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A mechanism for robust circadian timekeeping via stoichiometric balance

2012; Springer Nature; Volume: 8; Issue: 1 Linguagem: Inglês

10.1038/msb.2012.62

1744-4292

Jae Kyoung Kim, Daniel B. Forger,

Plant Molecular Biology Research

Article4 December 2012Open Access A mechanism for robust circadian timekeeping via stoichiometric balance Jae Kyoung Kim Jae Kyoung Kim Department of Mathematics, University of Michigan, Ann Arbor, MI, USA Search for more papers by this author Daniel B Forger Corresponding Author Daniel B Forger Department of Mathematics, University of Michigan, Ann Arbor, MI, USA Center for Computational Medicine and Bioinformatics, University of Michigan, Ann Arbor, MI, USA Search for more papers by this author Jae Kyoung Kim Jae Kyoung Kim Department of Mathematics, University of Michigan, Ann Arbor, MI, USA Search for more papers by this author Daniel B Forger Corresponding Author Daniel B Forger Department of Mathematics, University of Michigan, Ann Arbor, MI, USA Center for Computational Medicine and Bioinformatics, University of Michigan, Ann Arbor, MI, USA Search for more papers by this author Author Information Jae Kyoung Kim1 and Daniel B Forger 1,2 1Department of Mathematics, University of Michigan, Ann Arbor, MI, USA 2Center for Computational Medicine and Bioinformatics, University of Michigan, Ann Arbor, MI, USA *Corresponding author. Department of Mathematics, University of Michigan, 2074 East Hall, 525 East University, Ann Arbor, MI 48109, USA. Tel.:+1 734 763 4544; Fax:+1 734 764 0335; E-mail: [email protected] Molecular Systems Biology (2012)8:630https://doi.org/10.1038/msb.2012.62 PDFDownload PDF of article text and main figures. Peer ReviewDownload a summary of the editorial decision process including editorial decision letters, reviewer comments and author responses to feedback. ToolsAdd to favoritesDownload CitationsTrack CitationsPermissions ShareFacebookTwitterLinked InMendeleyWechatReddit Figures & Info Circadian (∼24 h) timekeeping is essential for the lives of many organisms. To understand the biochemical mechanisms of this timekeeping, we have developed a detailed mathematical model of the mammalian circadian clock. Our model can accurately predict diverse experimental data including the phenotypes of mutations or knockdown of clock genes as well as the time courses and relative expression of clock transcripts and proteins. Using this model, we show how a universal motif of circadian timekeeping, where repressors tightly bind activators rather than directly binding to DNA, can generate oscillations when activators and repressors are in stoichiometric balance. Furthermore, we find that an additional slow negative feedback loop preserves this stoichiometric balance and maintains timekeeping with a fixed period. The role of this mechanism in generating robust rhythms is validated by analysis of a simple and general model and a previous model of the Drosophila circadian clock. We propose a double-negative feedback loop design for biological clocks whose period needs to be tightly regulated even with large changes in gene dosage. Synopsis An accurate mathematical model of the mammalian circadian clock provides novel insights into the mechanisms that generate 24-h rhythms. A double-negative feedback loop design is proposed for biological clocks whose period needs to be tightly regulated. A 1–1 stoichiometric balance and tight binding between activators (PER–CRY) and repressors (BMAL1–CLOCK/NPAS2) is required for sustained rhythmicity. Stoichiometry is balanced by an additional negative feedback loop consisting of a stable activator. Our detailed model can explain more experimental data than previous models. Mathematical analysis of a simple model supports our claims. Introduction Circadian (∼24 h) clocks time many physiological and metabolic processes. When these clocks were first discovered, three basic properties were identified (Dunlap et al, 2004). (1) Rhythms need to be autonomous. (2) Rhythms need to be capable of adjusting in response to external signals. (3) Rhythms need to persist over a wide range of temperatures. More recently, the biochemical mechanisms of circadian timekeeping have been identified (Ko and Takahashi, 2006). In particular, interlocked transcription–translation feedback loops (TTFLs) have been discovered as the basic mechanism of rhythm generation in many organisms (Novak and Tyson, 2008). With this discovery, recent experimentation has identified another property of circadian rhythms in higher organisms. Circadian rhythms persist with a 24-h period even in the presence of large changes in the expression of the components of these TTFLs (Ko and Takahashi, 2006; Dibner et al, 2009). While mechanisms for rhythm generation with a flexible period have been identified (Stricker et al, 2008; Tsai et al, 2008; Tigges et al, 2009), mechanisms for this robustness of period to gene dosage