American Institute of Mathematical Sciences

Advanced
2011, 8(4): 1117-1133. doi: 10.3934/mbe.2011.8.1117

A dynamic model describing heterotrophic culture of chorella and its stability analysis

Yan Zhang 1, , Wanbiao Ma 1, , Hai Yan 2, and  Yasuhiro Takeuchi 3,

1. 

Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
2. 

Department of Biological Science and Technology, School of Chemical and Biological Engineering, University of Science and Technology Beijing, Beijing 100083, China
3. 

Graduate School of Science and Technology, Faculty of Engineering, Shizuoka University, Hamamatsu 432-8561

Received  October 2009 Revised  October 2010 Published  August 2011

Chlorella is an important species of microorganism, which includes about 10 species. Chlorella USTB01 is a strain of microalga which is isolated from Qinghe River in Beijing and has strong ability in the utilization of organic compounds and was identified as Chlorella sp. (H. Yan etal, Isolation and heterotrophic culture of Chlorella sp., J. Univ. Sci. Tech. Beijing, 2005, 27:408-412). In this paper, based on the standard Chemostat models and the experimental data on the heterotrophic culture of Chlorella USTB01, a dynamic model governed by differential equations with three variables (Chlorella, carbon source and nitrogen source) is proposed. For the model, there always exists a boundary equilibrium, i.e. Chlorella-free equilibrium. Furthermore, under additional conditions, the model also has the positive equilibria, i.e., the equilibira for which Chlorella, carbon source and nitrogen source are coexistent. Then, local and global asymptotic stability of the equilibria of the model have been discussed. Finally, the parameters in the model are determined according to the experimental data, and numerical simulations are given. The numerical simulations show that the trajectories of the model fit the trends of the experimental data well.
Keywords: carbon source, differential equation, Chlorella, nitrogen source, stability..
Mathematics Subject Classification: Primary: 34K20; Secondary: 92C3.
Citation: Yan Zhang, Wanbiao Ma, Hai Yan, Yasuhiro Takeuchi. A dynamic model describing heterotrophic culture of chorella and its stability analysis. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1117-1133. doi: 10.3934/mbe.2011.8.1117
References:
[1]

E. Beretta and Y. Takeuchi, Qualitative properties of chemostat equations with time delays,, Diff. Equ. Dyn. Sys., 2 (1994), 19. Google Scholar

[2]

G. J. Butler and G. S. K. Wolkowicz, A mathematical model of the chemostat with a general class of functions describing nutrient uptake,, SIAM J. Appl. Math., 45 (1985), 138. doi: 10.1137/0145006. Google Scholar

[3]

F. Chen and Y. Jiang, "Microalgal Biotechnology,", Chinese Light Industry Press, (1999). Google Scholar

[4]

L. Chen, "Nonlinear Biological Dynamical Systems,", Science Press, (1993). Google Scholar

[5]

A. Cunningham and P. Maas, Time lag and nutrient storage effects in the transient growth response of Chlamydomonas reinhardii in nitrogen-limited batch and continuous culture,, J. Gen. Microbiol., 104 (1978), 227. Google Scholar

[6]

A. Cunningham and R. M. Nisbet, Time lag and co-operativity in the transient growth dynamics of microalgae,, J. Theor. Biol., 84 (1980), 189. doi: 10.1016/S0022-5193(80)80003-8. Google Scholar

[7]

S. F. Ellermeyer, S. S. Pilyugin and Ray Redheffer, Persistence criteria for a chemostat with variable nutrient input,, J. Diff. Eq., 171 (2001), 132. Google Scholar

[8]

H. Endo, H. Hosoya and T. Koibuchi, Growth yields of Chlorella regularis in dark-heterotrophic continuous cultures using acetate,, J. Ferment. Technol., 55 (1977), 369. Google Scholar

[9]

J. K. Hale, "Ordinary Differential Equations," Second edition,, Robert E. Krieger Publishing Company, (1980). Google Scholar

[10]

