Development and validation of computational models for mammalian circadian oscillators.
ABSTRACT Circadian rhythms are endogenous rhythms with a cycle length of approximately 24 h. Rhythmic production of specific proteins within pacemaker structures is the basis for these physiological and behavioral rhythms. Prior work on mathematical modeling of molecular circadian oscillators has focused on the fruit fly, Drosophila melanogaster. Recently, great advances have been made in our understanding of the molecular basis of circadian rhythms in mammals. Mathematical models of the mammalian circadian oscillator are needed to piece together diverse data, predict experimental results, and help us understand the clock as a whole. Our objectives are to develop mathematical models of the mammalian circadian oscillator, generate and test predictions from these models, gather information on the parameters needed for model development, integrate the molecular model with an existing model of the influence of light and rhythmicity on human performance, and make models available in BioSpice so that they can be easily used by the general community. Two new mammalian models have been developed, and experimental data are summarized. These studies have the potential to lead to new strategies for resetting the circadian clock. Manipulations of the circadian clock can be used to optimize performance by promoting alertness and physiological synchronization.
- SourceAvailable from: Francesca Damiola[show abstract] [hide abstract]
ABSTRACT: The treatment of cultured rat-1 fibroblasts or H35 hepatoma cells with high concentrations of serum induces the circadian expression of various genes whose transcription also oscillates in living animals. Oscillating genes include rper1 and rper2 (rat homologs of the Drosophila clock gene period), and the genes encoding the transcription factors Rev-Erb alpha, DBP, and TEF. In rat-1 fibroblasts, up to three consecutive daily oscillations with an average period length of 22.5 hr could be recorded. The temporal sequence of the various mRNA accumulation cycles is the same in cultured cells and in vivo. The serum shock of rat-1 fibroblasts also results in a transient stimulation of c-fos and rper expression and thus mimics light-induced immediate-early gene expression in the suprachiasmatic nucleus.Cell 07/1998; 93(6):929-37. · 31.96 Impact Factor
- Nature 08/1999; 400(6743):410-1. · 38.60 Impact Factor
- [show abstract] [hide abstract]
ABSTRACT: Although several detailed models of molecular processes essential for circadian oscillations have been developed, their complexity makes intuitive understanding of the oscillation mechanism difficult. The goal of the present study was to reduce a previously developed, detailed model to a minimal representation of the transcriptional regulation essential for circadian rhythmicity in Drosophila. The reduced model contains only two differential equations, each with time delays. A negative feedback loop is included, in which PER protein represses per transcription by binding the dCLOCK transcription factor. A positive feedback loop is also included, in which dCLOCK indirectly enhances its own formation. The model simulated circadian oscillations, light entrainment, and a phase-response curve with qualitative similarities to experiment. Time delays were found to be essential for simulation of circadian oscillations with this model. To examine the robustness of the simplified model to fluctuations in molecule numbers, a stochastic variant was constructed. Robust circadian oscillations and entrainment to light pulses were simulated with fewer than 80 molecules of each gene product present on average. Circadian oscillations persisted when the positive feedback loop was removed. Moreover, elimination of positive feedback did not decrease the robustness of oscillations to stochastic fluctuations or to variations in parameter values. Such reduced models can aid understanding of the oscillation mechanisms in Drosophila and in other organisms in which feedback regulation of transcription may play an important role.Biophysical Journal 12/2002; 83(5):2349-59. · 3.67 Impact Factor
OMICS A Journal of Integrative Biology
Volume 7, Number 4, 2003
© Mary Ann Liebert, Inc.
Development and Validation of Computational Models for
Mammalian Circadian Oscillators
DANIEL B. FORGER,1DENNIS A. DEAN II,2KATHERINE GURDZIEL,2
JEAN-CHRISTOPHE LELOUP,3CHOOGON LEE,4CHARLOTTE VON GALL,4
JEAN-PIERRE ETCHEGARAY,4RICHARD E. KRONAUER,2
ALBERT GOLDBETER,3CHARLES S. PESKIN,1,5MEGAN E. JEWETT,2
and DAVID R. WEAVER4
Circadian rhythms are endogenous rhythms with a cycle length of approximately 24 h.
