The Model Accurately Describes the Growth and Division of Wild-Type Cells
As described in MATERIALS AND METHODS, we solve the equations in , given the parameter values and initial conditions in for a wild-type budding yeast cell. In , we illustrate how cell size, cyclin concentration, and other components vary during repetitive cycling of daughter cells. The computed properties of the model agree reasonably well with the observation of Brewer et al.
): for a wild-type diploid A364A D5 strain, growing at MDT = 90 min, the daughter cell cycle time is 97.5 min (computed value = 101.2 min), G1 length is 42 (36) min, and S/G2/M length is 57 (64) min, whereas the mother cell cycle time is 81 (80) min, G1 length is 22 (28) min, and S/G2/M is 59 (52) min.
Figure 2. Wild-type cell cycle. Numerical solution of the differential equations in , for the parameter values in . The MDT for an asynchronous culture is 90 min. We show the cycle of a daughter cell (cycle time, 101 min; duration of G1, 36 min). (more ...)
Furthermore, the relative amounts of certain groups of proteins are in rough quantitative agreement with recent measurements by Cross et al.
) and Archambault et al.
). The ratios, for an asynchronous culture with MDT = 90 min, are as follows:
The predicted ratios are determined by integrating each variable over a full cycle of both a mother cell and a daughter cell, dividing these numbers by the corresponding cycle times of mother and daughter cells, and then averaging these two numbers (because there are roughly equal numbers of mother and daughter cells in rapidly growing populations).
The most significant discrepancy between the model and experiment is the ratio of [Sic1] to ([Clb1] + [Clb2] + [Clb5] + [Clb6], 1:11 in experiment and 1:3 in model. Cross's measurements indicate that rapidly growing cells have much less Sic1, on average, than would be expected from the model. By decreasing the rate of synthesis of Sic1, we could get better agreement with the experiments for wild-type cells. But the model must satisfy many other constraints implied by the phenotypes of various mutants. In particular, in the model the triple-cln deletion strain (cln1Δ cln2Δ cln3Δ) would become viable, contrary to observations, if the rate of synthesis of Sic1 were reduced by half. In our simulation of the triple-cln mutant, if Sic1 concentration is too low, then Clb5 activates its own synthesis via SBF/MBF, and the simulated cell is perfectly viable by all our criteria. We do not have any explanation for this discrepancy in Sic1 level between the model and the experiment.
The parameter values in have been chosen taking into account the properties of wild-type and mutant cells, and they represent a compromise of many, often competing, observations. To this end we must settle on a “data set” of mutants () that will serve as a testing ground for the sufficiency of the model. The parameter values proposed in have been selected by a painstaking process of trial-and-error to provide a suitable fit to the full data set.
The Model Conforms to the Phenotypes of >100 Mutant Strains
The wiring diagram in has been composed from evidence provided by the phenotypes of dozens of budding yeast mutants that have been constructed and characterized by deleting or overexpressing each genetic component singly and in multiple combinations. It remains an informal “cartoon” of the molecular regulatory system until it is converted into a precise mathematical model and demonstrated to be consistent with most of the facts about budding yeast mitotic division. In , we list 131 mutants that have been used to test the model.
For each mutant simulation, we use exactly the same equations () and parameter values (), and we are allowed to change only those parameters that are governed by the nature of the mutation, as described in MATERIALS AND METHODS. We compare the computed behavior of the model with the observed phenotype of the cells. For example, if the mutant is inviable, at what phase of the cell cycle is it blocked? If viable, in what subtle ways does it differ from wild-type: size at onset of DNA synthesis, size at bud emergence, size at division, and duration of G1 phase?
In most cases, 120 of 131 mutants in , the model agrees well with observations; the 11 exceptions are marked with asterisks in the table.
The Model Is Fully Described on a Web Page
At our Web site, http://mpf.biol.vt.edu
, full details about the model and all simulations can be found. On this site, we summarize the basic experimental results on which the wiring diagram () is based. We present simulations of all mutants in , including the precise parameter values used in each case. We provide facilities for the user to repeat any simulations by using our parameter set or any new choice of parameter values. We also give instructions for how one may modify the wiring diagram, build a revised model, and run new simulations.
