The analysis of the detailed model for the Cdk network driving the mammalian cell cycle indicates that when growth factors (GF) exceed a critical value, repetitive activation of the cyclin/Cdk complexes occurs in the form of self-sustained oscillations (Gérard and Goldbeter, 2009). The ordered activation of the cyclin/Cdk complexes corresponds to the passage through the successive phases of the cell cycle. Indeed, cyclin D/Cdk4–6 and cyclin E/Cdk2 allow progression in G1 and elicit the G1/S transition, cyclin A/Cdk2 ensures progression in S and G2, while cyclin B/Cdk1 brings about the G2/M transition. Sustained Cdk oscillations in the course of time (Figure 2A) correspond, in the concentration space, to the evolution toward a limit cycle; this closed trajectory is unique and can be reached regardless of initial conditions (Figure 2B). In the detailed model for the mammalian cell cycle, the kinase Wee1 can inhibit Cdk1 and to a lesser extent Cdk2. Here, the initial linear increase in cyclin B/Cdk1 is due to the inactivation of Cdh1, resulting in the accumulation of cyclin B; while the subsequent abrupt peak of cyclin B/Cdk1 is due to the inactivation of Wee1 (see red curve in Figure 2A). The model indicates that the period of the cell cycle as well as the duration of the G2 phase increase with the level of Wee1. The model also predicts that high levels of Wee1 could produce endoreplication cycles, in which large-amplitude oscillations of Cdk2 are not accompanied by large-amplitude variations in Cdk1 (see section: “Endoreplication and tetraploidy”), while a further increase in the level of Wee1 generates an arrest of the cell cycle (results not shown).

The dynamic behavior of the Cdk network is controlled by a fine-tuned balance between factors that promote (oncogenes) or impede (tumor suppressors) progression in the cell cycle (Hanahan and Weinberg, 2000; Harbour and Dean, 2000; Chau and Wang, 2003). Here, we illustrate the antagonistic effects between two such factors, the transcription factor E2F, which promotes, and the tumor suppressor pRB, which prevents progression in the cell cycle. The dynamical behavior of the cell cycle is represented in Figure 3 as a function of the rate of synthesis of E2F, v_se2f, and the rate of synthesis of pRB, v_spRB. The Cdk network either evolves to a stable steady state, corresponding to cell cycle arrest and cellular quiescence, or to sustained oscillations, which correspond to cell proliferation.

The domain of sustained oscillations is divided into two sub-domains: oscillations do or do not depend on the presence of growth factor. Thus, at a given level of E2F, oscillations can occur spontaneously in the Cdk network, in the absence of GF, at sufficiently low levels of pRB, but they require the presence of suprathreshold amounts of GF when the level of pRB increases. Similarly, at a given level of pRB, Cdk oscillations cease to require the presence of GF when the rate of E2F synthesis is sufficiently large. The model therefore allows us to distinguish between two types of oscillatory behavior in the Cdk network: cell proliferation that depends on GF might correspond to normal, healthy cell proliferation, while cell proliferation independent of GF might define a “cancer-like” cell (Hanahan and Weinberg, 2000).

The detailed model for the Cdk network (Gérard and Goldbeter, 2009) is particularly useful for analyzing different ways to induce the transition between quiescence and proliferation. Such transition may result from overexpression or deletion of key proteins involved in the control of the mammalian cell cycle. Understanding the mechanisms responsible for the transition from quiescence to proliferation is of paramount importance, given that deregulation of the cell cycle as a result of overexpression of oncogenes (which activate the cell cycle) or deletion of tumor suppressors (which inhibit the cell cycle) might lead to cancer (Hanahan and Weinberg, 2000).

