Accurate chromosome segregation by probabilistic self-organization
SSaka et al.
Accurate chromosome segregation byprobabilistic self-organization
Yasushi Saka † , Claudiu V Giuraniuc and Hiroyuki Ohkura † * Correspondence:[email protected] Institute of Medical Sciences,School of Medical Sciences,University of Aberdeen,Foresterhill, AB25 2ZD Aberdeen,UKFull list of author information isavailable at the end of the article † Correspondence should beaddressed to YS or HO.
AbstractBackground:
For faithful chromosome segregation during cell division, correct attachmentsmust be established between sister chromosomes and microtubules from opposite spindle polesthrough kinetochores (chromosome bi-orientation). Incorrect attachments of kinetochoremicrotubules (kMTs) lead to chromosome mis-segregation and aneuploidy, which is oftenassociated with developmental abnormalities such as Down syndrome and diseases includingcancer. The interaction between kinetochores and microtubules is highly dynamic with frequentattachments and detachments. However, it remains unclear how chromosome bi-orientation isachieved with such accuracy in such a dynamic process.
Results:
To gain new insight into this essential process, we have developed a simplemathematical model of kinetochore-microtubule interactions during cell division in general, i.e.both mitosis and meiosis. Firstly, the model reveals that the balance between attachment anddetachment probabilities of kMTs is crucial for correct chromosome bi-orientation. With theright balance, incorrect attachments are resolved spontaneously into correct bi-orientedconformations while an imbalance leads to persistent errors. In addition, the model explainswhy errors are more commonly found in the first meiotic division (meiosis I) than in mitosis andhow a faulty conformation can evade the spindle assembly checkpoint, which may lead to achromosome loss.
Conclusions:
The proposed model, despite its simplicity, helps us understand one of theprimary causes of chromosomal instability—aberrant kinetochore-microtubule interactions. Themodel reveals that chromosome bi-orientation is a probabilistic self-organization, rather than asophisticated process of error detection and correction.
Keywords: chromosome segregation; kinetochore; microtubule; mitosis; meiosis; Markovchain; self-organization a r X i v : . [ q - b i o . S C ] J un aka et al. Page 2 of 39
Background
Accurate segregation of chromosomes during cell division is fundamental to life. Errors inthis process result in cell death or aneuploidy. Chromosome segregation is usually veryaccurate. However, mis-segregation occurs at a much higher frequency in cancer cells andoocytes, which is a contributing factor to cancer progression [1] and also a major cause ofinfertility, miscarriages and birth defects such as Down syndrome [2].The key event for chromosome segregation is the establishment of chromosome bi-orientation, in which sister chromatids in mitosis or homologous chromosomes in meiosisattach to the microtubules from opposite spindle poles by kinetochores [3]. Each kineto-chore consists of more than one hundred different proteins assembled on each centromericDNA sequence, many of which are involved in the interaction with microtubules [4]. Chromo-some bi-orientation is a very dynamic process with frequent attachments and detachmentsof microtubules [5, 6, 7, 8].For proper segregation of chromosomes, all kinetochores need to attach to spindle micro-tubules while erroneous attachments must be eliminated before anaphase onset. It is knownthat attachment errors are more frequent in meiosis I (especially in oocytes) than in mitosis[5, 6, 7, 2]. Yet it has not been understood why it is so. Unattached kinetochores act as signalgenerators for the spindle assembly checkpoint, which delays chromosome segregation untilproper bi-orientation is established for all chromosomes [9]. It remains unclear, however,whether improperly attached kMTs are also detected and corrected by the spindle assemblycheckpoint or by an independent mechanism [10].The precise mechanism of chromosome bi-orientation has been under intense investiga-tions. However, it is not yet possible to observe the dynamics of individual microtubules invivo in real time. Mathematical modeling provides a powerful means to study the chromo-some bi-orientation process. Since the discovery of the dynamic instability of microtubules[11], a number of theoretical analyses have provided important insights into the interac-tion between microtubules and kinetochores (for example, [12, 13]). The so-called search-and-capture model explains how dynamically unstable microtubules capture chromosomes[14, 15, 16, 17]. However, the original search-and-capture model did not concern events aftercapture, in particular, erroneous attachments of kMTs and their correction. To address this,Paul et al. put forward a modified search-and-capture model with explicit correction mech- aka et al.
Page 3 of 39 anisms [18]. Gay et al. proposed a stochastic model of kinetochore-microtubule attachmentsin fission yeast mitosis, which reproduced correct chromosome bi-orientation and segregationin simulations [19]. In addition to the kinetochore-microtubule interaction, Silkworth et al. showed that timing of centrosome separation also plays a crucial role for accurate chromo-some segregation [20]; using experimental and computational approaches they demonstratedthat cells with incomplete spindle pole separation have higher rate of kMT attachment errorsthan those with complete centrosome separation. Yet, the question remains unanswered asto how the cell can discriminate between correct and incorrect kMT attachments as theirmodels assumed an explicit bias based on the discrimination of correct versus incorrectconnections.A major impediment to fully understanding the mechanism of chromosome bi-orientationis the lack of a universal theoretical framework that covers the chromosome bi-orientationprocess during eukaryotic cell divisions in general, including both mitosis and meiosis. Herewe present such a universal model of chromosome bi-orientation, which is simple yet appli-cable to any eukaryotic cell division. Firstly, the model reveals that the balance betweenattachment and detachment probabilities of kMTs is crucial for correct chromosome bi-orientation. With the right balance, incorrect attachments are resolved spontaneously intocorrect bi-oriented conformations while an imbalance leads to persistent errors. Therefore,the superficially complex process, chromosome bi-orientation, is in fact a probabilistic self-organization. It implies that the cell does not need to discriminate between correct andincorrect kMT attachments. Moreover, the model explains why errors are more frequent inmeiosis I than in mitosis and how a faulty conformation can evade the spindle assemblycheckpoint by a gradual increase of the number of kMTs. Despite its simplicity, the modelis consistent with a number of experimental observations and provides theoretical insightsinto the origins of chromosomal instability and aneuploidy.
Results and discussion
A probabilistic model of kinetochore-microtubule interaction
A single kinetochore can bind randomly to microtubules from either left or right pole (Fig.1A). We assume a single kinetochore can accommodate up to n microtubules. The processof microtubule attachment/detachment can be represented as a discrete-time Markov chain[21] (Figs. 1B and S1). Each pair of sister chromatids in mitosis has two kinetochores (k aka et al. Page 4 of 39 and k in Fig. 1C). In meiosis I, a pair of sister kinecotochores are physically connected side-by-side and act as one [22, 23]. Therefore, in our model, a bivalent (a pair of homologouschromosomes connected by chiasma) also has two kinetochores in meiosis I. We assumethese two kinetochores interact with microtubules independently. Hence, the state of thekinetochores is represented as r n ( i , j , i , j ), which can be classified into one of five classesaccording to the pattern of microtubule attachments (Fig. 1D). State transitions occur ina stereotypical manner among these classes irrespective of the value of n ≥ r n ( i , j , i , j ) n − i − j n p −→ r n ( i + 1 , j , i , j ) , (1) r n ( i , j , i , j ) i q −→ r n ( i − , j , i , j ) , (2)where 0 ≤ p ≤ / ≤ q ≤ / n . 2 × p is the association probability of a singlemicrotubule to a free kinetochore; q is the dissociation probability of a single kMT.Experimental evidence strongly suggests that tension stabilises the spindle attachment tothe kinetochores in amphitelic states (class 5) [25, 26, 27]. The stabilisation by tension isbrought about by suppression of Aurora B kinase activity towards kinetochore substrates[28, 29, 27, 30] as well as by mechanical ’catch-bonds’ [31, 32]. We model this stabilisationby scaling the transition probabilities of states in class 5 by detachment with the parameter0 ≤ β ≤ ≤ α ≤ α = 0 for simplicity. For mitosis we aka et al. Page 5 of 39 introduce an additional parameter 0 ≤ γ ≤ α = β = 0, transitions out of class 5 are effectively blocked; hence, this Markov processalways ends up in class 5. For additional details of the model, see Supplementary Information(SI) Text. This simple model, which has only six parameters and is exactly solvable, providesa number of analytical insights into how correct chromosome bi-orientation is achieved. Dynamics of chromosome bi-orientation process
The model predicts how long it takes to reach class 5 (amphitelic) from class 1 (free), i.e.mean first passage time [33] (see SI Text). For a given value of q , the mean first passagetime (which is independent of α and β because they only affect transitions out of class 5) isshortest when p is roughly equal to q (Figs. 2A and S3A-D). Thus, the relative dissociationrate ( q/p ratio) of kMTs needs to be balanced for efficient chromosome bi-orientation.The model also predicts the dynamics of the system (Fig. 2B-D for meiosis I and E-G for mitosis). Note that the q/p ratio dictates the dynamics of the Markov chain (Fig.S5). For both mitosis and meiosis in an ideal condition ( p = q = 0 . , α = β = 0; Fig.2B, E), the probability of class 5 steadily increases, asymptotically reaching 1. Notably, inmeiosis I, class 4 (merotelic), and class 3 (syntelic) to a lesser extent, become transientlyprominent (Fig. 2B). Merotelic attachments are indeed frequently observed in prophase toprometaphase of meiosis I in mouse oocytes [7]. By contrast, in mitosis, class 2 (monotelic)becomes predominant before replaced by class 5, although minor fractions of class 3 and4 also appear briefly (Fig. 2E). Together, it explains why meiosis I is more error-pronethan mitosis; it is attributed to the parameter γ — the back-to-back conformation of sisterkinetochores, which biases the kinetochore orientation.If there is no bias in meiosis I (random condition; α = β = 1; Fig. 2C, see also Fig. S4),the probability of class 5 stays low while that of class 4 (merotelic) reaches nearly one halfat steady states. This is because class 4 is by far the largest among the five classes (Fig S2Aand S2B). In mitosis, when the spindle tension is lacking ( β = 1; Fig. 2F), the model predictshigh probability of errors, mainly monotelic (class 2) states, as well as the correct amphitelic(class 5) ones at steady states. When kinetochore-microtubule attachment is stabilised byreducing q , merotelic errors (class 4) persists in both meiosis and mitosis (Fig. 2D, G). Class aka et al. Page 6 of 39 p = 0 . , α = β = 0,the mean first passage times to class 5 are ∼ q = 0 .
