The simulation takes place within a sphere that represents the fission-yeast nucleus. Two SPBs are embedded in the nuclear envelope but free to move on the surface of the sphere (although we also consider effects of allowing SPBs to move radially due to a soft nuclear envelope in one variant of the model, as discussed below). Each SPB nucleates 14 MTs, with their minus-ends tethered to the SPBs by a spring and which undergo dynamic instability at their plus-ends. Steric interactions are mediated by short-range hard repulsion between MTs, SPBs, and the nuclear envelope (Figure 1A,B, Appendix 1).

Three classes of motors and crosslinkers assemble the spindle (Figure 1A,B). Kinesin-5 motors (representing Cut7) move bidirectionally on MTs (Edamatsu, 2014; Edamatsu, 2016; Britto et al., 2016; Singh et al., 2018), with plus-end directed movement on antiparallel MTs exerting force to slide apart the SPBs. Kinesin-14 motors (representing Pkl1 and Klp2) crosslink MTs and one head walks toward the MT minus-ends, aligning MTs and exerting force that shortens the spindle (Pidoux et al., 1996; Troxell et al., 2001; Chen et al., 2012; Olmsted et al., 2014; Hepperla et al., 2014; Yukawa et al., 2015; Yukawa et al., 2018). Crosslinkers (representing Ase1) preferentially bind antiparallel MTs (Yamashita et al., 2005; Loïodice et al., 2005; Janson et al., 2007; Kapitein et al., 2008; Courtheoux et al., 2009; Fu et al., 2009) and stabilize MT overlaps when crosslinking near the end of an MT, an effect which mimics the recruitment of stabilizing proteins such as CLASP (Bratman and Chang, 2007) to MT ends.

We represent the multiple outer kinetochore proteins involved in MT binding (Sacristan and Kops, 2015; Musacchio and Desai, 2017) by a single attachment factor that can be bound or unbound to an MT. Because fission-yeast kinetochores can bind up to 3 MTs (Ding et al., 1993), each kinetochore has three attachment factors in the model separated by 40 nm along the kinetochore plate (Figure 1C, Appendix 1—figure 1). Attachments are constrained so that no more than one attachment factor can bind to the same MT plus-end. The attachment factor is a 54-nm-long spring that exerts force on the MT and kinetochore when stretched or compressed (Tables 4 and 5). Attachment factors can make both lateral and end-on attachments to MTs, with different binding kinetics that favor end-on attachment. Importantly, the model includes tip tracking: a tip-bound attachment factor tracks MT ends by maintaining end-on attachment during MT growth and shrinking. The attachment factor also includes a plus-end-directed kinetochore motor, representing the measured contribution of kinetochore-localized dimeric Cut7 to chromosome alignment (Akera et al., 2015). End-on attachment alters MT dynamic instability and is force-dependent, as measured previously (Akiyoshi et al., 2010; Miller et al., 2016).

Physically each kinetochore is a rectangular plate of length 150 nm, width 50 nm, and zero thickness (Figure 1C; Ding et al., 1993) with a steric repulsion with MTs. Sister kinetochores are linked via springs that resist stretching and rotation, to maintain the distance and alignment of the kinetochores (Figure 1C, Appendix 1—figure 1; Mary et al., 2015; Smith et al., 2016). The pericentric DNA is modeled as a spherocylinder of length 200 nm and diameter 75 nm, which has a soft repulsion with MTs that allows MT-chromatin overlap with an energy penalty (Appendix 1).

With these ingredients, the model can achieve both correct and erroneous kinetochore-MT attachment states (Figure 1D). To achieve error correction and persistent biorientation, we found three key model ingredients were required: progressive restriction of attachment (Figure 1E), destabilization of misaligned attachment (Figure 1F), and stabilization of attachment by force (Figure 1G, Appendix 1). With these mechanisms, the model exhibits both spindle assembly and chromosome biorientation (Figure 1H, Video 1).

