J. theor. Biol. (2001) 210, 401}406
doi:10.1006/jtbi.2000.2226, available online at http://www.idealibrary.com on

Robust Oscillations within the Interlocked Feedback Modelof Drosophila Circadian Rhythm
HIROKI R. UEDA*-?A#, MASATOSHI HAGIWARA-? AND HIROAKI KITANO*-***ERA¹O Kitano Symbiotic Systems Group, Suite 6A, M31, 6-31-15 Jinguumae, Shibuya,¹okyo 150-0001, Japan, ?Department of Functional Genomics, Medical Research Institute,¹okyo Medical and Dental ;niversity, >ushima, ¹okyo 113-8510, Japan, AMolecular Medicine Research¸aboratories, >amanouchi Institute for Drug Discovery Research, 21 Miyukigaoka, ¹sukuba,
Ibaraki 305, Japan, and #Department of Pharmacology, Graduate School of Medicine,¹he ;niversity of ¹okyo, Bunkyo-ku, ¹okyo 113, Japan **Sony Computer
Science ¸aboratories, Inc., ¹okyo, Japan; ¹he Systems Biology Institute,¹okyo, Japan(Received on 6 June 2000, Accepted in revised form on 28 October 2000)

A mechanism for generating circadian rhythms has been of major interest in recent years. Afterthe discovery of per and tim, a model with a simple feedback loop involving per and tim has been
proposed. However, it is recognized that the simple feedback model cannot account forphenotypes generated by various mutants. A recent report by Glossop, Lyons & Hardin [Science
286, 766 (1999)] on Drosophila suggests involvement of another feedback loop by dClk that isinterlocked with per}tim feedback loop. In order to examine whether interlocked feedback loops
can be a basic mechanism for circadian rhythms, a mathematical model was created andexamined. Through extensive simulation and mathematical analysis, it was revealed that theinterlocked feedback model accounts for the observations that are not explained by the simple
feedback model. Moreover, the interlocked feedback model has robust properties in oscillations.
 2001 Academic Press

Introduction
Along evolution, organisms ranging from cyanobacteria to mammals have developed in themselves molecular clocks of circadian (about 24 hr)
period (Dunlap, 1999). In Drosophila like otherorganisms, several genes are involved in sustaining circadian rhythm, namely period (per), timeless (tim), Drosophila Clock (dClk), Cycle (Cyc)
and doubletime (dbt) (Hardin et al., 1990; Sehgal
et al., 1994; Darlington et al., 1998; Allada et al.,
1998; Rutila et al., 1998; Price et al., 1998; Kloss-E-mail: hiro@m.u-tokyo.ac.jp; m.hagiwara.end@mri. tmd.
ac.jp; kitano@symbio.jst.go.jp0022}5193/01/120401#06 $35.00/0

et al., 1998). Among them, three genes are rhythmically expressed: per, tim and dClk. per and tim
mRNA levels peak early in the evening whereas
the dClk mRNA level peaks late at night to earlyin the morning (Hardin et al., 1990; Darlington
et al., 1998; Sehgal et al., 1995; Bae et al., 1998).
Analysis of per and tim oscillatory expression
has revealed a negative feedback loop, &&per}tim
feedback loop''. Transcription of per and tim is
activated through E-box elements in per and tim
promoters by a heterodimer of basic helix-loophelix transcription factors, dCLK and CYC(Darlington et al., 1998; Allada et al., 1998; Rutila
et al., 1998; Hao et al., 1997). Peaks of PER and
TIM levels delay several hours from peaks of per
 2001 Academic Press

