Minimal model for collective kinetochore– microtubule dynamics Edward J. Banigana,b, Kevin K. Chioub, Edward R. Ballisterc,1, Alyssa M. Mayoc, Michael A. Lampsonc, and Andrea J. Liub,2 a Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208; bDepartment of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104; and cDepartment of Biology, University of Pennsylvania, Philadelphia, PA 19104

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

| metaphase | chromosome oscillations | error correction |

M

icrotubule (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. www.pnas.org/cgi/doi/10.1073/pnas.1513512112

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. 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.

CELL BIOLOGY

microtubules Aurora B

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.

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. 1

Present address: Department of Neurobiology, University of Manchester, Manchester M139PT, United Kingdom.

2

To whom correspondence should be addressed. Email: [email protected].

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. 1073/pnas.1513512112/-/DCSupplemental.

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PHYSICS

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.

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+ ðFÞ, or a shrinking state, in which it stochastically depolymerizes at force-dependent rate k− ðFÞ. Each MT can stochastically switch from growing to shrinking at force-dependent catastrophe rate kc ðFÞ, and from shrinking to growing at force-dependent rescue rate kr ðFÞ (Fig. 1B). 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 κ 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 Nκ m ≈ 30 pN/nm (Fig. S1).

A

B

C

MTs randomly detach from the kinetochore at force-independent rate kd,+ while growing and kd,− while shortening. Incorporating force dependence (4, 6) does not alter our main results (Supporting Information). MT tips a distance δ < ℓa from the kinetochore attach at force-independent rate ka. MTs are pulled away from their kinetochores at poleward flux velocity vp (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, Fext (Insets to Fig. 1C). To study chromosome pair oscillations, we connect two kinetochores with a spring with constant κ k < Nκ m (Fig. 1A). 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:

v = ðk+ kr − k− kc Þ=ðkr + kc Þ.

[1] pffiffiffiffi Deviations from this average are small (decreasing as 1= 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 = ðhk+ ihkr i − hk− ihkc iÞ=ðhkr i + hkc iÞ,

[2]

where h · i 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.

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, ℓi , may be in shrinking (flared MTs) or growing (pointed MTs) states. Kinetochores at xL and xR are connected by a chromatin spring. (B) A growing (+) MT with n tubulin subunits can add a subunit at rate k+ ðFÞ or become a shrinking (−) MT at rate kc ðFÞ. A shrinking MT can lose a subunit at rate k− ðFÞ or switch to the growing state at rate kr ðFÞ. (C) The velocity, v, of a kinetochore under an external force, Fext (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− < Fext < F+, both collectively growing and shrinking MT states are stable. The kinetochore exhibits hysteresis depending on its loading history.

12700 | www.pnas.org/cgi/doi/10.1073/pnas.1513512112

In living cells, kinetochores and their N ≈ 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, Fext (Fig. 1C). Large tensions (Fext > 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 (Fext < 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. Banigan et al.

Table 1. Model parameters

When the magnitude of Fext 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− < Fext < 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. 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). 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 (antipolewardmoving) 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). Banigan et al.

k0+ k−0 kr0 kc0 kr0 kc0 kd,+ kd,− ka vp c+ c− cr cc κm κk ζ

Value (ref.) 0.7 s−1 20 s−1 0.02 s−1 0.003 s−1 0.005 s−1 0.005 s−1 10−4 s−1 8 × 10−4 s−1 0.02 s−1 0.15 μm/min 0.18 −0.24 0.32 −0.72

pN−1 pN−1 pN−1 pN−1

(6) (6) (6) (6) (20) (21) (4, 6) (4, 6) (6, 18, 19) (30–32) (6) (6) (6) (6)

0.04 pN/nm (5, 12, 22, 54) 0.04 pN/nm (22, 55, 56) 4 × 10−6 kg/s (57–59)

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 singlekinetochore force–velocity relation with lower kr and higher kc (Fig. 3). 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 Fext = 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. In the experiment, after addition of rapamycin (+Rap) at metaphase, the dynamics are initially superficially similar to those without rapamycin (−Rap) (Movies S1 and S2). However, in under 10 min, +Rap kinetochore pair alignment is disrupted (Fig. 4A) compared with −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. In the model, kinetochore pairs oscillate, as in the experiment (Fig. S6). For both phosphorylated (+Rap) and unphosphorylated (−Rap) conditions, the width of the metaphase plate PNAS | October 13, 2015 | vol. 112 | no. 41 | 12701