To constrain model parameters, we developed multiple tests of simulation performance based on live-cell imaging, electron microscopy, and biorientation. First, we quantified the dynamics of spindle length and kinetochore position by confocal fluorescence light microscopy (Figure 2; Gergely et al., 2016; Blackwell et al., 2017a). Cells with low-level labeling of MTs with mCherry-atb2 (Yamagishi et al., 2012; Blackwell et al., 2017a) and the cen2-GFP marker on the centromeric DNA of chromosome 2 (Yamamoto and Hiraoka, 2003) allowed imaging of spindle length and centromere position (Appendix 1). The Cen2 marker is displaced only 125 nm on average from the kinetochore (Gay et al., 2012), allowing quantification of the position of a single pair of sister kinetochores. We measured spindle length and kinetochore position by fitting Gaussian spots and lines to detect features, and then tracked spindle length and kinetochore position over time using previous methods (Appendix 1; Jaqaman et al., 2008). Second, we used previously published electron tomographic reconstructions of fission yeast spindles (Grishchuk and McIntosh, 2006; McIntosh et al., 2013) to measure spindle structure (Blackwell et al., 2017a). Third, we quantified how successfully the models biorient chromosomes, measured by the fraction of simulation time during which all the chromosomes are bioriented and the average number of end-on attachments.

We combined these measures of simulation performance in a fitness function which quantifies the overall success of each simulation run with a set of model parameters. We then varied poorly constrained model parameters to maximize the fitness function. The optimized parameters defined the reference model (Appendix 1).

To understand the mechanisms most important for proper chromosome alignment on the spindle, we developed a computational model of fission-yeast mitosis (Figure 1) that includes spindle MTs nucleated from SPBs, crosslinking motors, passive crosslinkers, pericentric chromatin, and kinetochores, all contained within a spherical nucleus (Materials and methods, Figure 1A,B). Kinetochore-MT binding occurs via attachment factors that represent MT-binding kinetochore proteins (Figure 1C), which can form both correct and erroneous MT-kinetochore attachments (Figure 1D). Kinetochore-MT attachments progressively restrict in angle as MTs bind (Figure 1E), a mechanism motivated by previous work on kinetochore geometry and chromosome rotation in error correction (Gregan et al., 2007; Rumpf et al., 2010; Paul et al., 2009; Magidson et al., 2015; Zaytsev and Grishchuk, 2015). In particular, work on the S. pombe monopolin complex has proposed that monopolin acts as a site-clamp that co-orients MTs bound to the same kinetochore (Gregan et al., 2007). To correct attachment errors, we included destabilization of improper attachments and tip-enhanced catastrophe (Figure 1F), mimicking the effectsof Aurora B kinase (DeLuca et al., 2006; Cimini et al., 2006; Gay et al., 2012) and recapture of lost kinetochores by MT depolymerization (Grishchuk and McIntosh, 2006; Franco et al., 2007; Gachet et al., 2008; Gao et al., 2010; Gergely et al., 2016). To maintain biorientation, we implemented force-dependent kinetochore-MT attachment kinetics (Figure 1G), based on previous work that demonstrated an increase in attachment lifetime with tension when kinetochores are attached to depolymerizing MTs (Akiyoshi et al., 2010; Miller et al., 2016). For further details of the construction of the model, see Materials and methods and Appendix 1. With these ingredients, the model is able to spontaneously assemble a bipolar spindle starting with side-by-side SPBs, form MT-kinetochore attachments, correct attachment errors, and biorient the chromosomes (Figure 1H, Video 1).

To refine and test the model, we measured spindle assembly and chromosome alignment in fission yeast (Figure 2, Materials and methods, Appendix 1). We quantified spindle length, SPB-kinetochore separation, and interkinetochore stretch from the onset of mitosis until chromosome segregation (Figure 2A–D) and used these data to adjust model parameters (Materials and methods, Appendix 1). After refinement, simulations of the reference model showed dynamics of SPB separation, kinetochore movement along the spindle, and interkinetochore stretch similar to the experimental data (Figure 2E–H, Video 2). As occurs in cells, the dynamics varied from simulation to simulation, but were similar on average (Figure 2I, Appendix 1—figure 2).