402

H. R. UEDA E¹ A¸.

and tim mRNA levels. PER and TIM proteins
form complex. This complex enters into the
nucleus (Saez & Young, 1996) and represses transcription of per and tim by forming a complex
with a heterodimer of transcriptional activators
dCLK and CYC (Darlington et al., 1998; Leeet al., 1999). PER proteins are periodicallyphosphorylated (Edery et al., 1994). This phosphorylation is mediated at least partly, if notentirely, by DOUBLETIME protein (DBT),
which is closely related to human casein kinase
Ie. PER phosphorylation by DBT is suggested
to regulate stability of PER proteins because in
the mutant (dbt.) abolishing most dbt expression,
PER proteins constitutively accumulate and remain hypophosphorylated whereas per mRNAlevels are similar to that in the wild type (Priceet al., 1998; Kloss et al., 1998).
Several models based on this delayed negative
feedback has been proposed (Darlington et al.,
1998; Sehgal et al., 1995; Leloup & Goldbeter,
1998), which are called here as &&simple feedback
model''. The simple feedback model shows thata delayed negative feedback loop produces spontaneous oscillation (Leloup & Goldbeter, 1998).
However, these models seem to contradict decrease of per and tim mRNA levels in mutants
lacking PER (per) or TIM (tim) function(Hardin et al., 1990; So & Rosbash, 1997) anddo not explain dClk oscillatory expression (Baeet al., 1998).
On the other hand, analysis of dClk expression
has revealed another negative feedback loop,&&dClk feedback loop''. dClk mRNA level is decreased in per and tim mutants, indicating that
PER and TIM function as positive regulators
probably in dClk transcription (Bae et al., 1998;
Glossop et al., 1999). The decreased level of dClk
mRNA in the per mutant is restored to thewild-type level in per; dClk( and per; Cycdouble mutants, which suggests that dCLK and
CYC have a direct or indirect repressive role indClk transcription (Glossop et al., 1999).
Based on these analyses of dClk expression,another type of model has been proposed, whichis called here as &&interlocked feedback model''.
The interlocked feedback model is composed of
two coupled negative feedback loops: a per}tim
feedback loop, which is activated by dCLK}CYC
and repressed by PER}TIM, and a dClk

feedback loop, which is repressed by dCLK}CYC
and derepressed by PER}TIM (Glossop et al.,1999).
Results and Discussions
To verify whether the interlocked feedback
model explains the observations which the simple
feedback model does not, we constructed amathematical model based on the interlocked
feedback model. The interlocked feedback modelis schematized in Fig. 1. To consider kinetics ofexpression of per, tim and dClk, we formulate
this scheme to ten-variable di!erential equations(Fig. 2).
These equations can be integrated numericallyonce numerical values for all system parameters
and initial conditions are speci"ed. Instead of
converging toward a stable steady state, the system shows sustained oscillations [Fig. 3(a) and(b)]. The system follows the same trajectories
regardless of initial conditions, indicating that
our model provides a molecular basis for circadian rhythm of the limit-cycle type. per and tim
oscillations are in phase [Fig. 3(c)], whereas thedClk oscillation is almost in antiphase to per andtim oscillations [Fig. 3(d)]. These results are consistent with observations that per and tim mRNAlevels oscillate in phase to one another (Sehgal
et al., 1995; Bae et al., 1998; So & Rosbash, 1997)
and that the dClk mRNA level oscillates in antiphase to per and tim mRNA levels (Bae et al.,1998).
Parameter values in Fig. 3 are chosen to yielda period close to 24 hr in constant darkness conditions. Identical kinetic constants are taken
for corresponding processes involving PER and
TIM to produce in-phase oscillations of per andtim mRNA levels. These values are a possible setof kinetic constants to yield a circadian period
oscillation and in-phase oscillations of per andtim mRNA levels. However, these values yieldappropriate phenotypes in the wild type: the correct time course of mRNA and protein levels andthe reasonable phase shifts by light. First, wemeasured the peak time of mRNA and protein
levels. per (tim) mRNA level peaks at zeitgeber
time (Z¹) 15.0 given that per mRNA level bottoms at Z¹"0. The cytoplasmic PER (TIM),
cytoplasmic PER}TIM and nuclear PER}TIM

FIG. 1. The interlocked feedback model of Drosophila circadian rhythm. per}tim feedback loop: per (Per ) and tim (¹im )KmRNA are transcribed in the nucleus and translated to PER (PerA) and TIM (¹imA) in cytoplasm. Degradationof PER Kis
promoted by kinase DBT (DbtA ). Translated PER and TIM form a cytoplasmic PER}TIM complex (P¹A) and then enter the
nucleus. A nuclear PER}TIM complex (P¹L) represses its own transcription activated by dCLK}CYC (CCL). dClk feedback
loop: dClk (ClkK,) mRNA is transcribed in the nucleus and translated to dCLK (ClkA) in cytoplasm. Translated dCLK forms
a cytoplasmic dCLK} CYC complex (CCA) with CYC (CycA) and then enters the nucleus. A nuclear dCLK}CYC complex(CCL) represses its own transcription activated by dPER}TIM (P¹L).

