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

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

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.
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), 19-40; 263-288.

[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), 138-151. doi: 10.1137/0145006.

[3]

F. Chen and Y. Jiang, "Microalgal Biotechnology," Chinese Light Industry Press, Beijing, 1999.

[4]

L. Chen, "Nonlinear Biological Dynamical Systems," Science Press, Beijing, 1993.

[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), 227-231.

[6]

A. Cunningham and R. M. Nisbet, Time lag and co-operativity in the transient growth dynamics of microalgae, J. Theor. Biol., 84 (1980), 189-203. doi: 10.1016/S0022-5193(80)80003-8.

[7]

S. F. Ellermeyer, S. S. Pilyugin and Ray Redheffer, Persistence criteria for a chemostat with variable nutrient input, J. Diff. Eq., 171 (2001), 132-147.

[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), 369-379.

[9]

J. K. Hale, "Ordinary Differential Equations," Second edition, Robert E. Krieger Publishing Company, Inc., Huntington, New York, 1980.

[10]

S. Han, Z. Zhang and H. Liu, Effects of Chlorella growth factor on physiological function, Chinese J. Biochem. Pharmaceutics, 25 (2004), 5-7.

[11]

S. R. Hansen and S. P. Hubbell, Single-nutrient microbial competition: Qualitative agreement between experimental and theoretically forecast outcomes, Science, 207 (1980), 1491-1493. doi: 10.1126/science.6767274.

[12]

F. Khacik, Process for isolation, purification, and recrystallization of lutein from saponified marigold oleoresin and uses thereof: US patent, 5382714, 1995-01-17.

[13]

J. T. Landrum and R. A. Bone, Lutein, zeaxanthin, and the macular pigment, Arch. Biochem. Biophys., 385 (2001), 28-40. doi: 10.1006/abbi.2000.2171.

[14]

J. A. Leon and D. B. Tumpson, Competition between two species for two complementary or substitutable resources, J. Theor. Biol., 50 (1975), 185-201. doi: 10.1016/0022-5193(75)90032-6.

[15]

L. W. Levy, Trans-xanthophyll ester concentrates of enhanced purity and method of making same: US patent, 6191293, 2001-02-20.

[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), 2058-2086. doi: 10.1137/S0036139999359756.

[17]

B. Li and H. L. Smith, Global dynamics of microbial competition for two resources with internal storage, J. Math. Biol., 55 (2007), 481-515. doi: 10.1007/s00285-007-0092-8.

[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), 81-86.

[19]

D. L. Madhavi and D. I. Kagan, Process for the isolation of mixed carotenoids from plants: US patent, 6380442, 2002-04-30.

[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), 169-206.

[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), 359-366. doi: 10.1023/A:1007981930676.

[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), 67-74. doi: 10.1023/A:1008011201437.

[23]

T. Philip, Purification of lutein-fatty acid esters from plant materials: US patent, 4048203, 1977-09-13.

[24]

S. S. Pilyugin and P. Waltman, Multiple limit cycles in the chemostat with variable yield, Math. Biosci., 182 (2003), 151-166. doi: 10.1016/S0025-5564(02)00214-6.

[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), 361-362. doi: 10.1007/BF00367123.

[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), 341-347. doi: 10.1016/S0032-9592(98)00101-0.

[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), 312-318. doi: 10.1016/S0141-0229(00)00208-8.

[28]

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

[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), 1113-1131. doi: 10.1137/S0036139993245344.

[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), 193-196.

[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), 37-40.

[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), 489-530. doi: 10.1007/s00285-004-0311-5.

[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), 487-502. doi: 10.1007/BF02186327.

[34]

H. Yan, C. Ye and C. Yin, Kinetics of phthalate esters biodegradation by Chlorella pyrenoidosa, Environ. Toxicol. Chem., 14 (1995), 931-938.

[35]

H. Yan and G. Pan, Toxicity and bioaccumulation of copper in three green microalgal species, Chemosphere, 49 (2002),471-476. doi: 10.1016/S0045-6535(02)00285-0.

[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), 408-412.

[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), 18-21.

[38]

H. Zhang, S. Sun, K. Mai and Y. Liang, Advances in the studies on heterotrophic culture of microalgae, Trans. Oceanology Limnology, (2000), 51-59.

[39]

L. Zhang, R. Yang and H. Xiao, The heterotrophic culture of Chlorella and the optimization of growth condition, Guihaia, 24 (2001), 353-357.

[40]

H. Zhou, W. Lin and T. Chen, The heterotrophy and applications of Chlorella, Amino Acids Biotic Resources, 27 (2005), 69-73.

show all references

References:
[1]

E. Beretta and Y. Takeuchi, Qualitative properties of chemostat equations with time delays, Diff. Equ. Dyn. Sys., 2 (1994), 19-40; 263-288.

