Chromosome loss rate in cells of different ploidy can be explained by spindle self-organization

The mitotic spindle segregates chromosomes and minimizes chromosome loss for the specific number of chromosomes present in an organism. In Saccharomyces cerevisiae, for example, haploid and diploid cells are part of the sexual life cycle and have a thousand times lower rate of chromosome loss than tetraploid cells. Currently it is unclear what constrains the number of chromosomes that can be segregated with high fidelity in an organism. Here we developed a mathematical model to study if different rates of chromosome loss in cells with different ploidy can arise from changes in (1) spindle dynamics and (2) a maximum duration of mitotic arrest, after which cells enter anaphase. Our model reveals how small increases in spindle assembly time can result in exponential differences in rate of chromosomes loss between cells of increasing ploidy and predicts the maximum duration of mitotic arrest.


INTRODUCTION
Chromosome segregation is an important, highly conserved cellular function. A complex network of interacting components segregates chromosomes with high precision. However, rare errors in chromosome segregation are observed. The error rate generally increases when the number of sets of chromosomes (ploidy) increases within the cell (Comai, 2005). Experimental studies suggest that the spindle is optimized to minimize chromosome loss for the number of chromosomes present in that specific organism (Nannas et al., 2014;Schulman and Bloom, 1993;Storchova et al., 2006). For example, the normal sexual life cycle of the budding yeast Saccharomyces cerevisiae includes haploid ‫ܥ(‬ ൌ 1 6 chromosomes) and diploid cells ‫ܥ(‬ ൌ 3 2 chromosomes). In this organism the effect of ploidy on the rate of chromosome loss is very pronounced: haploid and diploid cells have chromosome loss rates around 10 -6 chromosomes per cell per cell division, whereas tetraploid cells have a loss rate around 10 -3 (Mayer and Aguilera, 1990;Storchova et al., 2006). Moreover, when ploidy levels are changed in laboratory yeast strains, the ploidy levels tend to return to the initial wild-type level in experimental evolution studies (Gerstein et al., 2006;Selmecki et al., 2015). In the wild, however the number of chromosomes per cell varies substantially between species, suggesting that cells can be optimized for different numbers of chromosomes (Otto and Whitton, 2000).
Chromosome segregation is driven by the mitotic spindle, a self-organized micro machine composed of microtubules and associated proteins. During spindle assembly, spindle poles nucleate microtubules, which grow in random directions or in a direction parallel with the central spindle (O'Toole et al., 1997;Winey et al., 1995). A microtubule that comes into the proximity of a kinetochore (KC), a protein complex at the sister chromatids, can attach to the KC and thus establish a link between chromatids and spindle poles (Akiyoshi et al., 2010;Gonen et al., 2012;Hill, 1985;Mitchison and Kirschner, 1985;Volkov et al., 2013). Theoretical studies have quantitatively shown that this process can contribute to spindle assembly (Kalinina et al., 2013;Paul et al., 2009;Vasileva et al., 2017;Wollman et al., 2005). Prior to chromosome separation, all connections between chromatids and the spindle pole must be established, and erroneous KC-microtubule attachments must be corrected for which several mechanisms are proposed (Tubman et al., 2017;Zaytsev and Grishchuk, 2015). These connections are monitored by the spindle assembly checkpoint (Li and Murray, 1991). Once KCs are properly attached and chromosomes congress to the metaphase plate (Gardner et al., 2008), the spindle assembly checkpoint is silenced and microtubules separate the sister chromatids. Cells have a limited time before progressing to anaphase, after which they enter anaphase regardless of erroneous connections, which we refer to as the maximum duration of mitotic arrest, and can result in chromosome loss (Rieder et al., 1994;Rudner and Murray, 1996). Even though a mechanistic picture of spindle assembly is emerging, it is an open question how changes in ploidy can result in a dramatic effect on the rates of chromosome loss.

Model for chromosome loss
In this report we introduce a model for chromosome loss in cells with different ploidy (For description see STAR Methods). We describe populations of cells in prometaphase, metaphase, and anaphase with either all KCs attached to the spindle, or with at least one unattached KC, which can become a lost chromosome ( Figure 1A). Transitions between these populations arise from the dynamics in spindle assembly: chromosome attachment, chromosome detachment, and silencing of the spindle assembly checkpoint ( Figure 1B). In addition, we introduce a function describing a maximum duration of mitotic arrest after which cell enter anaphase regardless whether all chromosomes are attached. This function allows for chromosome loss in our model.

Time course of chromosome attachment and progression to anaphase
To illustrate how chromosome loss occurs during the transition from prometaphase to anaphase, we numerically solve our model first for cells with only one chromosome,

‫ܥ‬
ൌ 1 , for parameters given in Figure 1C. We plot the time course for different populations of cells. Initially, cells have no chromosome attached to the spindle, ݊ ൌ 1 . In prometaphase, when spindle assembly starts and KCs attach to the spindle, the fraction of cells in this population decreases, while the fraction of cells in other populations increases (compare light and dark purple lines in Figure 1D). After an initial increase, the fraction of cells in prometaphase start decreasing as more KCs attach and cells switch to metaphase (compare purple and black lines in Figure 1D). Finally, cells switch to anaphase. The fractions of cells in anaphase increase and asymptotically approach a limit value because the model does not describe cells leaving anaphase (orange and blue lines in Figure 1D). In this case with only one chromosome, the fraction of cells with a lost KC is very low.

