Skip to main content
SpringerLink
Log in

Parameter identification for a nonlinear enzyme-catalytic dynamic system with time-delays

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

In this paper, we consider a nonlinear enzyme-catalytic dynamical system with uncertain system parameters and state-delays for describing the process of batch culture. Some important properties of the time-delay system are discussed. Taking account of the difficulty in accurately measuring the concentrations of intracellular substances and the absence of equilibrium points for the time-delay system, we define quantitatively biological robustness of the intracellular substance concentrations for the entire process of batch culture to identify the uncertain system parameters and state-delays. Taking the defined biological robustness as a cost function, we establish an identification model subject to the time-delay system, continuous state inequality constraints and parameter constraints. By a penalty approach, this model can be converted into a sequence of nonlinear programming submodels. In consideration of both the difficulty in finding analytical solutions and the complexity of numerical solution to the nonlinear system, based on an improved simulated annealing, we develop a parallelized synchronous algorithm to solve these nonlinear programming submodels. An illustrative numerical example shows the appropriateness of the optimal system parameters and state-delays as well as the validity of the parallel algorithm.

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
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Richard, J.P.: Time-delay systems: an overview of some recent advances and open problems. Automatica 39(10), 1667–1694 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Denis-Vidal, L., Jauberthie, C., Joly-Blanchard, G.: Identifiability of a nonlinear delayed-differential aerospace model. IEEE Trans. Autom. Control 51(1), 154–158 (2006)

    Article  MathSciNet  Google Scholar 

  3. Chai, Q.Q., Loxton, R., Teo, K.L., Yang, C.: A class of optimal state-delay control problems. Nonlinear Anal Real 14(3), 1536–1550 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Anguelova, M., Wennberg, B.: State elimination and identifiability of the delay parameter for nonlinear time-delay systems. Automatica 44(5), 1373–1378 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Loxton, R., Teo, K.L., Volker, R.: An optimization approach to state-delay identification. IEEE Tras. Autom. Control 55(9), 2113–2119 (2010)

    Article  Google Scholar 

  6. Witt, U., Miiller, R.J., Augusta, J., Widdecke, H., Deckwer, W.D.: Synthesis, properties and biodegradability of polyesters based on 1,3-propanediol. Macromol. Chem. Phys. 195(2), 793–802 (1994)

    Article  Google Scholar 

  7. Menzel, K., Zeng, A.P., Deckwer, W.D.: High concentration and productivity of 1,3-propanediol from continuous fermentation of glycerol by Klebsiella pneumoniae. Enzyme Microb. Technol. 20(2), 82–86 (1997)

    Article  Google Scholar 

  8. Ashoori, A., Moshiri, B., Sedigh, A.K., Bakhtiari, M.R.: Optimal control of a nonlinear fed-batch fermentation process using model predictive approach. J. Process Control 19(7), 1162–1173 (2009)

    Article  Google Scholar 

  9. Wang, G., Feng, E.M., Xiu, Z.L.: Modeling and parameter identification of microbial bioconversion in fed-batch cultures. J. Process Control 18(5), 458–464 (2008)

    Article  MathSciNet  Google Scholar 

  10. Ye, J.X., Feng, E.M., Lian, H.S., Xiu, Z.L.: Existence of equilibrium points and stability of the nonlinear dynamical system in microbial continuous cultures. Appl. Math. Comput. 207(2), 307–318 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gtinzel, B.: Mikrobielle Herstellung Von 1,3-Propandiol Durch Clostridium Butyricum und Adsorptive abtremutng von diolen. TU Braunschweig, Germany (1991)

    Google Scholar 

  12. Yuan, J.L., Zhang, X., Zhu, X., Yin, H.C., Feng, E.M., Xiu, Z.L.: Identification and robustness analysis of nonlinear multi-stage enzyme-catalytic dynamical system in batch culture. Comp. Appl. Math. (2014). doi:10.1007/s40314-014-0160-9

  13. Jiang, Z.G., Yuan, J.L., Feng, E.M.: Robust identification and its properties of nonlinear bilevel multi-stage dynamic system. Appl. Math. Comput. 219(12), 6979–6985 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Wang, J., Ye, J.X., Yin, H.C., Feng, E.M., Wang, L.: Sensitivity analysis and identification of kinetic parameters in batch fermentation of glycerol. J. Comput. Appl. Math. 236(9), 2268–2276 (2012)

    Article  MathSciNet  Google Scholar 

  15. Yuan, J.L., Zhu, X., Zhang, X., Yin, H.C., Feng, E.M., Xiu, Z.L.: Robust identification of enzymatic nonlinear dynamical systems for 1, 3-propanediol transport mechanisms in microbial batch culture. Appl. Math. Comput. 232, 150–163 (2014)

