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Modeling the temporal evolution of the spindle assembly checkpoint and role of Aurora B kinase

Communicated by Avner Friedman, Ohio State University, Columbus, OH, November 3, 2008 (received for review March 11, 2008)
Abstract
Faithful separation of chromosomes prior to cell division at mitosis is a highly regulated process. One family of serine/threonine kinases that plays a central role in regulation is the Aurora family. Aurora B plays a role in the spindle assembly checkpoint, in part, by destabilizing the localization of BubR1 and Mad2 at centrosomes and responds to changes in tension caused by aberrant microtubule kinetochore attachments. Aurora B is overexpressed in a subset of cancers and is required for mitosis, making it an attractive anticancer target. Here, we use mathematical modeling to extend a current model of the spindle assembly checkpoint to incorporate all signaling kinetochores within a cell rather than just one and the role of Aurora B within the resulting model. We find that the current model of the spindle assembly checkpoint is robust to variation in its key diffusionlimited parameters. Furthermore, when Aurora B inhibition is considered within the model, for a certain range of inhibitor concentrations, a prolonged prometaphase/metaphase is observed. This level of inhibitor concentrations has not yet been studied experimentally, to the authors' best knowledge. Therefore, experimental verification of the results discussed here could provide a deeper understanding of how kinetochores and Aurora B cooperate in the spindle assembly checkpoint.
The faithful separation of chromosomes prior to cell division at mitosis is essential for maintaining genomic integrity. Failure to do so correctly may lead to genomic instability, aneuploidy, and cancer (1–,3). Chromosome segregation requires the formation of a microtubule network that connects the spindle poles located at either end of the cell, originating in centrosomes, with kinetochores (protein structures located at the centromeres of each chromosome) (,4). This is a highly regulated process involving the interactions between multiple protein complexes and signaling pathways (,3, ,5). One family of serine/threonine kinases that plays a central role in regulation is the Aurora family consisting of 3 forms in metazoans: Aurora A, Aurora B, and Aurora C. In bakers' yeast and budding yeast only one homologue is found (Ark1 and Ipl1, respectively) (,6–,8). Aurora C is only expressed in germ cells, where as Aurora A and Aurora B are found in all somatic cells (,6). Significantly, all 3 kinases are overexpressed in a variety of cancers, suggesting that a growth advantage is gained by deregulating Aurora kinases (,5, ,9). Conversely, severe inhibition of Aurora kinase activity leads to a fatally flawed mitosis (,7, ,10–,16). Hence, this form of inhibition provides a possible mechanism for selective removal of replicating cells and thus has led to the development of Aurora kinase inhibitors as possible anticancer drugs (,13–,17)
Aurora B is active throughout mitosis with protein levels peaking at G_{2}/M phase of the cell cycle (6). Aurora B forms a complex with INCENP and survivin that regulate its activity and localization throughout mitosis (,18–,20). Proteins of the Aurora B complex are “chromosome passengers” localized to the centromeres from prophase until the metaphase–anaphase transition where Aurora B relocates to the spindle midzone and equatorial cell cortex as well as the microtubules (,19). During telophase Aurora B localizes to the midbody throughout cytokinesis (,6–,8). Aurora B regulates chromosome congression, segregation, and cytokinesis (,6, ,21). In budding yeast Ipl1 can promote correct spindle assembly by destabilizing syntelic attachments (,22, ,23). This has also been demonstrated in mammalian cells for syntelic attachments (,14, ,24) and merotelic attachments (,25). Aurora B promotes turnover of microtubules at the kinetochores (,21) possibly by regulating Hec 1 (,26). Aurora B is also required for cytokinesis where it phosphorylates and regulates several substrates (,7, ,8). Characterization of Aurora B inhibitors has suggested that Aurora B plays a role in the spindle assembly checkpoint, in part, by destabilizing the localization of BuBR1, Mad2, and CenpE at centrosomes (,14, ,27) and responds to changes in tension to promote biorientation (,14, ,22, ,28, ,29). Treatment of cells with Aurora B inhibitors cause chromosome alignment problems, spindle checkpoint override, and cytokinesis failure, broadly consistent with data generated by other methods (,14–,16).
