Bioprocess and Biosystems Engineering

, Volume 41, Issue 5, pp 641–655 | Cite as

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

  • Ali Nikdel
  • Richard D. Braatz
  • Hector M. Budman
Research Paper
  • 175 Downloads

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).

Keywords

Dynamic flux balance analysis Flux balance analysis Metabolic networks Metabolic engineering Bioprocess modeling 

Abbreviations

S

Stoichiometric matrix

\({{\varvec{v}}_{\varvec{k}}}\)

Vector of fluxes

\(n\)

Number of reactions

\(k\)

Time instance

\(\psi\)

Concentration

\(~{{\varvec{J}}_\mathbf{T}}\)

Fluxes that satisfy tight constraints

\(~{{\varvec{J}}_\mathbf{R}}\)

Fluxes that satisfy relaxed constraints

\({{\varvec{u}}_\mathbf{w}}\)

Weight of sum of squared errors

I

Identity matrix

\(w_{i}^{{{\text{sc}}}}\)

Time-varying values of the weights for all the metabolites

\(w_{i}^{{\text{u}}}\)

Weight of upper bound

\({{W}^{\text{u}}}\)

Maximum allowable value for \(w_{i}^{{{\text{sc}}}}\)

\(w_{i}^{{\text{l}}}\)

Weight of lower bound

\({N_{\text{C}}}\)

Number of the objective functions’ candidates

\({{\varvec{n}}_\mathbf{g}}\)

Estimated noise in the growth rate

\({N_{{\text{SC}}}}\)

Total number of metabolites

\({N_{\text{m}}}\)

Number of measured metabolites

\({V_{i,{\text{max}}}}\)

Maximum rate

\({K_i}\)

Half saturation concentration

\(\varepsilon\)

Measurement error

\({X_k}\)

The biomass value at time \(~k\)

\({w_{{{\text{c}}_i}}}\)

The weight coefficients of the objective function candidates

Notes

Acknowledgements

The authors would like to thank Natural Science and Engineering Research Council (NSERC).

References

  1. 1.
    Schilling CH, Letscher D, Palsson BO (2000) Theory for the systemic definition of metabolic pathways and their use in interpreting metabolic function from a pathway-oriented perspective. J Theor Biol 203(3):229–248CrossRefGoogle Scholar
  2. 2.
    Jaqaman K, Danuser G (2006) Linking data to models: data regression. Nat Rev Mol Cell Biol 7(11):813–819CrossRefGoogle Scholar
  3. 3.
    Orth JD, Thiele I, Palsson BO (2010) What is flux balance analysis?. Nat Biotechnol 28(3):245–248CrossRefGoogle Scholar
  4. 4.
    Foguet C et al (2016) HepatoDyn: a dynamic model of hepatocyte metabolism that integrates C-13 isotopomer data. PLoS Comput Biol 12(4):e1004899Google Scholar
  5. 5.
    Llaneras F, Sala A, Picó J (2012) Dynamic estimations of metabolic fluxes with constraint-based models and possibility theory. J Process Control 22(10):1946–1955CrossRefGoogle Scholar
  6. 6.
    Varma A, Palsson BO (1995) Parametric sensitivity of stoichiometric flux balance models applied to wild-type Escherichia coli metabolism. Biotechnol Bioeng 45(1):69–79CrossRefGoogle Scholar
  7. 7.
    Mahadevan R, Edwards JS, Doyle FJ (2002) Dynamic flux balance analysis of diauxic growth in Escherichia coli. Biophys J 83(3):1331–1340CrossRefGoogle Scholar
  8. 8.
    Sanchez CEG, Saez RGT (2014) Comparison and analysis of objective functions in flux balance analysis. Biotechnol Prog 30(5):985–991CrossRefGoogle Scholar
  9. 9.
    Pramanik J, Keasling JD (1997) Stoichiometric model of Escherichia coli metabolism: incorporation of growth-rate dependent biomass composition and mechanistic energy requirements. Biotechnol Bioeng 56(4):398–421CrossRefGoogle Scholar
  10. 10.
    Schuetz R, Kuepfer L, Sauer U (2007) Systematic evaluation of objective functions for predicting intracellular fluxes in Escherichia coli. Mol Syst Biol 3:119CrossRefGoogle Scholar
  11. 11.
    Knorr AL, Jain R, Srivastava R (2007) Bayesian-based selection of metabolic objective functions. Bioinformatics 23(3):351–357CrossRefGoogle Scholar
  12. 12.
    Burgard AP, Maranas CD (2003) Optimization-based framework for inferring and testing hypothesized metabolic objective functions. Biotechnol Bioeng 82(6):670–677CrossRefGoogle Scholar
  13. 13.
    Savinell JM, Palsson BO (1992) Network analysis of intermediary metabolism using linear optimization.1. Development of mathematical formalism. J Theor Biol 154(4):421–454CrossRefGoogle Scholar
  14. 14.
    Gianchandani EP et al (2008) Predicting biological system objectives de novo from internal state measurements. BMC Bioinform 9:43Google Scholar
  15. 15.
    Llaneras F, Picó J (2008) Stoichiometric modelling of cell metabolism. J Biosci Bioeng 105(1):1–11CrossRefGoogle Scholar
  16. 16.
    Nikdel A, Budman H (2016) Identification of active constraints in dynamic flux balance analysis. Biotechnol Prog 33(1):26–36Google Scholar
  17. 17.
    Maranas CD, Zomorrodi AR, Wiley Online Library (2016) Optimization methods in metabolic networks, Wiley, Hoboken (1 online resource) CrossRefGoogle Scholar
  18. 18.
    Borchers S et al (2013) Identification of growth phases and influencing factors in cultivations with AGE1.HN cells using set-based methods. PLoS One 8(8):e68124Google Scholar
  19. 19.
    Shuler ML, Kargi F (1992) Bioprocess engineering: basic concepts. Prentice Hall international series in the physical and chemical engineering sciences, Prentice Hall, Englewood Cliffs, pp 16+479 sGoogle Scholar
  20. 20.
    Thalen M et al (2006) Fed-batch cultivation of Bordetella pertussis: metabolism and Pertussis Toxin production. Biologicals 34(4):289–297CrossRefGoogle Scholar
  21. 21.
    Budman H et al (2013) A dynamic metabolic flux balance based model of fed-batch fermentation of Bordetella pertussis. Biotechnol Prog 29(2):520–531CrossRefGoogle Scholar
  22. 22.
    Weiss JN (1997) The Hill equation revisited: uses and misuses. Faseb J 11(11):835–841CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Ali Nikdel
    • 1
  • Richard D. Braatz
    • 2
  • Hector M. Budman
    • 1
  1. 1.Department of Chemical EngineeringUniversity of WaterlooWaterlooCanada
  2. 2.Department of Chemical EngineeringMassachusetts Institute of TechnologyCambridgeUSA

Personalised recommendations