A model of the cell-autonomous mammalian circadian clock

Abstract

Circadian timekeeping by intracellular molecular clocks is evident widely in prokaryotes and eukaryotes. The clockworks are driven by autoregulatory feedback loops that lead to oscillating levels of components whose maxima are in fixed phase relationships with one another. These phase relationships are the key metric characterizing the operation of the clocks. In this study, we built a mathematical model from the regulatory structure of the intracellular circadian clock in mice and identified its parameters using an iterative evolutionary strategy, with minimum cost achieved through conformance to phase separations seen in cell-autonomous oscillators. The model was evaluated against the experimentally observed cell-autonomous circadian phenotypes of gene knockouts, particularly retention of rhythmicity and changes in expression level of molecular clock components. These tests reveal excellent de novo predictive ability of the model. Furthermore, sensitivity analysis shows that these knockout phenotypes are robust to parameter perturbation.

The circadian clock is an endogenous biological oscillator found widely throughout the tree of life (1). Circadian rhythms impact many behavioral and physiological processes, allowing for the anticipation of daily changes in the environment, the coordination of events that need to occur simultaneously or sequentially, and the segregation of activities that should be separated. At the organism level, these 24-hour cycles are manifested diversely, occurring in the movement of leaves and the opening of flowers in Arabidopsis (2), spore formation in Neurospora (3), and blood pressure and body temperature, as well as hunting, foraging, mating, and sleeping behavior in mammals (4). At the genome level, it is estimated that ≈10% of genes in a given tissue type exhibit a circadian pattern (5–8). Clearly, because of both the widespread occurrence of the circadian clock and its effects at all levels of organization, an understanding of this system is an essential component of biological knowledge.

Although circadian behaviors had been observed macroscopically for hundreds of years, it was not known whether these behaviors were the result of environmental induction or whether they had an internal cause (9). In a series of classic experiments, rodents were kept isolated and in hermetic conditions (i.e., constant temperature and darkness) for an extended period, while their wheel-running activity was being recorded. It was found that the onset of daily locomotion would occur with astounding precision, generally within several minutes from day-to-day and with a period near 24 hours that varied by species. Therefore, it became apparent that rodents possess an endogenous and precise timekeeper. Furthermore, application of external stimuli, typically light, allowed for resetting of the clock to attune it to the daily rhythm (10).

Meanwhile, an elucidation of the mammalian circadian system proceeded anatomically (11, 12). These studies showed that the “master clock” resides within the suprachiasmatic nuclei (SCN) of the anterior hypothalamus, with putative dispersed “slave clocks” located elsewhere in the body. The SCN lie to either side of the brain midline in the hypothalamus, each containing ≈8,000–10,000 neuronal cells of varying physiology. Light, the principal entraining agent, is channeled to the SCN via the retinohypothalamic tract (4, 13, 14).

In the last 20 years, researchers have turned their attention to the internal chemistry of the cells in the SCN. We now know that the clock is formed by a delayed negative feedback loop comprised of the 3 Period (Per1, Per2, and Per3) and 2 Cryptochrome (Cry1 and Cry2) genes and their products (4, 15). These genes are perpetually activated by CLOCK/BMAL1 (CLK/BMAL1) through the E-box enhancer elements. The various PER and CRY proteins heterodimerize and directly bind to the CLK/BMAL1 activation complex to repress E-Box-mediated transcription of their own genes. Therefore, rising levels of PER and CRY lead to repression and subsequently lower levels of PER and CRY, giving rise to oscillatory gene expression. In addition to the feedback loop involving the Pers, Crys, Clk, and Bmal1, there exist other interconnected ancillary loops. For instance, REV-ERBα and RORc repress or activate, respectively, the transcription of genes with a ROR/REV-ERB binding element (RORE) in their promoters (e.g., Clk, Bmal1, Cry1, and Rorc). Although Rev-erb and Ror genes are not necessary for cellular rhythm generation, they play important roles in regulating phase and amplitude of gene expression (7, 16–20).

