Downloaded by guest on September 26, 2021 a n tG at ing anticancer the to possible led as has inhibitors (13–17). thus drugs kinase and Aurora cells of replicating development 10– of mechanism (7, removal possible a selective provides flawed for inhibition fatally of form a this to by Hence, leads 16). gained activity inhibition is kinase severe advantage Aurora Conversely, vari- of 9). growth a (5, a in kinases somatic that Aurora overexpressed all deregulating suggesting are in , kinases found of 3 are ety all B Aurora Significantly, germ and (6). in A cells expressed Aurora only as (Ark1 is where found C cells, is Aurora homologue (6–8). bak- one respectively) 3 In only Ipl1, of C. and yeast consisting Aurora budding and family and B, yeast Aurora Aurora ers’ A, the Aurora is metazoans: regulation in forms in plays that role kinases central serine/threonine signaling of a and family One complexes 5). (3, protein pathways the multiple involving process each between regulated kine- of highly interactions with a is centrosomes, the This in at (4). ) originating located located cell, structures poles (protein the spindle tochores of the end connects either that at network of formation a the requires and segregation , instability, Chromosome genomic (1–3). to lead may correctly so do to hoooecnrsin ergto,adctknss(,21). (6, cytokinesis and segregation, regulates B congression, Aurora (6–8). chromosome cytokinesis as throughout cortex midbody localizes cell the B Aurora to equatorial telophase During and (19). midzone the as spindle well the to relocates Aurora where B transition – from the centromeres until the prophase to localized passengers” localiza- complex “chromosome B and are Aurora the activity of Proteins its (18–20). mitosis regulate throughout that tion survivin and INCENP with T cycle cell | metazoan | | mitosis | mathematical B Aurora and checkpoint. assembly spindle how the of in pro- cooperate understanding could here deeper discussed a results vide the been of yet verification Therefore, not experimental knowledge. best has authors’ concentrations the to experimentally, inhibitor studied of of level range This certain observed. is a prometaphase/metaphase for prolonged model, a concentrations, the inhibitor within considered is Aurora inhibition when Furthermore, B cur- parameters. the diffusion-limited key that variation to its find robust in is We checkpoint assembly model. spindle resulting the of the model rent the within and B one Aurora just than incorporate of rather role to cell a a checkpoint extend within assembly kinetochores anti- to signaling spindle all modeling attractive the mathematical an of use model it current we making Here, mitosis, target. for cancer required of is subset a and in overexpressed cancers is B Aurora microtubule attachments. centrosomes aberrant by kinetochore at caused tension Mad2 in changes and to responds BubR1 and by of part, in localization family. checkpoint, assembly the Aurora spindle the destabilizing the serine/threonine is in role regulation of a in plays family B role Aurora central One a process. plays mito- regulated that at kinases highly division a cell is to prior sis of separation Faithful kinase B Mistry Aurora B. Hitesh of role and spindle checkpoint the assembly of evolution temporal the Modeling w.nsog/ci/di/1.03/pnas.0810706106 / 10.1073 / doi / cgi / www.pnas.org omnctdb ve remn hoSaeUiest,Clmu,O,Nvme ,20 rcie o eiwMrh1,2008) 11, March review for (received 2008 3, November OH, Columbus, University, State Ohio Friedman, Avner by Communicated iiino ahmtc,Uiest fDne,Dne D H,Soln;and Scotland; 4HN, DD1 Dundee Dundee, of University Mathematics, of Division uoaBi cietruhu ioi ihpoenlvl peak- levels protein with mitosis throughout active is B Aurora ioi sesnilfrmitiiggnmcitgiy Failure integrity. genomic maintaining for at essential division cell is to mitosis prior chromosomes of separation faithful he 2 Mpaeo h elcce() uoaBfrsacomplex a forms B Aurora (6). cycle cell the of phase /M a ai .MacCallum E. David , b oetC Jackson C. Robert , b akA .Chaplain J. A. Mark , 08b h ainlAaeyo cecso h USA the of Sciences of Academy National The by 2008 at © online information supporting 0810706106/DCSupplemental. contains article This reyaalbeoln hog h NSoe cesoption. access open PNAS 1 the through online available interest. Freely of conflict no declare authors The paper. the and wrote F.A.D. tools; and reagents/analytic new D.E.M., contributed H.B.M., M.A.J.C. and per- F.A.D. R.C.J., and D.E.M., H.B.M. research; research; formed designed F.A.D. and M.A.J.C., R.C.J., contributions: Author ute niie opee fCc0 hc r ifrn nthat in of different