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March  2020, 16(2): 579-599. doi: 10.3934/jimo.2018168

Identification and robustness analysis of nonlinear hybrid dynamical system of genetic regulation in continuous culture

Qi Yang 1,2, , Lei Wang 3,, , Enmin Feng 3, , Hongchao Yin 2, and  Zhilong Xiu 4,

1. 

School of Mathematical Science, Dalian University of Technology, Dalian, Liaoning 116024, China
2. 

School of Energy and Engineering, Dalian University of Technology, Dalian, Liaoning 116024, China
3. 

School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China
4. 

School of Life Science and Biotechnology, Dalian University of Technology, Dalian, Liaoning 116024, China

* Corresponding author. E-mail address: wanglei@dlut.edu.cn

Received  March 2018 Revised  July 2018 Published  November 2018

Fund Project: This work was supported by the National Natural Science Foundation of China (Grant Nos. 11771008, 11171050 and 11371164), the National Science Foundation for the Youth of China (Grant Nos. 11301051, 11301081 and 11401073), the Provincial Natural Science Foundation of Fujian (Grant Nos. 2014J05001), the Fundamental Research Funds for Central Universities in China (Grant DUT15LK25) and the China Scholorship Council (CSC, No. 201506060121). The authors acknowledge the Supercomputer Center of Dalian University of Technology for providing computing resources

In this paper, we present a framework to infer the possible transmembrane transport of intracellular substances. Considering four key enzymes, a modified fourteen-dimensional nonlinear hybrid dynamic system is established to describe the microbial continuous culture with enzyme-catalytic and genetic regulation. A novel quantitative definition of biological robustness is proposed to characterize the system's resilience when system parameters were perturbed. It not only considers the expectation of system output data after parameter disturbance but also considers the influence of the variance of these data. In this way, the definition can be used as an objective function of the system identification model due to the lack of data on the concentration of intracellular substances. Then, we design a parallel computing method to solve the system identification model. Numerical results indicate that the most likely transmembrane mode of transport is active transport coupling with passive diffusion for glycerol and 1, 3-propanediol.

Keywords: Pathway identification, nonlinear hybrid dynamical system, robustness analysis, parameter identification, continuous culture, parallel optimization.
Mathematics Subject Classification: Primary: 90B30, 65C20; Secondary: 35B20.
Citation: Qi Yang, Lei Wang, Enmin Feng, Hongchao Yin, Zhilong Xiu. Identification and robustness analysis of nonlinear hybrid dynamical system of genetic regulation in continuous culture. Journal of Industrial & Management Optimization, 2020, 16 (2) : 579-599. doi: 10.3934/jimo.2018168
References:
[1]

G. Ardizzon, G. Cavazzini and G. Pavesi, Adaptive acceleration coefficients for a new search diversification strategy in particle swarm optimization algorithms, Information Sciences, 299 (2015), 337-378.  doi: 10.1016/j.ins.2014.12.024.  Google Scholar

[2]

H. Biebl, K. Menzel, A. P. Zeng and W. D. Deckwer, Microbial production of 1, 3-propanediol, Applied Microbiology and Biotechnology, 52 (1999), 289-297.  doi: 10.1007/s002530051523.  Google Scholar

[3]

M. R. Bonyadi and Z. Michalewicz, A locally convergent rotationally invariant particle swarm optimization algorithm, Swarm Intelligence, 8 (2014), 159-198.  doi: 10.1007/s11721-014-0095-1.  Google Scholar

[4]

A. Bryson and Y. C. Ho, Applied Optimal Control, Halsted Press, New York, 1975.  Google Scholar

[5]

G. Cheng, L. Wang, R. Loxton and Q. Lin, Robust optimal control of a microbial batch culture process, Journal of Optimization Theory and Applications, 167 (2015), 342-362.  doi: 10.1007/s10957-014-0654-z.  Google Scholar

[6]

C. X. Gao, Y. Lang, E. M. Feng and Z. L. Xiu, Nonlinear impulsive system of microbial production in fed-batch culture and its optimal control, Journal of Applied Mathematics and Computing, 19 (2005), 203-214.  doi: 10.1007/BF02935799.  Google Scholar

[7]

K. K. Gao, X. Zhang, E. M. Feng and Z. L. Xiu, Sensitivity analysis and parameter identification of nonlinear hybrid systems for glycerol transport mechanisms in continuous culture, Journal of Theoretical Biology, 347 (2014), 137-143.  doi: 10.1016/j.jtbi.2013.12.025.  Google Scholar

[8]

Y. J. Guo, E. M. Feng, L. Wang and Z. L. Xiu, Complex metabolic network of 1, 3-propanediol transport mechanisms and its system identification via biological robustness, Bioprocess and Biosystems Engineering, 37 (2014), 677-686.  doi: 10.1007/s00449-013-1037-9.  Google Scholar

[9]