Different ways to induce the transition from quiescence to proliferation are illustrated in Figure 4. In the detailed model for the Cdk network this transition is associated with the switch from a stable steady state to sustained oscillations around an unstable steady state. Thus, the model shows in (A) that the transition from a stable steady state, corresponding to quiescence, to a sustained oscillatory regime of the Cdk network, corresponding to cell proliferation, can be triggered by overexpression (starting in t = 120 h) of the protein AP1 involved in the synthesis of cyclin D that follows growth factor signaling. A similar transition can follow from overexpression of the transcription factor E2F involved in the synthesis of cyclin proteins (B), overexpression of Skp2, involved in the degradation of cyclin E (C), and from an increase in the level of the phosphatase Cdc25 involved in the activation of cyclin A/Cdk2 (D). Similarly, the model shows that a decrease in the tumor suppressor Cdh1 involved in the degradation of cyclin B (E), or a deletion of p53 (F), a factor that promotes the synthesis of the Cdk inhibitor p21, also elicit the transition from quiescence to proliferation.

The model therefore suggests that abusive transitions from quiescence to proliferation may have multiple causes converging to the same effect. The switch from steady state to sustained oscillations in the Cdk network might be responsible for the tumorigenic effect of an overexpression of oncogenes such as AP1 (Smith et al., 1999) or the phosphatase Cdc25 (Parsons, 1998; Ray and Kiyokawa, 2008), of a rise in the levels of the transcription factor E2F (Gala et al., 2001) or the protein Skp2 (Yokoi et al., 2004), or a deletion of tumor suppressors such as the protein Cdh1 (Garcia-Higuera et al., 2008) or p53; the decrease in p53 is associated with a drop in the level of p21 (Nigro et al., 1989).

In mammals and, more generally, in most pluricellular organisms, cells divide and proliferate in a certain undifferentiated state. When appropriate conditions are met, cells initiate a program of cell differentiation (Quaroni et al., 2000; Buttitta et al., 2007), which involves regulatory mechanisms that are not fully understood yet. However, a first step on the way to cell differentiation is to stop cell proliferation. This arrest is often achieved through overexpression of the Cdk inhibitor p21/p27 (Evers, 1999; Quaroni et al., 2000; Ilyin et al., 2003; Hara et al., 2011).

The detailed model for the cell cycle incorporates p21 and allows us to test the effect of its increase on the dynamics of the Cdk network. Overexpression of p21/p27 (Figure 5A) or of pRB (Figure 5B) can block the progression in the cell cycle, which stops in a state characterized by a high level of cyclin D/Cdk4–6 and a low level of the other cyclin/Cdk complexes, which corresponds to the G1 phase of the cell cycle. Such an arrest of the cell cycle might represent a first step in the process of cell differentiation. Further studies are needed to clarify the coupling between cell proliferation and cell differentiation (Hara et al., 2011).

The question arises as to why the system reaches a steady state characterized by higher levels of cyclin D/Cdk4–6 and lower levels of Cdk2 and Cdk1 when p21 or pRB are overexpressed. Cyclin D/Cdk4–6 in the time series represents the total active form of the kinase Cdk4–6, which is the sum of free cyclin D/Cdk4–6 and the complex between cyclin D/Cdk4–6 and p21/p27. The Cdk inhibitor p21/p27 does not inhibit the activity of cyclin D/Cdk4–6, but inhibits the activity of Cdk1 and Cdk2. Because we assume that cyclin D/Cdk4–6 is degraded only in the free state, it is protected by its binding to p21/p27. This explains why the total level of active cyclin D/Cdk4–6 increases when p21/p27 is overexpressed, while the level of Cdk2 and Cdk1 is low (see Figure 5A). On the other hand, pRB inhibits the synthesis of cyclin D/Cdk4–6, cyclin E/Cdk2 and cyclin A/Cdk2. Thus, it is counterintuitive to reach in Figure 5B a stable steady state with a high level of cyclin D/Cdk4–6 relative to Cdk2 and Cdk1 when pRB is overexpressed. While the synthesis of cyclin E/Cdk2 and cyclin A/Cdk2 is only regulated by E2F and pRB, the synthesis of cyclin D/Cdk4–6 is also regulated by GF. Two terms are present in the synthesis of cyclin D/Cdk4–6: the first depends on GF and the second depends on pRB and E2F. In our model, the rate of synthesis that depends on GF is ten times higher than the rate of synthesis that depends on pRB and E2F. Thus, overexpression of pRB will not decrease the level of cyclin D/Cdk4–6.

