Mathematical details on the circadian clock transcription circuit with

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Supporting Document
Bayesian Experts in Exploring Reaction Kinetics of Transcription Circuits
- On Transcriptional Circuit Model for Mammalian Circadian Clock -
Yoshida, R., Saito, M.M., Nagao, H. and Higuchi, T.
The Institute of Statistical Mathematics, ROIS
Mammalian Clock and Clock-Controlled Genes in the Model
According to the TF promoter analysis of Ueda et al. (2005), we have developed a model on
transcription factor regulatory networks, involving 19 clock-related state variables in below:
Clock-Regulating
8 Transcription Factors
8 mRNAs
3 TF complexes
Clock
Bmal1 (Arntl or Mop3)
Helix-loop-helix-PAS transcription factors
Per1
Per2
Cry1
RevErbA α (Nr1d1)
Orphan nuclear hormone receptors
Rora (Ror αlpha)
Dbp
bZip-family (transcription factor)
clock
bmal1 (arntl or mop3)
per1
per2
cry1
revErbA α (nr1d1)
rora (ror αlpha)
dbp
Clock/Bmal1
Per1/Cry1
Per2/Cry1
Network Diagram of the Transcriptional Circuits (Hybrid Functional Petri Net)
All the diagrams relevant to the baseline and degradation are omitted.
dbp
per1
Dbp
Per1
clock
Clock
Per1/Cry1
per2
Per2
Bmal1/Clock
bmal1
Bmal1
Per2/Cry1
cry1
Cry1
rev-erbA α
Ror(a)
Normal arc
Rev-ErbA α
ror(a)
Inhibitory arc
Test arc
Continuous place
Transition
The Mathematical Form of ODE Model
The circuit model describes a regulatory mechanism of the endogenous variables based on the
following 19 ODEs.
Notations involving mRNA
mRNA
Three types of regulatory elements in promoters
E/E’ box
Notations involving Protein and Protein Complex
Protein or Protein complex
TF binding sites on promoters
Activators or Inhibitor?
E/E’ box
A R
D box
RRE
D box
RRE
The Mathematical Form of ODE Model
Variable: per1; Description: Transcription; Variable ID: 1
Regulatory elements in promoter: E/E’ box, D box
Activators: Bmal1/Clock, Dbp
Inhibitors: Per1/Cry1, Per2/Cry2
d x p e r 1 (t )
dt
s
= - f 1x p e r 1 (t ) + q 1
s
x D b p (t )
´
K
s
1 ,4
1 ,4
x B m a l 1 /C lo c k (t )
K
s
1 ,1
1 ,1
1 ,1
E/E’ box
1
s
+ x B m a l 1 /C lo c k (t )
1 ,1
1
1 + (x P e r 1 /C r y 1 (t ) K
s
1 ,2 )
1 ,2
dt
s
s
x B m a l 1 /C lo c k (t )
= - f 2 x p e r 2 (t ) + q 2
K
s
´
K
s
2 ,4
2 ,4
s
1 ,3 )
1 ,3
+ m1
1 ,4
+ x D b p (t )
x D b p (t )
1 + (x P e r 2 /C r y 1 (t ) K
1 ,4
Variable : per2; Description: Transcription; Variable ID: 2
Regulatory elements in promoter: E/E’ box, D box
Activators: Bmal1/Clock, Dbp, Npas2
Inhibitors: Per1/Cry1, Per2/Cry2, Per3, Dec1
d x p e r 2 (t )
D box
s
2 ,1
2 ,1
s
2 ,4
1
s
+ x B m a l 1 /C lo c k (t )
2 ,4
+ x D b p (t )
2 ,1
+ m2
E/E’ box
2 ,1
1 + (x P e r 1 /C r y 1 (t ) K
D box
1
s
2 ,2 )
2 ,2
1 + (x P e r 2 /C r y 1 (t ) K
s
2 ,3 )
2 ,3
The Mathematical Form of ODE Model