The effects of Aurora B inhibition on the spindle assembly checkpoint are of particular interest because this checkpoint is crucial to prevent onset of anaphase without proper alignment of chromosomes and correctly attached spindles (30). A wait signal is generated by kinetochores that inhibits the activation of the anaphasepromoting complex (APC/C). Even one unattached kinetochore is thought to be enough to prevent the onset of anaphase (,31). How this process occurs has yet to be fully defined, although a number of models have been proposed (,30, ,32, ,33).
Recently, mathematical modeling of the spindle assembly checkpoint, both in open and closed mitosis, has provided some mechanistic understanding of how the checkpoint could operate (34, ,35). Doncic et al. (34) considered various simplified models of how a single unattached kinetochore could generate a diffusive signal that could prevent the onset of anaphase in yeast. These models were based on cells that undergo closed mitosis, where the nuclear envelope is still intact even at the onset of anaphase. Therefore, the diffusive signal is only required to propagate throughout the nuclear volume. However, metazoan cells undergo open mitosis, where the nuclear envelope is no longer intact while the spindle assembly checkpoint is active. Therefore, the diffusive signal generated at a single unattached kinetochore now has to propagate throughout the entire cell rather than just the nucleus. One possible mechanism describing how this could be achieved was recently proposed by Sear and Howard (,35). They considered a 2step mechanism in which a stimulant (Cdc20) of the APC/C is inhibited: step 1, Cdc20 is assumed to interact with an unattached kinetochore to form an inhibited complex (candidates include CMad2Cdc20 and BubR1Cdc20, where CMad2 represents the closed form of Mad2); step 2, these inhibited complexes are assumed to be able to catalyze the production of further inhibited complexes of Cdc20, which are different in that they cannot themselves act as catalysts. This type of signal amplification ensures that the inhibitory signal is only amplified by molecules that interact with an unattached kinetochore. Furthermore, 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 evolution 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, resulting in 46 pairs of chromosomes, thus, 92 kinetochores. Each kinetochore 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 mechanochemical reactions (,31) occurring throughout prometaphase. It is not the aim in this article to investigate the details of the attachment 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. However, 3 types of incorrect attachments can occur that lead to a missegregation of chromosomes: (i) unattached; (ii) syntelic attachment—both sister kinetochores are bound to microtubules from the same pole; (iii) merotelic attachment—one sister kinetochore is bound to microtubules from both spindle poles, (see Fig. 1A). One further type of attachment (monotelic) is common in the very early stages of prometaphase. However, they exist only as brief, transitory states in the subsequent attachment process (37–,39). Therefore, we do not consider them further.
Syntelic and merotelic attachments are observed throughout prometaphase predominantly in the earlier stages (14, ,40). The correction process of these attachment types can be summarized as follows. Syntelically attached kinetochores are drawn toward the corresponding spindle pole. Subsequently, the incorrect attachments are released via the action of Aurora B kinase (,14, ,15) and, after relocating to the metaphase plate, it is common that this kinetochore pair form amphitelic attachments (,38). Merotelically attached kinetochores are aligned at the spindle equator, where Aurora B destabilizes the aberrant attachments, thus allowing the resultant pair of sister kinetochores to almost immediately become amphitelically attached (,37).
Attachment Transitions.
