Modeling and Compositional Analysis of Genetic Regulatory Networks

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HAL Id: inria-00071211

https://hal.inria.fr/inria-00071211

Submitted on 23 May 2006

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Modeling and Compositional Analysis of Genetic Regulatory Networks

Gregor Gössler

To cite this version:

Gregor Gössler. Modeling and Compositional Analysis of Genetic Regulatory Networks. [Research

Report] RR-5801, INRIA. 2006. �inria-00071211�

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ISRN INRIA/RR--5801--FR+ENG

a p p o r t

d e r e c h e r c h e

Thème COM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Modeling and Compositional Analysis of Genetic Regulatory Networks

Gregor Gössler

N° 5801

Janvier 2006

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Unité de recherche INRIA Rhône-Alpes

655, avenue de l’Europe, 38334 Montbonnot Saint Ismier (France)

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4\RF28

not

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(11)

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f`jlM^XlVWK5_`~d~]`¨d~cZ\GJMQXlMLcXMZ\G{fZZlGJM.XZ\fZ\M^Xlp{f`]LM^fgc{t@XaJXMLZ\XkZlGJM;jlM;_g¯fgj\Mnj\MLpJj\M;XlMLc=ZlMDtf`XkpJjlMDtYde]LfZ\M;X;¨jhfZ\GJMLj¦ZlG{f`cWV

M;c=a{KM;j\fgZld~cJZlGJMXZ\fgZlMDXL

S Ä _ Ä ??Ä= 7

not

tr = V

i=1,...,n reg i ∨ inc i (eq _i ⁶ ) ∧ dec i (eq ^> _i ) ∨ eq ^< _i ∧ eq ^> _i

^E

MZ

P(X )

tYM;cJ_gZ\MQZlGJMXlMZn_`pJj\M;tYde]LfgZlMDXH_`c

X

4À65b= ??Ä=HG #I9

C(M)

^< )V P[+# H,T4j % #T2$1

M = (X, U, Y, θ, µ, ν)

Bb46QD>6"dREQ,HmEah^DF

C(M) = (

evels, A, A ^c , G, F ) = (B 1 kB 2 k...kB n )/V

^'F04a ^E

• ∀i = 1, ..., n . B i = counter (D i )

^B ^#

• V : A → P (D(θ))

^.,TRR

V (B i .inc) = V _i ^>

V (B i .dec) = V _i ^<

^{4j dR}

V _i ^< =reg i ∧ not

tr ∧ eq ^< _i ∨

^µ59·

switch i ∧ inc i (not

tr ∧ eq ⁶ _i )

^µ ^·

V _i ^> =reg i ∧ not

tr ∧ eq ^> _i ∨

^µ«w`·

switch i ∧ dec i (not

tr ∧ eq _i ^> )

^µE·

4\RF28

(D i )

%&ZGT./DTZ,H./K2QD>6H

D i (θ)

G$hdRE&528E [$KF

(D i ) = {level i }, {inc _i , dec i }, {G ^inc ⁱ = level i 6 θ ^p _i ⁱ , G ^dec ⁱ = level i > θ ¹ _i }, {F ^inc ⁱ = succ i (level i ), F ^dec ⁱ = prec i (level i )}

^4\RF2T

{inc i , dec i }

^28g,H280G0 ^128F5 E",HE52T H *D,F '

x i ∈ U

^128F5 ^E

x i ∈ Y

^E

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(Q, )

^GT&dRE ^{Fh ,F} ^'

C(M)

^E

q]Zld~_`c{X

inc i

µ

dec i

level i

_g

pJj\_gZ\MLd~c

i

QIn_`Zlde]MbZlG{fgZQ(MtJ_@cJ_`Z.fXlXlaJK"MaJc]_`cX[Z\j\f`dcJMDt}fgj\d~f`J~M;X^Zl_@5M]_c{X[Zhfgc=Z;1FHGJMpJj\M;tYde]LfgZlMDX

V _i ^<

^fgc{t

V _i ^>

XlpMD]dVGJM;cf¬Z\j\f`c{XdZld~_`ctYM;]LjlM;K"MLc=Zld~cJ

x i

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x i

¨gjlMDXp5M;]Zld~}`M;V¨d~XoM;c{fgJ~M;t¯/P_jlM(pJj\M;]Ld~XlML~V`¨g~d~cJM;X

(

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(

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Z\GJM^Zhfgj\`MLZ¦M;¼=aJd~d~Jj\daJK _`»ZlGJMQX_aJj\]LM^tY_K"f`dc dcJMDX

( )

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)

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N_c{tYdZld~_`c

not

tr

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D ∈ D ^s

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D

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D

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D

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V _i ^< ∧ V _i ^>

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tY_Kfgd~c{XHKf¥VG{f¥}`MQK"_`j\MQZlG{f`c_cJMQZ\f`jlMZM;¼=aJd~d~Jj\da{K

