J. theor. Biol. (2002) 216, 501–512
doi:10.1006/yjtbi.3000, available online at http://www.idealibrary.com on

Intercellular Coupling Mechanism for Synchronized and Noise-ResistantCircadian Oscillators
Hiroki R. Ueda*wz, Kenzo Hirosew and Masamitsu Iinow
wDepartment of Pharmacology, Graduate School of Medicine, The University of Tokyo, Bunkyo-ku,
Tokyo 113, Japan and zMolecular Medicine Research Laboratories, Institute for Drug Discovery
Research, Yamanouchi Pharmaceutical Co., Ltd., 21 Miyukigaoka, Tsukuba-shi,
Ibaraki 305-8585, Japan(Received on 13 June 2001, Accepted in revised form on 22 February 2002)

The circadian clock in multicellular organisms consists of multiple autonomous single-cell
oscillators. These individual oscillator cells produce coherent oscillations even in the presenceof internal noise associated with rhythm-generating reaction rates and in the absence of
external time cues such as light and temperature. Thus, an intercellular coupling mechanism
must synchronize the cells to induce coherent circadian oscillations. We propose the roles of asynchronizing factor that is secreted from individual cells during subjective day to induce
light-pulse-type phase shifts in the neighboring cells or, alternatively, a factor that is secreted
during subjective night to induce dark-pulse-type phase shifts. Here, we present ourmulticellular stochastic model of Drosophila circadian rhythms that emulates the intercellular
coupling mechanism and suggest that the mechanism facilitates the constancy of thecircadian rhythm with possible functional redundancy among different synchronizing factors.
r 2002 Elsevier Science Ltd. All rights reserved.

Introduction
The Drosophila circadian clock is one of thebest characterized circadian systems (Dunlap,
1999; Pittendrigh, 1993; Williams & Sehgal,
2001). In Drosophila, like many other organisms,
several genes are involved in sustaining thecircadian rhythm, namely period ( per) (Konopka& Benzer, 1971), timeless (tim) (Sehgal et al.,1994), Drosophila Clock (dClk) (Allada et al.,1998), Cycle (Cyc) (Rutila et al., 1998) anddouble-time (dbt) (Price et al., 1998) (Fig. 1A).
*Corresponding author. Department of Pharmacology,
Graduate School of Medicine, The University of Tokyo,
Bunkyo-ku, Tokyo 113, Japan. Tel.: +81-3-5841-3414; fax:
+81-3-5841-3390.
E-mail address: hiro@m.u-tokyo.ac.jp (H.R. Ueda).
0022-5193/02/ $35.00/0

Among them, three genes are rhythmically
expressed: per (Hardin et al., 1990), tim (Sehgal
et al., 1995) and dClk (Bae et al., 1998). per andtim mRNAs peak early in the evening, whereas
dClk mRNA peaks at night to early in themorning. Analyses of per and tim oscillatory
expression have revealed a negative per–tim
feedback loop, in which per and tim expressions
are repressed by PERIOD (PER) and TIMELESS (TIM) heterodimers (PER–TIM) (Darlington et al., 1998). On the other hand, analysesof dClk periodic expression have uncovered
another inhibitory dClk feedback loop, in which
dClk expression is repressed by two helix-loophelix-PAS transcription factors, Drosophila
CLOCK (dCLK) and CYCLE (CYC) (Baeet al., 1998; Darlington et al., 1998; Glossop
r 2002 Elsevier Science Ltd. All rights reserved.

502

H. R. UEDA ET AL.

et al., 1999). The per–tim feedback loop
and dClk feedback loop are interlockedwith each other because the per–tim feedback
loop is activated by dCLK and CYC
heterodimers (dCLK–CYC) and the dClk feedback loop is derepressed by PER–TIM (Glossop
et al., 1999). A single cellular deterministic
model based on this structure (Fig. 1A) has
been shown to be sufﬁcient to reproduce
wild-type and several mutant phenotypes (Ueda
et al., 2001).
The circadian clock in multicellular organisms
consists of multiple autonomous single-cell
oscillators (Plautz et al., 1997; Welsh et al.,1995). Importantly, these individual cells
produce coherent oscillations (Yamazaki et al.,2000) even in the presence of internal
noise, which results from the stochastic natureof reaction events when there are few molecules
in a cell (Barkai & Leibler, 2000; McAdams &Arkin, 1999) and in the absence of external
time cues such as light and temperature. Moreover, Takahashi’s group reported a detailed
rhythm analysis on chimeric Clock/WT mutant
mice that strongly support the importance ofintercellular connections of neuronal clock cells(Low-Zeddies & Takahashi, 2001). In orderto address how multiple cells produce
coherent rhythms in the presence of internal
noise without referring to external time cues,
we extended the single cellular deterministic
model on Drosophila circadian rhythms(Ueda et al., 2001) to a three-dimensional 100cellular stochastic model. We hypothesized theexistence of a synchronizing factor that isregulated by the circadian clock, secreted from"———————————————————————————————————————————
Fig. 1. Molecular mechanisms of Drosophila circadian rhythms. (A) Interlocked feedback mechanism of Drosophila
circadian rhythms. per–tim feedback loop: Transcription of per and tim is activated by dCLK–CYC transcriptional
activators. PER is destabilized by DBT-dependent phosphorylation. PER and TIM dimerize with each other. PER–TIM
heterodimers enter the nucleus. Nuclear PER–TIM heterodimers repress transcription of per and tim by forming complexes
with dCLK–CYC heterodimers. dClk feedback loop: dClk expression is activated by PER–TIM heterodimers and repressedby dCLK–CYC heterodimers. per–tim and dClk feedback loops are interlocked with each other to produce anti-phaseoscillations of per (tim) and dClk mRNA. (B) Time courses of per and tim expression simulated with the interlocked
feedback model. per and tim mRNA (blue), PER and TIM monomers (light blue), cytoplasmic (green) and nuclear (lightgreen) PER–TIM heterodimers peak at circadian time (CT) 15.0, 17.3, 17.9 and 21.6, respectively. CT ¼ 0 and CT ¼ 12 areusually deﬁned as dawn and dusk, respectively. In the simulations, we deﬁned CT ¼ 0 as bottom of per mRNA expression.
Expression patterns of per and tim mRNA (blue) or PER and TIM monomer (green) are overlapped because identicalparameter values are used for per and tim expression. (C) Time courses of dClk expression. dClk mRNA (red), dCLK (pink),
cytoplasmic (orange) and nuclear (yellow) dCLK–CYC heterodimers peak at CT ¼ 4.0, 7.5, 8.2 and 13.4, respectively.