01 versus ∼
47 for q = 0 . β = 1) makes amphitelic states (class 5) unstable[25, 26, 27]. Conversely, inhibition or depletion of Aurora B kinase, which over-stabilizeskMT attachments (by reducing q ), caused errors in chromosome alignment and segregation[34, 27, 30, 35, 7]. These observations are consistent with our model predictions in whichimbalance of the q/p ratio causes persistent errors in kMT attachments (Fig. 2). Probability distribution of the number of kMTs over time
Next, we calculated the probability distribution of the number of kMTs over time in differentconditions (Figs. 3A-C and S6 for meiosis I; Fig. S7 for mitosis). We found qualitativelysimilar kMT distributions in mitosis and meiosis I, except the difference in the predictedphenotype in various conditions (Fig. 2B-G). The model predicts that in ’normal’ conditions( p = q = 0 . , α = β = 0) the number of kMTs increases steadily in class 5 while it remainslow in the other classes as their total probability diminishes (Figs. 3A and S7A). This isin agreement with experimental evidence suggesting the gradual increase of kMTs duringprometaphase to metaphase in mitosis [36] and in meiosis I [26]. With smaller q , the numberof kMTs increases not only in class 5 but also in class 4 (Figs. 3B and S7B). This explains whyerrors persists in this condition. Note that when β = 0, the number of kMTs approaches n .Increasing number of kMTs may also switch off the spindle assembly checkpoint in merotelic(class 4) states over time.These model predictions on the probability distribution of the number of kMTs havean important implication in the regulation of spindle assembly checkpoint. Experimentalevidence suggests that intrakinetochore stretching (or kinetochore deformation), which isbrought about by kMT attachments, has a role in relieving the spindle assembly checkpoint[37, 38, 39]. Therefore the predicted gradual increase of kMTs in amphitelic (class 5) states(Figs. 3A and S7A) may switch off the spindle assembly checkpoint progressively. The sameargument applies to merotelic (class 4) states, the probability of which increases when q/p ratio is small (Fig. 2D, G); stabilisation of kMTs (Figs. 3B and S7B) may also inactivatethe spindle assembly checkpoint in merotelic states over time. This provides the explanation aka et al. Page 7 of 39 as to why merotelic orientation evades the spindle assembly checkpoint [40], leading toaneuploidy. Intrakinetochore stretching by kMT attachment, however, does not allow thecell to discriminate between correct (amphitelic; class 5) versus incorrect (non-amphitelic;class 1-4) kMT attachments [10] — the cell does not need to do so because chromosomebi-orientation occurs by probabilistic self-organisation as our model indicates.We also examined how kMT number changes in amphitelic states under low spindle tension( β = 1; Figs. 3C and S7C). Regardless of the classes, the distribution of kMT numberremains low, which makes the transition of the process from one class to another morefrequent. Similar probability distributions of kMT number in meiosis I were obtained when α = β = 1 (Fig. S6A) and α = 1 , β = 0 (Fig. S6B).The exact probability distribution of kMT number at steady states can be derived in thespecial case when α = β = γ = 1: its mean is ¯ N = nρ/ ( n + ρ ) where ρ = 2 p/q ( ¯ N = 5 / p = q, n = 10). We also obtained an analytical approximation of the kMT numberdistribution in class 5 when α = 0:¯ N = ¯ ρ (cid:0) ¯ ρn + 2 (cid:1) n − (cid:0) ¯ ρn + 2 (cid:1) n − n , (3)where ¯ ρ = ρ/β = 2 p/ ( βq ) (Figs. 3D and S8A, B). This formula is valid for both mitosis andmeiosis and provides an analytical explanation as to how tension ( β ) alters the stability ofkMTs by modulating the q/p ratio. Dynamics of multiple chromosomes
The above results concern the behaviour of a single pair of homologous chromosomes. It isnatural to ask how multiple pairs in the cell are bi-oriented simultaneously—we call thisevent ’synchrony’ to distinguish it from anaphase onset. We assumed the system consistsof k independent Markov processes. Let θ t be the probability of a process being in class 5(amphitelic) at time T = t , then the probability of synchrony at T = t is θ tk (see SI Text).The timing of synchrony delays as k increases (Figs. 4A and S3E, solid lines). If thebalance of q/p ratio is broken by reducing q (Figs. 4A and S3E, dashed lines), the timingof synchrony is delayed further (see also Fig. S9). The probabilities of synchrony, however,eventually approach 1 in all of these conditions with β = 0. This implies that delayingthe onset of anaphase could reduce the chromosome mal-orientation and mis-segregation. aka et al. Page 8 of 39
Consistently, Cimini et al. showed that prolonging metaphase significantly reduced laggingchromosomes in anaphase (indicating incorrect kMT attachments) in mitosis [41].We next examined the contribution of α and β to the establishment of synchrony. Fig. 4Bshows the steady-state probability of synchrony in meiosis I as a contour plot. It indicatesthat, to achieve a synchrony reliably at steady states, α and β have to be relatively small. Itis conceivable that, to progress into anaphase, synchrony has to be maintained for a sufficienttime to relieve the spindle assembly checkpoint [10]. Fig. 4C depicts the half-life of synchronyin meiosis I as a contour plot (see also Fig. S3F for mitosis). The half-life increases steeplytowards the small values of α and β . These data suggest that α and β need to be tightlyregulated for efficient chromosome bi-orientation and segregation accuracy. Error correction of kMT attachments in meiosis I
Finally, we asked how many rounds of error correction of kMT attachments occur in meiosisI before the establishment of correct bi-orientation (see SI Text for methods). We calculatedthe number of bi-orientation attempts per bivalent, i.e. the mean number of transitions fromclass 2 or 4 to class 5 before the kinetochore is fully occupied ( r n ( n, , , n ) and r n (0 , n, n, β = 0) (Fig. 4D). It suggests that the larger α , the more bi-orientation attempts. Wealso found the number of bi-orientation attempts decreases as q (detachment probability)reduces (Fig. 4D, see also Fig. S10). Consistent with this, Kitajima et al. observed thenumber of attempts reduced from ∼ Conclusions
Our simple discrete-time Markov chain model captures the prominent features of chromo-some bi-orientation process. It provides a unified account of two modes of divisions, mitosisand meiosis I, under a single theoretical framework; the model reveals where the differencesin the bi-orientation process come from. It explains why errors are very frequent in thefirst meiotic division, which are major causes of infertility, miscarriages and birth defects inhumans.One of our key findings in this study is that the system dynamics (including the typeand frequency of transient kMT attachment errors) is dictated by the q/p ratio (relativedetachment rate) of kMTs. An imbalance of q/p ratio causes persistent attachment errors aka et al.