    Article  MathSciNet  Google Scholar 

  16. Wang, L., Feng, E.M., Xiu, Z.L.: Modeling nonlinear stochastic kinetic system and stochastic optimal control of microbial bioconversion process in batch culture. Nonlinear Anal. Model. 18(1), 99–111 (2013)

    MathSciNet  MATH  Google Scholar 

  17. Yuan, J.L., Zhang, X., Zhu, X., Yin, H.C., Feng, E.M., Xiu, Z.L.: Modelling and pathway identification involving the transport mechanism of a complex metabolic system in batch culture. Commun. Nonlinear Sci. Numer. Simul. 19(16), 2088–2103 (2014)

    Article  MathSciNet  Google Scholar 

  18. Zhu, X., Feng, E.M.: Joint estimation in batch culture by using unscented kalman filter. Biotechnol. Bioprocess Eng. 17(6), 1238–1243 (2012)

    Article  MathSciNet  Google Scholar 

  19. Liu, C.Y.: Modelling and parameter identification for a nonlinear time-delay system in microbial batch fermentation. Appl. Math. Model. 37(10–11), 6899–6908 (2013)

    Article  MathSciNet  Google Scholar 

  20. Kitano, H.: Biological robustness. Nat. Rev. Genet. 5(11), 826–837 (2004)

    Article  Google Scholar 

  21. Kitano, H.: Violations of robustness trade-offs. Mol. Syst. Biol. 6, (Article ID 384)(2010). doi:10.1038/msb.2010.40

  22. Stelling, J., Sauer, U., Szallasi, Z., Doyle, F.J., Doyle, J.: Robustness of cellular functions. Cell 118(6), 675–685 (2004)

    Article  Google Scholar 

  23. Perc, M., Marhl, M.: Sensitivity and flexibility of regular and chaotic calcium oscillations. Biophys. Chem. 104(2), 509–522 (2003)

    Article  Google Scholar 

  24. Perc, M., Marhl, M.: Noise enhances robustness of intracellular \(Ca^{2+}\) oscillations. Phys. Lett. A. 316(5), 304–310 (2003)

    Article  Google Scholar 

  25. Barkai, N., Leibler, S.: Robustness in simple biochemical networks. Nature 387(6636), 913–917 (1997)

    Article  Google Scholar 

  26. Kitano, H.: Towards a theory of biological robustness. Mol. Syst. Biol. 3, (Article ID 137)(2007)

  27. Alon, U., Surette, M.G., Barkai, N., Leibler, S.: Robustness in bacterial chemotaxis. Nature 397(6715), 168–171 (1999)

    Article  Google Scholar 

  28. Callaway, D.S., Newman, M.E.J., Strogatz, S.H., Watts, D.J.: Network, robustness and fragility: percolation on random graphs. Phys. Rev. Lett. 85(25), 5468–5471 (2000)

    Article  MATH  Google Scholar 

  29. Zhang, Y.D., Feng, E.M., Xiu, Z.L.: Robust analysis of hybrid dynamical systems for 1,3-propanediol transport mechanisms in microbial continuous fermentation. Math. Comput. Model. 54(11–12), 3164–3171 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Yan, H.H., Zhang, X., Ye, J.X., Feng, E.M.: Identification and robustness analysis of nonlinear hybrid dynamical system concerning glycerol transport mechanism. Comput. Chem. Eng. 40, 171–180 (2012)

    Article  Google Scholar 

  31. Gao, Y., Lygeros, J., Quincampoix, M.: On the reachability problem for uncertain hybrid systems. IEEE Trans. Autom. Control 52(9), 1572–1586 (2007)

    Article  MathSciNet  Google Scholar 

  32. Gao, Y., Lygeros, J., Quincampoix, M., Seube, N.: On the control of uncertain impulsive systems: approximate stabilization and controlled invariance. Int. J. Control 77(16), 1393–1407 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  33. Sun, Y.Q., Qi, W.T., Teng, H., Xiu, Z.L., Zeng, A.P.: Mathematica modeling of glycerol fermentation by Klebsiella pneumoniae: concerning enzyme-catalytic reductive pathway and transport of glycerol and 1,3-propanediol across cell membrane. Biochem. Eng. J. 38, 22–32 (2008)

