METABOLIC NETWORK THE STOICHIOMETRY MATRIX

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III

METABOLIC NETWORK THE STOICHIOMETRY MATRIX

Cours Métabolisme -3

Jean – Pierre Mazat Septembre 2008

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FROM GENOME TO PROTEOME AND TO METABOLOME

ÎProteome : Proteines ÎGenome : ADN

ÎTranscriptome : ARN

ARNm ARNt ARNi

……..

Sequencing

DNA ships

Transcription

Traduction

2D Gels, MS,…

ÎMetabolome/ Fluxome X 1

E 2

X 2

E 3

P

E 1 Flux measurements

ÎInteractome / Regulations Protein interactions measurements E 1

E 2 E 3

J.-P. Mazat Cours Bioinfo 2006

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WHAT IS A METABOLIC NETWORK ?

X 1 = 0,1 mM X 1 = 0,3 mM

= 0 X 1 S 0

V 1

X 1 P

V 2

X 1

2 enzymatic reactions

F

S 0 X

1 P

V 1 V

2

Metabolic network

X 1 concentrations are different in both reactions

V 1 = V 1 (X 1 , p 1 )

V 2 = V 2 (X 1 , p 2 ) [X 1 ]different

V 1 V 2

[X 1 ] value is the same in both reactions.

V 1 = V 1 (X 1 , p 1 ) V 2 = V 2 (X 1 , p 2 ) et

[X

1

] has the same value in both functions.

One gets a relationship between V

1

and V

2

eliminating X

1

between both equations

ÎIn (V

1

;V

2

) space, the rate space is a curve

V, set of acceptable rates. V 1

V 2

V (p)

V (p)

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STEADY STATE Example 1 (1)

= v 1 – v 2 = 0 Î v 1 = v 2 dx 1

dt

v

X 1

F ϑ ( ) μ 0

X 1 o

v

v 1 2

V Max

v 1 = v 1 (x 1 , p 1 )

v 2 = v 2 (x 1 , p 2 ) v 1 (x 1 , p 1 ) = v 2 (x 1 , p 2 ) Î x 1 o

Flux : F 1 = v 1 (x 1 o , p 1 ) F 2 = v 2 (x 1 o , p 2 ) F 1 = F 2 = F

F

S 0 X

1 P

V 1 V

2

V (p)

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STEADY STATE Exemple 1 (2)

= v 1 – v 2 = 0 Î v 1 = v 2 dx 1

dt

v

x 1

F ϑ ( ) μ 0

x 1 o

v

2

v

1

v 1 = k 1 (a - x 1 ) v 2 = k 2 x 1

k 1 (a - x 1 ) = k 2 x 1 Î x 1 o = k 1 a k 1 + k 2

Flux : F = F 1 = F 2 = k 1 k 2 a k 1 + k 2

Later on : Geometric interpretation N.V = 0 (2) ⇔ v 1 = v 2

v

2

v

1

F 2

ϑ ( ) μ 0

v

1

= v

2

(Ker [N])

F 1

F

S 0 X

1 P

V 1 V

2

k

1

a k

2

a

V (p)

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F

S 0 X

1 P

V 1 V

2

IS THERE ALWAYS A STEADY STATE? N0

v

X 1 v

2

v

1

Î Accumulation of X 1

Pathological cases of the inborn errors of metabolism when step 2 is largely Decreased.

Î accumulation intermediate metabolites which, in general, are toxic.

- One can also obtain oscillations (see chapter II). JP Mazat -

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A SLIGHTLY MORE COMPLEX EXAMPLE THE CONCEPT OF STOICHIOMETRY MATRIX

Cours Métabolisme -3

Jean – Pierre Mazat Septembre 2008

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STOICHIOMETRY MATRIX EXAMPLE 4

dX

dt = N . V with :

= V 1 – V 2 – V 4 dX 1

dt

= V 2 + V 6 – V 3 dX 2

dt

= V 4 – V 5 – V 6 dX 3

dt

dX

dt =

dX 1 dt dX 2

dt dX 3

dt

V 1 V 2 V 3

V = =

V 4 V 5 V 6 X 1

V 1

V 4 V 2

X 2

X 3

V 6 V 3

V 5

X 3 0 0 0 1 -1 -1 N = X 2 0 1 -1 0 0 1

and

X 1 1 -1 -0 -1 0 0 V 1 V 2 V 3 V 4 V 5 V 6

JP Mazat

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STOICHIOMETRY MATRIX - DEFINITION

The elements of the stoichiometry matrix N are the stoichiometries of the metabolites X 1 , X 2 … X m (rows) in the reactions V 1 , V 2 … V r (columns) With an arbitrary orientation.

It is easy to show that a metabolic network corresponds to a unique stoichiometry matrix and vice-versa.

X 1

V 1

V 4 V 2

X 2

X 3

V 6 V 3

V 5 X 3 0 0 0 1 -1 -1

N = X 2 0 1 -1 0 0 1 X 1 1 -1 -0 -1 0 0

V 1 V 2 V 3 V 4 V 5 V 6

C. REDER, (1988) : J. Theor. Biol (1988) 135: 175-201

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EXAMPLES OF STOICHIOMETRIC MATRICES

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EXEMPLE 1

X 1

V 1 V 2

N = 1 -1

Réaction 1

Réaction 2

X 1

[ ]

MATRICE RESEAU METABOLIQUE

X 1 V 1

V 2

V 3 N = 1 -1 -1 X 1

Réaction 1

Réaction 2

Réaction 3

[ ]

EXEMPLE 2

EXAMPLES OF STOICHIOMETRIC MATRICES (2)

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MATRICE RESEAU METABOLIQUE

X 2

X 1 X 3

(Rang 2) EXEMPLE 3

X 2 X 3

X 1 + + = Cte

Somme des lignes = 0

V 1 V 2

V 3

Réaction 1

Réaction 2

Réaction 3

N =

-1 0 1 1 -1 0 0 1 -1

X 1 X 2 X 3

EXAMPLES OF STOICHIOMETRIC MATRICES (3)

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R6i : Pyr + CO2 + ATP = OAA + Pi + ADP . R7i : Pyr + NAD + CoA = ACoA + NADH2 + CO2 . R8i : OAA + ACoA + H2O = Cit + CoA .