production are the which catalyze Cdc20, to of complexes able inhibited be further to inhibited these assumed 2, are step complex Mad2); complexes of inhibited form closed interact the an C- represents to where Mad2 form BubR1-Cdc20, assumed and to is C-Mad2-Cdc20 Cdc20 include kinetochore (candidates 1, unattached step (35). an inhibited: Howard with is and APC/C (Cdc20) Sear the stimulant a of which by in mechanism proposed 2-step could a recently considered this They was how just describing than achieved mechanism rather be possible cell entire One the nucleus. throughout the kinetochore propagate unattached to single has a now at longer Therefore, generated active. no signal is diffusive is checkpoint the envelope assembly spindle nuclear the the while of cells intact where metazoan onset mitosis, However, the open volume. at undergo nuclear prop- even the to required throughout intact only yeast. agate mitosis, is still signal closed in diffusive is the undergo anaphase Therefore, envelope anaphase. that of nuclear cells onset the on the where based diffu- were prevent a models generate could These could that kinetochore signal unattached sive single a how of operate could Doncic some checkpoint 35). provided the (34, has how mitosis, of closed understanding and mechanistic open in of both 33). 32, checkpoint, onset (30, proposed the been have prevent defined, models fully of be to number to yet a enough has although occurs be process this How to of (31). anaphase activation thought the is inhibits unattached kinetochore one that Even (APC/C). kinetochores complex anaphase-promoting sig- by the wait alignment generated A (30). proper is spindles without nal attached is anaphase correctly and checkpoint of chromosomes this onset of because prevent interest to particular crucial of are checkpoint by generated (14–16). data methods with other consistent override, broadly biorientation failure, checkpoint cytokinesis cause spindle promote and inhibitors B problems, to Aurora with alignment cells tension 27) chromosome of Treatment (14, in 29). centrosomes 28, changes 22, at (14, to Cenp-E responds and in Mad2, and role BuBR1, a local- of plays the destabilizing ization B by part, Aurora in checkpoint, that assembly spindle suggested the has of inhibitors Characterization B 8). Aurora (7, phospho- substrates it several where regulates cytokinesis for and required rylates Hec also is regulating B by Aurora possibly (26). of 1 (21) turnover kinetochores promotes the B at been Aurora microtubules (25). also attachments has merotelic 24) (14, and This by attachments syntelic assembly 23). for cells spindle mammalian (22, in demonstrated correct attachments promote syntelic can destabilizing Ipl1 yeast budding In b owo orsodnesol eadesd -al [email protected]. E-mail: addressed. be should correspondence whom To ylclLd,Dne ehooe udeD15J Scotland 5JJ, DD1 Dundee Technopole, Dundee Ltd., Cyclacel eety ahmtclmdln ftesideassembly spindle the of modeling mathematical Recently, assembly spindle the on inhibition B Aurora of effects The PNAS tal. et eebr2,2008 23, December a n odc .Davidson A. Fordyce and , 3)cniee aiu ipie models simplified various considered (34) o.105 vol. www.pnas.org/cgi/content/full/ o 51 no. a,1 b; 20215–20220

APPLIED BIOPHYSICS MATHEMATICS they cannot themselves act as catalysts. This type of signal ampli- fication ensures that the inhibitory signal is only amplified by molecules that interact with an unattached kinetochore. Further- more, because there is no autocatalytic step, the inhibitory signal switches off rapidly once the final kinetochore has been captured. However, the authors were concerned that, to work effectively, their model required certain system parameters to be chosen at the upper end of their expected range. Also, the model in ref. 35 did not encompass the temporal signaling dynamics of a complete set of kinetochores within a mammalian mitotic cell. In this article we embed the spindle assembly checkpoint model proposed by Sear and Howard (35) into a qualitative mathematical description of the temporal evolution of kinetochore–microtubule attachments. Thus, the model presented here considers the evo- lution of all signaling kinetochores rather than just one. The resultant model not only provides a more complete qualitative description of the wait signal dynamics, but also shows greater robustness to variations in certain system parameters, allaying concerns expressed in ref. 35. Moreover, by also incorporating the role of Aurora B within the resulting model, the principal action of Aurora B activity and the effects of inhibiting such activity are studied.