This advancing state of biological knowledge facilitates the creation of predictive mathematical models of the circadian system. The potential benefits of quantitative models are multifold: They permit rapid in silico prediction of experiments that could take molecular biologists many months to perform; they permit determination of sensitivities that can help to identify points of strength and fragility in the system, lending insight into the way biology is organized and helping to identify targets for therapeutics; and, when the models fail, they reveal where our understanding needs improvement.

Not surprisingly, then, mathematical models of circadian clocks have been proposed in virtually every organism for which sufficient knowledge of molecular biology exists. One can find models for cyanobacteria (21), Arabidopsis (22, 23), Neurospora (24), Drosophila (25, 26), and mouse (26–28). These models vary considerably in sophistication, with the most complex being the 16- and 19-state mouse models (26), the 73-state mouse model (27), and its corresponding stochastic version (28). All of these models produce sustained rhythms with 24-hour periodicity but are deficient in several important respects.

First, it is now known that circadian phenotypes in the SCN and at the behavioral levels are not always cell-autonomous. The SCN in vivo is a network of neurons that are synchronized through cell–cell communication mediating intercellular coupling, and consequently, the SCN ensemble is robustly rhythmic (13, 14). However, dispersed SCN neurons or peripheral tissues/cells lack functional intercellular coupling and thus display independently phased, cell-autonomous rhythms (14, 29, 30). Compared with coupled oscillators in the SCN, there is considerably more dispersion in the period lengths of cell-autonomous oscillators and an increase in cycle-to-cycle variability. Even more significantly, in mutants, oscillations of the SCN clock and its regulated behavior are not necessarily cell-autonomous. The effect of knockout mutations on phenotype can be radically different depending on whether one assesses at the cellular level or at the SCN tissue or organismal levels. For example, SCN explants of both Per1 and Cry1 knockouts retain rhythmicity, whereas dispersed SCN neurons of both knockout types (cell-autonomous) are largely arrhythmic (29). Thus, intercellular coupling confers robustness to the SCN clock network but can mask cell-autonomous circadian phenotypes. In this context, previous models were developed in part by fitting to the SCN tissue-level and/or behavioral knockout phenotypes and do a good job at matching these phenotypes. However, because cell-autonomous and behavioral phenotypes may differ, previous models are sometimes inaccurate as cell-level models. Equally important, while intercellular coupling comprises a special attribute of the SCN, the SCN clockwork is similar to oscillators in peripheral tissues at the molecular level. The model developed here makes exclusive use of cell-level data for both parametric fitting and validation.

Second, previous models do not include all essential molecular components, whereas we developed our model with 8 genes (Per1, Per2, Cry1, Cry2, Clk, Bmal1, Rev-erbα, and Rorc) and thereby provide a more complete network and offer greater opportunities for validation. The 73-state model (27) includes Per1, Per2, Cry1, Cry2, and Rev-erbα, but with CLK and BMAL1 present only implicitly at constitutively high levels and REV-ERBα present only at extremely low levels. The 16-state model (26) contains only Bmal1, Per, and Cry (plus Rev-erbα in its 19-state form). This model incorporates Rev-erbα but includes neither Rorc nor Clk, and it does not distinguish between the several types of Per and Cry. Most importantly, our model addresses the overlapping but differential functions of CRY1 and CRY2 in the clock mechanism: They antagonistically regulate period length and differentially control rhythm persistence and amplitude (29, 31, 32).