How the Model Represents APC Phosphorylation and the FEAR Pathway
In earlier models of cell cycle regulation in frog eggs (Novak and Tyson, 1993
) and fission yeast (Novak et al., 2001
), we proposed the existence of an intermediary enzyme (IE) whose purpose was to introduce a time delay between the activation of cyclin B-dependent kinase at the onset of M phase and the activation of Cdc20 (Fizzy in frog eggs and Slp1 in fission yeast) at the metaphase-to-anaphase transition. There are good reasons to suspect that IE is the APC core complex itself (Cross, 2003
). First, deletion of IE would cause cells to arrest in metaphase, as is the case for most components of the APC. Rudner and Murray (2000
) showed that three components of the APC core, Cdc16, Cdc23, and Cdc27, are phosphorylated by mitotic Cdc28 kinase and that only the phosphorylated forms associate with Cdc20 effectively. They also showed that APC-A
mutant cells (where all the possible Cdc28-phosphorylation sites in Cdc16/23/27 are removed by substituting alanine for serine/threonine residues) are viable, with a delay in mitotic exit. This phenotype is consistent with the identification of IE with APC, if we assume that the unphosphorylated form of APC retains some intrinsic ability to activate Cdc20 (simulations not shown). Hence, in the present model, the phosphorylated, active form of IE in our earlier models (IEP) is replaced by APC-P, the phosphorylated state of APC that is necessary for full activity in conjunction with Cdc20. Notice that APC need not be phosphorylated to function in conjunction with Cdh1 (Kramer et al., 2000
; Rudner and Murray, 2000
). The present model is not fully correct in that IEP/APC-P is treated as if it were an enzymatic activator of Cdc20 instead of a binding partner. This problem will be corrected in future versions of the model, by revising the wiring diagram and the differential equations. These changes will undoubtedly require minor revisions of the rate constants, but we do not expect any significant changes in the conclusions reached in this article.
The sequential activation of Cdc20 and Cdh1 suggests that Cdc20/APC-P degrades, directly or indirectly, an inhibitor of Cdh1 (Morgan, 1999
). The inhibitor must act before Cdc14 release from the nucleolus. If the Cdc20-sensitive inhibitor were to act downstream of Cdc14 release, then in the cdc20
Δ strain, Cdc14 would be released by MEN action as soon as the spindle assembly checkpoint is satisfied, which is contrary to observation (Shirayama et al., 1999
). Pds1 is an obvious candidate for this inhibitor: it is degraded by Cdc20 (Yamamoto et al., 1996b
), and overexpression of nondegradable Pds1 delays Clb2 degradation for several hours (Cohen-Fix and Koshland, 1999
). But how might Pds1 degradation relate to Cdc14 release? Cdc14 is released from the nucleolus in two stages (Visintin et al., 2003
): an early, transient stage, via the FEAR pathway (Uhlmann et al., 1999
; Stegmeier et al., 2002
; Yoshida et al., 2002
), involving Esp1 and Cdc5; and a late, sustained stage, involving Cdc15 and other MEN components. By binding and inhibiting Esp1, Pds1 inhibits the FEAR pathway, preventing Cdc14 release (FEAR stands for Cdc-f
At the time we were formulating this model, the information just cited was not known. In its place, we hypothesized a phosphatase (PPX) that inhibits Cdc14 release by opposing the action of Cdc15 on Net1. The role of Pds1 was to keep PPX level high by inhibiting its degradation by Cdc20/APC-P. Hence, in our model, Pds1 up-regulates PPX, an inhibitor of Cdc14 release. According to the FEAR story, Pds1 down-regulates activators of Cdc14 release. With appropriate choice of kinetic constants, the dynamic consequences of the two formulations should be nearly identical. In a future version of the model, we will replace PPX by a representation of the FEAR pathway.
The Model Is Based on Alternative Stable Steady States Corresponding to G1 and S/G2/M Phases of the Cell Cycle
Although the rigor and precision of the model are essential attributes in its favor, the sheer magnitude of information that comes out of the computer can overwhelm the user. To make sense of this information, we need intuitive ways to understand the model's behavior—an intuition disciplined by precise numerical simulations of the equations. We rely on the scheme in (described briefly in Chen et al., 2000
). Fundamental to cell cycle control in budding yeast are the antagonistic relations between B-type cyclins (Clb1–6, in association with Cdc28), which promote DNA synthesis and mitosis, and G1-stabilizers (Cdh1, Sic1, and the cyclin-dependent kinase inhibitor [CKI]-role of Cdc6), which oppose cell proliferation by degrading Clbs and stoichiometrically inhibiting Clb/Cdc28 complexes (For the CKI-role of Cdc6, see Greenwood et al., 1998
and Calzada et al., 2001
). We shall refer to Sic1 and Cdc6 together as the CKIs.