remain unexplained, even by mathematical models (Dibner et al, 2009). Two interlocked negative feedback loops have been identified in the TTFL networks generating circadian rhythms in higher organisms (Figure 1) (Blau and Young, 1999; Glossop et al, 1999; Benito et al, 2007; Liu et al, 2008). A ‘core’ negative feedback loop consists of repressors (PERIOD and TIMELESS in Drosophila or PERIOD1–3 and CRYPTOCHROME1–2 in mammals), which inactivate activators (CYCLE and CLOCK in Drosophila and BMAL1–2 and CLOCK in mammals) of their own transcription. An additional negative feedback loop controls the expression of the activators, which inactivate their own transcription through Vrille (Drosophila) or the Rev-erbs genes (Mammals) (Blau and Young, 1999; Preitner et al, 2002). While other feedback loops have also been identified, these two negative feedback loops seem to predominate (Blau and Young, 1999; Glossop et al, 1999; Benito et al, 2007; Liu et al, 2008; Bugge et al, 2012; Cho et al, 2012). Figure 1.Schematic of the detailed mammalian circadian clock model. (A) Only some of the relevant species are shown. Circles refer to transcripts and squares are proteins, possibly in complex. Small circles refer to phosphorylation states that are color coded by the kinases that perform the phosphorylation. See section ‘Description of the detailed model’ in Supplementary information for details. (B) The detailed model consists of a core negative feedback loop and an additional negative feedback loop (the NNF structure). The repressors (PER1–2 and CRY1–2) inactivate the activators (BMALs and CLOCK/NPAS2) of their own transcription expression through the core negative feedback loop. The activators inactivate their own transcription expression by inducing the Rev-erbs through the secondary negative feedback loop. Download figure Download PowerPoint Near 24-h oscillations persist even when the components of the TTFLs of the circadian clock are over or under expressed. Heterozygous mutations of clock genes never abolish rhythmicity, and their period phenotypes are either indistinguishable from the wild-type (WT) phenotypes or much smaller than mutations that affect post-translational modifications (Baggs et al, 2009; Etchegaray et al, 2009; Lee et al, 2009). Abolishing rhythmicity through single gene knockout is surprisingly difficult (Baggs et al, 2009; Ko et al, 2010). Moreover, the mammalian circadian clock is also resistant to global changes in transcription rates (Dibner et al, 2009). These results all suggest that gene dosage may not be important for circadian timekeeping in higher organisms. Gene dosage, however, is not completely unimportant for timekeeping. Knockdown of clock genes causes increased expression in similar components through paralog compensation, which may help restore gene dosage and indicates that gene dosage needs to be tightly regulated (Baggs et al, 2009). Population rhythmicity in mouse embryonic fibroblasts shows much lower amplitude than in liver, which might be due to the fact that the ratio of repressors to activators is significantly lower in fibroblasts than that found in liver (Lee et al, 2001, 2011). A 1–1 stoichiometric binding occurs between the activators and repressors driving rhythms in Drosophila (Menet et al, 2010), although not in Neurospora (He et al, 2005; Huang et al, 2007). Here, we propose a mechanistic explanation for the robustness to gene dosage in the circadian clock of higher organisms through mathematical modeling. We develop the most detailed mathematical model of the mammalian circadian clock available, which should be useful in many future studies. Our model reproduces a surprising amount of experimental data on the mammalian circadian clock including the time courses and relative concentrations of key transcripts and proteins, the effects of mutations of key clock genes, and the effects of changes in gene dosage. With this model, we show that proper stoichiometric balance between activators (BMAL–CLOCK/NPAS2) and repressors (PER1–2/CRY1–2) is key to sustained oscillations. Furthermore, we find that an additional slow negative feedback loop, in which activators indirectly inactivate themselves, improves the regulation of the stoichiometric balance and sustains oscillations with a nearly constant period over a large change in gene expression level. Tight binding between activators and repressors is also predicted to be crucial for rhythm generation. These mechanisms are also validated by mathematical analysis of a simplified mathematical model of the mammalian circadian clock, and simulations of a previously published Drosophila model. We here propose a novel design for biological oscillators where maintaining period is crucial: a core negative feedback loop with repression by protein sequestration, with an additional negative feedback loop, which controls a relatively stable activator. Results Mathematical modeling of the mammalian circadian clock We