In the following, we define a pair of sister kinetochores as a unit. We assume that both kinetochores within the unit have all their binding sites occupied or both kinetochores have binding sites available (37–,39). We assume further that these 2 sets can be divided as follows: a unit displaying full occupancy can be in either a merotelic (ME) or amphitelic (C) state; otherwise it is deemed to be in a syntelic (S) or unattached (U) state (37–,39). Following the above arguments a unit state transition network can be constructed (see ,Fig. 1B). The parameters, k_{i} (i = 1,…,5) in Fig. 1B, represent the rates at which a unit undergoes the corresponding state change within a single cell. Note that for ease of reference below, the parameters k_{4}(B^{s}) and k_{5}(B^{s}) are defined to be functions of another variable, B^{s}, which represents the steadystate fraction of active concentration of Aurora B. Later on we will consider how this fraction can be altered by inhibition. However, in the meantime, we simply assume that the parameters k_{4}(B^{s}) and k_{5}(B^{s}) are constants.
Let X = (X_{U}, X_{S}, X_{ME}, X_{C}) denote the state vector of random variables of unit type X_{i}, i ∈{U,S,ME,C}$ within a single cell, at any given time t. Define P(X_{i} = x_{i},t) to be the probability that there will be x_{i} ∈ {0,1,2,…,46} units in a state i, at time t. Let us assume that we can choose a Δ t sufficiently small such that, at most, 1 state transition can occur during the time interval, (t,t + Δ t). There are 5 possible state transitions, the probabilities for which can be defined as follows: (Correspondingly, each X_{i} remains unaltered during Δ t, with the naturally assigned probability.) The temporal evolution of the attachment transition probabilities is governed by the master equation and the time derivative of the mean (first moment) (41), for each X_{i}, is given by the following system of ordinary differential equations (ODEs). [See supporting information (SI) Appendix for a full derivation.] where 〈X_{i}〉 corresponds to the expected mean number of kinetochore units in a state i in any given cell within a large population of genetically identical cells. In fact, one of these equations can be eliminated as the total number of kinetochore units is conserved, i.e., for all times t ≥ 0, 〈X_{U}(0)〉 = 46 = 〈X_{U}(t)〉 + 〈X_{S}(t)〉 + 〈X_{ME}}(t)〉 + 〈X_{C}(t)〉. Finally, we define A(t) = 〈X_{U}(t)〉 + 〈X_{S}(t)〉 + 〈X_{ME}(t)〉, to be the expected mean number of aberrant units at any given time, t.
Parameter Constraint for Attachment Transitions.
We now consider how the above system of ODEs can be parameterized. Quantitative data regarding aberrant attachments during early stages of mitosis are, at present, very limited (14, ,40 and are insufficient to parameterize the system of ODEs ,Eq. 2 fully. We therefore appeal to a simpler mathematical argument, which will provide constraints on the system parameters. This simplified approach is based on the observations that kinetochore–microtubule attachment is a stochastic process (,42, ,43). This in turn has been interpreted as leading to a compound Poisson process governing the state transitions (,34) and is the methodology we adopt here.
Consider now a simplified state transition process in which only 2 states can occur: the amphitelic state and the unattached state. Define Y_{C} and Y_{U} to be random variables that represent the total number of units within each state. Similarly to above we can derive an ODE governing the temporal evolution of 〈 Y_{C}(t)〉, namely, where we have used 46 = 〈Y_{C}(t)〉 + 〈Y_{U}(t)〉 and the rate k_{c} can be approximated as follows. We assume the time taken for a unit to move from the unattached to the amphitelic state is Poisson distributed with a rate Y_{U}k_{c} (34). Thus, the transition of all 46 units is a compound Poisson process, where the average time to the next event is 1/Y_{U}k_{c}. Taking the total time for the transition of all 46 units to be 20 min (44), then,
We reasonably assume that the simplified arguments just derived and the more detailed state transition dynamics given in Eq. 2 are consistent in their prediction of the temporal evolution of the relevant expected means. That is, we insist, for some $ε⋘1. Setting ε = 0.01 and by comparing numerical integrations of Eq. 2 and the graph of 〈Y_{C}(t)〉, an optimal set of system parameters can be obtained, which ensures condition Eq. 5 holds. For the results discussed below, the following set of parameter values were chosen: k_{1} = 0.04 min^{−1} and k_{2} = 0.02 min^{−1}, k_{3} = 0.178 min^{−1}, k_{4}(B^{s}) = 0.3 min^{−1} and k_{5}(B^{s}) = 0.2 min^{−1}. This set was generated by first setting k_{1} and k_{2} to be of an appropriate order with k_{1} > k_{2} (37). The remaining parameters were selected to be again of an appropriate order with the values determined by ,Eq. 5. Extensive numerical investigations revealed that, providing condition ,Eq. 5 holds, altering these parameter values does not change the qualitative properties of the results discussed below.