S. Honda, T. Toraya and S. Fukui, In situ reactivation of glycerol-inactivated coenzyme B12-dependent enzymes, glycerol dehydratase and diol dehydratase, Journal of Bacteriology, 143 (1980), 1458-1465.   Google Scholar

[10]

C. Karakuzu, Fuzzy controller training using particle swarm optimization for nonlinear system control, ISA Transactions, 48 (2009), Page 245. doi: 10.1016/j.isatra.2008.11.003.  Google Scholar

[11]

J. Kennedy and R. C. Eberhart, Particle swarm optimization, in: Proceedings of the IEEE International Conference on Neural Networks, IEEE Press, Australia, (1995), 1942–1948. doi: 10.1109/ICNN.1995.488968.  Google Scholar

[12]

H. Kitano, Biological robustness, Nature Reviews Genetics, 5 (2004), 826-837.  doi: 10.1038/nrg1471.  Google Scholar

[13]

H. Kitano, Towards a theory of biological robustness, Molecular Systems Biology, 3 (2007), 137. doi: 10.1038/msb4100179.  Google Scholar

[14]

H. S. Lian, E. M. Feng, X. F. Li, J. X. Ye and Z. L. Xiu, Oscillatory behavior in microbial continuous culture with discrete time delay, Nonlinear Analysis: Real World Applications, 10 (2009), 2749-2757.  doi: 10.1016/j.nonrwa.2008.08.014.  Google Scholar

[15]

L. S. Jennings and K. L. Teo, A computational algorithm for functional inequality constrained optimization problems, Automatica, 26 (1990), 371-375.  doi: 10.1016/0005-1098(90)90131-Z.  Google Scholar

[16]

C. Y. Liu, Optimal control of a switched autonomous system with time delay arising in fed-batch processes, IMA Journal of Applied Mathematics, 80 (2015), 569-584.  doi: 10.1093/imamat/hxt053.  Google Scholar

[17]

C. Y. Liu, Z. H. Gong, K. L. Teo, R. Loxton and E. M. Feng, Bi-objective dynamic optimization of a nonlinear time-delay system in microbial batch process, Optimization Letters, 12 (2018), 1249-1264.  doi: 10.1007/s11590-016-1105-6.  Google Scholar

[18]

C. Y. Liu, Z. Gong, K. L. Teo, J. Sun and L. Caccetta, Robust multi-objective optimal switching control arising in 1, 3-propanediol microbial fed-batch process, Nonlinear Analysis: Hybrid Systems, 25 (2017), 1-20.  doi: 10.1016/j.nahs.2017.01.006.  Google Scholar

[19]

C. Y. Liu, Z. H. Gong and K. L. Teo, Robust parameter estimation for nonlinear multistage time-delay systems with noisy measurement data, Applied Mathematical Modelling, 53 (2018), 353-368.  doi: 10.1016/j.apm.2017.09.007.  Google Scholar

[20]

L. Liu, W. X. Liu and D. A. Cartes, Particle swarm optimization based parameter identification applied to permanent magnet synchronous motors, Engineering Applications of Artificial Intelligence, 21 (2008), 1092-1100.  doi: 10.1016/j.engappai.2007.10.002.  Google Scholar

[21]

X. H. Li, E. M. Feng and Z. L. Xiu, Stability and optimal control of microorganisms in continuous culture, Journal of Applied Mathematics and Computing, 22 (2006), 425-434.  doi: 10.1007/BF02896490.  Google Scholar

[22]

X. H. Li, J. J. Li, E. M. Feng and Z. L. Xiu, Discrete optimal control model and bound error for microbial continuous fermentation, Nonlinear Analysis: Real World Applications, 11 (2010), 131-138.  doi: 10.1016/j.nonrwa.2008.10.043.  Google Scholar

[23]

Q. Lin, R. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey, Journal of Industrial and Management Optimization, 10 (2014), 275-309.  doi: 10.3934/jimo.2014.10.275.  Google Scholar

[24]

M. Morohashi, A. E. Winn, M. T. Borisuk, H. Bolouri, J. Doyle and H. Kitano, Robustness as a measure of plausibility in models of biochemical networks, Journal of Theoretical Biology, 216 (2002), 19-30.  doi: 10.1006/jtbi.2002.2537.  Google Scholar

[25]

M. Perc and M. Marhl, Sensitivity and flexibility of regular and chaotic calcium oscillations, Biophysical Chemistry, 104 (2003), 509-522.  doi: 10.1016/S0301-4622(03)00038-3.  Google Scholar

[26]

J. F. Schutte, J. A. Reinbolt, B. J. Fregly, R. T. Haftka and A. D. George, Parallel global optimization with the particle swarm algorithm, International Journal for Numerical Methods in Engineering, 61 (2004), 2296-2315.  doi: 10.1002/nme.1149.  Google Scholar

[27]

