stochastic modelling of chromosomal segregation: errors

Bull Math Biol (2014) 76:1590–1606DOI 10.1007/s11538-014-9969-z

ORIGINAL ARTICLE

Stochastic Modelling of Chromosomal Segregation:Errors Can Introduce Correction

Anastasios Matzavinos · Blerta Shtylla ·Zachary Voller · Sijia Liu · Mark A. J. Chaplain

Received: 19 April 2012 / Accepted: 17 April 2014 / Published online: 13 May 2014© Society for Mathematical Biology 2014

Abstract Cell division is a complex process requiring the cell to have many internalchecks so that division may proceed and be completed correctly. Failure to dividecorrectly can have serious consequences, including progression to cancer. Duringmitosis, chromosomal segregation is one such process that is crucial for successfulprogression. Accurate segregation of chromosomes during mitosis requires regulationof the interactions between chromosomes and spindle microtubules. If left uncorrected,chromosome attachment errors can cause chromosome segregation defects which haveserious effects on cell fates. In early prometaphase, where kinetochores are exposedto multiple microtubules originating from the two poles, there are frequent errors inkinetochore-microtubule attachment. Erroneous attachments are classified into twocategories, syntelic and merotelic. In this paper, we consider a stochastic model for apossible function of syntelic and merotelic kinetochores, and we provide theoreticalevidence that merotely can contribute to lessening the stochastic noise in the time forcompletion of the mitotic process in eukaryotic cells.

A. Matzavinos (B)Division of Applied Mathematics, Brown University,182 George Street, Providence, RI 02912, USAe-mail: [email protected]

B. ShtyllaDepartment of Mathematics, Pomona College, Claremont, CA 91711, USA

Z. Voller · S. LiuDepartment of Mathematics, Iowa State University, Ames, IA 50011, USA

M. A. J. ChaplainDivision of Mathematics, University of Dundee, Dundee, DD1 4HN, UK

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Stochastic Modelling of Chromosomal Segregation 1591

Keywords Stochastic noise · Cell division · Chromosome segregation · Coefficientof variation · Markov jump process · Absorption time · Stochastic chemical kinetics

1 Introduction

Accurate segregation of chromosomes during mitosis requires regulation of the inter-actions between chromosomes and microtubules, which are the core components ofthe complex mitotic spindle segregation machine (Karsenti and Vernos 2001; Mitchi-son and Salmon 2001; Meunier and Vernos 2012). Equal partitioning of the geneticmaterial is achieved by ensuring that mitotic spindle forces align all sister chromatidsat the cell equator. At the centromere of each sister chromatid of a chromosome,kinetochore complexes provide a scaffold for chromosome/microtubule interactions(Cheeseman and Desai 2002). Microtubules during mitosis typically nucleate fromtwo sites, called spindle poles, and subsequently undergo frequent random transitionsbetween states of growth and shortening in search of kinetochores using a proposed“search and capture” mechanism (Kirschner and Mitchison 1986). In many cells, asingle kinetochore is capable of establishing connections with multiple microtubules.Precise genome partitioning requires stable attachment of each sister kinetochore tomicrotubules connected to opposite spindle poles. This kind of attachment, calledbipolar attachment, generates mechanical tension between the two sister chromatidscaused by the pulling force of the mitotic spindle.

In early prometaphase, where kinetochores are exposed to multiple microtubulesoriginating from the two poles, there are frequent errors in kinetochore-microtubuleattachment (Ault and Rieder 1992a; Tanaka 2008). There are various ways in whicha chromosome might become erroneously attached to microtubules (Gregan et al.2011); here we focus on two main categories, syntelic and merotelic attachments. Achromosome is said to be syntelically attached if both sister kinetochores are attachedto microtubules nucleated from the same spindle pole, whereas when a kinetochoreis attached to each spindle pole at the same time, a chromosome is considered to bemerotelically attached. In contrast to the above described two cases, an amphitelicattachment is established when a chromosome has each sister kinetochore attachedto separate spindle poles. Note that the chromosome is correctly connected to thespindle if it is amphitelically attached, since this kind of connection can ensure thateach chromatid is then properly guided to the appropriate spindle pole during anaphase.