Spindle Assembly Checkpoint.
The spindle assembly checkpoint model, proposed by Sear and Howard (35) and discussed earlier, concerns the wait signal generated by a single unattached kinetochore. This model can be embedded into the state transition process detailed above, as follows. From the above discussions it is reasonable to assume that a wait signal is generated from each U and Stype unit, the reaction processes being given by, Here, species e is associated with free Cdc20. We assume that units in either the U or S state can generate inhibited complexes of Cdc20, denoted by e*, at a diffusion limited rate, k_{k}(B^{s}) = 40 μm^{3}s^{−1}. (This is 2 times the corresponding rate used for a single kinetochore in ref. 35. For ease of exposition, we have indicated that this rate constant is a function of B^{s}, the steadystate concentration of Aurora B. As above, we simply take this to be a constant at present.) Note, to the authors' best knowledge, the production rate of e* is state independent. Also, all parameter values are taken from Sear and Howard (35) unless stated otherwise. It is assumed that the inhibited complex e* can catalyze the production of further inhibited complexes of e denoted by c*, at a diffusionlimited protein–protein interaction rate, k = 10^{−2} μ m^{3}s^{−1}. As in ref. 35 it is assumed that these complexes cannot catalyze further complexation. We use α^{−1} = α_{e}^{−1} = 5 min.
In ref. 35 only the final unattached kinetochore is considered and thus the initial data for the reaction system considered there are set by computing the steady states of the respective species. We cannot adopt a similar approach here because our model represents events occurring throughout prometaphase/metaphase. However, it is known that at the onset of prometaphase all Cdc20 molecules are in an inhibited state (,45, ,46. Furthermore, these inhibitory complexes of Cdc20 are known to be generated by a different mechanism from the one discussed here (,45). Therefore, it is necessary to include another different inhibited complex of Cdc20 in our model that is denoted here by g*. We assume that this complex decays releasing the Cdc20 at a rate, α_{g}^{−1} = 5 min, which in lieu of further experimental evidence, is chosen to be equal to the decay rate for the other inhibited complexes.
Finally, define N_{x} to be the total number of molecules of species x ∈ {e,e*,c*,g*}. The total number of molecules is conserved, so that, N = N_{e} + N_{e*} + N_{c*} + N_{g*} [N = 800,000 (35)]. By using this relationship, the reaction system ,Eq. 6 and following the approach in ref. ,35, we model the dynamics of the spindle assembly checkpoint as system ,Eq. 2 combined with the following system of ODEs, where, v_{c} = 6,000 μm^{3} is the cytoplasmic volume (35.
Recall each unit of type i comprises 2 kinetochores with the expected value 〈X_{i}〉 ∈ [0,46]. However, if 〈X_{i}〉 < 0.5, then this expected value corresponds to the presence of less than a single kinetochore. Therefore, in system Eq. 7 we assume that, if 〈X_{i}〉 < 0.5, then 〈X_{i}〉 ≡ 0. This leads to the definition t_{f} := inf{t ≥ 0〈X_{S}(t)〉 < 0.5 & 〈 X_{U}(t)〉 < 0.5} which best represents the time at which the last kinetochore is sensed by system Eq. 7. (This corresponds to t = 0 in ref. 35.) Furthermore, we define the time to anaphase (length of prometaphase/metaphase) as t_{a} := sup{t ≥ 0 N_{e}(t) < 200,000} (35).