Rhythmic production of specific proteins within pacemaker structures is the basis for these
physiological and behavioral rhythms. Prior work on mathematical modeling of molecular
circadian oscillators has focused on the fruit fly, Drosophila melanogaster. Recently, great
advances have been made in our understanding of the molecular basis of circadian rhythms
in mammals. Mathematical models of the mammalian circadian oscillator are needed to piece
together diverse data, predict experimental results, and help us understand the clock as a
whole. Our objectives are to develop mathematical models of the mammalian circadian os-
cillator, generate and test predictions from these models, gather information on the para-
meters needed for model development, integrate the molecular model with an existing model
of the influence of light and rhythmicity on human performance, and make models avail-
able in BioSpice so that they can be easily used by the general community. Two new mam-
malian models have been developed, and experimental data are summarized. These studies
have the potential to lead to new strategies for resetting the circadian clock. Manipulations
of the circadian clock can be used to optimize performance by promoting alertness and phys-
logically and to model it mathematically, leading to the generation of experimentally testable hypotheses
HE MAMMALIAN CIRCADIAN TIMING SYSTEM is responsible for the regulation of approximately 24-h rhyth-
micity in many physiological functions. The objective of this effort is to understand this system bio-
1Courant Institute, New York University, New York, New York.
2Biomathematical Modeling Unit, Division of Sleep Medicine, Brigham & Women’s Hospital and Harvard Medical
School, Boston, Massachusetts.
3Unité de Chronobiologique Théorique, Université Libre de Bruxelles, Brussels, Belgium.
4Department of Neurobiology, University of Massachusetts Medical School, Worcester, Massachusetts.
5Center for Neural Science, New York University, New York, New York.
and the potential for development of strategies to manipulate the circadian pacemaker. Our work focuses
on the 24-h biochemical oscillations that occur within individual cells. The suprachiasmatic nuclei (SCN)
of the anterior hypothalamus contain the master circadian pacemaker in mammals. Long-term recordings
of electrical activity rhythms from SCN neurons indicate that individual cells can maintain circadian time-
keeping (Herzog et al., 1998; Liu et al., 1997; Welsh et al., 1995). Furthermore, mutations that influence
rhythmicity in the whole animal have a corresponding effect on individual “clock cells” in vitro (Herzog
et al., 1998; Liu et al., 1997). Thus, in terms of fundamental brain mechanisms, the circadian system is
among the most tractable models for providing a complete understanding of the cellular and molecular
events connecting genes to behavior. A second focus of investigation is identification of the biological mech-
anisms by which light influences the mammalian circadian clock, and incorporation of the influence of light
into our mathematic models. We also intend to compare the influence of light in the molecular model with
our existing model of the influence of light and rhythmicity on human performance.
Basics of circadian rhythms
Numerous aspects of physiology and behavior vary over the course of the 24-h day. The most widely
appreciated daily rhythm is the sleep-wake cycle. Fluctuations in alertness, sleep latency, and other func-
tions occur with a cycle length of approximately 24 h. These fluctuations persist even in the absence of
sleep, or in experimental protocols where sleep is allowed but is distributed evenly (in brief segments) over
the 24-h cycle.
A second, widely appreciated aspect of rhythmicity is revealed by the phenomenon called “jet lag.” Af-
ter flight across several time zones, there is often a period of several days during which the individual is
“out of synch” with their environment. During this period, there is a discrepancy between the body’s bio-
logical time and the actual local time. A period of several days of re-adjustment follows, characterized by
altered sleep-wake cycles and numerous (but minor) physiological and psychological complaints. After sev-
eral days in the new environment, the body becomes resynchronized to the local environment. Unfortu-
nately, this often occurs just in time for the return trip home.
The body “knows what time it is” as a result of a biological timekeeping system. This system can be
thought of as consisting of a pacemaker (or clock), along with the input pathways that allow information
to reach it, and the output pathways that lead to overt rhythms in physiology and behavior (Fig. 1). A crit-
ical defining feature of circadian rhythms is that rhythmicity persists with a cycle length of approximately
(circa) one day (diem, thus the term circadian), even in constant environmental conditions. A second fea-
ture is that the clock mechanism can be reset by environmental stimuli. Light is the most effective agent
for synchronizing the circadian clock to the 24-h day in most species.
While rhythms in physiology and behavior are often studied at the whole-organism or tissue level, the
underlying circadian oscillatory machinery can reside within individual cells. Indeed, circadian rhythms
have been described in unicellular organisms, including cyanobacteria (Dunlap et al., 1999) and in indi-
vidual neurons isolated from the mammalian SCN (Herzog et al., 1998; Liu et al., 1997; Welsh et al., 1995).
While the master clock regulating behavior is located in the SCN of the anterior hypothalamus (Reppert
and Weaver, 2002), there are also circadian oscillators in other tissues (Abe et al., 2002; Balsalobre et al.,
1998; Reppert and Weaver, 2002; Yagita et al., 2000; Yamazaki et al., 2000; Zylka et al., 1998). The same
genes involved in the intracellular SCN clock mechanism are rhythmically expressed in other brain areas
and in peripheral organs, both in vivo and in vitro. These extra-SCN oscillators apparently can sustain
24-h oscillations for only a few days without input from the SCN, while SCN oscillations persist for weeks
to months in cells and tissues in vitro (Abe et al., 2002; Balsalobre et al., 1998; Yamazaki et al., 2000).
