青霉素发酵过程的模型化研究
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摘要
青霉素是人类历史上发现的第一种能够用于治疗疾病的抗生素,其出现开创了抗生素治疗的新时代,至今在临床治疗上仍有广泛的应用。从青霉素开始,微生物发酵逐渐成为制药业的支柱。本文针对青霉素工业生产实际过程,进行了以下几个方面的研究:
1.青霉素补料分批发酵过程代谢网络研究
详细阐述了存在于一般生物体内的三大代谢途径:糖酵解途径,TCA循环途径以及氧化磷酸化途径。对复杂的青霉素补料分批发酵代谢网络进行研究,分析了青霉素代谢网络中的主要代谢途径以及氮源代谢。
2.生物发酵过程模型的研究
回顾了生物过程发酵过程的模型化发展,从最开始的非结构化模型到结构化模型,再到控制论模型的应用。非结构化模型将细胞看成一个整体,具有一致的行为,同时也不考虑细胞内部的化学反应。只考虑外部环境对细胞生长的影响。结构化模型相对来说是一种机理模型,考虑到细胞内部的代谢途径以及相关的化学反应,通过建立化学计量学平衡模型来描述细胞的生长。控制论模型类似于灰箱模型,在已经分析出的代谢网络基础上,通过已知的变量来求解未知变量。也是一种机理的描述。不过由于一些变量在实际工业生产过程中不可测量,所以这些变量不能用于验证模型的有效性。
3.青霉素发酵过程的控制论模型
根据控制论模型的建模步骤以及建模框架,在文献中已有的青霉素发酵过程的代谢网络的基础上,提出了简化的青霉素代谢途径。并且使用大量工业生产数据(来自国内某大型制药企业)对青霉素发酵过程的控制论模型进行了验证。工业实测数据值与模型仿真值吻合性较好。
4.讨论与总结
以骨髓瘤细胞为例,探讨了动物细胞的控制论模型的建立方法。与青霉素控制论模型建模手段类似,简化的骨髓瘤细胞代谢网络也包括三大主要代谢路径。同时也对结构化模型与控制论模型进行了相关的比较。就优越性而言,控制论模型可以根据外部环境下的即时扰动,特别针对有补料的情况作出及时响应。
Penicillin is the first antibiotic found in history. It initiates a new age of antibiotic treatment and is still widely used in clinic. Accompanying with the birth of penicillin, microorganism fermentation has become a major part of pharmaceutical industry from then on. Aiming at the real process of industrial penicillin fermentation, the following aspects are focused on:
1. Research of the metabolic network of Penicillium chrysogenum in fed-batch fermentation
The major three metabolic pathways existing in the common organisms, including glycolysis, TCA cycle, as well as the oxidative phosphorylation pathways, are illustrated. Research into the metabolic network of complex penicillin fed-batch fermentation and analyze the main metabolic pathways in the metabolic network, as well as the nitrogen metabolism.
2. Research of the biological fermentation model
The development of modeling of biological fermentation process is retrospected, from the original unstructured model to structured model, then to the cybernetic model. For the unstructured model, the cell is considered as a whole and the behavior is cosistent. In the unstructured model, the reactions take place in the cell are not considered. The structured model is a kind of mechanism model, what it concern is the metabolic pathways and the corresponding chemical reactions. According to the established stoichiometric balance model, cell growth is described. Cybernetic model, belonging to the gray model, on the basis of the analyzed metabolic network, solve the unknown variables through the known variables. While, in the practically industrial process, some variable are not mesearable, and these variables can not be used to the validation of the model.
3. Cybernetic model of the penicillin fermentation process
According to the modeling steps and framework of cybernetic approach, on the basis of the proposed metabolic network of penicillin in the literature, the abstracted metabolic pathway of penicillin is derived. The industrial data are used to validate the model.
4. Discussion
Taking the myeloma cell as the example of animal cell, the establishing approach of cybernetic model for the animal cell is discussed. Similar to the penicillin modeling method, the abstracted myeloma cell metabolic network also includes three metabolic pathway.