Third, previous models have not captured the precise phase relationships among molecular components of the circadian clockwork revealed in recent experimental work at the intracellular level, which reflect the complex and often combinatorial regulation of the circadian genes (16, 20). Specifically, Rev-erbα mRNA leads Per1 and Per2 mRNAs by 4 hours; Per1 and Per2 mRNAs lead Cry1, Cry2, and Rorc mRNAs by 4 hours; Cry1, Cry2, and Rorc mRNAs lead Clk and Bmal1 mRNAs by 8 hours; and Clk and Bmal1 mRNAs lead Rev-erbα mRNA by 8 hours. Correct operation of the circadian system is in many respects a matter of correct phasing (10). Although the constituents of the prior models are correctly phased in a general sense, they do not incorporate the recently revealed subtleties because of combinatorial gene regulation. The 73-state model (27) places all components approximately in-phase, whereas the 16- and 19-state models (26) have Per and Cry in-phase and Bmal1 in anti-phase.

Thus, the need arises for a new mammalian circadian model that takes into consideration the complex transcriptional regulatory network and cell-autonomous clock phenotypes that more precisely define gene function. The new model must match phase subtleties and predict cell-autonomous knockout phenotypes at the cellular level. Although a network diagram can be constructed from biological data, the kinetics of the individual reactions is insufficiently characterized to parameterize a model. However, given the phase relationships in the system and an awareness of their fundamental importance to the correct operation of the clock, it is possible to use iterative computation, with an evolutionary strategy, to develop a parameter set (33). This approach was used previously to good effect in the development of a model for the circadian clock in Arabidopsis (22). The model can then be tested by using it to predict a host of additional behaviors for which experimental data exist, including the ability of the system to retain rhythmicity when one or more genes are removed, as well as the resulting changes in expression levels of clock components.

Results

Model Development.

The gene regulatory scheme for the mouse circadian clock, depicted in Fig. 1A, was used to construct the network diagram shown in Fig. 1B. This network was then translated into a system of 21 ordinary differential equations [supporting information (SI) Appendix] containing 132 (unknown) parameters. Michaelis–Menten kinetics was assumed for transcriptional rates, and mass action kinetics was assumed for all other rates (e.g., mRNA and protein degradations, translation, complex formation and dissociation, etc.). For simplicity, the redundant function of REV-ERBα and β repressors is represented in the model by REV-ERBα, and that of the ROR activators by RORc. Each transcriptional rate (except that for Rev-erbα) was defined as the sum of a basal rate and all activating contributions (i.e., any contributions from CLK/BMAL1 activation and/or RORc activation) multiplied by any inhibition terms (i.e., any contributions from PER1/CRY1, PER2/CRY1, PER1/CRY2, PER2/CRY2 inhibition, and/or REV-ERBα inhibition). The transcriptional rate for Rev-erbα did not include a basal rate, because it is known that its mRNA level drops to near zero in the absence of an activator (16, 34, 35). Because degradation of PER/CRY and CLK/BMAL complexes proceeds at a much slower rate than for the individual proteins, we do not provide a degradation term for the complexes (36–39).

Fig. 1.

Model structure. (A) Gene regulation scheme for the mouse circadian clock. Per1, Per2, and Rev-erbα are activated at E-Boxes by CLK/BMAL1 and deactivated when PER/CRY also binds. Clk and Bmal1 are activated by RORc and deactivated by REV-ERBα at RORE. Cry1, Cry2, and Rorc have both an E-Box and a RORE and are regulated accordingly. (B) Schematic of the mouse circadian clock model incorporating genetic regulation at Per1, Per2, Cry1, Cry2, Rev-erbα, Clk, Bmal1, and Rorc, the degradation of mRNAs, the translation of mRNAs to protein, the formation and dissociation of PER/CRY and CLK/BMAL1, and the degradation of PER1, PER2, CRY1, CRY2, REV-ERBα, CLK, BMAL1, and RORc. The loops involving Rev-erbα and Rorc are shown in red.

The number of genes incorporated and the complexity of the genetic regulation are both greater in this model than in previous circadian models, but the number of posttranslationally modified species has been kept intentionally small. Although it is generally known that posttranslational modifications affect protein stability, turnover, subcellular localization, and abundance and activity in the nucleus, the precise mechanisms of action remain largely unknown, are sometimes conflicting (40), and are difficult to model with confidence. Therefore, we have restricted our protein dynamics to the degradation of species and the formation and dissociation of CLK/BMAL1 and PER/CRY complexes.

