A systematic approach for finding the objective function and active constraints for dynamic flux balance analysis

Abstract

Dynamic flux balance analysis (DFBA) has become an instrumental modeling tool for describing the dynamic behavior of bioprocesses. DFBA involves the maximization of a biologically meaningful objective subject to kinetic constraints on the rate of consumption/production of metabolites. In this paper, we propose a systematic data-based approach for finding both the biological objective function and a minimum set of active constraints necessary for matching the model predictions to the experimental data. The proposed algorithm accounts for the errors in the experiments and eliminates the need for ad hoc choices of objective function and constraints as done in previous studies. The method is illustrated for two cases: (1) for in silico (simulated) data generated by a mathematical model for Escherichia coli and (2) for actual experimental data collected from the batch fermentation of Bordetella Pertussis (whooping cough).

Appendices

Appendix I

J₁:

39.43 Ac + 35O₂ $~ \to ~$ X

J₂:

9.46Glc + 12.92O₂ $~ \to ~$ X

J₃:

9.84 Glc + 12.73O₂ $~ \to ~$ 1.24Ac + X

J₄:

19.23Glc $~ \to ~$ 12.12Ac + X

Appendix II

J₁:

Pyruvate $~ \to ~$ PEP

J₂:

Pyruvate + CoA $\to ~$ Acetyl-coA + CO₂

J₃:

Acetyl-CoA + H₂O + oxaloacetate $\to$ citrate + CoA

J₄:

Citrate $\to \alpha$-ketoglutarate + CO₂

J₅:

$\alpha$-ketoglutarate + enz-N6 $~ \to$ succinyl transf. + CO₂

J₆:

Succinyl transf. + CoA $\to ~$ succinyl-CoA + enz-N6

J₇:

Succinyl-CoA + phosphate $\to$ CoA + succinate

J₈:

Succinate + acceptor $\to$ fumarate + reduction acceptor

J₉:

Fumarate + H₂O $~ \to$ malate

J₁₀:

Malate $~ \to$ oxaloacetate

J₁₁:

Glutamate + NH₃ $~ \to ~$ phosphate + glutamine

J₁₂:

2 Glutamate $~ \leftarrow ~$ glutamine + $~\alpha$-ketoglutarate

J₁₃:

Glutamate + H₂O $~ \to \alpha$-ketoglutarate + NH₃

J₁₄:

Proline + 2H₂O $~ \to$ glutamate

J₁₅:

Oxaloacetate + glutamate $\to ~$ aspartate + $\alpha$-ketoglutarate

J₁₆:

Aspartate + NH₃ $~ \leftarrow ~$ aspargine

J₁₇:

PEP + ${\text{HCO}}_{3}^{ - }~ \to ~$ phosphate + oxaloacetate*

J₁₈:

Lactate $~ \to ~$ pyruvate

J₁₉:

Acetate + CoA $~ \leftarrow ~$ acetyl-CoA + phosphate

J₂₀:

2 Acetyl-CoA $~ \to ~$ CoA + acetoacetyl-CoA

J₂₁:

Acetoacetyl-CoA $~ \to ~$ PHB

J₂₂:

Glucose-6-phosphate + 3 glyceraldehyde-3-phosphate $~ \to ~$ 3 ribose-5-phosphate

J₂₃:

Acetyl-CoA + carrier-protein $\to ~$ acetoacetyl-carrier + CoA + CO₂

J₂₄:

Threonine + 3 pyruvate + 2 glutamate + acetyl-CoA + H₂O $\to ~$ NH₃ + 3CO₂ + 2H₂O + isoleucine + $~\alpha$-ketoglutarate + valine + CoA + leucine

J₂₅:

Serine + tetrahydrofolate $\leftarrow \to$ 5,10-methylenetetrahydrofolate + glycine + H₂O

J₂₆:

Serine $~ \leftarrow \to$ pyruvate + NH₃

J₂₇:

Threonine $~ \leftarrow \to ~$ glycine + acetaldehyde

J₂₈:

Aspartate $\to$ threonine + phosphate

J₂₉:

Hydrogen sulfide + acetyl-CoA + serine $\to ~$ CoA + cysteine + acetate

J₃₀:

Glutamate + pyruvate $\leftarrow \alpha$-ketoglutarate + alanine

J₃₁:

Aspartate + pyruvate + glutamate + succynyl-CoA $~ \to ~$ phosphate + $~\alpha$-ketogultarate + succinate + lysine + CO₂ + CoA

J₃₂:

Malate $~ \to ~$ pyruvate + CO₂

J₃₃:

Amino acids $~ \to ~$ biomass

J₃₄:

Amino acids $~ \to ~$ pertactin

J₃₅:

PEP $~ \to$ glyceraldehyde 3-P + phosphate

J₃₆:

2GAP + H₂O $\to$ glucose 6-P + phosphate

J₃₇:

J ₃

J₃₈:

Amino acids $~ \to ~$ Pertussis toxin

J₃₉:

Amino acids $~ \to ~$ fimbria

J₄₀:

Amino acids $\to$ FHA

J₄₁:

Inverse of J14

J₄₂:

Inverse of J27†

J₄₃:

Inverse of J30

J₄₄:

2 Glutamate $+{\text{~}}$ aspartate $\to$ fumarate $+{\text{~}}\alpha$-ketoglutarate $~+$ arginine

J₄₅:

Glucose $~+$ 6-P GAP $+{\text{~}}$ 2 PEP $~+{\text{~}}$ glutamate $\to {\text{~}}$tyrosine $+$ d-xylose $+{\text{~}}$ α-ketoglutarate

J₄₆:

Inverse of J26

J₄₇:

Valine $+{\text{~}}$ α-ketoglutarate $\to$ glutamate $~+$ 4 methyl-2-oxopentanoate

J₄₈:

2 pyruvate $+$ serine $+$ aspartate $+$ cysteine $\to$ CoA $+$ 3 CO₂ $+$ glycine

J₄₉:

Inverse of J18

Stoichiometry of purines

Adenine: pyruvate $~+{\text{~}}$ 2 glutamine $+$ 2 aspartate $+{\text{~}}$ glycine $\to$ adenine $+{\text{~}}$ 2 glutamate $+{\text{~}}$ fumarate

Guanine: pyruvate $+$ 3 glutamine $+$ aspartate $+$ glycine $\to {\text{~}}$ guanine $+$ 3 glutamate $+$ fumarate

Stoichiometry of pyrimidines

UMP (Uridylic acid): glutamine $+$ ${\text{HCO}}_{3}^{ - }+$ aspartate $+$ ribose-5-phosphate $\to$ UMP  $+$  CO₂  $+$  glutamate

CMP (Cytidylic acid): glutamine  $+{\text{HCO}}_{3}^{ - }$ þ aspartate  $+$  ribose-5-phosphate  $+$  NH₃  $\to$  CMP  $+$  CO₂ $+$ glutamate

TMP (Thymidylic acid): ${\text{HCO}}_{3}^{ - }$ $+$ serine $+$ glutamine $+$ aspartate $+$ ribose-5-phosphate $\to$ glycine $+$ TMP $+$ glutamate $+$ CO₂

Metabolite	Value
Ala	− 1.028497997
Arg	− 0.023
Asp	− 0.209047
Asn	− 0.029
Gln	0.6234
Glu	− 3.623564
Gly	− 0.635784
His	− 0.173987138
Ileu	− 0.253078
Leu	− 0.26459
Lys	− 0.232495
Meth	− 0.45017
Phe	− 0.028945
Pro	− 1.532302
Ser	− 2.423
Thr	− 0.02846
Try	0
Tyr	− 0.00823
Val	− 0.92
Cys	0
Pyr	0
AcCoA	0
GAP	0
Lac	7.125
Ammonia	6.2675
Co2	5.243
Α-Ketoglutarate	0
Biomass	200
Pertactin	0.4171
Citrate	0.50043
Succ-transferase	0
Succ CoA	0
Succinate	0
Fumarate	0
Malate	0
Oxaloacetate	0.005
Ribose	8
G6P	0.4184
PEP	0
AcAcCoA	0
PHB	0
FFA	0.234
Pt toxin	0.33
Fimbria	0.31243
FHA	0.334125
Xylose	0
CoA	0
ATP	0