Figure 3. Logic of cell cycle transitions in budding yeast. (A) Antagonistic interactions between the G1-stabilizers (Cdh1 and CKI) and the Clb/Cdc28 kinases create two coexisting stable steady states, G1 and S/G2/M. Transitions between these states are called (more ...)
Because Clb-dependent kinases can inactivate Cdh1 (for review, see Zachariae and Nasmyth, 1999
) and destabilize CKIs (Verma et al., 1997
), these two classes of proteins are mutual antagonists (). The model is designed to have two coexisting, self-maintaining steady states: a G1 state, when Clbs are scarce and their antagonists (Cdh1 and CKIs) are abundant; and an S/G2/M state, when the reverse is true (Nasmyth, 1996
; Novak and Tyson, 1997
; Novak et al., 1998
; Tyson and Novak, 2001
; Tyson et al., 2001
). When yeast cells are proliferating, the control system is undergoing periodic transitions from the G1 state to the S/G2/M state and back again. These transitions (called start
) are irreversible and alternating. Once a cell has executed start
, it does not normally slip back into G1 phase and does start
again. Rather, it must execute a distinctly different transition (finish
) to return to G1. Likewise, a cell that has executed finish
does not slip back into mitosis and try to separate its chromosomes a second time. There are exceptions to these rules (endoreplication and meiosis), but they do not nullify the central role played by irreversible, alternating start
transitions in the cell cycle.
To a first approximation, we view the budding yeast cell cycle as an alternation between these two stable steady states generated by the antagonism between Clb kinases and G1-stabilizers. From the simulation of the wild-type cycle (), one can see how the control mechanism shifts from one state to the other, and how the transitions are carried out.
transition is facilitated by Cln1,2/Cdc28 complexes, which can phosphorylate and inactivate Cdh1 and CKIs, but they themselves are unopposed by the G1-stabilizers (for reviews, see Schwob et al., 1994
and Peters, 1998
). This transition occurs when the cell has grown large enough to accumulate a critical concentration of Cln3-dependent kinase in the nucleus (Miller and Cross, 2000
; Cross et al., 2002
). Cln3 kinase and a back-up (Bck2) activate SBF and MBF, the transcription factors for Cln1,2 and Clb5,6, so their levels increase. Clb5,6-dependent kinases are inactive due to inhibition by the CKIs, but Cln1,2-dependent kinases are not so inhibited. Cln-dependent kinases depress Cdh1 and CKIs enough to allow the Clb-dependent kinases to assert themselves, switching the control system into the stable S/G2/M state. Once the transition is made, Clb kinases by themselves are able to depress their antagonists without the help of Cln1,2 kinases. Rising activity of Clb1,2/Cdc28 turns off Cln1,2 synthesis, causing Cln-dependent kinase activity to drop. Hence, after doing their job, the start
Cdc20/APC facilitates the finish
transition. Cdc20 transcription is activated in G2/M phases by the transcription factor complex Mcm1/Fkh2/Ndd1 (Spellman et al., 1998
; Zhu et al., 2000
; Simon et al., 2001
), which is activated in turn by Clb1,2 kinase activity. In addition, APC core proteins (Cdc16, 23, and 27) are phosphorylated by Clb1,2 kinase, which facilitates APC binding with Cdc20 to form an active complex (Rudner and Murray, 2000
). Cdc20/APC-P depresses Clb kinase activity by labeling Clbs for degradation; it also initiates activation of a phosphatase, Cdc14, which reverses the inhibitory effects of Clb/Cdc28 on Cdh1 and CKIs, so the latter two can overpower the Clb kinases. As the G1-stabilizers extinguish Clb kinase activity, the transcription factor Mcm1 turns off and Cdc20 synthesis ceases. Because Cdc20 is an unstable protein, it quickly disappears, preparing the cell for the subsequent start
To consider G1 and S/G2/M as two alternative steady states, however, is an oversimplification. After start, Clb-dependent Cdc28-kinase activity rises in at least two stages. First, a moderate activity, dependent primarily on Clb5,6, is responsible for initiating DNA synthesis and inactivating Cdh1. Then, a high activity, dependent primarily on Clb1,2, is responsible for driving congression of replicated chromosomes to the metaphase plate. During finish, Clb-dependent Cdc28-kinase activity drops in at least two stages. The initial drop in activity is dependent primarily on Cdc20-dependent degradation of Clb1–6, and it coincides with Pds1 degradation and subsequent separation of sister chromatids (anaphase). As mentioned in the previous section, the disappearance of Pds1 also results in the degradation of PPX (or equivalently, activation of the FEAR pathway), which results in activation of Cdh1 and a second, deeper drop in Clb1,2. The drop in Clb-dependent Cdc28-kinase activity, together with the sustained release of Cdc14-phosphatase from RENT complexes, seems to be responsible for the final stages of exit from mitosis: nuclear division (telophase), cell division, and relicensing origins of DNA replication.
