Skip to main content
SpringerLink
Log in

A Novel Fractional Microbial Batch Culture Process and Parameter Identification

  • Original Research
  • Published:
Differential Equations and Dynamical Systems Aims and scope Submit manuscript

Abstract

This paper considers the microbial batch culture for producing 1,3-propanediol(1,3-PD) via glycerol disproportionation. Due to the nature of the fractional order operations, a novel fractional order model, which is based upon the original ordinary differential dynamic system, is introduced to describe the complex bioprocess in a more accurate manner. Existence and uniqueness of solutions to the novel fractional order system and the continuity of solutions with respect to the parameters are discussed respectively. In addition, a parameter identification problem of the system is presented, and a particle swarm optimization algorithm is constructed to solve it. Finally, the conclusion is drawn by numerical simulations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Liouville, J.: Mmoire sur quelques questions de gomtrie et de mcanique, et sur un nouveau genre de calcul pour rsoudre ces questions, pp. 1-69. (1832)

  2. Letnikov, A.V.: Theory of differentiation with an arbitrary index. Math. Sb 3, 1–66 (1868)

    Google Scholar 

  3. Hardy, G.H., Littlewood, J.E.: Some properties of fractional integrals. I. Math. Z. 34(1), 565–606 (1928)

    Article  MathSciNet  MATH  Google Scholar 

  4. Hardy, G.H., Littlewood, J.E.: Some properties of fractional integrals. II. Math. Z. 34(1), 403–439 (1932)

    Article  MathSciNet  MATH  Google Scholar 

  5. Caputo, M.: Linear models of dissipation whose Q is almost frequency independent II. Geophys. J. Int. 13(5), 529–539 (1967)

    Article  MathSciNet  Google Scholar 

  6. Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin Heidelberg (2010)

    Book  MATH  Google Scholar 

  7. Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265(2), 229–248 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Diethelm, K., Ford, N.J., Freed, A.D.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29(1), 3–22 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Sabatier, J., Agrawal, O.P., Machado, J.A.T.: Advances in Fractional Calculus. Springer, Dordrecht (2007)

    Book  MATH  Google Scholar 

  10. Assaleh, K., Ahmad, W.M.: Modeling of speech signals using fractional calculus, In: Signal Processing and Its Applications, ISSPA. 9th International Symposium on IEEE, pp. 1–4 (2007)

  11. Kulish, V.V., Lage, J.L.: Application of fractional calculus to fluid mechanics. J. Fluids Eng. 124(3), 803–806 (2002)

    Article  Google Scholar 

  12. Magin, R.L.: Modeling the cardiac tissue electrode interface using fractional calculus. J. Vib. Control 14(1), 1431–1442 (2008)

    Article  MATH  Google Scholar 

  13. Magin, R.L.: Fractional Calculus in Bioengineering. Begell House, Redding (2006)

    Google Scholar 

  14. Surez, J.I., Vinagre, B.M., Caldern, A.J., et al.: Using fractional calculus for lateral and longitudinal control of autonomous vehicles. Comput. Aided Syst. Theory EUROCAST 2003 2809, 337–348 (2003)

    Article  Google Scholar 

  15. Toledo-Hernandez, R., Rico-Ramirez, V., Iglesias-Silva, G.A., et al.: A fractional calculus approach to the dynamic optimization of biological reactive systems. Part I: fractional models for biological reactions. Chem. Eng. Sci. 117(1), 217C228 (2014)

    Google Scholar 

  16. Zeng, A.P., Ross, A., Biebl, H., et al.: Multiple product inhibition and growth modeling of Clostridium butyricum and Klebsiella pneumoniae in glycerol fermentation. Biotechnol. Bioeng. 44(8), 902–911 (1994)

    Article  Google Scholar 

  17. Zeng, A.P., Biebl, H.: Chemicals from Biotechnology: The Case of 1,3-Propanediol Production and the New Trends, Tools and Applications of Biochemical Engineering Science, 74th edn. Springer, Berlin Heidelberg (2002)

    Google Scholar 

  18. Gong, Z., Liu, C., Feng, E., et al.: Research article: computational method for inferring objective function of glycerol metabolism in Klebsiella pneumoniae. Comput. Biol. Chem. 33(1), 1–6 (2009)