Y. Q. Sun, W. T. Qi, H. Teng, Z. L. Xiu and A. P. Zeng, Mathematical modeling of glycerol fermentation by Klebsiella pneumoniae: Concerning enzyme-catalytic reductive pathway and transport of glycerol and 1, 3-propanediol across cell membrane, Biochemical Engineering Journal, 38 (2008), 22-32.  doi: 10.1016/j.bej.2007.06.002.  Google Scholar

[28]

Y. Q. Sun, J. X. Ye, X. J. Mu, H. Teng, E. M. Feng, A. P. Zeng and Z. L. Xiu, Nonlinear mathematical simulation and analysis of $\, dha\, $ regulon for glycerol metabolism in Klebsiella pneumoniae, Chinese Journal of Chemical Engineering, 20 (2012), 958-970.  doi: 10.1016/S1004-9541(12)60424-8.  Google Scholar

[29]

K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Long Scientific Technical, Essex, 1991.  Google Scholar

[30]

J. Wang, J. X. Ye, E. M. Feng, H. C. Yin and B. Tan, Complex metabolic network of glycerol fermentation by Klebsiella pneumoniae and its system identification via biological robustness, Nonlinear Analysis: Hybrid Systems, 5 (2011), 102-112.  doi: 10.1016/j.nahs.2010.10.002.  Google Scholar

[31]

J. Wang, J. X. Ye, H. C. Yin, E. M. Feng and L. Wang, Sensitivity analysis and identification of kinetic parameters in batch fermentation of glycerol, Journal of Computational and Applied Mathematics, 236 (2012), 2268-2276.  doi: 10.1016/j.cam.2011.11.015.  Google Scholar

[32]

L. Wang, G. M. Cheng, E. M. Feng, T. Su and Z. L. Xiu, Analysis and application of biological robustness as performance index in microbial fermentation, Applied Mathematical Modelling, 39 (2015), 2048-2055.  doi: 10.1016/j.apm.2014.10.022.  Google Scholar

[33]

L. Wang, J. L. Yuan, C. Wu and X. Wang, Practical algorithm for stochastic optimal control problem about microbial fermentation in batch culture, Optimization Letters, (2017), 1-15.  doi: 10.1007/s11590-017-1220-z.  Google Scholar

[34]

J. M. Whitacre, Biological robustness: Paradigms, mechanisms, and system principles, Frontiers in Genetics, 3 (2012), 1-15.  doi: 10.3389/fgene.2012.00067.  Google Scholar

[35]

U. Witt, R. J. Miiller, J. Augusta, H. Widdecke and W. D. Deckwer, Synthesis, properties and biodegradability of polyesters based on 1, 3-propanediol, Macromolecular Chemistry and Physics, 195 (1994), 793-802.  doi: 10.1002/macp.1994.021950235.  Google Scholar

[36]

Z. L. Xiu, A. P. Zeng and W. D. Deckwer, Multiplicity and stability analysis of microorganisms in continuous culture: Effects of metabolic overflow and growth inhibition, Biotechnology and Bioengineering, 57 (1998), 251-261.  doi: 10.1002/(SICI)1097-0290(19980205)57:3<251::AID-BIT1>3.0.CO;2-G.  Google Scholar

[37]

Z. L. Xiu, A. P. Zeng and A. J. Li, Mathematical modeling of kinetics and research on multiplicity of glycerol bioconversion to 1, 3-propanediol, Journal of Dalian University of Technology, 40 (2000), 428-433.  doi: 10.3846/13926292.2016.1142481.  Google Scholar

[38]

H. H. Yan, X. Zhang, J. X. Ye and E. M. Feng, Identification and robustness analysis of nonlinear hybrid dynamical system concerning glycerol transport mechanism, Computers and Chemical Engineering, 40 (2012), 171-180.  doi: 10.1016/j.compchemeng.2012.01.001.  Google Scholar

[39]

J. X. Ye, E. M. Feng, H. S. Lian and Z. L. Xiu, Existence of equilibrium points and stability of the nonlinear dynamical system in microbial continuous cultures, Applied Mathematics and Computation, 207 (2009), 307-318.  doi: 10.1016/j.amc.2008.10.046.  Google Scholar

[40]

J. X. Ye, A. Li and J. G. Zhai, A measure of concentration robustness in a biochemical reaction network and its application on system identification, Applied Mathematical Modelling, 58 (2018), 270-280.  doi: 10.1016/j.apm.2017.07.026.  Google Scholar

[41]

J. L. Yuan, X. Zhang, X. Zhu, E. M. Feng, H. C. Yin and Z. L. Xiu, Pathway identification using parallel optimization for a nonlinear hybrid system in batch culture, Nonlinear Analysis: Hybrid Systems, 15 (2015), 112-131.  doi: 10.1016/j.nahs.2014.08.004.  Google Scholar

[42]