1.青霉素补料分批发酵过程代谢网络研究
详细阐述了存在于一般生物体内的三大代谢途径:糖酵解途径,TCA循环途径以及氧化磷酸化途径。对复杂的青霉素补料分批发酵代谢网络进行研究,分析了青霉素代谢网络中的主要代谢途径以及氮源代谢。
2.生物发酵过程模型的研究
回顾了生物过程发酵过程的模型化发展,从最开始的非结构化模型到结构化模型,再到控制论模型的应用。非结构化模型将细胞看成一个整体,具有一致的行为,同时也不考虑细胞内部的化学反应。只考虑外部环境对细胞生长的影响。结构化模型相对来说是一种机理模型,考虑到细胞内部的代谢途径以及相关的化学反应,通过建立化学计量学平衡模型来描述细胞的生长。控制论模型类似于灰箱模型,在已经分析出的代谢网络基础上,通过已知的变量来求解未知变量。也是一种机理的描述。不过由于一些变量在实际工业生产过程中不可测量,所以这些变量不能用于验证模型的有效性。
3.青霉素发酵过程的控制论模型
根据控制论模型的建模步骤以及建模框架,在文献中已有的青霉素发酵过程的代谢网络的基础上,提出了简化的青霉素代谢途径。并且使用大量工业生产数据(来自国内某大型制药企业)对青霉素发酵过程的控制论模型进行了验证。工业实测数据值与模型仿真值吻合性较好。
4.讨论与总结
以骨髓瘤细胞为例,探讨了动物细胞的控制论模型的建立方法。与青霉素控制论模型建模手段类似,简化的骨髓瘤细胞代谢网络也包括三大主要代谢路径。同时也对结构化模型与控制论模型进行了相关的比较。就优越性而言,控制论模型可以根据外部环境下的即时扰动,特别针对有补料的情况作出及时响应。
Penicillin is the first antibiotic found in history. It initiates a new age of antibiotic treatment and is still widely used in clinic. Accompanying with the birth of penicillin, microorganism fermentation has become a major part of pharmaceutical industry from then on. Aiming at the real process of industrial penicillin fermentation, the following aspects are focused on:
1. Research of the metabolic network of Penicillium chrysogenum in fed-batch fermentation
The major three metabolic pathways existing in the common organisms, including glycolysis, TCA cycle, as well as the oxidative phosphorylation pathways, are illustrated. Research into the metabolic network of complex penicillin fed-batch fermentation and analyze the main metabolic pathways in the metabolic network, as well as the nitrogen metabolism.
2. Research of the biological fermentation model
The development of modeling of biological fermentation process is retrospected, from the original unstructured model to structured model, then to the cybernetic model. For the unstructured model, the cell is considered as a whole and the behavior is cosistent. In the unstructured model, the reactions take place in the cell are not considered. The structured model is a kind of mechanism model, what it concern is the metabolic pathways and the corresponding chemical reactions. According to the established stoichiometric balance model, cell growth is described. Cybernetic model, belonging to the gray model, on the basis of the analyzed metabolic network, solve the unknown variables through the known variables. While, in the practically industrial process, some variable are not mesearable, and these variables can not be used to the validation of the model.
3. Cybernetic model of the penicillin fermentation process
According to the modeling steps and framework of cybernetic approach, on the basis of the proposed metabolic network of penicillin in the literature, the abstracted metabolic pathway of penicillin is derived. The industrial data are used to validate the model.
4. Discussion
Taking the myeloma cell as the example of animal cell, the establishing approach of cybernetic model for the animal cell is discussed. Similar to the penicillin modeling method, the abstracted myeloma cell metabolic network also includes three metabolic pathway.
引文
1.杜捷.浅谈抗生素发展.科技创新导报,2008;0201-01
2.王磊,任东明.青霉素—从发现到应用.生物学通报,2006;41-12
3.李鹏,李广州.青霉素结构的探究.南京师范大学化学与环境科学学院,2007;7
4.刘毅,王海清.采用最小二乘支持向量机的青霉素发酵过程建模研究.生物工程学报,2006;22-1
5. Wang SQ , Yuan YJ . Automatic Technologyof Biochemical Process, Beijing : Chemical Industry Press, 1999
6. Nomikos P , MacGregor JF. Monitoring batch processes using multiway principal component analysis. AIChE J , 1994 , 40 ( 8) :1361 - 1375
7. Massimo CD , Montague GA , Willis MJ et al . Towards improvedpenicillin fermentation via artificial neural networks. Computers andChemical Engineering ,1992 ,16 (4) :283– 291
8.葛爱冬,隋青美,王斌鹏.混合神经网络建模方法在青霉素发酵过程中的应用.山东轻工业学院学报,2006;20-3
9.肖占中.战争与青霉素的命运.国防科技,2006; 85-87.