The SCN synchronizes the timing of extra-SCN (“peripheral” or “slave”) oscillators (for review, see Schi-
bler et al., 2003). Synchronized slave oscillators, in turn, regulate local (tissue-specific) rhythms in physi-
ology and behavior. From an experimental viewpoint, the existence of peripheral oscillators allows us to
characterize molecular rhythms in normal and mutant mice using peripheral tissues (e.g., liver, skeletal mus-
cle, and cell lines) that are much more abundant than the very small SCN. We have made extensive use of
peripheral tissues as a surrogate for the SCN, with the belief that the molecular components of the intra-
cellular feedback loops are similar across tissues.
FORGER ET AL.
Molecular basis of the mammalian circadian clock
At the molecular level, circadian oscillations are based on the rhythmic expression of clock genes. The
intracellular clock mechanism in the mouse SCN is based primarily on a transcriptional-translational neg-
ative feedback loop (Figs. 2 and 3). Molecular elements that comprise this system in mammals have been
identified recently (Fig. 3). Two basic helix-loop-helix (bHLH)/PAS-containing transcription factors,
CLOCK and BMAL1 (MOP3) provide the basic drive to the system by activating transcription of negative
regulators through E box enhancer elements (Bunger et al., 2000; Gekakis et al., 1998; Hogenesch et al.,
1998; Jin et al., 1999; King et al., 1997; Vitaterna et al., 1994). CLOCK:BMAL1 heterodimers drive tran-
scription of three Period genes (mPer1-3) and two Cryptochrome genes (mCry1-2) (Reppert and Weaver,
2002). mPER and mCRY proteins form complexes, translocate to the nucleus, and interact with
CLOCK:BMAL1 heterodimers to inhibit transcription, closing the feedback loop (Kume et al., 1999; Lee
et al., 2001; van der Horst et al., 1999; Vitaterna et al., 1999).
Analysis of mice with targeted disruption of the circadian-relevant genes described above reveals the im-
portance of these genes in circadian rhythmicity (Fig. 3; for review, see Reppert and Weaver 2002). Mu-
tations leading to transcriptionally non-functional CLOCK or BMAL1 leads to loss of rhythmicity in con-
stant darkness. While the Drosophila cryptochrome homolog functions as a circadian photoreceptor, mouse
mCRY1 and mCRY2 are redundant, and are essential components of the negative limb of the clock feed-
back loop (van der Horst et al., 1999; Vitaterna et al., 1999). Mice deficient in either mPER1 or mPER2
have altered period length and delayed loss of rhythmicity following transfer to constant darkness, while
mice lacking both mPER1 and mPER2 show a more severe phenotype, immediate arrhythmicity on place-
ment in constant conditions (Bae et al., 2001; Zheng et al., 2001). mPER3 does not have a critical role in
the maintenance of the core clock feedback loops (Bae et al., 2001; Shearman et al., 2000a).
COMPUTATIONAL MODELS FOR MAMMALIAN CIRCADIAN OSCILLATORS
in its cage is represented by vertical deflections (bars) on each horizontal line. The horizontal lines represent 48 h of
continuous recording; data are “double-plotted,” with day 2 of data being shown both to the right and below day 1.
Note that running wheel activity occurs at night in mice housed in a light-dark cycle (LD, light cycle shown across the
top of the panel, with light period indicated by the open portion of the bar). When the animal is placed into constant
darkness (DD), it continues to express rhythmicity of behavior, but with a cycle length of less than 24 h. Thus, rhythms
are regulated by an endogenous system and are not simply a passive response to environment. However, light is a crit-
ical stimulus for synchronizing the endogenous rhythmicity to environmental conditions. The circadian timing system
regulates rhythmicity in many aspects of physiology and behavior. (Modified from Bae et al., 2001.)
Critical features of circadian rhythms, illustrated by mouse locomotor activity. Activity of a mouse in a wheel
A “positive loop” regulates the rhythmic expression of Bmal1 (Preitner et al., 2002; Shearman et al.,
2000b). Bmal1 RNA levels are rhythmic, and the rhythms are in antiphase to those for the mPer and mCry
genes. Rhythmic Bmal1 transcription is due to rhythmic repression by REVERB-alpha (Preitner et al., 2002;
Ueda et al., 2002). RevErb-alpha RNA is regulated through E box elements, and has a phase of production
similar to that of mPers and mCrys. mPER2 may also play a role in stimulating Bmal1 expression (Shear-
man et al., 2000b; Zheng et al., 1999). This positive feedback loop regulating the expression of Bmal1 ap-
pears to be less important than the negative feedback loop in the mammalian system. Studies of REVERB-
alpha–deficient mice reveal that Bmal1 RNA levels in the SCN are constantly high and not rhythmic,
consistent with the proposal that rhythmically produced REVERB-alpha normally acts to repress Bmal1
transcription (Preitner et al., 2002). Nevertheless, REVERB-alpha–deficient mice maintain circadian loco-
motor activity rhythms in constant conditions, indicating that rhythmic Bmal1 expression is not necessary
for circadian clock function. We have extended this observation in several ways.