Variable: cry1; Description: Transcription; Variable ID: 3
Regulatory elements in promoter: E/E’ box, RRE
Activators: Bmal1/Clock, Rora,
Inhibitors: Per1/Cry1, Per2/Cry2, Rev/ErbA
d x c r y 1 (t )
dt
s
x B m a l 1 /C lo c k (t )
= - f 3 x c r y 1 (t ) + q 3
K
s
x R o r a (t )
´
K
s
3 ,4
s
3 ,1
3 ,1
3 ,1
1
s
+ x B m a l 1 /C lo c k (t )
3 ,4
4 ,1
+ x R o r a (t )
3 ,4
1 + (x R e v / E r b A (t ) K
s
3 ,5
)
3 ,5
s
3 ,2
)
3 ,2
dt
s
x B m a l 1 /C lo c k (t )
= - f 4 x r e v / e r b A (t ) + q 4
K
´
s
4 ,1
4 ,1
1
1 + (x P e r 2 /C r y 1 (t ) K
s
4 ,4
)
4 ,4
s
3 ,3
E/E’ box
s
4 ,1
x D b p (t )
s
+ x B m a l 1 /C lo c k (t )
+ m4
1 + (x P e r 2 /C r y 1 (t ) K
)
3 ,3
+ m3
Variable: rev/erbA; Description: Transcription; Variable ID: 4
Regulatory elements in promoter: E/E’ box D box
Activators: Bmal1/Clock, Dbp
Inhibitors: Per1/Cry1, Per2/Cry1
d x r e v / e r b A (t )
RRE
1
1 + (x P e r 1 /C r y 1 (t ) K
1
s
E/E’ box
4 ,1
K
s
4 ,2
4 ,2
4 ,2
D box
1
s
+ x D b p (t )
4 ,2
1 + (x P e r 1 /C r y 1 (t ) K
s
4 ,3
)
4 ,3
The Mathematical Form of ODE Model
Variable: rora; Description: Transcription; Variable ID: 5
Regulatory elements in promoter: D box
Activators: Dbp
Inhibitors:
d x r o r a (t )
dt
s
x D b p (t )
= - f 5 x r o r a (t ) + q 5
K
s
5 ,1
5 ,1
D box
5 ,1
s
+ x D b p (t )
5 ,1
Variable: clock; Description: Transcription; Variable ID: 6
Regulatory elements in promoter: RRE
Activators: Rora
Inhibitors: Rev/ErbA
d x c lo c k (t )
dt
RRE
s
= - f 6 x c lo c k (t ) + q 6
x R o r a (t )
1
1 + (x R e v / E r b A (t ) K
s
6 ,1
)
6 ,1
K
s
6 ,2
6 ,2
s
+ x R o r a (t )
6 ,2
The Mathematical Form of ODE Model
Variable: bmal1; Description: Transcription; Variable ID: 7
Regulatory elements in promoter: RRE
Activators: Rora, Per1/Cry1, Per2/Cry1
Inhibitors:, Rev/ErbA
d x b m a l 1 (t )
dt
RRE
s
= - f 7 x b m a l 1 (t ) + q 7
s
x P e r 1 /C r y 1 (t )
´
K
s
7 ,3
x R o r a (t )
1
1 + (x R e v / E r b A (t ) K
s
)
7 ,1
s
7 ,3
x P e r 2 /C r y 1 (t )
s
+ x P e r 1 /C r y 1 (t )
7 ,3
K
s
7 ,4
7 ,1
K
s
7 ,2
s
+ x R o r a (t )
dt
s
x B m a l 1 /C lo c k (t )
s
+ x P e r 2 /C r y 1 (t )
= - f 8 x d b p (t ) + q 8
K
s
8 ,1
8 ,1
8 ,1
7 ,4
+ m7
E/E’ box
1
s
+ x B m a l 1 /C lo c k (t )
8 ,1
7 ,2
7 ,4
Variable: dec1; Description: Transcription; Variable ID: 8
Regulatory elements in promoter: E/E’box
Activators: Bmal1/Clock
Inhibitors: Per1/Cry1, Per2/Cry2
d x d b p (t )
7 ,2
1 + (x P e r 1 /C r y 1 (t ) K
1
s
8 ,2 )
8 ,2
1 + (x P e r 2 /C r y 1 (t ) K
s
8 ,3 )
8 ,3
The Mathematical Form of ODE Model
Variable: Per1; Description: Translation, Biding; Variable ID: 9
Bind to: Cry1
d x P e r 1 (t )
= - f 9 x P e r 1 (t ) + p 9 x p e r 1 (t ) - l
dt
P e r 1 /C r y 1
Bind
x P e r 1 (t )x C r y 1 (t )
Variable : Per2; Description: Translation, Binding; Variable ID: 10