10.郭勇.生物制药技术M .北京:中国轻工业出版社,2001.
11.金晓华.β-内酰胺类抗生素在医药市场的发展前景.商场现代化,2006; 47: 107.
12. Borovkov A A. Asymptotic Methodsin Queuing Theory. translated by D.Newton,John Wiley & Sons,1984
13. Chaudhry M L& Templeton J G C. Afirst Course in Bulk Queues. JohnWiley & Sons,1984
14. Ronald Bentley. Journal of Chemical Education ,2004 ,81 ( 10) :1462 - 1470
15. Roland Reiner.蒋天蓉译.抗生素入门.合肥:安徽科学技术出版社,1985 :37– 46
16.沈萍.微生物学M.北京:高等教育出版社,2000.
17. Chen, L., Bastin, G., van Breusegem, V. A case study of adaptive nonlinear regulation of fed-batch biological reactors. Automatica 1995, 31, 55-65.
18. Montague, G., Morris, J. 1994 Nerual network contribution in biotechnology. Trends in Biotechnology 1994, 12, 312-323.
19. Horiuchia, J.I., Kishimotob, M. App;ication of fuzzy control to industrial bioprocesses in Japan.Fuzzy Sets and Systems 2002, 128, 117-124.
20.赵娟平,陈健,姜长洪.青霉素发酵过程建模研究.计算机仿真,2008;25-2
21. Bailey, J.E. Toward a science of metabolic engineering. Scienc 1991, 252, 1668-1675.
22. Stephanopoulos, G., Vallino, J.J. Network rigidity and metabolic engineering in metabolite overproduction. Science 1991, 252, 1675-1681.
23. Granstroem, T., Aristos, A.A., Leisola, M. Metabolic flux analysis of Candida tropcalis growing on xylose in an oxygen-limited chemostat. Metabolic Engineering 2002, 4, 248-256.
24. Lee, K., Berthiaume, F., Stephanopoulos, G.N., Yarmush, D.M., Yarmush, M.L. Metabolic flux analysis of postburn hepatic hypermetabolism. Metabolic Engineering 2000, 2, 312-327.
25. Zhao, J., Shimizu, K. Metabolic flux analysis of Excherichia coli K12 grown on 13Clabeled acetate and glucose using GC-MS and powerful flux calculation method. Journal of Biotechnology 2003, 101, 101-117.
26. Bailey, J.E. Mathematical modeling and analysis in biochemical engineering: Past accomplishments and future opportunities. Biotechnology Progress 1998, 14, 8-20.
27. Herwig, C., von Stockar, U. A small metabolic flux model to identify transient metabolic regulations in Saccharomyces cerevisia. Bioprocess and Biosystems Engineering 2002, 24, 395-403.
28. Yu, J., Wang, J.P. Metabolic flux modeling of detoxification of acetic acid by Ralstonia europha at slightly alkaline pH levels. Biotechnology and Bioengineering 2001, 73, 458-464.
1. Jφrgensen H, Nielsen J, Villadsen J. Metabolic flux distributions in Penicillium chrysogenum during fed-batch cultivations. Biotechnology and Bioengineering 1995; 46: 117-131.
2. Blumenthal HJ. Carbohydrate metabolism. 1. Glycolysis. In: Ainsworth GC and Sussman (eds.) AS. The fungi, vol. I. New York: Academic Press; 1965. p. 229-268.
3. Sih CJ, Hamilton PB, Knight SG. Demonstration of the pentose cycle reactions in Penicillium chrysogenum. J. Bacteriol 1957; 73: 447-451.
4. Niederpruem DJ. Carbohydrate metabolism. 2. Tricarboxylic acid cycle. In: Ainsworth GC and Sussman (eds.) AS. The fungi, vol. I. New York: Academic Press; 1965. p. 269-300.
5. Godzeski C, Stone RW. Dehydrogenases of the tricarboxylic acid cycle in Penicillium chrysogenum. Arch. Biochem 1955; 59: 132-144.
