S2.

Probability distributions of MT distances from kinetochore. At $F_{\text{ext}}=0$, due to the bistability of the system, the MT bundle can be in either a shrinking state (red) comprised primarily of depolymerizing MTs or a growing state (green) comprised primarily of polymerizing MTs. Each of these states has a distinct length distribution of filaments, which are shown here in the reference frame of the kinetochore. An MT at a positive distance from the kinetochore is under tension while one at a negative distance is under compression. The widths of the distribution of compressed MTs are approximately given by the minimum of the two lengths, ${\lambda }_{+}={\left(c_{+}{\kappa }_m\right)}^{-1}$ and ${\lambda }_c={\left(c_c{\kappa }_m\right)}^{-1}$. The widths of the distributions of MTs under tension are approximately given by the minimum of the two lengths, ${\lambda }_{-}={\left(c_{-}{\kappa }_m\right)}^{-1}$ and ${\lambda }_r={\left(c_r{\kappa }_m\right)}^{-1}$. Both widths have small corrections due to the velocity of the kinetochore.

Bistability is very different from the dynamic instability of a single MT, which switches between its two unstable states stochastically. The MT bundle cannot switch between growth and shortening stochastically; it requires a large tension, $F>F_{+}$, or compression, $F<F_{-}$, to switch states.

Bistable Dynamics Result in Kinetochore Oscillations.

In vivo, kinetochores exhibit two oscillation modes: CM oscillations, in which the midpoint between the two kinetochores oscillates, and breathing oscillations, in which interkinetochore distance oscillates (7–12). Similarly, our two-kinetochore model (Fig. 1A) exhibits complex dynamics (Fig. 2A, Top), with both CM and breathing oscillations (red and purple, respectively, in Fig. 2A, Bottom). These dynamics can be mechanistically understood through the bistable single-kinetochore force–velocity relation (Fig. 1C).

Fig. 2.

Two coupled kinetochores exhibit CM and breathing oscillations. (A) The positions of the left (blue) and right (black) kinetochores oscillate over time, producing CM (red) and breathing (purple) oscillation modes. The CM trajectory is offset by 1.55 μm for viewing convenience. The system is shown schematically at times labeled 1–4 in B. The kinetochores both move to the right in 1. The right kinetochore, which is in the shrinking state, moves more rapidly than the left kinetochore, which is in the growing state; this stretches the chromatin spring. Due to the tension, the right kinetochore switches to the growing state as in 2. The kinetochores move toward each other as in 3. High compression leads to one of the kinetochores switching to the shrinking state as in 1 or 4, and the cycle repeats.

Suppose the kinetochores move in the same direction (Fig. 2B1) so that one MT bundle rapidly shrinks as the other slowly grows. Due to the difference of speeds, the trailing (antipoleward-moving) kinetochore (Left in Fig. 2B1) falls increasingly far behind the leading (poleward-moving) kinetochore (Right in Fig. 2B1), beginning a breathing oscillation. The interkinetochore spring stretches, and the tension between kinetochores increases. When the tension is large enough, the MTs of the leading kinetochore switch to the collectively growing state (Fig. 2B2), so that the kinetochores move toward each other, completing the breathing oscillation. This builds a compressive spring force (Fig. 2B3), which induces one of the kinetochores to switch into a shrinking state. In Fig. 2B4, the last switch continues the CM oscillation. As indicated by the arrow from Fig. 2B3 to Fig. 2B1, the right kinetochore could switch instead. This occurs with nearly equal probability because kinetochore–MT dynamics depend only weakly on spatial position in the model (Supporting Information).

Phosphomimetic Changes in MT Rescue and Catastrophe Rates Alter Bistability and Lead to Error Correction.

Aurora B kinase is required for reliable correction of syntelic attachment errors (13, 14, 36). In vitro experiments with phosphomimetic mutations of Aurora B phosphorylation sites in kinetochore proteins show that phosphorylation decreases rescue and enhances catastrophe for single MTs (20, 21). To model the effect of Aurora B, we calculated the single-kinetochore force–velocity relation with lower $k_r$ and higher $k_c$ (Fig. 3).

Fig. 3.

The bistable region for phosphorylated kinetochores is shifted to higher tensions. With MT rescue and catastrophe rates altered as in single-MT experiments with phosphomimetic kinetochore proteins (red), the force–velocity curve shifts relative to the curve for unphosphorylated kinetochores (blue). Phosphorylated kinetochores are bistable only for $F_{\text{ext}}>0$; at $F_{\text{ext}}=0$, only the shrinking state is stable.

Eq. 2 suggests that the rate changes due to phosphorylation should favor the shrinking state. Indeed, the force regime of MT bistability shifts to higher tension (red lines in Fig. 3) so that, at zero force, kinetochore motion is poleward.

The shift of the bistability region suggests a mechanism for the MT dynamics observed during syntelic error correction, when MTs shrink while maintaining kinetochore attachment (14). Our unphosphorylated system is bistable at $F_{\text{ext}}=0$; motion may be poleward or antipoleward at zero tension, so that persistent syntelic MT–kinetochore attachments are possible. Under phosphorylated conditions, however, only the collectively shrinking MT state is viable at zero tension.

Phosphorylation Disrupts the Metaphase Plate in Experiments and Simulations.

To challenge the model with a new experimental perturbation, we turned to bioriented metaphase kinetochores, where Aurora B substrates are normally unphosphorylated (17). Small-molecule inhibitors and RNAi have been widely used to inhibit Aurora B. However, to test our model, we wanted to increase Aurora B activity at these unphosphorylated kinetochores. Therefore, we designed a novel in vivo experiment in which Aurora B is recruited to the Mis12 complex in metaphase kinetochores by chemically induced dimerization using the small-molecule rapamycin (Fig. 4A and Figs. S3 and S4; Materials and Methods). We compared the experiment to the two-kinetochore model with the rates corresponding to phosphorylated conditions described above.

