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2015, 5(1): 59-69. doi: 10.3934/naco.2015.5.59

## Optimal dilution strategy for a microbial continuous culture based on the biological robustness

 1 School of Mathematics and Statistics Science, Ludong University, Yantai, Shandong 264025, China, China 2 School of Mathematics and Computer Science, Fujian Normal University, Fuzhou, Fujian 350108, China

Received  January 2015 Revised  March 2015 Published  March 2015

A robust optimal parameter selection model is proposed to formulate the microbial continuous culture process of glycerol bio-dissimilation to 1,3-propanediol (1,3-PD), in which the dilution rate and the glycerol concentration in feed medium are taken as the optimization variables. In consideration of the uncertain factors that some system parameters may be changed with the change of the optimization variables, we establish a novelty mathematical model which is represented by an eight-dimensional nonlinear dynamical system with the unknown parameters. On the basis of biological robustness, we give a quantitative definition of robustness index. The concentration of 1,3-PD at the approximately steady-state time together with the robustness index are taken as the cost functional in consideration of uncertain metabolic mechanisms. A parallel particle swarm optimization -- optimal parameter selection algorithm (PPSO-OPSA) is constructed to find the optimal dilution rate and the feeding glycerol concentration. Numerical results show that, by employing the obtained optimal input strategy, not only the concentration of 1,3-PD at the approximately steady-state time can be increased considerably compared with the previous experimental results, but also the obtained optimal parameters are robust for the dynamical system.
Citation: Jingang Zhai, Guangmao Jiang, Jianxiong Ye. Optimal dilution strategy for a microbial continuous culture based on the biological robustness. Numerical Algebra, Control and Optimization, 2015, 5 (1) : 59-69. doi: 10.3934/naco.2015.5.59
##### References:
 [1] R. Eberhart and Y. Shi, Particle swarm optimization: Developments, applications and resources, in Proceedings of the 2001 Congress on Evolutionary Computation, 1 (2001), 81-86. [2] C. Karakuzu, Fuzzy controller training using particle swarm optimization for nonlinear system control, ISA transactions, 47 (2008), 229-239. [3] J. Kennedy and R. Eberhart, Particle swarm optimization, in Proceedings of IEEE International Conference on Neural Networks, 4 (1995), 1942-1948. [4] H. Kitano, Biological robustness, Nature Reviews Genetics, 5 (2004), 826-837. [5] H. Kitano, Towards a theory of biological robustness, Molecular systems biology, 3 (2007), Article 137. [6] B. I. Koh, A. D. George, R. T. Haftka and B. J. Fregly, Parallel asynchronous particle swarm optimization, International Journal for Numerical Methods in Engineering, 67 (2006), 578-595. [7] L. A. Laffend, V. Nagarajan and C. E. Nakamura, Bioconversion of a fermentable carbon source to 1,3-propanediol by a single microorganism, 1997, US Patent 5,686,276. [8] X. Li, E. Feng and Z. Xiu, Stability and optimal control of microorganisms in continuous culture, Journal of Applied Mathematics and Computing, 22 (2006), 425-434. doi: 10.1007/BF02896490. [9] H. Lian, E. Feng, X. Li, J. Ye and Z. 