Erroneous attachments are established early in mitosis (Ault and Rieder 1992b;Cimini et al. 2003; Tanaka 2008); however, most of them are corrected during thefinal stages of division (Cimini et al. 2003; Lampson et al. 2004). If left uncorrected,chromosome attachment errors can cause chromosome segregation defects which haveserious effects on cell fates (Cimini and Degrassi 2005). To this end, the cell hasdeveloped error-correcting mechanisms which ensure that all chromosomes becomeamphitelically attached before the anaphase separation signal is released. Aurora B,a member of serine/threonine protein kinsases, has been identified as the principalcomponent of error correcting mechanisms (Carmena and Earnshaw 2003; Biggins etal. 1999; Cheeseman et al. 2002; Francisco et al. 1994; Tanaka et al. 2002). Aurora Bis part of the chromosomal passenger complex (CPC), which also includes the inner

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1592 A. Matzavinos et al.

centromere protein (INCEP), Survivin and Borealin (Bolton et al. 2002; Ruchaud etal. 2007). The CPC is thought to contribute to the localization of the kinase at theinner centromeric regions close to kinetochores, in close proximity to attached micro-tubules. The inhibition of Aurora B in vertebrate cells leads to the stabilization ofincorrect attachments (Lampson et al. 2004). Conversely, when Aurora B is activatedby removal of its inhibitors, it is observed that erroneous attachments are selectivelydestabilized (Hauf et al. 2003; Kallio et al. 2002; Ditchfield et al. 2003). There areseveral proposed models for how Aurora B might regulate erroneous attachments(for a review see Maresca and Salmon 2010; Lampson and Cheeseman 2011); how-ever, the common theme is that Aurora B phosphorylates kinetochore substrates inthe absence of mechanical tension between sister kinetochores, which destabilizesincorrect attachments until amphitelic attachments are achieved.

Eukaryotic cells employ a highly sensitive surveillance mechanism, called thespindle-assembly checkpoint (SAC), which works to delay anaphase until the lastchromosome has established amphitelic attachments (Musacchio and Salmon 2007;Maresca and Salmon 2010). The signaling molecules that comprise the SAC are local-ized and regulated at kinetochores, and Aurora B has been suggested to play a role inthe SAC. Anaphase initiation requires the activation of the anaphase-promoting com-plex /cyclosome (APC/C), which targets cyclin B and securin for degradation. TheSAC pathway generates a signal that blocks the activation of the APC/C. More specif-ically, the SAC negatively regulates the ability of an APC/C cofactor to activate thepolyubiquination of cyclin B and securin (Musacchio and Salmon 2007), which thencan prolong metaphase until all chromosomes are correctly attached. At a molecularlevel, APC/C inhibition is achieved with the help of a four protein complex, knownas the mitotic checkpoint complex (MCC), which is composed of checkpoint proteinsMad2, BubR1, Bub3, and Cdc20 (Musacchio and Salmon 2007). In this complicatednetwork of surveillance reactions, Aurora B affects MCC formation by destabilizingthe localization of BubR1, Mad2, and Cenp-E at kinetochores (Ditchfield et al. 2003;Morrow et al. 2005). Many details of the SAC have been uncovered; however, the fullpicture of how these reactions check the state of a kinetochore is still the source ofintense debate (it is not clear whether lack of tension or simple microtubule occupancytriggers the SAC Musacchio and Salmon 2007).

Models which involve Aurora B have been proposed in an effort to clarify the roleof various kinetochore signaling proteins in anaphase onset control (see, e.g., Mistryet al. 2008 and references therein). Despite the various correcting mechanisms thatcells employ to ensure correct chromosome attachments, a central question in the fieldof mitosis is how hundreds of MTs experiencing constant shortening and growth intens of seconds are able to correctly engage with multiple chromosomal kinetochoresin surprisingly short timespans of ≈20 min (Rieder and Maiato 2004). Mathematicalmodeling that considers each chromosome individually and its transition betweendifferent attachment states is helpful in providing an answer to this important question,since it is still difficult to tease apart individual microtubule/kinetochore interactionsin experimental settings (Paul et al. 2009; Gay et al. 2012). Merotelic and syntelicattachments are established as errors inherent to the stochastic nature of the search-and-capture mechanism when each kinetochore is inevitably exposed to microtubules

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from both spindle poles. Thus, a quantitive approach for this process needs to bestochastic in nature.

