hypergraphs, metabolic networks, bioreaction systems.trelat/gdt/confs/georges...computation of...

Hypergraphs, Metabolic Networks, Bioreaction Systems.

G. Bastin

2

PART 1 : Metabolic flux analysis and minimal bioreaction modelling

PART 2 : Dynamic metabolic flux analysis of underdetermined networks

PART 1 : Metabolic flux analysis and minimal bioreaction modelling

CELLSSubstrates

ProductsLactate

AlanineGlucose

CO2Glutamine NH4

Growth of CHO cells in a serum-free medium

The basic issue

CELLSSubstrates

ProductsLactate

AlanineGlucose

CO2Glutamine NH4

Growth of CHO cells in a serum-free medium

Question : What is the minimal set of input-output bioreactions ?

• consistent with cell metabolism • explaining measurements in the culture medium

Outline

1. Metabolic network and stoichiometric matrix

2. Convex basis and elementary pathways

3. From elementary pathways to bioreactions

4. Experimental data and metabolic flux analysis

5. Computation of minimal sets of bioreactions

6. Final remarks

Metabolic network

inputs

outputs

intracellular intermediate metabolites

Intracellular biochemical reaction

inputs

outputs

intracellular intermediate metabolites

Remark 1 : for the sake of clarity, this is a simplified example representing only the metabolism of « energetic substrates » Glucose and Glutamine. Metabolism of amino-acids is not considered.

But the methodology applies to more complex networks as we shall see in Part II.

Metabolic network

inputs

outputs

intracellular intermediate metabolites

Remark 2 : Only internal nodes that are at « branching points » (no loss of generality).

Metabolic network

Glycolysis

metabolic flux

stoichiometric coefficient

Metabolic network

Stoichiometric matrix N

1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 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 0 0 -2 0 0 1 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 -1 0 0 0 0 0 1 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 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 -1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -2 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 -1 1 3 0 0 0 1 0 0 0 0 1 0 1 1 0 0 1 0 0 0 -1 -1

Glucose 6-P Dihydro-Ac-3P

Ribose-5-P Glyc-3-P Pyruvate

Acetyl-coA Citrate

Alpha-ketoglut Fumarate

Malate Oxaloacetate

Aspartate Glutamate

CO2

(= incidence matrix of the network)

internal nodes

Metabolic fluxes

Stoichiometric matrix N

1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 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 0 0 -2 0 0 1 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 -1 0 0 0 0 0 1 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 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 -1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -2 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 -1 1 3 0 0 0 1 0 0 0 0 1 0 1 1 0 0 1 0 0 0 -1 -1

Glucose 6-P Dihydro-Ac-3P

Ribose-5-P Glyc-3-P Pyruvate

Acetyl-coA Citrate

a-ketoglutarate Fumarate

Malate Oxaloacetate

Aspartate Glutamate

CO2

(= incidence matrix of the network)

1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 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 0 0 -2 0 0 1 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 -1 0 0 0 0 0 1 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 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 -1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -2 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 -1 1 3 0 0 0 1 0 0 0 0 1 0 1 1 0 0 1 0 0 0 -1 -1

Stoichiometric matrix N

The set of non-negative vectors of the kernel of N is a polyhedral cone in the non-negative orthant

Convex basis1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 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 0 0 -2 0 0 1 0 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 -1 0 0 0 0 0 1 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 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 -1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 -2 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 -1 1 3 0 0 0 1 0 0 0 0 1 0 1 1 0 0 1 0 0 0 -1 -1

Stoichiometric matrix

The convex basis is the set of

edges of the cone

matrix E

Elementary pathway Metabolic

interpretation of convex basis

Remark 3. In the literature, « Elementary pathways » are also called : « Elementary (flux) modes » or « Extreme pathways ».

Elementary pathway

C3H6O3C6H12O6 �! 2

Elementary pathway

Another example …

From Convex basis to Bioreaction system

CELLSSubstrates

ProductsLactate

AlanineGlucose

CO2Glutamine NH4

Nucleotides

Experimental data(batch)

Glucose

Lactate

NH4

Glutamine

Alanine

Cell density

Glucose

Lactate

NH4

Glutamine

Alanine

Green dots = exponential growth

Cell density

Experimental data(batch)

Cell density

Computation of specific consumption and production rates by linear regression

Glucose Lactate

NH4

Glutamine

Alanine

Metabolic flux analysis

Metabolic flux analysis

Non-negative decomposition of in the convex basis

Metabolic flux analysis

The 12 equivalent minimal decompositions

Exactly five non-zero coefficients in each vector !