FIG. 3. Simulation of the wild type. (a)}(d) Time courses of per and tim expression (a) and dClk expression (b) are simulated.
tim (c) and dClk (d) mRNA levels are plotted against per mRNA levels. tim and per mRNAs reach peak or bottom levels
simultaneously (c) whereas dClk mRNA approaches the peak level when the per mRNA reaches the bottom level (d). Parametervalues are: CycA"1 nM, DbtA"1 nM. C"C"C"0 nM hr\, S"S"1.45 nM hr\, S"S"0.48 hr\,S"1.63 nM hr\,S" 0.47 hr\, r"4, R"R"1.02 nM, R"0.89 nM, a"1, A" A"0.45 nM, A"0.8 nM,
B"B"0, B"0.6, <"1.45 nM\ hr\, <"1.45 hr\, <"1.63 nM\ hr\, <"1.63 hr\, ¹"1.73 nM hr\,¹"0.72 nM hr\, ¹"1.63 nM hr\, ¹"0.52 nM hr\, K"2 nM, K"2 nM, K"2 nM, K"2 nM,
D "D "0.94 nM hr\, D "D "0.44 nM hr\, D "0.44 nM hr\, D" 0.29 nM hr\, D"0.54 nM hr\,D"0.6 nM hr\, D"0.6 nMhr\,D"0.3 nM hr\, ¸"¸"0.3 nM, ¸"¸
"0.2 nM, ¸"0.2 nM, ¸"0.2 nM,¸"0.13 nM, ¸"0.2 nM, ¸"0.2 nM, ¸"0.2 nM and D"0.012 hr\. (e) Phase shifts by light pulse, which is emulated
by increasing TIM degradation rate (D) by two- (blue), four- (light blue), six- (green), eight- (yellow), ten- (red) folds for 1 hr.
Phase shifts are plotted along subjective circadian time, at which the perturbation (increase of TIM degradation rate) is applied.

FIG. 2. A mathematical formulation ofinterlocked feedback model. We constructed ten-variable di!erential equations toinvestigate kinetics of concentration of per(PerK), tim (¹imK) and dClk (ClkK,) mRNA,
PER (PerA), TIM (¹imA) and dCLK (ClkA)
monomers, cytoplasmic (P¹A) and nuclear(P¹L) PER}TIM heterodimers and cytoplasmic (CCA) andnuclear (CCL)
dCLK}CYC heterodimers. Concentrations of DBT (DbtA.) and CYC (CycA) aresupposed to be constant. To describe transcription, we used Hill equations withslight modi"cation, which are characterized by six parameters representing the
maximum velocity SG (i"1, 3, 5), two
DNA binding constants of an activator(AG , i"1, 2, 3) and a repressor (RG , i"1, 2,3), two Hill coe!cients for activation (a)and repression (r) and one constant
BG (i"1, 2, 3), which indicates transcriptional activation by other transcription
factors than PER}TIM or dCLK}CYC.
Translation rate is proportional to themRNA concentration with the proportional constant SG (i"2, 4, 6). Associationand dissociation of PER}TIM anddCLK}CYC complex are subject to law ofmass actions. These processes are characterized with the rate constant <G (i"1,2, 2). Degradation and nuclear transportation are supposed to be mediated bydegradation enzymes and transporters, respectively. We described these processes
with Michaelis}Menten-type equations
characterized by the maximum velocity
DG (i"1, 2, 2) and ¹H ( j"1, 2, 2), andthe binding constant ¸G (i"1, 2, 2) andKH ( j"1, 2, 2). Nonspeci"c degradation
terms are proportional to each variablewith the proportional constant D . For
rescue experiments (described below), we
introduced transcriptional activation fromthe constitutive promoter, which is characterized by the synthesis rate CG (i"1, 2, 3).