In this paper, we investigate how chromosome attachment errors and error-correcting mechanisms influence the noise in the time for completion of the mitoticprocess. The mathematical model we present considers a simplified scenario of thevarious states that chromosomes may experience during mitosis in dividing cells,focussing attention on the two most important cases of erroneous attachments forbioriented chromosomes. We assume that the transitions between these states arenot reversible (in the interest of simplicity) and further we include a direct transi-tion between an unattached chromosome to a correctly biorented state. The analyticaland computational results of our model provide theoretical evidence that a tightly con-trolled strictly positive rate of erroneous attachments facilitates a more accurate timingof the mitotic process than in the complete absence of attachment errors. The plan ofthe paper is the following. In Sect. 2, we briefly introduce the model of Mistry et al.(2008) on the dynamics of merotelic and syntelic kinetochore attachments. In Sect. 3,we present a complete analytical derivation of the dependence of the (stochastic) timeT at which all kinetochores become amphitelically attached on the system parameters.Finally, we conclude with a discussion of our results in Sect. 4.

2 The Mathematical Model

Following Mistry et al. (2008), we consider a continuous time Markov jump processthat is associated with the graphical representation of the possible kinetochore attach-ment transitions shown in Fig. 1. The vertices of this graph correspond to the possibleattachment states of the kinetochore pairs with directed edges connecting two verticesif kinetochore pair attachments can transition between the corresponding states.

As seen in Fig. 1, the possible kinetochore attachment transitions are:

Uk1−→ M, U

k2−→ S, Uk3−→ C, S

k4−→ C, and Mk5−→ C,

where the parameters k1, k2, . . . , k5 correspond to the rate constants at which a kine-tochore pair transitions between the corresponding attachment states. We denote thestate vector at time t > 0 as

X (t) = (XU (t), X S(t), X M (t), XC (t)) , (1)

where XU , X S, X M , and XC are integer-valued stochastic processes representingthe number of the 46 possible kinetochore pairs in the unattached (U), syntelic (S),merotelic (M), and amphitelic (C) attachment states, respectively. Throughout thepaper, the initial value of the state vector is given by X (0) = (46, 0, 0, 0).

We define P(x, t) to be the joint probability that X (t) = x at time t . Hence,P(x, t) denotes the probability that the state of the system at time t is given by x =

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Fig. 1 Network diagram showing the possible states each kinetochore unit could be in at any given timeand how they can move from state to state. In particular, each kinetochore pair can be in one of the followingstates: unattached (U), syntelic (S), merotelic (M), or amphitelic (C)

(x1, x2, x3, x4), where for each i ≤ 4, xi is a non-negative integer and∑4

i=1 xi = 46.In this context, the master Eq. (Gardiner 1985) that governs the evolution of P(x, t) is

ddt

P(x, t) =5∑

i=1

ai (x − νi ) · P(x − νi , t) −5∑

j=1

a j (x) · P(x, t), (2)

where ai (x − νi ) can be understood as the probability per unit time of a transitionfrom state x − νi to state x . In the chemical literature, the rate functions ai are usuallyreferred to as the propensities of the corresponding transitions (see, e.g., Gillespie2007). In general, the relation between the propensities ai and the deterministic kineticrate constants ki depends on the order of the corresponding reaction schemes (Higham2008). In the case of the first-order transitions shown in Fig. 1, we have that ai (x−νi ) =ki Ni (x − νi ), where Ni (x − νi ) is the number of kinetochores acting as the substrateof the i-th reaction scheme. The “stoichiometric vector” νi represents the change inpopulation that occurs when the i-th reaction fires (Gillespie 2007). Specifically, inwhat follows we consider the five transitions shown in Fig. 1 given by

ν1 = (−1, 0, 1, 0), ν2 = (−1, 1, 0, 0), ν3 = (−1, 0, 0, 1),

ν4 = (0,−1, 0, 1), and ν5 = (0, 0,−1, 1). (3)

The master Eq. (2) along with the specified “connectivity” of the system providedby (3) uniquely determines a continuous time Markov jump process, the analysis ofwhich is the main focus of this paper. More precisely, in what follows we analyzethe dependence of the distribution of the time T at which all kinetochores reach theamphitelic state (C) on the kinetic parameters of the model. We subsequently use theseresults to understand how stochastic fluctuations of T are controlled by the system.

3 Results

We focus on the question of how fluctuations in the random time T at which allkinetochores become amphitelically attached are controlled by the system parameters.