S/G2/M is not a unitary state; mutants reveal states of intermediate and high activities of Clb-dependent Cdc28 kinase. For example, clb1Δ clb2Δ cells arrest in G2 phase with moderate Clb-dependent kinase activity; CLB2dbΔ cells (Clb2 protein lacks “destruction box” sequences necessary for Cdc20-mediated proteolysis) arrest in telophase with very high Clb kinase activity; and cdc14ts cells (which activate Cdc20 but not Cdh1) arrest in telophase with an intermediate activity of Cdc28 kinase, due mostly to Clb1,2.
How Can Mutant Cells Lacking start or finish Facilitators Be Rescued?
The transitions from G1 to S/G2/M and back again, which we refer to as start and finish, are induced by helper proteins Cln2 and Cdc20 as described above. The helpers are involved in negative feedback loops, i.e., the processes they induce lead to their own demise. Rising Clb2 activity after start turns off Cln2 synthesis, and falling Clb2 activity after finish turns off Cdc20 synthesis. Because Cln2 and Cdc20 are unstable proteins, they disappear rapidly after their job is done. We describe next how the inviability of various mutants lacking the helpers is related to this central scheme and how they can be rescued.
Mutations involving the facilitators of start
have striking phenotypes. For example, in the absence of Cln-dependent kinase activity (cln1
” for short), start
cannot occur and cells arrest in G1 phase (Wittenberg et al., 1990
). In the simulation (), start
is delayed many hours, but eventually the cell grows large enough (in the model) for Clb5-dependent kinase activity to drive [ORI] to 1. This is a serious problem for the model. To account for inviability of the triple-cln
mutant, in this model we assume that a cell must reach [ORI] = 1 before a wild-type cell in the same growth medium has divided twice; if not, the cell is considered G1 arrested. Although this rule is introduced specifically to account for the phenotype of triple-cln
cells, it is applied uniformly to all mutants to decide whether the simulated cells are viable or G1 arrested.
Figure 4. Mutations that interfere with the start and finish transitions. (A) Deletion of all three CLN genes arrests cells in G1 because the start-facilitators are missing. (B and C) The triple-cln mutant is rescued by further deletion of SIC1, but not by deletion (more ...)
In the model, deletion of either SIC1
mutant cells to undergo start
() before transgressing the just mentioned rule about G1 arrest. The former construct is viable in the model and in reality (Tyers, 1996
). The latter, though able to replicate its DNA (just barely, according to the “rule”), gets stuck in telophase. Schwab et al.
) studied the response of cdh1
Δ cells to pheromone treatment (analogous to triple-cln
deletion). Although the cells were sensitive to pheromone (i.e., they did not form colonies), they did not arrest uniformly in G1. A fraction of the cell population proceeded through S phase and arrested with 2C DNA content. Considering that the simulated cells just barely satisfy our condition for entering S phase, we might expect, in a population of real cells that vary around the mean simulated behavior, some cells will arrest with 1C DNA content and some with 2C DNA content, as observed.