Models of molecular circadian oscillators
Previous work on mathematical modeling of molecular circadian oscillators has focused on Drosophila
(Goldbeter, 1995; Leloup and Goldbeter, 1998, 1999; Smolen et al., 2001, 2002; Tyson et al., 1999; Ueda
et al., 2001). From a biological perspective, homologs of most of the genes involved in the mammalian cir-
FORGER ET AL.
drive transcription of mPer1, mPer2, mCry1, and mCry2 through E box elements. These transcripts are then translated
to proteins that are phosphorylated (by casein kinase), dimerize, and enter the nucleus. These large multiprotein com-
plexes then interact with promoter-bound CLOCK:BMAL1 heterodimers to turn off transcription. A feedback loop in-
volving the rhythmic regulation of Bmal1, through activity of the repressor, REVERB-alpha, is not shown. (Adapted
from Reppert and Weaver, 2002.)
Schematic representation of the intracellular circadian clock in mammals. CLOCK and BMAL1 dimerize and
cadian clockwork were first identified in the fly circadian clock, and the general core clock mechanism of
interacting transcriptional feedback loops is similar between flies and mice (Reppert and Weaver, 2000;
Young and Kay, 2001). There has been a shuffling of functions between the components, however, and
gene duplication in mammals has increased the complexity of the mammalian feedback loops. Neverthe-
less, interaction between groups studying Drosophila and mammalian oscillators, or the mathematical mod-
els that describe them, provide fertile grounds for information transfer. Within the BioCOMP program, the
Byrne group is working on Drosophila circadian models. Mathematical models of the Drosophila oscilla-
tor have provided the starting point for one mammalian model, developed with partial support from the
BioCOMP program (Leloup and Goldbeter, 2003). A second, more detailed, strictly mammalian model of
the mammalian circadian oscillator has been developed based, to the extent possible, on experimental data
collected as a part of this research program (Forger and Peskin, 2003).
METHODS AND RESULTS
Experimental group cumulative results
Posttranslational mechanisms in the circadian clock. While the critical structure of the circadian feed-
back loop is known to be a transcriptional-translational feedback loop, little was known regarding the pro-
COMPUTATIONAL MODELS FOR MAMMALIAN CIRCADIAN OSCILLATORS
Clock Protein Families
Clock Gene Mutations
Gene Type Behavioral Phenotype
Long period to arrhythmic
Var. period to arrhythmic
Var. period to arrhythmic
Short period (tau hamster)
to be important for circadian rhythms in mammals. Within the basic helix-loop-helix-PAS family, CLOCK and BMAL1
are most important for behavioral rhythms, while their close homologs, MOP4 (also called NPAS2) and MOP9 (also
called BMAL2) appear less important within the central clock. The Period gene family has three members, mPer1,
mPer2, and mPer3. The Cryptochromefamily has two members, mCry1 and mCry2. Two casein kinases, CK1 epsilon
and CK1 delta, are capable of phosphorylating PER proteins as well as other circadian proteins. Signature structural
domains for each gene family are indicated. (Right) Mutations alter circadian behavioral rhythms in mice. For the Pe-
riod and Crytochrome gene families, partial redundancy of function is apparent. No mutations of CK1 delta have been
reported. (Adapted from Reppert and Weaver, 2002.)
Molecular components of the mammalian circadian clockwork. (Left) Four families of proteins are known
teins and their post-transcriptional modifications. Efforts to characterize circadian protein rhythms have fo-
cused on liver tissue, and have required generation of specific antisera for mouse proteins (Lee et al., 2001).
These antisera have been used for three types of experiments: Western blot analysis (detection of protein
levels and molecular size), immunoprecipitation (isolation of protein complexes by precipitation with an
antibody, then identification of co-precipitating proteins by Western blot analysis), and chromatin im-
munoprecipitation (isolation of chromatin fragments by precipitation with an antibody to protein bound to
DNA, followed by assessment of the levels of specific DNA sequences in the precipitated material).
