In Figure , we propose a realistic mechanism for regulating the cell division cycle in budding yeast. Its components are Cln1 and 2 (lumped together), Cln3 and Bck2, Clb1 and 2 (lumped), Clb5 and 6 (lumped), Sic1, Hct1 (=Cdh1), and Cdc20. (Cdc28, the kinase subunit that combines with the cyclins, is present in excess, so we need not keep track of its fluctuations.) In addition, the model tracks the relative activities of three transcription factors, Swi4/Swi6 (=SBF), Mcm1/SFF, and Swi5, which determine the rates of synthesis of Cln2, Clb2, and Sic1, respectively. At present, we assume that MBF, the transcription factor for Clb5, is regulated coordinately with SBF. In the model, overall cell growth is exponential, and the basic events of the yeast division cycle (DNA synthesis, budding, and spindle assembly) are driven by the integrated activities of cyclin-dependent kinases. These assumptions lead to a mathematical model (Table ) consisting of 10 nonlinear, ordinary differential equations (for mass, the cyclins, and their consorting proteins), three algebraic functions for transcription factors, three “integrators” to trigger DNA synthesis, budding, and spindle assembly, and a simple rule for separating mother and daughter cells at division.
The kinetic model introduces ~50 parameters (rate constants, binding constants, thresholds, relative efficiencies, etc.) that need to be determined by fitting specific experimental observations. For the present, we do this by trial and error (Appendix A), so we can only claim that our model equations and parameter set are sufficient to account for many properties of cell cycle control in budding yeast. Because we fit the model to the properties of dozens of different genotypes, we have enough data to fix the parameters and to provide meaningful confirmation of the mechanism in Figure .
Table is in no sense an optimal parameter set, nor can we quantify how robust is the system, although our experience suggests that the model is quite hardy. Currently we are working on computational methods of parameter optimization and sensitivity analysis and hope to address these problems in a later publication.
Bistability and Hysteresis
The crucial idea behind our model of the budding yeast cell cycle is Nasmyth's (1996)
hypothesis that G1 and S/M are alternative, self-maintaining states, generated by mutual antagonism between Clb-dependent kinases and their opponents, Sic1 and Hct1. In theoretical terms, the molecular regulatory system exhibits bistability and hysteresis (Figure ). In its “neutral” condition (no Cln2 or Cdc20), the control system can persist in either the stable G1 state or the stable S/M state. Transitions between these alternative steady states can be driven by changes in Cln2 and Cdc20 that push the control system past the “fold” points in Figure (Novak et al., 1998
Figure 9 Bistability and hysteresis (schematic). Steady-state level of total Clb-dependent kinase activity depends on the expression of CLN2 and CDC20. When [Cln2] is large and [Cdc20] is small, the Clb1–6 regulatory system (more ...)
At Start, Cln2-dependent kinase activity rises abruptly and pushes the cell from G1 to S/M by inactivating Hct1 and promoting Sic1 degradation (Figure , stage a). The Clns can drive this transition because they are neither degraded by Hct1 nor inhibited by Sic1. After Clb2 appears, Cln2 is removed, but the cell remains in S/M because the Clbs can now keep Hct1 and Sic1 in abeyance without further help from Clns (stage b). This effect, called hysteresis, makes the Start transition irreversible.
Cdc20, activated at metaphase, pushes the cell from S/M to G1 (Finish) by activating Hct1 and promoting Sic1 accumulation (Figure , stage c). Cdc20 can drive this transition, because it is not opposed by Clb-dependent kinase activities; indeed, Clb2 promotes Cdc20 accumulation and activation. As Cdc20 is destroyed in G1 (stage d), the control system does not flip back to the S/M state, because Hct1 and Sic1 can now keep the Clbs in abeyance without further help from Cdc20. The S-shaped curve in Figure accounts for the characteristic irreversibility of entry into S phase and exit from mitosis.
In this picture, not merely may the “pushers” be removed and the control system will not revert, but they must be removed to make a repeated sequence of properly regulated Start and Finish transitions. For instance, once Start is accomplished, Cln2 must disappear; otherwise it will work against the Finish transition. Higher concentrations of Cdc20 will be required to trigger Finish. Furthermore, after the cell leaves mitosis, as Cdc20 disappears, Sic1 and Hct1 will not be able to hold the cell in G1. For these reasons, although GAL-CLN2 mutants are viable, they have short G1 and long S/M periods.