Endoreplication corresponds to uncoupling DNA replication from mitosis: the cell undergoes multiple rounds of DNA replication without entering into mitosis (Edgar and Orr-Weaver, 2001). Such a situation can also be accounted for by the model for the mammalian cell cycle. Indeed, this model predicts that the Cdk network contains at least four oscillatory circuits (see Gérard and Goldbeter, 2009, 2010 and Figure 6), which are all able to generate by themselves sustained oscillations. Two of these oscillatory circuits can produce sustained oscillations of Cdk2, which is responsible for DNA replication, without involving the kinase Cdk1 that is responsible for the occurrence of mitosis. The two other circuits involve the kinases Cdk2 and Cdk1 whose regulatory properties are at the core of the “classical” cell cycle in which each round of DNA replication is followed by entry into mitosis.

The model shows that, depending on their degree of interconnection, these oscillatory circuits can produce simple periodic oscillations of the Cdk network that correspond to the mitotic cell cycle, to tetraploidy, or to endoreplication. One peak of Cdk2 is generally followed by one peak of Cdk1 per cycle, which case corresponds to the progression through the “classical” mitotic cell cycle (Figure 7A). Starting from this situation, a decrease in the rate of inhibition of the protein Cdh1 that is responsible for cyclin B degradation promotes the occurrence of endoreplication cycles: sustained oscillations of Cdk2 occur without significant oscillations of Cdk1 (Figure 7B). Reducing Cdh1 inhibition leads to a decrease in the level of cyclin B; such a decrease, with similar effects, can be obtained more directly by decreasing the rate of cyclin B synthesis. Thus, from the conditions yielding the mitotic cycle in Figure 7A, a first decrease in the rate of synthesis of cyclin B, v_sb, can elicit tetraploidy (Figure 7C), where two peaks of Cdk2 are present for one peak of Cdk1. A further decrease in v_sb generates again endoreplication cycles (Figure 7D). Tetraploidy will occur for the first cycle characterized by two peaks of Cdk2. For the subsequent cycles with two peaks of Cdk2, ploidy may increase if parameter conditions are maintained. Thus, for tetraploidy to occur, we must assume that a temporary change in some parameter value leads to inhibition of the Cdk1 module, and that this module may operate again when the altered parameters recover their original values.

These results suggest that the transition from a “normal” cell cycle to endoreplication or tetraploidy may be promoted in multiple ways, for example through overactivation of Cdh1 (Sorensen et al., 2000) or a decrease in cyclin B synthesis. In all cases, the model shows that the occurrence of endoreplication cycles may be readily achieved by inhibiting the activity of the Cdk1 module that controls the G2/M transition; this prediction is in agreement with experimental observations (Larkins et al., 2001).

The interactions between the different oscillatory circuits present in the Cdk network can also lead to complex dynamical behaviors such as complex periodic oscillations, quasi-periodic oscillations, and chaos (Gérard and Goldbeter, 2010). Numerical simulations based on a thorough, though limited, exploration in parameter space suggest that these complex modes of oscillatory behavior are nevertheless less frequent than the evolution toward simple periodic behavior. This conclusion holds with the view that the cell cycle is driven by simple periodic Cdk oscillations, and that such oscillations correspond to the physiological mode of dynamical behavior of the Cdk network.