Fig. 2. Multicellular stochastic models of Drosophila circadian rhythms. (A) Simulated time course of per mRNA levelin the single cellular deterministic model. (B–D) Simulated time course of per mRNA level in the 100-cellular stochastic
model. Amplitude of average per mRNA oscillations is reduced to less than 5% of the amplitude in the deterministic model(B) due to loss of coherence. Despite the identical initial conditions (C), the introduction of 10% noise completely abolishes
coherent per mRNA oscillations eventually (D). Different colors represent different cells.

Fig. 3. Synchronization mechanisms for coherent oscillations. (A) Magnitude of phase shifts induced by 18 possible
synchronizing factors found in ‘‘day’’ systems, in which the synchronizing factors are secreted during subjective day under
repression by cytoplasmic PER proteins (inset). (B) Magnitude of phase shifts induced by 20 possible synchronizing factors
found in ‘‘night’’ systems, in which the synchronizing factors are secreted during subjective night under activation by nuclear
dCLK–CYC heterodimers (inset). (C) Secretion patterns of seven synchronizing factors found in ‘‘light-pulse’’ systems, inwhich a synchronizing factor induces light-pulse-type phase shifts (inset). (D) Secretion patterns of 10 synchronizing factors
found in ‘‘dark-pulse’’ systems, in which a synchronizing factor induces dark-pulse-type phase shifts (inset). Lines with anasterisk represent two overlapping lines. (E) Average per mRNA oscillation in a synchronized system. In this system, thesecretion of the factor is activated by cytoplasmic PER proteins and the secreted factor promotes degradation of nuclear
PER–TIM heterodimers in the neighboring cells. Even from totally different initial conditions, the presence of asynchronization factor restores large and reproducible amplitude of average per mRNA oscillation. The time courses of permRNA level in individual cells show coherent oscillations (inset). (F) Average per mRNA oscillations in an unsynchronizedsystem. In this system, the secretion of the factor is activated by cytoplasmic PER proteins and the secreted factor represses
degradation of nuclear PER–TIM heterodimers in the neighboring cells. per mRNA levels in individual cells show totally
incoherent oscillations (inset).

doi. 10:1006 jtbi.2002.3000
H. R. UEDA ET AL.

doi: 10.1006 jtbi2002.3000
H. R. UEDA ET AL.

COUPLING MECHANISM CIRCADIAN OSCILLATORS

the cells, and received by the neighboring cells toaffect the rate of one of the reactions in thecircadian system. Through extensive simulation
and analysis, we determined the roles of asynchronizing factor that is secreted fromindividual cells during subjective day to induce
light-pulse-type phase shifts in the neighboring
cells or, alternatively, a factor that is secreted
during subjective night to induce dark-pulsetype phase shifts. We also found that theintercellular coupling mechanism facilitatesnoise resistance of the circadian system withpossible functional redundancy among different
synchronizing factors.
Results
STOCHASTIC SIMULATION OF MULTIPLE
CIRCADIAN OSCILLATORSThe rates of reactions in the circadian clock
are affected by stochastic noise because limited
copy numbers of molecules are involved in thereaction steps within individual cells (Barkai &Leibler, 2000; McAdams & Arkin, 1999). Suchinherent noise is expected to perturb thesynchrony among clock cells. To assess
the effect of noise on the circadian clock, we
extended the single-cellular deterministic modelof Drosophila circadian rhythms to a 100-cellularstochastic model. The per mRNA level of thesingle-cellular deterministic model shows oscillation with a constant period and a large

503

amplitude (Fig. 2A). The multicellular stochastic
model shows a dramatically different behavior
from the single-cellular deterministic model.
Despite the identical initial conditions (Fig. 2C),
the introduction of 10% noise to each reaction
rate (see Appendix A) completely abolishes
coherence in per mRNA oscillations in different
cells (Fig. 2D) and the averaged amplitude
is reduced to less than 5% of that of thedeterministic model (Fig. 2B). In reality, multiple clock cells (Plautz et al., 1997; Welsh et al.,1995) exhibit synchronized rhythm even in theabsence of external time cues such as light andtemperature (Brandes et al., 1996; Yamazaki
et al., 2000). Therefore, our model predicts thatcircadian systems in multicellular organisms
require intrinsic mechanisms for synchronization. The basic idea of synchronizing clocks waspreviously proposed as the ‘‘clock-shop’’ modelby Winfree (1975). However, a detailed molecular mechanism of synchronization has not yet
been described.
SEARCH FOR SYNCHRONIZATION FACTORS

To elucidate the molecular mechanism ofsynchronization, we hypothesized the existenceof a synchronizing factor that is: (1) locateddownstream of one of the components of thecircadian clock (e.g. activated by dCLK–CYC
transcription factors), (2) secreted from thecells and transported through diffusion, and(3) received by the neighboring cells to affect3———————————————————————————————————————————
Fig. 4. Noise tolerance induced by a synchronizing factor. Temporal autocorrelation function of per mRNA oscillationsin the system with (B) or without (A) a synchronizing factor. The system without synchronization (A) loses autocorrelation
rapidly and exponentially whereas the system with synchronization (B) loses autocorrelation slowly and linearly. Envelopes(peaks and bottoms connected, respectively) of the autocorrelation functions of the system with (D) or without (C)synchronization in individual cells (red lines) and data averaged over 100 cells (black line). The system without
synchronization maintains half-correlation only for 490 hr on average with large variance whereas the system withsynchronization holds half-correlation for 7400 hr on average with small variance. Due to the ﬁnite duration of the model(20 000 hr), the envelope of the autocorrelation function of a perfect system declines linearly (dotted lines).

Fig. 5. Schematic explanation of the synchronization mechanism. For simpliﬁcation, two circadian oscillators having a
5-hour difference in phase are considered (A). Light-pulse-type phase shifts of the delayed (A) and advanced (D) cells, andphotic secretion of a synchronizing factor from the delayed (B) and advanced (C) cells are plotted against the subjective
circadian time. The advanced cell secretes the factor during subjective day (C), which advances the delayed cell during
subjective late night to early day (A). On the other hand, the delayed cell secretes the factor during subjective day (B), which
delays the advanced cell during subjective late day to early night (D). These processes reduce the phase difference between
cells.