Results
We first analyze the temporal evolution of the wait signal generated by all kinetochore units, described by systems Eq. 2 and ,Eq. 7. Subsequently, the effects of Aurora B inhibition are studied.
Temporal Evolution of the Wait Signal.
The temporal evolution of N_{e}, N_{e*}, N_{c*}, and N_{g*}, illustrated in Fig. 2, can be split into 2 distinct phases: For 0 ≤ t < t_{f}, e molecules are under tight inhibition (N_{e}/N ≤ 0.05); at t = t_{f}, inhibition is lifted and N_{e} increases rapidly. These distinct phases can be explained as follows. In the first phase, recall that 〈X_{U}(0)〉 = 46 and initially all of these units contribute to the production of e* molecules (see Eq. 7a). Subsequently, 〈 X_{U}(t)〉 decreases, 〈X_{S}(t)〉 increases, but 〈X_{U}〉 + 〈X_{S}〉 decreases (data not shown). However, throughout this phase the production of e* dominates the production of e resulting from the decay of g*, c*, and e*. At t = t_{f} the production of e* is set to zero (see above discussion). Subsequently, the rapid decay rates (short lifetimes) of g*, e*, and c* ensure a rapid increase in N_{e}.
Sear and Howard (35) focused on the signal generated by a single unattached kinetochore. Essentially, they considered this to be the only signal source for a time period in our [0,t_{f}]. Doing so allowed them to compute “steady state” values for N_{e}, N_{e*}, and N_{c*}, which were subsequently used as initial conditions (at a time corresponding to our t = t_{f}) for the secondphase dynamics discussed above. The authors in ref. 35 were concerned that to work effectively, their model required certain system parameters to be chosen at the upper end of their expected range. This has implications for the robustness of the model in application.
The model introduced here not only provides a more complete description of the wait signal dynamics, but also provides greater robustness to variations in the key diffusionlimited parameters, k and k_{k}(B^{s}) (see Fig. 3). The model maintains tight inhibition of e molecules for t < t_{f} for a large range of biological realistic values for , i.e., k ≈ 10^{−3} to 10^{−2} μm^{3}s^{−1} and k_{k}(B^{s})≈ 6 to 60 μm^{3}s^{−1}. Remarkably, this parameter variation does not significantly alter the necessary rapid release of inhibition in the second phase (data not shown).
We were concerned that the attachment transition (and associated signaling) process detailed above would be too crude to detect the subtle differences between the alternative signaling models proposed by and investigated in refs. 34 and ,35. However, on replacing ,Eq. 6 with systems corresponding to the direct inhibition and emitted inhibition models, respectively, and embedding them appropriately in Eq. 7, it transpired that the attachment state transition model system is indeed able to differentiate these models. With either of these replacements, sufficiently tight inhibition of N_{e} was not observed for t ∈ [0,t_{f}] (see SI Appendix).
Aurora B Inhibition
Aurora B plays a central role in many processes involved in mitosis. As a first step toward understanding this complex role, we assume that the concentration of Aurora B affects the 2 main components of our model in the following way. The role Aurora B plays in releasing aberrant attachments for subsequent repair (14, ,24) is modeled by reasonably assuming that the rate constants k_{4} and k_{5} in Eq. 2 are positively correlated to the value of B^{s}. In lieu of further experimental evidence we assume this relationship to be linear, i.e., k_{4}(B^{s})  k_{4}′B^{s}min^{−1}, k_{5}(B^{s})  k_{5}′B^{s}min^{−1}, for constants k_{4}′ and k_{5}′. The role of Aurora B in facilitating the recruitment of key checkpoint proteins (14, ,24) is modeled similarly by allowing k_{k}(B^{s})  k_{k}′B^{s}μm^{3}s^{−1} in Eq. 7, for some constant k_{k}′. (The values used for the results illustrated above are k_{4}′  0.3min^{−1}, k_{5}′  0.2min^{−1} and k_{k}′  40μm^{3}s^{−1}.)