In triple-cln sic1Δ cells, what molecules initiate the start transition? These cells contain modest amounts of Clb5 in G1 phase ([Clb5] = k′s,b5/k′d,b5 = 0.08 au). Because Sic1 is missing and Cdc6, we assume, is a poor inhibitor of Clb5/Cdc28, Clb5-dependent kinase activity is not inhibited. Together with Bck2, Clb5 activates SBF and MBF when cells are large enough. At this point, Clb5-kinase activity rises sharply and initiates DNA synthesis. Clb5 also inhibits Cdh1, enabling Clb2 to accumulate and the division cycle to progress normally.
In the case of cln1
Δ, all the possible helpers for start
(except for Cln3) are eliminated, and the cell arrests in G1 (Schwob and Nasmyth, 1993
). In the model, this mutant can be rescued only be restoring CLN2
. Deletion of Sic1 (or CKI altogether) is not enough for rescue, because Cdh1 is active, and it keeps the cell arrested in G1. Deletion of Cdh1 (or addition of GAL-CLB2
) allows DNA synthesis to initiate and exit of mitosis to occur, but these quintuple mutant cells are predicted to be inviable because they do not form buds (See Figure S1 in Online Supplement A).
Exit from mitosis is initiated by Cdc20, and, as expected, cdc20
Δ is lethal (Lim et al., 1998
; Shirayama et al., 1998
), with cells arrested in metaphase. Cdc20 plays a role in the degradation of Clb2, Clb5, Pds1 (Yamamoto et al., 1996a
; Shirayama et al., 1999
; Yeong et al., 2000
) and our hypothetical PPX. The double mutants cdc20
Δ and cdc20
Δ arrest in telophase and metaphase, respectively (), but the triple mutant cdc20
Δ is viable (Shirayama et al., 1999
What is the helper for the finish
transition of the triple mutant cdc20
Δ? Shirayama et al.
) have shown that APC/Cdh1 activity is essential for the viability of this mutant. In the model, the negative feedback loop responsible for exiting mitosis is the following. Clb2 activates chromosome alignment on the mitotic spindle (SPN). When SPN = 1, Bub2 is inhibited and Cdc15 is activated, which in turn activates Cdc14 and Cdh1, causing Clb2 level to decrease. The CKIs are restored, and the cell returns to G1. This interpretation predicts that bub2
Δ would render cdc20
Δ inviable: because the negative feedback is broken, the quadruple mutant would arrest in the G1 phase.
How Can Mutant Cells Lacking G1-Stabilizers Be Rescued?
Removing G1-stabilizers (Cdh1 and/or CKIs) causes problems at start and finish.
Δ and sic1
Δ are both viable, they show synthetic lethality with viable mutations that have higher than normal Clb2-kinase activity. For example, both cdh1
Δ and sic1
Δ show synthetic lethality with cdc14ts
at 29°C (Yuste-Rojas and Cross, 2000
Deletion of all three G1-stabilizers (sic1
” for short) abolishes the stable G1 steady state. cdc6
encodes a truncated Cdc6 protein that has lost its role as a CKI but retains its role as a DNA licensing factor. In simulations, the triple-antagonist
arrests in telophase, with intermediate activity of Clb2-kinase (). However, in reality, although these cells are inviable (Cross, 2003
), they are not arrested in telophase (Archambault et al., 2003
). In the latter report, the authors showed that these cells have an unusual phenotype, they are able to undergo DNA synthesis and nuclear division but not cell division, and arrest as a “four-cell body object” with 4C DNA content. (We will describe in more detail below the problems of our model with this mutant.)
Although the model is not able to describe the phenotype of the triple-antagonist
correctly, it is able to simulate the rescue of this mutant by overproduction of Cdc20 (with GALL-CDC20
) as reported by Cross (2003
) (their Supplementary Figure 4). In this case, Cdc20 serves as the G1-stabilizer. Because it is synthesized constitutively, Cdc20 does not disappear from these cells as Clb2 is degraded during finish
(). SBF and MBF turn on as usual, but Clb5 does not accumulate because it continues to be degraded by Cdc20. The cell is delayed in G1 until Clb5 kinase eventually triggers DNA synthesis. At that point, Cdc20 is inactivated by Mad2, allowing Clb2 to rise and drive the cell into mitosis. When chromosomes are aligned, [SPN] = 1, the inhibition on Cdc20 is removed, and Clb2 is degraded, returning the cell to G1. This interpretation predicts that mad2
Δ would render sic1
The Model Predicts Phenotypes of Novel Mutants
As we have already pointed out with regard to (cln1Δ cln2Δ clb5Δ clb6Δ cdh1Δ), (cdc20Δ pds1Δ clb5Δ bub2Δ), and (sic1Δ cdc6Δ2-49 cdh1Δ GALL-CDC20 mad2Δ), we can use the model to predict phenotypes of mutants that have not yet been investigated experimentally, to test crucial features of the model.