Because kinetic parameters are not reliably known from experimental data, an optimal, numerically derived parameter set was identified (Table S1) using an iterative evolutionary procedure. The model was evaluated to see whether the desired phase separations were achieved. The results are shown graphically in Fig. 2, where the blue and red lines indicate mRNAs and proteins, respectively, the blue dots indicate the desired locations of mRNA peaks (relative to Rev-erbα mRNA peaks), and the red dots indicate the desired locations of protein peaks (relative to the corresponding mRNA peaks). The desired and achieved values are also given in Table S2. Phase separation is as desired for most species. Note that all phase relationships derive from interactions between the derived regulatory scheme and the optimized parameter set.

Fig. 2.

Time courses for the mouse circadian model. mRNA time courses are given in blue, and protein time courses are given in red. Blue dots indicate desired positions for mRNAs, relative to the position of Rev-erbα mRNA. Red dots indicate desired positions for proteins, relative to the corresponding mRNA.

Model Validation Against Knockout Phenotypes.

The true value of any model lies in its predictive capabilities and its ability to lend insight into biological questions. A first test of the model was its ability to predict the phenotypes of various knockout mutations. Knockouts were achieved in silico by setting the transcription rate(s) of the knocked-out gene(s) to zero. The specific parameters set to zero for each mutant are shown in Table S3.

Table S3 shows the experimentally determined cell-autonomous phenotypes associated with 7 single knockout mutations and 2 double knockout mutations (29), along with the results of knockout simulations. In all 9 cases, the model predicts the correct gross phenotypes (i.e., rhythmicity vs. arrhythmicity), gives the correct periods (i.e., a normal 23.7-h period with Rev-erbα and Rorc deletion and a long period with Cry2 deletion). The time courses for all knockouts with maintained rhythmicity (i.e., Cry2, Rev-erbα, and Rorc knockouts) are provided in Fig. S1A–C. Note in particular the extremely low-amplitude rhythms of Bmal1 and Clk mRNAs and BMAL1 and CLK proteins in Rev-erbα knockouts. These rhythms are far below the level of experimental detectability and were reported to be flat in a previous report (16).

Next, the effects of various knockout mutations on mRNA levels were tested in silico. Twelve knockout-caused changes in concentration were evaluated against experimental data (16, 29). Ten simulated results, shown in Fig. 3 and Fig. S2, are in accord with these data. Specifically, the model correctly predicts the increase in Per2 and the decrease in Bmal1 mRNA levels caused by the Cry2 knockout or the Cry1/Cry2 double knockout. However, the model fails to predict the increase in Per2 and the decrease in Bmal1 mRNA levels caused by knockout of Cry1 (29, 41). The model predicts increased amplitude of Per2 mRNA when Cry2 is deleted, consistent with experimental data (29). The model also correctly predicts the increase in Cry1, Clk, and Rorc mRNA levels, the reduction in Per1 and Per2 mRNA levels, and the elimination of Rev-erbα mRNA caused by deletion of the Bmal1 gene.

Fig. 3.

Time courses for Per2 mRNA in wild-type (solid) and Cry2 knockout (dashed) mice. Note the increase in amplitude in the knockout.

Model Validation Against Constitutive Phenotypes.

We model constant transcription of Bmal1 by removal, in Eq. 7, of the Michaelis–Menten terms governing regulated transcription, so that mRNA formation occurs at the fixed rate v_0,Bmal1 + v_1,Bmal1 (SI Appendix). Recent experimental data show that the mouse clock continues to produce intracellular oscillations when Bmal1 mRNA (16) is constitutively produced. The model correctly predicts this result (Fig. S1D).

We similarly model constitutive transcription of Cry2. Our model predicts that constitutive Cry2 can also sustain circadian oscillation (Fig. S1E), consistent with experimental data (42), and that the period of oscillation is inversely related to the rate of mRNA production (Fig. S1F), consistent with experimental data obtained in flies (43). Note that this result implies that the period of oscillation can therefore be tuned to the wild-type value of 23.7 h.