Page 9 of 39 leading to chromosome mis-segregations. The gradual increase of kMTs may help turn offthe spindle assembly checkpoint in normal conditions but can promote a faulty conformation(merotelic attachments) to evade the checkpoint.In summary, our study revealed that the chromosome bi-orientation is a probabilistic self-organization, rather than a sophisticated process of error detection and correction. Althoughour model omits many potentially important factors for chromosome bi-orientation, such asthe spatial arrangement of centrosomes, it allowed us to examine analytically all possibleoutcomes with different parameters (i.e. the whole parameter space), revealing what is fun-damental to accurate chromosome segregation. The proposed model, which is based on afirm mathematical foundation, gives valuable insights that help us understand one of theprimary causes of chromosomal instability—aberrant kMT dynamics.
Methods
The model and its analysis are explained in detail in SI Text (Additional File 1). Theanalysis of discrete-time Markov chains were performed according to [21, 33, 42]. We used
Mathematica (cid:114) (version 10, Wolfram Research) for the analysis of the model, with a standardlaptop (or desktop) computer. The
Mathematica codes used in this study are provided inAdditional File 2.
Competing interests
The authors declare that they have no competing interests.
Author’s contributions
YS and HO designed the model; YS wrote the computer codes and analysed the model;CVG analysed the model; and all authors wrote the paper.
Acknowledgements
We thank G. Bewick, C. Grebogi, S. Hoppler, A. Lorenz, C. McCaig, F. Perez-Reche, R.Sekido, M. Thiel and E. Ullner for helpful discussions and critical reading of the manuscript.YS and CG are supported by Scottish Universities Life Sciences Alliance (SULSA) and HOby Wellcome Trust (grant number 098030 and 092076). aka et al.
Page 10 of 39
Author details Institute of Medical Sciences, School of Medical Sciences, University of Aberdeen, Foresterhill, AB25 2ZD Aberdeen, UK. Wellcome Trust Centre for Cell Biology, University of Edinburgh, Michael Swann Building, Max Born Crescent, EH9 3BFEdinburgh, UK.
References
1. Bakhoum, S.F., Thompson, S.L., Manning, A.L., Compton, D.A.: Genome stability is ensured by temporal control ofkinetochore-microtubule dynamics. Nat Cell Biol (1), 27–35 (2009). doi:10.1038/ncb18092. Jones, K.T., Lane, S.I.R.: Molecular causes of aneuploidy in mammalian eggs. Development (18), 3719–30 (2013).doi:10.1242/dev.0905893. Tanaka, T.U.: Kinetochore-microtubule interactions: steps towards bi-orientation. EMBO J (24), 4070–82 (2010).doi:10.1038/emboj.2010.2944. Cheeseman, I.M.: The kinetochore. Cold Spring Harb Perspect Biol (7), 015826 (2014). doi:10.1101/cshperspect.a0158265. Dietz, R.: Multiple geschlechchromosomen bei den cypriden ostracoden, ihre evolution und ihr teilungsverhalten.Chromosoma , 359–440 (1958)6. Nicklas, R.B., Koch, C.A.: Chromosome micromanipulation. 3. spindle fiber tension and the reorientation of mal-orientedchromosomes. J Cell Biol (1), 40–50 (1969)7. Kitajima, T.S., Ohsugi, M., Ellenberg, J.: Complete kinetochore tracking reveals error-prone homologous chromosomebiorientation in mammalian oocytes. Cell (4), 568–81 (2011). doi:10.1016/j.cell.2011.07.0318. Bakhoum, S.F., Compton, D.A.: Kinetochores and disease: keeping microtubule dynamics in check! Curr Opin Cell Biol (1), 64–70 (2012). doi:10.1016/j.ceb.2011.11.0129. Chen, R.H., Waters, J.C., Salmon, E.D., Murray, A.W.: Association of spindle assembly checkpoint component xmad2 withunattached kinetochores. Science (5285), 242–6 (1996)10. Khodjakov, A., Pines, J.: Centromere tension: a divisive issue. Nat Cell Biol (10), 919–23 (2010).doi:10.1038/ncb1010-91911. Mitchison, T., Kirschner, M.: Dynamic instability of microtubule growth. Nature (5991), 237–42 (1984)12. Hill, T.L.: Theoretical problems related to the attachment of microtubules to kinetochores. Proc Natl Acad Sci U S A (13), 4404–8 (1985)13. Zaytsev, A.V., Sundin, L.J.R., DeLuca, K.F., Grishchuk, E.L., DeLuca, J.G.: Accurate phosphoregulation ofkinetochore–microtubule affinity requires unconstrained molecular interactions. The Journal of Cell Biology (1), 45–59(2014). doi:10.1083/jcb.201312107. http://jcb.rupress.org/content/206/1/45.full.pdf+html14. Kirschner, M., Mitchison, T.: Beyond self-assembly: from microtubules to morphogenesis. Cell (3), 329–42 (1986)15. Holy, T.E., Leibler, S.: Dynamic instability of microtubules as an efficient way to search in space. Proc Natl Acad Sci U S A (12), 5682–5 (1994)16. Wollman, R., Cytrynbaum, E.N., Jones, J.T., Meyer, T., Scholey, J.M., Mogilner, A.: Efficient chromosome capturerequires a bias in the ’search-and-capture’ process during mitotic-spindle assembly. Curr Biol (9), 828–32 (2005).doi:10.1016/j.cub.2005.03.01917. Gopalakrishnan, M., Govindan, B.S.: A first-passage-time theory for search and capture of chromosomes by microtubules inmitosis. Bull Math Biol (10), 2483–506 (2011). doi:10.1007/s11538-011-9633-918. Paul, R., Wollman, R., Silkworth, W.T., Nardi, I.K., Cimini, D., Mogilner, A.: Computer simulations predict thatchromosome movements and rotations accelerate mitotic spindle assembly without compromising accuracy. Proc Natl AcadSci U S A (37), 15708–13 (2009). doi:10.1073/pnas.090826110619. Gay, G., Courtheoux, T., Reyes, C., Tournier, S., Gachet, Y.: A stochastic model of kinetochore-microtubule attachmentaccurately describes fission yeast chromosome segregation. J Cell Biol (6), 757–74 (2012). doi:10.1083/jcb.20110712420. Silkworth, W.T., Nardi, I.K., Paul, R., Mogilner, A., Cimini, D.: Timing of centrosome separation is important for accuratechromosome segregation. Mol Biol Cell (3), 401–11 (2012). doi:10.1091/mbc.E11-02-009521. Norris, J.R.: Markov Chains, 1st pbk. ed edn. Cambridge University Press, Cambridge, UK (1998)22. Goldstein, L.S.: Kinetochore structure and its role in chromosome orientation during the first meiotic division in male d.melanogaster. Cell (3), 591–602 (1981) aka et al. Page 11 of 39
23. Watanabe, Y.: Geometry and force behind kinetochore orientation: lessons from meiosis. Nat Rev Mol Cell Biol (6),370–82 (2012). doi:10.1038/nrm334924. Mogilner, A., Craig, E.: Towards a quantitative understanding of mitotic spindle assembly and mechanics. Journal of CellScience (20), 3435–3445 (2010). doi:10.1242/jcs.06220825. Nicklas, R.B., Ward, S.C.: Elements of error correction in mitosis: microtubule capture, release, and tension. J Cell Biol (5), 1241–53 (1994)26. King, J.M., Nicklas, R.B.: Tension on chromosomes increases the number of kinetochore microtubules but only withinlimits. J Cell Sci