For example, we assume that the Cdc6 is a much weaker inhibitor of Clb5,6 kinases than is Sic1 and that Clb5,6 kinases are not able to phosphorylate and destabilize Cdc6 as efficiently as Clb1,2 kinases. The first assumption that Cdc6 is a weak inhibitor of Clb5 kinase is based on the following observations. 1) sic1
Δ cells have very short G1 period (Schneider et al., 1996
), initiating DNA synthesis before SBF/MBF activation and budding. For DNA replication to occur early in those mutant cells, the high concentration of Cdc6 in G1 must not be able to inhibit even the low level of Clb5,6 kinases generated by MBF-independent transcription. Hence, Cdc6 cannot be a strong inhibitor of Clb5,6. 2) GAL-CLB5
cells do not show advancement in DNA synthesis (Schwob et al., 1994
), indicating that Clb5/Cdc28 must be effectively inhibited by Sic1. This assumption was made independently of experiments published recently by Archambault et al.
), who showed that Sic1 coimmunoprecipitates with Clb5 but Cdc6 does not.
The second assumption, that Clb5 kinase is less able to phosphorylate and inactivate Cdc6, is inferred from the large size of cln1Δ cln2Δ cln3Δ sic1Δ cells. As described in 1), Clb5,6 kinases are able to initiate DNA synthesis early and to inactivate Cdh1 in the quadruple mutant, but Clb2 is still inhibited by Cdc6. The cell has to grow to very large size to accumulate enough Clb5 to inactivate Cdc6. Only then can Clb2 kinase activity rise, driving the cell into mitosis.
If it is true that Cdc6 is a weaker inhibitor to Clb5 kinase than is Sic1, then whenever the activity of Clb5 kinase is important, sic1
Δ will have very different effects than cdc6
. For example, triple-cln sic1
Δ is viable (Tyers, 1996
), but the model predicts that triple-cln cdc6
will remain arrested in G1. Similarly, the model predicts (, row i) that the inviability of GAL-CLB5-db
Δ (Jacobson et al., 2000
) can be rescued by overproducing Sic1 but not Cdc6. On the other hand, because Sic1 and Cdc6 are both strong inhibitors of Clb2, the inviability of Clb2-db
Δ (Wasch and Cross, 2002
) should be rescued by overproduction of either Sic1 or Cdc6 (, row ii), as confirmed recently by Archambault et al.
Viability of the triple mutant cdc20
Δ (Shirayama et al., 1999
) depends on a feedback loop that activates Cdh1 at the end of the cycle. If we perturb elements in this loop (, row iii), we are likely to render cdc20
Δ inviable. On the other hand, the viability of cdc20
Δ does not depend on CKIs.
As shown in , Cdc20 helps the finish
transition by degrading Clb kinases and by activating the CKIs and Cdh1 (through Cdc14). Hence, in row iv (), the model predicts that inviability of the double mutant cdc20
Δ (telophase arrest, with Cdc14 released from the nucleolus; Shirayama et al., 1999
) could be rescued by an extra copy of genomic CDC15
. The double mutant also can be rescued by TAB6-1
(a dominant Cdc14 mutation where Cdc14 binding to NET1 is reduced; Shou et al., 2001
). In TAB6-1
cells, less Cdc14 is sequestered in the nucleolus, making it easier for the triple mutant (cdc20
) to exit from mitosis even in the presence of Clb5.
Row v () shows when Cdc20 is crippled, as in APC-A mutants, Cdh1 is more important than the CKIs in getting out of M phase. Row vi () suggests that ability of Cdc20 overexpression to rescue the lethality of the triple-antagonist critically depends on the checkpoint mechanism. Row vii () indicates that net1ts can retain viability in nocodazole if Clbs are overexpressed but not if CKIs are deleted.