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We address this question by deriving analytically the probability distribution of T inSect. 3.1, and then in Sect. 3.2, we discuss some important biological implications.

3.1 The Distribution of the Absorption Time T

In this section, we derive analytically the probability distribution of the time it takesthe process to get absorbed at the state (0, 0, 0, 46). The latter corresponds to theconfiguration of the system where all 46 kinetochores are amphitelically attached. Asspecified in Sect. 2, we assume that the system is initially at state (46, 0, 0, 0) withprobability one. In addition to the state vector given by (1), we define the auxiliaryrandom vector

γ(t) = (γ1(t), γ2(t), . . . , γm(t)),

where m = 46 is the number of chromosomes and each random variable γi (t) ∈{U, M, S, C} gives the attachment state of the corresponding kinetochore pair at timet ≥ 0.

The amount of time it takes for all 46 kinetochore pairs to reach the amphitelicattachment state is formally defined as

T = inf {t > 0 : X (t) = (0, 0, 0, 46)} ,

where we use the standard convention that the infimum over an empty set is infinity.The hitting time T can also be represented as

T = supi≤m

Ti (C),

where

Ti (state) = inf {t > 0 : γi (t) = state}

with state ∈ {U, M, S, C}. In the following, for the sake of notational simplicity, weset Ti = Ti (C) for all i ≤ m. Correspondingly, for the distribution function of T , wehave

P(T ≤ t) = P(Ti ≤ t,∀ i = 1, 2, . . . , m).

Using the fact that the kinetochore pairs are assumed to be identical and act inde-pendently, this yields

P(T ≤ t) =[P(T1 ≤ t)

]m =[1 − P(T1 > t)

]m

Hence,

P(T > t) = 1 −[1 − P(T1 > t)

]m, (4)

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which gives the probability that not all kinetochore pairs have reached the amphitelicattachment state by time t . Therefore, we only need to calculate P(T1 > t), theprobability that the first kinetochore pair has not reached the amphitelic attachmentstate by time t > 0, in order to determine the tail of the probability distribution of thehitting time T .

In order to compute the distribution of T1, we introduce the events that the first kine-tochore pair reaches the amphitelic attachment state through the merotelic attachmentstate as

A = {T1(M) < ∞, T1(S) = ∞},

through the syntelic attachment state as

B = {T1(M) = ∞, T1(S) < ∞},

and by going directly to the amphitelic attachment state as

C = {T1(M) = ∞, T1(S) = ∞}.

Hence, the tail of the probability distribution of the hitting time T1 is given by

P(T1 > t) = P(T1(M) + T1 − T1(M) > t | A

)· P(A)

+ P(T1(S) + T1 − T1(S) > t | B

)· P(B) + P

(T1 > t | C

)· P(C)

= P(T1(M) + T1 − T1(M) > t | A

)· k1

k0

+ P(T1(S) + T1 − T1(S) > t | B

)· k2

k0+ P

(T1 > t | C

)· k3

k0, (5)

where

k0 = k1 + k2 + k3.

Notice that T1(M) and T1 − T1(M) are independent under the conditional mea-sure P(· | A). Similarly, T1(S) and T1 − T1(S) are independent under the conditionalmeasure P(· | B). We now consider the conditional probability,

P(T1(M) > t | A

)= P

(T1(M) > t,A

)

P(A),

which determines the probability that the first kinetochore pair transitions to themerotelic attachment state at some moment in time after time t given event A occurs.

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Since, P(A) = k1/k0 and the event A occurs when the reaction scheme U → M firesbefore U → S and U → C , we obtain

P(T1(M) > t | A

)= k0

k1

∞∫

t

k1e−k1ada

∞∫

a

k2e−k2bdb

∞∫

a

k3e−k3cdc

= k0

k1

∞∫

t

k1e−k0 ydy = e−k0t .

Following a similar derivation as above,

P(T1(M) > t | A

)= P

(T1(S) > t | B

)= P

(T1 > t | C

)= e−k0t . (6)

We note that (6) is a manifestation of a general property of a collection of independentexponential random variables. Therefore,

P(T1 > t | A) = P(T1(M) + T1 − T1(M) > t | A

)=

∞∫

t

fa(τ )dτ, (7)

where the continuous probability density fa(t) is determined by the convolution inte-gral:

fa(t) =t∫

0

k0e−k0τ k5e−k5(t−τ )dτ = k0k5e−k5t

t∫

0

e−(k0−k5)τ dτ

= k0k5

k0 − k5

(e−k5t − e−k0t). (8)

It follows from Eqs. (7) and (8) that

P(T1 > t | A) = k0

k0 − k5e−k5t − k5

k0 − k5e−k0t .