FIG. 4. Simulation of single and double mutants. Relative
levels of per (blue) and dClk (red) mRNA are shown in thewild type (a) and per (b), dClk(PI (c) per; dClk(PI (d)mutants. The peak levels of per and dClk mRNA in the wild
type are set to 100%. In per mutants (b), sustaining oscillations are abolished and per and dClk mRNA levels are
decreased to about 42 and 46% of the peak levels in the wild
type. In dClk(PI (c) and per; dClk(PI (d) mutants, sustaining
oscillations are also abolished and the per mRNA level isseverely decreased whereas the dClk mRNA level is restored
to 124% of the peak level in the wild type. Parameter values
are as in Fig. 3(a)}(d), except for S "0 hr\ for per,S "0 hr\ for dClk(PI and S "S "0 hr\ for per;
dClk(PI.

FIG. 5. Simulation of rescue experiments in single-mutant backgrounds. (a) per (blue), tim (green) and dClk (red) oscillationsin the wild type. (b}d) Restored oscillations in transfected c-per (b), c-tim (c), c-dClk (d) single mutants. Rescued per oscillations(blue), tim oscillation (green) and dClk oscillation (red) are shown in Per }P¹ , ¹im }P¹ and Clk }CC phase planes. (e)
K
L
K
L
K
L
Simulated time courses of di!erent forms of PER proteins in the c-per mutant. Parameter values are as in Fig. 3(a)}(d), except
for S "0 nM hr\, C "0.846 nM hr\ for c-per mutant, S "0 nM hr\, C "0.846 nM hr\ for c-tim mutant andS "0 nM hr\, C "0.558 nM hr\ for c-dClk mutant.
FIG. 6. Simulation of rescue experiments in double-mutant backgrounds. (a) per (blue), tim (green) and dClk (red)oscillations in the wild type. (b}d) Restored oscillations in transfected c-tim; c-dClk (b), c-per; c-dClk (c), c-per; c-tim (d) double
mutants. Parameter values are as in Fig. 3(a)}(d) except for S "S "0 nM hr\, C "0.846 nM hr\, C "0.558 nM hr\for c-tim; c-dClk mutant, S "S " 0 nM hr\, C "0.846 nM hr\, C "0.558 nM hr\ for c-per-; c-dClk mutant andS "S "0 nM hr\, C " C "0.846 nM hr\ for c-per; c-tim mutant.
ROBUST OSCILLATIONS IN DROSOPHI¸A CIRCADIAN RHYTHM

levels peak at Z¹"17.3, 17.9 and 21.6, respectively, while total amounts of PER (TIM) peak at
Z¹"18.8. On the other hand, the dClk mRNAlevel peaks at Z¹"4.0. These results are consistent with observations that per and tim mRNAs
reach peak levels early in the evening (Z¹"13}16) (Hardin et al., 1990; Sehgal et al., 1994,
1995), that PER and TIM levels do not peak untillate evening (Z¹"18}24) (Zerr et al., 1990;
Edery et al., 1994) and that the dClk mRNA levelpeaks late at night to early in the morning (Z¹ 23
to Z¹ 4) (Darlington et al., 1998; Bae et al., 1998).
Next, we investigated phase shifts by light pulse.
It has been reported that TIM is rapidly reducedby light pulse (Hunter-Ensor et al., 1996; Zenget al., 1996; Myers et al., 1996; Lee et al., 1996).
Thus, we emulated light pulse by increasing
the rate of TIM degradation by two-, four-, six-,eight-, or ten-fold for 1 h. We applied the same
perturbation (promotion of TIM degradation)
to the system from 100 di!erent circadian time
points and measured the following peak time of
per mRNA. By comparing it with the peak timeof per mRNA without perturbation, we can obtain phase shifts. Obtained phase shifts are plotted along circadian time at which perturbation isapplied [Fig. 3(e)]. They show no phase shift inthe middle of subjective day, maximum phasedelays in early subjective night, and maximum
phase advances in late subjective night. Shapes ofthe phase shifts}circadian time relationship are
maintained in various intensity of perturbation
although the magnitude of phase shifts becomeslarger along the intensity. These results are also
consistent with the observation that light-pulses
delay the phase of the circadian activity rhythms
during early subjective night and advance thephase during late subjective night whereas lightpulses tend to cause minimal or no phase shifts
during the subjective day (Hall & Rosbash, 1987).
To verify our model by comparing with phenotypes in reported mutants, we create mathematical &&mutants'' which correspond to per, tim,dClk( and Cyc single mutants and per;dClk( and per; Cyc double mutants. In per
mutants, sustaining oscillations are abolished[Fig. 4(b)]. per, tim and dClk mRNA levels are
decreased to about 42, 42 and 46% of the peaklevels in the wild type. tim mutant shows similar
phenotype to a per mutant. These results are403