Fig. 4.

Experiments and simulations with Aurora B recruited to the outer kinetochore. (A) Images of Mis12-GFP show kinetochores at 2 and 17 min. after addition of rapamycin. (Scale bars: 5 μm.) (B) The metaphase plate broadens more rapidly over time in $+\text{Rap}$ (phosphorylated; red) systems than in $-\text{Rap}$ (unphosphorylated; blue) systems in experiments (open circles) and simulations (solid points). (C) The difference, $p_{+\text{Rap}}\left(v/v_{\text{avg}}\right)-p_{-\text{Rap}}\left(v/v_{\text{avg}}\right)$, of probability distributions of normalized kinetochore speeds in the $+\text{Rap}$ and $-\text{Rap}$ systems shows that high speeds ($v\gtrsim v_{\text{avg}}$) are more likely in $+\text{Rap}$ experiments (dashed line) and simulations (solid line).

Fig. S3.

Schematic showing rapamycin-induced dimerization of Mis12-GFP-FKBP and FRB-INbox-mCherry and recruitment of FRB-INbox-mCherry to kinetochores. Aurora B binds INbox and is also recruited to kinetochores.

Fig. S4.

Images of cells expressing Mis12-GFP-FKBP and FRB-INbox-mCherry. Rapamycin ($+\text{Rap}$) or DMSO ($-\text{Rap}$) were added at $t=0$. (Scale bars: 5 μm.)

In the experiment, after addition of rapamycin ($+\text{Rap}$) at metaphase, the dynamics are initially superficially similar to those without rapamycin ($-\text{Rap}$) (Movies S1 and S2). However, in under 10 min, $+\text{Rap}$ kinetochore pair alignment is disrupted (Fig. 4A) compared with $-\text{Rap}$ kinetochore alignment (Fig. S4). To quantify the width of the metaphase plate, we measured the SD of kinetochore positions (open circles in Fig. 4B; also see Fig. S5) as in ref. 10.

Fig. S5.

Mean absolute deviation of kinetochores from the center of the metaphase plate for $+\text{Rap}$ (red) and $-\text{Rap}$ (blue) experiments. The position, $x_c$, of the center of the metaphase plate is estimated as the average position of the kinetochores. Early time data are used to estimate the orientation of the spindle. The mean absolute deviation, $\langle x_k-x_c\rangle$, of kinetochore positions, $x_k$, from the center grows rapidly for kinetochores in $+\text{Rap}$ experiments but remains small for kinetochores in $-\text{Rap}$ experiments.

In the model, kinetochore pairs oscillate, as in the experiment (Fig. S6). For both phosphorylated ($+\text{Rap}$) and unphosphorylated ($-\text{Rap}$) conditions, the width of the metaphase plate increases with time, indicating increasing disorder (red and blue solid points, in Fig. 4B). The phosphorylated kinetochores disperse more rapidly and to a greater degree than unphosphorylated kinetochores, in accord with the experiments.

Fig. S6.

Trajectories for kinetochores in simulated $+\text{Rap}$ system. In the simulated phosphorylated system, sister kinetochores oscillate as they do in the simulation with unphosphorylated kinetochores (Fig. 2A in the main text).

To analyze these data, we consider the effective diffusion constant $D\sim v^2\tau$ from the characteristic kinetochore speed, v, and oscillation period, τ. The metaphase plate disruption in $+\text{Rap}$ experiments and phosphorylated simulations correlates with a shift in the kinetochore speed distribution (Fig. 4C). In the $+\text{Rap}$ system, the high-speed tail of the distribution is elevated compared with the $-\text{Rap}$ system. We attribute this effect to phosphorylation biasing MTs toward the shrinking state, which has a larger speed than the growing state. Thus, phosphorylation increases D and broadens the metaphase plate.

Discussion

Collective Bistability as an Underlying Mechanism for Metaphase Chromosome Motions.

We have developed a model for collective MT dynamics based on a minimal set of assumptions drawn from in vitro single-MT experiments (2–4, 6, 20, 21). Our model demonstrates how individual MTs, coupled by their interactions with the kinetochore, may cooperate due to the force-dependent rates that govern their behavior.

The coupling of force-dependent MTs leads to a bistable force–velocity relationship (Figs. 1C and 3), in which stable growing and shrinking collective MT states exist at the same applied force. This behavior arises because tension stabilizes individual filaments while compression destabilizes them.

When many MTs are attached to the kinetochore, rescues or catastrophes of individual MTs have little effect on the collective state. The stability of the collective state despite individual variation is consistent with electron microscopy images showing that steadily growing or shrinking MT fibers have mixed populations of MTs (37). The model is also consistent with in vitro experiments observing collective catastrophes of MTs (5).

Bistability is the engine driving dynamical behavior of the model. It is responsible for bioriented kinetochore oscillations, as MTs attached to the poleward-moving kinetochore collectively shrink while MTs attached to the antipoleward-moving kinetochore collectively grow (Fig. 2 B1 and B4). With one additional ingredient—that the collective shortening speed exceeds the collective growing speed—we find that the leading (poleward-moving) kinetochore switches its direction first, in agreement with experimental observations (11, 12).