Similarly,

P(T1 > t | B) = k0

k0 − k4e−k4t − k4

k0 − k4e−k0t ,

and

P(T1 > t | C) = e−k0t

from Eq. (6).

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Thus, in lieu of (5), we obtain the probality distribution

P(T1 > t) = k1

k0 − k5e−k5t + k2

k0 − k4e−k4t

−(

k1k5

k0(k0 − k5)+ k2k4

k0(k0 − k4)− k3

k0

)e−k0t .

Using this expression in (4) yields an explicit expression for the distribution functionof T . The biological implications of the latter are delineated in the following section.

3.2 Biological Implications

We now employ the analytical results of Sect. 3.1 to compute the dependence of the firsttwo moments of the all-amphitelic hitting time T on the kinetic parameters associatedwith kinetochore transitions. While it would be of interest to explore the relationshipsbetween all five of the parameters k1, k2, k3, k4, and k5, we note that we have narrowedour focus to the relationship between the parameters k1 and k3 for the purpose of thispaper. We have allowed the parameter that governs transitions from the unattached tomerotelic attachment states, k1, to take on values in the range 0.01–0.25 min−1, and theparameter that governs transitions from the unattached to amphitelic attachment states,k3, to take on values in the range 0.01–0.5 min−1. The rest of the kinetic parametershave been set to the values adopted by Mistry et al. (2008).

The analytically derived expression for P(T > t) was used to compute the meanµ and standard deviation σ of the all-amphitelic hitting time T . Moreover, for eachchoice of values for k1 and k3, µ and σ were used to quantify the stochastic fluctua-tions in the time at which the system reaches the all-amphitelic state. Two measurescommonly employed in the stochastic chemical kinetics literature for quantifyingnoise/fluctuations include the coefficient of variation (CV) (see, e.g., Laurenzi 2000;Kepler and Elston 2001; Elowitz et al. 2002; Swain et al. 2002) and the Fano factor(Thattai and van Oudenaarden 2001; Ozbudak et al. 2002; Blake et al. 2003). Theformer is defined as the standard deviation divided by the mean, whereas the latter isdefined as the variance σ 2 divided by the mean. More detailed information on boththe CV and the Fano factor can be found in Appendix 2.

Figure 2a shows the values of the coefficient of variation over the discussed rangeof values for the parameters k1 and k3. The results show that noise in the all-amphitelichitting time for this system is minimized when k1 ≈ 0.15 min−1 and k3 ≈ 0.07 min−1.The model also predicts that the noise in the hitting time is much more dependent onthe values of the parameter k3. This can be seen in the much steeper increase in valuesof the CV along the k3 axis versus the k1 axis. This can also be achieved numericallyas shown in Fig. 2b by applying the Gillespie method to Eq. (2) (see Appendix 1).

Figure 3a, b show the dependence of the CV and Fano factor, respectively, onparameter k3 when k1 = 0.1 min−1. As can be seen, both measures predict a nonlineardependence of noise on the rate of transition from the unattached to the amphitelicstate, albeit they provide a different value of k3 for which the noise is minimized andthe timing of the process is most accurate. This is expected, as it has been pointed out

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Fig. 2 Plots showing the dependence of the coefficient of variation for the hitting time T on k1 and k3, ascomputed a analytically by the approach taken in Sect. 3.1 and b numerically by the Gillespie method (seeAppendix 1)

Fig. 3 Plots showing the dependence on parameter k3 of a the CV and b the Fano factor for the hittingtime T

by various authors that the CV and the Fano factor can lead to different conclusionswith respect to noise (Swain et al. 2002; Gadgil et al. 2005) (see also Appendix 2).Nonetheless, both measures predict that merotely can contribute to lessening the noisein the time for completion of the mitotic process.