6. Smith EL, Hill RL, Lehman IR, Lerkowit RJ, Handler P, White A. Biological oxidations. In: Laufer RS, Warren E, Mclvor (eds.) D. Principles of Biochemistry: General Aspects. USA: Donnelley RR and Sons Company press; 1983. p. 316-357.
1. Menezes, JC, Alves, SS, Lemos, JM, Feyo S, Azevedo de, Mathematical modelling of industrial pilot-plant penicillin-G fed-batch fermentations. Journal of Chemical Technology & Biotechnology 1994; 61: 123-138.
2. Paul, GC, Thomas, CR, A Structured Model for Hyphal Differentiation and Penicillin Production Using Penicillium chrysogenum. Biotechnology and Bioengineering 1996; 51: 558-572.
3. Wang ZF, Lauwerijssen MJC, Yuan JQ. Combined age and segregated kinetic model forindustrial-scale penicillin fed-batch cultivation. Biotechnology and Bioprocess Engineering 2005; 10: 142-148.
4. Bailey JE. Toward a science of metabolic engineering. Science 1991; 252: 1668-1675.
5. Smith EL, Hill RL, Lehman IR, Lerkowit RJ, Handler P, White A. Biological oxidations. In: Laufer RS, Warren E, Mclvor (eds.) D. Principles of Biochemistry: General Aspects. USA: Donnelley RR and Sons Company press; 1983. p. 316-357.
6. Bellgardt K-H. Modellbildung des Wachstums von Saccharomyces cerevisiae in Ruehrkesselreaktoren. Germany: University of Hannover; PhD Dissertation; 1983.
7. Monod J. The growth of bacterial culture. Ann. Rev. Microbiol. 1949; 3: 371-372.
8. Luedeking R, Piret EL. A kinetic study of the lactic acid fermentation. Batch process at controlled pH. Journal of Biochemical and Microbiological Technology and Engineering 1959; 1: 393-412.
9. Nielsen J, J?rgensen HS. Metabolic control analysis of the penicillin biosynthetic pathway in a high-yielding strain of Penicillium chrysogenum. Biotechnology Progress 1995; 11: 299-305.
10. Kompala DS, Ramkrishna D, Jansen NB, Tsao GT (1986) Investigation of bacterial growth on mixed substrates: experimental evaluation of cybernetic models. Biotechnol Bioeng 28: 1044-1055
11. Ramkrishna D (2003) On modeling of bioreactors for control. J Proc Cont 13: 581-589
12. Ramkrishna D (1982) A cybernetic perspective of microbial growth, In: foundations of biochemical engineering: Kinetics and Thermodynamics in Biological systems. American chemical society Publication
13. Paul GC, Thomas CR (1996) A Structured Model for Hyphal Differentiation and Penicillin Production Using Penicillium chrysogenum. Biotechnol Bioeng 51: 558-572
14. Nielsen J, J?rgensen HS (1996) A Kinetic model for the penicillin biosynthetic pathwway in Penicillium Chrysogenum. Control Eng 4: 765-771
15. Varner J, Ramkrishna D (1999) The non-linear analysis of cybernetic models. Guidelines for model formulation. J Biotechnol 71: 67-104
16. Straight JV, Ramkrishna D (1990) Regulation of complex growth dynamics-substitutable and complementary process, In: Abstracts of Papers of the American Chemical Society 200,4-BIOT
1. Straight JV, Ramkrishna D (1994) Cybernetic modeling and regulation of metabolic pathways growth on complementary nutrients. Biotechnol Prog 10:574-587
2. Straight JV, Ramkrishna D (1990) Regulation of complex growth dynamics-substitutable and complementary process, In: Abstracts of Papers of the American Chemical Society 200, 4-BIOT
3. Ramakrishna R, Ramkrishna D, Konopka AE (1996) Cybernetic modeling of growth in mixed, substitutable environments: Preferential and simulation utilization. Biotechnol Bioeng 52, 141-151.