Mathematical Framework Kinetochore Microtubule Attachments. A mammalian cell contains 23 pairs of chromosomes, which are duplicated during mitosis, result- ing in 46 pairs of chromosomes, thus, 92 kinetochores. Each kine- tochore is known to have between 20 and 30 microtubule binding sites (36). Formation of kinetochore–microtubule attachments is a very complex process involving a cascade of mechanochemi- cal reactions (31) occurring throughout prometaphase. It is not the aim in this article to investigate the details of the attach- ment process. Rather, we will simply concern ourselves with the temporal evolution of “attachment type” as we now discuss. Proper chromosome segregation requires sister kinetochores to form an amphitelic attachment, where 2 sister kinetochores are bound to microtubules from opposite spindle poles. How- ever, 3 types of incorrect attachments can occur that lead to a mis-segregation of chromosomes: (i) unattached; (ii) syntelic attachment—both sister kinetochores are bound to microtubules from the same pole; (iii) merotelic attachment—one sister kine- tochore is bound to microtubules from both spindle poles, (see Fig. 1A). One further type of attachment (monotelic) is common Fig. 1. Kinetochore microtubule attachment schematic. (A) Table showing in the very early stages of prometaphase. However, they exist only the different types of attachments possible, how each type of attachment as brief, transitory states in the subsequent attachment process corresponds to variables within the model framework, and their effect on (37–39). Therefore, we do not consider them further. the spindle assembly checkpoint. (B) Network diagram showing the possible Syntelic and merotelic attachments are observed throughout states each kinetochore unit could be in at any given time and how they can prometaphase predominantly in the earlier stages (14, 40). The move from state to state. (See text for detailed description.) correction process of these attachment types can be summarized as follows. Syntelically attached kinetochores are drawn toward the corresponding spindle pole. Subsequently, the incorrect attach- ments are released via the action of (14, 15) and, after relocating to the metaphase plate, it is common that this Fig. 1B, represent the rates at which a unit undergoes the corre- kinetochore pair form amphitelic attachments (38). Merotelically sponding state change within a single cell. Note that for ease of s s attached kinetochores are aligned at the spindle equator, where reference below, the parameters k4(B ) and k5(B ) are defined to Aurora B destabilizes the aberrant attachments, thus allowing the be functions of another variable, Bs, which represents the steady- resultant pair of sister kinetochores to almost immediately become state fraction of active concentration of Aurora B. Later on we will amphitelically attached (37). consider how this fraction can be altered by inhibition. However, s in the meantime, we simply assume that the parameters k4(B ) and s Attachment Transitions. In the following, we define a pair of sister k5(B ) are constants. kinetochores as a unit. We assume that both kinetochores within Let X = (XU , XS, XME, XC) denote the state vector of random the unit have all their binding sites occupied or both kinetochores variables of unit type Xi, i ∈{U, S, ME, C} within a single cell, have binding sites available (37–39). We assume further that these at any given time t. Define P(Xi = xi, t) to be the probability 2 sets can be divided as follows: a unit displaying full occupancy can that there will be xi ∈{0, 1, 2, ...,46} units in a state i, at time be in either a merotelic (ME) or amphitelic (C) state; otherwise it t. Let us assume that we can choose a t sufficiently small such is deemed to be in a syntelic (S) or unattached (U) state (37–39). that, at most, 1 state transition can occur during the time interval, Following the above arguments a unit state transition network can (t, t + t). There are 5 possible state transitions, the probabilities be constructed (see Fig. 1B). The parameters, ki (i = 1, ...,5)in for which can be defined as follows:

