 Research Article
 Open Access
 Published:
Accurate chromosome segregation by probabilistic selforganisation
BMC Biologyvolume 13, Article number: 65 (2015)
Abstract
Background
For faithful chromosome segregation during cell division, correct attachments must be established between sister chromosomes and microtubules from opposite spindle poles through kinetochores (chromosome biorientation). Incorrect attachments of kinetochore microtubules (kMTs) lead to chromosome missegregation and aneuploidy, which is often associated with developmental abnormalities such as Down syndrome and diseases including cancer. The interaction between kinetochores and microtubules is highly dynamic with frequent attachments and detachments. However, it remains unclear how chromosome biorientation is achieved with such accuracy in such a dynamic process.
Results
To gain new insight into this essential process, we have developed a simple mathematical 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 and detachment probabilities of kMTs is crucial for correct chromosome biorientation. With the right balance, incorrect attachments are resolved spontaneously into correct bioriented conformations while an imbalance leads to persistent errors. In addition, the model explains why errors are more commonly found in the first meiotic division (meiosis I) than in mitosis and how a faulty conformation can evade the spindle assembly checkpoint, which may lead to a chromosome loss.
Conclusions
The proposed model, despite its simplicity, helps us understand one of the primary causes of chromosomal instability—aberrant kinetochore–microtubule interactions. The model reveals that chromosome biorientation is a probabilistic selforganisation, rather than a sophisticated process of error detection and correction.
Background
Accurate segregation of chromosomes during cell division is fundamental to life. Errors in this process result in cell death or aneuploidy. Chromosome segregation is usually very accurate. However, missegregation occurs at a much higher frequency in cancer cells and oocytes, which is a contributing factor to cancer progression [1] and also a major cause of infertility, miscarriages and birth defects such as Down syndrome [2].
The key event for chromosome segregation is the establishment of chromosome biorientation, in which sister chromatids in mitosis or homologous chromosomes in meiosis, attach to the microtubules from opposite spindle poles by kinetochores [3]. Each kinetochore consists of more than 100 different proteins assembled on each centromeric DNA sequence, many of which are involved in the interaction with microtubules [4]. Chromosome biorientation is a very dynamic process with frequent attachments and detachments of microtubules [5–8].
For proper segregation of chromosomes, all kinetochores need to attach to spindle microtubules while erroneous attachments must be eliminated before the onset of anaphase. It is known that attachment errors are more frequent in meiosis I (especially in oocytes) than in mitosis [2, 5–7]. Yet it is not understood why this is so. Unattached kinetochores act as signal generators for the spindle assembly checkpoint, which delays chromosome segregation until proper biorientation is established for all chromosomes [9]. It remains unclear, however, whether improperly attached kinetochore microtubules (kMTs) are also detected and corrected by the spindle assembly checkpoint or by an independent mechanism [10].
The precise mechanism of chromosome biorientation has been under intense investigations. However, it is not yet possible to observe the dynamics of individual microtubules in vivo in real time. Mathematical modelling provides a powerful means to study the chromosome biorientation process. Since the discovery of the dynamic instability of microtubules [11], a number of theoretical analyses have provided important insights into the interaction between microtubules and kinetochores (for example, [12, 13]). The socalled searchandcapture model explains how dynamically unstable microtubules capture chromosomes [14–17].
However, the original searchandcapture model did not concern events after capture, in particular, erroneous attachments of kMTs and their correction. To address this, Paul et al. put forward a modified searchandcapture model with explicit correction mechanisms [18]. Gay et al. proposed a stochastic model of kinetochore–microtubule attachments in fission yeast mitosis, which reproduced correct chromosome biorientation and segregation in 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 chromosome segregation [20]; using experimental and computational approaches, they demonstrated that cells with incomplete spindle pole separation have a higher rate of kMT attachment errors than those with complete centrosome separation. Yet, the question remains unanswered as to how the cell can discriminate between correct and incorrect kMT attachments as their models assumed an explicit bias based on the discrimination of correct versus incorrect connections.
A major impediment to understanding fully the mechanism of chromosome biorientation is the lack of a universal theoretical framework that covers the chromosome biorientation process during eukaryotic cell divisions in general, including both mitosis and meiosis. Here we present such a universal model of chromosome biorientation, which is simple yet applicable to any eukaryotic cell division. Firstly, the model reveals that the balance between attachment and detachment probabilities of kMTs is crucial for correct chromosome biorientation. With the right balance, incorrect attachments are resolved spontaneously into correct bioriented conformations while an imbalance leads to persistent errors. Therefore, the superficially complex process, chromosome biorientation, is in fact a probabilistic selforganisation. It implies that the cell does not need to discriminate between correct and incorrect kMT attachments. Moreover, the model explains why errors are more frequent in meiosis I than in mitosis and how a faulty conformation can evade the spindle assembly checkpoint by a gradual increase of the number of kMTs. Despite its simplicity, the model is consistent with a number of experimental observations and provides theoretical insights into 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. 1 a). We assume a single kinetochore can accommodate up to n microtubules. The process of microtubule attachment/detachment can be represented as a discretetime Markov chain [21] (Fig. 1 b and Additional file 1: Figure S1).