develop a new mathematical model of the intracellular mammalian circadian clock. This model contains key genes, mRNAs and proteins (PER1, PER2, CRY1, CRY2, BMAL1/2, NPAS2, CLOCK, CKIε/δ, GSK3β, Rev-erbα/β) that have been found to be central to mammalian circadian timekeeping (Figure 1A). While greatly expanded, the model is largely based on our previous model, which has made surprising predictions about mammalian timekeeping that have been subsequently verified experimentally (Forger and Peskin, 2003; Gallego et al, 2006; Ko et al, 2010; Yamada and Forger, 2010). Modifications and extensions of the model are described in the Materials and methods, Supplementary information and Supplementary Tables 1 and 2. The parameters of the model are estimated using experimental data and a simulated annealing method (a global stochastic parameter searcher) (Gonzalez et al, 2007) (see Materials and methods, Supplementary information and Supplementary Table 3 for details). In particular, we incorporated experimentally determined rate constants (Supplementary Table 3) (Kwon et al, 2006; Siepka et al, 2007; Chen et al, 2009; Suter et al, 2011), fit the time courses of both mRNA and proteins (Figure 2A and B) (Lee et al, 2001; Reppert and Weaver, 2001; Ueda et al, 2005) and fit the relative abundance of proteins (Figure 2C) (Lee et al, 2001). Figure 2.Validation of the detailed model. (A) Predicted mRNAs time courses in SCN (Ueda et al, 2005). Time courses were normalized so that the peak value is 1, matching experimental data. (B) Predicted protein time courses in SCN (Reppert and Weaver, 2001). As had been done previously, we normalize the protein time courses so that the maximum is 1 and the minimum is 0. (C) Model comparison of the relative abundance of proteins in liver and fibroblast (Lee et al, 2001, 2009, 2011). All of the values were normalized so that the maximum abundance of the CRY1 protein is 1. For the CKIε/δ, CKIε maximal expression is ∼22.5% of the maximum abundance of CRY1 in the liver (Lee et al, 2001) and CKIδ is two times more abundant than CKIε in the fibroblast (Lee et al, 2009). From this, we assumed that total CKIε/δ would be ∼67.5% of the maximum value of CRY1 in mice liver and fibroblast. Download figure Download PowerPoint Our model accurately predicts the phenotype of known mutations of genes in the central circadian clock (suprachiasmatic nuclei, SCN) (Yoo et al, 2005; Baggs et al, 2009; Ko et al, 2010), which other models do not predict (Table I) (Forger and Peskin, 2003; Leloup and Goldbeter, 2003; Mirsky et al, 2009; Relógio et al, 2011). Interestingly, our model shows opposite phenotypes for Cry1−/− and Cry2−/− matching experimental data (Liu et al, 2007). There are two differences between CRY1 and CRY2 in our model. First, Cry1 transcription is delayed through repression by Rev-erbα and Rev-erbβ (Preitner et al, 2002; Liu et al, 2008; Ukai-Tadenuma et al, 2011). Additionally, Cry1 mRNA is more stable than Cry2 mRNA and CRY1 protein is more stable than CRY2 protein (Busino et al, 2007; Siepka et al, 2007; Chen et al, 2009). Since a longer half-life causes rhythms to be delayed, and delayed rhythms cause a longer period (Forger, 2011; Ukai-Tadenuma et al, 2011), removing CRY1 shortens the period and removing CRY2 lengthens the period. The opposite phenotypes of Clock−/− (null mutation) and ClockΔ19/+ (dominant-negative mutation) are also correctly simulated in the model for the first time (Vitaterna et al, 1994; Herzog et al, 1998; Debruyne et al, 2006). Moreover, our model also predicts the mutant phenotypes of isolated SCN neurons, which are different from the SCN slices (Liu et al, 2007). We note that SCN slices have significantly higher gene expression of per1 and per2 through CREB/CRE pathway than isolated SCN neurons (Yamaguchi et al, 2003). Interestingly, when we reduced per1 and per2 expression about 60% in our model, our model was able to accurately reproduce the phenotypes of isolated SCN neurons (Table II). Table 1. Comparison of model predictions with experimental data and previous model predictions on the phenotypes of circadian mutations Gene SCN Animal New model Relógio et al (2011) Mirsky et al (2009) Leloup and Goldbeter (2003) Forger and Peskin (2003) Cry1−/− Short Short −1 Long AR Short WT Cry2−/− Long Long +1.6 Long Long Short Long Per1−/− WT AR AR Short Long Per1ldc WT Short/AR Per2−/− AR AR AR Short Short Per2ldc Short/AR Bmal1−/− SR** AR AR AR AR AR Bmal1−/+ WT* +0.1 AR NA AR Long Clock−/− WT Short −0.2 Long AR AR AR ClockΔ19/Δ19 AR* Long AR Long NA NA NA ClockΔ19/+ Long* +1.1 Long NA NA NA Npas2−/− WT Short WT NA NA NA NA Rev-erbα−/− Short −0.2 AR Short NA WT CK1εtau/tau Short Short −3 NA NA Short Short Here we indicate whether the phenotype predicted by our model, or seen in experimental data is WT, stochastically rhythmic (SR), arrhythmic (AR) or shows a change in period in hours. Experimental data can be found in Baggs et al (2009) as well as references