4. Liu YH, Yuan JQ (2008) A cybernetic approach based dynamic model for myeloma cell cultivation. Appl Math Comput 205: 84-97
5. Guardia MJ, Gambhir A, Europa AF (1986) Cybernetic modeling and regulation of metabolic pathways in multiple steady states of hybridoma cells. Biotechnol Bioeng 28: 927-937
6. Namjoshi AA, Hu WS, Ramkrishna D (2002) Unveiling steady-state multiplicity in hybridoma cultures: The cybernetic approach. Biotechnol Bioeng 81: 79-91
7. Varner J, Ramkrishna D (1999) Metabolic engineering from a cybernetic perspective: 1, theoretical preliminaries. Biotechnol Proc 15: 407-425
1. Ren HT, Yuan JQ, Bellgardt K-H, Macrokinetic model for methylotrophic Pichia pastoris based on stoichiometric balance, Journal of Biotechnology, 2003;106: 53-68.
2.周海英,袁景淇,邓建慧等.重组毕氏酵母发酵过程的结构模型.上海交通大学学报,2002, 36 (10): 1443-1447.
2.王磊,任东明.青霉素—从发现到应用.生物学通报,2006;41-12
3.李鹏,李广州.青霉素结构的探究.南京师范大学化学与环境科学学院,2007;7
4.刘毅,王海清.采用最小二乘支持向量机的青霉素发酵过程建模研究.生物工程学报,2006;22-1
5. Wang SQ , Yuan YJ . Automatic Technologyof Biochemical Process, Beijing : Chemical Industry Press, 1999
6. Nomikos P , MacGregor JF. Monitoring batch processes using multiway principal component analysis. AIChE J , 1994 , 40 ( 8) :1361 - 1375
7. Massimo CD , Montague GA , Willis MJ et al . Towards improvedpenicillin fermentation via artificial neural networks. Computers andChemical Engineering ,1992 ,16 (4) :283– 291
8.葛爱冬,隋青美,王斌鹏.混合神经网络建模方法在青霉素发酵过程中的应用.山东轻工业学院学报,2006;20-3
9.肖占中.战争与青霉素的命运.国防科技,2006; 85-87.
10.郭勇.生物制药技术M .北京:中国轻工业出版社,2001.
11.金晓华.β-内酰胺类抗生素在医药市场的发展前景.商场现代化,2006; 47: 107.
12. Borovkov A A. Asymptotic Methodsin Queuing Theory. translated by D.Newton,John Wiley & Sons,1984
13. Chaudhry M L& Templeton J G C. Afirst Course in Bulk Queues. JohnWiley & Sons,1984
14. Ronald Bentley. Journal of Chemical Education ,2004 ,81 ( 10) :1462 - 1470
15. Roland Reiner.蒋天蓉译.抗生素入门.合肥:安徽科学技术出版社,1985 :37– 46
16.沈萍.微生物学M.北京:高等教育出版社,2000.
17. Chen, L., Bastin, G., van Breusegem, V. A case study of adaptive nonlinear regulation of fed-batch biological reactors. Automatica 1995, 31, 55-65.
18. Montague, G., Morris, J. 1994 Nerual network contribution in biotechnology. Trends in Biotechnology 1994, 12, 312-323.
19. Horiuchia, J.I., Kishimotob, M. App;ication of fuzzy control to industrial bioprocesses in Japan.Fuzzy Sets and Systems 2002, 128, 117-124.
20.赵娟平,陈健,姜长洪.青霉素发酵过程建模研究.计算机仿真,2008;25-2
21. Bailey, J.E. Toward a science of metabolic engineering. Scienc 1991, 252, 1668-1675.
22. Stephanopoulos, G., Vallino, J.J. Network rigidity and metabolic engineering in metabolite overproduction. Science 1991, 252, 1675-1681.
23. Granstroem, T., Aristos, A.A., Leisola, M. Metabolic flux analysis of Candida tropcalis growing on xylose in an oxygen-limited chemostat. Metabolic Engineering 2002, 4, 248-256.
24. Lee, K., Berthiaume, F., Stephanopoulos, G.N., Yarmush, D.M., Yarmush, M.L. Metabolic flux analysis of postburn hepatic hypermetabolism. Metabolic Engineering 2000, 2, 312-327.
25. Zhao, J., Shimizu, K. Metabolic flux analysis of Excherichia coli K12 grown on 13Clabeled acetate and glucose using GC-MS and powerful flux calculation method. Journal of Biotechnology 2003, 101, 101-117.
26. Bailey, J.E. Mathematical modeling and analysis in biochemical engineering: Past accomplishments and future opportunities. Biotechnology Progress 1998, 14, 8-20.