Each pair of sister chromatids in mitosis has two kinetochores (k _{1} and k _{2} in Fig. 1 c). In meiosis I, a pair of sister kinetochores are physically connected sidebyside and act as one [22, 23]. Therefore, in our model, a bivalent (a pair of homologous chromosomes connected by chiasma) also has two kinetochores in meiosis I. We assume these two kinetochores interact with microtubules independently. Hence, the state of the kinetochores is represented as r _{ n }(i _{1},j _{1},i _{2},j _{2}), which can be classified into one of five classes according to the pattern of microtubule attachments (Fig. 1 d). State transitions occur in a stereotypical manner among these classes irrespective of the value of n≥2 (Fig. 1 e and Additional file 1: Figure S2E; refer to Table 1 for a summary of the parameters herein). Notably, the only possible transitions out of class 5 (amphitelic, i.e. correct conformation) is to class 2 (monotelic) or 4 (merotelic) (red and green arrows in Fig. 1 e). Note also that this transition scheme is similar to the kinetic error correction model (a deterministic ordinary differential equation model) proposed by Mogilner and Craig [24]; their scheme is a limiting case—only two kMT attachments per kinetochore are allowed and transitions out of amphitelic states are prohibited.
We assume the association probability is proportional to the available surface area of the kinetochore while the dissociation probability is independent, as illustrated below:
where 0≤p≤1/4 and 0≤q≤1/2n. 2×p is the association probability of a single microtubule to a free kinetochore; q is the dissociation probability of a single kMT.
Experimental evidence strongly suggests that tension stabilises the spindle attachment to the kinetochores in amphitelic states (class 5) [25–27]. The stabilisation by tension is brought about by suppression of Aurora B kinase activity towards kinetochore substrates [27–30] as well as by mechanical catchbonds [31, 32]. We model this stabilisation by scaling the transition probabilities of states in class 5 by detachment with the parameter 0≤β≤1 (Fig. 1 f). This rule also reduces the probability of transitions from class 5 to class 2 states (Fig. 1 e, red arrow). Similarly, the probability of class 5 (amphitelic) to class 4 (merotelic) transitions, which occur by attachment of a microtubule but not by detachment (Fig. 1 e, green arrow), scales with 0≤α≤1 (Fig. 1 f). This is due to the physical constraint imposed in amphitelic states in meiosis I [6, 7] or the kinetochore geometry (backtoback position of sister kinetochores) in mitosis [3]. In mitosis, α=0 for simplicity. For mitosis we introduce an additional parameter 0≤γ≤1 to scale the transition probabilities from class 2 (monotelic) to class 3 (syntelic) or 4 (merotelic) (Fig. 1 e blue arrows). This is because the biased orientation of sister kinetochores hinders those transitions (Fig. 1 f). Note that when α=β=0, transitions out of class 5 are effectively blocked; hence, this Markov process always ends up in class 5. For additional details of the model, see Additional file 1. This simple model, which has only six parameters and is exactly solvable, provides a number of analytical insights into how correct chromosome biorientation is achieved.
Dynamics of chromosome biorientation process
The model predicts how long it takes to reach class 5 (amphitelic) from class 1 (free), i.e. the mean first passage time [33] (see Additional file 1). For a given value of q, the mean first passage time (which is independent of α and β because they only affect transitions out of class 5) is shortest when p is roughly equal to q (Fig. 2 a and Additional file 1: Figure S3A–D). Thus, the relative dissociation rate (q/p ratio) of kMTs needs to be balanced for efficient chromosome biorientation.
The model also predicts the dynamics of the system (Fig. 2 b–d for meiosis I and e–g for mitosis). Note that the q/p ratio dictates the dynamics of the Markov chain (Additional file 1: Figure S5). For both mitosis and meiosis in an ideal condition (p=q=0.05,α=β=0; Fig. 2 b, e), the probability of class 5 steadily increases, asymptotically reaching 1. Notably, in meiosis I, class 4 (merotelic), and class 3 (syntelic) to a lesser extent, become transiently prominent (Fig. 2 b). Merotelic attachments are indeed frequently observed in the prophase to prometaphase of meiosis I in mouse oocytes [7]. By contrast, in mitosis, class 2 (monotelic) becomes predominant before being replaced by class 5, although minor fractions of classes 3 and 4 also appear briefly (Fig. 2 e). Together, this explains why meiosis I is more errorprone than mitosis; it is attributed to the parameter γ—the backtoback conformation of sister kinetochores, which biases the kinetochore orientation.