cited therein, except those marked with * which can be found in Yoo et al (2005) and ** which can be found in Ko et al (2010). See Materials and methods for details. Bold represents different phenotype prediction of previous models from the new model. NA represents not available. For the Leloup–Goldbeter model, first parameter set of the model is used. Table 2. Comparison of modified model predictions with experimental data of single SCN neurons on the phenotypes of circadian mutations Gene dSCN Model Cry1−/− AR AR Cry2−/− Long +2.3 Per1−/− AR Per1ldc AR Bmal1−/− AR* AR Here, we indicate whether the phenotype predicted by our model, or seen in experimental data is arrhythmic (AR) or shows a change in period in hours. Experimental data can be found in Liu et al (2007), except those marked with * which can be found in Ko et al (2010). See Materials and methods for details. We also conducted a sensitivity analysis to look at what parameters determine the period of our model. Four of the top five high parameters, in our sensitivity analysis, were also in the top five found in a previous sensitivity analysis with the original Forger and Peskin model and which was used to conclude that PER2 plays a dominant role in period determination (Wilkins et al, 2007) (see Supplementary Figure 1). Proper stoichiometric balance between activators and repressors is crucial to sustained rhythms Since our mathematical model can accurately predict the phenotype of known mutations of the mammalian circadian clock, we next looked for a mechanism that could explain why some phenotypes were rhythmic, while others were not. We found that stoichiometry plays a key role in determining which mutations showed rhythmic phenotypes. Here, we define stoichiometry as the average ratio between the concentration of repressors (all forms of PER and CRY in the nucleus) to that of activators (all forms of BMAL–CLOCK/NPAS2 in the nucleus) over a period. Moreover, we specifically refer to repressors and activators of E/E’-boxes when discussing stoichiometry. We found that mutations that caused the stoichiometry to be too high or too low, yielded arrhythmic phenotypes (Figure 3A). So long as the mutations allowed the stoichiometry to be around a 1–1 ratio, relatively high amplitude oscillations were seen. Thus, we predict that stoichiometry provides a unifying principle to determine the rhythmicity of mutations of the mammalian circadian clock. To further test this principle, we constitutively expressed either the Per2 gene (the dominant repressor gene) or the Bmal and Clock genes (the dominant activator genes) at different levels. Interestingly, within a range centered near a 1–1 stoichiometry, the model shows sustained oscillations with high amplitude (Figure 3B). However, if the stoichiometry was too high or too low, rhythms are dampened or completely absent (Figure 3B). This matches a recent experimental study showing that the amplitude and sustainability of population rhythms increase when the level of PER–CRY is increased closer to that of BMAL1–CLOCK in mouse fibroblasts (Lee et al, 2011). Figure 3.Proper stoichiometry between activators and repressors is the key to sustained oscillations. (A) Our detailed mathematical model accurately predicts the phenotype of the known mutations in circadian genes (Table I). We plot the stoichiometry predicted by our model in these mutants with the relative amplitude of Per1 mRNA rhythms (or Per2 mRNA when considering the Per1−/−). Here, an amplitude of zero means rhythms are not sustained. These results indicate that the phenotype of the mutants can be predicted by their effects on stoichiometry. (B) The stoichiometry between repressors and activators is changed by constitutively expressing either the Per2 gene or the Bmals and Clock genes at different levels. Note that the model is rhythmic only when the stoichiometry is near 1–1. The relative amplitude of the Per1 mRNA is measured. (C) Schematic of a simplified model based on the Goodwin oscillator. Instead of a Hill-type equation, the sequestration of the activator (A) by the repressor (P) is used to describe repression of the gene. (D) Oscillations are seen around a 1–1 stoichiometry as the level of activator is changed. The range of the stoichiometry widens as the dissociation constant (Kd) decreases or the binding between the activator and the repressor tightens. Download figure Download PowerPoint We defined the stoichiometry as the average ratio between the total concentrations of repressors to that of activators over a period. However, recent work has shown that CRY1 has stronger repressor activity than CRY2. The underlying biochemical mechanisms for this result have not been fully identified (Khan et al, 2012). If the difference is due to a different post-translational mechanism (e.g., binding between PER and CRY, which could affect the repressor concentration in the nucleus), the current definition of stoichiometry can be kept. Otherwise, a