504

H. R. UEDA ET AL.

the rate of one of the reactions in thecircadian system (e.g. affecting the rate of TIM
degradation).
First, in order to determine the appropriate
target reaction controlled by the synchronizing
factors, we ﬁxed the secretion time course of thepotential synchronizing factor and scanned thereaction steps that are controlled by the factor(see Appendix B). We performed two types ofsimulations: a ‘‘day’’ system, in which a factor issecreted during subjective day under repression
by PER (Fig. 3A inset), and a ‘‘night’’ system,
in which a factor is secreted during subjective
night under activation by nuclear dCLK–CYC(Fig. 3B inset). For each system, we generated 96
variant models, in which the factor either
up- or down-regulates one of the 48 reactionsin the circadian clock. We found that some ofthe variant models restored coherent per mRNA
oscillations (Fig. 3E), whereas others failed to doso (Fig. 3F). As a result, we found 18 and 20
possible target reactions regulated by a synchro-nizing factor in the day and night systems,
respectively. In the ‘‘day’’ system, 18 possible
reactions controlled by the synchronizing factor
result either in a decrease of nuclear PER–TIM
heterodimers or in an increase of nuclear dCLK–CYC heterodimers. On the other hand, 20
possible reactions regulated by the factor in the‘‘night’’ system lead to an increase of nuclear
PER–TIM heterodimers or in a decrease of
nuclear dCLK–CYC heterodimers (Table 1).
To elucidate the common characteristics of thefound target reaction steps, we analysed alteration of the phase, period and amplitude of permRNA oscillation induced by the synchronization factor. The period and the amplitude of permRNA oscillations are slightly changed (periods
range from 22.1 to 25.5 hr, and amplitudes range
from 108.8 to 95.5% of that without synchronizing factors). However, no common features ofthe effects on the period and the amplitude wereobserved. On the other hand, the phase shifts
induced by the regulation of the target reaction

Table 1
Reactions regulated by synchronizing factors secreted during subjective day or night‘‘Day’’ systemn

PER translation (S2 ; i)
TIM translation (S4 ; i)
Cytoplasmic PER–TIM degradation (D5 ; p)
Nuclear PER–TIM degradation (D6 ; p)
PER–TIM nuclear import (T1 ; i)
PER–TIM unbinding from nuclear importer (K1 ; p)
PER–TIM nuclear export (T2 ; p)
PER–TIM unbinding from nuclear exporter (K2 ; i)
PER–TIM unbinding from the dClk promoter (A3 ; p)dCLK–CYC unbinding from the dClk promoter (R3 ; i)dCLK–CYC association (V3 ; p)dCLK–CYC disassociation (V4 ; i)dCLK–CYC nuclear import (T3 ; p)dCLK–CYC nuclear export (T4 ; i)dCLK–CYC unbinding from nuclear importer (K3 ; i)dCLK degradation (D8 ; i)
Cytoplasmic dCLK–CYC degradation (D9 ; i)
Nuclear dCLK–CYC degradation (D10 ; i)‘‘Night’’ systemw
PER degradation (D2 ; i)
TIM degradation (D4 ; i)
Cytoplasmic PER–TIM degradation (D5 ; i)
Cytoplasmic PER–TIM unbinding from degradation enzymes(L5 ; p)
Nuclear PER–TIM degradation (D6 ; i)
Nuclear PER–TIM unbinding from degradation enzymes(L6 ; p)
PER–TIM association (V1 ; p)
PER–TIM dissociation (V2 ; i)
PER–TIM nuclear export (T2 ; i)
PER–TIM unbinding from nuclear exporter (K2 ; p)
dCLK translation (S6 ; i)dCLK degradation (D8 ; p)dCLK–CYC association (V3 ; i)dCLK–CYC dissociation (V4 ; p)
Cytoplasmic dCLK–CYC degradation (D9 ; p)
Nuclear dCLK–CYC degradation (D10 ; p)dCLK–CYC nuclear import (T3 ; i)
Cytoplasmic dCLK–CYC unbinding from nuclear importer(K3 ; p)dCLK–CYC nuclear export (T4 ; p)
Nuclear dCLK–CYC binding to nuclear exporter (K4 ; i)*In the ‘‘day’’ system, a synchronizing factor is secreted during subjective day under repression by PER proteins.
wIn the ‘‘night’’ system, a synchronizing factor is secreted during subjective night under activation by nuclear dCLK–CYC
heterodimers. The parameter regulated by the synchronizing factor is indicated in parentheses (for meaning of parameter, see
Appendix A). (i) and (p) in parentheses represent inhibition and promotion, respectively, of the reaction by the synchronizing
factor.

505

COUPLING MECHANISM CIRCADIAN OSCILLATORS

steps by the synchronizing factor showed aninteresting feature. We calculated the phase shiftby changing the rate of each target reaction for
1 hr at different circadian time points andmeasuring the shift of the subsequent peak ofper mRNA oscillations as compared with thatwithout perturbation (see Appendix C). The
magnitude of the phase shift is plotted against
the circadian time point at which the reaction
rate was controlled (Fig. 3). In 18 cases of the
day system, the phase shifts show transition
from delay to advance during subjective night,
and transition from advance to delay during
subjective day (Fig. 3A). On the other hand, 20
phase shifts in the night system show inverse
patterns, i.e., transition from delay to advanceduring subjective day and transition fromadvance to delay during subjective night(Fig. 3B). Interestingly, these phase shifts in the
day and night systems resemble those induced by
the light pulse and the dark pulse, respectively(Takahashi et al., 1989).
In the above analyses, we ﬁxed the secretiontime course of the potential synchronizing factor
and scanned the reaction steps that are affectedby the factor. Next, in order to elucidate theappropriate upstream regulation of the synchronizing factor, we also carried out a converse
analysis, i.e., we ﬁxed the reaction step that is
affected and altered the time course of thesecretion of the synchronizing factor. We ﬁrst
chose the factor that induces light-pulse-typephase shifts by promoting nuclear PER–TIM
degradation (Fig. 3C inset). We then scanned thetime course of the factor such that one of the 10components of the circadian clock either up- or
down-regulates its secretion. Seven samples
among 20 possible variant models restored
coherent per mRNA oscillations (Fig. 3E),
whereas others failed to do so (Fig. 3F). In
another set of analyses, we chose the factor thatinduces dark-pulse-type phase shifts by promoting nuclear dCLK–CYC degradation (Fig. 3D
inset). When we scanned the time course of thefactor, we found 10 possible synchronizing
factors in the dark-pulse system (Table 2). All
the synchronizing factors in the light-pulse
system peak during subjective day (Fig. 3C)
while those in the dark-pulse system peak during
subjective night or dawn (Fig. 3D). Based on

Table 2
Regulators of synchronizing factors inducing
light-pulse- or dark-pulse-type phase shifts‘‘Light-pulse’’ systemn