The Model Predicts Effective Values of Rate Constants
makes ~100 predictions about the rates of component biochemical processes involved in cell cycle control, e.g., synthesis and degradation rates, phosphorylation and dephosphorylation rates, relative activities of cyclins with overlapping substrate specificities. Some of these rates are well established experimentally (e.g., protein half-lives are easily measured), whereas most others were completely unknown. As biochemists seek to measure these rate constants and binding constants directly, our estimates will serve as landmarks for relating in vitro measurements to in vivo activities. Some of these predictions are described in detail in Online Supplement A, along with a few supporting observations.
Inconsistencies between Simulations and Observations Identify Problems in the Model
Although the model gives a quantitatively accurate representation of many aspects of the budding yeast cell cycle, it fails to account for certain properties of 11 mutants in our test series (). Although these failures () may reflect faulty parameter settings () or mistranslations of the mechanism into equations (), careful consideration of the inconsistencies (see Online Supplement B) indicates that there are problems in the wiring diagram itself (). Identifying these problems suggests ways to improve the model in the future.
nconsistencies between simulations and observations
The most troublesome mutants for the model are the triple mutant cdh1Δ sic1Δ cdc6Δ2-49 and the double mutant swi5Δ cdh1Δ strains, which have very similar phenotypes ().
, where all three G1-stabilizers are deleted) is inviable (Archambault et al., 2003
), but the cells are not telophase arrested. When a triple-antagonist GAL-SIC1
(integrated) strain is transferred from galactose to glucose medium, the mutant cells reproducibly accumulate as four-cell body objects that are resistant to sonication. The cells are able to undergo DNA synthesis and nuclear division but not cell division, and they arrest as binucleate cells with 4C DNA content. In our simulation, this mutant would be unbudded, arrested in telophase with 2C DNA content.
We do not know how to interpret the phenotype of the triple-antagonist and how to model it properly. There are three problems. First, how can the mutant exit from mitosis? What drives Clb2 activity below the threshold for nuclear division? Degradation by Cdc20 is not enough, because cdc14Δ (which would be equivalent to the triple-antagonist) arrests in telophase. The observed phenotype of the triple-antagonist suggests that our requirement for nuclear division is incorrect and that Cdc14 must have additional roles besides activating CKI and Cdh1.
Perhaps it is the rise in Cdc14 phosphatase activity, rather than (or in addition to) the fall of Clb2 kinase activity, that triggers nuclear division (see Online Supplement C). Suppose that nuclear division is triggered when the phosphorylation state of a target protein drops below a critical value. If the target protein is phosphorylated by Clb2 kinase and dephosphorylated by Cdc14 phosphatase, then the triple-antagonist cells, which have high Cdc14 activity, could exit from mitosis. However, there are other problems.
Even if they could execute nuclear division, they would have trouble forming a bud in the next cell cycle. The persistent high Clb2 kinase activity after nuclear division would prevent SBF activation and Cln2 accumulation, hence in simulation [BUD]max = 0.25 (it never reaches 1). However, triple-antagonist cells do form buds. It may be that Clb2 kinases are able to trigger bud formation, albeit at very low efficiencies compared with Cln2 kinases.
A third problem for the triple-antagonist cells is that high Clb2 kinase activity after the first nuclear division would presumably prevent reactivation of the licensing factors. Surprisingly, triple-antagonist cells are able to replicate their DNA but block in G2/M for some unknown reason. Maybe high Clb activity is not able to block origin relicensing totally, allowing these mutants to enter S phase from a few licensed origins. In this case, S phase may not be properly completed, causing cells to arrest in a G2/M state with seemingly replicated DNA.
In Online Supplement C, we simulate the model with a target protein (as described above). Ignoring the bud problem, we can find a basal parameter set that describes the phenotypes of triple-antagonist cells and the double mutant swi5Δ cdh1Δ (except their deficiency in cytokinesis). The same basal parameter set can explain the phenotype of cdh1Δ sic1Δ cells as well, without causing additional problems for the other mutants in .
A drawback of this “target hypothesis” is its lack of robustness; it is very sensitive to small changes of certain parameter values. However, as described in Online Supplement C, if proper checkpoint mechanisms are added, then the extended model is expected to be suitably robust.