J. L. Yuan, Y. D. Zhang, J. X. Ye, J. Xie and K. L. Teo, Robust parameter identification using parallel global optimization for a batch nonlinear enzyme-catalytic time-delayed process presenting metabolic discontinuities, Applied Mathematical Modelling, 46 (2017), 554-571.  doi: 10.1016/j.apm.2017.01.079.  Google Scholar

[43]

A. P. Zeng and W. D. Deckwer, A kinetic model for substrate and energy consumption of microbial growth under substrate-sufficient condition, Biotechnology Progress, 11 (1995), 71-79.  doi: 10.1021/bp00031a010.  Google Scholar

[44]

J. G. Zhai, J. X. Ye, L. Wang, E. M. Feng, H. C. Yin and Z. L. Xiu, Pathway identification using parallel optimization for a complex metabolic system in microbial continuous culture, Nonlinear Analysis: Real World Applications, 12 (2011), 2730-2741.  doi: 10.1016/j.nonrwa.2011.03.018.  Google Scholar

show all references

References:
[1]

G. Ardizzon, G. Cavazzini and G. Pavesi, Adaptive acceleration coefficients for a new search diversification strategy in particle swarm optimization algorithms, Information Sciences, 299 (2015), 337-378.  doi: 10.1016/j.ins.2014.12.024.  Google Scholar

[2]

H. Biebl, K. Menzel, A. P. Zeng and W. D. Deckwer, Microbial production of 1, 3-propanediol, Applied Microbiology and Biotechnology, 52 (1999), 289-297.  doi: 10.1007/s002530051523.  Google Scholar

[3]

M. R. Bonyadi and Z. Michalewicz, A locally convergent rotationally invariant particle swarm optimization algorithm, Swarm Intelligence, 8 (2014), 159-198.  doi: 10.1007/s11721-014-0095-1.  Google Scholar

[4]

A. Bryson and Y. C. Ho, Applied Optimal Control, Halsted Press, New York, 1975.  Google Scholar

[5]

G. Cheng, L. Wang, R. Loxton and Q. Lin, Robust optimal control of a microbial batch culture process, Journal of Optimization Theory and Applications, 167 (2015), 342-362.  doi: 10.1007/s10957-014-0654-z.  Google Scholar

[6]

C. X. Gao, Y. Lang, E. M. Feng and Z. L. Xiu, Nonlinear impulsive system of microbial production in fed-batch culture and its optimal control, Journal of Applied Mathematics and Computing, 19 (2005), 203-214.  doi: 10.1007/BF02935799.  Google Scholar

[7]

K. K. Gao, X. Zhang, E. M. Feng and Z. L. Xiu, Sensitivity analysis and parameter identification of nonlinear hybrid systems for glycerol transport mechanisms in continuous culture, Journal of Theoretical Biology, 347 (2014), 137-143.  doi: 10.1016/j.jtbi.2013.12.025.  Google Scholar

[8]

Y. J. Guo, E. M. Feng, L. Wang and Z. L. Xiu, Complex metabolic network of 1, 3-propanediol transport mechanisms and its system identification via biological robustness, Bioprocess and Biosystems Engineering, 37 (2014), 677-686.  doi: 10.1007/s00449-013-1037-9.  Google Scholar

[9]

S. Honda, T. Toraya and S. Fukui, In situ reactivation of glycerol-inactivated coenzyme B12-dependent enzymes, glycerol dehydratase and diol dehydratase, Journal of Bacteriology, 143 (1980), 1458-1465.   Google Scholar

[10]

C. Karakuzu, Fuzzy controller training using particle swarm optimization for nonlinear system control, ISA Transactions, 48 (2009), Page 245. doi: 10.1016/j.isatra.2008.11.003.  Google Scholar

[11]

J. Kennedy and R. C. Eberhart, Particle swarm optimization, in: Proceedings of the IEEE International Conference on Neural Networks, IEEE Press, Australia, (1995), 1942–1948. doi: 10.1109/ICNN.1995.488968.  Google Scholar

[12]

H. Kitano, Biological robustness, Nature Reviews Genetics, 5 (2004), 826-837.  doi: 10.1038/nrg1471.  Google Scholar

[13]

H. Kitano, Towards a theory of biological robustness, Molecular Systems Biology, 3 (2007), 137. doi: 10.1038/msb4100179.  Google Scholar

[14]

H. S. Lian, E. M. Feng, X. F. Li, J. X. Ye and Z. L. Xiu, Oscillatory behavior in microbial continuous culture with discrete time delay, Nonlinear Analysis: Real World Applications, 10 (2009), 2749-2757.  doi: 10.1016/j.nonrwa.2008.08.014.  Google Scholar

[15]

L. S. Jennings and K. L. Teo, A computational algorithm for functional inequality constrained optimization problems, Automatica, 26 (1990), 371-375.  doi: 10.1016/0005-1098(90)90131-Z.  Google Scholar

[16]