À YB 7f ,H

k

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B i

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B 1 kB 2 k...kB _n

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Q(M)

4À65b= ??Ä=HG 7 9

C(M) ˆ

^< )V PMh H,HF4j %Z #T2h$1

M

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C(M) = (D, A ˆ

^B

A ^c , G, F) = (B 1 kB 2 k...kB n )/V

^'F04a ^E

• ∀i = 1, ..., n . B i = counter (D _i )

^B ^#

• V : A → P (D)

^.,TRR

V (B i .inc) = V _i ^>

V (B i .dec) = V _i ^<

^{4j dR}

V ˆ _i ^< =reg i ∧ not

tr ∧ eq ^< _i ∨ switch i ∧ inc i (eq ⁶ _i )

V ˆ _i ^> =reg i ∧ not

tr ∧ eq ^> _i ∨ switch i ∧ dec i (eq _i ^> )

(12)

\F

(Q, ) ˆ

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C(M) ˆ

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a

b

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µ a =

20

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0 6 x a < θ ² _a ∧ 0 6 x b < θ _b ¹

0

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µ b =

20

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0 6 x a < θ ¹ _a ∧ 0 6 x b < θ _b ²

0

^{dRE24j %}

4j dR

θ _a ¹ = θ _b ¹ = 4

θ _a ² = θ _b ² = 8

^E Rh12T5 O28KF (' GTRh#28KF 28h46$

2

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C(M )

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C(M) ˆ

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θ ² _b

θ ¹ _b

θ _a ² θ _a ¹

x b

0 x a

θ ² _b

θ ¹ _b

θ ¹ _a θ ² _a x a

0 x b

§/d~`aJj\Mnz`ª»ikG{f`XlM¦Xpf`]M¤_`WZlGJMpJdMD]M;d~XlM¤d~cJMDfgjXlVYX[Z\MLK dZ\G1jlM;~fgZld~}`M¤p5_XldZ\d_cQ_`WZ\fgj\`MLZ¯M;¼=aJd~d~Jj\d~f1µ©MLZh·fgc{tQZ\j\f`c{XdZld~_`c

XlVYX[Z\MLK#_`Z\GJM¼=a{fg~dZhfZ\d}MQK"_YtYM;

Q(M)

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x b

θ ² _b

θ ¹ _b

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θ ¹ _b

θ ¹ _a θ _a ² x a

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C(M)

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C(M ˆ )

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C(M)

Q(M)

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K"_YtYMLd~¯5M1jl_Ja{XZHM;cJ_`a{`GrX_f`XZ\_fg~_¹_jdc=Z\MLj\MDf¥}=d~cJbZljhfgc{XldZld~_`c{XHd~c{XZlM;ft@_`oXlVWc{]hGJj\_`cJ_a{X(X[Z\MLp{X;

; ] ÀWÄ À 7'

D D ⁰

^'F2

D, D ⁰ ∈ D

^BRF

D → D ⁰

^E

(13)

FHG{fgZd~X;¨Jd

C(M)

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Q(M )

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D D ⁰

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D ⁰ _k < D k

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∀i 6= k . D ⁰ _i = D i

pMtYdeX[Z\dcJaJdeXGZ[(_"];f`XlM;X;

³

switch(D _k ⁰ )

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V _k ^< (D)

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(not

tr ∧ eq _k ^< )(D)

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→

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D → D ⁰

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inc k (eq _k ⁶ )

(D)

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eq _k ⁶ (D ⁰ )

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D ⁰⁰

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i = 1, ..., n

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switch i ∧ dec i (eq _i ⁶ )

(D)

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D _k ⁰ = D ⁰⁰ _k

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(D ⁰⁰ ) ∧ dec i (eq _i ^< )

(D ⁰⁰ )

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eq ^< _i

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→

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D → D ⁰⁰

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D → D ⁰

; ] ÀWÄ À G

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C(M)

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switching(D) ⊂ switching(D ⁰ )

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switching(D ⁰ ) ⊂ switching(D)

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(not

tr∧ eq _k ^< )(D)

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(D)

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Q(M)

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deXp5_X\Xd~J~MQd~c

C(M) ˆ

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S Ä _ Ä ??Ä=IG '

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switching(D ⁰ ) ⊂ switching(D)

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I = {i | D i 6= D _i ⁰ }

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V

i∈I eq _i ⁶ (D ⁰ )

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(i 1 , i 2 , . . . , i k )

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I

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tY_Kfgd~c{X

D ⁱ ⁰ , D ⁱ ¹ , D ⁱ ² , . . . , D ⁱ ^k = D ⁰

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j ∈ {1, ..., k}

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V ˆ ^<

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Q(M)

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4À65b= ??Ä=HG G 9

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^< ^\

D ⁰¹

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i∈{1,...,n}

^

j∈{1,...,n} r {i}

reg j

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D ⁰¹

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C(M)