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. [10] C. Liu, Z. Gong, E. Feng and H. Yin, Optimal switching control of a fed-batch fermentation process, Journal of Global Optimization, 52 (2012), 265-280. doi: 10.1007/s10898-011-9663-8. [11] C. Liu, Z. Gong, B. Shen and E. Feng, Modelling and optimal control for a fed-batch fermentation process, Applied Mathematical Modelling, 37 (2013), 695-706. doi: 10.1016/j.apm.2012.02.044. [12] L. Liu, W. 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. [13] Y. Ma, Z. Xiu, L. Sun and E. Feng, Hopf bifurcation and chaos analysis of a microbial continuous culture model with time delay, International Journal of Nonlinear Sciences and Numerical Simulation, 7 (2006), 305-308. [14] C. E. Nakamura and G. M. Whited, Metabolic engineering for the microbial production of 1,3-propanediol, Current opinion in biotechnology, 14 (2003), 454-459. [15] B. Shen, C. Liu, J. Ye, E. Feng and Z. Xiu, Parameter identification and optimization algorithm in microbial continuous culture, Applied Mathematical Modelling, 36 (2012), 585-595. doi: 10.1016/j.apm.2011.07.031. [16] 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. [17] J. Wang, Q. Sun and E. Feng, Modelling and properties of a nonlinear autonomous switching system in fed-batch culture of glycerol, Communications in Nonlinear Science and Numerical Simulation, 11 (2012), 4446-4454. doi: 10.1016/j.cnsns.2012.03.031. [18] J. Wang, J. Ye, E. Feng, H. 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. [19] L. Wang, E. Feng, J. Ye and Z. Xiu, Modeling and properties of nonlinear stochastic dynamical system of continuous culture, Complex Sciences, 4 (2009), 458-466. [20] L. Wang, Z. Xiu, Z. Gong and E. Feng, Modeling and parameter identification for multistage simulation of microbial bioconversion in batch culture, International Journal of Biomathematics, 5 (2012), 177-188. doi: 10.1142/S179352451100174X. [21] S. Wang and E. Feng, Stability of nonlinear microbial bioconversion system concerning glycerol's active transport and 1,3-pd's passive transport, Nonlinear Analysis: Real World Applications, 11 (2010), 3501-3511. doi: 10.1016/j.nonrwa.2009.12.011. [22] Z. L. Xiu, A. P. Zeng and L. J. An, 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. [23] J. Ye, E. Feng, H. Lian and Z. 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. [24] J. Ye, E. Feng, H. Yin and Z. Xiu, Modelling and well-posedness of a nonlinear hybrid system in fed-batch production of 1,3-propanediol with open loop glycerol input and ph logic control, Nonlinear Analysis: Real World Applications, 12 (2011), 364-376. doi: 10.1016/j.nonrwa.2010.06.022. [25] J. Ye, Y. Zhang, E. Feng, Z. Xiu and H. Yin, Nonlinear hybrid system and parameter identification of microbial fed-batch culture with open loop glycerol input and ph logic control, Applied Mathematical Modelling, 36 (2012), 357-369. doi: 10.1016/j.apm.2011.05.059. [26] A. P. Zeng, A. Ross, H. Biebl, C. Tag, B.Günzel and W. D. Deckwer, Multiple product inhibition and growth modeling of clostridium butyricum and klebsiella pneumoniae in glycerol fermentation, Biotechnology and bioengineering, 44 (1994), 902-911. [27] A. P. Zeng and H. Biebl, Bulk chemicals from biotechnology: The case of 1,3-propanediol production and the new trends, Tools and Applications of Biochemical Engineering Science, 74 (2002), 239-259. [28] J. Zhai, J. Ye, L. Wang, E. Feng, H. Yin and Z. 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. [29] Y. Zhang, E. Feng and Z. Xiu, Robust analysis of hybrid dynamical systems for 1,3-propanediol transport mechanisms in microbial continuous fermentation, Mathematical and Computer Modelling, 54 (2011), 3164-3171. doi: 10.1016/j.mcm.2011.08.010.