113 Pt 21 , 3815–23 (2000)27. Dewar, H., Tanaka, K., Nasmyth, K., Tanaka, T.U.: Tension between two kinetochores suffices for their bi-orientation onthe mitotic spindle. Nature (6978), 93–7 (2004). doi:10.1038/nature0232828. Biggins, S., Murray, A.W.: The budding yeast protein kinase ipl1/aurora allows the absence of tension to activate thespindle checkpoint. Genes Dev (23), 3118–29 (2001). doi:10.1101/gad.93480129. Tanaka, T.U., Rachidi, N., Janke, C., Pereira, G., Galova, M., Schiebel, E., Stark, M.J.R., Nasmyth, K.: Evidence that theipl1-sli15 (aurora kinase-incenp) complex promotes chromosome bi-orientation by altering kinetochore-spindle poleconnections. Cell (3), 317–29 (2002)30. Cimini, D., Wan, X., Hirel, C.B., Salmon, E.D.: Aurora kinase promotes turnover of kinetochore microtubules to reducechromosome segregation errors. Curr Biol (17), 1711–8 (2006). doi:10.1016/j.cub.2006.07.02231. Akiyoshi, B., Sarangapani, K.K., Powers, A.F., Nelson, C.R., Reichow, S.L., Arellano-Santoyo, H., Gonen, T., Ranish, J.A.,Asbury, C.L., Biggins, S.: Tension directly stabilizes reconstituted kinetochore-microtubule attachments. Nature (7323),576–9 (2010). doi:10.1038/nature0959432. Sarangapani, K.K., Asbury, C.L.: Catch and release: how do kinetochores hook the right microtubules during mitosis?Trends Genet (4), 150–9 (2014). doi:10.1016/j.tig.2014.02.00433. Bertsekas, D., Tsitsiklis, J.: Introduction to Probability, ed. 2 edn. Athena Scientific, Nashua, NH (2008)34. Hauf, S., Cole, R.W., LaTerra, S., Zimmer, C., Schnapp, G., Walter, R., Heckel, A., van Meel, J., Rieder, C.L., Peters,J.-M.: The small molecule hesperadin reveals a role for aurora b in correcting kinetochore-microtubule attachment and inmaintaining the spindle assembly checkpoint. J Cell Biol (2), 281–94 (2003). doi:10.1083/jcb.20020809235. Kelly, A.E., Funabiki, H.: Correcting aberrant kinetochore microtubule attachments: an aurora b-centric view. Curr OpinCell Biol (1), 51–8 (2009). doi:10.1016/j.ceb.2009.01.00436. McEwen, B.F., Heagle, A.B., Cassels, G.O., Buttle, K.F., Rieder, C.L.: Kinetochore fiber maturation in ptk1 cells and itsimplications for the mechanisms of chromosome congression and anaphase onset. J Cell Biol (7), 1567–80 (1997)37. Uchida, K.S.K., Takagaki, K., Kumada, K., Hirayama, Y., Noda, T., Hirota, T.: Kinetochore stretching inactivates thespindle assembly checkpoint. J Cell Biol (3), 383–90 (2009). doi:10.1083/jcb.20081102838. Maresca, T.J., Salmon, E.D.: Intrakinetochore stretch is associated with changes in kinetochore phosphorylation andspindle assembly checkpoint activity. J Cell Biol (3), 373–81 (2009). doi:10.1083/jcb.20080813039. Nannas, N.J., Murray, A.W.: Tethering sister centromeres to each other suggests the spindle checkpoint detects stretchwithin the kinetochore. PLoS Genet (8), 1004492 (2014). doi:10.1371/journal.pgen.100449240. Cimini, D., Howell, B., Maddox, P., Khodjakov, A., Degrassi, F., Salmon, E.D.: Merotelic kinetochore orientation is a majormechanism of aneuploidy in mitotic mammalian tissue cells. J Cell Biol (3), 517–27 (2001)41. Cimini, D., Moree, B., Canman, J.C., Salmon, E.D.: Merotelic kinetochore orientation occurs frequently during early mitosisin mammalian tissue cells and error correction is achieved by two different mechanisms. J Cell Sci (Pt 20), 4213–25(2003). doi:10.1242/jcs.0071642. Kemeny, J.G., Snell, J.L.: Finite Markov Chains. The University series in undergraduate mathematics. Van Nostrand,Princeton, N.J. (1960) aka et al. Page 12 of 39
Figures AC i j Left pole(L) Right pole(R) i i j j Kinetochorek k L RL R DB (0,0)(1,0) (1,1) (0,1)(0,2)(2,0)( n- n- n ,0) (0, n )(1, n- n- E Class 1 (free)Class 2(monotelic)Class 3(syntelic)Class 4(merotelic)Class 5 (amphitelic) k k LR RL 00 00 0 j i i j j j j j i j j i
00 00 0 i i i j i j i i j j i i j i j i j i j i ClassKinetochore
Transition probabilityscaled by: α β γ γ F Mitosis-specific scaling
Fig.1
Figure 1 A discrete-time Markov chain model of kMT dynamics. (A) Schematic diagram of the interactionbetween a kinetochore (orange) and microtubules (green) from either left (L) or right (R) pole. i and j indicate the number of kMTs. (B) Kinetochore-microtubule interactions as a Markov chain. The maximalnumber of kMTs per kinetochore is n . (C) Schematic diagram of kMT dynamics during cell division. A pairof kinetochores (k1 and k2) are connected by bivalent chromatids in meiosis I or centromere chromatins(blue). (D) States of kinetochore-microtubule complex are defined with r n ( i , j , i , j ) . Every state can beclassified into one of five classes in the table. Schematic diagrams of each class are shown on the right. (E)Transition diagram among classes. A subset of states in the Markov chain categorised in (D) can move fromone class to another according to this diagram. To increase the probability of class 5 states, transitions outof class 5 (red and green arrows) must be reduced, the probabilities of which are scaled with parameters α (for green arrow) and β (for red arrow) in the model. In mitosis, transitions from class 2 to class 3 or 4 arescaled with γ (blue arrows). (F) Schematic diagram of the scaling by parameters α , β and γ . Probabilities ofstate transition by attachment or detachment (arrowheads) are scaled by the indicated parameters.aka et al. Page 13 of 39 pq ABDC
Time Time P r obab ili t y P r obab ili t y P r obab ili t y p = q = 0.05, α = β = 0 p = q = 0.05, α = β = 0 p = q = 0.05, α = 0, β = 1 p = 0.05, q = 0.01, α = β = 0 p = 0.05, q = 0.01, α = β = 0 p = q = 0.05, α = β = 1 EFG
MitosisMeiosis I
Fig.2
Figure 2 Dynamics of kinetochore-microtubule interaction. (A) Contour plot of mean first passage time toclass 5 starting from class 1 in meiosis I. (B-G) Probabilities of each class over time for meiosis I (B-D) andmitosis (E-G). n = 10 for all panels. γ = 1 for meiosis I and γ = 0 . for mitosis. Other parameters are asindicated for each panel. (B, E) An ’ideal’ condition. The probability of class 5 approaches 1. (C, F) A’random’ condition with no bias towards class 5. The probability of class 4 (merotelic) becomes predominantin meiosis I (C) while class 2 (monotelic) is as prevalent as class 5 (amphitelic) in mitosis (F). Note thatclass 3 and class 5 have identical probabilities by symmetry in (C). (D, G) A condition in which q/p ratio islow. Class 4 persists both in meiosis I and in mitosis.aka et al. Page 14 of 39 p = 0.05 q = 0.05β = 0 p = 0.05 q = 0.05β = 1 p = 0.05 q = 0.01β = 0 Class 5Class 1 to Class 4Class 5Class 1 to Class 4Class 5Class 1 to Class 4 0 max i + j i + j
10 50 100 200 300Time
ABCD n = 40 n = 20 n = 10 Fig.3
Figure 3 Probability distribution of the number of kMTs over time. (A-C) Probability density plots of thenumber of kMTs in meiosis I in 2D ( i + j vs. i + j ; see Fig. 1C) at the indicated time points.Parameters are indicated on the left. α = 0 , n = 10 for all panels. Probabilities are decomposed into class 5and the rest (class 1 to 4) at each time point. Total probabilities are indicated on each panel. The densitiesare scaled from 0 to the maximal for each panel. (D) Mean number of microtubules ( ± s.d. ) attached to akinetochore derived by the approximation formulae (Eqs. (3) and (10)). Plots for n = 10 , and areshown. For details, see SI Text.aka et al. Page 15 of 39 k q P r obab ili t y Time A q B i o r i en t a t i on a tt e m p t s DB C -4 -3 -2 α αβ Fig.4
Figure 4 Dynamics of multiple chromosomes in meiosis I. (A) Probabilities of synchrony over time. k = number of chromosomes; p = 0 . , α = β = 0 . (B) Contour plot of probability of synchrony at steadystates. (C) Contour plot of half-life of synchrony at steady states. In (B) and (C), p = q = 0 . , k = 5 . (D)Number of biorientation attempts before absorption. p = 0 . , β = 0 . n = 10 for all panels.aka et al. Page 16 of 39
Tables
Table 1
Model parameters.Parameter for Range of value Biological meaning n Maximal number of kMTs ≤ n Maximal number of kMTs that can be accommodated on a singleper kinetochore kinetochore. n is proportional to the size of a kinetochore. p Association probability ≤ p ≤ / × p is the association probability of a single microtubule to a freekinetochore in each discrete time step. Upper limit of p is 1/4because total probability ≤ . q Dissociation probability ≤ q ≤ / n Dissociation probability of a single kMT in each discrete time step. α Scaling factor of p ≤ α ≤ Scaling applies to transitions from amphitelic (class 5) to merotelic(class 4) states; reflecting the physical constraint imposed inamphitelic states (meiosis I) or the back-to-back position of sisterkinetochores (mitosis). α = 0 in mitosis for simplicity. β Scaling factor of q ≤ β ≤ Scaling applies to transitions in/from amphitelic states (class 5);reflecting the kMT stabilization by tension. γ Scaling factor of p ≤ γ ≤ Scaling applies to transitions from monotelic (class 2) to syntelic(class 3) or merotelic (class 4) states in mitosis; reflecting thebiased orientation of sister kinetochores in monotelic states.aka et al.
