Supporting Information
A mathematical model of the mammalian circadian oscillatory network for comparing the
expression phases of two Per genes (Model1).
The mRNA expression phases of Period1 (Per1) and Period2 (Per2) are calculated using the
mathematical model schematized in Figure 1A, which is based on the model proposed by Leloup and
Goldbeter in 2003 [1]. The original model incorporates the following cellular processes: transcription,
translation, and degradation of Period (Per), Cryptochrome (Cry), Bmal1 and Rev-erb mRNAs;
reversible formation of unphosphorylated PER-CRY complex and PER-CRY-BMAL1 complex;
reversible phosphorylation of PER, CRY, PER-CRY and BMAL1; and reversible nuclear entry of
unphosphorylated PER-CRY, BMAL1 and REV-ERB. PER, CRY, REV-ERB, BMAL1, PER-CRY
complex, and BMAL1-PER-CRY complex are denoted by P, C, R, B, PC, and BPC, respectively, in
schematics and equations. In addition, the subscript letters C, N, CP, and NP denote cytoplasmic
protein, nuclear protein, phosphorylated cytoplasmic protein, and phosphorylated nuclear protein,
respectively. For simplicity, the Cry1 and Cry2 genes are represented by the single Cry gene. Because
CLOCK protein is constitutively expressed at a high level, BMAL1 protein is assumed to form a
complex with CLOCK immediately and maintain this complex. The PER1 and PER2 proteins are
represented by the single PER protein. The cellular abundances of Per1 and Per2 mRNAs are
calculated by the same differential equation as for the original Per mRNA. All kinetics parameters are
available in Table S1 (Model1), and all kinetic equations for the 20 variables are described below:
dPer1
Bn
Per1
 vsP1 n N n  vmP1
 kdmp Per1
dt
K AP  BN
K mP  Per1
(S1)
dPer2
Bn
Per2
 vsP2 n N n  vmP2
 kdmp Per2
dt
K AP  BN
K mP  Per2
(S2)
dCry
Bn
Cry
 vsC n N n  vmC
 kdmcCry
dt
K AC  BN
K mC  Cry
(S3)
1/5
dRev - erb
Bh
Rev - erb
 vsR h N h  vmR
 kdmr Rev - erb
dt
K AR  BN
K mR  Rev - erb
(S4)
dBmal1
Km
Bmal1
 vsB m IB m  vmB
 kdmb Bmal1
dt
K IB  RN
K mB  Bmal1
(S5)
dPC
PC
PCP
 ksP * (Per1  Per2)  k3 PC CC  k4 PCC  V1P
 V2P
 kdn PC
dt
K p  PC
K dp  PCP
(S6)
dCC
CC
CCP
 ksC * Cry  k3 PC CC  k4 PCC  V1C
 V2C
 kdnCC
dt
K p  CC
K dp  CCP
(S7)
dPCP
PC
PCP
PCP
 V1P
 V2P
 vdPC
 kdn PCP
dt
K p  PC
K dp  PCP
K d  PCP
(S8)
dCCP
CC
CCP
CCP
 V1C
 V2C
 vdCC
 kdnCCP
dt
K p  CC
K dp  CCP
K d  CCP
(S9)
dPCC
PCC
PCCP
 k3 PC CC  k4 PCC  k1PCC  k2 PCN  V1PC
 V2PC
 kdn PCC
dt
K p  PCC
K dp  PCCP
(S10)
dPCN
PCN
PCNP
 k1PCC  k2 PCN  k7 BN PCN  k8 BPCN  V3PC
 V4 PC
 kdn PCN
dt
K p  PCN
K dp  PCNP
(S11)
dPCCP
PCC
PCCP
PCCP
 V1PC
 V2PC
 vdPCC
 kdn PCCP
dt
K p  PCC
K dp  PCCP
Kd  PCCP
(S12)
dPCNP
PCN
PCNP
PCNP
 V3PC
 V4 PC
 vdPCN
 kdn PCNP
dt
K p  PCN
K dp  PCNP
K d  PCNP
(S13)
dRC
RC
 ksR * Rev - erb  k9 RC  k10 RN  vdRC
 kdn RC
dt
K d  RC
(S14)
dRN
RN
 k9 RC  k10 RN  vdRN
 kdn RN
dt
K d  RN
(S15)
dBC
BC
BCP
 ksB * Bmal1  k5 BC  k6 BN  V1B
 V2B
 kdn BC
dt
K p  BC
K dp  BCP
(S16)
dBN
BN
BNP
 k5 BC  k6 BN  k7 BN PCN  k8 BPCN  V3B
 V4B
 kdn BN
dt
K p  BN
K dp  BNP
(S17)
dBCP
BC
BCP
BCP
 V1B
 V2B
 vdBC
 kdn BCP
dt
K p  BC
K dp  BCP
K d  BCP
(S18)
2/5
dBNP
BN
BNP
BNP
 V3B
 V4B
 vdBN
 kdn BNP
dt
K p  BN
K dp  BNP
K d  BNP
(S19)
dBPC N
BPC N
 k7 BN PC N  k8 BPC N  vdIN
 kdn BPC N
dt
K d  BPC N
(S20)
A modified mathematical model introduced a nuclear PER and CRY monomer to simulate the
expression phase difference between Per1 and Per2 mRNA expressions (Model2).