R9 : Cit = Isocit .

R10i : Isocit + NAD = Akg + NADH2 + CO2 . R11i : Akg + NAD + CoA = SucCoA + NADH2 + CO2 . R12 : SucCoA + Pi + ADP = Succ + CoA + ATP . R13 : Succ + FAD = Fum + FADH2 .

R14 : Fum + H2O = Mal .

R15 : Mal + NAD = OAA + NADH2 . R16 : Akg + NADH2 = Glu + NAD .

R17 : Ala + NAD + H2O = Pyr + NH3 + NADH2 . R18 : OAA + Glu = Asp + Akg .

R20i : 2 ATP + NH3 + CO2 + H2O = 2 ADP + Pi + CarbamoylP . R21 : CarbamoylP + Ornit = Pi + Citrulline .

R25i : 2 ACoA = CoA + AcetoACoA .

R26 : ACoA + H2O + AcetoACoA = HmethylGlutCoA + CoA . R27 : HmethylGlutCoA = ACoA + Acetoacetate .

R28 : Acetoacetate + NADH2 = Hbutanoate + NAD . R1i : NADH2 + 10 H = NAD + 10 H_ext .

R2i : FADH2 + 6 H = FAD + 6 H . R3 : ADP + Pi + 3 H_ext = ATP + 3 H . R4i : H_ext = H .

R30 : Acylcarnitine + CoA = Carnitine + AcylCoA .

R31i : AcylCoA + 7 FAD + 7 NAD + 7 CoA + 7 H2O = 7 NADH2 + 7 FADH2 + 8 ACoA . T1 : Cit + H + Mal_ext = Mal + Cit_ext + H_ext .

T2 : AKG_ext + Mal = Mal_ext + Akg .

T3 : AcylC_ext + Carnitine = Carnitine_ext + Acylcarnitine . T4 : ADP_ext + ATP + H_ext = ADP + ATP_ext + H .

T5 : Pi_ext + H_ext = Pi + H . T6 : Pyr_ext + H_ext = Pyr + H . T7 : Mal + Pi_ext = Pi + Mal_ext .

T8 : Citrulline + Ornit_ext = Citru_ext + Ornit . T9 : Mal + Asp_ext = Mal_ext + Asp .

T10 : Hbutanoate = HB_ext . T11 : AA_ext = Acetoacetate .

T12 : Asp + Glu_ext + H_ext = Asp_ext + Glu + H . T13 : Mal + Succ_ext = Mal_ext + Succ .

T14 : Asp + Succ_ext = Asp_ext + Succ . T15 : Asp + AKG_ext = Asp_ext + Akg . T16 : Asp + Pi_ext = Asp_ext + Pi . T17 : Succ + AKG_ext = Succ_ext + Akg . T18 : Succ + Pi_ext = Succ_ext + Pi . T19 : Akg + Pi_ext = AKG_ext + Pi . T20 : Glu_ext + H_ext = Glu + H .

REACTIONS OF

MITOCHONDRIAL ENERGY

METABOLISM

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[0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 1, -1, 0, 0, 0, -2, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, -1, 0;

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, -1, -1, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, -1, 0, 0, 0, 0, -2, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0;

0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

-1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 7, 0, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, -1, 1, 0, -1, 0, 1, -7, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

1, 0, 0, 0, -1, 0, -1, -1, 0, 0, -7, 0, 0, 0, 0, 0, -1, 1, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -7, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, -1, -1, -1, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1;

10, 6, -1, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1]

STOICHIOMETRIC MATRIX OF THE

MITOCHONDRIAL ENERGY METABOLISM

matrix dimension r31 x c45

The following line indicates reversible (0) and irreversible reactions (1)

1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 rows and columns are sorted as declared in the inputfile

N =

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RETURN ON EXAMPLE 4

THE STEADY-STATE

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STEADY STATE

dX

dt = N . V = 0

= V 1 – V 2 – V 4 = 0 dX 1

dt

= V 2 + V 6 – V 3 = 0 dX 2

dt

= V 4 – V 5 – V 6 = 0 dX 3

dt X 1

V 1

V 4 V 2

X 2

X 3

V 6 V 3

V 5

JP Mazat

Equivalent of Kirchoff’s laws in electricity

The steady state solutions belong to the kernel kernel ( (null null- -space space) ) of the matrix N : Ker [N].

(2)

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EXEMPLE 4 ETAT STATIONNAIRE (2)

N.V = 0 (2)

= V 1 – V 2 – V 4 = 0 dX 1

dt

= V 2 + V 6 – V 3 = 0 dX 2

dt

= V 4 – V 5 – V 6 = 0 dX 3

dt

Les V i sont des fonctions des X j : V i = V i (X 1 , X 2 , X 3 ).