Model Validation Against Constitutive CLK/BMAL1 Activation.

Experimental data suggest that CLK/BMAL1 is constitutively bound to E-Boxes (36), perhaps because it exists at high levels (44). A further test of the model, then, would be to evaluate the Michaelis–Menten terms associated with CLK/BMAL1 activation on the various genes to see if these activation terms are close to unity, indicating perpetual activation. The curves can be determined for Per1, Per2, Cry1, and Cry2 through application of Eqs. 1–4, respectively (SI Appendix). In each case, we allowed CLK/BMAL1 to range from 0 nM to 10 nM, sufficient to drive each curve to saturation. We then superimposed the portion of each curve actually traversed given the simulated levels of CLK/BMAL1. For all 4 curves, CLK/BMAL1 levels lead to complete perpetual activation of the gene (Fig. S3 A–D). Perpetual activation also explains how CLK/BMAL1 regulates target gene expression at multiple phases that span most of the circadian cycle (from Rev-erbα, to Per1 and Per2, and to Cry1 and Cry2) during which Bmal1 expression levels range from high to low (16).

Knockout Phenotype Robustness to Parameter Perturbation.

We next explored the ability of the wild-type and gene-knockout models to maintain their gross phenotypes in the face of parameter fluctuation. We created each new parameter value by randomly drawing from a normal probability distribution with a mean equal to the nominal parameter value. New parameter sets were created by repeating the procedure for each parameter in the model. To determine an appropriate SD for the probability distributions, we produced groups of 10,000 parameter sets, each group developed with an arbitrarily selected SD, simulated the wild-type model, and assessed rhythmicity. Classification as “rhythmic” required a display of true limit-cycle behavior, with all states oscillating at the same period. Setting the SD at 10% of each parameter's nominal value produced rhythmic behavior in 98% of the simulations. Above this value, retention of rhythmicity declined, as expected. The process was then repeated for each of the gene-knockout models, in each case using a SD of 10% of each parameter's value. For all knockout models, as in the wild-type model, fidelity of phenotype was well conserved (Table S4).

Insights into Clock Kinetics.

We next sought to gain some insight into the biology of the clock. One curiosity is the contrasting phenotypes resulting from knockout of Cry1, Cry2, Per1, or Per2. Why is Cry2 knockout rhythmic while the other knockouts are arrhythmic? We examined the Michaelis–Menten curves associated with the Cry1, Cry2, Per1, and Per2 genes in each of these mutants to gain some understanding. All 4 genes can be activated by CLK/BMAL1 and repressed by the remaining PER/CRY complexes. Within the model, the Michaelis–Menten terms associated with each are multiplied together to give a fraction of total transcriptional activity that can take place at each gene. Provided CLK/BMAL1 and PER/CRY levels are such that none of the terms is “0” (the gene is not always “off”) and the terms do not all reside within a part of the Michaelis–Menten curve that is “flat” (the output of the gene is not constant), limit-cycle behavior is obtainable. Fig. 4 A–C shows the Michaelis–Menten curves associated with Per2 transcription in Cry2 knockout mutants, where the red indicates the portion of the curve traversed because of variations in activator or repressor level. When these curves are multiplied together, a variety of activation levels of the Per2 gene is possible, thus allowing limit-cycle behavior. In contrast, Fig. 4 D–F shows the Michaelis–Menten curves associated with Per2 transcription in Cry1 knockout mutants. CLK/BMAL1 activation is complete and inhibition because of PER1/CRY2 is negligible. However, the PER2/CRY2 level is sufficiently high so that it perpetually represses the Per2 gene. The Per2 transcription rate is the product of the CLK/BMAL1 activation rate and the factor because of each repressor (PER1/CRY2 and PER2/CRY2). Because the factor because of PER2/CRY2 repression is close to zero, the overall product is close to zero (and nearly constant). Therefore, the transcription rate at the Per2 gene is close to zero (and nearly constant). This rate would remain close to zero even if the activator and inhibitor levels varied somewhat. Thus, the model suggests why limit-cycle behavior is possible in the absence of Cry2, but not in the absence of Cry1.