Page 17 of 39
Supplementary Information1 A basic Markov chain model of kinetochore-microtubule interactions(Model I)
The interaction of a single kinetochore with microtubules is modeled as a birth/death(discrete-time) Markov process. First, we consider a kinetochore that can bind up to n microtubules. The possible states are M = { , , , ... , n } . Transition probability from state i to j is p i,j = P ( X t +1 = j | X t = i ) , i, j ∈ M , where X t is the state at time t . As statedin the main text, we assume the association probability is proportional to the surface areaof a kinetochore available for microtubule attachment. Therefore, the association (birth)probability is p k,k +1 = ( n − k ) b/n, ≤ k ≤ n − b is the association probabilityof a single microtubule to a free kinetochore. The dissociation (death) probability for state k is p k,k − = k d, ≤ k ≤ n , because each microtubule bound to a kinetochore ( k micro-tubules in total) has the same dissociation probability d . The self-transition probability is p k,k = 1 − ( n − k ) b/n − k d, ≤ k ≤ n . As an example, consider a kinetochore that can bindup to 2 microtubules. The possible states are { , , } . The transition probability matrix is R = p , p , p , p , p , p , p , p , p , = − b b d − b − d b d − d . Fig. S1A shows a diagram of this Markov chain. This model is a variation on the M/M/squeue [21]. The Markov chain consists of a single aperiodic recurrent class. Let u k be thesteady-state probabilities of state k and U be the row vector of u k , k = 0 , , , ... , n . Applyingthe steady-state convergence theorem, then U = U R n . This is equivalent to a local balanceequation: (1 − k/n ) b u k = ( k + 1) d u k +1 . Let ρ = b/d then,(1 − k/n ) ρu k = ( k + 1) u k +1 , k = 0 , , ... , n − . (4)The normalization equation (the sum of all probabilities equals to 1) is n (cid:88) k =0 u k = 1 . (5) aka et al. Page 18 of 39
Eqs. (4) and (5) yield a unique solution: u k = (cid:18) nk (cid:19) ( ρ/n ) k (1 + ρ/n ) − n , k = 0 , , , ... , n, (6)where (cid:0) nk (cid:1) is the binomial coefficient ( " n choose k " ). When n is large, u k approaches thePoisson distribution e − ρ ρ k (cid:14) k !. The mean and variance of u k , derived from Eq. (6), are nρ/ ( n + ρ ) and n ρ (cid:14) ( n + ρ ) , respectively. Fig. S1B shows an example of the probabilitydistribution of u k (the number of attached microtubules) for n = 20. As illustrated in thisexample, the stability of kinetochore-microtubule interaction can be controlled by ρ (i.e. d/b ratio) alone. Now we consider the interaction of a kinetochore with bipolar spindles (Fig. 1B in the maintext). There are ( n + 1)( n + 2) / n = 2. We assigna unique index number to each state denoted as s n ( i, j ): s n ( i, j ) (cid:55)−→ i + 1 + ( i + j + 1)( i + j )2 , ≤ i + j ≤ n. (7)We use these indices to construct the probability transition matrix in Mathematica codes.Using the same argument for model I, the state transition probabilities are s n ( i, j ) n − i − jn p −→ s n ( i + 1 , j ) ,s n ( i, j ) n − i − jn p −→ s n ( i, j + 1) , (8) s n ( i, j ) i q −→ s n ( i − , j ) ,s n ( i, j ) j q −→ s n ( i, j − , aka et al. Page 19 of 39 where p , q are parameters with 0 ≤ p ≤ / , ≤ q ≤ /n . For example, the transitionmatrix P n with n = 2 is P = − p p p q − p − q p p q − p − q p p q − q q q − q
00 0 2 q − q . Model II is fundamentally the same as Model I: b in Model I is equal to the combinedprobability of a microtubule binding to a free kinetochore (= 2 × p ) in Model II. Dissociationprobability of a single kMT is the same ( d = q ). Hence ρ = 2 p/q . The model of kinetochore-microtubule interactions in meiosis and mitosis is built from ModelII, which we call Model III. This model describes the state of a pair of kinetochores physicallyconnected by a centromere chromatin (in mitosis and meiosis II) or a bivalent (in meiosisI) of homologous chromosomes, which is defined by r n ( i , j , i , j ). Note that 0 ≤ p ≤ / ≤ q ≤ / n in Model III because the total transition probability from a given stateincluding self-transition is 1. Also note that there is no direct transition from class 1 (free)to class 5 (amphitelic, i.e. correct conformation).As briefly mentioned in the main text, spindle tension stabilises the kMT attachments inamphitelic states (class 5), which is represented by the scaling with the parameter β . Thisapplies to both mitosis and meiosis. The scaling with the parameter β is exemplified by r n ( i , , , j ) i β q −→ r n ( i − , , , j ) . This rule also reduces the probability of transitions from class 5 to class 2 states (red arrowin Fig. 1E; with i = 1 in the above example).The scaling of the probability of class 5 (amphitelic) to class 4 (merotelic) transitions withthe parameter α is based on the experimental evidences. In amphitelic states in mitosis,the kinetochore geometry in mitotic chromosomes prevents each sister kinetochore from aka et al. Page 20 of 39 interacting with the microtubules from the opposite pole [3]. Therefore class 5 (amphitelic)to class 4 (merotelic) transitions are effectively eliminated in mitosis, i.e. α = 0. In meiosis I,Nicklas suggested that the stability of amphitelic conformation is also gained by the alignedposition of kinetochores with the pole-to-pole axis, with each kinetochore pointing at a pole[6]. A recent study of meiosis I in mouse oocyte indeed revealed the restricted movementof kinetochores in amphitelic states (see supplemental movies in Kitajima et al [7]). Thescaling with the parameter α is exemplified by r n ( i , , , j ) n − i n α p −→ r n ( i , , , j ) . With a similar reason transitions from class 2 (monotelic) to class 3 (syntelic) or 4 (merotelic)are reduced in mitosis because the attached sister kinetochore are facing towards the polefrom which the kMT emanates, while the other unattached sister kinetochore are facing theopposite pole. Thus, these transitions (blue arrows in Fig. 1E) are scaled by the parameter0 ≤ γ ≤
1. For example, r n ( i , , , γ p −→ r n ( i , , , ,r n ( i , , , n − i n γ p −→ r n ( i , , , . These scaling of the transitions by γ are unique to mitosis (and meiosis II); for meiosis I, γ = 1.With sufficiently small α and β , class 5 becomes stable; when α = β = 0, transitions outof class 5 are not possible. That means class 5 is an absorbing class in the Markov chain.This bias towards class 5 underpins the probabilistic self-organisation of the system. Bycontrast, when α = β = 1 there is no bias towards class 5, that is, amphitelic states areunstable. Note that when α (cid:54) = 0 , β = 0, the process eventually ends up in either r n ( n, , , n )or r n (0 , n, n, To calculate the steady-state PMF in Model II, consider it as a process of choosing thenumber of microtubules per kinetochore and distributing them to left and right poles. Let k ( ≤ n ) be the total number of microtubules attached to the kinetochore and φ n ( i, j ) be thePMF for state s n ( i, j ), then k (cid:80) i =0 φ n ( i, j ) = u k , i + j = k. φ n ( i, j ) is derived by distributing aka et al. Page 21 of 39 u k according to the binomial distribution: (cid:18) i + ji (cid:19) (1 / i (1 / j = (cid:18) ki (cid:19) (1 / k , ≤ i ≤ k. Hence, using Eq. (6), φ n ( i, j ) = u k × (cid:18) ki (cid:19) (1 / k = (cid:16) ρn (cid:17) − n (cid:16) ρ n (cid:17) i + j n ! i ! j !( n − i − j )! . (9)Let Φ n = ( φ n (0 , , φ n (0 , , φ n (1 , , φ n (0 , , φ n (1 , , ... , φ n ( n − , , φ n ( n, n .P n = Φ n , which is consistent with the equilibrium atsteady states. The size of a Markov chain in Model II (total number of states) corresponds to the maximumof the s n ( i, j ) indices according to Eq. (7), which is ( n +1)( n +2) /