20216 www.pnas.org / cgi / doi / 10.1073 / pnas.0810706106 Mistry et al. Downloaded by guest on September 26, 2021 Downloaded by guest on September 26, 2021 where eie n h oedtie tt rniindnmc ie in given dynamics transition state detailed more the and derived ttscnocr h mhtlcsaeadteuatce state. unattached the and state amphitelic Define the occur: can states state 2 the here. interpreted adopt governing we been process methodology has the Poisson is turn compound and (34) a in transitions to This leading 43). a as (42, is on attachment process based kinetochore–microtubule is stochastic approach that simplified observations This the parameters. constraints system provide the will on which para- argument, to mathematical insufficient simpler Eq. are a ODEs and of 40) system (14, the limited meterize very Quantitative present, parameterized. mitosis at of are, be stages early can during attachments ODEs aberrant regarding data of system above the oefo h ntahdt h mhtlcsaei oso dis- Poisson is state to amphitelic rate unit a the with a to tributed for unattached taken the time from the move assume We follows. as approximated hr ehv sd46 used have we where Crepnigy each (Correspondingly, X opudPisnpoes hr h vrg iet h next the to time 1/Y average is the event where process, Poisson compound a X each the of tion evolution the by governed temporal is probabilities The transition attachment probability.) assigned naturally aaee osritfrAtcmn Transitions. Attachment for Constraint Parameter Mistry qain OE) [See (ODEs). equations o ulderivation.] full a for nOEgvrigtetmoa vlto of evolution temporal derive the can governing we above ODE to an Similarly state. each within units of number nt ob 0mn(4,then, (44), min 20 be to units .. o l times conserved, is all units for kinetochore of be i.e., number can total equations these the of as one eliminated fact, In cells. identical genetically of state a in units chore ie time, given eraoal sueta h ipie ruet just arguments simplified the that assume reasonably We osdrnwasmlfidsaetasto rcs nwihonly which in process transition state simplified a now Consider ME ME n h iedrvtv ftema fis oet 4) for (41), moment) (first mean the of derivative time the and (t (t X tal. et P X ob h xetdma ubro bratuisa any at units aberrant of number mean expected the be to ), )+ d i Y sgvnb h olwn ytmo riaydifferential ordinary of system following the by given is , (X d X d d C P i X P X dt  X dt ME dt dt (X ME and (X 20 orsod oteepce ennme fkineto- of number mean expected the to corresponds t U C U . X S C     S k → C = c → Y = =− = = (t → aigtettltm o h rniino l 46 all of transition the for time total the Taking . U X ial,w define we Finally, ). k 1 k k k t X c ME ob admvralsta ersn h total the represent that variables random be to X 1 2 3 ⇒ Y (k X X C X  n=0 S ≥ 45 U i 1 + + + k U U U naygvncl ihnalrepopulation large a within cell given any in + Y 0, d = c −k +k −k 46 1; 1; 1; X Y C 3) hs h rniino l 6uisis units 46 all of transition the Thus, (34). dt k i X (t t t t 1 2 Y C − ean nlee during unaltered remains uprigifrain(SI) information supporting + + + )=46(1  + C U 5 4 4 n (t (B (B (B (0) t t t = k , )+ 3 s s s k ) ) ) )X =46 )X )X )X c = = = ⇒ 2 Y U ul.W hrfr pelto appeal therefore We fully. ME S + S Y k k k U k +k , − , 3 1 2 U A(t , c k X X X , (t e 5 ≈ U U U (B −k ) ) = t t t 5 .2min 0.22 enwcnie how consider now We c = (B s n h rate the and t )X ), , , + s X Y X )X ME U k U C 4 (t t (B (t (t ME )+ −1 )+ namely, ), . s , )X atrequa- master t . S ihthe with , Appendix k t X X c S S a be can (t (t [2d] [2b] [1b] [3b] [3a] [1a] [2a] )+ [2c] [1c] )+ [4] hsstwsgnrtdb rtsetting first by generated was set This iie rti–rti neato rate, interaction protein–protein limited k ope easrlaigteCc0a rate, a at Cdc20 the releasing decays complex ocnrto fArr .A bv,w ipytk hst ea be of to pro- rate the this knowledge, duction best take authors’ the simply to Note, we present.) above, at constant As B. Aurora of concentration x ee species Here, ae htti aecntn safnto of function a is indi- constant