Nuclear dCLK–CYC(r, 3.3)
per mRNA (r, 4.4)
tim mRNA (r, 4.4)
PER protein (r, 6.5)
TIM protein (r, 6.5)
dClk mRNA (a, 7.1)
cytoplasmic PER–TIM(r, 8.2)‘‘Dark-pulse’’ systemw
Nuclear dCLK–CYC (a, 16.2)
per mRNA (a, 17.9)
tim mRNA (a, 17.9)
PER protein (a, 20.3)
TIM protein (a, 20.3)
dClk mRNA (r, 20.7)
Cytoplasmic PER–TIM(a, 21.2)
dCLK protein (r, 22.5)
Cytoplasmic dCLK–CYC(r, 23.2)
Nuclear PER–TIM (a, 0.54)*In the ‘‘light-pulse’’ system, a synchronizing factor
induces light-pulse-type phase shifts by promoting PER
degradation.
wIn the ‘‘dark-pulse’’ system, a synchronizing factor
induces dark-pulse-type phase shifts by promoting nuclear
dCLK–CYC degradation. (a) and (r) represent an activatorand repressor, respectively, on the secretion of thesynchronizing factor. Numerical values in the parentheses
indicate peak secretion time of the synchronizing factor.

these results, we propose that a factor thatsatisﬁes the following characteristics can synchronize the clock cells: a factor that is secreted
from individual cells during subjective day to
induce light-pulse-type phase shifts in the neighboring cells. Factors with their secretion andphase shift patterns shifted along the circadiantime by the same magnitude may also synchronize the clock cells, an example of which is afactor that is secreted during subjective night toinduce dark-pulse-type phase shifts.
FUNCTIONAL REDUNDANCY AMONG DIFFERENTSYNCHRONIZING FACTORS

The intercellular coupling mechanism proposed here predicts that there are several
potential factors that produce coherent oscillations. We examined whether one synchronizing
factor would conﬂict with another. Coherent
oscillations are produced in a system having two
synchronizing factors, one secreted during subjective day under repression by PER andinducing light-pulse-type phase shifts by promoting nuclear PER–TIM degradation, and the506

H. R. UEDA ET AL.

other secreted during subjective night underactivation by nuclear dCLK–CYC heterodimers
and inducing dark-pulse-type phase shifts bypromoting nuclear dCLK–CYC degradation(data not shown). These two synchronizing
factors are functionally redundant because only
one of the factors can synchronize oscillators(Fig. 3A and B, Table 1).
NOISE RESISTANCE BY THE
SYNCHRONIZING MECHANISMInternal noise effects on the reaction ratesof the circadian clock may also perturb theconstancy of the circadian rhythm, although an
ability to function reliably in the presence ofinternal noise is required for the circadian system(Barkai & Leibler, 2000). We examined whether
the synchronizing mechanism presented here
would satisfy this constraint. We calculated the
autocorrelation functions of time series data in
the 100-cellular system with or without intercellular coupling (Fig. 4). The system without
intercellular coupling loses correlation rapidly(Fig. 4A), whereas the system with intercellular
coupling retains correlation for a much longer
time (Fig. 4B). The system with intercellular
coupling maintains correlation for about 7400 hr(Fig. 4D), which is about 15 times longer thanthat of the system without intercellular coupling(Fig. 4C). Therefore, owing to a synchronizing
factor, the neighboring cells correct each other’sdeviation to stay in the proper rhythm even in
the presence of internal noise. Thus, the intercellular coupling mechanism is important not
only for synchronization among multiple clock
cells but also for constancy of the circadianrhythm.
Discussion
INTUITIVE EXPLANATIONS FOR THE
SYNCHRONIZING MECHANISM

The scheme depicted in Fig. 5 provides asimpliﬁed explanation to help us understandhow the intercellular coupling mechanism synchronizes multiple circadian oscillators. Let us
consider two cells having different phases, onewith an advanced phase and the other witha delayed phase, so that there is a 5-hr timedifference. The advanced cell secretes a synchronizing factor during subjective day (Fig. 5C).
The secreted factor is received by the delayed cell
during subjective late night to early day becauseof the phase difference and advances its phase(Fig. 5A). The delayed cell also secretes thesynchronizing factor during subjective day(Fig. 5B). The advanced cell receives the secreted
factor during late day to early night because of
the phase difference and its phase is delayed(Fig. 5D). Thus, the advanced cell advances thedelayed cell while the delayed cell delays theadvanced cell (Fig. 5E). These processes reduce
the phase difference between cells. The same
explanation holds for a synchronizing factor
secreted during subjective night if the time axis isshifted by 12 hr. To verify that the intercellular
coupling mechanism proposed here is modelindependent, we implemented a 100-cellularstochastic model based on Leloup & Goldbeter’s(1998) model and obtained similar results (datanot shown).

SYNCHRONIZATION IN UNICELLULAR ORGANISMS

Unicellular organisms have cell division cycle
periods that are shorter than 24 hr. Cell division
cycle greatly affects global changes in transcription and translation rates. Unicellular organisms
such as cyanobacteria show clear circadianrhythms in transcription even though the cells
divide for less than 24 hr (Kondo et al., 1997;
Mori et al., 1996). The intercellular couplingmechanisms proposed here provide a solution to
reduce noise and to maintain constant circadianperiodicity despite global changes in the state of
the cell. Actually, the work done by Hastings’
group provided suggestive evidence for synchronization in the unicellular organism Gonyaulax
polyedra (Broda et al., 1986). They also extracted
a substance of low molecular weight, gonyauline,from Gonyaulax polyedra and showed that
gonyauline shortens the circadian clock period(Roenneberg et al., 1991). Although it is notknown whether gonyauline is secreted from thecell, when it is secreted or what type of phaseshift it induces, it is interesting to examine thepossibility that gonyauline is a synchronization
factor.