In addition to computing the dependence of the coefficient of variation and Fanofactor on k3, we investigated whether the variation of these quantities depends onvariations of the mean hitting time only, i.e., whether or not there is an essentialdependence of the standard deviation on k3. Figure 4 clearly shows that both meanand standard deviation of the hitting time distribution decrease as the parameter k3increases, and hence neither the mean nor the standard deviation alone is the dominantfactor for variations in the level of noise when varying parameter values.

One of the main conclusions from these computations is that, in the context of themodel of Mistry et al. (2008), the process of merotely (i.e., erroneous attachment) cancontribute to a lessening of the noise in the time for completion of the mitotic process.Specifically, our analytical results clearly demonstrate that a tightly controlled strictly

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Fig. 4 Plots showing the dependence of a the mean hitting time and b the standard deviation of the hittingtime on parameter k3

positive rate of erroneous attachments facilitates a more accurate timing of the mitoticprocess than in the complete absence of attachment errors.

4 Discussion

Cell division is a complex process requiring the cell to have many internal checks so thatdivision may proceed and be completed correctly. Failure to divide correctly can haveserious consequences, including progression to cancer. During mitosis, chromosomalsegregation is one such process that is crucial for successful progression. Accuratesegregation of chromosomes during mitosis requires regulation of the interactionsbetween chromosomes and spindle microtubules. These chromosome/microtubuleinteractions are controlled by kinetochore complexes which provide a scaffold at thecentromere of each sister chromatid, and progression through mitosis depends uponthe stable attachment of each sister kinetochore to opposite spindle poles. This processof kinetochore-microtubule attachment is prone to errors and erroneous attachmentsare classified into two categories—syntelic and merotelic. In the former category, achromosome is said to be syntelically attached if both sister kinetochores are attachedto chromosomes nucleated from the same spindle pole. In the latter category, a chromo-some is said to be merotelically attached if a kinetochore is attached to each spindlepole at the same time (in contrast to the above described two cases, an amphitelicattachment is established when a chromosome has each sister kinetochore attached toone spindle pole).

In this paper, we have developed a stochastic model of kinetochore attachmenttransitions. Following Mistry et al. (2008), we considered a continuous time Markovjump process that is associated with all possible kinetochore attachment transitionsand derived a system of stochastic differential equations governing the evolution intime of the probability of being in one of the possible attachment states of kinetochorepairs i.e., unattached (U), syntelic (S), merotelic (M), and amphitelic (C). Our modelconsidered a simplified scenario of the various states that chromosomes may experi-

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ence during mitosis in dividing cells, while focussing on the two most important casesof erroneous attachments for bioriented chromosomes. We assumed that the transi-tions between these states were not reversible (in the interest of simplicity) and furtherwe included a direct transition between an unattached chromosome to a correctlybiorented state. Such a transition can be justified in light of chromosome cooperativemechanisms (Tanaka et al. 2005) and chromosome-based microtubule nucleation path-ways (Tulu et al. 2006; Goshima et al. 2005). Furthermore, here we did not attempt toinclude the role of Aurora B error-correcting feedbacks explicitly in our model. Manysuch mechanisms remain highly speculative. However, our modeling framework is agood starting point for studying such pathways, since stochastic effects are significant.In the model of (Paul et al. 2009) for chromosome attachment to the spindle duringprometaphase, a fully attached spindle could be reached within physiological timesif chromosomes could reorient to shield kinetochores from errors and microtubuledynamics were appropriately adjusted. In their simulations, cooperative effects werenot included. With our network structure, we however observe that spindle attachmentcan be optimized with the presence of erroneous initial attachments. It is possible thatour transition to direct amphitelic orientation might be contributing to the creation ofthis feature, and we interpret this transition within the framework of cooperativity.

Having developed the model, we derived specific expressions for the dependence ofthe distribution of the time T , at which all kinetochores reach the amphitelic state, onthe kinetic parameters of the model. One of the key results from our model is that theprocess of merotely (i.e., erroneous attachment) can contribute to a lessening of thenoise in the time for completion of the mitotic process. We note that since merotelicand syntelic states are structurally indistinguishable in the model, similar results holdfor the latter. Specifically, our analytical results show that a tightly controlled strictlypositive rate of erroneous attachments facilitates a more accurate timing of the mitoticprocess than in the complete absence of attachment errors.