C. Y. Liu, Optimal control of a switched autonomous system with time delay arising in fed-batch processes, IMA Journal of Applied Mathematics, 80 (2015), 569-584.  doi: 10.1093/imamat/hxt053.  Google Scholar

[17]

C. Y. Liu, Z. H. Gong, K. L. Teo, R. Loxton and E. M. Feng, Bi-objective dynamic optimization of a nonlinear time-delay system in microbial batch process, Optimization Letters, 12 (2018), 1249-1264.  doi: 10.1007/s11590-016-1105-6.  Google Scholar

[18]

C. Y. Liu, Z. Gong, K. L. Teo, J. Sun and L. Caccetta, Robust multi-objective optimal switching control arising in 1, 3-propanediol microbial fed-batch process, Nonlinear Analysis: Hybrid Systems, 25 (2017), 1-20.  doi: 10.1016/j.nahs.2017.01.006.  Google Scholar

[19]

C. Y. Liu, Z. H. Gong and K. L. Teo, Robust parameter estimation for nonlinear multistage time-delay systems with noisy measurement data, Applied Mathematical Modelling, 53 (2018), 353-368.  doi: 10.1016/j.apm.2017.09.007.  Google Scholar

[20]

L. Liu, W. X. Liu and D. A. Cartes, Particle swarm optimization based parameter identification applied to permanent magnet synchronous motors, Engineering Applications of Artificial Intelligence, 21 (2008), 1092-1100.  doi: 10.1016/j.engappai.2007.10.002.  Google Scholar

[21]

X. H. Li, E. M. Feng and Z. L. Xiu, Stability and optimal control of microorganisms in continuous culture, Journal of Applied Mathematics and Computing, 22 (2006), 425-434.  doi: 10.1007/BF02896490.  Google Scholar

[22]

X. H. Li, J. J. Li, E. M. Feng and Z. L. Xiu, Discrete optimal control model and bound error for microbial continuous fermentation, Nonlinear Analysis: Real World Applications, 11 (2010), 131-138.  doi: 10.1016/j.nonrwa.2008.10.043.  Google Scholar

[23]

Q. Lin, R. Loxton and K. L. Teo, The control parameterization method for nonlinear optimal control: A survey, Journal of Industrial and Management Optimization, 10 (2014), 275-309.  doi: 10.3934/jimo.2014.10.275.  Google Scholar

[24]

M. Morohashi, A. E. Winn, M. T. Borisuk, H. Bolouri, J. Doyle and H. Kitano, Robustness as a measure of plausibility in models of biochemical networks, Journal of Theoretical Biology, 216 (2002), 19-30.  doi: 10.1006/jtbi.2002.2537.  Google Scholar

[25]

M. Perc and M. Marhl, Sensitivity and flexibility of regular and chaotic calcium oscillations, Biophysical Chemistry, 104 (2003), 509-522.  doi: 10.1016/S0301-4622(03)00038-3.  Google Scholar

[26]

J. F. Schutte, J. A. Reinbolt, B. J. Fregly, R. T. Haftka and A. D. George, Parallel global optimization with the particle swarm algorithm, International Journal for Numerical Methods in Engineering, 61 (2004), 2296-2315.  doi: 10.1002/nme.1149.  Google Scholar

[27]

Y. Q. Sun, W. T. Qi, H. Teng, Z. L. Xiu and A. P. Zeng, Mathematical modeling of glycerol fermentation by Klebsiella pneumoniae: Concerning enzyme-catalytic reductive pathway and transport of glycerol and 1, 3-propanediol across cell membrane, Biochemical Engineering Journal, 38 (2008), 22-32.  doi: 10.1016/j.bej.2007.06.002.  Google Scholar

[28]

Y. Q. Sun, J. X. Ye, X. J. Mu, H. Teng, E. M. Feng, A. P. Zeng and Z. L. Xiu, Nonlinear mathematical simulation and analysis of $\, dha\, $ regulon for glycerol metabolism in Klebsiella pneumoniae, Chinese Journal of Chemical Engineering, 20 (2012), 958-970.  doi: 10.1016/S1004-9541(12)60424-8.  Google Scholar

[29]

K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Long Scientific Technical, Essex, 1991.  Google Scholar

[30]

J. Wang, J. X. Ye, E. M. Feng, H. C. Yin and B. Tan, Complex metabolic network of glycerol fermentation by Klebsiella pneumoniae and its system identification via biological robustness, Nonlinear Analysis: Hybrid Systems, 5 (2011), 102-112.  doi: 10.1016/j.nahs.2010.10.002.  Google Scholar

[31]

J. Wang, J. X. Ye, H. C. Yin, E. M. Feng and L. Wang, Sensitivity analysis and identification of kinetic parameters in batch fermentation of glycerol, Journal of Computational and Applied Mathematics, 236 (2012), 2268-2276.  doi: 10.1016/j.cam.2011.11.015.  Google Scholar