S. Han, Z. Zhang and H. Liu, Effects of Chlorella growth factor on physiological function,, Chinese J. Biochem. Pharmaceutics, 25 (2004), 5. Google Scholar

[11]

S. R. Hansen and S. P. Hubbell, Single-nutrient microbial competition: Qualitative agreement between experimental and theoretically forecast outcomes,, Science, 207 (1980), 1491. doi: 10.1126/science.6767274. Google Scholar

[12]

F. Khacik, Process for isolation, purification, and recrystallization of lutein from saponified marigold oleoresin and uses thereof: US patent,, 5382714, (): 1995. Google Scholar

[13]

J. T. Landrum and R. A. Bone, Lutein, zeaxanthin, and the macular pigment,, Arch. Biochem. Biophys., 385 (2001), 28. doi: 10.1006/abbi.2000.2171. Google Scholar

[14]

J. A. Leon and D. B. Tumpson, Competition between two species for two complementary or substitutable resources,, J. Theor. Biol., 50 (1975), 185. doi: 10.1016/0022-5193(75)90032-6. Google Scholar

[15]

L. W. Levy, Trans-xanthophyll ester concentrates of enhanced purity and method of making same: US patent,, 6191293, (): 2001. Google Scholar

[16]

B. Li, G. S. K. Wolkowicz and Y. Kuang, Global asymptotic behavior of a Chemostat model with two perfectly complementary resources and distributed delay,, SIAM J. Appl. Math., 60 (2000), 2058. doi: 10.1137/S0036139999359756. Google Scholar

[17]

B. Li and H. L. Smith, Global dynamics of microbial competition for two resources with internal storage,, J. Math. Biol., 55 (2007), 481. doi: 10.1007/s00285-007-0092-8. Google Scholar

[18]

S. Liu, H. Meng, S. Liang, J. Yin and P. Mai, High-density heterotrophic culture of Chlorella Vulgaris in bioreactor,, J. South China Univ. Tech., 28 (2000), 81. Google Scholar

[19]

D. L. Madhavi and D. I. Kagan, Process for the isolation of mixed carotenoids from plants: US patent,, 6380442, (): 2002. Google Scholar

[20]

A. Narang and S. S. Pilyugin, Towards an integrated physiological theory of microbial growth: From subcellular variables to population dynamics,, Math. Biosci. Eng., 2 (2005), 169. Google Scholar

[21]

J. C. Ogbonna, H. Masui and H. Tanaka, Sequential heterotrophic / autotrophic cultivation - An efficient method of producing Chlorella biomass for health food and animal feed,, J. Appl. Phycol., 9 (1997), 359. doi: 10.1023/A:1007981930676. Google Scholar

[22]

J. C. Ogbonna, S. Tomiyama and H. Tanaka, Heterotrophic cultivation of Euglena gracilis Z for efficient production of $\alpha$-tocopherol,, J. Appl. Phycol., 10 (1998), 67. doi: 10.1023/A:1008011201437. Google Scholar

[23]

T. Philip, Purification of lutein-fatty acid esters from plant materials: US patent,, 4048203, (1977). Google Scholar

[24]

S. S. Pilyugin and P. Waltman, Multiple limit cycles in the chemostat with variable yield,, Math. Biosci., 182 (2003), 151. doi: 10.1016/S0025-5564(02)00214-6. Google Scholar

[25]

K. Sasaki, K. Watanabe, T. Tanaka, Y. Hotta and S. Nagai, 5-aminolevulinic acid production by Chlorella sp. during heterotrophic cultivation in the dark,, World J. Microbiol. Biotech., 11 (1995), 361. doi: 10.1007/BF00367123. Google Scholar

[26]

X. Shi, H. Liu, X. Zhang and F. Chen, Production of biomass and lutein by Chlorella protothecoides at various glucose concentrations in heterotrophic cultures,, Process Biochem., 34 (1999), 341. doi: 10.1016/S0032-9592(98)00101-0. Google Scholar

[27]

X. Shi, X. Zhang and F. Chen, Heterotrophic production of biomass and lutein Chlorella protothecoides on various nitrogen sources,, Enzyme Microb. Technol., 27 (2000), 312. doi: 10.1016/S0141-0229(00)00208-8. Google Scholar