Acknowledgments This research has been supported in part by the Mathematical Biosciences Instituteand the National Science Foundation under Grant DMS 0931642. The research of Sijia Liu has beensupported in part by an Alberta Wolfe Research Fellowship from the Iowa State University Mathematicsdepartment. The research of Blerta Shtylla has been supported in part by the National Science Foundationunder Grant NSF DMS 1358932. MAJC gratefully acknowledges the support of the ERC Advanced Inves-tigator Grant 227619, “M5CGS - From Mutations to Metastases: Multiscale Mathematical Modelling ofCancer Growth and Spread”.

Appendix 1: Numerical Computations

Extensions of the analysis presented in this paper for more complex models thatinclude explicitly the effect of Aurora B, such as the models discussed by Mistry etal. (2008), are highly non-trivial, and hence numerical approaches become crucial. Inthis appendix, we employ the Gillespie simulation algorithm (Gillespie 1976, 2007)to compute an ensemble of independent realizations of the master Eq. (2). This allowsus, among other things, to numerically approximate the distribution of the hitting timeT of the amphitelic state, i.e., of the state where all kinetochores are amphitelicallyattached. As we will see, the results of the numerical computations corroborate theanalytical derivation of Sect. 3.1.

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Fig. 5 Plot showing 10,000 realizations of the time it takes for all sister kinetochores to form amphitelicattachments. The red line represents the empirical mean of these realizations and can serve as an estimationof the mean hitting time T (see text for details) (Color figure online)

As explained in Sect. 2, because of the small numbers of chemically interactingspecies in the system, we treat the evolution of the latter as a Markovian stochasticprocess, rather than as a deterministic process (Higham 2008), and use the standardmaster equation formalism (Gardiner 1985; van Kampen 2007). In this context, MonteCarlo methods are commonly used to generate realizations of the underlying stochasticprocess. For chemical reaction networks this is frequently done using one of the twostochastic simulation algorithms, called the direct method and first reaction method,respectively, developed by Gillespie (1976).

Simulations were conducted by choosing a value in the range discussed for the para-meters k1 and k3 and performing 10,000 realizations using Gillespie’s direct methodon these parameter values. Following the assumptions outlined in Mistry et al. (2008),each realization started with all 46 kinetochore pairs in the unattached state and thenproceeded by updating the populations of the kinetochore attachment types in theprevious time step according to Gillespie’s direct method, with possible transitionsfollowing the network diagram in Fig. 1. A realization was then deemed to have fin-ished when all 46 kinetochore pairs reached the amphitelic state. The amount of timerequired to reach this state, i.e., the hitting time for the specific realization, was thenrecorded. This process was repeated for each of the 10,000 realizations. The resultsfor all realizations for a specific choice of parameter values are shown in Fig. 5.

The data in Fig. 5 (and similar data sets for other combinations of parameter values)were used to compute the empirical mean µ and empirical standard deviation σ ofthe all-amphitelic hitting time distribution. Ensembles of 10,000 realizations weregenerated for all possible combinations of values in the range of the parameters k1and k3 described in Sect. 3, and the respective coefficients of variation were calculatedfor all possible pairs of parameter values. The results of these simulations are plottedin Fig. 2b. This can be compared with the plot in Fig. 2a which shows the values of

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the CV over the same range of values for the parameters k1 and k3 as computed byusing the analytically derived probability distribution for the hitting time, discussed inSect. 3.1. The comparison of Fig. 2a, b clearly shows that both the stochastic simulationapproach based on Gillespie’s direct method and the analytical derivation of Sect. 3.1return similar results.

Appendix 2: On the Coefficient of Variation and the Fano Factor

A significant number of cellular processes involve reactions whose components arepresent in low concentrations, and this is especially true for the reactions involved ingene expression (Rasner and O’Shea 2004). The low concentrations of the componentsnaturally give rise to stochastic fluctuations, typically called noise, in the number ofmolecules produced by these reaction (Eldar and Elowitz 2010). Two quantities arecommonly used in the literature to measure the noise of the molecular concentrations:the CV, defined as the standard deviation divided by the mean, and the Fano factor(F), defined as the variance divided by the mean (Rasner and O’Shea 2004). Whilethe CV is largely accepted as the standard measurement of noise, the Fano factor is avalid measure of noise under specific circumstances (Gadgil et al. 2005). Furthermore,inappropriate application of the Fano factor can lead to misinterpret the role of noisebiological processes (Swain et al. 2002; Gadgil et al. 2005), and the following literaturereview is provided to clarify the meaning, utility, and appropriate application of theFano factor.