    Article  Google Scholar 

  34. Lin, Q., Loxton, R., Teo, K.L.: The control parameterization method for nonlinear optimal control: a survey. J. Ind. Manag. Optim. 10(1), 275–309 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Lin, Q., Loxton, R., Teo, K.L., Wu, Y.H., Yu, C.J.: A new exact penalty method for semi-infinite programming problems. J. Comput. Appl. Math. 261, 271–286 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wang, L.: Determining the transport mechanism of an enzyme-catalytic complex metabolic network based on biological robustness. Bioprocess Biosyst. Eng. 36(4), 433–441 (2013)

    Article  Google Scholar 

  37. Wang L.: Modelling and Regularity of nonlinear impulsive switching dynamical system in fed-batch culture. Abstr. Appl. Anal. 15 (2012) [Article ID 295627]. doi:10.1155/2012/295627

  38. Polak, E.: Optimization algorithms and consistent approximations. Springer, New York (1997)

    MATH  Google Scholar 

  39. Ahonen, H., de Alvarenga, A.G., Amaral, A.R.S.: Simulated annealing and tabu search approaches for the corridor allocation problem. Eur. J. Oper. Res. 232(1), 221–233 (2014)

    Article  MATH  Google Scholar 

  40. Aarts, E., Jan, K., Wil, M.: Simulated annealing. Search methodologies. Springer, Berlin (2005)

    Google Scholar 

  41. Romeijn, H.E., Robert, L.S.: Simulated annealing for constrained global optimization. J. Global Optim. 5(2), 101–126 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  42. Gelatt, C.D., Vecchi, M.P.: Optimization by simulated annealing. Science 220(4598), 671–680 (1983)

    Article  MathSciNet  Google Scholar 

  43. Koakutsu, S., Sugai, Y., Hirata, H.: Block placement by improved simulated annealing based on genetic algorithm. Syst. Model. Optim. 180, 648–656 (1992)

    Article  Google Scholar 

  44. Ishibuchi, H., Misaki, S., Tanaka, H.: Modified simulated annealing algorithms for the flow shop sequencing problem. Eur. J. Oper. Res. 81(2), 388–398 (1995)

    Article  MATH  Google Scholar 

  45. Onbasoǧlu, E., Özdamar, L.: Parallel simulated annealing algorithms in global optimization. J. Global Optim. 19(1), 27–50 (2001)

    Article  MathSciNet  Google Scholar 

  46. Ferreiro, A.M., García, J.A., López-Salas, J.G., Vázquez, C.: An efficient implementation of parallel simulated annealing algorithm in GPUs. J. Global Optim. 57(3), 863–890 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  47. Czapiński, M.: Parallel Simulated Annealing with Genetic Enhancement for flowshop problem with \(C_{sum}\). Comput. Ind. Eng. 59(4), 778–785 (2010)

    Article  Google Scholar 

  48. Goberna, M.A., López, M.A.: Semi-infinite Programming Recent Advances. Kluwer, Dordrecht (2001)

    Book  MATH  Google Scholar 

  49. Ellermeyer, A., Hendrix, J., Ghoochan, N.: A theoretical and empirical investigation of delayed growth response in the continuous culture of bacteria. J. Theor. Biol. 222(4), 485–494 (2003)

    Article  MathSciNet  Google Scholar 

  50. Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, Berlin (1993)

    Book  MATH  Google Scholar 

  51. Chai, Q.Q., Loxton, R., Teo, K.L., Yang, C.H.: A unified parameter identification method for nonlinear time-delay systems. J. Ind. Manag. Optim. 9(2), 471–486 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  52. Nocedal, J., Wright, S.J.: Numerical optimization. Springer, New York (1999)

    Book  MATH  Google Scholar 

Download references

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11171050, 11101262 and 11371164), the National Natural Science Foundation for the Youth of China (Grant Nos. 11301051, 11301081 and 11401073) and Provincial Natural Science Foundation of Fujian (Grant Nos. 2014J05001).

Author information

Authors and Affiliations

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

    Jinlong Yuan, Lei Wang, Xu Zhang & Enmin Feng

  2. School of Energy and Power Engineering, Dalian University of Technology, Dalian, People’s Republic of China

    Jinlong Yuan & Hongchao Yin

  3. School of Life Science and Biotechnology, Dalian University of Technology, Dalian, People’s Republic of China

    Zhilong Xiu

Authors
  1. Jinlong Yuan
    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. Xu Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Enmin Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Hongchao Yin
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Zhilong Xiu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jinlong Yuan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yuan, J., Wang, L., Zhang, X. et al. Parameter identification for a nonlinear enzyme-catalytic dynamic system with time-delays. J Glob Optim 62, 791–810 (2015). https://doi.org/10.1007/s10898-014-0245-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-014-0245-4

Keywords

Search

Navigation