After developing the reference model, we verified that single model perturbations recapitulate results from fission-yeast genetics. Kinesin-5 motors are essential for spindle assembly in S. pombe, and temperature-sensitive mutants of the kinesin-5/Cut7 fail to separate spindle-pole bodies (Hagan and Yanagida, 1990; Hagan and Yanagida, 1992; Yukawa et al., 2018; Toda et al., 2018). Consistent with this, when we remove kinesin-5 from the model, SPBs do not separate (Figure 2J). Similarly, the microtubule-associated protein CLASP is essential for spindle assembly in fission yeast, where it is recruited to MT antiparallel overlaps by Ase1 and stabilizes MT dynamics (Bratman and Chang, 2007). When the stabilization of dynamics of crosslinked MTs is turned off in the model, SPBs do not separate (Figure 2K). Chromosome biorientation is abolished in models where the SPBs do not separate (Figure 2L, Video 2).

We further studied combined perturbations (Figure 2—figure supplement 1) by varying kinesin-5 and crosslinker number in the absence of kinesin-14 (Figure 2—figure supplement 1A) and by varying kinesin-5 and −14 number in the absence of crosslinkers (Figure 2—figure supplement 1B). Kinesin-14 in our models combines the functions of fission-yeast Pkl1 and Klp2, neglecting the anchoring of MT minus-ends to SPBs by Pkl1 previously measured (Olmsted et al., 2014; Syrovatkina and Tran, 2015; Yukawa et al., 2015; Yukawa et al., 2018). Experimentally, cells lacking Klp2 or both Pkl1 and Klp2 do not show altered average spindle length (Syrovatkina et al., 2013; Troxell et al., 2001). Consistent with this, model spindles formed and bioriented chromosomes in the absence of kinesin-14, and spindle length depended on the ratio of kinesin-5 to crosslinkers.

In fission yeast, Ase1 deletion cells assemble spindles (Yamashita et al., 2005; Syrovatkina et al., 2013; Yukawa et al., 2019). To test if our model correctly reproduced these results, we removed the crosslinker from the model and varied the number of kinesin-5 and kinesin-14 molecules present (Figure 2—figure supplement 1B). Removing crosslinkers in the reference model abolished spindle assembly because spindles cannot maintain robust antiparallel MT overlaps. However, in the reference model the kinesin-14 motors are highly sensitive to force-dependent unbinding: the characteristic distance that quantifies this is 3.2 times larger for kinesin-14 motors than kinesin-5 motors. This leads to kinesin-14 motors that unbind relatively easily under force, and they fail to maintain microtubule antiparallel overlaps necessary for bipolar spindle assembly. When we model the kinesin-14 motors with the same force sensitivity to unbinding as for the kinesin-5 motors, spindle formation and chromosome biorientation are rescued (Figure 2—figure supplement 1C).

Most of our simulations represent the nuclear envelope as a rigid sphere with the SPBs constrained to move on the surface of this sphere. However, constraining SPBs to a fixed radius alters force balance on the spindle and may alter spindle length. Therefore, we tested a model of a soft nuclear envelope by allowing the SPBs to move radially in a potential that mimics the energy required to deform the nuclear envelope (Rincon et al., 2017; Lamson et al., 2019) (Materials and methods, Appendix 1). The results show that a soft nuclear envelope leads to slightly longer spindles (Figure 2—figure supplement 1D, Video 3), but for a physically realistic nuclear envelope force of around 17 pN, spindle length remains near 3 μm, as measured experimentally.

Our simulations start in a state mimicking early mitosis with monotelic chromosomes, then spontaneously assemble a bipolar spindle and biorient chromosomes. Biorientation requires the model to correct attachment errors and maintain correct attachments. This occurs in the simulations primarily through progressive restriction of attachment angle, misaligned destabilization, and force-dependent kinetochore-MT attachment.

To facilitate correct initial attachment of MTs to kinetochores, the model progressively restricts the angle at which binding can occur as more MTs bind (Figure 1E). This is motivated by previous work demonstrating that kinetochore geometry and chromosome rotation play an important role in promoting correct kinetochore-MT attachment and correcting errors (Gregan et al., 2007; Rumpf et al., 2010; Paul et al., 2009; Magidson et al., 2015; Zaytsev and Grishchuk, 2015). We have extended previous work to include both multiple MT binding sites per kinetochore and changes in kinetochore geometry upon binding. In our model, unattached kinetochores have a wide angular range over which attachments can form (modeled as an angular spring constant for binding, represented by the three wide cones in Figure 1E left). Each attachment formed narrows the angle allowed for the subsequent attachment, favoring attachment to MTs that are more perpendicular to the kinetochore plate (represented by the narrower cones in Figure 1E right). Attachments exert an alignment force/torque on kinetochores and MTs based on the stiffness of this angular spring.