12 equivalent minimal (sub)sets of bioreactions

34

Density Glucose

LactateGlutamine

Alanine Ammonia

PART 2 : Dynamic metabolic flux analysis of underdetermined

networks

35

Case study : Hybridoma cells for production

of Immunoglobuline

36

CELLSSubstratesProductsLactate

AlanineGlucose

CO2Glutamine

NH4

37

CELLSSubstratesProductsLactate

AlanineGlucose

CO2Glutamine

NH4

Glutamate Serine

Arginine Asparagine

Aspartate Histidine

Leucine Isoleucine

Lysine Methionine

Phenylalanine Threonine

Tryptophan Valine

Tyrosine (Proline)

(Cysteine)

Glycine Immunoglobuline

38

Central Metabolism

39

Amino-Acids Metabolism

40

41

Antibody Synthesis

Biomass Synthesis

42

Perfusion culture

43

dX

dt= µX � ↵DX

dS

dt= ��SX +D(Sin � S)

dP

dt= �PX �DP

biomass

substrate

product

44

DATA batch perfusion

45

46

r = 70 reactions n = 44 internal metabolites p = 22 measurements

Underdetermined system !

Metabolic Flux Analysis ProblemN�(t) = 0

Nm�(t) = �m(t)

0 �(t)

n x r

p x r

dX

dt= µX � ↵DX

dS

dt= ��SX +D(Sin � S)

dP

dt= �PX �DP

biomass

substrate

product

✓��S�P

◆= �m

N�(t) = 0

Nm�(t) = �m(t)

0 �(t)

Again the set of solutions is a pointed polyhedral cone in the positive orthant

� =qX

i=1

!ifi⇣!i � 0,

qX

i=1

!i = 1⌘

edges of the cone

N�(t) = 0

Nm�(t) = �m(t)

0 �(t)

Again the set of solutions is a pointed polyhedral cone in the positive orthant

� =qX

i=1

!ifi⇣!i � 0,

qX

i=1

!i = 1⌘

edges of the cone

� =

0

BBB@

�1�2...�r

1

CCCA

Solution intervals

�min

i �i �max

i

�min

i = min

�fki, k = 1, . . . , q

,

�max

i = max

�fki, k = 1, . . . , q

Results : Glycolysis fluxes

49

Results :TCA-cycle fluxes

50

Final remarks

2. « Minimal » must be understood as « data compression »: there is no loss of information in the « minimal model » with respect to the initial metabolic network.

3. Combinatorial explosion. In « Zamorano et al., an example of CHO cells where the network involves 84930 elementary pathways and leads to 88926 equivalent minimal models ! …. but each minimal model includes only 22 bioreactions and the polyhedral cone for the interval metabolic flux analysis has only 32 edges.

Combinatorial explosion of the number of elementary pathways is not an issue because these 22 bioreactions and 32 edges can be computed directly !

1. Dynamical model under pseudo-steady-state assumption of metabolic flux analysis = order reduction by singular perturbation.

References

A. Provost, G. Bastin, "Dynamical metabolic modelling under the balanced growth condition", Journal of Process Control, Vol. 14(7), 2004, pp. 717 - 728.

F. Zamorano, A. Vande Wouwer, G. Bastin, "A detailed metabolic flux analysis of an undetermined network of CHO cells", Journal of Biotechnology, Vol.150(4), 2010, pp. 497 - 508.

R.M. Jungers, F. Zamorano, V.D. Blondel, A. Vande Wouwer, G. Bastin, "Fast computation of minimal elementary decompositions of metabolic flux vectors", Automatica, Special issue on Systems Biology, Vol. 47(7), 2011, pp. 1255 - 1259.

S. Fernandes de Souza, G.Bastin, M. Jolicoeur, A. Vande Wouwer, "Dynamic metabolic flux analysis using a convex analysis approach: application to hybridoma cell cultures in perfusion", Biotechnology and Bioengineering, 2016, in press.

52

Thank You !

53