In our model, bistability is the mechanism for the first stage of syntelic error correction—MT retraction and poleward chromosome motion. Phosphomimetic changes in single-MT rescue and catastrophe rates shift the force–velocity relation (Fig. 3) so that MTs shrink and misoriented kinetochores stably move poleward at zero tension. Without this shift, MTs could instead grow at zero tension, inhibiting error correction, as in experiments with Aurora B inhibitors (14, 36). Increased detachment rate, as observed with phosphomimetic Ndc80 in ref. 21, cannot by itself induce error correction in our model; enhanced catastrophe, also observed in ref. 21, is needed. To describe the $+\text{Rap}$ system, we must also suppress rescue, following in vitro measurements in ref. 20. Thus, we predict that phosphorylation by Aurora B enhances catastrophe and suppresses rescue. However, we cannot rule out that detachment occurs in our $+\text{Rap}$ experiments; moreover, partial detachment leading to imbalances in the numbers of MTs attached to sister kinetochores, further amplifies kinetochore dispersion (Fig. S7A).

Fig. S7.

Behavior of the two-kinetochore system with different detachment kinetics. (A) When the detachment rate is increased (red), the width of the metaphase plate increases more rapidly than it does in the control case (blue). (B) Including exponential force dependence of the detachment rate as found in in vitro experiments (6) does not dramatically alter kinetochore dynamics. However, the trajectories of sister kinetochores (blue and black) and CM (red) and breathing (pink) oscillations are smoother than in the case of force-independent detachment rates. The CM trajectory is shifted upward by 1 μm.

Our results suggest that tension and phosphorylation may jointly regulate error correction. Phosphorylation could induce poleward motion at zero tension. Tension may regulate this process because even under phosphorylated conditions in our model, error correction does not reliably occur for kinetochores under tensions of the order of piconewtons per MT. This is consistent with defects in syntelic error correction observed when tension is maintained by overexpression of the chromokinesin NOD (38).

An experiment looking for bistability would directly test our model. One possibility is a set of in vitro experiments similar to those of Akiyoshi et al. (6), but with multiple MTs attached.

Model Results Are Consistent with Experimental Perturbations.

Our model provides a framework for understanding experimental perturbations via their effects on MT rates and force sensitivities. To model Aurora B recruitment to the kinetochore ($+\text{Rap}$), for example, we alter the rescue and catastrophe rates. Because the oscillation amplitude is $A\sim v\tau$, the enhanced kinetochore speeds lead to larger oscillations and decreased kinetochore alignment (Fig. 4 B and C), consistent with previous in vivo results (10, 18). Our finding is also consistent with experiments showing decreased oscillation speed and amplitude when phosphorylation by Aurora B is suppressed (16).

Our model is consistent with experiments with the kinesin Kif18A, which increases the catastrophe rate (39, 40). In our model, enhanced catastrophe slows the trailing kinetochore but does not affect the already shrinking MTs of the leading kinetochore. Thus, tension between the kinetochores increases more rapidly. This leads to a shorter time between directional switches and, thus, smaller oscillation amplitudes (Fig. S8), as in experiments modulating Kif18a levels (10, 39, 40).

Fig. S8.

Stabilization of metaphase plate with increased catastrophe rate. When the catastrophe rate is increased (pink), the width of the metaphase plate increases less rapidly than it does in the control case (blue).

Experiments also show that the interkinetochore connection plays a role in regulating chromosome motions (10, 41). When the chromatin spring is weakened by depleting the condensin I subunits CAP-D2 (10) or SMC2 (41), the oscillation period increases. Similarly, in our model, with a weaker interkinetochore spring, the spring must stretch (compress) to a longer (shorter) length before reaching the force at which shrinking (growing) MTs collectively undergo rescue (catastrophe), leading to larger oscillation amplitude and period.

A Minimal Model as a Foundation for Additional Complexity.

Our model provides a framework for incorporating additional complexity. Factors that regulate MTs in vivo can be included for a better quantitative description of kinetochore dynamics. These variables can alter dynamics by shifting the bistable force–velocity relation (Figs. 1C and 3). Changes to MT force sensitivities alter the threshold force for directional switches, which affects oscillation amplitudes and periods. MT rate changes can alter oscillation speeds, amplitudes, and periods (Fig. 4 B and C, and Figs. S7–S9). These effects may be subtle (Fig. 4C) but can strongly perturb kinetochore motions (Fig. 4B).

Fig. S9.

Behavior of the two-kinetochore system with a smaller asymmetry in average polymerization and depolymerization rates. Here, $k_{+}^0=7$ s⁻¹, $k_{-}^0=18$ s⁻¹, $c_{+}=0.24$ pN⁻¹, $c_r=0.58$ pN⁻¹, $c_c=-0.60$ pN⁻¹, ${\kappa }_k=$ 0.05 pN/nm, and $\mathrm{\Delta }{x}_0=$ 2,500 nm; all other parameters are as listed in Table 1 in the main text and Table S1. The two kinetochores (blue and black) exhibit CM oscillations (red) with poleward and antipoleward velocities that are similar. Breathing oscillations (pink) are present as well. Here, the CM trajectory is shifted by 3 μm for viewing convenience.

In contrast to other models (22–24), polar ejection is unnecessary to obtain bioriented oscillations in our model. Bistability underlies kinetochore dynamics in our model, whereas polar ejection forces dominate in other models (22–24). Because polar ejection is present in vivo, an important future experimental question is whether the dynamics are primarily regulated by collective bistability or polar ejection. The nonlinear dynamics of the bistability mechanism suggests that kinetochore and collective MT motions may be tunable through subtle changes to MT rates and force sensitivities. We note, however, that model oscillations are not necessarily centered and are not as regular as those observed experimentally, and poleward and antipoleward speeds are highly asymmetric. Spatial cues such as polar ejection, length-dependent rates, and chemical gradients could rectify these issues (7, 34, 38–40, 42, 43). These cues could be included in our model to study phenomena such as oscillations in monopolar spindles and congression (35, 39, 42, 43).

