Skip to main content
SpringerLink
Log in

Identification and robustness analysis of nonlinear multi-stage enzyme-catalytic dynamical system in batch culture

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this paper, based on biological phenomena of different characters at different stages, we propose a nonlinear multi-stage enzyme-catalytic dynamic system with unknown time and system parameters. Such system starts at different initial conditions for formulating batch culture process of glycerol bio-dissimilation to 1,3-propanediol. Some properties of the nonlinear system are discussed. In view of the difficulty in accurately measuring the concentration of intracellular substances and the absence of equilibrium points for the nonlinear system, we quantitatively define biological robustness for the entire process of batch culture instead of one for the approximately stable state of continuous culture. Taking the biological robustness of the intracellular substances together with the relative error between the experimental data and the computational values of the extracellular substances as the cost function, we formulate an identification problem subject to the nonlinear system, continuous state inequality constraints and parameter constraints. Analytical solution to system is not naturally available, therefore, a huge number of numerical computations of the proposed system and the proposed biological robustness make solving the identification problem by a serial computer a very complicated task. To improve computational efficiency, we develop an effective parallelized optimization algorithm, based on the constraint transcription and smoothing approximation techniques, for seeking the optimal time and system parameters. Compared with previous work, we assert that the optimal time and system parameters together with the corresponding nonlinear multi-stage dynamical system can reasonably describe batch fermentation at different initial conditions.

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
Fig. 5

Similar content being viewed by others

References

  • Alon U, Surette MG, Barkai N, Leibler S (1999) Robustness in bacterial chemotaxis. Nature 397:168–171

    Article  Google Scholar 

  • Antonio JK, Tsai WK, Huang GM (1991) A highly parallel algorithms for multistage optimization problems and shortest path problems. J Parallel Distrb Comput 12:213–222

    Article  MATH  MathSciNet  Google Scholar 

  • Babu BV, Angira R (2006) Modified differential evolution (MDE) for optimization of non-linear chemical processes. Comput Chem Eng 30:989–1002

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Bonnans JF, Gilbert JC, Lemaréchal C, Sagastizábal CA (2006) Numerical optimization: yheoretical and practical aspects. Springer, Berlin

  • Callaway DS, Newman MEJ, Strogatz SH, Watts DJ (2000) Network: Robustness and fragility: percolation on random graphs. Phys Rev Lett 85:5468–5471

    Article  Google Scholar 

  • Canto EB, Vassiliadis VS, Banga JR (2005) Dynamic optimization of single- and multi-Stage systems using a hybrid stochastic-deterministic method. Ind Eng Chem Res 44:1514–1523

    Article  Google Scholar 

  • Gao JG, Shen BY, Feng EM, Xiu ZL (2013) Modelling and optimal control for an impulsive dynamical system in microbial fed-batch culture. Comput Appl Math 32:275–290

    Article  MATH  MathSciNet  Google Scholar 

  • Garca-Raffia LM, Romagueraa S, Schellekens MP (2008) Applications of the complexity space to the general probabilistic divide and conquer algorithms. J Math Anal Appl 348:346–355

    Article  MathSciNet  Google Scholar 

  • Gong ZH, Liu CY, Feng EM, Wang L, Yu YS (2011) Modelling and optimization for a switched system in microbial fed-batch culture. Appl Math Model 35:3276–3284

    Article  MATH  MathSciNet  Google Scholar 

  • Gtinzel B (1991) Mikrobielle Herstellung von 1,3-propandiol durch Clostridium butyricum und adsorptive Abtremutng von Diolen, Ph.D.Dissertation, TU Braunschweig, Germany

  • Holland JH (1975) Adaptaion in natural and artificial systems. The University of Michigan Press, Ann Arbor

    Google Scholar 

  • Jiang ZG, Yuan JL, Feng EM (2013) Robust identification and its properties of nonlinear bilevel multi-stage dynamic system. Appl Math Comput 219:6979–6985

    Article  MATH  MathSciNet  Google Scholar 

  • Karci A (2005) Generalized parallel divide and conquer on 3D mesh and torus. J Syst Architect 51:281–295

    Article  Google Scholar 

  • Kirkpatrick S, Gelatt CD Jr, Vecchi MP (1983) Optimization by simulated annealing, American Association for the advancement of science. Science 220(4598):671–680

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  • Kitano H (2007) The theory of biological robustness and its implication in cancer. Syst Biol 61:69–88

    Article  Google Scholar 

  • Kitano H (2007) Towards a theory of biological robustness. Mol Syst Biol 3:137

    Article  Google Scholar 

  • Knuth DE (1998) The art of computer programming: volume 3, sorting and searching, 2nd edn. Posts and Telecom Press, Peking, China

  • Liu CY (2013) Sensitivity analysis and parameter identification for a nonlinear time-delay system in microbial fed-batch process. Appl Math Model 38:1449–1463