have rate we this exposition, of that ease For cated 35. ref. in kinetochore single 40 ffrhrihbtdcmlxsof complexes inhibited further of r ae rmSa n oad(5 nessae tews.I is It complex otherwise. inhibited stated the unless that (35) assumed Howard and Sear from taken are fterlvn xetdmas hti,w insist, we is, That means. expected relevant the of Eq. e.3 ti sue htteecmlxscno aayefurther catalyze cannot complexes use these We that complexation. assumed is it 35 ref. osntcag h ulttv rpriso h eut discussed results the of below. properties qualitative the change not does Eq. condition providing deter- values the with Eq. order by appropriate mined an of again be to selected sraoal oasm htawi inli eeae rmeach from generated is signal wait it a U transition discussions that above assume state to the the From reasonable follows. is into kine- as embedded above, unattached be detailed single process can a model by This generated earlier, tochore. signal discussed wait and (35) the Howard concerns and Sear by proposed model, raeodrwith order priate ti eesr oicueaohrdfeetihbtdcmlxof complex by here inhibited denoted is different that model another Therefore, our include in (45). Cdc20 to here necessary discussed is one it the dif- a these from by generated Furthermore, mechanism be 46). to ferent known (45, are Cdc20 state of Cdc20 inhibited complexes all inhibitory an prometaphase of in onset are the at molecules that prometaphase/metaphase. known is throughout it However, rep- species. occurring model respective our events the because here of resents approach states similar a steady there adopt the cannot considered We system computing reaction by the set for are data initial the thus and ee auswr chosen: were values meter Eq. condition ensures which 5 obtained, be can parameters system pnl sebyCheckpoint. Assembly Spindle nle ffrhreprmna vdne scoe ob qa to complexes. equal inhibited be other to the chosen for rate is decay evidence, the experimental further of lieu in that, eainhp h ecinsse Eq. system reaction the relationship, fCc0 eoe by denoted Cdc20, of the either in units o some for nertoso Eq. of integrations 3 od.Frterslsdsusdblw h olwn e fpara- of set following the below, discussed results the For holds. ∈{ and - nrf 5ol h nluatce ieohr sconsidered is kinetochore unattached final the only 35 ref. In ial,define Finally, = μm 2 e, N r osseti hi rdcino h eprlevolution temporal the of prediction their in consistent are .7 min 0.178 3 e S = s ∗ −1 tp nt h ecinpoessbiggvnby, given being processes reaction the unit, -type , c N Ti s2tmstecrepnigrt sdfra for used rate corresponding the times 2 is (This . ∗ e ,

g + |X xesv ueia netgtosrvae that, revealed investigations numerical Extensive 5. ∗ e h oa ubro oeue scnevd so conserved, is molecules of number total The }. PNAS e .Setting 1. N −1 ∗ sascae ihfe d2.W suethat assume We Cdc20. free with associated is C N e ssaeidpnet lo l aaee values parameter all Also, independent. state is ∗ , k (t e e e e U x 1 k ∗ + 2 )− + + + ob h oa ubro oeue fspecies of molecules of number total the be to 4 → > or α (B n h rp of graph the and N S e U e α eebr2,2008 23, December ∗ e c k S −1 s ∗ k e, ∗ 5 −→ ) k → 2 k Y −→ tadfuinlmtdrate, limited diffusion a at , k + k tt a eeaeihbtdcomplexes inhibited generate can state (B od,atrn hs aaee values parameter these altering holds, (B C = 3) h eann aaeeswere parameters remaining The (37). = k s = N (t h pnl sebycheckpoint assembly spindle The c 1 s ) c ) ∗ α )| . min 0.3 ∗ e g = e ∗ e + ∗ −1 .1adb oprn numerical comparing by and 0.01 → ∗ α [N + .4min 0.04 < + e = e ∗ e, S , U e = eoe by denoted , min. 5 6 ∗ , −1 , k a aayeteproduction the catalyze can n olwn h approach the following and 0,0 3).B sn this using By (35)]. 800,000 1 g Y ∀t and ∗ and −1 k → C o.105 vol. α g ∈[ (t g = ∗ and k easm htthis that assume We . k notmlstof set optimal an ), e. ,20 0, B 2 α 5 10 (B s g −1 ob fa appro- an of be to c h steady-state the , k −2 ∗ s 2 tadiffusion- a at , ], ) = μm = o 51 no. = i,which min, 5 .2min 0.02 3 . min 0.2 s k −1 k (B .Asin s 20217 [6d] [6b] [6a] [6c] ) [5] −1 −1 = , .