To illustrate the importance of progressive restriction, we removed it, making the angular range identical for all three kinetochore-MT attachment events (Figure 3A, Video 4). Doing this nearly abolishes biorientation in the model: the fraction of simulation time for which all three chromosomes are bioriented is below 10%, independent the value of the angular spring constant from 1 kB⁢T (almost any angle of attachment is allowed) to 100 kB⁢T (attachment is highly restricted in angle). These failures occur for different reasons as the angular spring constant varies. When attachment angle is most permissive, merotelic attachments form and are not corrected sufficiently rapidly to biorient the chromosomes. When the attachment angle is highly restricted, attachments are unlikely to form at all. Overall, this result shows that in our model progressive restriction of attachment is essential for biorientation.

The progressive restriction model requires that the first binding event be relatively permissive in angle, the second more restricted, and the third highly restricted. To study this, we varied the angular spring constant of each attachment independently (Figure 3B,C, Figure 3—figure supplement 1, Video 4). The model achieves a high fraction of simultaneous biorientation around 70% when the first attachment is maximally permissive (spring constant is 1 kB⁢T); an increase in this spring constant restricts the angle and decreases simultaneous biorientation to below 20% (Figure 3B). This means that for the first attachment, promoting kinetochore binding to any MT is important: initial attachments should be established easily, even if erroneous. By contrast, biorientation is increased when the third (final) binding event is highly restricted (Figure 3C): chromosomes are bioriented in the model <10% of the time when the third attachment is most permissive, but the fraction of simultaneous biorientation increases with the angular stiffness of the third binding site. The second value of the angular potential for progressive restriction was less important (Figure 3—figure supplement 1A): varying it did not significantly change the fraction of simultaneous biorientation.

Because of the importance of progressive restriction in our model, we additionally examined whether varying the number of allowed kinetochore-MT attachments might affect how easily biorientation is achieved, but found no significant effect (Figure 3—figure supplement 1B). In these simulations, we chose how to vary the angular spring stiffness as the number of attachment sites varies. For fewer attachment sites, we chose the lower values of angular spring stiffnesses for progressive restriction that matched the reference stiffness. For increased number of attachments, the later attachments were fixed at an upper limit of 100 kB⁢T. In all cases, chromosome biorientation was not compromised.

Progressive restriction of attachment reduces but does not eliminate erroneous kinetochore-MT attachments. Previous experimental work has shown that merotelic attachments are common in early mitosis and are corrected over time (Cimini et al., 2003) due to increased turnover of kinetochore MTs from the activity of Aurora B kinase (DeLuca et al., 2006; Cimini et al., 2006; Gay et al., 2012). To study this, we considered two different error correction models: biorientation-dependent stabilization and force-dependent stabilization. First, we implemented the rule-based model of misaligned destabilization by accelerating the detachment of kinetochore-MT attachments that are not amphitelic (Figure 1F). Because experimental work has demonstrated a decrease in kinetochore MT turnover by up to a factor of 65 in the presence of Aurora B inhibitors (Cimini et al., 2006), we varied the misaligned destabilization factor in the model, which quantifies the increased turnover of incorrect attachments, over a similar range from 1 to 100 (Figure 3D, Video 4). Consistent with experimental results, biorientation is nearly eliminated in the absence of misaligned destabilization. Biorientation time in the model is maximum when the misaligned destabilization factor is 70, comparable to the experimental value. This demonstrates the importance of error correction in the model.

The biorientation-dependent model has the disadvantage that it cannot test any mechanisms by which incorrect attachments are destabilized. We therefore additionally tested a force-dependent error correction model, based on previous results that kinetochore-MT attachments are stabilized by force (Nicklas and Koch, 1969; Cane et al., 2013). We modeled the kinetics of kinetochore-MT attachments as a function of interkinetochore tension, with the rates decreasing with force (Figure 3E, Video 5), controlled by a a characteristic force for significant stabilization.

