# Minimal model for collective kinetochore–microtubule dynamics

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Edited by Timothy J. Mitchison, Harvard Medical School, Boston, MA, and approved August 28, 2015 (received for review July 13, 2015)

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## 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. 1*A*), 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 *B*).

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 *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

MTs randomly detach from the kinetochore at force-independent rate *Supporting Information*). MT tips a distance

MTs are pulled away from their kinetochores at poleward flux velocity

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

## 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:**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:**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 *Supporting Information*) attached MTs under an external force, *C*).

Large tensions (**2** we expect stable collective MT growth (*C*. Large compressive forces (*C*.

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 *C*). For *v* is determined by the initial state. If most of the MTs are initially shrinking (growing), then

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,

### 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. 1*A*) exhibits complex dynamics (Fig. 2*A*, *Top*), with both CM and breathing oscillations (red and purple, respectively, in Fig. 2*A*, *Bottom*). These dynamics can be mechanistically understood through the bistable single-kinetochore force–velocity relation (Fig. 1*C*).

Suppose the kinetochores move in the same direction (Fig. 2*B1*) 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. 2*B1*) falls increasingly far behind the leading (poleward-moving) kinetochore (*Right* in Fig. 2*B1*), 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. 2*B2*), so that the kinetochores move toward each other, completing the breathing oscillation. This builds a compressive spring force (Fig. 2*B3*), which induces one of the kinetochores to switch into a shrinking state. In Fig. 2*B4*, the last switch continues the CM oscillation. As indicated by the arrow from Fig. 2*B3* to Fig. 2*B1*, 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

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

### 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. 4*A* 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.

In the experiment, after addition of rapamycin (*A*) compared with *B*; also see Fig. S5) as in ref. 10.

In the model, kinetochore pairs oscillate, as in the experiment (Fig. S6). For both phosphorylated (*B*). The phosphorylated kinetochores disperse more rapidly and to a greater degree than unphosphorylated kinetochores, in accord with the experiments.

To analyze these data, we consider the effective diffusion constant *v*, and oscillation period, *τ*. The metaphase plate disruption in *C*). In the *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. 1*C* 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 *A*).

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 (*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).

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. 1*C* 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. 4*C*) but can strongly perturb kinetochore motions (Fig. 4*B*).

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 *x* is the kinetochore position,

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

where *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. S7*A*) 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. S7*B*).

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. 2*B* 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),

### 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, **S3** can be rewritten as follows:**S4**, we have the following:

### Force-Dependent MT Tip Velocity.

Average MT length,

Using Eqs. **S1** and **S2** to substitute for **S9** comes by rearranging and combining terms in the summation and the third equality comes from noting that

Next, we note that there is a dynamic balance of catastrophes of growing filaments and rescues of shrinking filaments:**S13**, it follows that:

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

## Number of MTs in the Model

In the model, we typically study

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

We study the

The discrete model reaches the large *N* master equation limit for *N*. We find that center-of-mass (CM) oscillations occur far less frequently at

## 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 *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:

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 (

#### 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

#### 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, *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

#### Stiffnesses of springs.

We estimate the spring constants by considering the force and length scales in the system. Individual MTs exert forces of order

For the results presented in this work, we use a slightly softer spring constant of

#### Kinetochore drag.

The kinetochore drag coefficient in the model is ^{−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

Estimates and experimental measurements of the interkinetochore spring constant, ^{−4} to 10^{−2} pN/nm in yeast (55, 56) to

### Numerical Simulation.

The kinetochore–MT system is evolved with an Euler algorithm. During each time step, **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

### 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

### 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

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

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

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

## Acknowledgments

We thank C. L. Asbury and N. S. Wingreen for helpful discussions and A. D. Stephens for critically reading the manuscript. We gratefully acknowledge the support of the National Science Foundation through Grants DMR-1206868 (to E.J.B.) and DMR-1104637 (to E.J.B., K.K.C., and A.J.L.) and the NIH through Grant GM083988 (to M.A.L.). This work was partially supported by a grant from the Simons Foundation (305547, to A.J.L.).

## Footnotes

↵

^{1}Present address: Department of Neurobiology, University of Manchester, Manchester M139PT, United Kingdom.- ↵
^{2}To whom correspondence should be addressed. Email: ajliu{at}physics.upenn.edu.

Author contributions: E.J.B., K.K.C., E.R.B., M.A.L., and A.J.L. designed research; E.J.B., K.K.C., E.R.B., and A.M.M. performed research; E.J.B. performed numerical calculations; K.K.C. performed numerical calculations and performed image analysis; E.R.B. and A.M.M. performed experiments; E.J.B. and K.K.C. analyzed data; and E.J.B., M.A.L., and A.J.L. wrote the paper.

Conflict of interest statement: K.K.C. is a former PhD student of Editorial Board Member Boris Shraiman.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1513512112/-/DCSupplemental.

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