Abstract
This paper considers the microbial batch culture for producing 1,3-propanediol(1,3-PD) via glycerol disproportionation. Due to the nature of the fractional order operations, a novel fractional order model, which is based upon the original ordinary differential dynamic system, is introduced to describe the complex bioprocess in a more accurate manner. Existence and uniqueness of solutions to the novel fractional order system and the continuity of solutions with respect to the parameters are discussed respectively. In addition, a parameter identification problem of the system is presented, and a particle swarm optimization algorithm is constructed to solve it. Finally, the conclusion is drawn by numerical simulations.
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Liouville, J.: Mmoire sur quelques questions de gomtrie et de mcanique, et sur un nouveau genre de calcul pour rsoudre ces questions, pp. 1-69. (1832)
Letnikov, A.V.: Theory of differentiation with an arbitrary index. Math. Sb 3, 1–66 (1868)
Hardy, G.H., Littlewood, J.E.: Some properties of fractional integrals. I. Math. Z. 34(1), 565–606 (1928)
Hardy, G.H., Littlewood, J.E.: Some properties of fractional integrals. II. Math. Z. 34(1), 403–439 (1932)
Caputo, M.: Linear models of dissipation whose Q is almost frequency independent II. Geophys. J. Int. 13(5), 529–539 (1967)
Diethelm, K.: The Analysis of Fractional Differential Equations. Springer, Berlin Heidelberg (2010)
Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265(2), 229–248 (2002)
Diethelm, K., Ford, N.J., Freed, A.D.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29(1), 3–22 (2002)
Sabatier, J., Agrawal, O.P., Machado, J.A.T.: Advances in Fractional Calculus. Springer, Dordrecht (2007)
Assaleh, K., Ahmad, W.M.: Modeling of speech signals using fractional calculus, In: Signal Processing and Its Applications, ISSPA. 9th International Symposium on IEEE, pp. 1–4 (2007)
Kulish, V.V., Lage, J.L.: Application of fractional calculus to fluid mechanics. J. Fluids Eng. 124(3), 803–806 (2002)
Magin, R.L.: Modeling the cardiac tissue electrode interface using fractional calculus. J. Vib. Control 14(1), 1431–1442 (2008)
Magin, R.L.: Fractional Calculus in Bioengineering. Begell House, Redding (2006)
Surez, J.I., Vinagre, B.M., Caldern, A.J., et al.: Using fractional calculus for lateral and longitudinal control of autonomous vehicles. Comput. Aided Syst. Theory EUROCAST 2003 2809, 337–348 (2003)
Toledo-Hernandez, R., Rico-Ramirez, V., Iglesias-Silva, G.A., et al.: A fractional calculus approach to the dynamic optimization of biological reactive systems. Part I: fractional models for biological reactions. Chem. Eng. Sci. 117(1), 217C228 (2014)
Zeng, A.P., Ross, A., Biebl, H., et al.: Multiple product inhibition and growth modeling of Clostridium butyricum and Klebsiella pneumoniae in glycerol fermentation. Biotechnol. Bioeng. 44(8), 902–911 (1994)
Zeng, A.P., Biebl, H.: Chemicals from Biotechnology: The Case of 1,3-Propanediol Production and the New Trends, Tools and Applications of Biochemical Engineering Science, 74th edn. Springer, Berlin Heidelberg (2002)
Gong, Z., Liu, C., Feng, E., et al.: Research article: computational method for inferring objective function of glycerol metabolism in Klebsiella pneumoniae. Comput. Biol. Chem. 33(1), 1–6 (2009)
Yuan, J., Zhang, X., Zhu, X., et al.: Modelling and pathway identification involving the transport mechanism of a complex metabolic system in batch culture. Commun. Nonlinear Sci. Numer. Simul. 19(6), 2088–2103 (2014)
Yuan, J., Zhang, X., Zhu, X., et al.: Pathway identification using parallel optimization for a nonlinear hybrid system in batch culture. Nonlinear Anal. Hybrid Syst. 15, 112–131 (2015)
Yuan, J., Zhang, X., Zhu, X., et al.: Identification and robustness analysis of nonlinear multi-stage enzyme-catalytic dynamical system in batch culture. Comput. Appl. Math. 34(3), 957–978 (2015)
Yin, H., Yuan, J., Zhang, X., et al.: Modeling and parameter identification for a nonlinear multi-stage system for dha regulon in batch culture. Appl. Math. Model. 40(1), 468–484 (2016)
Li, X.H., Feng, E.M., Xiu, Z.L.: Optimal control and property of nonlinear dynamic system for microorganism in batch culture. OR Trans. 9(4), 67–79 (2005)
Gao, C., Feng, E., Wang, Z., et al.: Parameters identification problem of the nonlinear dynamical system in microbial continuous cultures. Appl. Math. Comput. 169(1), 476–484 (2005)
Xiu, Z., Zeng, A., An, L.: Mathematical modeling of kinetics and research on multiplicity of glycerol bioconversion to 1, 3-propanediol. J. Dali. Univ. Technol. 40(4), 428–433 (2000)
Wang, L.: Determining the transport mechanism of an enzyme-catalytic complex metabolic network based on biological robustness. Bioprocess Biosyst. Eng. 36(4), 433–441 (2012)
Wang, L., Ye, J., Feng, E., et al.: An improved model for multistage simulation of glycerol fermentation in batch culture and its parameter identification. Nonlinear Anal. Hybrid Syst. 3(4), 455–462 (2009)
Wang, L., Feng, E.: An improved nonlinear multistage switch system of microbial fermentation process in fed-batch culture. J. Syst. Sci. Complex. 28(3), 580–591 (2015)
Wang, L., Xiu, Z., Gong, Z., et al.: Modeling and parameter identification for multistage simulation of microbial bioconversion in batch culture. Int. J. Biomath. 5(4), 1250034 (2012). doi:10.1142/S179352451100174X
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Yverdon (1993)
Kilbas, A.A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science Limited, Amsterdam (2006)
Podlubny, I.: Geometric and physical interpretation of fractional integration and fractional differentiation. Fract. Calc. Appl. Anal. 5(4), 367–386 (2004)
Heymans, N., Podlubny, I.: Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives. Rheol. Acta 45(5), 765–771 (2006)
Zhou, Y., Wang, J., Zhang, L.: Basic theory of fractional differential equations. World Scientific (2016)
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This work was supported by the National Natural Science Foundation for the Youth of China (Grant No. 11401073), and the Fundamental Research Funds for Central Universities in China (Grant DUT15LK25).
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Mu, P., Wang, L., An, Y. et al. A Novel Fractional Microbial Batch Culture Process and Parameter Identification. Differ Equ Dyn Syst 26, 265–277 (2018). https://doi.org/10.1007/s12591-017-0381-7
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DOI: https://doi.org/10.1007/s12591-017-0381-7