Le système (2) est donc un système de 3 équations aux 3 inconnues X 1 , X 2 , X 3 . Les solutions X° 1 , X° 2 , X° 3 (il peut y en avoir plusieurs ou ne pas y en avoir) sont les concentrations de X° 1 , X° 2 , X° 3 à l’état stationnaire (qui vérifient (2)).

F i = V i (X° 1 , X° 2 , X° 3 ) sont les flux correspondants à l’état stationnaire.

Rappel : Les flux F = V à l’état stationnaire vérifient (2) et donc appartiennent au noyau de N : Ker(N) (C’est la définition du noyau d’une application N).

Il est donc capital de bien déterminer ce noyau.

(2)

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EXEMPLE 4 ETAT STATIONNAIRE (3)

V 1 V 2 V 3

V = =

V 4 V 5 V 6 N = 0 1 -1 0 0 1

1 -1 -0 -1 0 0 0 0 0 1 -1 -1

toutes les solutions V appartiennent au noyau de N)

N est de dimension 3 (colonnes 1, 2 et 4 par exemple).

Î Im(N) est donc de dimension 3.

L’espace des vitesses est de dimension 6.

Î Le noyau est de dimension 6 - 3 = 3.

Il suffit de trouver 3 vecteurs indépendants du noyau pour exprimer toutes les solutions.

X 1

V 1

V 4 V 2

X 2

X 3

V 6 V 3

V 5

dX

dt = N . V Î N . V = 0

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EXEMPLE 4 : BASE DU NOYAU DE N

N . V = 0

V 1 V 2 V 3

V = =

V 4 V 5 V 6

Î toutes les solutions V appartiennent au noyau de N sont de la forme :

1 1

K 1 = = 1 0 0 0

1 0

K 2 = = 0 1 1 0

K 3 = =

0 -1 0 1 0 1

V 1 V 2 V 3

V =

V 4 V 5 V 6

= a 1 K 1 + a 2 K 2 + a 3 K 3

= V 1 – V 2 – V 4 = 0 dX 1

dt

= V 2 + V 6 – V 3 = 0 dX 2

dt

= V 4 – V 5 – V 6 = 0 dX 3

dt

3 2

V = 1

1 2 -1

= 1 K 1 + 2 K 2 - 1 K 3

Exemple :

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LES VECTEURS INDÉPENDANTS DU NOYAU COMME VOIES MÉTABOLIQUES DE BASE (1)

X 1

V 1

V 2

X 2

X 3

V 3

1 1

K 1 = = 1 0 0 0

1 0

K 2 = = 0 1 1 0

X 1

V 1

V 4

X 2 X 3

V 5

0 -1

K 3 = = 0 1 0 1

X 1

V 4 V 2

X 2 X 3

V 6

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LES VECTEURS INDÉPENDANTS DU NOYAU COMME VOIES MÉTABOLIQUES DE BASE (2)

X 1

V 1

V 2

X 2

X 3

V 3

1 1

K 1 = = 1 0 0 0

1 0

K 2 = = 0 1 1 0

X 1

V 1

V 4

X 2 X 3

V 5

0 -1

K 3 = = 0 1 0 1

X 1

V 4

X 2 X 3

V 6

1 0

K 4 =K 1 + K 3 = 1 1 0 1

V 1

V 3 Prise en compte du sens des réactions irréversibles

K 3 ne convient pas Î

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CONCLUSION

Les vecteurs de base du noyau de N sont eux-mêmes des chemins possibles du réseau métabolique à l’état stationnaire.

Ils sont indépendants et permettent de construire l’ensemble des chemins possibles du réseau métabolique par combinaison linéaire.

Peut-on faire la liste de tous les chemins possibles du réseau métabolique ?

Mais ne sont pas uniques

Î Les modes élémentaires de flux

L’ensemble des vitesses des réactions du réseau métabolique à un instant donné définit un vecteur vitesse V, élément de l’espace vectoriel des vitesses à r dimensions.

Soit un réseau métabolique comportant r réactions et m métabolites

V

1

V

2

. V = . .

V

r

Les vecteurs vitesses d’un réseau métabolique à l’état stationnaire

appartiennent au noyau de la matrice de stoechiométrie N : V ∈ Ker[N]

Ker[N] est un espace vectoriel à (r – rg[N]) dimensions que l’on

peut définir par des vecteurs de base.

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IV

ÉLÉMENTARY MODES

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Common idea : fluxes follow the metabolic pathways drawn in the academic textbooks

Ex: Linear glycolysis, circular Krebs cycle.

METABOLIC PATHWAYS ELEMENTARY MODES

Glycolysis and Krebs cycle involve a lot of branching points.

The drawing of metabolic pathways does not take into account coenzymes and more generally the fact that most of the reactions are bimolecular.

.

Elementary modes : minimal pathways in a metabolic network compatible

with the thermodynamics and a steady state.