Poleward flux (30–32) is included in our model but is unnecessary for oscillations and error correction. However, higher poleward flux induces higher tension across kinetochore pairs and suppresses oscillations because it moves the system away from the bistable region. This result is consistent with observations in Xenopus extract spindles (44).

Several essential features of the model in ref. 24, such as linker viscosity, multiphasic detachment rates, and sharp thresholds for stalling MT growth/shortening, are not in our model. These effects may lead to a better quantitative description if added to our model, but they are secondary to bistability.

In our model, collective MT dynamics are sufficient to drive complex chromosome motions. Bistability arises from the force dependence of the rates regulating MTs and the coupling between MTs attached to the same kinetochore. Bistability may be regulated by biophysical and biochemical factors. These factors, which control essential metaphase chromosome motions in vivo, can be incorporated into the model via their effects on the rates. Thus, our model provides a framework for understanding cell biological observations of chromosome motions through the physics of collective MT dynamics.

Materials and Methods

Additional Model Details.

A growing MT of length $\ell$ can polymerize, increasing its length to $\ell +\sigma$, or undergo catastrophe, switching it to the shrinking state. A shrinking MT of length $\ell$ can depolymerize, decreasing its length to $\ell -\sigma$, or be rescued, switching it to the growing state. Tubulin concentration is assumed to be high; growth is reaction limited, and $k_{+}\left(F\right)$ is a pseudo–first-order rate constant. Force dependences are exponential: $k_{\alpha }\left(F\right)=k_{\alpha }^0{e}^{c_{\alpha }F}$, where $k_{\alpha }^0$ is the zero-force rate. $F>0$ is a tensile force and $F<0$ is a compressive force. For an attached MT, $F={\kappa }_m\left(x-{\ell }_i\right)$, where x is the kinetochore position, ${\ell }_i$ is the MT tip position, and ${\kappa }_m$ is the spring constant. A detached MT is never under tension but, if sufficiently long, can be compressed by the kinetochore. The overdamped equation of motion for a single kinetochore is the following:

$$\zeta \dot{x}=-{\kappa }_m{\displaystyle {\sum }_{i,\underset{\text{or}\hspace{0.17em}{\ell }_i>x}{\text{attached}}}\left(x-{\ell }_i\right)+F_{\text{ext}}}.$$ [3]

The equation of motion for the left kinetochore in the two-kinetochore system is the following:

$$\zeta {\dot{x}}_{\text{L}}=-{\kappa }_m{\displaystyle {\sum }_{i,\underset{\text{or}\hspace{0.17em}{\ell }_i>x_{\text{L}}}{\text{attached}}}\left(x_{\text{L}}-{\ell }_i\right)-{\kappa }_k\left(x_{\text{L}}-x_{\text{R}}-\mathrm{\Delta }{x}_0\right)},$$ [4]

where $\mathrm{\Delta }{x}_0$ is the rest length of interkinetochore spring and the summation runs over MTs originating at the left pole. There is a similar equation for $x_{\mathrm{R}}$. Integration of the equations and estimations of $k_a,{\kappa }_m,{\kappa }_k,\zeta$, and ${\ell }_a$ are described in Supporting Information.

Stable Cell Line for Rapamycin Inducible Dimerization.

Aurora B activity at kinetochores was manipulated using rapamycin-inducible dimerization (45–48) in a stable cell line expressing Mis12-GFP-FKBP, mCherry-INbox-FRB, and shRNA against endogenous FKBP. FKBP and FRB are dimerization domains that bind rapamycin. Endogenous FKBP depletion improves rapamycin dimerization efficiency (48). Full-length human Mis12 (an outer-kinetochore protein) was used to localize FKBP to kinetochores throughout mitosis. INbox is a C-terminal fragment of INCENP (amino acids 818–918 of human INCENP) that binds and activates Aurora B (49–53). GFP and mCherry were included to visualize kinetochores and the INbox:Aurora B complex, respectively. In this cell line, Mis12-GFP-FKBP and the FKBP shRNA are constitutively expressed; mCherry-INbox-FRB is inducibly expressed using doxycycline (Tet-ON). Additional experimental procedures are provided in Supporting Information.

Effects of Changing Attachment/Detachment Details

Surprisingly, the details of microtubule (MT)–kinetochore attachment and detachment rates do not qualitatively affect the dynamics. For instance, the inclusion of force dependence in the detachment rate smooths kinetochore oscillations (Fig. S7A) but does not qualitatively change the force–velocity relation. We observe error correction even without increasing detachment due to phosphorylation (18–21). Nonetheless, we find that a higher detachment rate further promotes error correction by narrowing the bistable force regime for syntelically attached pairs, making the shrinking state more robust to tension fluctuations near zero force. For bioriented kinetochore pairs in the model, an elevated detachment rate (compared with the normal detachment rate) broadens the metaphase plate even further (Fig. S7B).

We note that enhancing detachment rates and their force sensitivities could play a role in regularizing kinetochore oscillations in our model (i.e., biasing the kinetochores to switch to state 4 in Fig. 2B in the main text instead of state 1). In principle, this would allow MTs to preferentially detach from antipoleward-moving kinetochores (poleward-moving kinetochores can overtake and reattach to detached shrinking MTs in our model). As a result, the MT bundle that has been growing for a longer duration would be more likely to undergo collective catastrophe at a smaller total force (because the force per individual MT is higher with fewer MTs).