2. The total number of statesin the full model (Model III) is thus ( n + 1) ( n + 2) (cid:14)
4, which grows rapidly as n increases(Fig. S2A). Note that class 4 becomes predominant as the system size gets larger (Fig.S2B). Consequently, the number of possible state transitions also increases exponentiallywith the system size (Fig. S2C), which corresponds to the number of non-zero entries in theprobability transition matrix. For a Markov chain of Model III, the mean first passage time f i to class 5 from state i isobtained as the solution of linear equations [33]: f i = (cid:80) j / ∈ class 5 p i,j f j , i / ∈ class 5 , , i ∈ class 5 .f in Fig. 2A was calculated by incrementing p and q by 0.005 (50 ×
10 points) and in Figs.S3A and S3D by 0.0001 (100 ×
100 points). For a given value of q (= 0 . f plateaus as n grows (Fig. S3B), although the number of states and transitions increaserapidly (Figs. S2A and S2C). For meiosis I, the q/p ratio for the minimum f approaches aka et al. Page 22 of 39 ∼ n increases (Fig. S3C). For mitosis, the optimal q/p ratio is somewhat skewed (Fig.S3D). r n ( i , j , i , j ) It is straightforward to calculate the PMF of r n ( i , j , i , j ) at each time point from thetransition probability matrix. We classified the PMFs according to Fig. 1D and calculatedthe sums for each class to obtain Fig. 2B to G. Fig. S4 shows the PMF time series in meiosisI with α = 0 , β = 1 (a) and α = 1 , β = 0 (b) ( n = 10 , p = q = 0 . n ; this is mainly because the structureof the Markov chain remains the same as the size of the chain grows (Fig. 1B and S2E). Wehave used n = 10 for most of the analysis as a representative value. We extensively exploredthe dynamics with different values of n and found fundamentally no diffence in the behaviorof the Markov chain by altering n (for example, n = 10 versus 15 in Figs. S5B, C, E and F). q/p ratio As long as q/p ratio (relative kMT dissociation rate) remains the same, the steady-stateprobabilities of r n ( i , j , i , j ) stay the same for all p, q pairs. The PMF time series arealso almost invariable (but with different time scales) as long as q/p ratio remains constant,illustrated by the examples shown in Fig. S5. Only the time scale changes, which is inverselyproportional to √ p q . Strictly speaking, although steady-state probabilities are identical,PMFs at any given moment are not exactly the same: these small differences come aboutby the assumption that only a single event happens in every state transition. Differences aresmall enough to be ignored when p and q are sufficiently small. From a biological perspective,the change in time scale with the same dynamics has a different meaning: the faster theassociation and dissociation of kMTs, the more efficient the chromosome biorientation. Seealso the section ’Biorientation attempts’ below. Class 5
Steady state probability distributions of the number of kMTs in class 5 for n = 10 , p =0 . , q = 0 .
05 are shown in Fig. S8B as density plots. Total probability of class 5 is indicatedfor each panel. Gray scale is normalized to the total probability of class 5. When α and aka et al. Page 23 of 39 β are sufficiently small, class 5 (amphitelic) states are stable, i.e. the number of kMTs areclose to the maximum. Otherwise, only a few microtubules on average are attached to eachkinetochore. Class 1 to 4
Steady state probability distributions of the number of kMTs in classes 1 to 4 for n = 10, p = 0 . q = 0 .
05 are shown in Fig. S8C as density plots. Although the total probabilitiesare greatly affected by the parameters α and β , the distribution of the number of kMTs ofnon-class 5 states barely changes. This is presumably because the size of the non-amphitelicclasses (mainly class 4) in total is significantly larger than that of class 5 (Fig. S2B), bufferingthe influence of class 5. Thus, ¯ N = nρ/ ( n + ρ ) (the exact solution in the random condition α = β = γ = 1) is also an approximate of the steady state distribution of kMTs in meiosisI for non-amphitelic states when α (cid:54) = 1 , < β (cid:54) = 1.
10 Number of kMTs at steady states in class 5 — an analyticalapproximation
When α = 0, the number of kMTs for states in class 5 can be estimated analytically withoutexplicitly calculating the PMF, which is computationally expensive for large n . The reasonwhy it is possible becomes apparent by looking at the Markov chain’s structure and transi-tions — when α = 0, transitions from class 5 to class 2 are still possible, but not to class 4anymore. Note that for large n the number of transitions between class 5 and class 4 is farlarger than the one between class 5 and class 2 (Fig. S2D).Because of its limited communication with other classes when α = 0, class 5 behaves as if itis a disjoint class at steady states. Its two sub-classes (e.g. left and right square grids in Fig.S2E) have the identical probability distribution by symmetry, which can be approximatedby ψ n ( i, j ) = φ n ( i, × φ n (0 , j )= 2 − ( i + j ) u i u j = (cid:18) ni (cid:19)(cid:18) nj (cid:19) (cid:16) ¯ ρ n (cid:17) i + j (cid:16) ρn (cid:17) − n , aka et al. Page 24 of 39 where 1 ≤ i, j ≤ n and ¯ ρ = ρ/β = pβq . We now compute the conditional expectation of thenumber of kMTs i (or j ) given the state s is in class 5: E ( i | s ∈ class 5) = n (cid:88) i =1 n (cid:88) j =1 ( ψ n ( i, j ) × i ) (cid:46) n (cid:88) i =1 n (cid:88) j =1 ψ n ( i, j )= ¯ N . After a lengthy algebra, it simplifies to Eq. (3) in the main text. Similarly, E (cid:0) i (cid:12)(cid:12) s ∈ class 5 (cid:1) = n (cid:88) i =1 n (cid:88) j =1 (cid:0) ψ n ( i, j ) × i (cid:1) (cid:46) n (cid:88) i =1 n (cid:88) j =1 ψ n ( i, j ) . After another lengthy calculation, variance of i is reduced to:Var( i | s ∈ class 5) = E (cid:0) i (cid:12)(cid:12) s ∈ class 5 (cid:1) − ( E ( i | s ∈ class 5) ) = ¯ ρ (cid:0) ¯ ρn (cid:1) n − (cid:0) (cid:0) ¯ ρn (cid:1) n − n (2 + ¯ ρ ) (cid:1)(cid:0)(cid:0) ¯ ρn (cid:1) n − n (cid:1) . (10)Eqs. (3) and (10) fit very well to the exact number of kMTs derived from the steady-statePMF (Fig. S8A). When n is small (e.g. n = 4), the approximation diverges a little from theexact values, but is still pretty good (not shown). The mean and variance approach n and0, respectively, as ¯ ρ → ∞ .