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NADH NAD

NAD NADH

NAD NADH

NAD NADH

NADH NAD FAD+GDP

FADH

2

+GTP

NAD

NADH NAD

NADH

FADH

2

FAD Lactate (c)

Glu (c) Triose

Phosphates (c) PEP (c)

Pyruvate (c)

Acétyl_CoA (m)

OAA (m) Pyruvate (m)

Citrate (m)

AKG (m) Mal (m)

Mal (c)

AKG (c)

OAA (c)

Glycérol-3- Phosphates (c)

O

2

ETS

ATP NADH

NAD

Flux-balance analysis of mitochondrial energy metabolism: consequences of systemic stoichiometric constraints Ramprasad Ramakrishna, Jeremy S. Edwards, Andrew McCulloch, and Bernhard O. Palsson

Am J Physiol Regul Integr Comp Physiol 280: R695-R704, 2001;

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ELEMENTARY MODES – AN EXAMPLE

X 1

V 1

V 2

X 2

X 3

V 3

X 1

V 1

V 4

X 2 X 3

V 5

X 1

V 4

X 2 X 3

V 6 V 1

V 3 All reactions irreversible

X 1

V 4

X 2

X 3

V 6

V 5 C B

R5 reversible ==> An extra elementary mode :

Elementary modes : minimal pathways able to maintain steady-state

X 1

V 1

V 4 V 2

X 2

X 3

V 6 V 3

V 5

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ELEMENTARY MODES – AN EXAMPLE (2)

X 1

V 1

V 2

X 2

X 3

V 3

X 1

V 1

V 4

X 2 X 3

V 5

X 1

V 4

X 2 X 3

V 6 V 1

V 3 All reactions reversible

X 1

V 4

X 2

X 3

V 6

V 5 C B

6 elementary modes + their reverse = 12 elementary modes.

Elementary modes : minimal pathways able to maintain steady state

X 1

V 1

V 4 V 2

X 2

X 3

V 6 V 3

V 5

X 1

V 4

X 2 X 3

V 6 V 1

V 3

X 1

V 4 V 2

X 2 X 3

V 6

(28)

R6i : Pyr + CO2 + ATP = OAA + Pi + ADP . R7i : Pyr + NAD + CoA = ACoA + NADH2 + CO2 . R8i : OAA + ACoA + H2O = Cit + CoA .

R9 : Cit = Isocit .

R10i : Isocit + NAD = Akg + NADH2 + CO2 . R11i : Akg + NAD + CoA = SucCoA + NADH2 + CO2 . R12 : SucCoA + Pi + ADP = Succ + CoA + ATP . R13 : Succ + FAD = Fum + FADH2 .

R14 : Fum + H2O = Mal .

R15 : Mal + NAD = OAA + NADH2 . R16 : Akg + NADH2 = Glu + NAD .

R17 : Ala + NAD + H2O = Pyr + NH3 + NADH2 . R18 : OAA + Glu = Asp + Akg .

R20i : 2 ATP + NH3 + CO2 + H2O = 2 ADP + Pi + CarbamoylP . R21 : CarbamoylP + Ornit = Pi + Citrulline .

R25i : 2 ACoA = CoA + AcetoACoA .

R26 : ACoA + H2O + AcetoACoA = HmethylGlutCoA + CoA . R27 : HmethylGlutCoA = ACoA + Acetoacetate .

R28 : Acetoacetate + NADH2 = Hbutanoate + NAD . R1i : NADH2 + 10 H = NAD + 10 H_ext .

R2i : FADH2 + 6 H = FAD + 6 H . R3 : ADP + Pi + 3 H_ext = ATP + 3 H . R4i : H_ext = H .

R30 : Acylcarnitine + CoA = Carnitine + AcylCoA .

R31i : AcylCoA + 7 FAD + 7 NAD + 7 CoA + 7 H2O = 7 NADH2 + 7 FADH2 + 8 ACoA . T1 : Cit + H + Mal_ext = Mal + Cit_ext + H_ext .

T2 : AKG_ext + Mal = Mal_ext + Akg .

T3 : AcylC_ext + Carnitine = Carnitine_ext + Acylcarnitine . T4 : ADP_ext + ATP + H_ext = ADP + ATP_ext + H .

T5 : Pi_ext + H_ext = Pi + H . T6 : Pyr_ext + H_ext = Pyr + H . T7 : Mal + Pi_ext = Pi + Mal_ext .

T8 : Citrulline + Ornit_ext = Citru_ext + Ornit . T9 : Mal + Asp_ext = Mal_ext + Asp .

T10 : Hbutanoate = HB_ext . T11 : AA_ext = Acetoacetate .

T12 : Asp + Glu_ext + H_ext = Asp_ext + Glu + H . T13 : Mal + Succ_ext = Mal_ext + Succ .

T14 : Asp + Succ_ext = Asp_ext + Succ . T15 : Asp + AKG_ext = Asp_ext + Akg . T16 : Asp + Pi_ext = Asp_ext + Pi . T17 : Succ + AKG_ext = Succ_ext + Akg . T18 : Succ + Pi_ext = Succ_ext + Pi . T19 : Akg + Pi_ext = AKG_ext + Pi . T20 : Glu_ext + H_ext = Glu + H .

REACTIONS OF

MITOCHONDRIAL ENERGETIC

METABOLISM

(29)

[0, 0, 0, 1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 1, -1, 0, 0, 0, -2, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, -1, 0;

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, -1, -1, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, -1, 0, 0, 0, 0, -2, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0;

0, 0, 0, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

-1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 7, 0, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, -1, 1, 0, -1, 0, 1, -7, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

1, 0, 0, 0, -1, 0, -1, -1, 0, 0, -7, 0, 0, 0, 0, 0, -1, 1, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -7, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0, -1, -1, -1, 0, 0, 0, 0;

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1;

10, 6, -1, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1]

LA MATRICE DE STOECHIOMÉTRIE DU

MÉTABOLISME ÉNERGÉTIQUE MITOCHONDRIAL

matrix dimension r31 x c45

The following line indicates reversible (0) and irreversible reactions (1)

1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 rows and columns are sorted as declared in the inputfile

N =

(30)

FLUX ELEMENTARY MODES OF MITOCHONDRIAL ENERGETIC

METABOLISM

Biological meaning ?