APPLIED BIOPHYSICS MATHEMATICS Fig. 3. Plot showing the level of inhibition, defined by, max{Ne(t)/N ∀t ∈ s [0, tf ]}, for a range of kk (B ) and k values. The regions R1, R1 + R2, and + + Fig. 2. Plot showing the temporal evolution of the fraction of mole- R1 R2 R3 correspond to the regions where, at most, only 5, 10, or 15% of the molecules are in the uninhibited e state, respectively. X marks the cules Ne/N (solid line), Ne∗ /N (dotted line), Nc∗ /N (dashed line), and Ng∗ /N values of k (Bs) and k implemented by Sear and Howard (35). (dash-dotted line). See text for definition of tf and ta. k

in ref. 35, we model the dynamics of the spindle assembly check- Sear and Howard (35) focused on the signal generated by a sin- point as system Eq. 2 combined with the following system of ODEs, gle unattached kinetochore. Essentially, they considered this to be the only signal source for a time period in our [0, tf ]. Doing so allowed them to compute “steady state” values for Ne, Ne∗ , s ∗ dNe∗ kk(B ) and Nc , which were subsequently used as initial conditions (at a =  +  − ∗ ( XU (t) XS(t) )Ne αeNe , [7a] time corresponding to our t = t ) for the second-phase dynamics dt vc f discussed above. The authors in ref. 35 were concerned that to dNc∗ k = Ne∗ Ne − αNc∗ , [7b] work effectively, their model required certain system parameters dt vc to be chosen at the upper end of their expected range. This has dNg∗ implications for the robustness of the model in application. =−α N ∗ , [7c] dt g g The model introduced here not only provides a more complete description of the wait signal dynamics, but also provides greater 3 where, vc = 6, 000 μm is the cytoplasmic volume (35). robustness to variations in the key diffusion-limited parameters, k s Recall each unit of type i comprises 2 kinetochores with the and kk(B ) (see Fig. 3). The model maintains tight inhibition of e expected value Xi∈[0, 46]. However, if Xi < 0.5, then this molecules for t < tf for a large range of biological realistic values s −3 −2 3 −1 s expected value corresponds to the presence of less than a sin- for k and kk(B ), i.e., k ≈ 10 to 10 μm s and kk(B ) ≈ 6to gle kinetochore. Therefore, in system Eq. 7 we assume that, if 60 μm3s−1. Remarkably, this parameter variation does not signifi- Xi < 0.5, then Xi≡0. This leads to the definition tf := cantly alter the necessary rapid release of inhibition in the second inf{t ≥ 0|XS(t) < 0.5 & XU (t) < 0.5} which best repre- phase (data not shown). sents the time at which the last kinetochore is sensed by system We were concerned that the attachment transition (and asso- Eq. 7. (This corresponds to t = 0 in ref. 35.) Furthermore, we ciated signaling) process detailed above would be too crude to define the time to anaphase (length of prometaphase/metaphase) detect the subtle differences between the alternative signaling as ta := sup{t ≥ 0|Ne(t) > 200,000} (35). models proposed by and investigated in refs. 34 and 35. However, on replacing Eq. 6 with systems corresponding to the direct inhi- Results bition and emitted inhibition models, respectively, and embedding We first analyze the temporal evolution of the wait signal gener- them appropriately in Eq. 7, it transpired that the attachment state ated by all kinetochore units, described by systems Eq. 2 and Eq. 7. transition model system is indeed able to differentiate these mod- Subsequently, the effects of Aurora B inhibition are studied. els. With either of these replacements, sufficiently tight inhibition of Ne was not observed for t ∈[0, tf ] (see SI Appendix). Temporal Evolution of the Wait Signal. The temporal evolution of Ne, Aurora B Inhibition. Aurora B plays a central role in many processes Ne∗ , Nc∗ , and Ng∗ , illustrated in Fig. 2, can be split into 2 distinct phases: For 0 ≤ t < t , e molecules are under tight inhibition involved in mitosis. As a first step toward understanding this com- f plex role, we assume that the concentration of Aurora B affects (Ne/N ≤ 0.05); at t = tf , inhibition is lifted and Ne increases rapidly. These distinct phases can be explained as follows. In the the 2 main components of our model in the following way. The role Aurora B plays in releasing aberrant attachments for sub- first phase, recall that XU (0)=46 and initially all of these units contribute to the production of e∗ molecules (see Eq. 7a). Sub- sequent repair (14, 24) is modeled by reasonably assuming that      +  the rate constants k4 and k5 in Eq. 2 are positively correlated sequently, XU (t) decreases, XS(t) increases, but XU XS s decreases (data not shown). However, throughout this phase the to the value of B . In lieu of further experimental evidence we ∗ s =  s −1 production of e dominates the production of e resulting from the assume this relationship to be linear, i.e., k4(B ) k4B min , ∗ ∗ ∗ = ∗ s =  s −1   decay of g , c , and e .Att tf the production of e is set to zero k5(B ) k5B min , for constants k4 and k5. The role of Aurora (see above discussion). Subsequently, the rapid decay rates (short B in facilitating the recruitment of key checkpoint proteins (14, 27) ∗ ∗ ∗ s =  s 3 −1 lifetimes) of g , e , and c ensure a rapid increase in Ne. is modeled similarly by allowing kk(B ) kkB μm s in Eq. 7,