Checkpoints ensure that cells progress in the next phase of the cell cycle only if the preceding phase is completed correctly (Hartwell and Weinert, 1989). Checkpoints are particularly crucial in the presence of cellular damage; if DNA damage is too severe, the p53 pathway can induce apoptosis (Levine, 1997). Even during normal cell cycle progression, the correct sequence of events must be highly regulated. To illustrate how checkpoints may affect the oscillatory dynamics of the Cdk network, we incorporated into the model for the Cdk network the endogenous checkpoint that blocks cell cycle progression during DNA replication (Gérard and Goldbeter, 2009). This checkpoint is mediated by the ATR/Chk1 pathway, which inhibits the phosphatases Cdc25 that activate Cdk2 and Cdk1.

At the G1/S transition, cyclin E/Cdk2 activates by phosphorylation the anchor factor Cdc45, which permits the binding of DNA polymerase α to DNA and the initiation of DNA replication (see Figure 8). The kinase ATR is activated upon binding the RNA primer synthesized by DNA polymerase α. ATR phosphorylates, and thereby activates, the kinase Chk1. Once activated, Chk1 inhibits the Cdc25 phosphatases; this inhibition blocks cell cycle progression by preventing the activation of Cdk2 and Cdk1 as long as DNA replication proceeds. Finally the decrease in Cdk2 activity, inherent to the oscillatory dynamics of the Cdk network, inhibits DNA polymerase at the end of the S phase (Dart et al., 2004). The subsequent inhibition of ATR and Chk1 relaxes the inhibition of the phosphatases Cdc25 and thereby permits the rise in the activity of cyclin B/Cdk1 that will elicit the G2/M transition [see Figure 8 and (Gérard and Goldbeter, 2009) for further details about the molecular regulatory mechanisms of the DNA replication checkpoint]. Incorporation of the DNA checkpoint involves the addition of five new variables, so that the extended, detailed model now counts 44 variables instead of 39.

The model shows that the DNA replication checkpoint does not alter qualitatively the oscillatory nature of the dynamics of the Cdk network. However, it slows down the progression in the cell cycle and allows for a better separation between DNA replication and mitosis (Figure 9). Indeed, when the checkpoint is inactivated, we observe an overlap between the peak of activity of DNA polymerase α, which is a marker of the S phase, and the peak of cyclin B/Cdk1, which corresponds to the M phase (Figure 9A). In the presence of mild (Figure 9B) or strong activation (Figure 9C) of the checkpoint, the progression in the cell cycle slows down: the period of the cell cycle passes from 19 h in (A) to 21.6 h in (B) and 31 h in (C). Moreover, the checkpoint allows for better separation between DNA replication and mitosis (compare panel C with panels B and, even more strikingly, A). This ensures that the cell completes DNA replication before entering mitosis.

The dynamics of the Cdk network is governed by multiple positive and negative feedback loops that involve the various cyclin/Cdk complexes. At the end of the network, the last Cdk module that controls the G2/M transition is regulated by a negative feedback loop between cyclin B/Cdk1 and APC/Cdc20 (see Figure 1). The earlier modules that control the progression from G1 to S, G2, and M harbor a cascade of bistable switches in the activation of the various cyclin/Cdk complexes. Indeed, as previously stressed in experimental as well as theoretical studies, PF loops control the dynamics of the G1/S as well as the G2/M transitions of the cell cycle (Hoffmann et al., 1994; Pomerening et al., 2003; Sha et al., 2003; Chen et al., 2004; Yao et al., 2008). PF is either direct, as in the activation by cyclin A/Cdk2 and cyclin B/Cdk1 of their activating Cdc25 phosphatase, or indirect, as in the inhibition by cyclin B/Cdk1 of its inhibitory kinase Wee1.

The dynamics of the Cdk network that governs progression from G1 to S, G2, and M is illustrated in Figure 10 in the detailed model for the Cdk network driving the mammalian cell cycle. First the time evolution of pRB and E2F is illustrated in (A). These antagonistic factors oscillate in antiphase. A high level of pRB and a low level of E2F characterize the G1 phase of the cell cycle. By eliciting the synthesis of the various cyclins, the rise in E2F promotes progression in the G1, S, and G2 phases of the cell cycle.