[28]

H. L. Smith and P. Waltman, "The Theory of the Chemostat. Dynamics of Microbial Competition,", Cambridge Studies in Mathematical Biology, 13 (1995). doi: 10.1017/CBO9780511530043. Google Scholar

[29]

H. L. Smith and P. Waltman, Competition for a single limiting resource in continuous culture: The variable-yield model,, SIAM J. Appl. Math., 54 (1994), 1113. doi: 10.1137/S0036139993245344. Google Scholar

[30]

L. V. Thinh and D. J. Griffiths, Amino-acid composition of autotrophic and heterotrophic cultures of emerson strain of Chlorella,, Plant Cell Physiol., 17 (1976), 193. Google Scholar

[31]

S. Wang, H. Yan, B. Zhang, L. Lv and H. Lin, Effects of various nitrogen sources and phytohormones on growth and content of lutin in Chlorella sp. USTB01,, Sci. Tech. Review, 23 (2005), 37. Google Scholar

[32]

H. Xia, G. S. K. Wolkowicz and L. Wang, Transient oscillation induced by delayed growth response in the chemostat,, J. Math. Biol., 50 (2005), 489. doi: 10.1007/s00285-004-0311-5. Google Scholar

[33]

K. Yamaguchi, Recent advances in microalgal bioscience in Japan, with special reference to utilization of biomass and metabolites: A review,, J. Appl. Phycol., 8 (1996), 487. doi: 10.1007/BF02186327. Google Scholar

[34]

H. Yan, C. Ye and C. Yin, Kinetics of phthalate esters biodegradation by Chlorella pyrenoidosa,, Environ. Toxicol. Chem., 14 (1995), 931. Google Scholar

[35]

H. Yan and G. Pan, Toxicity and bioaccumulation of copper in three green microalgal species,, Chemosphere, 49 (2002), 471. doi: 10.1016/S0045-6535(02)00285-0. Google Scholar

[36]

H. Yan, J. Zhou, H. He, Y. Wei and J. Sun, Isolation and heterotrophic culture of Chlorella sp.,, J. Univ. Sci. Tech. Beijing, 27 (2005), 408. Google Scholar

[37]

H. Yan, B. Zhang, S. Wang, Y. Li, S. Liu and S. Yang, Advances in the heterotrophic culture of Chlorella sp.,, Modern Chem. Indust., 27 (2007), 18. Google Scholar

[38]

H. Zhang, S. Sun, K. Mai and Y. Liang, Advances in the studies on heterotrophic culture of microalgae,, Trans. Oceanology Limnology, (2000), 51. Google Scholar

[39]

L. Zhang, R. Yang and H. Xiao, The heterotrophic culture of Chlorella and the optimization of growth condition,, Guihaia, 24 (2001), 353. Google Scholar

[40]

H. Zhou, W. Lin and T. Chen, The heterotrophy and applications of Chlorella,, Amino Acids Biotic Resources, 27 (2005), 69. Google Scholar

show all references

References:
[1]

E. Beretta and Y. Takeuchi, Qualitative properties of chemostat equations with time delays,, Diff. Equ. Dyn. Sys., 2 (1994), 19. Google Scholar

[2]

G. J. Butler and G. S. K. Wolkowicz, A mathematical model of the chemostat with a general class of functions describing nutrient uptake,, SIAM J. Appl. Math., 45 (1985), 138. doi: 10.1137/0145006. Google Scholar

[3]

F. Chen and Y. Jiang, "Microalgal Biotechnology,", Chinese Light Industry Press, (1999). Google Scholar

[4]

L. Chen, "Nonlinear Biological Dynamical Systems,", Science Press, (1993). Google Scholar

[5]

A. Cunningham and P. Maas, Time lag and nutrient storage effects in the transient growth response of Chlamydomonas reinhardii in nitrogen-limited batch and continuous culture,, J. Gen. Microbiol., 104 (1978), 227. Google Scholar