20218 www.pnas.org / cgi / doi / 10.1073 / pnas.0810706106 Mistry et al. 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Eq. operating mechanism checkpoint the However, larger. becomes constant some for 4. Fig. Mistry observed is units unattached of fraction small at a now but out, level oeicesstefato of fraction the increases dose X eeaigawi inl h ecinEq. reaction the signal, wait a generating otedfeetpae fihbto setx o oedetails). more for text (see inhibition of phases different the to (see reactant corresponding the hr ute nrae h iet anaphase to time the increase, further short h feto uoaBihbto on inhibition B Aurora of effect The (Inset) line). (dotted and f ME and nteD oigitra,tefatoso ntsae at states unit of fractions the interval, dosing D3 the In tteD/2budr,amre hnei vdn.Ti point This evident. is change marked a boundary, D1/D2 the At eaepicplyitrse nteefcso uoaBinhi- B Aurora of effects the in interested principally are We hogotD,tedsrbto fui ttsi setal unal- essentially is states unit of distribution the D1, Throughout 4. Fig. in summarized are inhibition B Aurora of results The t k = t k (t f vntog hr r agrnme fuiscpbeof capable units of number larger a are there though even : /t tal. et a t )/A(0 k 7 t a(B a a k h time The . anpo hw htfato of fraction what shows plot Main snwsvrl opoie n h oe rdcsthat predicts model the and compromised severely now is r olwi hsitra httewi inlgenerated signal wait the that interval this in low so are ) s sosre n a eepanda olw.Frthese For follows. as explained be can and observed is =1) k 4  [B ds-otdline). (dash-dotted sldline), (solid = − decreases ], . min 0.3 B t s k a .. nraigtecnetaino inhibitor, of concentration the increasing i.e., , k  lolvl u tasgicnl eue value reduced significantly a at out levels also Tevle sdfrterslsillustrated results the for used values (The . X −1 S , (t t k f a IAppendix SI )/A(0 5  B = [·] = s (see t S 0 a . min 0.2 Thus, . and - ersnsteiiilcnetainof concentration initial the represents ds-otdln) and line), (dash-dotted IAppendix SI .D,D,D,adD correspond D4 and D3, D2, D1, ). ME t −1 a S A(t nraigteinhibitor the Increasing . k tp nt r o pre- now are units -type 7 and tp nt n fe a after and units -type 4 a , sisfcetystrong insufficiently is t )/A(0 f k t and a 5 k k Consequently, . erae rapidly. decreases k ).  4 t , a = e /t t k dse ie are line) (dashed a ∗ 5 a(B , endabove defined 40 and , c s ∗ =1) X and , μm U 2 sldline) (solid (t B k 7 3 scom- is a k S s s )/A(0 t sstill is cho- , g −1 -type (and = ∗ t a . .) t is t a f unl,temdlpeit htlrenmeso ntahdunits at unattached of appear numbers large that predicts model the quently, nieyb h ea aeo h niioymolecule inhibitory the of rate decay the by entirely et hteprmna vdnespotn hs assumptions these sug- supporting here evidence described model experimental the that from gests output damage The cell model. derived be the can and from onset anaphase biomarkers to time and measurable attachments) (aberrant between of relationship concentration qualitative verifica- the the invites regarding yet therefore Predictions not and has tion. experimentally concentration studied inhibitor fully authors’ of been the level To this anaphase. knowledge, at best present attachments syntelic merotelic with and associated particular, prometaphase/metaphase in prolonged surprising, a somewhat the were with inhibition anaphase regarding of predictions levels enter The lower to attachments. aberrant check- cells of spindle causing number the large abolished, severe data: is regarding experimental signal with Predictions point line inhibition. in B model are Aurora inhibition the of differ- Thus, levels to response model. ent wait qualitative the the and regarding transition of predictions state reac- provided unit components the the generation and both in activity signal assumed B was Aurora rates here. between tion presented model relationship the simple in A considered also was checkpoint 3). bly Fig. more (see far predicted fact 35 in ref. is than have model variations we core parameter transitions, to this robust that state demonstrate unit to the able of signaling at been description this values embedding our take By into to range. model chosen expected their were of parameters extreme key the concerns certain 35 if ref. only work did in but proposed model Second, signaling alternative framework. the that our raised were in unaltered inhibition mod- are these emitted of els and applicability the direct regarding the conclusions The namely, models. 