We are principally interested in the effects of Aurora B inhibition. Here, inhibition of Aurora B is modeled as a simple mass action process, which mediates the steadystate fraction of active Aurora B, B^{s}, i.e., increasing the concentration of inhibitor, denoted by [B^{−}], decreases B^{s} (see SI Appendix).
The results of Aurora B inhibition are summarized in Fig. 4. It is difficult to draw precise quantitative conclusions from the results, but the model predicts that inhibition can be separated into 4 qualitatively distinct phases as we now discuss.
Throughout D1, the distribution of unit states is essentially unaltered and, for low dosing fractions, the times t_{f} and t_{a} defined above also change little. This provides further evidence that the model is not sensitive to small changes in the values of k_{4}, k_{5}, and k_{k}, chosen to represent the uninhibited state. However, as the inhibitor fraction is increased to the upper end of D1, a rapid change in t_{f} and t_{a} is observed and can be explained as follows. For these higher dosing levels, the repair function modeled by Eq. 2 is compromised by the reduction in the rates k_{4}, k_{5}. Consequently, t_{f} becomes larger. However, the checkpoint mechanism Eq. 7 is still operating efficiently. Hence, the latter detects the lingering Stype units, continues to produce a wait signal, and consequently t_{a} is also extended.
At the D1/D2 boundary, a marked change is evident. This point is marked by the equality t_{f}t_{a}. Thus, Stype units are now predicted to exist at the time to anaphase t_{a}. Increasing the inhibitor dose increases the fraction of S and MEtype units and after a short further increase, the time to anaphase t_{a} decreases rapidly. This rapid decrease is in response to the corresponding reduction in k_{k}: even though there are a larger number of units capable of generating a wait signal, the reaction Eq. 7 is insufficiently strong to counter the decay of the inhibitory complexes e^{*}, c^{*}, and g^{*}.
In the D3 dosing interval, the fractions of unit states at tt_{a} level out, but now a small fraction of unattached units is observed at t  t_{a}. The time t_{a} also levels out at a significantly reduced value compared with that of the uninhibited state. Values of B^{s} (and hence k_{k}) are so low in this interval that the wait signal generated by Eq. 7 is now severely compromised and the model predicts that significant numbers of aberrant attachments are present at the time to anaphase, which is itself now determined essentially by the decay rate of the inhibitory molecules.
Finally, in D4 a catastrophe in the dynamics of Eq. 7 occurs and the time to anaphase becomes decoupled from the state transition model ,Eq. 2. Hence, t_{a} drops discontinuously and is determined entirely by the decay rate of the inhibitory molecule g^{*}. Consequently, the model predicts that large numbers of unattached units appear at t t_{a}.
Discussion
In eukaryotic cells, the spindle assembly checkpoint is central for accurate chromosome segregation in mitosis 30. The checkpoint prevents chromosome missegregation and aneuploidy; consequently, checkpoint failure has been implicated in tumourigenesis ,5. One of the kinases that play a key role in the fidelity of the checkpoint and chromosome segregation is Aurora B ,6, ,21. Aurora B inhibition overrides the checkpoint and causes polyploidy, this acute inhibition kills cancer cells ,15, ,16, but is not thought to induce tumourigenesis. This has led Aurora B to be selected as a possible target for cancer therapy ,13–,16. The presence of several Aurora kinase inhibitors in clinical trials suggests that this is an interesting target area ,13–,16. However, despite considerable available experimental information, the task of constructing a mechanistic understanding of the underlying biological processes by which these drugs operate remains a major challenge. A fuller understanding of the mechanisms involved may assist in the rapid development of effective Aurora B kinase inhibitors.