CELL BIOLOGY

Rate constants Zero force polymerization rate Zero force depolymerization rate Zero force rescue rate Zero force catastrophe rate Zero force rescue rate (+Rap) Zero force catastrophe rate (+Rap) Detachment rate in growing state Detachment rate in shrinking state Attachment rate Poleward flux velocity Force dependence of rates Force sensitivity of polymerization Force sensitivity of depolymerization Force sensitivity of rescue Force sensitivity of catastrophe Mechanical properties MT–kinetochore spring stiffness Interkinetochore spring stiffness Kinetochore drag

Symbol

PHYSICS

Parameter description

B

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.

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. To analyze these data, we consider the effective diffusion constant D ∼ v2 τ from the characteristic kinetochore speed, v, and oscillation period, τ. The metaphase plate disruption in +Rap experiments and phosphorylated simulations correlates with a shift in the kinetochore speed distribution (Fig. 4C). In the +Rap system, the high-speed tail of the distribution is elevated compared with the −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.

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 +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 +Rap experiments; moreover, partial detachment leading to imbalances in the numbers of MTs attached to sister kinetochores, further amplifies kinetochore dispersion (Fig. S7A). 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).

0

Unphosphorylated Phosphorylated

growing shrinking

v (microns / min)

A

-0.4

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. 12702 | www.pnas.org/cgi/doi/10.1073/pnas.1513512112

-0.8

compression tension -1

0

1

Average Fext per MT (pN)

2

Fig. 3. The bistable region for phosphorylated kinetochores is shifted to higher tensions. With MT rescue and catastrophe rates altered as in singleMT 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 Fext > 0; at Fext = 0, only the shrinking state is stable.

Banigan et al.

p+Rap - p-Rap

C 0.1 0

-0.1 0

Experiment Simulation 1

2

3

v / vavg

4

5

-Rap expt +Rap expt -Rap sim +Rap sim 2

1

0 0

5

10

15

20

Time (min)

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 +Rap (phosphorylated; red) systems than in −Rap (unphosphorylated; blue) systems in experiments (open circles) and simulations (solid points). (C) The difference, p+Rap ðv=vavg Þ − p−Rap ðv=vavg Þ, of probability distributions of normalized kinetochore speeds in the +Rap and −Rap systems shows that high speeds (vJvavg) are more likely in +Rap experiments (dashed line) and simulations (solid line).

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 (+Rap), for example, we alter the rescue and catastrophe rates. Because the oscillation amplitude is A ∼ vτ, 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). 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). Banigan et al.

Materials and Methods Additional Model Details. A growing MT of length ℓ can polymerize, increasing its length to ℓ + σ, or undergo catastrophe, switching it to the shrinking state. A shrinking MT of length ℓ can depolymerize, decreasing its length to ℓ − σ, or be rescued, switching it to the growing state. Tubulin concentration is assumed to be high; growth is reaction limited, and k+ ðFÞ is a pseudo–first-order rate constant. Force dependences are exponential: kα ðFÞ = kα0 ecα F , where kα0 is the zero-force rate. F > 0 is a tensile force and F < 0 is a compressive force. For an attached MT, F = κm ðx − ℓi Þ, where x is the kinetochore position, ℓi is the MT tip position, and κ 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: ζ x_ = −κ m

X i, attached

ðx − ℓi Þ + Fext .

[3]

or  ℓi > x

The equation of motion for the left kinetochore in the two-kinetochore system is the following: ζ x_ L = −κ m

X i, attached

ðxL − ℓi Þ − κk ðxL − xR − Δx0 Þ,

[4]

or  ℓi > xL

where Δx0 is the rest length of interkinetochore spring and the summation runs over MTs originating at the left pole. There is a similar equation for xR. Integration of the equations and estimations of ka , κ m , κ k , ζ, and ℓa 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

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CELL BIOLOGY

3

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.

PHYSICS

B

+17 min

+2 min

Kinetochore position stdev ( m)

Mis12-GFP

A

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.

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.).

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12704 | www.pnas.org/cgi/doi/10.1073/pnas.1513512112

Banigan et al.

Minimal model for collective kinetochore-microtubule dynamics.

Chromosome segregation during cell division depends on interactions of kinetochores with dynamic microtubules (MTs). In many eukaryotes, each kinetoch...
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