The force-stabilization model of error correction that we implemented experiences the initial problem of biorientation (IPBO): a bioriented attachment that has just formed is not under tension, and therefore is not stable (Zhang et al., 2013; Kalantzaki et al., 2015; Tubman et al., 2017). Consistent with this, we found implementing force-dependent stabilization alone did not lead to biorientation. Recent work has suggested that the IPBO may be solved by initial syntelic-like attachments that are end-on between the kinetochore face near a pole, and lateral to the kinetochore farther from that same pole (Kuhn and Dumont, 2017). Therefore, we varied parameters in the model that might facilitate tension generation before biorientation, including the angular spring constants of the interkinetochore spring, the characteristic angular factor for binding high angles to the kinetochore plate, the effective concentration for binding laterally, and the number of kinesin-5 motors, which affect overall spindle force generation. We were able to achieve long-lived biorientation in the force-dependent error correction model with model parameters that favored end-on over lateral attachments, inhibited attachments at high angle, and allowed sister kinetochores to more easily reorient (Table 6).

In this version of the model, we then varied the characteristic force that controls how much attachments are stabilized by force (Figure 3E, Video 5). The characteristic force is the value of the interkinetochore force at which attachments are stabilized by a factor of two, so a small value reflects rapid variation of attachment stability with force, while an infinite value means that attachments are force independent. We found that the model is sensitive to the value of this characteristic force, with best performance of the model at a characteristic force of 1.67 pN. Higher or lower values decrease cumulative biorientation by up to a factor of two.

Once amphitelic kinetochore-MT attachments are formed, they must be maintained for biorientation to persist. Attachments between single MTs and purified budding-yeast kinetochores were altered by force applied to the kinetochore, even in the absence of Aurora kinase (Akiyoshi et al., 2010; Miller et al., 2016). In particular, the kinetochore-MT attachment lifetime increased with tension when kinetochores were attached to depolymerizing MTs, an effect dependent on a TOG protein (Akiyoshi et al., 2010; Miller et al., 2016). Consistent with this, we implemented force dependence of attachments in the model (Figure 1G). This effect is required to maintain biorientation: if we eliminate the force dependence of attachment kinetics, biorientation is nearly abolished in the model (Figure 3F, Video 4). To understand which force-dependent rate is most important for this effect, we added them back to the model one at a time. The increase in attachment lifetime of a kinetochore bound to a shrinking MT is the key force-dependent rate, because making this the only force-dependent lifetime in the model restores nearly all biorientation compared to the model with all rates force-dependent (Figure 3F). This demonstrates that maintenance of biorientation requires kinetochore-MT attachments to persist during MT depolymerization.

Spindle length regulation (Dumont and Mitchison, 2009; Goshima and Scholey, 2010; Syrovatkina et al., 2013; Hepperla et al., 2014; Nannas et al., 2014; Rizk et al., 2014) can be understood using the force-balance model of Saunders and Hoyt in which plus-end directed sliding motors produce outward force, and minus-end directed sliding motors and chromosomes produce inward force (Saunders and Hoyt, 1992; Nabeshima et al., 1998; Goshima et al., 1999; Severin et al., 2001; Tolić-Nørrelykke et al., 2004; Bouck and Bloom, 2007; Stephens et al., 2013; Syrovatkina et al., 2013; Costa et al., 2014; Zheng et al., 2014; van Heesbeen et al., 2014; Syrovatkina and Tran, 2015). The force-balance model has been used in mathematical models of spindles in yeast (Gardner et al., 2005; Gardner et al., 2008; Chacón et al., 2014; Hepperla et al., 2014; Ward et al., 2015; Blackwell et al., 2017a; Rincon et al., 2017; Lamson et al., 2019), and Drosophila (Cytrynbaum et al., 2003; Cytrynbaum et al., 2005; Wollman et al., 2008; Civelekoglu-Scholey and Scholey, 2010) cells. This work has focused on spindle length at steady state, not dynamic changes. However, some fission-yeast mutants exhibit large fluctuations in spindle length in which the bipolar spindle assembles, but then shortens or falls apart, known as spindle collapse (Bratman and Chang, 2007; Griffiths et al., 2008; Choi et al., 2009; Hsu and Toda, 2011; Masuda et al., 2013; Wälde and King, 2014; Syrovatkina et al., 2013; Gergely et al., 2016). Remarkably, fission-yeast double mutants can have wild-type average metaphase spindle length, but much larger fluctuations than wild-type (Syrovatkina et al., 2013). The underlying mechanisms of large spindle length fluctuations have remained unclear, in part because apparently contradictory changes can cause it. For example, deletion of proteins known either to stabilize (Bratman and Chang, 2007) or destabilize MTs (Gergely et al., 2016) can both lead to large spindle length fluctuations. In recent work we examined how deletion of the kinesin-8 motor proteins could contribute to large spindle length fluctuations in fission yeast (Gergely et al., 2016), but a general understanding of this phenomenon is lacking. Therefore, we sought to understand what mechanisms might lead to large length fluctuations.