Derivation of Steady-State Velocity

Equations of Motion.

In the limit of many MTs bound to the kinetochore, we can write equations of motion for the concentrations (unnormalized distributions), ${\rho }_g\left(\ell \right)$ and ${\rho }_s\left(\ell \right)$, of growing and shrinking MTs of length $\ell$, respectively. For simplicity, we neglect attachment/detachment dynamics in this calculation. In the single-kinetochore system, the equations of motion are as follows:

$$\begin{array}{c}{\dot{\rho }}_g\left(\ell \right)=k_{+}\left(\ell -\sigma \right){\rho }_g\left(\ell -\sigma \right)-\left(k_{+}\left(\ell \right)+k_c\left(\ell \right)\right){\rho }_g\left(\ell \right)\\ +k_r\left(\ell \right){\rho }_s\left(\ell \right),\end{array}$$ [S1]

$$\begin{array}{c}{\dot{\rho }}_s\left(\ell \right)=k_c\left(\ell \right){\rho }_g\left(\ell \right)-\left(k_{-}\left(\ell \right)+k_r\left(\ell \right)\right){\rho }_s\left(\ell \right)\\ +k_{-}\left(\ell +\sigma \right){\rho }_x\left(\ell +\sigma \right),\end{array}$$ [S2]

for each population of MTs, and

$$\dot{x}=-\frac{{\kappa }_m}{\zeta }\left({\displaystyle \sum _{\ell }\left({\rho }_g\left(\ell \right)+{\rho }_s\left(\ell \right)\right)\left(x-\ell \right)}\right)+\frac{F_{\text{ext}}}{\zeta }$$ [S3]

for the kinetochore. In our model, the rates are $k_{\alpha }\left(\ell \right)=k_{\alpha }^0{e}^{c_{\alpha }{\kappa }_m\left(x-\ell \right)}$. For the force-independent case, $c_{\alpha }=0$ so that the rates are independent of $\ell$. However, the following results hold for arbitrary force dependences of rates.

The Kinetochore Velocity Is the Mean Microtubule Tip Velocity.

To see that the kinetochore velocity must be identical to the mean MT tip velocity, consider the steady state in which all of the average filament length changes at a constant rate, ${\partial }_t\langle \ell \rangle \equiv V$. First, Eq. S3 can be rewritten as follows:

$$\dot{x}=-\frac{{\kappa }_m}{\zeta }{\rho }_T\left(x-\langle \ell \rangle \right)+\frac{F_{\text{ext}}}{\zeta }.$$ [S4]

Immediately, we see that, instantaneously, the kinetochore is effectively connected by a spring of strength ${\kappa }_m{\rho }_T$ to a MT bundle tip located at $\langle \ell \rangle$. Then, by taking the time derivative of Eq. S4, we have the following:

$$\ddot{x}=-\frac{{\kappa }_m}{\zeta }\left(\dot{x}-V\right),$$ [S5]

which has a general solution of the following:

$$\dot{x}=V+v_0{e}^{-{\kappa }_m{\rho }_{T}t/\zeta },$$ [S6]

where $v_0$ is an arbitrary constant. Thus, in the steady-state solution ($t\to \infty$ limit), the kinetochore velocity is the mean MT tip velocity: $\dot{x}=V$.

Force-Dependent MT Tip Velocity.

Average MT length, $\langle \ell \rangle$, is given by the following:

$$\langle \ell \rangle =\frac{1}{{\rho }_T}{\displaystyle \sum _{\ell }\ell \left({\rho }_g\left(\ell \right)+{\rho }_s\left(\ell \right)\right)},$$ [S7]

where ${\rho }_T={\displaystyle {\sum }_{\ell }\left({\rho }_g\left(\ell \right)+{\rho }_s\left(\ell \right)\right)}$. Thus, the average tip velocity, ${\partial }_t\langle \ell \rangle$ is given by the following:

$${\partial }_t\langle \ell \rangle =\frac{1}{{\rho }_T}{\displaystyle \sum _{\ell }\ell \left({\dot{\rho }}_g\left(\ell \right)+{\dot{\rho }}_s\left(\ell \right)\right)}.$$ [S8]

Using Eqs. S1 and S2 to substitute for ${\dot{\rho }}_g\left(\ell \right)$ and ${\dot{\rho }}_s\left(\ell \right)$, we have the following:

$$\begin{array}{c}{\partial }_t\langle \ell \rangle =\frac{1}{{\rho }_T}{\displaystyle \sum _{\ell }\ell (k_{+}\left(\ell -\sigma \right){\rho }_g\left(\ell -\sigma \right)-k_{+}\left(\ell \right){\rho }_g\left(\ell \right)}\\ -k_{-}\left(\ell \right){\rho }_s\left(\ell \right)+k_{-}\left(\ell +\sigma \right){\rho }_s\left(\ell +\sigma \right))\\ =\frac{\sigma }{{\rho }_T}{\displaystyle \sum _{\ell }\left(k_{+}\left(\ell \right){\rho }_g\left(\ell \right)-k_{-}\left(\ell \right){\rho }_s\left(\ell \right)\right)}\\ =\frac{\sigma }{{\rho }_T}{\displaystyle \sum _{\ell }\left(\langle k_{+}\rangle {\displaystyle \sum _{\ell }{\rho }_g\left(\ell \right)}+\langle k_{-}\rangle {\displaystyle \sum _{\ell }{\rho }_s\left(\ell \right)}\right)},\end{array}$$ [S9]

where $\langle k_{+}\rangle$ and $\langle k_{-}\rangle$ are, respectively, the polymerization and depolymerization rates averaged over the distribution of MT lengths in the steady state (e.g., see Fig. S2). The second equality in Eq. S9 comes by rearranging and combining terms in the summation and the third equality comes from noting that

$$\langle k_{+}\rangle =\frac{{\displaystyle {\sum }_{\ell }{k}_{+}\left(\ell \right){\rho }_g\left(\ell \right)}}{{\displaystyle {\sum }_{\ell }{\rho }_g\left(\ell \right)}}$$ [S10]

and

$$\langle k_{-}\rangle =\frac{{\displaystyle {\sum }_{\ell }{k}_{-}\left(\ell \right){\rho }_s\left(\ell \right)}}{{\displaystyle {\sum }_{\ell }{\rho }_s\left(\ell \right)}}.$$ [S11]