COUPLING MECHANISM CIRCADIAN OSCILLATORSHETEROGENEOUS ORGANIZATION AND
COORDINATION OF CIRCADIAN SYSTEM

In multicellular organisms, cell-type dependent circadian characteristics are known. In
Drosophila, a limited number of lateral neuron
cells direct a long-term-sustaining locomotion
activity, while in other peripheral tissues different molecular mechanisms have been suggested(Krishnan et al., 2001). In mammals, cultured
suprachiasmatic nucleus (SCN) cells showed arobust circadian rhythm that persisted for up to
32 days in vitro (Yamazaki et al., 2000), whileliver, lung and skeletal muscle expressed circadian rhythms, which became damped after twoor seven cycles in vitro. Evidence for theheterogeneous organization of the circadiansystem raises at least two questions. Is there adifference in molecular mechanism between the
peripheral damped oscillators and the central
pace-making oscillator? How are these heterogeneous oscillators organized and coordinated toconstruct circadian systems? Although the detailed molecular mechanisms have not been fully
elucidated, it is important to raise these issues,
which are important subjects for future studies.
CANDIDATES FOR SYNCHRONIZING FACTORS

What do we know about the candidate
substances for the synchronizing factors? In
Drosophila, it is known that the pdf geneproduct, a humoral peptide hormone (PDF),
accumulates in a circadian manner during
subjective day in lateral neurons and is positively
regulated by the dClk gene product (Blau &Young, 1999; Park et al., 2000). Ectopic expression of PDF alters circadian activity rhythms
without abolishing the circadian per/tim expression levels (Helfrich-Forster et al., 2000).
Although the phase-shift effects caused by PDF
on circadian rhythms in Drosophila are unknown, it is important to examine the possibility
that PDF may be the synchronization factor. If
PDF is a synchronization factor, the phase shift
induced by PDF should resemble the light-pulsetype pattern.
In vertebrates, several agonists, including
glutamate, GABA, melatonin, serotonin anddopamine, have been reported to be involvedin phase shifts induced by external time cues,507such as light pulses, and can directly induce
phase shifts in circadian clocks (Adachi et al.,
1998; Liu & Reppert, 2000; McArthur et al.,
1991; Prosser et al., 1990; Shibata et al., 1994).
Furthermore, these agonists are secreted periodically in phase with the circadian period evenin the absence of external time cues (Adachi
et al., 1998; Aguilar-Roblero et al., 1993; Dudley
et al., 1998; Hamada et al., 1999; Ikeda et al.,
1997; Shinohara et al., 1998). Interestingly,
dopamine and melatonin in the pigeon eye show
secretion patterns and phase shifts that are quite
similar to the ‘‘day-type’’ and ‘‘night-type’’
synchronizing factors presented here: dopamine
peaks in subjective day and induces light-pulsetype phase shifts, whereas melatonin peaks insubjective night and induces dark-pulse-typephase shifts (Adachi et al., 1998). However, thephysiological roles of these agonists remain
elusive. Our results suggest possible roles into
these agonists in synchronization and noise
resistance of multiple circadian oscillators andencourage further investigation of the roles ofthese agents. One caution drawn from thepresent work is that, due to the possible presenceof redundant factors, the blockage or elimination of one putative synchronizing factor may
not result in the abolition of synchrony.
We thank H. Iwasaki, T. Kondo, M. Hasegawa, A.
Adachi, G. A. Oda, B. Ermentrout for useful
information, H. Kitano, M. Hagiwara, S. Onami
for critical advice, K. M. Kyoda, M. Morohashi, S.
Hamahashi for technical advice, S. Hashimoto, T.
Takasugi, T. Horiguchi, S. Iwase for useful discussion, A. Kimura for comment on the manuscript.
This work was performed as part of the research anddevelopment project of Industrial Science and Technology Program supported by NEDO.

REFERENCES
Adachi, A., Nogi, T. & Ebihara, S. (1998). Phaserelationship and mutual effects between circadianrhythms of ocular melatonin and dopamine in the pigeon.
Brain Res. 792, 361–369.
Aguilar-Roblero, R., Verduzco-Carbajal, L., Rodriguez, C., Mendez-Franco, J., Moran, J. & de laMora, M. P. (1993). Circadian rhythmicity in the
GABAergic system in the suprachiasmatic nuclei of therat. Neurosci. Lett. 157, 199–202.
Allada, R., White, N. E., So, W. V., Hall, J. C. &Rosbash, M. (1998). A mutant Drosophila homolog of508

H. R. UEDA ET AL.

mammalian clock disrupts circadian rhythms and transcription of period and timeless. Cell 93, 791–804.
Bae, K., Lee, C., Sidote, D., Chuang, K. Y. & Edery, I.
(1998). Circadian regulation of a Drosophila homolog ofthe mammalian Clock gene: PER and TIM function as
positive regulators. Mol. Cell Biol. 18, 6142–6151.
Barkai, N. & Leibler, S. (2000). Circadian clocks limited
by noise. Nature 403, 267–268.
Blau, J. & Young, M. W. (1999). Cycling vrille expression
is required for a functional Drosophila clock. Cell 99,
661–671.
Brandes, C., Plautz, J. D., Stanewsky, R., Jamison,
C. F., Straume, M., Wood, K. V., Kay, S. A.
& Hall, J. C. (1996). Novel features of Drosophila
period transcription revealed by real-time luciferase
reporting. Neuron 16, 687–692.
Broda, H., Brugge, D., Homma, K. & Hastings, J. W.
(1986). Circadian communication between unicells?
Effects on period by cell-conditioning of medium. Cell
Biophys. 8, 47–67.
Darlington, T. K., Wager-Smith, K., Ceriani, M. F.,
Staknis, D., Gekakis, N., Steeves, T. D. L., Weitz, C.
J., Takahashi, J. S. & Kay, S. A. (1998). Closing thecircadian loop: CLOCK-induced transcription of its own
inhibitors per and tim. Science 280, 1599–1603.
Dudley, T. E., DiNardo, L. A. & Glass, J. D. (1998).
Endogenous regulation of serotonin release in thehamster suprachiasmatic nucleus. J, Neurosci. 18,
5045–5052.
Dunlap, J. C. (1999). Molecular bases for circadian clocks.
Cell 96, 271–290.
Gillespie, D. T. (1977). Exact stochastic simulation of
coupled chemical reactions. J. Phys. Chem. 81,
2340–2361.
Glossop, N. R., Lyons, L. C. & Hardin, P. E. (1999).
Interlocked feedback loops within the Drosophila circadian oscillator. Science 286, 766–768.
Hamada, T., Ootomi, M., Horikawa, K., Niki, T.,
Wakamatu, H. & Ishida, N. (1999). The expressionof the melatonin synthesis enzyme: arylalkylamine
N-acetyltransferase in the suprachiasmatic nucleus ofrat brain. Biochem. Biophys. Res. Commun. 258, 772–777.
Hardin, P. E., Hall, J. C. & Rosbash, M. (1990).
Feedback of the Drosophila period gene product on
circadian cycling of its messenger RNA levels. Nature
343, 536 –540.
Helfrich-Forster, C., Tauber, M., Park, J. H.,
Muhlig-Versen, M., Schneuwly, S. & Hofbauer, A.
(2000). Ectopic expression of the neuropeptide pigmentdispersing factor alters behavioral rhythms in Drosophilamelanogaster. J. Neurosci. 20, 3339–3353.
Ikeda, M., Azuma, S. & Inoue, S. (1997). Vitamin B12enhances GABA content but reduces glutamate content
in the rat suprachiasmatic nucleus. Am. J. Physiol. 273,
R359–R363.
Kondo, T., Mori, T., Lebedeva, N. V., Aoki, S., Ishiura,
M. & Golden, S. S. (1997). Circadian rhythms in rapidly
dividing cyanobacteria. Science 275, 224–227.
Konopka, R. J. & Benzer, S. (1971). Clock mutants of
Drosophila melanogaster. Proc. Natl Acad. Sci. U.S.A.
68, 2112–2116.
Krishnan, B., Levine, J. D., Lynch, M. K., Dowse,
H. B., Funes, P., Hall, J. C., Hardin, P. E. & Dryer,