[32]

L. Wang, G. M. Cheng, E. M. Feng, T. Su and Z. L. Xiu, Analysis and application of biological robustness as performance index in microbial fermentation, Applied Mathematical Modelling, 39 (2015), 2048-2055.  doi: 10.1016/j.apm.2014.10.022.  Google Scholar

[33]

L. Wang, J. L. Yuan, C. Wu and X. Wang, Practical algorithm for stochastic optimal control problem about microbial fermentation in batch culture, Optimization Letters, (2017), 1-15.  doi: 10.1007/s11590-017-1220-z.  Google Scholar

[34]

J. M. Whitacre, Biological robustness: Paradigms, mechanisms, and system principles, Frontiers in Genetics, 3 (2012), 1-15.  doi: 10.3389/fgene.2012.00067.  Google Scholar

[35]

U. Witt, R. J. Miiller, J. Augusta, H. Widdecke and W. D. Deckwer, Synthesis, properties and biodegradability of polyesters based on 1, 3-propanediol, Macromolecular Chemistry and Physics, 195 (1994), 793-802.  doi: 10.1002/macp.1994.021950235.  Google Scholar

[36]

Z. L. Xiu, A. P. Zeng and W. D. Deckwer, Multiplicity and stability analysis of microorganisms in continuous culture: Effects of metabolic overflow and growth inhibition, Biotechnology and Bioengineering, 57 (1998), 251-261.  doi: 10.1002/(SICI)1097-0290(19980205)57:3<251::AID-BIT1>3.0.CO;2-G.  Google Scholar

[37]

Z. L. Xiu, A. P. Zeng and A. J. Li, Mathematical modeling of kinetics and research on multiplicity of glycerol bioconversion to 1, 3-propanediol, Journal of Dalian University of Technology, 40 (2000), 428-433.  doi: 10.3846/13926292.2016.1142481.  Google Scholar

[38]

H. H. Yan, X. Zhang, J. X. Ye and E. M. Feng, Identification and robustness analysis of nonlinear hybrid dynamical system concerning glycerol transport mechanism, Computers and Chemical Engineering, 40 (2012), 171-180.  doi: 10.1016/j.compchemeng.2012.01.001.  Google Scholar

[39]

J. X. Ye, E. M. Feng, H. S. Lian and Z. L. Xiu, Existence of equilibrium points and stability of the nonlinear dynamical system in microbial continuous cultures, Applied Mathematics and Computation, 207 (2009), 307-318.  doi: 10.1016/j.amc.2008.10.046.  Google Scholar

[40]

J. X. Ye, A. Li and J. G. Zhai, A measure of concentration robustness in a biochemical reaction network and its application on system identification, Applied Mathematical Modelling, 58 (2018), 270-280.  doi: 10.1016/j.apm.2017.07.026.  Google Scholar

[41]

J. L. Yuan, X. Zhang, X. Zhu, E. M. Feng, H. C. Yin and Z. L. Xiu, Pathway identification using parallel optimization for a nonlinear hybrid system in batch culture, Nonlinear Analysis: Hybrid Systems, 15 (2015), 112-131.  doi: 10.1016/j.nahs.2014.08.004.  Google Scholar

[42]

J. L. Yuan, Y. D. Zhang, J. X. Ye, J. Xie and K. L. Teo, Robust parameter identification using parallel global optimization for a batch nonlinear enzyme-catalytic time-delayed process presenting metabolic discontinuities, Applied Mathematical Modelling, 46 (2017), 554-571.  doi: 10.1016/j.apm.2017.01.079.  Google Scholar

[43]

A. P. Zeng and W. D. Deckwer, A kinetic model for substrate and energy consumption of microbial growth under substrate-sufficient condition, Biotechnology Progress, 11 (1995), 71-79.  doi: 10.1021/bp00031a010.  Google Scholar

[44]

J. G. Zhai, J. X. Ye, L. Wang, E. M. Feng, H. C. Yin and Z. L. Xiu, Pathway identification using parallel optimization for a complex metabolic system in microbial continuous culture, Nonlinear Analysis: Real World Applications, 12 (2011), 2730-2741.  doi: 10.1016/j.nonrwa.2011.03.018.  Google Scholar