35, previously ref. models to signal in enough core sensitive investigated the still in is differences it significant yet detect kinetochores, notable signaling 2 evolu- all temporal in the of resulted of tion account has takes This model the obtained. First, been conclusions. of has evolution the signal of wait description the attachment qualitative complete kinetochore–microtubule more a all units, of evolution the assembly of temporal spindle behavior mean the expected of the considering model By current model (35). checkpoint a This core its prometaphase/metaphase. at has within framework processes key for inhibitors. kinase B in Aurora assist effective may of involved development mechanisms rapid the the of understanding challenge. fuller major a A remains operate drugs these which by biological processes underlying the despite of understanding con- However, mechanistic of a (13–16). task structing the area information, target experimental interesting available suggests considerable an trials clinical is in this inhibitors be that pres- kinase to The Aurora (13–16). several B therapy of not Aurora cancer ence for is led target but possible has a 16), This as (15, selected tumourigenesis. poly- cells induce causes cancer to kills and thought inhibition of checkpoint acute fidelity the this the overrides 21). ploidy, in (6, role inhibition B key Aurora B a is Aurora segregation play chromosome that and kinases checkpoint the the of One tumourigen- in (5). implicated esis been has failure checkpoint con- central aneuploidy; sequently, and is check- missegregation The chromosome checkpoint (30). prevents mitosis point assembly in segregation spindle chromosome accurate the for cells, eukaryotic In Discussion h iet npaebcmsdculdfo h tt transition state Eq. the from model decoupled becomes anaphase to time the by essentially molecules. determined inhibitory now the the of itself rate at is decay present the which are anaphase, attachments to aberrant time of numbers significant h eta oeo h iae uoaB ntesideassem- spindle the in B, Aurora kinase, the of role central The model mathematical tractable a proposed we article this In ial,i 4actsrpei h yaiso Eq. of dynamics the in catastrophe a D4 in Finally, t Hence, 2. = t a PNAS . t a eebr2,2008 23, December rp icniuul n sdetermined is and discontinuously drops o.105 vol. o 51 no. 7 cusand occurs g ∗ Conse- . 20219

APPLIED BIOPHYSICS MATHEMATICS could be found by titrating the inhibitor and recording the corre- Finally, given the complex nature of the role of Aurora kinase sponding time to anaphase and/or counting aberrant attachments activity within the , it seems unlikely that a straightfor- at anaphase onset. ward PK/PD model with the PD component being of standard Clearly, there are many additions and improvements that could response-function type, will yield significant practical informa- be made to the model proposed here, from a more detailed tion. Developing mechanistic models of the type proposed here description of the attachment process to a better representation of may provide a step toward a better understanding of the action the role of Aurora B. For example spatial distribution of Aurora of Aurora kinase inhibitors. The strength of these models is their B within the cell is ignored in the present model but may play ability to (theoretically) link drug concentration to intracellular a role in the processes described here. Furthermore, although response (as is done here) and through further modeling processes attempts have been made to include the stochasticity of the search relate cell response to cell fate, cell fate to measurable biomarkers, and capture process of the kinetochores, it is assumed here that and thus potentially predict drug efficacy. Experimentally verified the spindle checkpoint mechanism is deterministic and it may be models could lead to improved dosing and scheduling in clinical that both of these processes have stochastic elements. Finally, the trials, and ultimately models may form part of new clinical tools process of search and capture of the kinetochores is very com- for patient-specific drug treatment. plex, and detailed modeling of these events may provide further refinement of the signaling mechanism modeled here. However, ACKNOWLEDGMENTS. We thank Dr. F. Scaerou, (Cyclacel Ltd.) for many these improvements constitute significant further work, not least interesting and helpful discussions. This work was supported by Engineering because of the lack of experimental evidence that could afford an and Physical Sciences Research Council by the EPSRC Grant EP/D04859 under accurate parameterization of more complex models. the Mathematics for Business Scheme.

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