A bifurcation diagram illustrating the dynamical behavior of cyclin A/Cdk2 as a function of the transcription factor E2F (considered as a parameter) is shown in (B). This second module of the network controls the progression from G1 to S and G2. In early G1, the level of E2F is low, which allows the maintenance of a low level of cyclin A/Cdk2. The rise in E2F during G1 elicits an abrupt increase in the level of cyclin A/Cdk2, as soon as E2F reaches the limit point of the bistable switch. Such an abrupt activation elicits progression in S and G2 and contributes to render the G1/S transition irreversible (B). The bistable behavior in the activation of Cdk2 results from the presence of the PF regulation of cyclin E/Cdk2 (not shown) and cyclin A/Cdk2, via the activation of their phosphatase Cdc25. Later, during G2, the level of E2F will decrease because cyclin A/Cdk2 promotes its degradation by phosphorylation (Gérard and Goldbeter, 2009; see also Figure 1). This regulation provides yet another illustration of the principles of cell cycle control: each module in the Cdk network activates the subsequent modules and inhibits the previous modules in the network.

The abrupt rise in the level of cyclin A/Cdk2 permits the activation of cyclin B/Cdk1 at the G2/M transition: the increase in cyclin A/Cdk2 pushes the Cdk network into a domain of sustained oscillations of Cdk1 (see the bifurcation diagram established in Figure 10C for cyclin B/Cdk1 as a function of cyclin A/Cdk2, considered as a parameter). The occurrence of oscillations results from the negative feedback loop between cyclin B/Cdk1 and Cdc20 (see Figure 1). However, the entry of Cdk1 in the domain of sustained oscillations in (C) is only transient. This is made clear in (D) where the projection of the limit cycle trajectory followed by the full Cdk network in the cyclin A/Cdk2 vs. cyclin B/Cdk1 plane is superimposed on the bifurcation diagram shown in (C). There is only one peak of Cdk1 when the last module enters the oscillatory domain: indeed, soon after this peak, cyclin A/Cdk2 begins to decrease because cyclin B/Cdk1 activates, through phosphorylation, the protein Cdc20 that triggers degradation of both cyclins A and B. As cyclin A/Cdk2 decreases, soon followed by a decrease in Cdk1, the Cdk1 module leaves its domain of sustained oscillations (C and D). This completes the M phase of the cell cycle and leads to a return to G1, characterized by low levels of activity of the various cyclin/Cdk complexes.

The detailed model for the Cdk network thus shows that the presence of multiple PF loops elicit a cascade of abrupt activation of the various cyclin/Cdk complexes, which controls the progression from G1 to S, G2, and M. At the G2/M transition, the cell enters transiently into an oscillatory regime; the pulsatile increase in Cdk1 at the same time triggers the M phase and resets the cycle. The cell division cycle can thus be viewed as a cascade of dominoes, controlled by PF loops, which drives the ordered transition along the successive phases of the cell cycle. The negative feedback loops between Cdk1 and Cdc20 at the end of the network turns the bistable behavior of the different Cdk modules into a global limit cycle oscillator. The reset brought about by the rise in Cdk1 allows the cell to start a new cycle spontaneously if appropriate conditions are met. The mammalian cell cycle thus behaves as a self-organized, oscillating cascade of dominoes. Such a conclusion reconciles two views of the cell cycle, as dominoes and clock (Murray and Kirschner, 1989).

Several molecular components of the cell cycle network are regulated in a circadian manner (Fu et al., 2002; Matsuo et al., 2003; Gréchez-Cassiau et al., 2008). In the Cdk network that governs progression in the cell cycle, the synthesis of the kinase Wee1, which inhibits the G2/M transition, is enhanced by the complex CLOCK/BMAL1 that plays a major role in the circadian clock network (Matsuo et al., 2003). Another component of the circadian network, REV-ERBα, inhibits the synthesis of the Cdk inhibitor p21 (Gréchez-Cassiau et al., 2008). Moreover, the synthesis of the oncogene c-Myc, which promotes G1 cyclin synthesis, is repressed by CLOCK/BMAL1 (Fu et al., 2002). By coupling the detailed model for the cell cycle (Gérard and Goldbeter, 2009) to a model for the circadian clock in mammals (Leloup and Goldbeter, 2003), we investigated the conditions in which the cell cycle can be entrained through these various modes of coupling to the circadian clock (Gérard and Goldbeter, 2012a).