S. E. (2001). A new role for cryptochrome in a Drosophila
circadian oscillator. Nature 411, 313–317.
Leloup, J. C. & Goldbeter, A. (1998). A model for
circadian rhythms in Drosophila incorporating theformation of a complex between the PER and TIM
proteins. J. Biol. Rhythms. 13, 70–87.
Liu, C. & Reppert, S. M. (2000). GABA synchronizes
clock cells within the suprachiasmatic circadian clock.
Neuron 25, 123–128.
Low-Zeddies, S. S. & Takahashi, J. S. (2001). Chimera
analysis of the clock mutation in mice shows thatcomplex cellular integration determines circadian behavior. Cell 105, 25–42.
McAdams, H. H. & Arkin, A. (1999). It’s a noisy business!
Genetic regulation at the nanomolar scale. Trends Genet.
15, 65–69.
McArthur, A. J., Gillette, M. U. & Prosser, R. A.
(1991). Melatonin directly resets the rat suprachiasmatic
circadian clock in vitro. Brain Res. 565, 158–161.
Mori, T., Binder, B. & Johnson, C. H. (1996). Circadiangating of cell division in cyanobacteria growing withaverage doubling times of less than 24 hours. Proc. Natl
Acad. Sci. U.S.A. 93, 10 183–10 188.
Park, J. H., Helfrich-Forster, C., Lee, G., Liu, L.,
Rosbash, M. & Hall, J. C. (2000). Differential
regulation of circadian pacemaker output by separate
clock genes in Drosophila. Proc. Natl Acad. Sci. U.S.A.
97, 3608–3613.
Pittendrigh, C. S. (1993). Temporal organization: reﬂections of a Darwinian clock-watcher. Annu. Rev. Physiol.
55, 16–54.
Plautz, J. D., Kaneko, M., Hall, J. C. & Kay, S. A.
(1997). Independent photoreceptive circadian clocks
throughout Drosophila. Science 278, 1632–1635.
Price, J. L., Blau, J., Rothenfluh, A., Abodeely, M.,
Kloss, B. & Young, M. W. (1998). double-time is a novel
Drosophila clock gene that regulates PERIOD protein
accumulation. Cell 94, 83–95.
Prosser, R. A., Miller, J. D. & Heller, H. C. (1990).
A serotonin agonist phase-shifts the circadian clockin the suprachiasmatic nuclei in vitro. Brain Res. 534,
336–339.
Roenneberg, T., Nakamura, H., Cranmer 3rd, L. D.,
Ryan, K., Kishi, Y. & Hastings, J. W. (1991).
Gonyauline: a novel endogenous substance shortening
the period of the circadian clock of a unicellular alga.
Experientia 47, 103–106.
Rutila, J. E., Suri, V., Le, M., So, W. V., Rosbash, M. &Hall, J. C. (1998). CYCLE is a second bHLH-PAS
clock protein essential for circadian rhythmicity andtranscription of Drosophila period and timeless. Cell 93,
805–814.
Sehgal, A., Price, J. L., Man, B. & Young, M. W.
(1994). Loss of circadian behavioral rhythms and per
RNA oscillations in the Drosophila mutant timeless.
Science 263, 1603–1606.
Sehgal, A., Rothenfluh-Hilfiker, A., Hunter-Ensor,
M., Chen, Y., Myers, M. P. & Young, M. W. (1995).
Rhythmic expression of timeless: a basis for promoting
circadian cycles in period gene autoregulation. Science
270, 808–810.
Shibata, S., Watanabe, A., Hamada, T., Ono, M. &Watanabe, S. (1994). N-methyl-D-aspartate induces

509

COUPLING MECHANISM CIRCADIAN OSCILLATORS

phase shifts in circadian rhythm of neuronal activity of
rat SCN in vitro. Am J Physiol 267, R360–R364.
Shinohara, K., Honma, S., Katsuno, Y., Abe, H. &Honma, K. (1998). Circadian release of amino acidsin the suprachiasmatic nucleus in vitro. Neuroreport 9,
137–140.
Takahashi, J. S., Murakami, N., Nikaido, S. S., Pratt,
B. L. & Robertson, L. M. (1989). The avian pineal, avertebrate model system of the circadian oscillator:
cellular regulation of circadian rhythms by light, second
messengers, and macromolecular synthesis. Recent. Prog.
Horm. Res. 45, 279–348.
Ueda, H. R., Hagiwara, M. & Kitano, H. (2001).
Robust oscillations within the interlocked feedback
model of Drosophila circadian rhythm. J. theor. Biol.
210, 401–406.
Welsh, D. K., Logothetis, D. E., Meister, M.
& Reppert, S. M. (1995). Individual neurons dissociatedfrom rat suprachiasmatic nucleus express independently phased circadian ﬁring rhythms. Neuron
14, 697–706.
Williams, J. A. & Sehgal, A. (2001). Molecularcomponents of the circadian system in Drosophila. Annu.
Rev. Physiol. 63, 729–755.
Winfree, A. T. (1975). Unclocklike behaviour of biological clocks. Nature 253, 315–319.
Yamazaki, S., Numano, R., Abe, M., Hida, A.,
Takahashi, R., Ueda, M., Block, G. D., Sakaki, Y.,
Menaker, M. & Tei, H. (2000). Resetting central andperipheral circadian oscillators in transgenic rats. Science
288, 682–685.

dPerci¼ S2 Permi  V1 Perci Timic þ V2 PTci
dt
 D2

Perci
 D0 Perci ;L2 þ PercidTimimðCCni =A2 Þa þ B2
¼ S3
dt
1 þ ðPTni =R2 Þr þ ðCCni =A2 Þa þ B2
 D3