more sophisticated definition of stoichiometry may be needed (e.g., one that gives more weight to concentration of CRY1 than that of CRY2). How stoichiometry generates rhythms To test the role of stoichiometry in sustaining oscillations, we developed a simple model by modifying the well-studied Goodwin model (Goodwin, 1965) to include an activator (A), which becomes inactive when bound by a repressor (P) (Figure 3C). Transcription is proportional to the fraction of free activator that is not bound by the repressor, f(P, A, Kd) (Buchler and Cross, 2009), matching experimental data from the mammalian circadian clock (Supplementary Figure 2) (Froy et al, 2002). mRNA (M) is translated to a repressor protein (Pc). The protein enters the nucleus (P) and binds and inhibits the activator (A). This generates a single-negative feedback loop (SNF) since the activator is constitutively expressed. The model is similar to a previously published mathematical model (Francois and Hakim, 2005); however, we allow for both association and dissociation of the activator and repressor (through a defined Kd), which turns out to be crucial for understanding the effects of stoichiometry. By nondimensionalization and setting the clearance rates of all species to be equal (to increase the chance of oscillations, see Forger, 2011), only two parameters remain: the activator concentration (A) and the dissociation constant (Kd) (see Supplementary information). When we changed the activator concentration, which changed the stoichiometry (average ratio between the level of repressor (P) to the level of activator (A)), sustained oscillations were only seen at around a 1–1 stoichiometry similar to our detailed model (Figure 3D). As the other parameter (Kd) decreased (indicating tight binding), the range of stoichiometry that permitted oscillation increased (Figure 3D). Interestingly, if the binding was too weak, the rhythms did not occur. The tight binding between activators and repressors is also found in the detailed model, and in the mammalian circadian clock (Lee et al, 2001; Froy et al, 2002; Sato et al, 2006). This indicates that the sustained rhythms require tight binding as well as balanced stoichiometry in the circadian clock. Many previous studies have argued that ultrasensitive responses (e.g., a large change in transcription rate for a small change in repressor or activator concentration) can cause oscillations in feedback loops (Kim and Ferrell, 2007; Buchler and Louis, 2008; Novak and Tyson, 2008; Forger, 2011). A previous study showed that an ultrasensitive response can be generated by tight binding of activators and repressors in a synthetic system (Buchler and Cross, 2009). Taken together, this provides a potential mechanism of rhythm generation. That is, when the total concentration of repressor is higher than that of activators, the repressor sequesters and buffers activator and inhibits transcription completely (Buchler and Louis, 2008). As the repressor is depleted, the excess free activators are no longer sequestered by repressors and are free to turn on the transcription. At this threshold, transcription of repressor shows an ultrasensitive response to the concentration of repressor or activator. Ultrasensitive responses amplify rhythms and prevent rhythms from dampening (Forger, 2011). In both our simple and our detailed model, we found ultrasensitive responses around a 1–1 stoichiometry (Supplementary Figure 3A). When the stoichiometry was not around 1–1, an ultrasensitive response was not seen, and both models did not show sustained rhythms. Over the course of a day, as levels of repressor and activator change, the stoichiometry and also sensitivity change as well. We found that the 1–1 average stoichiometry is required to generate the ultrasensitive response, which causes rhythms through mathematical analysis, confirming our simulation results (Figure 3D). That is, via both local and global stability analysis, we derived an approximate range of the stoichiometries (〈S〉) that permit oscillations (see Supplementary information). In agreement with our simulations shown in Figure 3D, this mathematical analysis also suggests that (1) oscillations are seen around a 1–1 stoichiometry; (2) the stoichiometry needs to be >8/9 for sustained rhythmicity; (3) as the binding between activators and repressors becomes tighter, the upper bound on stoichiometry increases; (4) if the binding is too weak (e.g., Kd=10−3), sustained oscillations do not occur. An additional negative feedback loop improves the regulation of stoichiometric balance If stoichiometry is key to sustained oscillation, are there mechanisms within circadian clocks that keep the stoichiometry of components balanced? Does the additional negative feedback loop of the negative–negative feedback loop (NNF) structure, found in circadian clocks, help balance stoichiometry? To test this structure, we added an additional negative feedback loop into