  • Liu CY (2013) Modelling and parameter identification for a nonlinear time-delay system in microbial batch fermentation. Appl Math Model 37:6899–6908

    Article  MathSciNet  Google Scholar 

  • Menzel K, Zeng AP, Deckwer WD (1997) High concentration and productivity of 1,3-propanediol from continuous fermentation of glycerol by Klebsiella pneumoniae. Enzyme Microb Technol 20:82–86

    Article  Google Scholar 

  • Price K, Storn R (1997) Differential evolution-a simple evolution strategy for fast optimization. Dr. Dobb’s J 22:18–24 (78)

    Google Scholar 

  • Rockfellar RT, Wets Roger JB (1998) Variational analysis. Springer, Berlin Heidelberg New York

    Book  Google Scholar 

  • Shen LJ, Wang Y, Feng EM, Xiu ZL (2007) Bilevel parameters identification for the multi-stage nonlinear impulsive system in microorganisms fed-batch cultures. Nonlinear Anay-Real 9:1068–1077

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  • Sun YQ, Qi WT, Teng H, Xiu ZL, Zeng AP (2008) Mathematica probleming 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

    Article  Google Scholar 

  • Teo KL, Goh CJ, Wong KH (1991) A unified computational approach to optimal control problems. Long Scientific Technical, Essex

  • Wang L (2012) Modelling and regularity of nonlinear impulsive switching dynamical system in fed-batch culture. Abstr Appl Anal. doi:10.1155/2012/295627 (15p, article ID 295627)

  • Wang L, Ye JX, Feng EM, Xiu ZL (2009) An improved problem for multistages imulation of glycerol fermentation in batch culture and its parameter identification. Nonlinear Anal Hybrid Syst 3:455–462

    Article  MATH  MathSciNet  Google Scholar 

  • Wang J, Ye JX, Yin HC, Feng EM, Wang L (2012) Sensitivity analysis and identification of kinetic parameters in batch fermentation of glycerol. J Comput Appl Math 236:2268–2276

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  • Wang L, Feng EM, Xiu ZL (2013) Modelling nonlinear stochastic kinetic system and stochastic optimal control of microbial bioconversion process in batch culture. Nonlinear Anal Model 18:99–111

    MATH  MathSciNet  Google Scholar 

  • Wardono B, Fathi Y (2004) A tabu search algorithm for the multi-stage parallel machine problem with limited buffer capacities. Eur J Oper Res 155:380–401

    Article  MATH  MathSciNet  Google Scholar 

  • Witt U, Miiller RJ, Augusta J, Widdecke H, Deckwer WD (1994) Synthesis, properties and biodegradability of ployesters based on 1,3-propanediol. Macromol Chem Phys 195:793–802

    Article  Google Scholar 

  • Yan HH, Zhang X, Ye JX, Feng EM (2012) Identification and robustness analysis of nonlinear hybrid dynamical system concerning glycerol transport mechanism. Comput Chem Eng 40:171–180

    Article  Google Scholar 

  • Yuan JL, Zhu X, Zhang X, Yin HC, Feng EM, Xiu ZL (2014a) Robust identification of enzymatic nonlinear dynamical systems for 1, 3-propanediol transport mechanisms in microbial batch culture. Appl Math Comput 232:150–163

  • Yuan JL, Zhang X, Zhu X, Yin HC, Feng EM, Xiu ZL (2014b) Modelling and pathway identification involving the transport mechanism of a complex metabolic system in batch culture. Commun Nonlinear Sci Numer Simulat 19:2088–2103

  • Zhang YD, Feng EM, Xiu ZL (2011) Robust analysis of hybrid dynamical systems for 1,3-propanediol transport mechanisms in microbial continuous fermentation. Math Comput Model 54:3164–3171

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  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 and 11301081).

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, Liaoning, People’s Republic of China

    Jinlong Yuan, Xu Zhang, Xi Zhu & Enmin Feng

  2. School of Energy and Power Engineering, Dalian University of Technology, Dalian, 116024, Liaoning, People’s Republic of China

    Jinlong Yuan & Hongchao Yin

  3. School of Science and Biotechnology, Dalian University of Technology, Dalian, 116024, Liaoning, People’s Republic of China

    Jinlong Yuan & Zhilong Xiu

  4. School of Mathematics and Statistics, Nanyang Normal University, Nanyang, 473061, Henan, People’s Republic of China

    Bing Tan

Authors
  1. Jinlong Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xu Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xi Zhu
    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

  7. Bing Tan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jinlong Yuan.

Additional information

Communicated by Domingo Alberto Tarzia.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yuan, J., Zhang, X., Zhu, X. et al. Identification and robustness analysis of nonlinear multi-stage enzyme-catalytic dynamical system in batch culture. Comp. Appl. Math. 34, 957–978 (2015). https://doi.org/10.1007/s40314-014-0160-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40314-014-0160-9

Keywords

Mathematics Subject Classification (2000)

Search

Navigation