Tissues Muscle Liver Yeast

Number of reactions 37 44 40

Internal metabolites 29 34 32

External metabolites 21 25 25

EFM number 6 I90 4 226 4 637

EFM minimal length 2 2 2

EFM maximal length 23 24 22

Classification

(Sabine Pérès, Marie Beurton-Aimar and Jean-Pierre Mazat. Classification des modes élémentaires :

Application au métabolisme énergétique mitochondrial. Technique et Science informatiques, 2007, 26 (1-2) pp 197-216.)

(31)

Pyr_ext

Pyr T

6

ACoA

Cit

Isocit

AKG Suc-CoA

Suc Fum

OAA Mal

R

7

R

12

R

13

R

14

R

9

R

8

R

10

R

11

R

15

Glu R

6

Asp

R

16

R

17

NADH

NAD FADH

2

FAD

H

ADP

ATP_ext

H

H

H Pi_M

H_ext

H_ext H_ext

H_ext Pi_M_ext ADP_ext

ATP

Pi_2M

Pi_2M_ext H_ext

R

2

R

1

R

3

R

5

AMP R

4

T

4

T

5

2 6 10

AcetoAcCoA R

24

2

AcetoAcetate

R

28

HB R

27

AA_ext

HB_ext T

11

T

10

T

3

Acylcarnitine_ext Acylcarnitine

Carnitine_ext

Carnitine AcylCoA

R

30

R

31

Cit_ext Mal_ext H_ext

T

1

T

2

Mal_ext AKG_ext Mal_ext

T

7

T

9

Asp_ext Glu_ext

H_ext T

12

Mal_ext Fum_ext

T

13

T

19

Glu_ext T

20

G3P_ext

DHAP_ext R

32

H_ext

MITOCHONDRIAL ENERGETIC NETWORK

MUSCLE EFM 1

JP Mazat-Munich Fev 2007

(32)

Pyr_ext

Pyr

59 T

6

ACoA

Cit

Isocit

AKG Suc-CoA

Suc Fum

OAA Mal

15 R

7

15 R

12

15 R

13

15 R

14

15 R

9

15 R

8

15 R

10

15 R

11

29 R

15

Glu

44 R

6

Asp

R

16

R

17

NADH

NAD FADH

2

FAD

H

ADP

ATP_ext

H

H

H Pi_M

H_ext

H_ext H_ext

H_ext Pi_M_ext ADP_ext

ATP

Pi_2M

Pi_2M_ext H_ext

15 R

2

16 R

1

73 R

3

44 R

5

AMP R

4

44 T

4

T

5

2 6 10

AcetoAcCoA R

24

2

AcetoAcetate

R

28

HB R

27

AA_ext

HB_ext T

11

T

10

T

3

Acylcarnitine_ext Acylcarnitine

Carnitine_ext

Carnitine AcylCoA

R

30

R

31

Cit_ext Mal_ext H_ext

T

1

T

2

Mal_ext AKG_ext Mal_ext

T

7

T

9

Asp_ext Glu_ext

H_ext T

12

Mal_ext Fum_ext

T

13

T

19

Glu_ext T

20

G3P_ext

DHAP_ext R

32

H_ext

MITOCHONDRIAL ENERGETIC METABOLIC NETWORK

MUSCLE EFM 6190

(17) [bl 1] (16 R1i) (15 R2i) (44 R6i) (15 R7i) (15 R8i) (15 R10i) (15 R11i) (15 R13i) (59 T6i) (73 R3) (44 R5) (15 R9) (15 R12) (15 R14) (-29 R15) (44 T4) (-44 T7) irreversible

44

44

JP Mazat-Munich Fev 2007

(33)

DECOMPOSITION IN ELEMENTARY MODES

- The fluxes through a metabolic network can be split into a linear combination of elementary modes with positive coefficients.

Problem :

Combinatorial explosion of the number of elementary modes - The decomposition is variable according to the tissues

and/or the conditions.

JP Mazat

- In general the decomposition is not unique.

(34)

LOGICIELS DE CALCUL DES MODES ELEMENTAIRES DE FLUX

METATOOL

(35)

LISTE DES LOGICIELS DE CALCUL DES MODES ELEMENTAIRES DE FLUX

- Metatool

- FluxAnalyzer

- Yana

(36)

METATOOL 4.3 FICHIER D’ENTREE

-ENZREV

-ENZIRREV

R1 R2 R3 R4 R5 R6

-METINT X1 X2 X3

-METEXT A B C

-CAT

R1 : A = X1 . R2 : X1 = X2 . R3 : X2 = B . R4 : X1 = X3 . R5 : X3 = C . R6 : X3 = X2 .

X 1

V 1

V 4 V 2

X 2

X 3

V 6 V 3

V 5

A

B

C

J.-P. Mazat Cours Bioinfo 2005

(37)

METATOOL OUTPUT (int) Version 4.3 (25 October 2002) D:\SABINE PERES\PETITS EXEMPLES\meta4.3_int.exe

INPUT FILE: ex-cr4.txt INTERNAL METABOLITES: 3 EXTERNAL METABOLITES: 3 REACTIONS: 6

3 int X1 3 int X2 3 int X3 1 external A 1 external B 1 external C

6 metabolites, 12 is the summarized frequency

METATOOL 4.3

FICHIER DE SORTIE (1)

X 1

V 1

V 4 V 2

X 2

X 3

V 6 V 3

V 5

A

B

C

edges frequency of nodes

1 3

3 3

freq_of_nodes = 3.0000 * edges^(+0.0000) Linear correlation coefficient r = -2.000000 The dependency is significant (p<0.001).