To introduce an additional mechanism of transcriptional regulation by a nuclear PER monomer, several
assumptions were introduced to the model described above. The dissociation of the nuclear PER-CRY
complex (PCN) was described in the same equation, and parameters such as cytoplasmic PER-CRY
(PCC) and nuclear PER (PN) and CRY (CN) were degraded in the same manner as the nuclear
PER-CRY complex instead of by the PER-CRY degradation processes. Nuclear CRY associates and
dissociates with nuclear CLOCK-BMAL1 (BN) in the same manner as nuclear PER-CRY. Changes in
five additionally introduced molecules (nuclear PER (PN), nuclear CRY (CN), nuclear phosphorylated
PER (PNP), nuclear phosphorylated CRY (CNP), and BMAL1-CRY complex (BCN)) were calculated by
the following equations:
dPN
PN
PNP
 k11PN CN  k12 PCN  V3P
 V4 P
 kdn PN
dt
K p  PN
K dp  PNP
(S21)
dCN
CN
CNP
 k11PN CN  k12 PCN  V3C
 V4C
 k13 BN CN  k14 BCN  kdnCN
dt
K p  CN
K dp  CNP
(S22)
dPNP
PN
PNP
PNP
 V3P
 V4 P
 vdPN
 kdn PNP
dt
K p  PN
K dp  PNP
K d  PNP
(S23)
dCNP
CN
CNP
CNP
 V3C
 V4C
 vdCN
 kdnCNP
dt
K p  CN
K dp  CNP
K d  CNP
(S24)
dBC N
BC N
 k13 BN C N  k14 BC N  vdBCN
 kdn BC N
dt
K d  BC N
(S25)
3/5
Accordingly, the kinetic equation of nuclear PER-CRY complex (PCN, Eq. S11) and nuclear BMAL1
(BN, Eq. S17) were replaced by Eq. S11’ and Eq. S17', respectively:
dPCN
PCN
PCNP
 k1PCc  k2 PCN  k11PN CN  k12 PCN  k7 BN PCN  k8 BPCN  V3PC
 V4 PC
 kdn PCN
dt
K p  PCN
Kdp  PCNP
(S11’)
dBN
BN
BNP
 k5 BC  k6 BN  k7 BN PCN  k8 BPCN  k13 BN CN  k14 BCN  V3B
 V4B
 kdn BN
dt
K p  BN
K dp  BNP
(S17’)
All kinetics parameters are available in Table S1 (Model2).
The Per2 positive feedback regulation model.
This model is based on the modified model described above (Eqs. S1-S25) and tests the hypothesis that
nuclear PER acts as a positive regulator of Per2 mRNA transcription. The positive feedback regulation
of Per2 transcription by nuclear PER (PN) is described in a mass action law. The Per2 mRNA kinetic
equation (Eq. S2) is replaced by Eq. S2a:
dPer2
Bn
Per2
 vsP2 n N n  kAP2 PN  vmP2
 kdmp Per2
dt
K AP  BN
K mP  Per2
(S2a)
where kAP2 denotes a rate coefficient of positive feedback regulation of Per2 transcription by PER. In
this model including the positive feedback regulation of Per2 transcription by PER proteins (kAP2 > 0),
bistability appeared as the fluctuating orbits kept stable oscillation or converged to a stable steady state
depending on the initial values of variables.
The Per1 positive feedback regulation model.
This model tests the hypothesis that nuclear PER2 acts as a positive regulator of Per1 mRNA
transcription. The positive feedback regulation of Per1 transcription by nuclear PER (PN) is described
in a mass action law. The Per1 mRNA kinetic equations (Eq. S1) is replaced by Eq. S1a:
dPer1
Bn
Per1
 vsP1 n N n  kAP1PN  vmP1
 kdmp Per1
dt
K AP  BN
K mP  Per1
4/5
(S1a)
where kAP1 denotes a rate coefficient of positive feedback regulation of Per1 transcription by PER. The
circadian oscillations are simulated when kAP1 is a value from 0.01 to 0.92 h-1. The full model is
governed by Eq. S1a and Eqs. S2-S25.
The Per2 negative feedback regulation model.
This model tests the hypothesis that nuclear PER acts as a negative regulator of Per2 mRNA
transcription, the negative feedback regulation of Per2 transcription by nuclear PER (PN) is introduced
into the original transcriptional term. The Per2 mRNA kinetic equation (Eq. S2) is replaced by Eq.
S2b:
dPer2
BNn
Per2
 vsP2 n
 vmP2
 kdmp Per2
n
n
dt
K AP  BN  (kRP2 PN )
K mP  Per2
(S2b)
where kRP2 denotes the strength of negative feedback regulation of Per2 transcription by PER. kRP2 can
be varied from 0 to more than 10000. The full model is governed by Eq. S1, Eq. S2b and Eqs. S3-S25.
The Per1 negative feedback regulation model.
This model is governed by tests the hypothesis that nuclear PER acts as a negative regulator of Per1
mRNA transcription. The negative feedback regulation of Per1 transcription by nuclear PER (PN) is
introduced into the original transcriptional term. The Per1 mRNA kinetic equation (Eq. S1) is replaced
by Eq. S1b:
dPer1
BNn
Per1
 vsP1 n
 vmP1
 kdmp Per1
n
n
dt
K AP  BN  (kRP1PN )
K mP  Per1
(S1b)
where kRP1 denotes the strength of negative feedback regulation of Per1 transcription by PER. kRP1 can
be varied from 0 to more than 10000. The full model is governed by Eq. S1band Eqs. S2-S25.
1.
Leloup JC, Goldbeter A (2003) Toward a detailed computational model for the mammalian
circadian clock. Proc Natl Acad Sci USA 100: 7051-7056.
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