[6]

A. Cunningham and R. M. Nisbet, Time lag and co-operativity in the transient growth dynamics of microalgae,, J. Theor. Biol., 84 (1980), 189. doi: 10.1016/S0022-5193(80)80003-8. Google Scholar

[7]

S. F. Ellermeyer, S. S. Pilyugin and Ray Redheffer, Persistence criteria for a chemostat with variable nutrient input,, J. Diff. Eq., 171 (2001), 132. Google Scholar

[8]

H. Endo, H. Hosoya and T. Koibuchi, Growth yields of Chlorella regularis in dark-heterotrophic continuous cultures using acetate,, J. Ferment. Technol., 55 (1977), 369. Google Scholar

[9]

J. K. Hale, "Ordinary Differential Equations," Second edition,, Robert E. Krieger Publishing Company, (1980). Google Scholar

[10]

S. Han, Z. Zhang and H. Liu, Effects of Chlorella growth factor on physiological function,, Chinese J. Biochem. Pharmaceutics, 25 (2004), 5. Google Scholar

[11]

S. R. Hansen and S. P. Hubbell, Single-nutrient microbial competition: Qualitative agreement between experimental and theoretically forecast outcomes,, Science, 207 (1980), 1491. doi: 10.1126/science.6767274. Google Scholar

[12]

F. Khacik, Process for isolation, purification, and recrystallization of lutein from saponified marigold oleoresin and uses thereof: US patent,, 5382714, (): 1995. Google Scholar

[13]

J. T. Landrum and R. A. Bone, Lutein, zeaxanthin, and the macular pigment,, Arch. Biochem. Biophys., 385 (2001), 28. doi: 10.1006/abbi.2000.2171. Google Scholar

[14]

J. A. Leon and D. B. Tumpson, Competition between two species for two complementary or substitutable resources,, J. Theor. Biol., 50 (1975), 185. doi: 10.1016/0022-5193(75)90032-6. Google Scholar

[15]

L. W. Levy, Trans-xanthophyll ester concentrates of enhanced purity and method of making same: US patent,, 6191293, (): 2001. Google Scholar

[16]

B. Li, G. S. K. Wolkowicz and Y. Kuang, Global asymptotic behavior of a Chemostat model with two perfectly complementary resources and distributed delay,, SIAM J. Appl. Math., 60 (2000), 2058. doi: 10.1137/S0036139999359756. Google Scholar

[17]

B. Li and H. L. Smith, Global dynamics of microbial competition for two resources with internal storage,, J. Math. Biol., 55 (2007), 481. doi: 10.1007/s00285-007-0092-8. Google Scholar

[18]

S. Liu, H. Meng, S. Liang, J. Yin and P. Mai, High-density heterotrophic culture of Chlorella Vulgaris in bioreactor,, J. South China Univ. Tech., 28 (2000), 81. Google Scholar

[19]

D. L. Madhavi and D. I. Kagan, Process for the isolation of mixed carotenoids from plants: US patent,, 6380442, (): 2002. Google Scholar

[20]

A. Narang and S. S. Pilyugin, Towards an integrated physiological theory of microbial growth: From subcellular variables to population dynamics,, Math. Biosci. Eng., 2 (2005), 169. Google Scholar

[21]

J. C. Ogbonna, H. Masui and H. Tanaka, Sequential heterotrophic / autotrophic cultivation - An efficient method of producing Chlorella biomass for health food and animal feed,, J. Appl. Phycol., 9 (1997), 359. doi: 10.1023/A:1007981930676. Google Scholar

[22]

J. C. Ogbonna, S. Tomiyama and H. Tanaka, Heterotrophic cultivation of Euglena gracilis Z for efficient production of $\alpha$-tocopherol,, J. Appl. Phycol., 10 (1998), 67. doi: 10.1023/A:1008011201437. Google Scholar

[23]

T. Philip, Purification of lutein-fatty acid esters from plant materials: US patent,, 4048203, (1977). Google Scholar

[24]