Timim
 D0 Timim ;L3 þ TimimdTimic¼ S4 Timim  V1 Perci Timic þ V2 PTci
dt
Timic
 D4
 D0 Timic ;L4 þ Timic

dPermiðCCni =A1 Þa þ B1
¼ S1
dt
1 þ ðPTni =R1 Þr þ ðCCni =A1 Þa þ B1
Permi
 D1
 D0 Permi ;ðA:1Þ
L1 þ Permi

ðA:4Þ

PTni
PTci

D
 D0 PTci ;5
K2 þ PTni
L5 þ PTci

Appendix A

We previously constructed the following
differential equations (see below, Ueda et al.,2001) with 48 system parameters and 10
variables
representing
concentrationsofper, tim and dClk mRNAs (Permi ; Timim
and Clkmi ; i indicates the index of cells),
and their respective protein products PER,
TIM and dCLK (Perci ; Timic and Clkci ); bothcytoplasmic and nuclear PER–TIM heterodimers (PTci and PTni ) as well as both cytoplasmic
and nuclear dCLK–CYC heterodimers (CCci andCCni ).

ðA:3Þ

dPTci
PTci
¼ V1 Perci Timic  V2 PTci  T1
dt
K1 þ PTciþ T2

Stochastic Simulation of Multiple
Circadian Oscillators

ðA:2Þ

ðA:5Þ
dPTni
PTci
PTni
¼ T1
 T2
i
dt
K1 þ PTc
K2 þ PTni
 D6

PTni
 D0 PTni ;L6 þ PTni

ðA:6Þ

dClkmiðPTni =A3 Þa þ B3
¼ S5
dt
1 þ ðCCni =R3 Þr þ ðPTni =A3 Þa þ B3
 D7

Clkmi
 D0 Clkmi ;L7 þ Clkmi

ðA:7Þ

dClkci¼ S6 Clkmi  V3 Clkci Cycc þ V4 CCci
dt
 D8

Clkci
 D0 Clkci ;L8 þ Clkci

ðA:8Þ

510

H. R. UEDA ET AL.

dCCci
CCci
¼ V3 Clkci Cyc  V4 CCci  T3
dt
K3 þ CCci
þ T4

CCni
CCci

D
 D0 CCci :9
K4 þ CCni
L9 þ CCci
ðA:9Þ

To describe transcription, we used Hill-type
equations that are characterized by six parameters representing the maximum velocity(Sl ; l ¼ 1; 3; 5), two DNA binding constants
for an activator and a repressor (Al andRl ; l ¼ 1y3), two Hill coefﬁcients for activation
and repression (a and r) and one constant(Bl ; l ¼ 1y3), which indicates transcriptional
activation by transcription factors other than
PER–TIM or dCLK–CYC. The translation rate
(Sl ; l ¼ 2; 4; 6) was assumed to be proportional
to mRNA concentration. Association and dissociation rates (Vl ; l ¼ 1y4) were made to obey
the law of mass action. Degradation and nuclear
transportation were supposed to be mediated bydegradation enzymes and transporters, respectively. We described these processes withMichaelis–Menten-type equations characterized
by the maximum velocity (for degradation,andfortransportation,
Dl ; l ¼ 1y10;
Tl ; l ¼ 1y4) and the binding constant (fordegradation, Ll ; l ¼ 1y10; and for transportation, Kl ; l ¼ 1y4). Non-speciﬁc degradation
terms are proportional to each variablewith proportionality constant D0 : We used
100 cellular models for multicellular models(i ¼ 1y100 in eqns (A.1)–(A.9)). Values ofall of the 48 system parameters are chosen
to reproduce per, tim, and dClk mRNA
time courses in wild-type and steady-statelevels of these mRNAs in mutants in theabsence of internal noise (Ueda et al.,2001). Parameter values are: Cycc ¼ 1 nM;
S2 ¼ S4 ¼ 0:48 hr1,
S1 ¼ S3 ¼ 1:45 nM hr1,
1
S6 ¼ 0:47 hr1 ; r ¼ 4;
S5 ¼ 1:63 nM hr ,R1 ¼ R2 ¼ 1:02 nM, R3 ¼ 0:89 nM, a ¼ 1;
A1 ¼ A2 ¼ 0:45 nM, A3 ¼ 0:8 nM, B1 ¼ B2 ¼ 0;
B3 ¼ 0:6; V1 ¼ 1:45 nM1 hr1, V2 ¼ 1:45 hr1,
V3 ¼ 1:63 nM1 hr1, V4 ¼ 1:63 hr1, T1 ¼ 1:73
nM hr1, T2 ¼ 0:72 nM hr1, T3 ¼ 1:63 nM hr1,
T4 ¼ 0:52 nM hr1, K1 ¼ 2 nM, K2 ¼ 2 nM,
K4 ¼ 2 nM,
D1 ¼ D3 ¼ 0:94
K3 ¼ 2 nM,

nM hr1, D2 ¼ D4 ¼ 0:44 nM hr1, D5 ¼ 0:44
nM hr1, D6 ¼ 0:29 nM hr1, D7 ¼ 0:54 nM
hr1, D8 ¼ 0:6 nM hr1, D9 ¼0:6 nM hr1, D10 ¼
0:3 nM hr1, L1 ¼ L3 ¼ 0:3 nM, L2 ¼ L4 ¼ 0:2
nM, L5 ¼ 0:2 nM, L6 ¼ 0:2 nM, L7 ¼ 0:13 nM,
L8 ¼ 0:2 nM, L9 ¼ 0:2 nM, L10 ¼ 0:2 nM and
D0 ¼ 0:012 hr1.
To introduce internal noise, the rates of all
reactions, Sl ðl ¼ 1y6Þ; Tl ðl ¼ 1y4Þ; Vl ðl ¼ 1y4Þ and Dl ðl ¼ 0y10Þ; were varied ateach time step of simulation (0.1 hr) in such a
way that their values are normally distributed
around their mean with a 10% standard
deviation. For example, S1 is normally
distributed around its mean (¼ 1:45) with astandard deviation (¼ 0:145). We used an Euler
scheme to integrate these stochastic differential
equations with multiplicative noise. We used
0.1 hr as the time interval. Multiplicative noise
associated with rates of reactions is used for
simulating the randomness inherent in biochemical reactions involving a small number ofmolecules.
An alternative way to treat internal noise isGillespie’s (1977) method, where the internal
noise of reaction is naturally calculated from thenumber of molecules participating in that reaction. However, Gillespie’s method requires fullknowledge of detailed molecular mechanismsthat are not available so far and much morecomputational power to simulate. Comparedwith Gillespie’s method, the main disadvantage
of our method is the incapability to determine
the magnitude of internal noise from ﬁrst
principles.
To compensate for this shortcoming, we canassess the magnitude of internal noise by using
previously reported experimental data. Although
no information is available regarding the magnitude of internal noise in Drosophila, we
assessed the magnitude of noise based on thefollowing consideration. Our simulation systemwith 10 or 30% noise exhibits per mRNA
oscillations with a period of 24.0 7 0.5 or 24.0
7 1.5 hr (mean 7 S.D., determined by a chisquare periodogram using 100-cell time series
data over 5 days). These variations are similar towhat was observed in the neural activity of single
mammalian neurons in culture (24.35 7 1.20 hr,
mean 7 S.D.) (Welsh et al., 1995). Preliminary