One key determinant of the magnitude of spindle length fluctuations is the lifetime of kinetochore-MT attachments (Figure 4, Video 6). We quantified the magnitude of length fluctuations by determining the standard deviation in spindle length over time after spindle elongation for each individual simulation of the model, then averaging that standard deviation over multiple model runs with the same parameters. This measure of length fluctuations increases with kinetochore-MT attachment lifetime: the longer the lifetime, the larger the fluctuations (Figure 4A–D).

To understand this result, note that for long-lived attachment, the force exerted by a stretched kinetochore can grow over time to a larger value: long-lived attachment allows multiple MTs to bind per kinetochore, exert greater force, and stretch apart the sisters. This allows larger inward force to be exerted on the spindle by attached kinetochores. Indeed, the average interkinetochore distance increases with kinetochore-MT attachment lifetime (Figure 4D). Thus, slow cycles of attachment and detachment lead to slowly varying force on the spindle that causes its length to fluctuate. In the opposite limit, short-lived kinetochore-MT attachment causes relatively quick turnover, limiting interkinetochore stretch, inward force, and variation in inward force.

Alteration in kinetochore-MT attachment lifetime could occur through multiple molecular mechanisms. To illustrate how this could occur, we considered two perturbations to the model that have downstream effects on both lifetime and length fluctuations (Figure 4E). The first perturbation is a restricted attachment model, in which the angular spring constant of attachment discussed above (Figure 3A) is set to 100 kB⁢T for all attachments. In this case, attachments rarely form and when formed, their lifetime is short (<0.05 min on average). As a result, the force produced by interkinetochore stretch is small and does not vary much, leading to small length fluctuations in the model (<0.01μ μm on average). The opposite limit can occur in a model in which the force-dependent rescue of kinetochore MTs is greatly reduced, by increasing the force constant from 6.4 pN to 12.8 pN (this reduces the force sensitivity of rescue, see Appendix 1). This causes kinetochore MTs to depolymerize for longer time, and because kinetochore-MT attachments are stabilized during depolymerization, this change dramatically increases the attachment lifetime to 0.2 min. As a result, interkinetochore stretch can increase, and length fluctuations correspondingly increase (0.3 μm).

This analysis suggests that altered kinetochore-MT attachment lifetime could be a downstream effect that may result from the diverse mutations observed to cause spindle length fluctuations in S. pombe. We note that the effect of lifetime may not be the only source of spindle length fluctuations: other mutations that lead to slow changes in force exerted on the spindle could have similar effects.

The force-balance model can explain why multiple perturbations alter steady-state spindle length, including mutation of motors and microtubule-associated proteins (Syrovatkina et al., 2013; Hepperla et al., 2014), and chromosome/kinetochore number and chromatin stiffness (Chacón et al., 2014; Nannas et al., 2014). However, it can be challenging to distinguish direct from indirect effects of altering force balance. For example, the force-balance model posits that minus-end-directed kinesin-14 motors contribute inward force that shortens the spindle, so their deletion would be expected to lead to longer spindles. However, in budding yeast, kinesin-14 deletion instead leads to shorter spindles, because kinesin-14 helps bundle spindle MTs, allowing kinesin-5 motors to generate greater outward force when kinesin-14 is present (Hepperla et al., 2014). Similarly, kinesin-8 deletion in fission yeast leads to longer spindles, but this is likely due to effects of this motor on MT dynamics rather than direct inward force generation by kinesin-8 (Syrovatkina et al., 2013; Gergely et al., 2016).

