Model

What is flux balance analysis (FBA)

Flux balance analysis (FBA) is a mathematical model which attempts to simulate metabolic distributions of a cell. Different from ODE, FBA does not need kinetic parameters, as well as the initial concentration of each metabolites. Therefore, simulations using FBA are computationally inexpessive and can process a large number of metabolic fluxes in a few seconds on a PC[1].

In the model, metabolic reactions are represented as a stoichiometric matrix. Negative coefficients are used to represent metabolites consumed, positive coefficients are used to represent metabolites produced, and the zero is used to represent those metabolites does not participate in a particular reaction.[2] Then a stoichiometric matrix that represents metabolic reacions can be constructed with these coefficients.

$$\frac{dX}{dt}=Sv\text{ (1)}$$

where X is the concentration of metabolites, S is a stoichiometric matrix, and v is the metabolic reaction. Since FBA calculates the change in concentration of metabolites at steady state, thera is

$$\frac{dX}{dt}=0\text{ (2)}$$

FBA seeks to maximize or minimize an objective function Z, which can be any linear combination of fluxes under the constraint conditions (1) and (2), as below

$$Z=c^Tv\text{ (3)}$$

where c is a vector of weights indicating how much each reaction contributes to the objective function.

FBA uses linear programming to solve it. We choose a powerful toolbox called COBRApy to implement FBA[3]. As a result, it is expected that the predicted metabolic flux distributions will explain many phenomenon in our project and then guide us to complete the project.

Feasibility Analysis

In principle, the more the amount of crRNA is, the higher the inhibitory rate against gltA is. Supposing that the decrease of the flux of citrate synthesis is proportional to the inhibitory rate against gltA, we model the flux of BIOMASS and the two competitive flux (CS and ACACT1r) with the increase of it. Our predictive result is as follow.

Figure 1. Feasibility Analysis.

As expected, with the increase of the inhibitory rate, the growth rate characterized by BIOMASS decreases, and the objective flux ACACT1r increases. It suggests that the suppression of gltA would redirect part of CS flux to the objective pathway and would inhibit the growth of E. coli.

Flux Distributions in Different Culture Medium

Our experiments show that crRNA inhibits the growth of E. coli at different extent in different culture medium. To explain the phenomenon, We model the metabolic flux distributions in different culture medium.

In our model, We can see that E. coli using glucose as sole carbon source has higher threshold of crRNA's inhibiting growth significantly than E. coli using glycerol as sole carbon source. It causes that E. coli using glycerol as sole carbon is easier to be inhibited by crRNA than that using glucose as sole carbon, which is matched by our experiment results.

Figure 2. E. coli using glucose as sole carbon source and E. coli using glycerol as sole carbon source.

The figure above shows the effects of adding yeast extraction to the media which uses glycerol as sole carbon. Compared with the predictive result of no addition of yeast extraction, it can be seen that the addition of yeast extraction eliminates the influences of crRNA to the growth rate, which is also matched by our experiment results.

Figure 3. The effects of adding yeast extraction to the media which uses glycerol as sole carbon.

Furthermore, in terms of adding the yeast extraction, supposing that the inhibitory rate of gltA is 80%, we analyze the flux distribution center on the acytel-CoA, which distributes part of flux to CS and ACACT1r. As we can see, even though the ratio of the objective flux - ACACT1r increases from 7% to 18%, the flux responsible for producing its precursor - acytel-CoA decreases from 51.18 to 18.60, causing the total flux of the objective flux essentially unchanged.

Figure 4. Flux distribution analysis (FBA) of E. coli growing in the media added glycerol and yeast extraction.

How to Optimize Our Project In the Future

According our human practice, we know that the medium component is very important to industrial production. To optimize our project in the future, we examine a variety of additional nutrient by production potential analysis.

Carbon sources analysis

The figure above shows the effect of using different carbon sources on the objective flux. We can see, theoretically, our objective pathway has the same flux with both glucose and fructose, both of which exceeds the others.

Figure 5. Carbon sources analysis. D-Glucose, D-Fructose, D-Xylose, Glycerol, Pyruvate, Acetate.

Nitrogen sources analysis

The figures above (Figure 6-30) show the effects of adding different nitrogen sources on the objective flux. There are many kinds of additional nitrogen sources that can increase ACACT1r flux.