J.-P. Mazat Cours Bioinfo 2005

(38)

STOICHIOMETRIC MATRIX matrix dimension r3 x c6 1 -1 0 -1 0 0

0 1 -1 0 0 1 0 0 0 1 -1 -1

The following line indicates reversible (0) and irreversible reactions (1) 1 1 1 1 1 1

rows and columns are sorted as declared in the inputfile

METATOOL 4.3

FICHIER DE SORTIE (2)

X 1

V 1

V 4 V 2

X 2 X 3

V 6 V 3

V 5

A

B

C

KERNEL

matrix dimension r3 x c6 1 1 1 0 0 0

1 0 0 1 1 0 0 -1 0 1 0 1

6 reactions (columns) are sorted in the same order as in the ENZREV ENZIRREV section.

K 1 K 2 K 3

X1 V1

V4 V2 X2

X3 V6 V3

V5 A

B

C

X1 V1

V4 V2 X2

X3 V6 V3

V5 A

B

C

X1 V1

V4 V2 X2

X3 V6 V3

V5 A

B

C

J.-P. Mazat Cours Bioinfo 2005

(39)

METATOOL 4.3

FICHIER DE SORTIE (3)

X 1

V 1

V 4 V 2

X 2 X 3

V 6 V 3

V 5

A

B

KERNEL C

matrix dimension r3 x c6 1 1 1 0 0 0

1 0 0 1 1 0 0 -1 0 1 0 1

6 reactions (columns) are sorted in the same order as in the ENZREV ENZIRREV section.

K 2 K 3

K 1

X1 V1

V4 V2 X2

X3 V6 V3

V5 A

B

C

X1 V1

V4 V2 X2

X3 V6 V3

V5 A

B

C

X1 V1

V4 V2 X2

X3 V6 V3

V5 A

B

C

enzymes

1: R1 R2 R3 irreversible 2: R1 R4 R5 irreversible 3: -R2 R4 R6 irreversible

overall reaction

1: A = B 2: A = C

3: no net transfomation of external metabolites

J.-P. Mazat Cours Bioinfo 2005

(40)

METATOOL 4.3

FICHIER DE SORTIE (4)

X 1

V 1

V 4 V 2

X 2 X 3

V 6 V 3

V 5

A

B

C

BLOCK DIAGONALISATION

Reaction blocks were found from nullspace matrix (KERNEL).

1. block:

R1 R2 R3 R4 R5 R6

SUBSETS OF REACTIONS

matrix dimension r6 x c6 1 0 0 0 0 0

0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

6 reactions (columns) are sorted in the same order as in the ENZREV ENZIRREV section.

enzymes

1: R1 irreversible 2: R2 irreversible 3: R3 irreversible 4: R4 irreversible 5: R5 irreversible 6: R6 irreversible

overall reaction

1: A = X1 2: X1 = X2 3: X2 = B 4: X1 = X3 5: X3 = C 6: X3 = X2

J.-P. Mazat Cours Bioinfo 2005

(41)

CONVEX BASIS

matrix dimension r3 x c6 1 1 1 0 0 0

1 0 0 1 1 0 1 0 1 1 0 1

enzymes

1: R1 R2 R3 irreversible 2: R1 R4 R5 irreversible 3: R1 R3 R4 R6 irreversible overall reaction

1: A = B 2: A = C 3: A = B

METATOOL 4.3

FICHIER DE SORTIE (5)

X 1

V 1

V 4 V 2

X 2 X 3

V 6 V 3

V 5

A

B

C

REDUCED SYSTEM with 3 branch point metabolites in 6 reactions (columns)

matrix dimension r3 x c6 1 -1 0 -1 0 0

0 1 -1 0 0 1 0 0 0 1 -1 -1

The following line indicates reversible (0) and irreversible reactions (1)

1 1 1 1 1 1

-> Branch metabolites are : met cons built reactions

X1 2 1 3 iii X2 1 2 3 iii X3 2 1 3 iii -> No branch metabolites are :

met cons built reactions

K 1 K 2

K 4 = K 1 + K 3

J.-P. Mazat Cours Bioinfo 2005

(42)

METATOOL 4.3

FICHIER DE SORTIE (6)

X 1

V 1

V 4 V 2

X 2 X 3

V 6 V 3

V 5

A

B

C

K 1 K 2 K 4 = K 1 + K 3

CONSERVATION RELATIONS - not found -

ELEMENTARY MODES

matrix dimension r3 x c6 1 1 1 0 0 0

1 0 0 1 1 0 1 0 1 1 0 1

6 reactions (columns) are sorted in the same order as in the ENZREV ENZIRREV section.