Model explains dramatic increase in rate of chromosome loss with increase in ploidy
To explore the relevance of our model for haploid, diploid, and tetraploid yeast cells, we further solve our model for ‫ܥ‬ ൌ 1 6 , 32, and 64 ( Figure 2A). We find that cells with an increasing number of chromosomes spend longer time in prometaphase and metaphase, though the general trend is similar to the case with ‫ܥ‬ ൌ 1 . However, as time progresses, there is a rapid decrease in the fraction of cells in prometaphase and metaphase which corresponds to reaching the maximum time of mitotic arrest,

‫ݐ‬ ൌ ‫ݐ‬
. Because populations of cell with more chromosomes have a larger time lag predominantly in prometaphase they also enter anaphase later ( Figure 2B). This time lag also results in an increasing fraction of cells in anaphase with at least one lost chromosome because these cells have a greater chance to proceed to anaphase without a completely formed spindle.

Spindle assembly time and the rate of chromosome loss increase with number of chromosomes
To explore which processes in spindle assembly are responsible for significant chromosome loss, we determine the relevance of parameters in our model. As our model describes both spindle formation and transition to anaphase, we separately analyze the contribution of each process. We introduce the average time of both prometaphase and metaphase, which we term the time of spindle assembly (STAR Methods). We find that the time of spindle assembly increases with the number of chromosomes, which we plot on a linear scale ( Figure 2C). Cells with more microtubules in the spindle and/or larger KC have shorter time of spindle assembly (Figure S1A, S1B). Next, we explored how ploidy affects chromosome loss. We find that haploid and diploid cells have the same order of magnitude in the fraction of the population with at least one lost chromosome ( Figure 2D). Interestingly, the fraction of cells with at least one lost chromosome increases dramatically for cells with higher ploidy, such as tetraploid cells (C=64). This behavior does not qualitatively change by changing parameters describing the duration of mitotic arrest ( Figure S2A, S2B). In conclusion, linear changes in spindle assembly time result in exponential differences in chromosome loss rate as soon as prometaphase time approaches the maximum time of mitotic arrest.

DISCUSSION
Here we introduced a model by which we explored chromosome loss taking into account key aspects of spindle assembly, including microtubule nucleation, KC attachment/detachment, and spindle geometry, together with a maximum time of mitotic arrest. Our theory provides a plausible explanation for experiments in yeast tetraploid cells where there is a thousand-fold increase in the rate of chromosome loss relative to haploid and diploid cells (Mayer and Aguilera, 1990). Our model may also be relevant to mammalian and plant cell types that also experience increased rates of chromosome mis-segregation with increasing ploidy (Hufton and Panopoulou, 2009).
In yeast cells of different ploidy, chromosome loss can occur for many reasons. Our model predicts that longer duration of spindle assembly, which occurs in cells with more chromosomes, increases chromosome loss. This prediction can be verified by measurements of average spindle assembly time in haploid, diploid, and tetraploid yeast cells. Importantly, key parameters of cytoplasmic microtubule dynamics were measured previously for diploid and tetraploid S. cerevisiae cells, including the rates of microtubule growth, shrinkage, catastrophe and rescue during G1 and mitosis (Storchova et al., 2006). We hypothesize that change in these parameters may cause an increase in the average spindle assembly time in a population of cells, but experimental validation in yeast is needed.
Laboratory tetraploid yeast cells have an increased rate of chromosome loss. However, a recent experimental evolution study with laboratory yeast found that some tetraploid cell lines could maintain their full chromosome complement ‫ܥ(‬ ൌ 6 4 ) for >1000 generations (Lu et al., 2016). The evolved, stable tetraploid cells had elevated levels of Sch9 protein, one of the major regulators downstream of TORC1, which is a central controller of cell growth. Interestingly, the evolved stable tetraploid cells also had increased resistance to the microtubule depolymerizing drug benomyl, indicating that increased Sch9 protein activity may at least in part rescue spindle formation defects observed in wild-type tetraploid cells (Lu et al., 2016;Storchova et al., 2006). This is consistent with our model, where chromosome stability in tetraploid cells can be obtained by increasing the rate of spindle assembly.

STAR+METHODS
Detailed methods are provided in the online version of this paper and include the following:

SUPPLEMENTAL INFORMATION
Supplemental Information includes two figures and can be found with this article online.

AUTHOR CONTRIBUTION
N.P., L.L. and A.M.S. conceived and supervised the project. N.P. and L.L. developed the model, I.J. solved the model. All authors wrote the paper.

REFERENCES
, S a r a n g a p a n i  m  o  s  o  m  a  l  a  t  t  a  c  h  m  e  n  t  s  s  e  t  l  e  n  g  t  h  a  n  d  m  i  c  r  o  t  u  b  u  l  e  n  u  m  b  e  r  i  n  t  h  e  S  a  c  c  h  a  r  o  m  y  c  e  s  c  e  r  e  v  i  s  i  a  e  m  i  t  o  t  i  c  s  p  i  n  d  l Figure 1C.