Figure 6. Nitrogen sources analysis. 1,5-Diaminopentane, 2',3'-Cyclic AMP, 2',3'-Cyclic CMP, 2',3'-Cyclic GMP, 2',3'-Cyclic UMP, 2,3-diaminopropionate, meso-2,6-Diaminoheptanedioate, 3'-AMP, 3'-cmp, 3'-GMP.
Figure 7. 3'-UMP, 4-Aminobutanoate, L-alanine-D-glutamate, L-alanine-D-glutamate-meso-2,6-diaminoheptanedioate, L-alanine-D-glutamate-meso-2,6-diaminoheptanedioate-D-alanine, L-alanine-L-glutamate, N-Acetyl-D-galactosamine, N-Acetyl-D-galactosamine 1-phosphate, N-Acetyl-D-glucosamine, N-Acetyl-D-glucosamine 1-phosphate.
Figure 8. Nitrogen sources analysis. N-Acetyl-D-mannosamine, N-Acetylmuramate, N-Acetylneuraminate, 4-Amino-4-deoxy-L-arabinose modified core oligosaccharide lipid A, O-Acetyl-L-serine, Adenine, Adenosine, Adenosylcobalamin, Agmatine, beta-Alanine.
Figure 9. Nitrogen sources analysis. D-Alanine, L-Alanine, D-Alanyl-D-alanine, Allantoin, AMP, N-Acetyl-D-glucosamine(anhydrous)N-Acetylmuramic acid, aerobactin minus Fe3, Aerobactin, L-Arginine, L-Asparagine.
Figure 10. Nitrogen sources analysis. L-Aspartate, Biotin, Cobinamide, Cob(I)alamin, Cys-Gly, Choline, N,N'-diacetylchitobiose, Chloramphenicol, CMP, core oligosaccharide lipid A.
Figure 11. Nitrogen sources analysis. core oligosaccharide lipid A diphosphate, coprogen, coprogen unloaded (no Fe(III)), L-Carnitine, D-Carnitine, Cytosine, Hydrogen cyanide, Cyanate, D-Cysteine, L-Cysteine.
Figure 12. Nitrogen sources analysis. Cytidine, Deoxyadenosine, dAMP, dCMP, Deoxycytidine, dGMP, Deoxyguanosine, dIMP, Deoxyinosine, Dopamine.
Figure 13. Nitrogen sources analysis. doxorubicin, dTMP, dUMP, Deoxyuridine, (enterobacterial common antigen)x4 core oligosaccharide lipid A, phosphoethanolamine KDO(2)-lipid (A), Enterochelin, Ethanolamine, ferric 2,3-dihydroxybenzoylserine, Fe(III)hydroxamate.
Figure 14. Nitrogen sources analysis. Fe(III)hydoxamate, Ferrichrome, Ferrichrome minus Fe(III), Fe-enterobactin, ferroxamine, ferroxamine minus Fe(3), fructoselysine, sn-Glycero-3-phosphocholine, sn-Glycero-3-phosphoethanolamine, Glycerophosphoserine.
Figure 15. Nitrogen sources analysis. D-Glucosamine, D-Glucosamine 6-phosphate, gamma-butyrobetaine, GDP, L-Glutamine, L-Glutamate, Glycine, Glycine betaine, GMP, Guanosine.
Figure 16. Nitrogen sources analysis. Oxidized glutathione, Reduced glutathione, GTP, Guanine, hepta-acylated core oligosaccharide lipid A (E. coli), hepta-acylated KDO(2)-lipid (A), L-Histidine, L-Homoserine, Hypoxanthine, L-Isoleucine.
Figure 17. Nitrogen sources analysis. IMP, Indole, Inosine, KDO(2)-lipid IV(A), L-Leucine, KDO(2)-lipid (A), cold adapted KDO(2)-lipid (A), L-Lysine, D-Methionine, L-Methionine.
Figure 18. Nitrogen sources analysis. L-methionine-R-sulfoxide, L-Methionine Sulfoxide, minocycline, S-Methyl-L-methionine, Nitrous oxide, Nicotinate, Ammonium, NMN, Nitric oxide, Nitrite.
Figure 19. Nitrogen sources analysis. Nitrate, novobiocin, (O16 antigen)x4 core oligosaccharide lipid A, Ornithine, Orotate, Phenethylamine, L-Phenylalanine, Protoheme, (R)-Pantothenate, L-Proline.
Figure 20. Nitrogen sources analysis. L-Prolinylglycine, psicoselysine, O-Phospho-L-serine, Putrescine, Pyridoxamine, Pyridoxal, Pyridoxine, rifampin, D-Serine, L-Serine.
Figure 21. Nitrogen sources analysis. Spermidine, Taurine, Thiocyanate, Thiamin, L-Threonine, L-Threonine O-3-phosphate, Thymine, Thymidine, trimethylamine, Trimethylamine N-oxide.
Figure 22. Nitrogen sources analysis. L-Tryptophan, tetracycline, Tyramine, L-Tyrosine, Phosphotyrosine, UDP-N-acetyl-D-glucosamine, UDP-N-acetyl-D-galactosamine, UDPglucose, UDPgalactose, UDP-D-glucuronate.
Figure 23. Nitrogen sources analysis. UMP, Uracil, Urea, Uridine, L-Valine, Xanthine, Xanthosine 5'-phosphate, Xanthosine.

Phosphorus sources analysis

The figures above show the effect of using different phosphorus sources on the objective flux. We can see that our objective flux is not sensitive to the additional phosphorus sources.