Bind to: Cry1
d x P e r 2 (t )
= - f
dt
10
x P e r 2 (t ) + p 10 x p e r 2 (t ) - l
P e r 2 /C r y 1
x P e r 2 (t )x C r y 1 (t )
Variable: Cry1; Description: Translation, Binding; Variable ID: 11
Bind to: Per1, Per2
d x C r y 1 (t )
= - f
dt
11
x C r y 1 (t ) + p 11x c r y 1 (t ) - l
P e r 1 /C r y 1
x P e r 1 (t )x C r y 1 (t ) - l
Variable: Rev/ErbA; Description: Translation; Variable ID: 12
d x R e v / E r b A (t )
dt
= - f
12
x R e v / E r b A (t ) + p 12 x r e v / e r b A (t )
Bind
P e r 2 /C r y 1
Bind
x P e r 2 (t )x C r y 1 (t )
RRE
R
The Mathematical Form of ODE Model
Variable: Clock; Description: Translation, Binding; Variable ID: 13
Bind to: Bmal1
d x C lo c k (t )
= - f
dt
13
x C lo c k (t ) + p 13 x c lo c k (t ) - l
B m a l 1 /C lo c k
x B m a l 1 (t )x C lo c k (t )
Variable: Bmal1; Description: Translation; Variable ID: 14
Bind to: Clock
d x B m a l 1 (t )
= - f
dt
14
x B m a l 1 (t ) + p 14 x b m a l 1 (t ) - l
Bind
B m a l 1 /C lo c k
Bind
x B m a l 1 (t )x C lo c k (t )
Variable: Rora; Description: Translation; Variable ID: 15
d x R o r a (t )
= - f
dt
15
dt
= - f
16
A
D box
A
x R o r a (t ) + p 15 x r o r a (t )
Variable: Dbp; Description: Translation, Biding; Variable ID: 16
d x D b p (t )
RRE
x D b p (t ) + p 16 x d b p (t )
The Mathematical Form of ODE Model
Variable: Per1/Cry1; Description: Binding; Variable ID: 17
d x P e r 1 /C r y 1 (t )
= - f
dt
17
x P e r 1 /C r y 1 (t ) + l
P e r 1 /C r y 1
= - f
dt
18
x P e r 2 /C r y 1 (t ) + l
P e r 2 /C r y 1
dt
= - f
19
x B m a l 1 /C lo c k (t ) + l
E/E’ box
R
E/E’ box
A
x P e r 2 (t )x C r y 1 (t )
Variable: Bmal1/Clock; Description: Binding; Variable ID: 19
d x B m a l 1 /C lo c k (t )
R
x P e r 1 (t )x C r y 1 (t )
Variable: Per2/Cry1; Description: Binding; Variable ID: 18
d x P e r 2 /C r y 1 (t )
E/E’ box
B m a l 1 /C lo c k
x B m a l 1 (t )x C lo c k (t )
A Principle of Transcriptional Activities in Circadian Clocks
Order of increase in daily transcriptional activities. Each gene is first categorized into the three
elements (in rows) according to its target binding element, and further classified (in columns)
according to its regulatory elements (combinations of the three regulatory sequences).
Ueda et al., Nature Genetics, 2005
about 24 hours
E/E’ box
RRE
E/E’ box
E/E’ box
D box
D box
RRE
RRE
Set of genes binding to E/E’ box elements in promoters (purple: repressors; green: activators)
Cloc
k
Bhlh
b2
Per1
Bhlh
b3
Per2
Per3
Cry1
Npas
2
Bmal
1
Set of genes binding to D box elements in promoters (purple: repressors; green: activators)
Nfil3
D box
Dbp
Set of genes binding to RRE elements in promoters (purple: repressors; green: activators)
RevE
rbAα
Rora
Rorc
Nr1d
2
Rorb
RRE
E/E’ box

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