To better understand direct and indirect changes in spindle length, we examined the force produced by spindle molecules as the spindle elongates, averaged over many simulation runs (Figure 5, Video 7). In this analysis, we considered each half-spindle separately, and calculated the total force exerted along the spindle axis produced by separate force-generating elements: outward force by kinesin-5 motors, and inward force by kinesin-14 motors, passive crosslinkers, and kinetochore-MT attachments (Figure 5A). We computed spindle length as a function of time (Figure 5B,E,H), force as a function of time (Figure 5C,F,I) and spindle length (Figure 5D,G,J) in the reference, restricted attachment, and weak rescue models.

Force generation by kinesin-5 motors, kinesin-14 motors, crosslinkers, and chromosomes changes significantly as the bipolar spindle assembles. For early time (up to 5 min) when spindles are short (up to 1 μm), motors and crosslinkers exert force that slowly increases in magnitude up to a few tens of pN, but chromosomes exert almost no force (Figure 5C,F,I, Video 7). Because chromosomes are not bioriented on the spindle during initial SPB separation, they do not exert significant inward force. This result is consistent with our previous work, which demonstrated that initial bipolar spindle assembly can occur in a model lacking chromosomes (Blackwell et al., 2017a; Rincon et al., 2017; Lamson et al., 2019).

The outward sliding force produced by kinesin-5 motors increases approximately linearly with spindle length, as the length of antiparallel MT overlaps increases during spindle assembly (Figure 5D,G,J). This agrees with the experimental result that the sliding force generated by kinesin-5 motors is proportional to overlap length (Shimamoto et al., 2015). The inward force generated by kinesin-14 motors is small, as in previous work that has shown that kinesin-14 is less effective at force generation that kinesin-5 (Hentrich and Surrey, 2010) and that in the spindle kinesin-14 may be more important to align spindle MTs than to generate force directly (Hepperla et al., 2014).

During initial spindle assembly, crosslinkers play the primary role of maintaining antiparallel MT overlaps in opposition to the sliding activity of kinesin-5. Remarkably, we find that the inward force generated by passive crosslinkers initially increases with spindle length to approximately 25 pN when the spindle is 0.75 μm long. Beyond this point, the crosslinker force steadily decreases, dropping to near zero within a few minutes (Figure 5C,F,I). This is consistent with previous results on force generation by the crosslinker Ase1, which found large force for small overlaps that drops significantly as overlaps become larger (Lansky et al., 2015). Therefore, our results support a picture of early spindle assembly in which high braking force by crosslinkers on short antiparallel MT overlaps oppose the outward force generated by kinesin-5. This highlights the key role of crosslinkers in early spindle assembly suggested previously (Blackwell et al., 2017a; Rincon et al., 2017; Lamson et al., 2019).

Once the spindle elongates sufficiently to separate SPBs by 1 μm, there is a transition in the primary contributer to spindle force. In this regime, chromosomes biorient and the inward force from interkinetochore stretch becomes significant, balancing outward force from kinesin-5 motors (Figure 5C,F,I). This balance is crucial to setting metaphase spindle length.

To perturb this force balance, we considered two additional models discussed above (Figure 4E) with restricted attachment and weak rescue. When attachment is restricted, chromosomes rarely biorient and the inward force from chromosomes is small for spindles of all length. This leads to unbalanced force from kinesin-5 motors and long spindles (Figure 5E–G, Video 7). When MT rescue is reduced, interkinetochore stretch is larger and the inward force from stretched sister kinetochores increases (Figure 5H–J, Video 7). This leads to shorter metaphase spindle length and a corresponding increase in force from stretched kinetochores.

After developing the model of spindle assembly and chromosome biorientation, we examined what additional mechanisms were required for the model to segregate chromosomes to the poles. Relatively few changes are required for robust chromosome segregation, suggesting that significant new mechanisms are not required in anaphase for chromosome segregation. The rules added to the model for anaphase A include severing the chromatin spring between kinetochores (based on cumulative time the chromosomes are bioriented), stabilization of kinetochore-MT attachment, and depolymerization of MTs (Table 7). With these additions to the model, simulations consistently segregate chromosomes to the poles (Figure 6A–D, Video 8). We compared our simulations to experimental measurements of chromosome segregation, and found similar speed of chromosome movement to the poles and separation of sisters (Figure 6E–G), as expected from the choice of MT depolymerization speed in the anaphase model.