Is the Model Robust?
The eukaryotic cell cycle engine must satisfy two basic conditions. 1) It must alternate between phases of DNA replication and sister chromatid segregation (Botchan, 1996
). 2) It must coordinate the DNA replication-segregation cycle with overall cell growth, so that the cycle times of mother and daughter cells (P
) and the mass doubling time of the culture (MDT = ln2/kg
) satisfy the relationship e-kgD
= 1 (Hartwell and Unger, 1977
; Fantes and Nurse, 1981
). Furthermore, it must be robust in the sense that these basic requirements are satisfied over a broad range of parameter values, so that cell viability is not delicately dependent on enzymatic activities. For a budding yeast cell, we have an operational definition of exactly how robust is its cell cycle engine, because the large set of phenotypically characterized mutants probes the parameter space of the control system. The viability/inviability of these mutants tells us exactly how large the region of robust control is in the budding yeast parameter space. Because the model is consistent with most mutant phenotypes, we can say that it is neither more nor less robust than warranted by observations.
We also have tested the robustness of the model systematically, as in Cross (2003
), by changing the parameters in the basal (wild-type) parameter set one at a time to determine the range over which cyclic behavior persists (see MATERIALS AND METHODS for the requirements of viability in simulations). We find that 72% of the parameters can be changed at least 10-fold in either direction (). In Online Supplement D and Table S2, those 35 parameters that do not have this flexibility are described. They identify fragile parts of the model. It remains to be determined whether these fragilities are all real features of the control system. Some may be artifacts of the model that need to be corrected.
Figure 5. Robustness assay of the wild-type and six mutant strains, cdh1Δ, ckiΔ, APC-A, cln2Δ, cln3Δ, and clb5Δ. For each strain, simulations were run with systematic variations in each parameter to determine the maximum (more ...)
For example, measurements of Ghaemmaghami et al.
) and Huh et al.
) suggest that [Net1]T
= 0.2, which lies outside the range required by the model, 1 < [Net1]T
< 2.8. Perhaps the model is incorrect in assuming 1:1 stoichiometric inhibition of Cdc14 by Net1. Furthermore, the model does not include the fact that Cdc14 and Cdc5 are involved in a homeostatic negative feedback loop: Cdc5 activates Cdc14, which activates Cdh1, which in turn degrades Cdc5 (Charles et al., 1998
; Shou et al., 1999
; Pereira et al., 2002
; Stegmeier et al., 2002
; Visintin et al., 2003
). This loop may buffer fluctuations in Cdc14 amount, making the model more robust to [Cdc14]T
. And the requirement of Cdc14 phosphorylation by Cdc5 for its release from the nucleolus and activation, as reported recently by Visintin et al.
), may help to explain how cells can have higher [Cdc14]T
The Phenotypes of Some Mutants Are Not as Robust as Wild Type
One would expect that most mutants will be more sensitive to parameter changes than wild-type cells. also shows robustness analyses of six mutants: cdh1Δ, ckiΔ (= sic1Δ cdc6Δ2-49), APC-A, cln2Δ, clb5Δ, and cln3Δ. The first five mutants are all less robust than wild type.
For example, in cdh1Δ, because SBF is turned off earlier due to higher Clb2 level, less Cln2 is made, and the cell is compromised in its ability to bud. Changes in other parameters that further decrease Cln2 activity causes the cdh1Δ mutant to fail our viability criteria by failing to bud.
The other example is the APC-A
mutant (Cross, 2003
). Because this mutant is compromised in its degradation of Clb2 by Cdc20/APC-P and depends on Cdh1 activity for viability (APC-A cdh1
Δ telophase arrest), parameter changes that further decrease Cdh1 activity tend to make APC-A
On the other hand, the cki
Δ mutant (or the single sic1
Δ mutant) seems to be very robust to parameter changes in our analysis (less robust than wild type, but more than cdh1
Δ or APC-A
). This may not be true in reality. Because we have not taken into account the fact that sic
Δ mutant cells have poor viability and show increased genome instability (Nugroho and Mendenhall, 1994
; Lengronne and Schwob, 2002
), the physiology of the mutant is not properly described by the model.