11 Probability of synchrony
Computing the probability of synchrony
We compute the probability of synchrony at time T = t , i.e. the probability that the processin every chain is in the same class (in particular class 5) at the same time. This is illustratedby an example below, which shows the state (class) transtions of each process (Ch.1 to 4)from T = 0 to 20. Synchrony in class 5 is highlighted in red, which occurs at T = 18.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Ch.1 1 2 1 1 1 1 1 1 2 3 4 4 4 4 4 2 2 2 5 5 5Ch.2 1 1 1 1 1 1 1 1 2 4 4 4 4 4 4 4 5 5 5 5 5Ch.3 1 2 1 1 1 1 1 2 3 3 2 5 5 5 5 5 5 5 5 5 5Ch.4 1 2 3 3 2 2 2 3 4 4 4 4 5 5 5 5 5 5 5 5 5Let s [ t ] be the state of a single Markov process at time T = t , a ( t ) j be the probability of s [ t ] = j and k be the total number of chains. In the above example, k = 4. Also let P ( t ) s aka et al. Page 25 of 39 and P ( t ) as be the probability of synchrony and asynchrony at T = t , respectively. Then, P ( t ) s = θ tk ,P ( t ) as = 1 − P ( t ) s , = 1 − θ tk , (11)where θ t = P ( s [ t ] ∈ class 5) = (cid:88) j a ( t ) j , j ∈ class 5 . We used Eq. (11) for Figs. 4A and S3E. Now we consider the probability of synchronyattempts. For this, we need some events and their probabilities defined. The probability ofbiorientation attempts P ( t )+ in a single process is P ( t )+ = P ( s [ t − / ∈ class 5 ∧ s [ t ] ∈ class 5)= (cid:88) i,j a ( t − i p i,j , i / ∈ class 5 , j ∈ class 5 , where p i,j is the transition probability from state i to j . Likewise, the probability of biori-entation loss P ( t ) − is P ( t ) − = (cid:88) i,j a ( t − j p j,i , i / ∈ class 5 , j ∈ class 5 . The probability of biorientation maintainance P ( t )0 of a process is P ( t )0 = P ( s [ t − ∈ class 5 ∧ s [ t ] ∈ class 5)= (cid:88) i,j a ( t − i p i,j , i ∈ class 5 , j ∈ class 5 . Let m be the number of Markov processes in class 5 at T = t . Note that a synchrony attemptoccurs only when all m processes that are in class 5 stay in class 5 and the remaining k − m processes undergo transition from non-class 5 to class 5 states. The probability of such asynchrony attempt, P ( t )as ,s , is P ( t ) as ,s = P (asynchrony at T = t − ∧ synchrony at T = t )= k − (cid:88) m =0 (cid:18) km (cid:19) (cid:16) P ( t )+ (cid:17) k − m (cid:16) P ( t )0 (cid:17) m , aka et al. Page 26 of 39 where (cid:0) km (cid:1) is the binomial coefficient. Note that Fig. S9A may help understand the followingderivation of formulae.The probability of synchrony maintenance P ( t ) s,s is P ( t ) s,s = P (synchrony at T = t − ∧ synchrony at T = t )= P ( t ) s − P ( t ) as ,s . Likewise, the probability of asynchrony maintenance P ( t ) as , as is P ( t ) as ,as = P (asynchrony at T = t − ∧ asynchrony at T = t )= P ( t ) as − P ( t ) s, as . The probability of synchrony loss P ( t ) s, as is obtained by P ( t ) s, as = P (synchrony at T = t − ∧ asynchrony at T = t )= P ( t − s − P ( t ) s,s = P ( t − s − (cid:16) P ( t ) s − P ( t ) as ,s (cid:17) = P ( t ) as ,s − (cid:16) P ( t ) s − P ( t − s (cid:17) . (12) P ( t ) s,as can also be obtained by P ( t ) s, as = k (cid:88) m =1 (cid:18) km (cid:19) (cid:16) P ( t ) − (cid:17) m (cid:16) P ( t )0 (cid:17) k − m . At steady states, the conditional probability of synchrony loss given the present state is insynchrony is P ( ∞ ) s, as /P ( ∞ ) s , therefore the mean duration (half-life) of synchrony is P ( ∞ ) s /P ( ∞ ) s, as .Fig. 4C was derived by this formula.Now we examine when the synchrony happens for the first time, i.e. the probability ofthe first synchrony at T = t , denoted by P ( t ) fs . With α = β = 0, once a process is in class5, it is trapped in the class (i.e. P ( t ) s, as = 0). Therefore P ( t ) fs = P ( t ) as ,s . When α and β aresmall, the majority of synchrony attempts are for the first time; in addition, as k gets larger(number of processes, i.e. pairs of sister chromatids in mitosis or bivalents in meiosis I),synchrony becomes a rarer event. Thus, the probability of synchrony loss P ( t ) s, as are small at aka et al. Page 27 of 39 any given moment for large k and small α and β . In such a condition, it is therefore possibleto approximate P ( t ) fs with ˜ P ( t ) fs :˜ P ( t ) fs = P (0) as × (cid:32) t − (cid:89) τ =1 P (asynchrony at T = τ − ∧ asynchrony at T = τ | asynchrony at T = τ − (cid:33) × P (asynchrony at T = t − ∧ synchrony at T = t | asynchrony at T = t − (cid:32) t − (cid:89) τ =1 P ( τ ) as , as P ( τ − as (cid:33) × P ( t ) as ,s P ( t − as , t = 2 , , . It is apparent that for α = β = 0 (therefore P ( t ) s, as = 0), ˜ P ( t ) fs = P ( t ) as ,s . Fig. S9B shows anexample of ˜ P ( t ) fs together with Monte Carlo simulation results (probability in 5,000 simula-tions), demonstrating a good fit of the approximation to the simulation result. Timing of first synchrony and q/p ratio
We asked how q/p ratio affects the timing of first synchrony. We also asked how efficientlysynchrony can be achieved in a slightly compromized condition, i.e. α = β = 0 .
05. Fig. S9Cshows the probability of first synchrony in meiosis I at each time point with decreasing q value ( n = 10 , p = 0 .
05 and the number of bivalents k = 5). When p = q = 0 .
05, firstsynchrony happens most frequently around T = 100; By T = 400 synchrony takes placeat least once in ∼ .
7% of cases (not shown). As q/p ratio declines, the timing of firstsynchrony spreads more and more over time, becoming unpredictable. Therefore, synchronydoes happen relatively efficiently with the right q/p ratio even in a slightly compromizedcondition with α = β = 0 .
05. For a fixed value of p = 0 .
05, the probability of synchrony atsteady state (at any give moment) is 0.66 with q = 0 .
05, but only 0.017 when q = 0 .
01. For k = 20, the probability declines to 0.19 with q = 0 .
05 and 8 . × − with q = 0 .
01. Thus,keeping the balance of q/p ratio is all the more important for the cell with a large numberof chromosomes. This principle applies to both mitosis and meiosis I.
12 Bi-orientation attempts
Probability of bi-orientation attempts
Probability of biorientation attempts at time T = t , µ t , is µ t = (cid:88) i,j a ( t ) i p i,j , i / ∈ class 5 , j ∈ class 5 , aka et al. Page 28 of 39 where a ( t ) i is the probability of the process in state i at time t and p i,j is the transitionprobability from state i to j . µ t can also be interpreted as the mean number of attempts tobiorientation at time t . Fig. S10A shows a plot of µ t by this formula (analytical solution) andsimulations (parameters: n = 5 , p = q = 0 . , α = β = 0 .
1; 10,000 simulations). Fig. S10Bshows an example of probability time series of biorientation attempts, with p = q = 0 . p = q = 0 .
01 ( n = 10, α = β = 0 . q/p ratio, their PMF timeseries are almost identical (not shown) if the time scale is adjusted. Because of this changeof time scale, the probability (i.e. frequency) of biorientation attempts also changes. In thisexample, the probability at steady states (at any time point) decreases from ∼ p = q = 0 .
05 to ∼ p = q = 0 . Mean number of biorientation attempts before absorption
The number of biorientation attempts before the onset of anaphase is equivalent in our modelto the total number of transitions to class 5 from either class 2 or class 4 before absorption(referred as ¯ M hereafter; Fig. 4D). This can be computed by first calculating the mean totalnumber of times the process is in each transient state before absorption, starting from class1. We denote this number as M ( i ), i ∈ (transient states). There are two absorbing stateswhen β = 0, so the total number of transient states is l = (total number of states) − M ( i )is obtained from the so-called fundamental matrix N defined as N = ( I − Q ) − , where I isthe l × l identity matrix and Q is the l × l submatrix of P (the transition probability matrix): P = Q RO S .Q defines the transition within the transient states. O is a 2 × l matrix with all 0’s; R concerns the transtion from transient to absorbing states; S is the 2 × N corresponds to M ( i ). The formula for fundamental matrices(and also for the mean and variances of absorption time) is according to Kemeny and Snell[42]. Computing the mean of ¯ M , (cid:10) ¯ M (cid:11) , is straightforward using M ( i ): (cid:10) ¯ M (cid:11) = (cid:88) i (cid:88) j M ( i ) p i,j , aka et al. Page 29 of 39 where i ∈ (class2 ∨ class4), j ∈ class 5 and p i,j is the transition probability from state i to j .Note that, when α = 0 , ¯ M = 1 for all n ≥ aka et al. Page 30 of 39
10 2 P r obab ili t y Number of microtubules ρ = 0.5ρ = 2ρ = 5ρ = 10ρ = 20ρ = 50ρ = 2005 10 15 200.60.50.40.30.20.1 AB Fig. S1. A basic Markov chain model of kinetochore-microtubule interactions. (A) The Markov chain for n = 2. (B) Probability distribution of the number of kMTs ( u k ) for n = 20 and ρ as indicated. See SI Text for details. aka et al. Page 31 of 39 n nn
Class 4Class 5 C l ass A BC
Total numberof states Number of statesNumber of transitions D Number of transitions Class 5 Class 2Class 5 Class 4 n
25 30 E Fig. S2. The size and structure of the Markov chain in the full model (Model III).