The following line indicates reversible (0) and irreversible reactions (1)

1 1 1 1 1 1

enzymes

# in () indicates # of enzymes used by the elementary mode

# in [] indicates the diagonal block of the kernel matrix to which the elementary mode belongs

1: ( 3) [bl 1] R1 R2 R3 irreversible 2: ( 3) [bl 1] R1 R4 R5 irreversible 3: ( 4) [bl 1] R1 R3 R4 R6 irreversible

overall reaction 1: A = B

2: A = C 3: A = B

The elementary modes ARE EQUAL to convex

basis. J.-P. Mazat Cours Bioinfo 2005

(43)

Pyruvate

ACoA

Citrate OAA

P

i

r

6

r

7

r

8

Isocitrate

α - Cétoglutarate r

10

r

9

Succinyl -CoA r

11

Succinate

P

i

Fumarate

Malate

r

12

r

13

r

14

r

15

Pyr

ext

P

i

P

i ext

Mal

ext

t

5

t

7

t

2

t

1

Cit

ext

Mal

ext

Mal

ext

AKG

ext

t

6

KREBS CYCLE – METATOOL (1) (Krebs1)

-ENZREV

R15 T1 T2 T5 T7 -ENZIRREV

R6i R7i R8i R9i R11i T6 -METINT

OAA ACoA Cit Akg Mal Pi Pyr -METEXT

Pyr_ext NAD NADH2 CoA ADP ATP H2O CO2 Mal_ext Cit_ext AKG_ext Pi_ext

-CAT

R6i : Pyr + CO2 + ATP = OAA + Pi + ADP . R7i : Pyr + NAD + CoA = ACoA + NADH2 + CO2 .

R8i : OAA + ACoA + H2O = Cit + CoA . R9i : Cit + NAD = Akg + NADH2 + CO2 . R11i : Akg + NAD + Pi + ADP = Mal + NADH2 + CO2 + ATP .

R15 : Mal + NAD = OAA + NADH2 . T1 : Cit + Mal_ext = Mal + Cit_ext . T2 : AKG_ext + Mal = Mal_ext + Akg . T5 : Pi_ext = Pi .

T6 : Pyr_ext = Pyr .

T7 : Mal + Pi_ext = Pi + Mal_ext .

(44)

KREBS CYCLE – METATOOL -2 (Krebs1)

STOICHIOMETRIC MATRIX matrix dimension r7 x c11 1 0 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 -1 0 0 0 0 0 1 -1 0 0 0 0 1 0 0 0 0 0 1 -1 0 -1 1 -1 0 -1 0 0 0 0 1 0 0 0 0 1 1 1 0 0 0 -1 0 0 0 0 0 0 -1 -1 0 0 0 1

The following line indicates reversible (0) and irreversible reactions (1) 0 0 0 0 0 1 1 1 1 1 1

rows and columns are sorted as declared in the inputfile

KERNEL

matrix dimension r4 x c11 -2 -1 0 -2 1 1 -1 -1 0 0 0

0 1 1 0 0 0 0 0 -1 0 0 0 0 1 1 0 0 0 0 0 1 0 -1 0 0 -2 1 1 0 0 0 0 1

11 reactions (columns) are sorted in the same order as in the ENZREV ENZIRREV section.

enzymes

1: (-2 R15) -T1 (-2 T5) T7 R6i -R7i -R8i irreversible 2: T1 T2 -R9i irreversible

3: T2 T5 R11i irreversible

4: -R15 (-2 T5) T7 R6i T6 irreversible overall reaction

1: 3 NADH2 + ATP + 2 CO2 + Cit_ext = 3 NAD + ADP + H2O + 2 Mal_ext + Pi_ext 2: NADH2 + CO2 + AKG_ext = NAD + Cit_ext

3: NAD + ADP + AKG_ext + Pi_ext = NADH2 + ATP + CO2 + Mal_ext

4: Pyr_ext + NADH2 + ATP + CO2 = NAD + ADP + Mal_ext + Pi_ext

(45)

KREBS CYCLE – METATOOL -3 (Krebs1)

CONSERVATION RELATIONS - not found -

ELEMENTARY MODES matrix dimension r16 x c11 0 -1 -1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 -1 0 1 0 0 0 0 1 1 0 1 1 0 0 0 0 1 1 0 0 1 1 0 -1 0 0 0 1 1 1 0 1 -1 0 0 -2 1 1 0 0 0 0 1 0 1 0 -2 1 1 1 1 0 0 2 0 0 -1 -2 1 1 1 1 1 0 2 1 0 0 1 0 0 1 1 1 1 1 0 0 0 -1 1 1 1 1 1 1 2 -1 0 2 0 1 1 0 0 0 2 1 -1 -2 0 0 1 1 0 0 2 2 1 1 0 0 0 1 1 2 2 2 2 3 0 1 2 0 1 1 1 1 0 2 2 0 -1 0 0 1 1 1 1 2 2 2 0 0 1 0 1 1 1 1 1 2 2

11 reactions (columns) are sorted in the same order as in the ENZREV ENZIRREV section.

The following line indicates reversible (0) and irreversible reactions (1) 0 0 0 0 0 1 1 1 1 1 1

enzymes

# in () indicates # of enzymes used by the elementary mode

# in [] indicates the diagonal block of the kernel matrix to which the elementary mode belongs 1: ( 3) [bl 1] -T1 -T2 R9i irreversible

2: ( 3) [bl 1] T2 T5 R11i irreversible 3: ( 4) [bl 1] -T1 T5 R9i R11i irreversible 4: ( 5) [bl 1] R15 T1 R7i R8i T6 irreversible 5: ( 6) [bl 1] R15 -T2 R7i R8i R9i T6 irreversible 6: ( 5) [bl 1] -R15 (-2 T5) T7 R6i T6 irreversible

7: ( 7) [bl 1] T1 (-2 T5) T7 R6i R7i R8i (2 T6) irreversible 8: ( 8) [bl 1] -T2 (-2 T5) T7 R6i R7i R8i R9i (2 T6) irreversible 9: ( 7) [bl 1] R15 T5 R7i R8i R9i R11i T6 irreversible