Fig. 4.

Michaelis–Menten curves (blue) for CLK/BMAL1 activation of Per2 with Cry2 knockout (A); PER1/CRY1 inhibition of Per2 with Cry2 knockout (B); PER2/CRY1 inhibition of Per2 with Cry2 knockout (C); CLK/BMAL1 activation of Per2 with Cry1 knockout (D); PER1/CRY2 inhibition of Per2 with Cry1 knockout (E); and PER2/CRY2 inhibition of Per2 with Cry1 knockout (F). The portions of the curves highlighted in red are the regions traversed, given the model-predicted concentrations of CLK/BMAL1 or the relevant PER/CRY. Curves in A–C, which pertain to Cry2 knockout, when multiplied together can yield a variety of gene activity levels, while curves in D–F, which pertain to Cry1 knockout, when multiplied together lead to near-inactivation of transcription.

A kinetic explanation may also be given for the increased amplitude of Per2 mRNA observed in Cry2 knockout cells, compared with that seen in wild type. The transcription rate at the Per2 gene in wild-type cells is given as the product of the maximum transcription rate and 5 Michaelis–Menten terms (1 term because of activation by CLK/BMAL1 and 4 because of repression by each of the 4 possible PER/CRY complexes). As discussed above and shown in Fig. S3B, CLK/BMAL1 activation at the Per2 gene is complete (i.e., the term is always “1”), and therefore its contribution can be ignored in the analysis. Similarly, the repression because of PER1/CRY1 and PER1/CRY2 complexes is negligible (i.e., the terms are always “1”), and therefore these contributions can also be ignored (Fig. S3 E–F). In wild-type cells, the transcription rate at any given time is simply the product of the maximum transcription rate and the Michaelis–Menten terms associated with PER2/CRY1 and PER2/CRY2 repression. This rate surface is shown in Fig. S3G, along with the region of the surface actually traversed as the system moves along its limit cycle.

A similar analysis was performed for the Cry2 knockout. Here, the transcription rate at the Per2 gene is given as the product of the maximum transcription rate and 3 Michaelis–Menten terms (1 because of activation by CLK/BMAL1 and 2 because of repression by each of the 2 possible PER/CRY complexes). In the Cry2 knockout system, activation at the Per2 gene by CLK/BMAL1 is not complete, and therefore the Michaelis–Menten term associated with CLK/BMAL1 cannot be ignored. Similarly, repression by PER2/CRY1 is not negligible and cannot be ignored. However, repression by PER1/CRY1 is negligible (Fig. 4B). In Cry2 knockout cells, then, the transcription rate at any given time is the product of the maximum transcription rate and the Michaelis–Menten terms associated with CLK/BMAL1 activation and PER2/CRY1 repression. This rate surface is shown in Fig. S3H, along with the region of the surface actually traversed as the system moves along its limit cycle.

Comparison of Fig. S3 G and H shows that the rate of Per2 production in the Cry2 mutant periodically spikes to a higher level than is ever experienced in wild type, providing an explanation for the higher amplitude of Per2 observed in the knockout. The shape of the curve suggests that these excursions to higher Per2 production rates are due primarily to relaxation of repression by PER2/CRY1 rather than enhanced activation by CLK/BMAL1.

Discussion

Using the latest gene regulation data for the mouse clock, we developed a mathematical model and used an iterative evolutionary algorithm, with a primary focus on the intracellular phase relationships among components, to fit a parameter set. We have shown that the resulting model describes well the molecular patterns observed in the clock at the molecular level, as each species in turn rises and falls in a precise pattern. The model also describes cell-autonomous circadian phenotypes in mutant cells.