For details of (A-D), see SI Text. (A) Total number of states. (B) Number of states in each class. Note that Class 3 and 5 have the same number of states. (C) Number of possible transitions. (D) Number of possible transitions in and out of class 5. (E) Graph of class 1, 2 and 5 for n = 5. All edges are bi-directional. Numbers indicate the indices of the states. Class 3 and class 4 are omitted. Class 5 (blue, red and orange) consists of two sub-classes corresponding to r n ( i , 0, 0, j ) and r n (0, j , i , 0). Two class 5 states (orange) correspond to those with max number of microtubules, i.e. r n (0, n , n , 0) and r n ( n , 0, 0, n ). Class 2 states are marked in green. Class 5 states can communicate with both class 2 and class 4. Note that every state in class 5 (red and blue) but two (orange) directly communicates with class 4. In contrast, only a minor fraction of class 5 (blue) can communicate directly to class 2, which are at the 'periphery' of the class 5 chain. Note that the amphitelic states in blue has i = 1 and/or j = 1, i.e. either of the kinetochores has only a single microtubule attached. As n grows, so do the two square lattices of the chain. But this graph structure remains the same. aka et al. Page 32 of 39 pq q / p nn AB M i n i m a l m ean f i r s t pa ss age t i m e C D pq E k q
00 100 200 300 40010.5 P r obab ili t y time F β H a l f - li f e o f sy n c h r on y Fig. S3. Chromosome bi-orientation in meiosis and mitosis. (A) A density plot of mean first passage time to class 5 in meiosis I. n =10. Green indicates >10,000. (B, C) The minimum mean first passage time in meiosis I (B) and the corresponding q / p ratio (C) for a fixed value of q (= 0.0005) plotted as a function of n . q / p approaches 1 while the minimum first passage time plateaus as n grows. (D) A density plot of the mean first passage time in mitosis. (E) Probabilities of synchrony in mitosis over time. k = number of pairs of sister chromatids, p = 0.05, β = 0. (F) Half-life of synchrony at steady states in mitosis. k = 5, p = q = 0.05. β needs to be small to maintain synchrony. In (D, E, F), n =10, α = 0 and γ = 0.1. aka et al. Page 33 of 39 p = q = 0.05, α = 0, β = 1 p = q = 0.05, α = 1, β = 0 P r obab ili t y P r obab ili t y AB Class 1 (free)Class 2 (monotelic)Class 3 (syntelic)Class 4 (merotelic)Class 5 (amphitelic)
Fig. S4. Probability change of each class over time in meiosis I. (A) α = 0, β = 1. (B) α = 1, β = 0. The other parameter values are n = 10, p = q = 0.05. aka et al. Page 34 of 39 P r obab ili t y P r obab ili t y P r obab ili t y AB n = 10, p = q = 0.05 n = 10, p = 0.05, q = 0.01 n = 10, p = 0.01, q = 0.002 n = 15, p = 0.01, q = 0.002 n = 10, p = q = 0.01 n = 15, p = q = 0.01 C DEF : Class 1 (free) : Class 2 (monotelic) : Class 3 (syntelic) : Class 4 (merotelic) : Class 5 (amphitelic)
Fig. S5. Invariant dynamics of the Markov process with a constant q / p ratio. Examples of PMF time series (meiosis I) are shown. α = β = 0 for all panels. Other parameters are as indicated. The dynamics are almost indistinguishable for a given q / p ratio besides the difference in time scale. Even with different n , they are qualitatively very similar (B versus C, or E versus F). See SI for details. aka et al. Page 35 of 39 α = 1β = 1α = 1β = 0 Class 5Class 1 to Class 4Class 5Class 1 to Class 4
10 50 100 200 300Time i + j i + j Fig. S6. Probability distribution of the number of kMTs over time in meiosis I.
The probability distributions of the kMT number are shown as density plots. Total probability of class 5 and class 1 to 4 are indicated on each panel. Parameters: n = 10, p = q = 0.05; α, β values are indicated on the left. kMT number distribution remains low (unstable) with α = 1. AB aka et al. Page 36 of 39 p = 0.05 q = 0.05α = 0β = 1 Class 5Class 1 to Class 4 0.8370.163 0.3720.628 0.3820.618 0.3820.618 0.3820.618 i + j i + j p = 0.05 q = 0.05α = 0β = 0 Class 5Class 1 to Class 4 0.2130.787 0.8220.178 0.0340.966 0.998 0.9990.0010.002 ABC p = 0.05 q = 0.01α = 0β = 0 Class 5Class 1 to Class 4 0.7730.227 0.8010.199 0.8730.127 0.118 0.8870.113 Fig. S7. Probability distribution of the number of kMTs over time in mitosis.
Probability density plots of kMT numbers in 2D ( i + j vs. i + j ; see Fig. 1C) at the indicated time points. Parameters are indicated on the left for each panel; n = 10 and γ = 0.1 for all panels. Probabilities are decomposed into class 5 and the rest (class 1 to 4) at each time point, Total probabilities are indicated on each panel. The densities are scaled from 0 to the maximal in each panel. aka et al. Page 37 of 39 αβ i + j i + j αβ i + j i + j -4 -6 CB Class 5 (amphitelic) Class 1 to 4 (non-amphitelic)Mean Variance ρ ρ A Fig. S8. Distribution of the number of kMTs at steady states. (A) An analytical approximation of kMT number in class 5 when α = 0 at steady states. Orange indicates the sample points of the exact kMT numbers (mean and variance) derived from the steady-state PMF and blue curves the functions of approximation according to Eqs. (3) and (10). See SI Text for the derivation of the approximation. Parameters: n =10, p = 0.05, q = 0.05. (B, C) The probability distributions of kMT number in meiosis I with the indicated parameters α and β are shown as density plots. Total probabilities of class 5 (B) and class 1 to 4 (C) are indicated on each panel. Parameters: n = 10, p = 0.1, q = 0.05. aka et al. Page 38 of 39
Probability Time
BAC
Probability TimeSimulationsAnalytical approximation P ( t ) fs ~0.010.0080.0060.0040.002 50 100 150 200 250 300100 200 300 400 5000.0070.0060.0050.0040.0030.0020.001 q = 0.05 q = 0.04 q = 0.03 q = 0.02 q = 0.01 Fig. S9. Approximation of the probability of first synchrony. (A) Diagram of probability change of synchrony and asynchrony in each time step. For details, see SI Text section ‘Probability of synchrony’. (B) Analytical approximation of the probability of first synchrony (yellow) together with Monte Carlo simulation results (blue dots; probability at each time point in 5,000 simulations) is shown. Parameters: n = 5, p = q = 0.05, α = β = 0.01. (C) Timing of the first synchrony with reducing q / p ratio. The plot shows the probability of the first synchrony at every time points in different conditions as indicated. As q / p ratio declines, the timing of first synchrony spreads more and more over time, becoming unpredictable. Parameter values: n = 10, p = 0.05, α = β = 0.05 and number of bivalents (Markov processes) k = 5; γ = 1 (meiosis I). T = t -1 T = t P s,as ( t ) P as,as ( t ) P as,s ( t ) P as ( t -1) P s,s ( t ) P s ( t -1) P s ( t ) P as ( t ) aka et al. Page 39 of 39
Probability TimeProbability Time AB p = q = 0.05 p = q = 0.01SimulationsAnalytical solution Fig. S10. Probability of bi-orientation attempts. (A) An analytical solution (yellow) and the probabilities of bi-orientation attempts obtained from simulations (blue dots; probability at each time point in 10,000 simulations) is shown. Parameters: n = 5, p = q = 0.01, α = β = 0.1. This demonstrates a good fit of the analytical solution to the data obtained by simulations. (B) Probability time series of bi-orientation attempts. Parameters: n = 10, α = β = 0.1. Bi-orientation attemps are more frequent with p = q = 0.05 than with p = q = 0.01. The probability of bi-orientation attempts at steady states (at any time point) decreases from ~0.033 with p = q = 0.05 to ~0.0066 with p = qq