Similarly, removal of Cdc20 after Finish is crucial for the next Start transition. Because the phosphatase (Cdc14) activated by Cdc20 can overwhelm all CDK activity at metaphase and thereby induce Finish, then it will be difficult to induce the next Start if Cdc20 activity does not disappear in G1 phase so that this phosphatase can be inactivated. In this regard, notice that mild overproduction of Cdc20 (GAL-CDC20
in 0.2% galactose) induces prolongation of G1 (Prinz et al., 1998
), and strong overproduction (3X GAL-CDC20
in 2% galactose) induces G1 arrest (Shirayama et al., 1998
also induces G1 arrest (Visintin et al., 1998
Direct experimental confirmation of bistability can be sought by holding the control system in neutral (Figure , position A/B) and then driving it between G1 and S/M by ectopic expression of Clb5 and Sic1 (Figure , vertical arrows). This experiment has been done in part by Dahmann et al. (1995)
. After arresting cells in mitosis with nocodazole, they induced transition to G1 (without nuclear or cell division) by ectopic expression of Sic1. When ectopic synthesis of Sic1 was repressed, their cells executed a second round of DNA synthesis, because endogenous production of Clns drove the cells through Start. To prevent autonomous reentry into S phase, we suggest that cells be blocked with α-factor as well as nocodazole.
We propose that a synchronous culture of MET-CLB5 TET-SIC1 cells (where MET = methionine-repressible promoter and TET = tetracyline-inducible promoter), about to execute Start and bud, be transferred from “growth” medium (containing methionine) to “arrest” medium (containing methionine, α-factor, and nocodazole). (Notice that the use of α-factor and nocodazole to arrest cells in neutral could be replaced by cln1–3Δ and clb1–4Δ, respectively.) Those cells that have not yet executed Start when the medium is changed will be kept in G1 phase by α factor (moving from a to G1 in Figure ), whereas those cells that have already executed Start will be arrested in M phase by nocodazole (moving from b to S/M in Figure ). The culture is now a mixed population of G1- and S/M-arrested cells, suggesting that, in this neutral position, there coexist two stable steady states of Clb activity. To prove the coexistence of these states, divide the culture into two batches. One batch is subjected to transient Clb5 synthesis by transferring the cells briefly to “Clb5” medium (α-factor + nocodazole) and then back to arrest medium. All cells in this batch are expected to arrest in the S/M state (in Figure , cells initially at G1 will be driven to S/M, whereas those initially at S/M will return there). Cells of the other batch, after brief exposure to “Sic1” medium (methionine + tetracycline + α-factor + nocodazole), are expected to arrest uniformly in G1 phase. Furthermore, the duration of the “brief” exposure is important: there should be threshold levels of exposure to Clb5 and Sic1 below which the transitions are not accomplished (see Figure ).
Note that, at the end of treatment, all cells are of uniform size and are exposed to arrest medium. Nonetheless, if our model is correct, individual cells will be in different phases of the division cycle, depending on how they were perturbed. Those cells initially in G1 will be pushed into S/M by a Clb5 perturbation but not by a Sic1 perturbation (Schwob and Nasmyth, 1993
) and vice versa for those cells initially in S/M (Dahmann et al., 1995
). This behavior would indicate that two stable states of Clb activity coexist (bistability) when the regulatory system is in neutral. By alternating treatment with Clb5 and Sic1, one should be able to induce multiple rounds of endoreplication in cln1–3
Reversibility of the SBF Switch
In contrast to the irreversibility of the Start and Finish transitions, the activation of SBF in our model is a reversible, ultrasensitive switch. To test this feature of the model, one could modify slightly the experimental design of Dirick et al. (1995)
. The strain cln1
Δ cln3ts MET-CLN2
is grown in the absence of methionine, so that newborn daughter cells are small. Small cells, transferred to methionine-containing medium at permissive temperature, will activate SBF (measured by expression of PCL1
mRNA, say) at wild-type size, but Sic1 degradation and Hct1 activation will be delayed to a much larger size. If, after SBF activation, the cells are transferred to restrictive temperature, then SBF should inactivate (i.e., this event is reversible), and the cells should remain in G1.
Autonomously Oscillating Versus Checkpoint-controlled Cell Cycles
Cell division cycles of budding yeast (and somatic cells in general) are blocked by drugs that inhibit DNA replication or spindle assembly (“checkpoint controls”), whereas early embryonic cell divisions are unrestrained by these same drugs (Murray and Hunt, 1993
). If somatic and embryonic cells use the same cell cycle control machinery, why do they behave so differently?