The analysis of circadian protein rhythms revealed several important findings (Lee et al., 2001). Absolute
levels of the mCRY proteins are significantly higher than PER levels. Cellular fractionation studies reveal
that mCRY is present in the cytoplasm throughout the circadian cycle. Nuclear entry of the mCRY and
mPER proteins occurs simultaneously, both in liver and in SCN (Hastings et al., 1999; Lee et al., 2001),
suggesting that the highly rhythmic production of mPER proteins is the key to regulating nuclear entry of
the complex consisting of the negative regulators (Lee et al., 2001).
An unexpected finding was that the levels of casein kinase I epsilon exceed PER levels throughout the
circadian cycle (Lee et al., 2001). This situation indicates that the enzyme is actually more abundant than
Consistent with previous work in Drosophila, analysis of mouse proteins revealed temporal changes in
phosphorylation state of circadian proteins, especially mPER1, mPER2, CLOCK, and BMAL1 (Lee et al.,
2001). These changes in phosphorylation state (detected by alterations in electrophoretic mobility) occur
coincident with changes in cellular localization and inter-molecular interactions, indicating an important
role for post-translational mechanisms in clock function.
Details of transcriptional mechanisms. The accessibility of DNA for interaction with transcription fac-
tors is strongly influenced by the interaction of DNA and histone proteins in nucleosomes. Post-transla-
tional alterations in histone H3 and histone H4 (e.g., phosphorylation, acetylation, and methylation) at spe-
cific residues in the amino-terminal tail of these proteins are highly correlated with alterations in
transcriptional activity of specific genes. In liver, a circadian rhythm in histone modification occurs at the
promoters of the mPer1, mPer2, and mCry1 genes (Etchegaray et al., 2003). The acetylation of histone H3
is rhythmic, and the rhythm parallels the rhythms in mPer1 RNA levels and in RNA polymerase II bind-
ing to the promoter. The rhythmic acetylation of histone H3 is likely accompanied by other covalent mod-
ifications (e.g., histone phosphorylation and methylation) that collectively alter chromatin structure. Thus,
the circadian rhythmicity in transcription appears to be regulated by a rhythm in chromatin remodeling
which prepares the promoter region for the activation/inhibition cycle. The mCRY proteins may repress
CLOCK:BMAL1-mediated transcription by inducing alterations in chromatin structure (e.g., disrupting a
coactivator complex or recruitment of a histone deacetylase activity).
Mechanisms of circadian entrainment in the SCN. A critical feature of the circadian timing system is the
ability to be entrained by the environmental light/dark cycle. Time-dependent responsiveness to light is a
characteristic feature of circadian clocks in diverse species (Dunlap, 1999). Exposure of mice to light early
in the night shifts the clock so that subsequent cycles begin at a later time. Exposure to light late in the
night, in contrast, advances the circadian clock.
Several lines of evidence suggest that mPER1 and mPER2 play important roles in mediating phase-shift-
ing responses to light (for review, see Reppert and Weaver, 2002; Bae and Weaver, 2003). Our studies of
mice with disruption of mPer genes indicate, however, that neither mPER1 nor mPER2 is absolutely re-
quired for phase-shifting by light (Bae and Weaver, 2003). This suggests that multiple molecular pathways
may be activated in response to light, and that effort should be spent on identifying these pathways. Re-
cently, induction of DEC gene expression in the SCN in response to light has been reported (Honma et al.,
2002); the functional importance of this event also should be investigated.
One alternative pathway for light-induced resetting of the circadian clock has been proposed involving
rapid, light-induced destruction of BMAL1 protein. (This would be an attractive mechanism, as light-in-
duced degradation of the TIM protein is critical for resetting of the Drosophila circadian pacemaker.) This
proposal is based on immunoblot studies conducted with rat SCN. The studies revealed a circadian rhythm
FORGER ET AL.
in BMAL1 protein, with peak levels occurring at night, and rapid degradation of BMAL1 after exposure to
light at night (Tamaru et al., 2000). We have recently reexamined this issue in mice. Our study (von Gall
et al., 2003) shows that BMAL1 and CLOCK proteins are continuously expressed at high levels in the
mouse SCN, supporting the hypothesis that rhythmic negative feedback plays the major role in rhythm gen-
eration in the mammalian pacemaker. Furthermore, light exposure did not lead to degradation of BMAL1
protein in the mouse SCN, as assessed by both immunocytochemistry and immunoblot analysis (von Gall
et al., 2003). These results indicate that rapid degradation of BMAL1 protein is not a consistent feature of
resetting mechanisms in rodents.
Experimental group work in progress. Many of our conclusions regarding the relative amounts of cir-
cadian proteins and their interactions are derived from studies in liver (Lee et al., 2001). We are working
on confirming these findings in other tissues using methods similar to those described previously. Deter-
mination of the accuracy of the parameter values from liver in at least one additional tissue is an impor-
tant, but time-consuming, effort.