Next, we note that there is a dynamic balance of catastrophes of growing filaments and rescues of shrinking filaments:

$${\displaystyle \sum _{\ell }{k}_c\left(\ell \right){\rho }_g\left(\ell \right)}={\displaystyle \sum _{\ell }{k}_r\left(\ell \right){\rho }_s\left(\ell \right)}.$$ [S12]

Writing these in terms of $\langle k_c\rangle$ and $\langle k_r\rangle$, we have the following:

$$\langle k_c\rangle {\displaystyle \sum _{\ell }{\rho }_g\left(\ell \right)}=\langle k_r\rangle {\displaystyle \sum _{\ell }{\rho }_s\left(\ell \right)}.$$ [S13]

From Eq. S13, it follows that:

$${\displaystyle \sum _{\ell }{\rho }_g\left(\ell \right)={\rho }_T\frac{\langle k_r\rangle }{\langle k_r\rangle +\langle k_c\rangle }},$$ [S14]

$${\displaystyle \sum _{\ell }{\rho }_s\left(\ell \right)={\rho }_T\frac{\langle k_c\rangle }{\langle k_r\rangle +\langle k_c\rangle }}.$$ [S15]

Combining Eqs. S9, S14, and S15, we have the following:

$${\partial }_t\langle \ell \rangle =\frac{\langle k_{+}\rangle \langle k_r\rangle -\langle k_{-}\rangle \langle k_c\rangle }{\langle k_r\rangle +\langle k_c\rangle },$$ [S16]

which is the force-dependent generalization of the force-independent mean MT tip velocity derived in ref. 33 and in agreement with the expression derived in ref. 26.

Number of MTs in the Model

In the model, we typically study $N=30$ MTs. This is approximately the number of MTs observed to interact with a kinetochore in electron microscopy images of mammalian cells (37, 60). Nonetheless, we also investigated the limits of small and large numbers of MTs.

When only a few MTs interact with the kinetochore, MTs still behave cooperatively but exhibit dynamically unstable behavior instead of stable or bistable behavior. This is in agreement with the in vitro experiments of Laan et al. (5), which find that bundles of up to 11 MTs grow and shrink stochastically in a coordinated manner. Additionally, we find that, for $N\lesssim 10$, it is possible for all of the MTs to detach from the single kinetochore at large tensions or in the bioriented kinetochore system. For $N\gtrsim 10$ MTs, we recover the results discussed in the main text.

We study the $N\to \infty$ limit using a master equation approach, with the various populations of MTs (growing, shrinking, attached, detached) represented in terms of a number density, which is a continuous variable.

The discrete model reaches the large N master equation limit for $N\gtrsim 100$. We find that the single-kinetochore force–velocity relation in this limit is similar to that of the $N=30$ case, with a bistable force regime separating regimes of stable MT shortening and stable MT growth. As expected, the bistable region is wider in the $N\to \infty$ limit. The dynamics of the two-kinetochore system are noticeably different when we increase N. We find that center-of-mass (CM) oscillations occur far less frequently at $N=\infty$ than in the $N=30$ case. Instead, breathing modes dominate the kinetochore motions.

Implicit Spatial Dependence in the Model

Although we do not explicitly include polar ejection forces or MT length-dependent effects in the model, kinetochore motions in the two-kinetochore model depend implicitly, but weakly, on spatial position within the simulated “spindle.”

MTs are confined to a box of $L=\mathrm{2,000}\sigma$, where σ is the length of a tubulin subunit. Thus, MTs cannot grow/shrink to lengths greater than L or less than 0; similarly, kinetochores cannot move to positions outside of the box.

Additionally, as calculated in ref. 33, force-independent MTs emanating from an adsorbed site have exponentially distributed lengths with a characteristic length scale of the following:

$$\lambda =\sigma {\left[\text{log}\left(\frac{k_{-}\left(k_c+k_{+}\right)}{k_{+}\left(k_{-}-k_r\right)}\right)\right]}^{-1}.$$ [S17]

Thus, detached MTs have a preferred average length (i.e., distance between MT tip and spindle pole).

Finally, as shown in Fig. S2, microtubules attached to the kinetochore have a characteristic distribution. This distribution, which is ≈100 nm wide, is deformed if the kinetochores approach one of the simulation boundaries.

SI Materials and Methods

Parameter Estimation.

In this section, we explain how we estimated the following parameters: attachment rate ($k_a$), MT–kinetochore interaction range (${\ell }_a$), MT–kinetochore spring constant (${\kappa }_m$), interkinetochore spring constant (${\kappa }_k$), and the kinetochore drag coefficient (ζ).

Attachment parameters.

The attachment rate was tuned to give high percentage of attached microtubules as observed in in vitro binding assays of kinetochore proteins (6, 18, 19). The attachment range was estimated based the schematic model of the kinetochore proposed in ref. 61, which suggests that the MT–kinetochore interface is a structure of order 100 nm in size.