511

COUPLING MECHANISM CIRCADIAN OSCILLATORS

results suggest that the basic conclusions (asynchronizing factor is secreted from individual
cells during subjective day to induce light-pulsetype phase shifts in the neighboring cells, or,alternatively, a factor is secreted during subjective night to induce dark-pulse-type phaseshifts) obtained from the system with 10% noise
also hold for the system with 30% noise. In thepresence of 30% noise and in the absence ofsynchronization, the averaged per mRNA oscillation is reduced to less than 5% within several
days, which is within the physiological range for
damped circadian oscillators.
Appendix B
Search for Synchronizing Factors
To synchronize circadian oscillators, we introduced a factor (X i ) into the system describedabove (eqns (A.1)–(A.9)). The kinetics of thesynchronizing factor is described as follows.
dX i
ðY i =AX ÞaX
Xi
¼ SX

D
X
a
dt
1 þ ðY i =AX Þ X
LX þ X i
X D0 X i þ Dc
ðX j  X i Þ:

ðB:1Þ

j

Secretion of the factor was supposed to be
activated or repressed by one of the 10 variables
described above ðY i AfPermi ; Timim ; Clkmi ; Perci ;Timic ; Clkci ; PTci ; PTni ; CCci ; CCni gÞ: To describe thekinetics of secretion (the ﬁrst term on r.h.s of eqn(B.1)), we used Hill-type equations that werecharacterized by three parameters: the maximum
velocity (SX ) and the DNA binding afﬁnity ofactivators or repressors (AX ), and the nonlinearity of activation or repression (jaX j). If Y i
activates the secretion of the factor, aX is positive(aX ¼ jaX j). On the other hand, if Y i represses thesecretion, aX is negative (aX ¼ jaX j). In thiscase, the ﬁrst term of eqn (B.1) can betransformed toSX1
1 þ ðY i =A

jaX j
XÞ

:

Degradation of the factor was supposed to be
mediated by degradation enzymes with the
maximum velocity (DX ) and the Michaelis
constant (LX ). The non-speciﬁc degradation termis proportional to X i with the proportionality
constant D0 : Synchronizing factors were supposed to spread through diffusion in threedimensional space with the diffusion constant(Dc ), and 100 cells were placed in a hexagonallattice [index j in eqn (B.1) indicates neighboring
cells]. Namely, we performed reaction–diffusion
simulations on a hexagonal lattice, each point of
which represents a cell. We used a straightforward discretization of the Laplacian operator
and each point (cell) in the hexagonal lattice issurrounded by 12 points (cells) on average. Someparameter values were: jaX j ¼ 4; DX ¼ 0:6 nM
hr1, LX ¼ 0:5 nM, Dc ¼ 0:5 hr1. Values of two
parameters (SX and AX ) were dependent on an
activator or a repressor of the factor. These
values are chosen so that the peak concentrationof the factor is 4.0 nM and the half-maximal
duration is 8.0 hr (see Table B1).
The factor was supposed to increase or
decrease the rate of one of the reactions inTable B1
Activator

SX (nM hr1)

AX (nM)

Repressor

SX (nM hr1)

AX (nM)

Permi
Perci
Timim
Timic
PTci
PTni
Clkmi
Clkci
CCci
CCni

1.53583.9962
1.53583.9962
1.12451.7796
3.9962
3.99622.87653.99623.45532.48133.45532.4813
1.72722.85296.71864.27842.88574.4730

Permi
Perci
Timim
Timic
PTci
PTni
Clkmi
Clkci
CCci
CCni

0.9417
1.10930.9417
1.10930.96461.01790.92650.9646
0.9646
1.06360.64480.48150.64480.48150.23060.47571.07780.49060.28220.8192

512

H. R. UEDA ET AL.

the circadian system. Parameters regulated by
the factor include Sl ðl ¼ 1y6Þ; Tl ðl ¼ 1y4Þ;Vl ðl ¼ 1y4Þ; Dl ðl ¼ 0y10Þ; Al ðl ¼ 1y3Þ; Rl ðl ¼ 1y3Þ; Bl ðl ¼ 1y3Þ; Kl ðl ¼ 1y4Þ and Ll ðl ¼ 1y10Þ:
For activation, we substituted the target
parameter P (e.g. S1 ) with the following logistic-type function:



M 1
1þ
P
1 þ exp½kðX i  Xh Þ¼M þ exp½kðX i  Xh ÞP:
1 þ exp½kðX i  Xh Þ

ð2aÞThis function was characterized by three parameters: maximum velocity M; threshold concentration Xh and nonlinearity for increase k:
For repression, we substituted the target
parameter P with the following function:
1 þ exp½kðX i  Xh ÞP:
M þ exp½kðX i  Xh Þ

ð2bÞThis function was characterized by three parameters:
minimal velocity 1=M; threshold concentration Xh

and nonlinearity for decrease k: Parameter values
were: M ¼ 2; Xh ¼ 2 nM and k ¼ 3:
Appendix C
Calculation of Phase Shifts
Phase shifts were calculated by increasing (ordecreasing) the target reaction rate by 2-, 4-, 6-,8-, or 10-fold for 1 hr from 100 different
circadian time points and measuring the peaktime of per mRNA oscillation, which wascompared with the peak time of per mRNA
oscillation without perturbation. Phase shifts
became larger as the intensity of the factor
became greater, but the shapes of the phaseshift–circadian time relationship remained similar. Ninety-six patterns of phase shifts wereobtained by increasing or decreasing all of the 48
system parameters that represent the rates ofreactions constituting the circadian clock. In
Fig. 3, phase shifts obtained by increasing (ordecreasing) the reaction rate targeted by synchronizing factors are depicted. Periods of permRNA oscillations are determined from the chisquare periodogram.