We continue to work with several cell lines with the objective of being able to induce rhythmicity un-
der controlled conditions and also to visualize rhythmicity within individual cells. Specifically, we have
generated a number of constructs with destabilized fluorescent reporter proteins under control of E-box con-
taining promoter elements that should oscillate in vitro after experimental perturbation (e.g., serum shock;
see Balsalobre et al., 1998; Yagita et al., 2001). However, we have had poor luck to date in generating sta-
ble cell lines that show the expected rhythmic expression patterns. We also have worked rather extensively
with stable human cell lines with inducible expression of constructs that direct expression of mouse circa-
dian proteins (O. Gildemeister, K. Bae, O. Froy, and D.R. Weaver, unpublished data). Additional charac-
terization of the cell lines is continuing, and we are testing additional manipulations for ability to induce
rhythmicity. Detection of rhythmicity in individual cells will allow us to gather important information on
the precision of oscillations within individual pacemakers, revealing both the cycle-to-cycle variability as
well as the variation between cells. Almost all existing data addresses the mean behavior of populations of
oscillators without addressing the behavior of individual oscillators. Analysis of oscillations in single cells
will impact our understanding of oscillator robustness and the importance of molecular noise. Our future
plans are to continue the work in progress as described above, and to test model predictions generated by
the Modeling Group regarding molecular shuttling and the influence of protein:protein interactions on pro-
tein localization and stability. We will also investigate mechanisms for cooperativity at E box elements in
the promoter of responsive genes.
Modeling group cumulative results
Two quite distinct molecular models of the mammalian circadian oscillator have been developed. There
are several features that one would expect to be achieved by a circadian model (Table 1) and both models
meet the major criteria. Within each model, there are terms for the processes influencing each molecular
entity (Fig. 4), including RNA production (transcription) and degradation, intracellular movement of RNA,
protein and protein/protein complexes, interactions between proteins and alteration in their localization,
stability, and activity as a result of these interactions. The major difference between the two models is the
level of biochemical detail which they represent.
A mammalian model developed as an adaptation of a Drosophila model. Jean-Christophe Leloup and
Albert Goldbeter have developed a deterministic model of the mammalian circadian pacemaker (Leloup
and Goldbeter, 2003). The modeling techniques used are similar to a previous model they developed of the
Drosophila circadian oscillator (Leloup and Goldbeter, 1998). The Leloup-Goldbeter (2003) mammalian
model does not distinguish among the products of the three mPer genes or the two mCry genes, and in-
stead uses single mPer and mCry entities (RNA and proteins), and with phosphorylation as a mechanism
controlling protein degradation. Enzyme-substrate interactions are modeled with Michaelis-Menten type ex-
pressions, and transcription regulation is modeled by Hill-type expressions. The model consists of 16 equa-
tions. By the addition of three additional equations and the modification of one other, the model can be
COMPUTATIONAL MODELS FOR MAMMALIAN CIRCADIAN OSCILLATORS
modified to include the influence of REV-ERB-alpha on Bmal1 transcription. Light input to the model is
achieved through elevation of mPer levels. Two benefits of this approach are the limited number of equa-
tions and that the model can also be used as a Drosophila model simply by renaming variables and chang-
ing the effects of light.
FORGER ET AL.
TABLE 1.DRAFT CRITERIA FOR ASSESSING CIRCADIAN OSCILLATORY MODELS
Criterion Expected model performance
Molecular oscillations occur with a free-running period (tau) of approximately 24 h.
(a) Molecular oscillations occur with appropriate phase relationships to each other in free-
running conditions (appropriate 5 consistent with experimental data or plausible).
(b) Molecular oscillations occur with appropriate phase relationships to each other and to
the light-dark cycle.
(a) Input repeated at 24-h intervals results in 24-h periodicity of the molecular oscillations.
(b) The molecular basis for the input to influence molecular oscillations should be based on
(a) Single stimuli lead to alterations in the phase of molecular oscillation.
(b) The response to a stimulus depends on the phase at which it is administered.
(c) The molecular basis for the input to influence molecular oscillations should be based on
Mutations affecting the level or activity of circadian-relevant genes in vivo should produce
similar effects on oscillations in silico.
4. Phase response
cillator are RNA transcription, transport and degradation, protein production, protein modification (by phosphorylation),
and interactions, degradation, and entry of multimeric complexes into the nucleus. This schematic represents steps for
mPER1 protein (square), including interaction with casein kinase (rectangle), leading to phosphorylation (addition of
small black “lollipops”), and interaction with mCRY proteins (triangle).