Figure 1.  Main pathway of 1, 3-propanediol biosynthesis in K. pneumoniae. [28]. Abbreviations: GDH-glycerol dehydrogenase; GDHt-glycerol dehydratase; DHAK Ⅰ-dihydroxyacetone kinases (ATP dependent); DHAK Ⅱ-dihydroxyacetone kinases (PEP dependent); PDOR-1, 3-propanediol oxydoreductase; HOR-hypothetical oxydoreductase; PDH-pyruvate dehydrogenase; PFL-pyruvate format lyase; DHA-dihydroxyacetone; DHAP-dihydroxyacetone phosphate; PEP-phosphoenolpyruvate; Pyr-pyruvate; HAc-acetate; EtOH-ethanol; 3-HPA-3-hydroxypropionaldehyde; 1, 3-PD-1, 3-propanediol; RP-regulatory protein
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Figure 2.  The simulated results of three extracellular concentrations ($x_{1}$-biomass, $x_2$-glycerol, $x_3$-1, 3-PD) under three different initial conditions, i.e. $(D, C_{S0}) = (0.08, 435), ~(D, C_{S0}) = (0.08, 152)$ and $(D, C_{S0}) = (0.23, 375.7)$
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Figure 3.  The simulated results of first three intracellular concentrations ($x_{6}$-glycerol, $x_7$-1, 3-PD, $x_8$-3-HPA) under three different initial conditions, i.e. $(D, C_{S0}) = (0.08, 435), ~(D, C_{S0}) = (0.08, 152)$ and $(D, C_{S0}) = (0.23, 375.7)$
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Figure 4.  The simulated results of six intracellular concentrations ($x_9$-mR, $x_{10}$-R, $x_{11}$-mGDHt, $x_{12}$-DGHt, $x_{13}$-mPDOR, $x_{14}$-PDOR) under three different initial conditions, i.e. $(D, C_{S0}) = (0.08, 435), ~(D, C_{S0}) = (0.08, 152)$ and $(D, C_{S0}) = (0.23, 375.7)$
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Table 1.  Transport mechanisms of glycerol and 1, 3-PD of metabolic system $NHDS(l_k)$, corresponding to parameter vector $l_k , k\in I_9$. Abbreviations: A, active transport; P, passive diffusion; AP, passive diffusion coupled with active transport
NHDS$(l_k)\rightarrow l_k$Glycerol1, 3-PD
NHDS$(l_1)\rightarrow l_1:(0, 1, 0, 1)$PP
NHDS$(l_2)\rightarrow l_2:(0, 1, 1, 0)$PA
NHDS$(l_3)\rightarrow l_1:(0, 1, 1, 1)$PAP
NHDS$(l_4)\rightarrow l_4:(1, 0, 0, 1)$AP
NHDS$(l_5)\rightarrow l_5:(1, 0, 1, 0)$AA
NHDS$(l_6)\rightarrow l_6:(1, 0, 1, 1)$AAP
NHDS$(l_7)\rightarrow l_7:(1, 1, 0, 1)$APP
NHDS$(l_8)\rightarrow l_8:(1, 1, 1, 0)$APA
NHDS$(l_9)\rightarrow l_9:(1, 1, 1, 1)$APAP
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NHDS$(l_k)\rightarrow l_k$Glycerol1, 3-PD
NHDS$(l_1)\rightarrow l_1:(0, 1, 0, 1)$PP
NHDS$(l_2)\rightarrow l_2:(0, 1, 1, 0)$PA
NHDS$(l_3)\rightarrow l_1:(0, 1, 1, 1)$PAP
NHDS$(l_4)\rightarrow l_4:(1, 0, 0, 1)$AP
NHDS$(l_5)\rightarrow l_5:(1, 0, 1, 0)$AA
NHDS$(l_6)\rightarrow l_6:(1, 0, 1, 1)$AAP
NHDS$(l_7)\rightarrow l_7:(1, 1, 0, 1)$APP
NHDS$(l_8)\rightarrow l_8:(1, 1, 1, 0)$APA
NHDS$(l_9)\rightarrow l_9:(1, 1, 1, 1)$APAP
Table 2.  The Parameters in Kinetic Models (16)-(18)
$i$$m_i$$Y_i$$\Delta q_i$ $K_i^*$
22.200.008228.5811.43
4-0.9733.075.7485.71
55.2611.66--
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$i$$m_i$$Y_i$$\Delta q_i$ $K_i^*$
22.200.008228.5811.43
4-0.9733.075.7485.71
55.2611.66--
Table 3.  The computational values of robustness index for system $NHDS$ corresponding to parameter vector $l_k , k\in I_9$
$k$123456789
$J(p^*, j, l_k)$1.748$+\infty$15.267 $+\infty$3.65215.689$+\infty$0.6570.1258
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$k$123456789
$J(p^*, j, l_k)$1.748$+\infty$15.267 $+\infty$3.65215.689$+\infty$0.6570.1258