    Article  MATH  Google Scholar 

  19. Yuan, J., Zhang, X., Zhu, X., et al.: Modelling and pathway identification involving the transport mechanism of a complex metabolic system in batch culture. Commun. Nonlinear Sci. Numer. Simul. 19(6), 2088–2103 (2014)

    Article  MathSciNet  Google Scholar 

  20. Yuan, J., Zhang, X., Zhu, X., et al.: Pathway identification using parallel optimization for a nonlinear hybrid system in batch culture. Nonlinear Anal. Hybrid Syst. 15, 112–131 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Yuan, J., Zhang, X., Zhu, X., et al.: Identification and robustness analysis of nonlinear multi-stage enzyme-catalytic dynamical system in batch culture. Comput. Appl. Math. 34(3), 957–978 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Yin, H., Yuan, J., Zhang, X., et al.: Modeling and parameter identification for a nonlinear multi-stage system for dha regulon in batch culture. Appl. Math. Model. 40(1), 468–484 (2016)

    Article  MathSciNet  Google Scholar 

  23. Li, X.H., Feng, E.M., Xiu, Z.L.: Optimal control and property of nonlinear dynamic system for microorganism in batch culture. OR Trans. 9(4), 67–79 (2005)

    Google Scholar 

  24. Gao, C., Feng, E., Wang, Z., et al.: Parameters identification problem of the nonlinear dynamical system in microbial continuous cultures. Appl. Math. Comput. 169(1), 476–484 (2005)

    MathSciNet  MATH  Google Scholar 

  25. Xiu, Z., Zeng, A., An, L.: Mathematical modeling of kinetics and research on multiplicity of glycerol bioconversion to 1, 3-propanediol. J. Dali. Univ. Technol. 40(4), 428–433 (2000)

  26. Wang, L.: Determining the transport mechanism of an enzyme-catalytic complex metabolic network based on biological robustness. Bioprocess Biosyst. Eng. 36(4), 433–441 (2012)

    Article  Google Scholar 

  27. Wang, L., Ye, J., Feng, E., et al.: An improved model for multistage simulation of glycerol fermentation in batch culture and its parameter identification. Nonlinear Anal. Hybrid Syst. 3(4), 455–462 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Wang, L., Feng, E.: An improved nonlinear multistage switch system of microbial fermentation process in fed-batch culture. J. Syst. Sci. Complex. 28(3), 580–591 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Wang, L., Xiu, Z., Gong, Z., et al.: Modeling and parameter identification for multistage simulation of microbial bioconversion in batch culture. Int. J. Biomath. 5(4), 1250034 (2012). doi:10.1142/S179352451100174X

    Article  MathSciNet  MATH  Google Scholar 

  30. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon (1993)

    MATH  Google Scholar 

  31. Kilbas, A.A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science Limited, Amsterdam (2006)

    MATH  Google Scholar 

  32. Podlubny, I.: Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calc. Appl. Anal. 5(4), 367–386 (2004)

    MathSciNet  MATH  Google Scholar 

  33. Heymans, N., Podlubny, I.: Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheol. Acta 45(5), 765–771 (2006)

    Article  Google Scholar 

  34. Zhou, Y., Wang, J., Zhang, L.: Basic theory of fractional differential equations. World Scientific (2016)

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation for the Youth of China (Grant No. 11401073), and the Fundamental Research Funds for Central Universities in China (Grant DUT15LK25).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, People’s Republic of China

    Pan Mu & Lei Wang

  2. School of Control Science and Engineering, Dalian University of Technology, Dalian, People’s Republic of China

    Yi An

  3. Department of Mathematics, Loyola Marymount University, Los Angeles, USA

    Yanping Ma

Authors
  1. Pan Mu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lei Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yi An
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yanping Ma
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lei Wang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mu, P., Wang, L., An, Y. et al. A Novel Fractional Microbial Batch Culture Process and Parameter Identification. Differ Equ Dyn Syst 26, 265–277 (2018). https://doi.org/10.1007/s12591-017-0381-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12591-017-0381-7

Keywords

Mathematics Subject Classification

Search

Navigation