The Fano factor, or the noise strength (e.g., Rasner and O’Shea 2004), was intro-duced by Thattai and van Oudenaarden (2001) in their analysis of gene expression inprokaryotes. It measures noise as a deviation away from the Poisson distribution, andas opposed to the coefficient of variation, the amount of noise measured by the Fanofactor does not diminish for increasing mean values (Ozbudak et al. 2002). Followingthe derivation of Thattai and van Oudenaarden (2001), let r and p denote the numberof mRNA and protein molecules, respectively, and define

kR = the rate mRNA molecules are produced from the template DNA strand

kP = the rate protein molecules are produced from the template mRNA strandγR = the rate mRNA molecules degradeγP = the rate protein molecules degrade

where kR and kP can be viewed as the rates for transcription and translation, respec-tively. Their first-order analysis of single gene expression at steady state determinedthat the protein numbers are given by a distribution with mean

<p>= kR · kp

γP · γR

and variance

δp2 =(

kP

γR + γP+ 1

)<p> .

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Thus the Fano factor is calculated as

F = δp2

<p>=

(b

1 + η

)+ 1 (9)

where η = γP/γR and b = kp/γR , typically called the burst size, is the averagenumber of protein molecules created from a strand of mRNA. However, the life spanof a mRNA strand is much shorter than the life span of a protein molecule (Thattaiand van Oudenaarden 2001). Therefore, η is typically small and (9) implies F ∼=b + 1, and the authors conclude that intrinsic noise is controlled at the translationallevel (Thattai and van Oudenaarden 2001). Subsequent work by van Oudenaardenand collaborators, (Ozbudak et al. 2002), varied the transcription and translation ratesindependently and experimentally verified the conclusion that noise in the proteinnumber is controlled at the translational level (Ozbudak et al. 2002; Hasty and Collins2002).

Subsequent research was conducted by Kepler and Elston (2001) modeling stochas-tic fluctuations in protein number arising from operator fluctuations, the addition orremoval of proteins bound to DNA which overlap the promoter region of RNA poly-merase. The results of those simulation demonstrated that the noise, as measured bythe CV, in protein number could arise from the regulation of transcription (Kepler andElston 2001; Hasty and Collins 2002). Additionally, research conducted by Blake etal. (2003) on the noise, as measured by the Fano factor, in eukaryotic gene expressionarise from transcriptional regulation, “providing an important distinction between thesources of noise in prokaryotic and eukaryotic cells.” While it was eventually shownthat the sources of noise in eukaryotic and prokaryotic gene expression are the same(see, e.g., Rasner and O’Shea 2004; Eldar and Elowitz 2010), the discussion aboveclearly demonstrates the lack of consensus concerning the validity of the Fano factoras a measurement of noise.

Research conducted by Swain et al. (2002) clearly demonstrated complications thatarise from using the Fano factor as the only measurement of noise. The source of thepotential complications stems from the fact that Fano factor removes the dependenceon transcription rate and scales with the translation rate (Swain et al. 2002; Rasnerand O’Shea 2004). As demonstrated by the in depth theoretical analysis of Gadgil etal. (2005), the Poisson distribution is the limiting distribution of all open conversionnetworks. Therefore, as stated in Gadgil et al. (2005):

The Fano factor as the measure of noise leads to the prediction that the fluctu-ations of all components in an open conversion reaction system are identical.Thus a species that has a mean of 10 molecules will exhibit the same amount ofnoise as a species that has a mean concentration of 1M.

Any usage of the Fano factor in this paper is appropriate. The network presentedin this paper is closed. Furthermore, any comparisons made between the Fano factorand the coefficient of variation are done with the same data.

In conclusion, we would like to thank the reviewers for their thoughtful criticism,especially the reviewer at whose request this appendix was written. Additionally, werefer interested readers to the following papers: Gadgil et al. (2005) contains thorough

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review of theoretical first-order analysis techniques, and Eldar and Elowitz (2010) pro-vides an excellent review for the role of noise in gene expression. Furthermore, Eldarand Elowitz (2010) contains numerous references to review papers covering mathe-matical representations for the sources of noise in gene expression, the analysis ofnoise in the context of dynamic circuits, and the advantages of noise on phenotypicvariability.

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stochastic modelling of chromosomal segregation: errors

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