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Accurate Chromosome Segregation by Probabilistic Self-Organisation Yasushi Saka1*, Claudiu V

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Accurate Chromosome Segregation by Probabilistic Self-Organisation Yasushi Saka1*, Claudiu V Saka et al. BMC Biology (2015) 13:65 DOI 10.1186/s12915-015-0172-y RESEARCH ARTICLE Open Access Accurate chromosome segregation by probabilistic self-organisation Yasushi Saka1*, Claudiu V. Giuraniuc1 and Hiroyuki Ohkura2* 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 bi-orientation). Incorrect attachments of kinetochore microtubules (kMTs) lead to chromosome mis-segregation 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 bi-orientation 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 bi-orientation. With the right balance, incorrect attachments are resolved spontaneously into correct bi-oriented 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 bi-orientation is a probabilistic self-organisation, rather than a sophisticated process of error detection and correction. Keywords: Chromosome segregation, Kinetochore, Microtubule, Mitosis, Meiosis, Markov chain, Self-organisation Background meiosis, attach to the microtubules from opposite spindle Accurate segregation of chromosomes during cell division poles by kinetochores [3]. Each kinetochore consists of is fundamental to life. Errors in this process result in cell more than 100 different proteins assembled on each cen- death or aneuploidy. Chromosome segregation is usually tromeric DNA sequence, many of which are involved very accurate. However, mis-segregation occurs at a much in the interaction with microtubules [4]. Chromosome higher frequency in cancer cells and oocytes, which is a bi-orientation is a very dynamic process with frequent contributing factor to cancer progression [1] and also a attachments and detachments of microtubules [5–8]. major cause of infertility, miscarriages and birth defects For proper segregation of chromosomes, all kineto- such as Down syndrome [2]. chores need to attach to spindle microtubules while The key event for chromosome segregation is the estab- erroneous attachments must be eliminated before the lishment of chromosome bi-orientation, in which sister onset of anaphase. It is known that attachment errors are chromatids in mitosis or homologous chromosomes in more frequent in meiosis I (especially in oocytes) than in mitosis [2, 5–7]. Yet it is not understood why this is so. *Correspondence: [email protected]; [email protected] Unattached kinetochores act as signal generators for the 1Institute of Medical Sciences, School of Medical Sciences, University of spindle assembly checkpoint, which delays chromosome Aberdeen, Foresterhill, AB25 2ZD Aberdeen, UK segregation until proper bi-orientation is established for 2Wellcome Trust Centre for Cell Biology, University of Edinburgh, Michael Swann Building, Max Born Crescent, EH9 3BF Edinburgh, UK all chromosomes [9]. It remains unclear, however, whether © 2015 Saka et al. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons. org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated. Saka et al. BMC Biology (2015) 13:65 Page 2 of 10 improperly attached kinetochore microtubules (kMTs) conformation can evade the spindle assembly checkpoint are also detected and corrected by the spindle assembly by a gradual increase of the number of kMTs. Despite checkpoint or by an independent mechanism [10]. its simplicity, the model is consistent with a number The precise mechanism of chromosome bi-orientation of experimental observations and provides theoretical has been under intense investigations. However, it is not insights into the origins of chromosomal instability and yet possible to observe the dynamics of individual micro- aneuploidy. tubules in vivo in real time. Mathematical modelling provides a powerful means to study the chromosome bi- Results and discussion orientation process. Since the discovery of the dynamic A probabilistic model of kinetochore–microtubule instability of microtubules [11], a number of theoretical interaction analyses have provided important insights into the inter- A single kinetochore can bind randomly to microtubules action between microtubules and kinetochores (for exam- from either left or right pole (Fig. 1a). We assume a single ple, [12, 13]). The so-called search-and-capture model kinetochore can accommodate up to n microtubules. The explains how dynamically unstable microtubules capture process of microtubule attachment/detachment can be chromosomes [14–17]. represented as a discrete-time Markov chain [21] (Fig. 1b However, the original search-and-capture model did and Additional file 1: Figure S1). not concern events after capture, in particular, erroneous Each pair of sister chromatids in mitosis has two kine- attachments of kMTs and their correction. To address tochores (k1 and k2 in Fig. 1c). In meiosis I, a pair of sister this, Paul et al. put forward a modified search-and-capture kinetochores are physically connected side-by-side and model with explicit correction mechanisms [18]. Gay et al. act as one [22, 23]. Therefore, in our model, a bivalent (a proposed a stochastic model of kinetochore–microtubule pair of homologous chromosomes connected by chiasma) attachments in fission yeast mitosis, which reproduced also has two kinetochores in meiosis I. We assume these correct chromosome bi-orientation and segregation two kinetochores interact with microtubules indepen- in simulations [19]. In addition to the kinetochore– dently. Hence, the state of the kinetochores is represented microtubule interaction, Silkworth et al. showed that tim- as rn(i1, j1, i2, j2), which can be classified into one of five ing of centrosome separation also plays a crucial role for classes according to the pattern of microtubule attach- accurate chromosome segregation [20]; using experimen- ments (Fig. 1d). State transitions occur in a stereotypical tal and computational approaches, they demonstrated manner among these classes irrespective of the value of that cells with incomplete spindle pole separation have n ≥ 2 (Fig. 1e and Additional file 1: Figure S2E; refer to a higher rate of kMT attachment errors than those Table 1 for a summary of the parameters herein). Notably, with complete centrosome separation. Yet, the question the only possible transitions out of class 5 (amphitelic, remains unanswered as to how the cell can discriminate i.e. correct conformation) is to class 2 (monotelic) or 4 between correct and incorrect kMT attachments as their (merotelic) (red and green arrows in Fig. 1e). Note also models assumed an explicit bias based on the discrimina- that this transition scheme is similar to the kinetic error tion of correct versus incorrect connections. correction model (a deterministic ordinary differential A major impediment to understanding fully the mech- equation model) proposed by Mogilner and Craig [24]; anism of chromosome bi-orientation is the lack of a their scheme is a limiting case—only two kMT attach- universal theoretical framework that covers the chro- ments per kinetochore are allowed and transitions out of mosome bi-orientation process during eukaryotic cell amphitelic states are prohibited. divisions in general, including both mitosis and meio- We assume the association probability is proportional sis. Here we present such a universal model of chro- to the available surface area of the kinetochore while mosome bi-orientation, which is simple yet applicable the dissociation probability is independent, as illustrated to any eukaryotic cell division. Firstly, the model reveals below: − − that the balance between attachment and detachment n i1 j1 n p probabilities of kMTs is crucial for correct chromosome rn(i1, j1, i2, j2) −→ rn(i1 + 1, j1, i2, j2),(1) bi-orientation. With the right balance, incorrect attach- i q r (i , j , i , j ) −→1 r (i − 1, j , i , j ),(2) ments are resolved spontaneously into correct bi-oriented n 1 1 2 2 n 1 1 2 2 conformations while an imbalance leads to persistent where 0 ≤ p ≤ 1/4and0 ≤ q ≤ 1/2n.2× p is the errors. Therefore, the superficially complex process, chro-
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