our simple model (Figure 4A). Previously, other studies suggested that an additional positive, rather than negative, feedback loop could sustain intracellular clocks (Barkai and Leibler, 2000; Stricker et al, 2008; Tsai et al, 2008; Tigges et al, 2009). We tested these structures by including an additional protein R (Rev-ERBs or RORs in the mammalian circadian clocks) that is transcribed in a similar way to P. R then represses (as in the Rev-erbs) or promotes (as in the Rors) the production of A in the negative–negative feedback loop (NNF) or the positive–negative feedback loop (PNF) structure, respectively (Figure 4A). Figure 4.The NNF structure maintains stoichiometry in balance by active compensation of both repressors and activators. (A) A negative or positive feedback controlling the activator is added to the original negative feedback controlling the repressor. (B) The relative sensitivity (% change in mean level of stoichiometry per % change in transcription rate of repressor) in the simple models with SNF, NNF and PNF structure were measured over a range of the transcription rates of repressor (see Supplementary Figure 4A). Then, we calculated the average of relative sensitivity over the range of parameters. On average, the relative sensitivity of the NNF model is about two-fold less sensitive than that of the SNF model, but that of the PNF model is about four-fold more sensitive than that of the SNF model (see Supplementary information and Supplementary Figure 4A–C for details). (C) The detailed model matches data from Gene Dose Network Analysis experiments (Baggs et al, 2009). After the knockout of a repressor gene (here, Cry1), the activity of the repressor promoters, controlled by an E-box, increases. This increases the expression of Rev-Erbs and reduces the activity of the activator promoter, controlled by a RORE. An opposite phenotype is seen when an activator (here, Clock) is knocked out. The activity of E-box decreases. This decreases the expression of Rev-Erbs and increases the activity of RORE. This active compensation through the NNF structure allows the stoichiometry to be balanced after the repressors or activators knockout. (D) The detailed model matches data from Rev-erbs−/− (Liu et al, 2008; Bugge et al, 2012; Cho et al, 2012). Rev-erbα−/− (50% reduction of transcription rate of the Rev-erbs due to the presence of Rev-erbβ) slightly shortens the period and has little effect on the expression level of Per2, Cry1 and Bmal1. Double knockout of the Rev-erbα and Rev-erbβ (100% reduction of transcription rate of the Rev-erbs) increases the expression level of Bmal1 and Cry1, but decreases that of Per2. All the values were normalized by the average of Per2 expression level in WT. Download figure Download PowerPoint We studied how the SNF, NNF and PNF structures effectively maintain the stoichiometric balance when model parameters (e.g., transcription rate) are changed. With both simulation and steady-state analysis, we found that the NNF structure is best at keeping stoichiometry balanced while the PNF structure is worst at keeping stoichiometry balanced, regardless which parameters are perturbed (see Supplementary information, Figure 4B and Supplementary Figure 4A–C). Moreover, our detailed model, which also follows the NNF structure, also carefully balanced the stoichiometry by controlling the expression of repressors and activators. Knockdown of the repressor Cry1 leads to higher expression of the repressors, which are controlled by E-boxes, and lower expression of the activators, which are controlled by a ROREs (Figure 4C). Opposite effects are seen when the activator CLOCK is removed (Figure 4C). This active control of repressors and/or activators via the NNF structure regulates the stoichiometric balance tightly (Supplementary Figure 4D) and matches experimental data on gene dosage (Baggs et al, 2009). Moreover, the detailed model (with the NNF structure) also correctly predicts the change of clock gene expression after the removal of the additional negative feedback loop (Rev-erbα,β−/−) (Figure 4D) (Liu et al, 2008; Bugge et al, 2012; Cho et al, 2012). In particular, knockout of the Rev-erbα,β decreases PER expression, but increase CRY1 expression. For our nominal set of parameters, oscillations are still possible when this additional negative feedback is removed. However, for other sets of parameters, where stoichiometry is not as well balanced, removal of this additional negative feedback stops rhythmicity (see below). This could explain the phenotype of the Rev-erbα,β−/−, which show some indications of rhythmicity (Bugge et al, 2012; Cho et al, 2012). Our model predicts that rhythm generation remains in cell types that have a near balanced stoichiometry, and a lack of rhythms in cell types without a balanced stoichiometry. A slow additional negative feedback loop improves the robustness of rhythms Our central hypothesis is that, as stoichiometry is more tightly regu