It may be argued that, given a sufficient number of parameters in a model, it would always be possible to meet an arbitrary set of criteria. This criticism is valid, and with 132 parameters in our system, the possibility that this assertion could be so was substantial. Consequently, we built our model on engineering principles, whereby the model was parameterized from 1 set of data (in our case, phase and, to a lesser extent, amplitude) and then validated against another set of data (in our case, intracellular knockout phenotypes). Because our model correctly predicts de novo the observed phenotypes of all 7 knockout mutations and 2 double-knockout mutations, there is substantial reason to believe that it describes the system with accuracy. This contention is bolstered by the additional correct predictions associated with expression level changes in these knockout mutants: 10 of 12 are correctly matched. Removal of the Bmal1 native promoter and replacement by a constitutive promoter leads to continued oscillations, even while the respective mRNA becomes flat — in accord with experimental results. Moreover, perpetual activation of Per1, Per2, Cry1, and Cry2 by CLK/BMAL1, deduced from experiment, is evident in an evaluation of our model kinetics.

By definition, no mathematical model is perfectly accurate. Aside from failure to predict the expression level changes in Per2 and Bmal1 mRNAs when the Cry1 gene is removed, our model exhibits a few other problems. First, experimental data point to a smaller amplitude for Cry2 than for Cry1 (7, 36, 45, 46), whereas in our model they are approximately equivalent. Tests also show that constitutive activation of the Bmal1 gene leads to flattening of both mRNA and protein levels; however, our simulations show only flattening of the Bmal1 mRNAs.

These deficiencies may have several causes. First, the architecture in our model may not be accurate because of a lack of complete understanding of the complex transcriptional regulation, or the model's mathematical representation may not be accurate. For instance, the Cry2 mRNA rhythm is of low amplitude in many tissues examined (7, 45, 46), but the mechanism for Cry2 transcription is not understood in detail. As a result, our model does not differentiate Cry2 from Cry1 in terms of their basic transcriptional regulation. Second, posttranslational events, which were minimized in our model, likely play a more significant role in the system than is accounted for here. Third, our decision to downplay concentration levels and amplitudes of mRNAs and proteins in the parameterization effort may account for the larger-than-desired Cry2 amplitude. Fourth, it is possible that the experimental observation of flat BMAL1 protein levels with constitutive Bmal1 promoters is incorrect; the amplitudes of the proteins may simply be smaller than detectable by experiment. Note in particular that our simulations with a constitutive Cry2 promoter lead to a substantial reduction in CRY2 protein amplitude, from ≈6 nM peak-to-trough in the wild-type model to ≈1 nM peak-to-trough in the mutant. This reduction suggests that it may be worthwhile for experimental biologists to reexamine these protein levels. In general, where there are discrepancies between the model and experimental data, biologists are challenged in their understanding of the system and may need to revise hypotheses in future studies.

By allowing each parameter in the wild-type and the 9 single- and double-knockout models to vary around their means so that each took a value drawn from a distribution with a 10% SD and then repeating the process 10,000 times for each model, we were able to assess the stability of each model's gross phenotype of rhythmicity or arrhythmicity within a local region of parameter space. Because all models exhibit stable behaviors, we may conclude that the models are robust in their basic performance and that our nominal parameter set is not in a “foxhole,” producing desired results in an impossibly narrow kinetic range. This stability is an essential criterion for model success, both because the actual kinetics of the system are unknown, so that reality probably does differ somewhat from the numerically derived values, and because the kinetic values do probably vary somewhat over time with factors such as local chemistry.

It must be reemphasized that this model was built from cell-autonomous clock data and is intended to describe that system. It does not, therefore, necessarily describe phenotypes observed in tissues or organisms, where the intracellular clocks are in communication with one another. This fact offers an opportunity for further model development, and it would be especially interesting to see whether the intracellular oscillators modeled here could be coupled in a way that gives rise to the altered higher-level phenotypes.