[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), 138-151. doi: 10.1137/0145006.

[3]

F. Chen and Y. Jiang, "Microalgal Biotechnology," Chinese Light Industry Press, Beijing, 1999.

[4]

L. Chen, "Nonlinear Biological Dynamical Systems," Science Press, Beijing, 1993.

[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), 227-231.

[6]

A. Cunningham and R. M. Nisbet, Time lag and co-operativity in the transient growth dynamics of microalgae, J. Theor. Biol., 84 (1980), 189-203. doi: 10.1016/S0022-5193(80)80003-8.

[7]

S. F. Ellermeyer, S. S. Pilyugin and Ray Redheffer, Persistence criteria for a chemostat with variable nutrient input, J. Diff. Eq., 171 (2001), 132-147.

[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), 369-379.

[9]

J. K. Hale, "Ordinary Differential Equations," Second edition, Robert E. Krieger Publishing Company, Inc., Huntington, New York, 1980.

[10]

S. Han, Z. Zhang and H. Liu, Effects of Chlorella growth factor on physiological function, Chinese J. Biochem. Pharmaceutics, 25 (2004), 5-7.

[11]

S. R. Hansen and S. P. Hubbell, Single-nutrient microbial competition: Qualitative agreement between experimental and theoretically forecast outcomes, Science, 207 (1980), 1491-1493. doi: 10.1126/science.6767274.

[12]

F. Khacik, Process for isolation, purification, and recrystallization of lutein from saponified marigold oleoresin and uses thereof: US patent, 5382714, 1995-01-17.

[13]

J. T. Landrum and R. A. Bone, Lutein, zeaxanthin, and the macular pigment, Arch. Biochem. Biophys., 385 (2001), 28-40. doi: 10.1006/abbi.2000.2171.

[14]

J. A. Leon and D. B. Tumpson, Competition between two species for two complementary or substitutable resources, J. Theor. Biol., 50 (1975), 185-201. doi: 10.1016/0022-5193(75)90032-6.

[15]

L. W. Levy, Trans-xanthophyll ester concentrates of enhanced purity and method of making same: US patent, 6191293, 2001-02-20.

[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), 2058-2086. doi: 10.1137/S0036139999359756.

[17]

B. Li and H. L. Smith, Global dynamics of microbial competition for two resources with internal storage, J. Math. Biol., 55 (2007), 481-515. doi: 10.1007/s00285-007-0092-8.

[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), 81-86.

[19]

D. L. Madhavi and D. I. Kagan, Process for the isolation of mixed carotenoids from plants: US patent, 6380442, 2002-04-30.

[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), 169-206.

[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), 359-366. doi: 10.1023/A:1007981930676.

[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), 67-74. doi: 10.1023/A:1008011201437.

[23]

T. Philip, Purification of lutein-fatty acid esters from plant materials: US patent, 4048203, 1977-09-13.

[24]

S. S. Pilyugin and P. Waltman, Multiple limit cycles in the chemostat with variable yield, Math. Biosci., 182 (2003), 151-166. doi: 10.1016/S0025-5564(02)00214-6.

[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), 361-362. doi: 10.1007/BF00367123.

[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), 341-347. doi: 10.1016/S0032-9592(98)00101-0.

[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), 312-318. doi: 10.1016/S0141-0229(00)00208-8.

[28]

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

[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), 1113-1131. doi: 10.1137/S0036139993245344.

[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), 193-196.

[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), 37-40.

[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), 489-530. doi: 10.1007/s00285-004-0311-5.

[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), 487-502. doi: 10.1007/BF02186327.

[34]

H. Yan, C. Ye and C. Yin, Kinetics of phthalate esters biodegradation by Chlorella pyrenoidosa, Environ. Toxicol. Chem., 14 (1995), 931-938.

[35]

H. Yan and G. Pan, Toxicity and bioaccumulation of copper in three green microalgal species, Chemosphere, 49 (2002),471-476. doi: 10.1016/S0045-6535(02)00285-0.

[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), 408-412.

[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), 18-21.

[38]

H. Zhang, S. Sun, K. Mai and Y. Liang, Advances in the studies on heterotrophic culture of microalgae, Trans. Oceanology Limnology, (2000), 51-59.

[39]

L. Zhang, R. Yang and H. Xiao, The heterotrophic culture of Chlorella and the optimization of growth condition, Guihaia, 24 (2001), 353-357.

[40]

H. Zhou, W. Lin and T. Chen, The heterotrophy and applications of Chlorella, Amino Acids Biotic Resources, 27 (2005), 69-73.

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