S. S. Pilyugin and P. Waltman, Multiple limit cycles in the chemostat with variable yield,, Math. Biosci., 182 (2003), 151. doi: 10.1016/S0025-5564(02)00214-6. Google Scholar

[25]

K. Sasaki, K. Watanabe, T. Tanaka, Y. Hotta and S. Nagai, 5-aminolevulinic acid production by Chlorella sp. during heterotrophic cultivation in the dark,, World J. Microbiol. Biotech., 11 (1995), 361. doi: 10.1007/BF00367123. Google Scholar

[26]

X. Shi, H. Liu, X. Zhang and F. Chen, Production of biomass and lutein by Chlorella protothecoides at various glucose concentrations in heterotrophic cultures,, Process Biochem., 34 (1999), 341. doi: 10.1016/S0032-9592(98)00101-0. Google Scholar

[27]

X. Shi, X. Zhang and F. Chen, Heterotrophic production of biomass and lutein Chlorella protothecoides on various nitrogen sources,, Enzyme Microb. Technol., 27 (2000), 312. doi: 10.1016/S0141-0229(00)00208-8. Google Scholar

[28]

H. L. Smith and P. Waltman, "The Theory of the Chemostat. Dynamics of Microbial Competition,", Cambridge Studies in Mathematical Biology, 13 (1995). doi: 10.1017/CBO9780511530043. Google Scholar

[29]

H. L. Smith and P. Waltman, Competition for a single limiting resource in continuous culture: The variable-yield model,, SIAM J. Appl. Math., 54 (1994), 1113. doi: 10.1137/S0036139993245344. Google Scholar

[30]

L. V. Thinh and D. J. Griffiths, Amino-acid composition of autotrophic and heterotrophic cultures of emerson strain of Chlorella,, Plant Cell Physiol., 17 (1976), 193. Google Scholar

[31]

S. Wang, H. Yan, B. Zhang, L. Lv and H. Lin, Effects of various nitrogen sources and phytohormones on growth and content of lutin in Chlorella sp. USTB01,, Sci. Tech. Review, 23 (2005), 37. Google Scholar

[32]

H. Xia, G. S. K. Wolkowicz and L. Wang, Transient oscillation induced by delayed growth response in the chemostat,, J. Math. Biol., 50 (2005), 489. doi: 10.1007/s00285-004-0311-5. Google Scholar

[33]

K. Yamaguchi, Recent advances in microalgal bioscience in Japan, with special reference to utilization of biomass and metabolites: A review,, J. Appl. Phycol., 8 (1996), 487. doi: 10.1007/BF02186327. Google Scholar

[34]

H. Yan, C. Ye and C. Yin, Kinetics of phthalate esters biodegradation by Chlorella pyrenoidosa,, Environ. Toxicol. Chem., 14 (1995), 931. Google Scholar

[35]

H. Yan and G. Pan, Toxicity and bioaccumulation of copper in three green microalgal species,, Chemosphere, 49 (2002), 471. doi: 10.1016/S0045-6535(02)00285-0. Google Scholar

[36]

H. Yan, J. Zhou, H. He, Y. Wei and J. Sun, Isolation and heterotrophic culture of Chlorella sp.,, J. Univ. Sci. Tech. Beijing, 27 (2005), 408. Google Scholar

[37]

H. Yan, B. Zhang, S. Wang, Y. Li, S. Liu and S. Yang, Advances in the heterotrophic culture of Chlorella sp.,, Modern Chem. Indust., 27 (2007), 18. Google Scholar

[38]

H. Zhang, S. Sun, K. Mai and Y. Liang, Advances in the studies on heterotrophic culture of microalgae,, Trans. Oceanology Limnology, (2000), 51. Google Scholar

[39]

L. Zhang, R. Yang and H. Xiao, The heterotrophic culture of Chlorella and the optimization of growth condition,, Guihaia, 24 (2001), 353. Google Scholar

[40]

H. Zhou, W. Lin and T. Chen, The heterotrophy and applications of Chlorella,, Amino Acids Biotic Resources, 27 (2005), 69. Google Scholar

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