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A two-warehouse probabilistic model with price discount on backorders under two levels of trade-credit policy
Identification and robustness analysis of nonlinear hybrid dynamical system of genetic regulation in continuous culture
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 |
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.
References:
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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. |
[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. |
[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. |
[4] |
A. Bryson and Y. C. Ho,
Applied Optimal Control, Halsted Press, New York, 1975. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
H. Kitano,
Biological robustness, Nature Reviews Genetics, 5 (2004), 826-837.
doi: 10.1038/nrg1471. |
[13] |
H. Kitano, Towards a theory of biological robustness,
Molecular Systems Biology, 3 (2007), 137.
doi: 10.1038/msb4100179. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[29] |
K. L. Teo, C. J. Goh and K. H. Wong,
A Unified Computational Approach to Optimal Control Problems, Long Scientific Technical, Essex, 1991. |
[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. |
[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. |
[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. |
[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. |
[34] |
J. M. Whitacre,
Biological robustness: Paradigms, mechanisms, and system principles, Frontiers in Genetics, 3 (2012), 1-15.
doi: 10.3389/fgene.2012.00067. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[4] |
A. Bryson and Y. C. Ho,
Applied Optimal Control, Halsted Press, New York, 1975. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
H. Kitano,
Biological robustness, Nature Reviews Genetics, 5 (2004), 826-837.
doi: 10.1038/nrg1471. |
[13] |
H. Kitano, Towards a theory of biological robustness,
Molecular Systems Biology, 3 (2007), 137.
doi: 10.1038/msb4100179. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[29] |
K. L. Teo, C. J. Goh and K. H. Wong,
A Unified Computational Approach to Optimal Control Problems, Long Scientific Technical, Essex, 1991. |
[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. |
[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. |
[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. |
[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. |
[34] |
J. M. Whitacre,
Biological robustness: Paradigms, mechanisms, and system principles, Frontiers in Genetics, 3 (2012), 1-15.
doi: 10.3389/fgene.2012.00067. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |




NHDS | Glycerol | 1, 3-PD |
NHDS | P | P |
NHDS | P | A |
NHDS | P | AP |
NHDS | A | P |
NHDS | A | A |
NHDS | A | AP |
NHDS | AP | P |
NHDS | AP | A |
NHDS | AP | AP |
NHDS | Glycerol | 1, 3-PD |
NHDS | P | P |
NHDS | P | A |
NHDS | P | AP |
NHDS | A | P |
NHDS | A | A |
NHDS | A | AP |
NHDS | AP | P |
NHDS | AP | A |
NHDS | AP | AP |
| | |||
2 | 2.20 | 0.0082 | 28.58 | 11.43 |
4 | -0.97 | 33.07 | 5.74 | 85.71 |
5 | 5.26 | 11.66 | - | - |
| | |||
2 | 2.20 | 0.0082 | 28.58 | 11.43 |
4 | -0.97 | 33.07 | 5.74 | 85.71 |
5 | 5.26 | 11.66 | - | - |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 1.748 | 15.267 | | 3.6521 | 5.689 | 0.657 | 0.1258 |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 1.748 | 15.267 | | 3.6521 | 5.689 | 0.657 | 0.1258 |
| |
(0.08, 435) | (81.7276, 4.97131, 1075.54, 99.9598, 1.89429, 23.9368, 0.362061, |
28.1892, 8.95164, | |
1.50452, 10.0365, | |
12.7944, 0.1133, 0.529596, 191.37, 1.75482, 20.1505, 9.64496, 14.0072, | |
0.8037, 13.9279 | |
(0.08, 152) | (89.7339, 2.31116, 680.762, 77.858, 2.64026, 68.5132, 0.296838, |
18.2558, 9.80988, | |
1.95972, 17.3442, | |
8.60437, 0.058844, 0.550928, 262.521, 1.69241, 25.8673, 5.83124, 9.62714, | |
22.5502, 11.171 | |
(0.23, 375.7) | (87.3566, 4.11554, 599.756, 86.7367, 1.47777, 94.5825, 0.269305, |
34.3741, 9.01509, | |
1.97522, 10.6856, | |
8.52086, 0.196155, 0.790479, 315.66, 1.84771, 18.6017, 6.09894, 8.43732, | |
34.01, 15.35 |
| |
(0.08, 435) | (81.7276, 4.97131, 1075.54, 99.9598, 1.89429, 23.9368, 0.362061, |
28.1892, 8.95164, | |
1.50452, 10.0365, | |
12.7944, 0.1133, 0.529596, 191.37, 1.75482, 20.1505, 9.64496, 14.0072, | |
0.8037, 13.9279 | |
(0.08, 152) | (89.7339, 2.31116, 680.762, 77.858, 2.64026, 68.5132, 0.296838, |
18.2558, 9.80988, | |
1.95972, 17.3442, | |
8.60437, 0.058844, 0.550928, 262.521, 1.69241, 25.8673, 5.83124, 9.62714, | |
22.5502, 11.171 | |
(0.23, 375.7) | (87.3566, 4.11554, 599.756, 86.7367, 1.47777, 94.5825, 0.269305, |
34.3741, 9.01509, | |
1.97522, 10.6856, | |
8.52086, 0.196155, 0.790479, 315.66, 1.84771, 18.6017, 6.09894, 8.43732, | |
34.01, 15.35 |
| | |
(0.08, 435) | (2.6, 0.22, 136, | (2.66078, 0.199219, 151.787, 56.1728, 204.64, |
55, 188 | 0.189992, 48.8683, 0.0906137, 3.18097, 0.00879915, | |
0.335116, 0.694374, 0.043648, 0.0648233 | ||
(0.08, 152) | (1.1, 0.09, 53, | (1.15502, 0.0720033, 53.1286, 24.2607, 88.8432, |
28, 63 | 0.0605281, 47.8053, 0.130965, 5.17053, 0.0108471, | |
0.405015, 1.77233, 0.073951, 0.148219 | ||
(0.23, 375.7) | (2.86, 0.91, 148.5, | (2.858, 0.893436, 150.942, 83.1966, 98.1893 |
65.7, 101.6 | 0.869124, 58.9767, 0.240652, 1.59739, 0.00399712, | |
0.327636, 0.962971, 0.0785403, 0.171447 |
| | |
(0.08, 435) | (2.6, 0.22, 136, | (2.66078, 0.199219, 151.787, 56.1728, 204.64, |
55, 188 | 0.189992, 48.8683, 0.0906137, 3.18097, 0.00879915, | |
0.335116, 0.694374, 0.043648, 0.0648233 | ||
(0.08, 152) | (1.1, 0.09, 53, | (1.15502, 0.0720033, 53.1286, 24.2607, 88.8432, |
28, 63 | 0.0605281, 47.8053, 0.130965, 5.17053, 0.0108471, | |
0.405015, 1.77233, 0.073951, 0.148219 | ||
(0.23, 375.7) | (2.86, 0.91, 148.5, | (2.858, 0.893436, 150.942, 83.1966, 98.1893 |
65.7, 101.6 | 0.869124, 58.9767, 0.240652, 1.59739, 0.00399712, | |
0.327636, 0.962971, 0.0785403, 0.171447 |
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