In theoretical terms, checkpoints correspond to stable steady states (G1 and S/M in the current model), and drugs that inhibit growth, DNA synthesis, or spindle assembly abort the signals that normally push the cell from one checkpoint to the next. The existence of these checkpoints depends on the mutual antagonism between cyclin B-dependent kinases and their opponents (Sic1 and Hct1 homologues).
In early Xenopus
embryos, there are no effective antagonists of cyclin B/Cdc2 kinase (also called MPF). 1) The only identified MPF inhibitors are p28Kix1
(Shou and Dunphy, 1996
) and p27Xic1
(Su et al., 1995
), but both are present at very low levels until the late gastrula stage. Furthermore, neither one inhibits MPF in vitro. 2) XFZR
homologue of HCT1
) is not translated before the midblastula transition. Instead, X-FZY (the Xenopus
homologue of Cdc20) is responsible for cyclin B degradation during early embryonic cell cycles (Lorca et al., 1998
). X-FZY, like Cdc20, seems to be activated rather than inhibited by MPF (Felix et al., 1990
). 3) Even the antagonistic relationship between MPF and Wee1 (a tyrosine kinase that inhibits Cdc2) seems to be ineffective, because Cdc2 shows very little tyrosine 15 phosphorylation during the early cycles of intact embryos (Ferrell et al., 1991
Without effective antagonists in early embryonic cells, MPF cannot establish the alternative steady states characteristic of checkpoint controls. The only remaining control is a time-delayed negative feedback loop, whereby MPF activates X-FZY, which degrades cyclin B and thereby destroys MPF activity. The sufficiency of this mechanism to generate autonomous oscillations in MPF activity was shown first by Goldbeter (1991)
and later by Novak and Tyson (1993)
. Because the early embryo lacks cyclin B antagonists, it supports rapid MPF oscillations that are insensitive to errors in DNA replication and spindle assembly; apparently the early embryo has sacrificed accuracy for speed. However, in frog egg extracts, checkpoint control can be elicited if a sufficient amount of unreplicated sperm DNA is added (Dasso and Newport, 1990
; Smythe and Newport, 1992
), which creates an effective antagonist by activating Wee1. If our hypothesis is right, one may be able to elicit checkpoint responses in Xenopus
embryos before the midblastula transition by injecting XFZR
mRNA (or protein) into the fertilized egg.
Models as Tools in Molecular Biology
Undoubtedly the genetic regulatory system of cell division in yeast and higher eukaryotes is even more complex than Figure . To understand regulatory systems of such complexity, we need analytical tools that can handle realistic biochemical control mechanisms. Our work confirms that modern methods of kinetic theory and computation are capable of connecting a realistic, multilayered, regulatory mechanism to the complex physiological behavior of cells.
In addition to its role in synthesizing molecular and physiological details about cell division, the model is a predictive tool. The rate constant estimates in Table can be tested by more direct kinetic measurements. Tables – specify many quantitative properties of mutant cells that have never been reported, and they predict phenotypes of several mutants yet to be examined.
One can learn as much from the failures of the model as from its successes. Where there are inconsistencies between the model and experiment, we are prompted, first of all, to look for a better parameter set. If that fails, we consider slight changes in the mechanism, which might bring the model in accord with observations. If that fails, and if the experimental community is convinced that the observations are reliable and significant, then we have identified an area that deserves closer scrutiny to resolve the discrepancies. If the mechanism proves insufficient, that does not invalidate our approach. Mathematical modeling, as a tool, is no more “falsifiable” than gel electrophoresis. The tool tells us what a mechanism can and cannot explain. When the model fails, the fault lies with the mechanism, not the tool.
The molecular mechanism of cell cycle control in budding yeast is an evolving hypothesis that must be continually examined, revised, and improved as new observations tell us more about the control system. We intend to extend the model in several directions. First, we will provide a more detailed description of Finish (Novak et al., 1999
), including roles for Cdc14 (Visintin et al., 1998
; Jaspersen et al., 1999
), RENT complexes (Shou et al., 1999
), and Pds1/Esp1 interactions (Ciosk et al., 1998
; Cohen-Fix and Koshland, 1999
; Tinker-Kulberg and Morgan, 1999
). The next step will be to connect mathematical representations of surveillance mechanisms to the underlying cell cycle engine. For instance, the mating factor pathway connects pheromone binding at the cell surface, through a protein kinase cascade, to the inhibition of Cln kinases, which arrests cells before Start (Wittenberg and Reed, 1996
; Posas et al., 1998
). Another important signal transduction pathway, through mad
gene products, arrests cells in mitosis, if the mitotic spindle is improperly assembled (Alexandru et al., 1999
; Taylor, 1999