If there is no bias in meiosis I (random condition; α=β=1; Fig. 2 c, see also Additional file 1: Figure S4), the probability of class 5 stays low while that of class 4 (merotelic) reaches nearly 1/2 at steady states. This is because class 4 is by far the largest among the five classes (Additional file 1: Figure S2A, B). In mitosis, when the spindle tension is lacking (β=1; Fig. 2 f), the model predicts a high probability of errors, mainly monotelic (class 2) states, as well as the correct amphitelic ones (class 5) at steady states. When kinetochore–microtubule attachment is stabilised by reducing q, merotelic errors (class 4) persist in both meiosis and mitosis (Fig. 2 d, g). Class 5 will eventually replace class 4 but only very slowly; in meiosis I with p=0.05,α=β=0, the mean first passage times to class 5 are ∼1631 for q=0.01 versus ∼47 for q=0.05.
A number of studies have demonstrated that experimental manipulations of kinetochore–microtubule interactions lead to accumulation of incorrect spindle attachments (classes 1–4) and aneuploidy [8]. Lack of tension (i.e. β=1) makes amphitelic states (class 5) unstable [25–27]. Conversely, inhibition or depletion of aurora B kinase, which overstabilises kMT attachments (by reducing q), causes errors in chromosome alignment and segregation [7, 27, 30, 34, 35]. These observations are consistent with our model predictions in which imbalance 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 different conditions (Fig. 3 a–c and Additional file 1: Figure S6 for meiosis I; Additional file 1: Figure S7 for mitosis). We found qualitatively similar kMT distributions in mitosis and meiosis I, except the difference in the predicted phenotype in various conditions (Fig. 2 b–g). The model predicts that in normal conditions (p=q=0.05,α=β=0) the number of kMTs increases steadily in class 5 while it remains low in the other classes as their total probability diminishes (Fig. 3 a and Additional file 1: Figure S7A). This is in agreement with experimental evidence suggesting the gradual increase of kMTs during the prometaphase to metaphase in mitosis [36] and in meiosis I [26]. With smaller q, the number of kMTs increases not only in class 5 but also in class 4 (Fig. 3 b and Additional file 1: Figure S7B). This explains why errors persist in this condition. Note that when β=0, the number of kMTs approaches n. An increasing number of kMTs may also switch off the spindle assembly checkpoint in merotelic states (class 4) over time.
These model predictions on the probability distribution of the number of kMTs have an important implication in the regulation of the spindle assembly checkpoint. Experimental evidence suggests that intrakinetochore stretching (or kinetochore deformation), which is brought about by kMT attachments, has a role in relieving the spindle assembly checkpoint [37–39]. Therefore, the predicted gradual increase of kMTs in amphitelic states (class 5) (Fig. 3 a and Additional file 1: Figure S7A) may switch off the spindle assembly checkpoint progressively. The same argument applies to merotelic states (class 4), the probability of which increases when the q/p ratio is small (Fig. 2 d, g); stabilisation of kMTs (Fig. 3 b and Additional file 1: Figure S7B) may also inactivate the spindle assembly checkpoint in merotelic states over time. This explains why merotelic orientation evades the spindle assembly checkpoint [40], leading to aneuploidy. Intrakinetochore stretching by kMT attachment, however, does not allow the cell to discriminate between correct (amphitelic; class 5) versus incorrect (nonamphitelic; classes 1–4) kMT attachments [10]—the cell does not need to do so because chromosome biorientation occurs by probabilistic selforganisation as our model indicates.
We also examined how kMT number changes in amphitelic states under low spindle tension (β=1; Fig. 3 c and Additional file 1: Figure S7C). Regardless of the classes, the distribution of kMT number remains low, which makes the transition of the process from one class to another more frequent. Similar probability distributions of kMT number in meiosis I were obtained when α=β=1 (Additional file 1: Figure S6A) and α=1,β=0 (Additional file 1: Figure S6B).
The exact probability distribution of kMT number at steady states can be derived in the special case when α=β=γ=1: its mean is \(\bar {N}=n\rho /(n+\rho)\) where ρ=2p/q (\(\bar {N}=5/3\) for p=q,n=10). We also obtained an analytical approximation of the kMT number distribution in class 5 when α=0:
where \(\bar {\rho }=\rho /\beta =2p/(\beta q)\) (Fig. 3 d and Additional file 1: Figure S8A, B). This formula is valid for both mitosis and meiosis and provides an analytical explanation as to how tension (β) alters the stability of kMTs by modulating the q/p ratio.
Dynamics of multiple chromosomes
The above results concern the behaviour of a single pair of homologous chromosomes. It is natural to ask how multiple pairs in the cell are bioriented simultaneously—we call this event synchrony to distinguish it from the onset of anaphase. We assumed the system consists of 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 \(\theta _{t}^{k}\) (see Additional file 1).