consistent with reported observations that inper and tim mutants, the per and tim transcription is constitutive and per, tim and dClk mRNAlevels are relatively low (Hardin et al., 1990;
Sehgal et al., 1994; Allada et al., 1998; Bae et al.,
1998; So & Rosbash, 1997; Glossop et al., 1999).
In dClk( and per; dClk( mutants, sustaining
oscillations are also abolished [Fig. 4(c) and (d)].
In these mutants, per and tim mRNA transcripts
severely decreased whereas the dClk mRNA levelis restored to 124% of the peak level in the wild
type. Cyc and per; Cyc mutants show similar
phenotypes to dClk( and per; dClk( mutants.
These results are also consistent with previous
observations that in dClk( and Cyc mutants
the per and tim mRNA levels are constant andsigni"cantly reduced (Allada et al., 1998; Rutila
et al., 1998) and that in dClk( and Cyc single
mutants and per; dClk( and per; Cycdouble mutants, dClk mRNA levels are almost
restored to the peak level in the wild type
(Glossop et al., 1999). The mathematical model
based on the interlocked feedback mechanism
reproduces decrease of per and tim mRNA levels
in mutants lacking PER (per) or TIM (tim)function, which seems to contradict the simple
feedback mechanism. An intuitive explanation
for these phenotypes is that PER and TIM protein knockout will inhibit the transcription ofdClk, which will inhibit the transcription of per
and tim resulting in low levels of per and tim
messages.
per genomic fragments lacking a promoter(Frisch et al., 1994) or driven by constitutively
active promoters (Ewer et al., 1988; Vosshall &Young, 1995) rescue locomotor activity rhythms
in per #ies. To test whether our model explainsthese observations, we create a mathematicalmutant which corresponds a per #y carrying
a transgene that constitutively expresses permRNA (termed as c-per). In the c-per mutant, tim
and dClk oscillations are restored [Fig. 5(b)]. We
also "nd that cytoplasmic PER, cytoplasmic
PER}TIM and total PER protein levels show
oscillations in the c-per mutant [Fig. 5(e)]. Oscillations in cytoplasmic PER and PER}TIM levels
occur because a tim feedback loop works andoscillatory expressed TIM proteins form PER}
TIM complexes with PER proteins. These oscillations produce the oscillation of the total PER