10: ( 8) [bl 1] -T5 T7 R6i R7i R8i R9i R11i (2 T6) irreversible 11: ( 6) [bl 1] -R15 (2 T2) T7 R6i (2 R11i) T6 irreversible

12: ( 7) [bl 1] -R15 (-2 T1) T7 R6i (2 R9i) (2 R11i) T6 irreversible

13: ( 8) [bl 1] R15 T7 R6i (2 R7i) (2 R8i) (2 R9i) (2 R11i) (3 T6) irreversible 14: ( 8) [bl 1] T1 (2 T2) T7 R6i R7i R8i (2 R11i) (2 T6) irreversible

15: ( 8) [bl 1] -T1 T7 R6i R7i R8i (2 R9i) (2 R11i) (2 T6) irreversible

16: ( 8) [bl 1] T2 T7 R6i R7i R8i R9i (2 R11i) (2 T6) irreversible

(46)

KREBS CYCLE – METATOOL - 4 (Krebs1)

enzymes

# in () indicates # of enzymes used by the elementary mode

# in [] indicates the diagonal block of the kernel matrix to which the elementary mode belongs 1: ( 3) [bl 1] -T1 -T2 R9i irreversible

2: ( 3) [bl 1] T2 T5 R11i irreversible 3: ( 4) [bl 1] -T1 T5 R9i R11i irreversible 4: ( 5) [bl 1] R15 T1 R7i R8i T6 irreversible 5: ( 6) [bl 1] R15 -T2 R7i R8i R9i T6 irreversible 6: ( 5) [bl 1] -R15 (-2 T5) T7 R6i T6 irreversible

7: ( 7) [bl 1] T1 (-2 T5) T7 R6i R7i R8i (2 T6) irreversible 8: ( 8) [bl 1] -T2 (-2 T5) T7 R6i R7i R8i R9i (2 T6) irreversible 9: ( 7) [bl 1] R15 T5 R7i R8i R9i R11i T6 irreversible

10: ( 8) [bl 1] -T5 T7 R6i R7i R8i R9i R11i (2 T6) irreversible 11: ( 6) [bl 1] -R15 (2 T2) T7 R6i (2 R11i) T6 irreversible

12: ( 7) [bl 1] -R15 (-2 T1) T7 R6i (2 R9i) (2 R11i) T6 irreversible

13: ( 8) [bl 1] R15 T7 R6i (2 R7i) (2 R8i) (2 R9i) (2 R11i) (3 T6) irreversible 14: ( 8) [bl 1] T1 (2 T2) T7 R6i R7i R8i (2 R11i) (2 T6) irreversible

15: ( 8) [bl 1] -T1 T7 R6i R7i R8i (2 R9i) (2 R11i) (2 T6) irreversible 16: ( 8) [bl 1] T2 T7 R6i R7i R8i R9i (2 R11i) (2 T6) irreversible overall reaction

1: NAD + Cit_ext = NADH2 + CO2 + AKG_ext

2: NAD + ADP + AKG_ext + Pi_ext = NADH2 + ATP + CO2 + Mal_ext 3: 2 NAD + ADP + Cit_ext + Pi_ext = 2 NADH2 + ATP + 2 CO2 + Mal_ext 4: Pyr_ext + 2 NAD + H2O + Mal_ext = 2 NADH2 + CO2 + Cit_ext 5: Pyr_ext + 3 NAD + H2O + Mal_ext = 3 NADH2 + 2 CO2 + AKG_ext 6: Pyr_ext + NADH2 + ATP + CO2 = NAD + ADP + Mal_ext + Pi_ext 7: 2 Pyr_ext + NAD + ATP + H2O = NADH2 + ADP + Cit_ext + Pi_ext

8: 2 Pyr_ext + 2 NAD + ATP + H2O = 2 NADH2 + ADP + CO2 + AKG_ext + Pi_ext 9: Pyr_ext + 4 NAD + ADP + H2O + Pi_ext = 4 NADH2 + ATP + 3 CO2

10: 2 Pyr_ext + 3 NAD + H2O = 3 NADH2 + 2 CO2 + Mal_ext

11: Pyr_ext + NAD + ADP + 2 AKG_ext + Pi_ext = NADH2 + ATP + CO2 + 3 Mal_ext 12: Pyr_ext + 3 NAD + ADP + 2 Cit_ext + Pi_ext = 3 NADH2 + ATP + 3 CO2 + 3 Mal_ext 13: 3 Pyr_ext + 7 NAD + ADP + 2 H2O + Pi_ext = 7 NADH2 + ATP + 5 CO2 + Mal_ext

14: 2 Pyr_ext + 3 NAD + ADP + H2O + 2 AKG_ext + Pi_ext = 3 NADH2 + ATP + 2 CO2 + 2 Mal_ext + Cit_ext 15: 2 Pyr_ext + 5 NAD + ADP + H2O + Cit_ext + Pi_ext = 5 NADH2 + ATP + 4 CO2 + 2 Mal_ext

16: 2 Pyr_ext + 4 NAD + ADP + H2O + AKG_ext + Pi_ext = 4 NADH2 + ATP + 3 CO2 + 2 Mal_ext

(47)

Pyruvate

ACoA

Citrate OAA

P

i

r 6 r 7

r 8

Isocitrate

α - Cétoglutarate

r 10 r 9

Succinyl-CoA

r 11

Succinate

P

i

Fumarate Malate

r 12 r 13

r 14

r 15

Pyr ext

P

i

P i ext

Mal ext

t 5

t 7

t 2

t 1

Cit ext Mal ext

Mal ext AKG ext

t 6

Figure

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Références

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