Importantly, changes of order $50\text{\%}$ in either of these parameters have little effect on the results described in the main text. The model is insensitive to $k_a$, because the detachment rate, $k_{d,\pm }$, is relatively small. The model is insensitive to ${\ell }_a$ because MTs in our model typically bind the kinetochore and remain within a few tens of nanometers of the kinetochore, in accord with reports that the MT–kinetochore interface may be as small as 50 nm (62).

Poleward flux velocity.

Our model parameters are based on data taken from both in vivo and in vitro experiments, and therefore our model does not always yield accurate quantitative results for these parameter combinations. The poleward flux velocity is a case in point.

The poleward flux velocity, $v_p$, is typically experimentally observed to be in the range of 0.3–0.7 μm/min (30–32). We tested several values of poleward flux velocity with the microtubule rates and force sensitivities given in Table 1. Collective bistability is a robust feature of our model and is only quantitatively affected by changes to $v_p$. However, kinetochore oscillations and dispersion of bioriented kinetochores in the phosphorylated model, depend more sensitively on $v_p$. At values $v_p\gtrsim$ 0.5 μm/min, the amplitude of kinetochore oscillations is very small. This is because MTs are stabilized by the rather large tension that is transmitted to their tips by the rapid poleward motion. Even though we still observe bistability for high values of $v_p$, it is insufficient to drive kinetochore oscillations because the bistable region of the force–velocity relation is primarily in the regime of large compressive forces. Suppression of kinetochore oscillations at high $v_p$ is in agreement with what has been observed in Xenopus extract spindles, which do not exhibit oscillations and have poleward flux velocities in excess of 1 μm/min (44).

It is possible to obtain the kinetochore–MT dynamics described in the main text with $v_p\gtrsim$ 0.5 μm/min, but the magnitudes of the force sensitivities, $c_{+},c_{-},c_r,$ and $c_c$, must be decreased accordingly. Our choice of $v_p=$ 0.15 μm/min, although slow, is of the correct order of magnitude, and illustrates that bistability-driven dynamics is compatible with poleward flux.

Stiffnesses of springs.

We estimate the spring constants by considering the force and length scales in the system. Individual MTs exert forces of order $f_m\approx$ 1 pN (5, 54). Because there are $N\approx 30$ MTs per kinetochore, we estimate that the total force on the kinetochore is several tens of piconewtons, which is consistent with the findings of ref. 5. We next note that Dumont et al. (12) observed kinetochore deformations of order ${\delta }_k\approx$ 10 nm during chromosome oscillations. Altogether, this suggests that the entire MT–kinetochore interface has a spring constant of approximately $N{f}_m/{\delta }_k\approx$ 3 pN/nm. Thus, individual MT–kinetochore springs are estimated to have a stiffness of $f_m/{\delta }_k\approx$ 0.1 pN/nm.

For the results presented in this work, we use a slightly softer spring constant of ${\kappa }_m=$ 0.04 pN/nm. However, the qualitative behavior of the model is insensitive to ${\kappa }_m$ up to individual MT–kinetochore spring constant ${\kappa }_m\approx$ 1 pN/nm or total MT–kinetochore interface spring constant $N{\kappa }_m\approx$ 30 pN/nm. As shown in Fig. S1, the single kinetochore force–velocity relation exhibits bistability for ${\kappa }_m=$ 0.4 pN/nm (10 times larger than the value used in the main text). Although ${\kappa }_m\approx$ 1 pN/nm is the approximate upper bound for bistability with the parameters given in Table 1 of the main text, bistability can be observed with stiffer MT–kinetochore springs if the magnitudes of the force sensitivities, $c_{\alpha }$, are decreased.

Kinetochore drag.

The kinetochore drag coefficient in the model is $\zeta =$ 4 × 10⁻⁶ kg/s, similar to an estimate in ref. 59 and the same order of magnitude as an estimate based on refs. 57 and 58. In refs. 57 and 58, the spindle viscosity is estimated as $\eta \approx$ 1 P, and $\zeta =3\pi \eta d_{\text{chr}}$ is the Stokes’ drag for a chromosome of diameter $d_{\text{chr}}\approx$ 1 μm. Although this is a rough estimate, the kinetochore is believed to be insensitive to the drag in the physiological regime (57). Indeed, in our model, we find that the kinetochore drag has little qualitative effect on our results. However, if we increase ζ by a factor of $\gtrsim 100$, the bistability region narrows.

Estimates and experimental measurements of the interkinetochore spring constant, ${\kappa }_k$, span several orders of magnitude, ranging from ${\kappa }_k\approx$ 10⁻⁴ to 10⁻² pN/nm in yeast (55, 56) to ${\kappa }_k\approx$ 0.1 pN/nm (22) in vertebrate cells. Thus, we explore a range of spring constants, ${\kappa }_k$. As discussed in the main text, the interkinetochore spring constant affects the period of oscillations. Additional effects are not noticeable for spring constants within a factor of ≈10 of ${\kappa }_k=$ 0.04 pN/nm, which we typically used in our model.

Numerical Simulation.

The kinetochore–MT system is evolved with an Euler algorithm. During each time step, $\mathrm{\Delta }t={10}^{-4}\hspace{0.17em}\text{s}$, forces are exerted on the kinetochore(s) according to Eqs. 3 and 4 in the main text. The force exerted by each MT is the force exerted on each MT. MT polymerization, depolymerization, rescue, catastrophe, attachment, and detachment are modeled as Poisson processes with force-dependent rates as described above. To incorporate poleward flux, each MT is shortened by a length $v_p\mathrm{\Delta }t$ at the end of each time step. We ran $n\gtrsim 20$ model realizations for all simulation data presented in the figures.