In this article we proposed a tractable mathematical model for key processes within prometaphase/metaphase. This model framework has at its core a current model of the spindle assembly checkpoint 35. By considering the expected mean behavior of the temporal evolution of all kinetochore—microtubule attachment units, a more complete qualitative description of the evolution of the wait signal has been obtained. This has resulted in 2 notable conclusions. First, the model takes account of the temporal evolution of all signaling kinetochores, yet it is still sensitive enough to detect significant differences in the core signal models previously investigated in ref.,35, namely, the direct and emitted inhibition models. The conclusions regarding the applicability of these models are unaltered in our framework. Second, in ref. ,35 concerns were raised that the alternative signaling model proposed did work but only if certain key parameters were chosen to take values at the extreme of their expected range. By embedding this signaling model into our description of the unit state transitions, we have been able to demonstrate that this core model is in fact far more robust to parameter variations than ref.,35 predicted (see Fig. 3).
The central role of the kinase, Aurora B, in the spindle assembly checkpoint was also considered in the model presented here. A simple relationship between Aurora B activity and the reaction rates was assumed in both the unit state transition and wait signal generation components of the model. Thus, the model provided predictions regarding the qualitative response to different levels of Aurora B inhibition. Predictions regarding severe inhibition are in line with experimental data: the spindle checkpoint signal is abolished, causing cells to enter anaphase with a large number of aberrant attachments. The predictions regarding lower levels of inhibition were somewhat surprising, in particular, the prolonged prometaphase/metaphase associated with syntelic and merotelic attachments present at anaphase. To the authors' best knowledge, this level of inhibitor concentration has not yet been fully studied experimentally and therefore invites verification. Predictions regarding the qualitative relationship between the concentration of measurable biomarkers and cell damage (aberrant attachments) and time to anaphase onset can be derived from the model. The output from the model described here suggests that experimental evidence supporting these assumptions could be found by titrating the inhibitor and recording the corresponding time to anaphase and/or counting aberrant attachments at anaphase onset.
Clearly, there are many additions and improvements that could be made to the model proposed here, from a more detailed description of the attachment process to a better representation of the role of Aurora B. For example spatial distribution of Aurora B within the cell is ignored in the present model but may play a role in the processes described here. Furthermore, although attempts have been made to include the stochasticity of the search and capture process of the kinetochores, it is assumed here that the spindle checkpoint mechanism is deterministic and it may be that both of these processes have stochastic elements. Finally, the process of search and capture of the kinetochores is very complex, and detailed modeling of these events may provide further refinement of the signaling mechanism modeled here. However, these improvements constitute significant further work, not least because of the lack of experimental evidence that could afford an accurate parameterization of more complex models.
Finally, given the complex nature of the role of Aurora kinase activity within the cell cycle, it seems unlikely that a straightforward PK/PD model with the PD component being of standard responsefunction type, will yield significant practical information. Developing mechanistic models of the type proposed here may provide a step toward a better understanding of the action of Aurora kinase inhibitors. The strength of these models is their ability to (theoretically) link drug concentration to intracellular response (as is done here) and through further modeling processes relate cell response to cell fate, cell fate to measurable biomarkers, and thus potentially predict drug efficacy. Experimentally verified models could lead to improved dosing and scheduling in clinical trials, and ultimately models may form part of new clinical tools for patientspecific drug treatment.
Acknowledgments
We thank Dr. F. Scaerou, (Cyclacel Ltd.) for many interesting and helpful discussions. This work was supported by Engineering and Physical Sciences Research Council by the EPSRC Grant EP/D04859 under the Mathematics for Business Scheme.
Footnotes
 ^{1}To whom correspondence should be addressed. Email: fdavidso{at}maths.dundee.ac.uk

Author contributions: R.C.J., M.A.J.C., and F.A.D. designed research; H.B.M. and F.A.D. performed research; D.E.M., R.C.J., and M.A.J.C. contributed new reagents/analytic tools; and H.B.M., D.E.M., and F.A.D. wrote the paper.

The authors declare no conflict of interest.

Freely available online through the PNAS open access option.

This article contains supporting information online at www.pnas.org/cgi/content/full/0810706106/DCSupplemental.
Freely available online through the PNAS open access option.
 © 2008 by The National Academy of Sciences of the USA
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 A quantitative systems view of the spindle assembly checkpoint