^f`c{t

Z\GJM1_}`M;j±´fgpJp{jl_¥¡Yd~K"fgZld~_`c

C(M ˆ )

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Modeling and Compositional Analysis of Genetic Regulatory Networks

HAL Id: inria-00071211

https://hal.inria.fr/inria-00071211

Submitted on 23 May 2006

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Modeling and Compositional Analysis of Genetic Regulatory Networks

Gregor Gössler

To cite this version:

Gregor Gössler. Modeling and Compositional Analysis of Genetic Regulatory Networks. [Research

Report] RR-5801, INRIA. 2006. �inria-00071211�

ISRN INRIA/RR--5801--FR+ENG

a p p o r t

d e r e c h e r c h e

Thème COM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Modeling and Compositional Analysis of Genetic Regulatory Networks

Gregor Gössler

N° 5801

Janvier 2006

Unité de recherche INRIA Rhône-Alpes

655, avenue de l’Europe, 38334 Montbonnot Saint Ismier (France)

˙

x = f (x, u) − g(x, u)x

x i

D r i = {[0, θ 1 i )} ∪ {(θ j i , θ i j+1 ) | 1 6 j < p i } ∪ {(θ p i i , max i ]}

D i s = {θ j i | 1 6 j 6 p i }

θ i j < θ j+1 i

1 6 j 6 p i

D i = D r i ∪ D i s

D = D 1 ×D 2 ×...×D n

D r = D 1 r ×D 2 r ×...×D r n

D s = D r D r

D ∈ D r

˙

x = µ D − ν D x

φ

φ i (D) = µ D i /ν D i

φ(D) ∈ D

D

D

µ

ν

D

X

X

P (X) = 2 X

B

(X, A, A c , G, F)

• X

• A

A u

A c

• G : A → P (X)

• F : A → X → X

G a

F a

G(a)

F (a)

B = (X, A, A c

G, F )

→: X × A × X

∀x, x 0 ∈ X ∀a ∈ A . x → a x 0 ⇐⇒ G a (x) ∧ x 0 = F a (x)

x → x 0

∃a ∈ A . x → a x 0

→ ∗

→

B = (X, A, A c , G, F)

a ∈ A

∃x 0 , x 00 . x → ∗ x 0 ∧x 0 → a x 00

G(B)

G(B ) = W

a∈A G a

B = (X, A, A c , G, F )

P ∈ P(X)

P

a

pre a (P ) = G a ∧ P[F a (X )/X]

P[F a (X )/X]

P

D ^r _i = {[0, θ ¹ _i )} ∪ {(θ ^j _i , θ _i ^j+1 ) | 1 6 j < p i } ∪ {(θ ^p _i ⁱ , max i ]}

D _i ^s = {θ ^j _i | 1 6 j 6 p i }

θ _i ^j < θ ^j+1 _i

D i = D ^r _i ∪ D _i ^s

D ^r = D ₁ ^r ×D ₂ ^r ×...×D ^r _n

D ^s = D r D ^r

D ∈ D ^r

x = µ ^D − ν ^D x

φ i (D) = µ ^D _i /ν ^D _i

P (X) = 2 ^X

(X, A, A ^c , G, F)

A ^u

A ^c

G ^a

F ^a

B = (X, A, A ^c

∀x, x ⁰ ∈ X ∀a ∈ A . x → â x ⁰ ⇐⇒ G â (x) ∧ x ⁰ = F â (x)

x → x ⁰

∃a ∈ A . x → ^a x ⁰

→ ^∗

B = (X, A, A ^c , G, F)

∃x ⁰ , x ⁰⁰ . x → ^∗ x ⁰ ∧x ⁰ → ^a x ⁰⁰

a∈A G ^a

B = (X, A, A ^c , G, F )

pre a (P ) = G ^a ∧ P[F ^a (X )/X]

P[F ^a (X )/X]

F ^a (X)

B = (X, A, A ^c , G, F )

a∈A ^c pre a (P) ∧ ¬ W

a∈A ^u pre a (¬P )

c ⁰ (P ) = P

c ⁱ⁺¹ (P ) = pre

c ⁱ (P )

B = (X, A, A ^c , G, F)

B = (X, A, A ^c , G, F)

B = (X, A, A ^c , G, F )

V = (U ^a ) a∈A ^c

B = (X, A, A ^c , G, F )

V = (U ^a ) _a∈A ^c

B/(P, V ) = (X, A, A ^c , G ⁰ , F )

a ∈ A ^c

(G ^a ) ⁰ =

G ^a ∧U ^a ∧pre a (P)

blocked(a) = G ^a ∧ ¬(G ^a ) ⁰

EA(x) = {a ∈ A | (G ^a ) ⁰ (x)}

A = A ^c ∪ A ^u

≺ ⊆ A ^c × A

A ⁰ ⊆ A

A ⁰ /≺ = {a ∈ A ⁰ | ¬∃a ⁰ ∈ A ⁰ . a ≺ a ⁰ }