404

H. R. UEDA E¹ A¸.

protein level because of di!erent degradation
rates in di!erent forms of PER and/or nonlinearity of degradation process. These results are
consistent with a previously reported observationthat the PER protein level oscillates in per "lescarrying a transgene that constantly expresses
per mRNAs (Cheng & Hardin, 1998).
Similar rescue experiments in other single- or
double-mutant backgrounds reveal robust oscillations within the interlocked feedback model.
First, we transfect tim or dClk( single mutant
with a transgene expressing tim or dClk mRNAs
constitutively (termed as c-tim and c-dClk). c-tim
mutant restores per and dClk oscillations[Fig. 5(c)] while c-dClk mutant restores per andtim oscillations [Fig. 5(d)]. Next, we transfecttim; dClk( , per; dClk( or per; tim double
mutants with two transgenes expressing tim/
dClk, per/dClk or per/tim mRNAs constantly(termed as c-tim; c-dClk, c-per; c-dClk and c-per;c-tim). In c-tim; c-dClk, c-per; c-dClk and c-per;c-tim mutants, per, tim and dClk oscillations arerestored, respectively [Fig. 6(b)}(d)].
The experimental results summarized in
Figs. 3}6 show that the interlocked feedback
model provides a possible mechanism of Drosophila circadian rhythms. First, the interlocked
feedback model provides the similar time courseof mRNA and protein levels in the wild type andsimilar mRNA levels in single or double mutants
to previously reported observations, including
antiphase oscillations of per (tim) and dClk
mRNAs and decreased per and tim mRNA levels
in per and tim mutants, which are not explained by one feedback model. The interlocked
feedback model also provides light-pulse-typephase shifts induced by temporal promotion of
TIM degradation, which is reported to be involved in light-pulse entrainment. Second, theinterlocked feedback model shows highly robust
oscillations in mutations, which abolish oscillations of one or two mRNAs among per, tim anddClk mRNAs.
One of the most attractive properties ofcircadian rhythms is the relative constancy of
circadian period in the temperature range over
103C, which is referred to as temperature compensation (Pittendrigh, 1954; Konopka et al.,1989). Temperature compensation poses an important constraint on models of the circadianrhythms. For, this phenomenon seemingly contradicts the intuitive prediction that increase of
temperature by 103C typically doubles the reaction rate (Segel, 1975), resulting in shortening theperiod of oscillations to half of the original one.
One way to satisfy this constraint has been proposed by Leloup & Goldbeter (1997). They havefound that the increase of one system parameterthat represents one reaction rate of the circadianoscillator can either increase or decrease the period of oscillation and that magnitude of periodchange di!ers with altered parameters. They utilized this dependence of period on parameters andshowed that if appropriate set of parameters ismore sensitive (or insensitive) to temperaturethen the period of the circadian oscillators remains constant in the temperature range over
103C. The interlocked feedback model presented
here also shows various dependences of period
on system parameters (unpublished data). Thus,
in a way similar to that by Leloup and Goldbeter,
the interlocked feedback model might show constancy of the period in some temperature range
with several possible sets of parameters which are
more sensitive to temperature than others. The
challenge is to determine the correct parameter
set among them and this remains for the future
work.
Along evolution, various organism have developed molecular clocks, which seem more complex than simple negative feedback oscillators. In
mammals, per homologues Per1, Per2, Per3 anddClk functional homologue Bmal1 mRNA levels
oscillate in antiphase (Honma et al., 1998; Shearman et al., 2000). It has been shown that CLOCK
and BMAL1 activate vasopressin gene transcription and that all three mouse PERs and TIM
repress this activation (Jin et al., 1999). Although
this simple negative feedback regulation may
explain oscillatory expression of Pers mRNA,
it does not explain how BMA¸1 mRNA levels
oscillate in antiphase to Pers. In Neurosopra,
FREQUENCY protein (FRQ) is periodically
expressed and represses its own transcription.
A positive regulator of frq transcription, WHITECOLOR 1 protein (WC-1) is also rhythmically
expressed in antiphase to FRQ from constantwc-1 mRNA by post-transcriptional regulation.
WC-1 expression is regulated directly or indirectly by FRQ because driven FRQ expression

ROBUST OSCILLATIONS IN DROSOPHI¸A CIRCADIAN RHYTHM

positively regulates WC-1 synthesis with similar
lag time to that seen in the wild type (Lee et al.,2000). Although there are some molecular details
which remain to be uncovered and some divergent twists among species, a conserved principal
more complicated than simple negative feedback
mechanism seems to exist. The mathematicalmodel presented here would provide a means toinvestigate the design principal of seemingly
complex molecular clocks.
This work was performed as a part of the researchand development project of Industrial Science andTechnology Program supported by NEDO.
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