In the presence of coupling to the circadian clock (see Figure 11, for t > 120 h), the model shows that autonomous periods of the cell cycle smaller, e.g., 20 h in (A), or larger than 24 h, e.g., 28 h in (B), can be entrained to oscillate at a circadian period (Gérard and Goldbeter, 2012a). This result demonstrates that the cell cycle can be brought to oscillate at a circadian period when its autonomous period prior to coupling is in an appropriate range. The model indicates that the combination of multiple modes of coupling does not necessarily facilitate entrainment of the cell cycle by the circadian clock. Furthermore, outside the range of entrainment, the coupling to the circadian clock may lead either to disconnected oscillations in the cell cycle and the circadian system, or to complex oscillatory dynamics of the cell cycle in the form of endoreplication, complex periodic oscillations, or even chaos (Gérard and Goldbeter, 2012a).

Previous studies have established the existence of a link between dysregulation of circadian rhythms and cancer (Filipski et al., 2002; Gauger and Sancar, 2005; Sahar and Sassone-Corsi, 2009). The existence of robust circadian rhythms is positively correlated with the health status of cancer patients (Innominato et al., 2009). Entrainment of the cell cycle by the circadian clock could thus play a crucial role in normal cell proliferation and might be deregulated in cancer cells (Fu and Lee, 2003; Pendergast et al., 2010).

The question arises as to whether the complexity of the detailed model for the Cdk network can be reduced without losing the insights provided by this model. We have shown that most of the results on the oscillatory dynamics can indeed be retained when the number of variables and parameters are significantly reduced in a skeleton model for the Cdk network. A necessary price must be paid for this reduction in complexity. We have to abandon a large number of biochemical details while trying to retain the logic of the regulatory wiring of the Cdk network. The advantage of reducing the number of variables is that the system becomes more amenable to a thorough numerical analysis of its dynamical properties, both in its deterministic and stochastic versions.

A first version of the skeleton network (Gérard and Goldbeter, 2011) containing only five variables did not incorporate Cdk regulation through phosphorylation-dephosphorylation. Without increasing the number of variables we recently extended this skeleton model by incorporating the control of Cdk2 and Cdk1 by phosphatases Cdc25 and kinase Wee1. This inclusion allows us to examine the effect of PF loops exerted on Cdk2 via its activation of phosphatase Cdc25 and on Cdk1 via its activation of Cdc25 and inhibition of kinase Wee1 (Gérard et al., 2012).

The extended skeleton model for the Cdk network is schematized in Figure 12 (see also Gérard et al., 2012). At the beginning of the cell cycle, growth factor GF activates directly the synthesis of the cyclin D/Cdk4–6 complex, which promotes progression in G1. This complex ensures the activation of E2F, which in turn activates the synthesis of cyclin E/Cdk2 and cyclin A/Cdk2. During the G1 phase, cyclin E/Cdk2 reinforces the activation of E2F. During the S phase of DNA replication, cyclin A/Cdk2 activates the degradation of cyclin E/Cdk2, which ensures that the latter complex will not be activated at the end of the cell cycle. Cyclin A/Cdk2 also permits exit of the S phase by allowing the inactivation of E2F and by promoting the synthesis of cyclin B/Cdk1, which brings about the G2/M transition. During mitosis, cyclin B/Cdk1 activates, by phosphorylation, the protein Cdc20. This protein creates two negative feedback loops by promoting the inactivation of cyclin A/Cdk2 and cyclin B/Cdk1, thereby ensuring the completion of the cell cycle. A new cell cycle starts if GF is present in sufficient amounts—see Figure 13 and (Gérard and Goldbeter, 2011) for further details about the regulatory mechanisms of the Cdk network in the skeleton, five-variable, model for the mammalian cell cycle.

