Hypergraphs, Metabolic Networks, Bioreaction Systems. G. Bastin
PART 1 : Metabolic flux analysis and minimal bioreaction modelling PART 2 : Dynamic metabolic flux analysis of underdetermined networks 2
PART 1 : Metabolic flux analysis and minimal bioreaction modelling Growth of CHO cells in a serum-free medium Substrates Glucose CELLS Lactate Alanine Products Glutamine NH4 CO2
The basic issue Growth of CHO cells in a serum-free medium Substrates Glucose CELLS Lactate Alanine Products Glutamine NH4 CO2 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
Metabolic network Remark 1 : for the sake of clarity, this is a simplified example representing only the metabolism of «energetic substrates» Glucose inputsand Glutamine. Metabolism of amino-acids is not considered. But outputs the methodology applies to more complex networks as we shall see in Part II. intracellular intermediate metabolites
Metabolic network Remark inputs2 : Only internal nodes that are at «branching points» (no loss of generality). outputs intracellular intermediate metabolites
Glycolysis
Metabolic network metabolic flux stoichiometric coefficient
Stoichiometric matrix N (= incidence matrix of the network) internal nodes Glucose 6-P Dihydro-Ac-3P Ribose-5-P Glyc-3-P Pyruvate Acetyl-coA Citrate Alpha-ketoglut Fumarate Malate Oxaloacetate Aspartate Glutamate CO2 Metabolic fluxes 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 (= incidence matrix of the network) Glucose 6-P Dihydro-Ac-3P Ribose-5-P Glyc-3-P Pyruvate Acetyl-coA Citrate a-ketoglutarate Fumarate Malate Oxaloacetate Aspartate Glutamate CO2 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 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 The set of non-negative vectors of the kernel of N is a polyhedral cone in the non-negative orthant
Stoichiometric matrix 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 Convex basis The convex basis is the set of edges of the cone matrix E
Elementary pathway Metabolic interpretation of convex basis
Elementary pathway Remark 3. In the literature, «Elementary pathways» are also called : «Elementary (flux) modes» or «Extreme pathways».
Elementary pathway C 6 H 12 O 6! 2 C 3 H 6 O 3
Another example
From Convex basis to Bioreaction system Nucleotides Substrates Glucose CELLS Lactate Alanine Products Glutamine NH4 CO2
Experimental data (batch) Glucose Cell density Lactate NH4 Glutamine Alanine
Experimental data (batch) Glucose Cell density Lactate Green dots = exponential growth NH4 Glutamine Alanine
Computation of specific consumption and production rates by linear regression Glucose Glutamine Lactate NH4 Alanine Cell density
Metabolic flux analysis
Metabolic flux analysis
Metabolic flux analysis Non-negative decomposition of in the convex basis
The 12 equivalent minimal decompositions Exactly five non-zero coefficients in each vector!
12 equivalent minimal (sub)sets of bioreactions
Density Glucose Glutamine Lactate Alanine Ammonia 34
PART 2 : Dynamic metabolic flux analysis of underdetermined networks Case study : Hybridoma cells for production of Immunoglobuline 35
Substrates Glucose Glutamine CELLS NH4 Lactate Alanine CO2 Products 36
Substrates Glucose Glutamine CELLS NH4 Lactate Alanine CO2 Products Glutamate Serine Arginine Asparagine Aspartate Histidine Leucine Isoleucine Lysine Methionine Phenylalanine Threonine Tryptophan Valine Tyrosine (Proline) (Cysteine) Glycine Immunoglobuline 37
Central Metabolism 38
Amino-Acids Metabolism 39
40
Biomass Synthesis Antibody Synthesis 41
42
Perfusion culture biomass substrate product dx dt = µx DX ds dt = SX + D(S in S) dp dt = P X DP 43
DATA batch perfusion 44
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Metabolic Flux Analysis Problem biomass substrate dx dt = µx DX ds dt = SX + D(S in S) n x r p x r N (t) =0 N m (t) = m (t) product dp dt = P X DP 0 apple (t) P S = m r = 70 reactions n = 44 internal metabolites p = 22 measurements Underdetermined system! 46
N (t) =0 N m (t) = m (t) 0 apple (t) = Again the set of solutions is a pointed polyhedral cone in the positive orthant qx! i f i! i 0, i=1 qx i=1! i =1 edges of the cone
N (t) =0 N m (t) = m (t) 0 apple (t) = Again the set of solutions is a pointed polyhedral cone in the positive orthant qx! i f i! i 0, i=1 qx i=1! i =1 edges of the cone = 0 B @ 1 2. 1 C A Solution intervals min i apple i apple i max r min i =min f ki,k=1,...,q, max i = max f ki,k=1,...,q
Results : Glycolysis fluxes 49
Results :TCA-cycle fluxes 50
Final remarks 1. Dynamical model under pseudo-steady-state assumption of metabolic flux analysis = order reduction by singular perturbation. 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!
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