Last, while our model is highly predictive of the mouse circadian clock, the ultimate goal of model development is to enhance the ability of the scientist to understand the system. By examining the Michaelis–Menten curves associated with the gene regulatory apparatus, our model can explain kinetically why removal of the Per1, Per2, or Cry1 genes leads to arrhythmicity, whereas removal of the Cry2 gene does not have this effect. Additionally, while it may not be surprising that knockout of the Cry2 gene leads to an increase in Per2 amplitude (because of decreased repression in the system), our analyses suggest that this effect is due mostly to rising and falling levels of repression from PER2/CRY1, with only a minor role because of CLK/BMAL1 activation and essentially no contribution because of repression from PER1/CRY1. These analyses open interesting avenues of exploration for experimentalists and would provide a further test for model validation.

Materials and Methods

The model was implemented in MATLAB (Mathworks) as a system of 21 ordinary differential equations (ODEs) (SI Appendix) and evaluated on a cluster of 47 × 2.8 GHz CPUs using the MATLAB Distributed Computing Toolbox. The model incorporates Hill-type and mass action kinetics. Because little experimentally derived kinetic data are available for this system, the model parameters were fit numerically within bounds of 0 and 5. Bounds must be set loose enough to explore a kinetically viable region of parameter space yet tight enough to produce results. Our first attempts used upper bounds of 10 but failed to find parameter sets that produced oscillations. Bounds were tightened until the region of parameter space searched became sufficiently constrained to produce oscillating systems with less than a day of computation. Ten initial parameter sets were retained, all producing limit-cycle oscillations. In addition, these initial parameter sets were required to produce limit-cycle oscillations on a reduced model consisting of Clk, Bmal1, Per1, Per2, Cry1, and Cry2 but lacking Rev-erbα and Rorc. This reduced model was obtained by elimination of the rate equations for Rev-erbα and Rorc mRNAs and proteins. It is known that the molecular clock continues to function without the presence of Rev-erbα or Rorc, so this additional test helped to ensure consistency in accord with known system behavior.

A cost function was developed (SI Appendix) based principally upon the desired phase relationships between mRNAs and proteins and among the various mRNAs. These desired phase relationships are shown in Table S2 and were derived from experimental results (16, 20, 45). Phase separations were measured at constituent maxima. Additionally, costs were developed for the maximum and minimum values for mRNA and protein levels (36), although existing experimental data were determined using peripheral tissues and are thought to be only approximately accurate for SCN neurons; therefore, concentrations were not weighted very highly in the cost function.

The selection of a weight associated with each term in the cost function was a somewhat subjective decision but not without justification. Because the clock was envisioned as a timekeeper that operates by turning on each gene product at specific points in the circadian cycle, the phase relationships among the mRNAs were considered of greatest significance and given weights of 10. Good data are available for the phase lags between mRNA peaks and protein peaks, and these phase lags were given weights of 5. The questionable concentration data were given weights of one-half.

Using the ODE model, the initial 10 “parent” parameter sets, and the cost function, the system was optimized via an evolutionary strategy (33), identifying a set of parameters that most closely matched the desired phase and concentration data. The evolutionary strategy randomly selects 2 of the parent parameter sets, randomly combines their individual parameters, alters the value of each parameter in the combined set, and then evaluates the cost of the combined set through the cost function. Each “child” thus generated was required to meet the principal constraints of limit-cycle behavior on models both including and excluding Rev-erbα and Rorc. The 10 parent parameter sets were thereby used to create 100 “child” parameter sets. The 10 with the lowest cost were retained and became the new parent parameter sets in a cycle that lasted through 100 generations.

The final parameter set was modified to bring the period of its limit cycle to 23.7 h. The final parameter set found was used to simulate the clock, and its period of oscillation was determined. Then, all rate parameters were multiplied by this period divided by 23.7. Consequently, a clock with a period <23.7 h, running too fast, would have all of its rates reduced, slowing the clock, and vice versa for a clock running too slow.