The skeleton, five-variable, model or the Cdk network is also capable of temporal self-organization in the form of sustained oscillations. As in the detailed model for the cell cycle, sustained oscillations of the various cyclin/Cdk complexes correspond to the evolution toward a limit cycle, which can be reached regardless of initial conditions (Gérard and Goldbeter, 2011; Gérard et al., 2012). Here we focus on the effect of ultrasensitivity and PF loops on the oscillatory dynamics of the Cdk network.

A number of experimental and theoretical studies, mostly devoted to the early cell cycles in amphibian embryos (Goldbeter, 1993; Novak and Tyson, 1993; Ferrell and Machleder, 1998; Pomerening et al., 2003; Sha et al., 2003) and to the yeast cell cycle (Chen et al., 2004) showed that PF contributes to the robustness of oscillatory behavior and make the cell cycle transitions irreversible (Novak et al., 2007). Based on a previous modeling study showing that PF loops increase the robustness of Cdk oscillations toward molecular noise (Gonze and Hafner, 2010; Gérard et al., 2012), we illustrate here the effect of PF loops on the dynamics of the cell cycle.

The time evolution of cyclin E/Cdk2 and cyclin B/Cdk1 in the extended version of the skeleton model is shown in the presence of mild (Figures 13A,C) or strong ultrasensitivity (Figures 13B,D), and in the absence (Figures 13A,B) or presence (Figures 13C,D) of 3 PF loops in the G1/S and G2/M transitions of the cell cycle. The presence of PF loops as well as the degree of ultrasensitivity in phosphorylation–dephosphorylation of Cdk1 and Cdk2 contribute to the robustness of oscillations in the Cdk network by augmenting their amplitude (see Gérard et al., 2012; and also Figure 13). The ultrasensitivity considered here originates from zero-order kinetics for the kinases and phosphatases involved in phosphorylation-dephosphorylation (Goldbeter and Koshland, 1981). The results likely extend to other sources of ultrasensitivity such as multiple phosphorylation or substrate competition (Gunawardena, 2005; Kim and Ferrell, 2007; Barik et al., 2010; Trunnell et al., 2011).

The presence of PF loops also increases the robustness of Cdk oscillations toward molecular noise (Gérard et al., 2012). The skeleton model is more amenable to stochastic simulations than the detailed version of the model for the Cdk network. The results of such stochastic simulations are shown in Figures 14 and 15. The time evolution of cyclin B/Cdk1 as well as the projection, in the cyclin A/Cdk2 vs. cyclin B/Cdk1 plane, of the stochastic limit cycle (black curve) together with the corresponding deterministic limit cycle (red curve) are shown in Figures 14 and 15, respectively, in the absence of PF (A), in the presence of 1 PF on G1/S via Cdc25 (B), 1 PF on G2/M via Cdc25 (C), 1 PF on G1/S and 1 PF on G2/M via Cdc25 (D), 2 PF on G2/M via Cdc25 and Wee1 (E), or 3 PF, i.e., 1 PF on G1/S via Cdc25 and 2 on G2/M via Cdc25 and Wee1 (F). To display many cycles the time series in Figure 14 are obtained for extended durations, so as to better illustrate the degree of irregularity of the oscillations in the presence of noise; shorter stochastic time series in which details of changes in Cdk1 activity are more visible can be found elsewhere (Gérard et al., 2012). The results of Figures 14 and 15 indicate that the presence of multiple PF loops reinforces the robustness of Cdk oscillations toward molecular noise. Moreover, the model predicts that the redundant PF loops on the G2/M transition through Cdc25 and Wee1 have a synergistic effect on the robustness of the cell cycle toward molecular noise (compare panels E, F with panels A–D in Figures 14 and 15).