The timing of synchrony delays as k increases (Fig. 4 a and Additional file 1: Figure S3E, solid lines). If the balance of the q/p ratio is broken by reducing q (Fig. 4 a and Additional file 1: Figure S3E, dashed lines), the timing of synchrony is delayed further (see also Additional file 1: Figure S9). The probabilities of synchrony, however, eventually approach 1 in all of these conditions with β=0. This implies that delaying the onset of anaphase could reduce the chromosome malorientation and missegregation. Consistently, Cimini et al. showed that prolonging the metaphase significantly reduced the number of lagging chromosomes in the anaphase (indicating incorrect kMT attachments) in mitosis [41].
We next examined the contribution of α and β to the establishment of synchrony. Figure 4 b shows the steadystate probability of synchrony in meiosis I as a contour plot. It indicates that, to achieve synchrony reliably at steady states, α and β have to be relatively small. It is conceivable that, to progress into anaphase, synchrony has to be maintained for a sufficient time to relieve the spindle assembly checkpoint [10]. Figure 4 c depicts the halflife of synchrony in meiosis I as a contour plot (see also Additional file 1: Figure S3F for mitosis). The halflife increases steeply for small values of α and β. These data suggest that α and β need to be tightly regulated for efficient chromosome biorientation 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 meiosis I before the establishment of correct biorientation (see Additional file 1 for methods). We calculated the number of biorientation attempts per bivalent, i.e. the mean number of transitions from class 2 or 4 to class 5 before the kinetochore is fully occupied (r _{ n }(n,0,0,n) and r _{ n }(0,n,n,0) when β = 0) (Fig. 4 d). It suggests that the larger α is, the more biorientation attempts are needed. We also found the number of biorientation attempts decreases as q (detachment probability) reduces (Fig. 4 d, see also Additional file 1: Figure S10). Consistent with this, Kitajima et al. observed the number of attempts reduced from ∼3 in untreated mouse oocytes to just one on average in those treated with hesperadin, an Aurora B kinase inhibitor [7].
Conclusions
Our simple discretetime Markov chain model captures the prominent features of the chromosome biorientation process. It provides a unified account of two modes of divisions, mitosis and meiosis I, under a single theoretical framework. The model reveals where the differences in the biorientation process come from and it explains why errors are very frequent in the first meiotic division, which are major causes of infertility, miscarriages and birth defects in humans.
One of our key findings in this study is that the system dynamics (including the type and frequency of transient kMT attachment errors) is dictated by the q/p ratio (relative detachment rate) of kMTs. An imbalance of the q/p ratio causes persistent attachment errors leading to chromosome missegregations. The gradual increase of kMTs may help turn off the spindle assembly checkpoint in normal conditions but can promote a faulty conformation (merotelic attachments) to evade the checkpoint.
In summary, our study revealed that chromosome biorientation is a probabilistic selforganisation, rather than a sophisticated process of error detection and correction. Although our model omits many potentially important factors for chromosome biorientation, such as the spatial arrangement of centrosomes, it allowed us to examine analytically all possible outcomes with different parameters (i.e. the whole parameter space), revealing what is fundamental for accurate chromosome segregation. The proposed model, which is based on a firm mathematical foundation, gives valuable insights that help us understand one of the primary causes of chromosomal instability—aberrant kMT dynamics.
Methods
The model and its analysis are explained in detail in Additional file 1. The analysis of discretetime Markov chains was performed according to [21, 33, 42]. We used Mathematica®; (version 10, Wolfram Research) to implement and analyse the model, with a standard laptop (or desktop) computer. The Mathematica codes used in this study are provided in Additional file 2.
Abbreviations
 kMT:

kinetochore microtubule
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Acknowledgments
We thank G Bewick, C Grebogi, S Hoppler, A Lorenz, C McCaig, F PerezReche, R Sekido, M Thiel and E Ullner for helpful discussions and critical reading of the manuscript. YS and CG were supported by Scottish Universities Life Sciences Alliance (SULSA) and HO by Wellcome Trust (grant numbers 098030 and 092076).
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Competing interests
The authors declare that they have no competing interests.
Authors’ 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. All authors read and approved the final manuscript.
Additional files
Additional file 1
Supplementary Information. Details of the model construction and analysis and Figures S1–S10. (PDF 930 kb)
Additional file 2
Mathematica codes. The Mathematica codes used in this study. To read the file, Wolfram CDF player (available free from https://www.wolfram.com/cdfplayer/) or Mathematica (Wolfram Research) is required. (ZIP 62 kb)
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Keywords
 Chromosome segregation
 Kinetochore
 Microtubule
 Mitosis
 Meiosis
 Markov chain
 Selforganisation