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##### References:
 [1] R. Eberhart and Y. Shi, Particle swarm optimization: Developments, applications and resources, in Proceedings of the 2001 Congress on Evolutionary Computation, 1 (2001), 81-86. [2] C. Karakuzu, Fuzzy controller training using particle swarm optimization for nonlinear system control, ISA transactions, 47 (2008), 229-239. [3] J. Kennedy and R. Eberhart, Particle swarm optimization, in Proceedings of IEEE International Conference on Neural Networks, 4 (1995), 1942-1948. [4] H. Kitano, Biological robustness, Nature Reviews Genetics, 5 (2004), 826-837. [5] H. Kitano, Towards a theory of biological robustness, Molecular systems biology, 3 (2007), Article 137. [6] B. I. Koh, A. D. George, R. T. Haftka and B. J. Fregly, Parallel asynchronous particle swarm optimization, International Journal for Numerical Methods in Engineering, 67 (2006), 578-595. [7] L. A. Laffend, V. Nagarajan and C. E. Nakamura, Bioconversion of a fermentable carbon source to 1,3-propanediol by a single microorganism, 1997, US Patent 5,686,276. [8] X. Li, E. Feng and Z. Xiu, Stability and optimal control of microorganisms in continuous culture, Journal of Applied Mathematics and Computing, 22 (2006), 425-434. doi: 10.1007/BF02896490. [9] H. Lian, E. Feng, X. Li, J. Ye and Z. 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. [10] C. Liu, Z. Gong, E. Feng and H. Yin, Optimal switching control of a fed-batch fermentation process, Journal of Global Optimization, 52 (2012), 265-280. doi: 10.1007/s10898-011-9663-8. [11] C. Liu, Z. Gong, B. Shen and E. Feng, Modelling and optimal control for a fed-batch fermentation process, Applied Mathematical Modelling, 37 (2013), 695-706. doi: 10.1016/j.apm.2012.02.044. [12] L. Liu, W. 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. [13] Y. Ma, Z. Xiu, L. Sun and E. Feng, Hopf bifurcation and chaos analysis of a microbial continuous culture model with time delay, International Journal of Nonlinear Sciences and Numerical Simulation, 7 (2006), 305-308. [14] C. E. Nakamura and G. M. Whited, Metabolic engineering for the microbial production of 1,3-propanediol, Current opinion in biotechnology, 14 (2003), 454-459. [15] B. Shen, C. Liu, J. Ye, E. Feng and Z. Xiu, Parameter identification and optimization algorithm in microbial continuous culture, Applied Mathematical Modelling, 36 (2012), 585-595. doi: 10.1016/j.apm.2011.07.031. [16] 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. [17] J. Wang, Q. Sun and E. Feng, Modelling and properties of a nonlinear autonomous switching system in fed-batch culture of glycerol, Communications in Nonlinear Science and Numerical Simulation, 11 (2012), 4446-4454. doi: 10.1016/j.cnsns.2012.03.031. [18] J. Wang, J. Ye, E. Feng, H. 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. [19] L. Wang, E. Feng, J. Ye and Z. Xiu, Modeling and properties of nonlinear stochastic dynamical system of continuous culture, Complex Sciences, 4 (2009), 458-466. [20] L. Wang, Z. Xiu, Z. Gong and E. Feng, Modeling and parameter identification for multistage simulation of microbial bioconversion in batch culture, International Journal of Biomathematics, 5 (2012), 177-188. doi: 10.1142/S179352451100174X. [21] S. Wang and E. Feng, Stability of nonlinear microbial bioconversion system concerning glycerol's active transport and 1,3-pd's passive transport, Nonlinear Analysis: Real World Applications, 11 (2010), 3501-3511. doi: 10.1016/j.nonrwa.2009.12.011. [22] Z. L. Xiu, A. P. Zeng and L. J. An, 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. [23] J. Ye, E. Feng, H. Lian and Z. 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. [24] J. Ye, E. Feng, H. Yin and Z. Xiu, Modelling and well-posedness of a nonlinear hybrid system in fed-batch production of 1,3-propanediol with open loop glycerol input and ph logic control, Nonlinear Analysis: Real World Applications, 12 (2011), 364-376. doi: 10.1016/j.nonrwa.2010.06.022. [25] J. Ye, Y. Zhang, E. Feng, Z. Xiu and H. Yin, Nonlinear hybrid system and parameter identification of microbial fed-batch culture with open loop glycerol input and ph logic control, Applied Mathematical Modelling, 36 (2012), 357-369. doi: 10.1016/j.apm.2011.05.059. [26] A. P. Zeng, A. Ross, H. Biebl, C. Tag, B.Günzel and W. D. Deckwer, Multiple product inhibition and growth modeling of clostridium butyricum and klebsiella pneumoniae in glycerol fermentation, Biotechnology and bioengineering, 44 (1994), 902-911. [27] A. P. Zeng and H. Biebl, Bulk chemicals from biotechnology: The case of 1,3-propanediol production and the new trends, Tools and Applications of Biochemical Engineering Science, 74 (2002), 239-259. [28] J. Zhai, J. Ye, L. Wang, E. Feng, H. Yin and Z. 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. [29] Y. Zhang, E. Feng and Z. Xiu, Robust analysis of hybrid dynamical systems for 1,3-propanediol transport mechanisms in microbial continuous fermentation, Mathematical and Computer Modelling, 54 (2011), 3164-3171. doi: 10.1016/j.mcm.2011.08.010.
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