Biochemical processes considered in mammalian model. Processes included in models of the circadian os-
Development of a mammalian model based on experimental data. Recent data on the mammalian circa-
dian oscillator provide a detailed view of transcription regulation, as well as post-translational events, in-
cluding phosphorylation. To model these data, the actual binding between kinases and substrates must be
used instead of Michaelis-Menten dynamics. In addition, data from the experimental group provides a more
detailed picture of transcription regulation than Hill-type expressions allows. A new mammalian model
(Forger and Peskin, 2003) has been developed which considers these details. The model incorporates a se-
ries of 73 equations. The molecular entities tracked in the model include: mPER1, mPER2, mCRY1, mCRY2,
and casein kinase I proteins, and their corresponding mRNAs. CLOCK and BMAL1 are constitutively ex-
pressed. PER3 is not included in the model, as it appears not to play a role in the core circadian feedback
loop (Bae et al., 2001; Shearman et al., 2000a). Where indicated by experimental data, the model consid-
ers multiple phosphorylation states of several of these proteins. Incorporating this level of detail allows us
to test the differential roles of mPER1 and mPER2 in phase resetting, simulate mutations in individual pro-
teins (e.g., mPER1 or mPER2), and study specific aspects of phosphorylation (e.g., the tau mutation) or
transcription regulation. This model achieves a good agreement with experimental data (Fig. 5).
Modeling group future directions. The major objectives of the Modeling Group in the future will be de-
velopment of a stochastic version of the Forger-Peskin mammalian model, sensitivity analysis of the model,
and mathematical reduction of the model to allow its integration as an input component to an existing model
of the human circadian pacemaker and its role in regulating neurobehavioral performance. More specifi-
cally, a model exists describing the influence of light on the human circadian pacemaker and on perfor-
mance (Forger et al., 1999; Jewett and Kronauer, 1998, 1999; Jewett et al., 1999; Kronauer et al., 1999).
In a previous modeling and analytical study, Forger and Kronauer (2002) showed how a biochemical cir-
cadian clock model (Goldbeter, 1995) had essentially the same behavior as a model describing the influ-
ence of light on the human circadian pacemaker (Forger et al., 1999). In line with this study, our project
objective is to develop and simplify, then merge the detailed molecular models with this existing perfor-
mance model. To do this, we will need to conduct extensive analyses to determine the most critical vari-
ables to consider in the simplified format. This analysis is ongoing. Our Software Group will participate
COMPUTATIONAL MODELS FOR MAMMALIAN CIRCADIAN OSCILLATORS
total mPER levels were generated from data on the mPER1 and mPER2 rhythms in whole cell extracts from mouse
liver, determined by quantitative western blotting (Lee et al., 2001). The model predictions are from the mammalian
model developed by D.B. Forger and C. Peskin (2003). Shown is model prediction (line) and experimental data (points)
in darkness after entrainment to a lighting cycle consisting of 12-h light and 12-h darkness per 24 h. The amplitude
and shape of the profile are correctly predicted, and the phase of entrainment of the model accurately reflects the ex-
Comparison of model predictions with experimental data. Experimental data (points) for the time profile of
through their knowledge of BioSpice tools that may be useful in simplifying the model structure. Addi-
tional areas that we intend to pursue in the future include comparison of the two mammalian models, in-
vestigation of model behavior in the presence or absence of the REV-ERB-alpha loop, and a more detailed
investigation of entrainment mechanisms.
Software group cumulative results
Software submissions in support of our BioCOMP program effort provide resources to allow the BioSpice
community, and eventually the public, to work with published circadian oscillator models. The models cur-
rently available through the BioCOMP website are listed in Table 2. Models have been submitted in sev-
eral formats, consistent with the moving target for development of models and their distribution. Most re-
cently, we have translated three models into SBML as part of our support for BioSPICE agents, in order
to allow the agents to pass Experimental Working Group data types or SBML structures between the fa-
cilitators or agents. We used both Jigcell and Cellerator (www.aig.jpl.nasa.gov/public/mls/cellerator/) to cre-
ate SBML-compatible implementations of three published Drosophila models (Goldbeter, 1995; Leloup and
Goldbeter, 1998; Ueda et al., 2001).
Current software efforts are focused on assisting model development and in preparing the newly devel-
oped mammalian models for submission in BioSpice-compatible format. We are also involved in develop-
ment of user interfaces for the models (Fig. 6).
Our future plans are to (a) continue to submit molecular models to the BioCOMP program according to
the current state of the art, and (b) investigate resources available within the BioCOMP program, (c) use
these resources to promote modeling efforts, and (d) interact effectively with other groups and developers
to identify areas where additional work is needed. As we begin integration of the molecular model with the
human performance model, a significant effort will be devoted to identification of BioCOMP resources that
will facilitate model development. Areas of major importance include automatic parameter fitting and iden-
tifying tools for determining the quality of fit for objective comparison of model predictions with experi-
The Software Group (Dennis Dean and Katherine Gurdziel) will remain instrumental in our efforts to
make circadian oscillatory models available in a BioSpice environment and in a user-friendly format. This
will include providing examples of the user interfaces and also gathering feedback from biologists on the
“accessibility” of BioCOMP program resources.