Figure 24. Phosphorus sources analysis. 2',3'-Cyclic AMP, 2',3'-Cyclic CMP, 2',3'-Cyclic GMP, 2',3'-Cyclic UMP, 3'-AMP, 3'-cmp, 3'-GMP, 3'-UMP, N-Acetyl-D-galactosamine 1-phosphate, N-Acetyl-D-glucosamine 1-phosphate.
Figure 25. Phosphorus sources analysis. 4-Amino-4-deoxy-L-arabinose modified core oligosaccharide lipid A, Adenosylcobalamin, AMP, Cob(I)alamin, CMP, core oligosaccharide lipid A, core oligosaccharide lipid A diphosphate, dAMP, dCMP, dGMP.
Figure 26. Phosphorus sources analysis. dIMP, dTMP, dUMP, (enterobacterial common antigen)x4 core oligosaccharide lipid A, phosphoethanolamine KDO(2)-lipid (A), D-Fructose 6-phosphate, D-Glucose 1-phosphate, sn-Glycero-3-phosphocholine, sn-Glycero-3-phosphoethanolamine, Glycerophosphoglycerol.
Figure 27. Phosphorus sources analysis. sn-Glycero-3-phospho-1-inositol, Glycerophosphoserine, D-Glucose 6-phosphate, alpha-D-Galactose 1-phosphate, D-Glucosamine 6-phosphate, GDP, D-Glucuronate 1-phosphate, Glycerol 2-phosphate, Glycerol 3-phosphate, GMP.
Figure 28. Phosphorus sources analysis. GTP, hepta-acylated core oligosaccharide lipid A (E. coli), hepta-acylated KDO(2)-lipid (A), IMP, KDO(2)-lipid IV(A), KDO(2)-lipid (A), cold adapted KDO(2)-lipid (A), D-Mannose 6-phosphate, myo-Inositol hexakisphosphate, NMN.
Figure 29. Phosphorus sources analysis. (O16 antigen)x4 core oligosaccharide lipid A, Phosphate, Phosphonate, O-Phospho-L-serine, alpha-D-Ribose 5-phosphate, L-Threonine O-3-phosphate, Phosphotyrosine, UDP-N-acetyl-D-glucosamine, UDP-N-acetyl-D-galactosamine, UDPglucose.
Figure 30. Phosphorus sources analysis. UDPgalactose, UDP-D-glucuronate, UMP, Xanthosine 5'-phosphate.

Sulfur sources analysis

The figures above (Figure 31-33) show the effects of adding different sulfur sources on the objective flux. We can see that our objective flux is not sensitive to the additional sulfur sources except reduced glutathione, which can be also regarded as nitrogen source.

Figure 31. Sulfur sources analysis. 5-Methylthio-D-ribose, Biotin, butanesulfonate, Cys-Gly, D-Cysteine, L-Cysteine, Dimethyl sulfide, Dimethyl sulfoxide, ethanesulfonate, Oxidized glutathione.
Figure 32. Sulfur sources analysis. Reduced glutathione, Hydrogen sulfide, Isethionic acid, Lipoate, D-Methionine, L-Methionine, L-methionine-R-sulfoxide, L-Methionine Sulfoxide, S-Methyl-L-methionine, methanesulfonate.
Figure 33. Sulfur sources analysis. sulfur dioxide, Sulfite, Sulfate, sulfoacetate, Taurine, Thiocyanate, Thiamin, Thiosulfate.

Ions analysis

The figures above (Figure 34-35) show the effects of adding different ions on the objective flux. We can see that our objective flux is not sensitive to the additional ions.

Figure 34. Ions analysis. Cu+, silver, Cu2+, magnesium, Co2+, Mn2+, Sodium, nickel, Zinc, Calcium.
Figure 35. Ions analysis. Cadmium, Chloride, Fe2+, Fe3+.

Amino acids analysis

The figures above (Figure 36-38) show the effects of adding different amino acids on the objective flux. These amino acids can be also regarded as nitrogen sources. There are many kinds of additional amino acids that can increase ACACT1r flux. It is worth mentioning that the addition of L-serine could reduce our objective flux.

Figure 36. Amino acids analysis. L-Tryptophan, L-Alanine, D-Alanine, beta-Alanine, Glycine, L-Arginine, L-Asparagine, L-Aspartate, D-Cysteine, L-Cysteine.
Figure 37. Amino acids analysis.L-Glutamine, L-Glutamate, L-Histidine, L-Isoleucine, L-Leucine, L-Lysine, D-Methionine, L-Methionine, L-Phenylalanine, L-Proline.
Figure 38. Amino acids analysis. D-Serine, L-Serine, L-Threonine, L-Tryptophan, L-Tyrosine, L-Valine.

In summary, the predictive results suggest that our objective pathway - ACACT1r is not sensitive to most of the phosphorus sources, sulfur sources and ions, while many nitrogen-related sources can increase our objective flux. In practical production, we should focus on selecting appropriate nitrogen sources.

References

[1] Gianchandani, Erwin P., A. K. Chavali, and J. A. Papin. "The application of flux balance analysis in systems biology." Wiley Interdisciplinary Reviews Systems Biology & Medicine 2.3(2010):372-382.
[2] Heshiki, Yoshitaro, "Optimization of Polyhydroxybutyrate Production in Recombinant Escherichia Coli Through Metabolic Modeling and Simulation" (2013). All Graduate Plan B and other Reports. 291.
[3] Ebrahim, Ali, et al. "COBRApy: COnstraints-Based Reconstruction and Analysis for Python." Bmc Systems Biology 7.1(2013):1-6.