Stable Cell Line Generation and Culture.

The stable HeLa cell line used in this study was generated by recombinase-mediated cassette exchange (RMCE) using the HILO RMCE system obtained from E. V. Makeyev (Nanyang Technological University, Singapore) (63). Cells were cultured in growth medium (DME with $10\text{\%}$ FBS and penicillin–streptomycin) at 37 °C in a humidified atmosphere with $5\text{\%}$ ${\text{CO}}_2$. Cells at ∼60% confluency in a single well of a six-well plate were transfected with 1 μg of donor plasmid plus 10 ng of Cre plasmid pEM784 (see below for plasmid details) using Fugene 6 (Promega). Two days after transfection, 1 μg/mL puromycin was added to the growth medium for selection; puromycin-sensitive cells were eliminated in ∼2 d at this concentration. Puromycin selection was maintained for ∼10 d, after which the cells were cryopreserved and maintained without puromycin. Doxycycline at 125 ng/mL was added to the growth medium 2 d before experiment to induce mCherry-INbox-FRB expression from the TRE Tet-responsive promoter.

Plasmids.

pEM784, expressing nuclear-localized Cre recombinase, and the donor cassette plasmid pEM791 were obtained from E. V. Makeyev (63). The donor cassette used for this study, pERB110, was derived from pEM791, which is designed for inducible expression of micro-RNA (miRNA)-based shRNA and a reporter gene. pEM791 contains the following: a Puro resistance gene (Pur) positioned for constitutive transcription from the $\text{EF}1\alpha$ promoter upstream of the LoxP site in the acceptor locus, the gene for reverse-tetracycline transactivator 3 (rtTA3) constitutively expressed from a UBC promoter, and a tetracycline-responsive element (TRE) promoter for inducible transcription of an artificial miRNA-based shRNA nested in an intron upstream of a GFP reporter gene (63, 64).

For this study, we modified pEM791 for constitutive expression of an additional miRNA and protein sequence. Between the LoxP site and Pur, we added the following: (i) a miRNA-based shRNA against the 3′-UTR of FKBP12, nested within an intron; (ii) Mis12-GFP-FKBPx3 (a tandem trimer of FKBP); and (iii) an internal ribosome entry sequence (IRES). These modifications allowed constitutive polycistronic coexpression of FKBP miRNA, Mis12-targeted FKBP, and the Puro resistance gene from the $\text{EF}1\alpha$ promoter in the acceptor locus. FKBP miRNA was designed using the miR RNAi function within the Block-iT RNAi Designer (rnaidesigner.thermofisher.com/rnaiexpress/). The following oligos were used for FKBP miRNA:

5′-TGCTGATATGGATTCATGTGCACATGGTTTTGGCCACTGACTGACCATGTGCATGAATCCATAT-3′; and

5′-CCTGATATGGATTCATGCACATGGTCAGTCAGTGGCCAAAACCATGTGCACATGAATCCATATC-3′.

mCherry-INbox-FRB was cloned in place of GFP downstream of TRE. No shRNA sequences were added to the empty miRNA backbone in the inducible transcript for this study. The final donor plasmid, pERB110, will be available through the Addgene plasmid repository. The genes for FRB and FKBP were cloned from plasmids pC4EN-F1, pC4M-F2E and pC4-RHE (obtained from Ariad Pharmaceuticals).

Rapamycin Treatment and Image Acquisition.

For live imaging, cells were plated on 22 × 22-mm glass coverslips (no. 1.5; Fisher Scientific) coated with poly-d-lysine (Sigma-Aldrich). Coverslips were mounted in magnetic chambers (Chamlide CM-S22-1; LCI) using l-15 medium without phenol red (Invitrogen) supplemented with $10\text{\%}$ FBS and penicillin/streptomycin. Temperature was maintained at ∼35 °C using an environmental chamber (Incubator BL; PeCon GmbH). All rapamycin experiments were done at a final working concentration of 500 nM rapamycin. Rapamycin was stored as a 500 mM stock in DMSO, then diluted in medium to a 1 mM intermediate dilution in prewarmed L-15 and directly added to cells in the imaging chamber on the microscope. In $-\text{Rap}$ control experiments, warm L-15 with DMSO only was added similarly.

All images were acquired with a spinning disk confocal microscope (DM4000; Leica) with a 100×, 1.4 N.A. objective, an XY Piezo-Z stage (Applied Scientific Instrumentation), a spinning disk (Yokogawa), an electron multiplier charge-coupled device camera (ImageEM; Hamamatsu Photonics), and a laser merge module equipped with 488- and 593-nm lasers (LMM5; Spectral Applied Research) controlled by MetaMorph software (Molecular Devices). The stage positions of several metaphase cells were recorded at the beginning of each experiment. After rapamycin (or vehicle control) treatment, cells were imaged sequentially for 2.5 min each for a total of ∼20 min. Each cell was imaged over a 7-μm z depth (15 slices, 0.5 μm/slice) in GFP, every 3 s for 2.5 min. mCherry was imaged at the first and last time points for each cell to monitor INbox:Aurora B complex recruitment to kinetochores.

Image Analysis.

Kinetochore identification and tracking was performed using ImageJ and the TrackMate extension. Statistics of kinetochore motion were extracted from image data by custom-written MATLAB code available upon request. Metaphase plate variance data were aggregated from 12 $+\text{Rap}$ and 9 $-\text{Rap}$ cells. Kinetochore speed statistics were computed from images with 3-s intervals of six $\pm \text{Rap}$ cells.