The circadian oscillator represents a useful test case for the principle underlying the BioCOMP program.
A relatively circumscribed set of molecular and genetic interactions leads to a biological event, oscillation,
with a period of approximately 24 h. The oscillations of individual cells are representative of oscillatory
mechanisms that occur at the organismal level. Furthermore, these oscillations have important functional
FORGER ET AL.
TABLE 2.CIRCADIAN MODELS IN THE NEURORHYTHMS AREA OF THE SANDBOX
Model Species Matlab SBML ParametersDocumentation
Leloup and Goldbeter, 1998
Ueda et al., 2001
Leloup and Goldbeter, 1999
Leloup and Goldbeter, 2003
Forger and Peskin, 2003
1, Information currently posted to the NeuroRhythms section of the BioSpice website.
11, Both Chaos and Birhythmicity parameter sets are included.
COMPUTATIONAL MODELS FOR MAMMALIAN CIRCADIAN OSCILLATORS
User Interface for the Goldbeter (1995) Drosophila model. (A) Parameter input screen. (B) Output screen. Robust circa-
dian rhythmicity in the abundance of the molecular species is present despite a slight change from the default para-
meter set. The only change from default is that k1, the transfer rate of PER into the cytoplasm, has been changed from
the default value of 2.0 to 1.9. (C) The amplitude of rhythmicity is greatly reduced within several cycles when K1, the
Michaelis constant of PER in the cytoplasm, is changed from 2.0 to 0.01. The user interface and models are accessible
in the Jewett Team submissions in the NeuroRhythms section of the Sandbox of the BioSPICE website (www.biospice.org).
(From D. Dean and K. Gurdziel, unpublished.)
Representative circadian model user interface created in Matlab. Representative “screenshots” depicting the
implications. Optimization of work schedules through attention to circadian principles is now more com-
mon in the workplace, and the use of timed light exposure to optimize performance and mission readiness
offers significant advantages over current pharmacological approaches, such as the use of alertness-pro-
moting stimulants. Thus, we have a cellular model system for experimental analysis and mathematical mod-
eling, with tangible implications for human performance.
Parallel development of mammalian models by our research team and of Drosophila models by the Byrne
group (Smolen et al., 2001, 2002) within the BioCOMP Program provides important opportunities for in-
teraction and exploration of the specific aspects of model behavior. The group of investigators led by Jim
Schwaber focuses on gene regulatory networks and signal transduction events, particularly those occurring
after angiotensin receptor activation, a stimulus that can initiate rhythmicity in cultured cells (Nonaka et
al., 2001). Environmental light information is conveyed to the SCN oscillator by neurochemical pathways,
leading to activation of signal transduction pathways that convey input from the cell surface into the os-
cillator, and leading to transcriptional responses. Thus, there is likely overlap in the cellular mechanisms
occurring in our experimental models. Furthermore, the Schwaber group has worked with mathematical
models of circadian rhythms (Zak et al., 2001). Finally, the Tyson group also has experience with model-
ing oscillators, including circadian oscillators (Tyson et al., 1999). These groups participate in the Neu-
roRhythms group, which can be found in the Sandbox on the BioSpice website (www.BioSpice.org), and
would be a good starting point for those seeking additional information about recent advances in this area
Our long-term objective is to achieve a comprehensive understanding of the circadian oscillator itself, as
well as how light exposure impacts the oscillator. This information should allow identification of novel
pharmacological approaches to either stop the clock or reset it with precision, regardless of ambient light-
ing conditions. Control over the circadian oscillator should promote adaptation to shift work and non–
24-h duty cycles, and minimize the physiological disturbance caused by jet lag.
This effort was sponsored by the Defense Advanced Research Projects Agency (DARPA) and the Air
Force Research Laboratory under agreement number F30602-01-2-0554. The U.S. Government is autho-
rized to reproduce and distribute reprints for Government purposes notwithstanding any copyright annota-
tion thereon. D.B. Forger was supported by an NSF pre-doctoral fellowship. J.-C. Leloup is Chargé de
recherches du Fonds National de la Recherche Scientifique (F.N.R.S., Belgium). Portions of the Experi-
mental Group work reported here were also supported by NIH grant NS39303 to S.M. Reppert and the
Emmy-Noether Programm of the Deutsche Forschungsgemeinschaft to C. von Gall. The contents of this
publication are solely the responsibility of the authors and do not necessarily represent the official views
of the awarding agencies.
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FORGER ET AL.