Table 4.  The computational values of the optimal parameters under three experiments
$(D, C_{s0})$ $p^*(l_9)$
(0.08, 435)(81.7276, 4.97131, 1075.54, 99.9598, 1.89429, 23.9368, 0.362061,
28.1892, 8.95164, $7.79472\times10^{-4}$, 40.415, 41.1295, 27.8876, 1.12982,
1.50452, 10.0365, $4.69745\times10^{-5}$, 949.377, 9.47578, 25.4648, 32.6392,
12.7944, 0.1133, 0.529596, 191.37, 1.75482, 20.1505, 9.64496, 14.0072,
0.8037, 13.9279$)^T$
(0.08, 152)(89.7339, 2.31116, 680.762, 77.858, 2.64026, 68.5132, 0.296838,
18.2558, 9.80988, $9.56009\times10^{-4}$, 40.1752, 49.5181, 23.1997, 1.22681,
1.95972, 17.3442, $8.18616\times10^{-5}$, 998.73, 13.4516, 23.1863, 42.6794,
8.60437, 0.058844, 0.550928, 262.521, 1.69241, 25.8673, 5.83124, 9.62714,
22.5502, 11.171$)^T$
(0.23, 375.7)(87.3566, 4.11554, 599.756, 86.7367, 1.47777, 94.5825, 0.269305,
34.3741, 9.01509, $8.30467\times10^{-4}$, 33.6429, 48.385, 27.4457, 2.66168,
1.97522, 10.6856, $4.51274\times10^{-5}$, 1012.51, 16.0382, 25.8691, 40.1752,
8.52086, 0.196155, 0.790479, 315.66, 1.84771, 18.6017, 6.09894, 8.43732,
34.01, 15.35$)^T$
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$(D, C_{s0})$ $p^*(l_9)$
(0.08, 435)(81.7276, 4.97131, 1075.54, 99.9598, 1.89429, 23.9368, 0.362061,
28.1892, 8.95164, $7.79472\times10^{-4}$, 40.415, 41.1295, 27.8876, 1.12982,
1.50452, 10.0365, $4.69745\times10^{-5}$, 949.377, 9.47578, 25.4648, 32.6392,
12.7944, 0.1133, 0.529596, 191.37, 1.75482, 20.1505, 9.64496, 14.0072,
0.8037, 13.9279$)^T$
(0.08, 152)(89.7339, 2.31116, 680.762, 77.858, 2.64026, 68.5132, 0.296838,
18.2558, 9.80988, $9.56009\times10^{-4}$, 40.1752, 49.5181, 23.1997, 1.22681,
1.95972, 17.3442, $8.18616\times10^{-5}$, 998.73, 13.4516, 23.1863, 42.6794,
8.60437, 0.058844, 0.550928, 262.521, 1.69241, 25.8673, 5.83124, 9.62714,
22.5502, 11.171$)^T$
(0.23, 375.7)(87.3566, 4.11554, 599.756, 86.7367, 1.47777, 94.5825, 0.269305,
34.3741, 9.01509, $8.30467\times10^{-4}$, 33.6429, 48.385, 27.4457, 2.66168,
1.97522, 10.6856, $4.51274\times10^{-5}$, 1012.51, 16.0382, 25.8691, 40.1752,
8.52086, 0.196155, 0.790479, 315.66, 1.84771, 18.6017, 6.09894, 8.43732,
34.01, 15.35$)^T$
Table 5.  The experimental data y and the numerical results $x^*$ for three experiments
$(D, C_{s0})$$y$ $x^*$
(0.08, 435)(2.6, 0.22, 136, (2.66078, 0.199219, 151.787, 56.1728, 204.64,
55, 188$)^T$0.189992, 48.8683, 0.0906137, 3.18097, 0.00879915,
0.335116, 0.694374, 0.043648, 0.0648233$)^T$
(0.08, 152)(1.1, 0.09, 53, (1.15502, 0.0720033, 53.1286, 24.2607, 88.8432,
28, 63$)^T$0.0605281, 47.8053, 0.130965, 5.17053, 0.0108471,
0.405015, 1.77233, 0.073951, 0.148219$)^T$
(0.23, 375.7)(2.86, 0.91, 148.5, (2.858, 0.893436, 150.942, 83.1966, 98.1893
65.7, 101.6$)^T$0.869124, 58.9767, 0.240652, 1.59739, 0.00399712,
0.327636, 0.962971, 0.0785403, 0.171447 $)^T$
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$(D, C_{s0})$$y$ $x^*$
(0.08, 435)(2.6, 0.22, 136, (2.66078, 0.199219, 151.787, 56.1728, 204.64,
55, 188$)^T$0.189992, 48.8683, 0.0906137, 3.18097, 0.00879915,
0.335116, 0.694374, 0.043648, 0.0648233$)^T$
(0.08, 152)(1.1, 0.09, 53, (1.15502, 0.0720033, 53.1286, 24.2607, 88.8432,
28, 63$)^T$0.0605281, 47.8053, 0.130965, 5.17053, 0.0108471,
0.405015, 1.77233, 0.073951, 0.148219$)^T$
(0.23, 375.7)(2.86, 0.91, 148.5, (2.858, 0.893436, 150.942, 83.1966, 98.1893
65.7, 101.6$)^T$0.869124, 58.9767, 0.240652, 1.59739, 0.00399712,
0.327636, 0.962971, 0.0785403, 0.171447 $)^T$
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