27. Herwig, C., von Stockar, U. A small metabolic flux model to identify transient metabolic regulations in Saccharomyces cerevisia. Bioprocess and Biosystems Engineering 2002, 24, 395-403.
28. Yu, J., Wang, J.P. Metabolic flux modeling of detoxification of acetic acid by Ralstonia europha at slightly alkaline pH levels. Biotechnology and Bioengineering 2001, 73, 458-464.
1. Jφrgensen H, Nielsen J, Villadsen J. Metabolic flux distributions in Penicillium chrysogenum during fed-batch cultivations. Biotechnology and Bioengineering 1995; 46: 117-131.
2. Blumenthal HJ. Carbohydrate metabolism. 1. Glycolysis. In: Ainsworth GC and Sussman (eds.) AS. The fungi, vol. I. New York: Academic Press; 1965. p. 229-268.
3. Sih CJ, Hamilton PB, Knight SG. Demonstration of the pentose cycle reactions in Penicillium chrysogenum. J. Bacteriol 1957; 73: 447-451.
4. Niederpruem DJ. Carbohydrate metabolism. 2. Tricarboxylic acid cycle. In: Ainsworth GC and Sussman (eds.) AS. The fungi, vol. I. New York: Academic Press; 1965. p. 269-300.
5. Godzeski C, Stone RW. Dehydrogenases of the tricarboxylic acid cycle in Penicillium chrysogenum. Arch. Biochem 1955; 59: 132-144.
6. Smith EL, Hill RL, Lehman IR, Lerkowit RJ, Handler P, White A. Biological oxidations. In: Laufer RS, Warren E, Mclvor (eds.) D. Principles of Biochemistry: General Aspects. USA: Donnelley RR and Sons Company press; 1983. p. 316-357.
1. Menezes, JC, Alves, SS, Lemos, JM, Feyo S, Azevedo de, Mathematical modelling of industrial pilot-plant penicillin-G fed-batch fermentations. Journal of Chemical Technology & Biotechnology 1994; 61: 123-138.
2. Paul, GC, Thomas, CR, A Structured Model for Hyphal Differentiation and Penicillin Production Using Penicillium chrysogenum. Biotechnology and Bioengineering 1996; 51: 558-572.
3. Wang ZF, Lauwerijssen MJC, Yuan JQ. Combined age and segregated kinetic model forindustrial-scale penicillin fed-batch cultivation. Biotechnology and Bioprocess Engineering 2005; 10: 142-148.
4. Bailey JE. Toward a science of metabolic engineering. Science 1991; 252: 1668-1675.
5. Smith EL, Hill RL, Lehman IR, Lerkowit RJ, Handler P, White A. Biological oxidations. In: Laufer RS, Warren E, Mclvor (eds.) D. Principles of Biochemistry: General Aspects. USA: Donnelley RR and Sons Company press; 1983. p. 316-357.
6. Bellgardt K-H. Modellbildung des Wachstums von Saccharomyces cerevisiae in Ruehrkesselreaktoren. Germany: University of Hannover; PhD Dissertation; 1983.
7. Monod J. The growth of bacterial culture. Ann. Rev. Microbiol. 1949; 3: 371-372.
8. Luedeking R, Piret EL. A kinetic study of the lactic acid fermentation. Batch process at controlled pH. Journal of Biochemical and Microbiological Technology and Engineering 1959; 1: 393-412.
9. Nielsen J, J?rgensen HS. Metabolic control analysis of the penicillin biosynthetic pathway in a high-yielding strain of Penicillium chrysogenum. Biotechnology Progress 1995; 11: 299-305.
10. Kompala DS, Ramkrishna D, Jansen NB, Tsao GT (1986) Investigation of bacterial growth on mixed substrates: experimental evaluation of cybernetic models. Biotechnol Bioeng 28: 1044-1055
11. Ramkrishna D (2003) On modeling of bioreactors for control. J Proc Cont 13: 581-589
12. Ramkrishna D (1982) A cybernetic perspective of microbial growth, In: foundations of biochemical engineering: Kinetics and Thermodynamics in Biological systems. American chemical society Publication
13. Paul GC, Thomas CR (1996) A Structured Model for Hyphal Differentiation and Penicillin Production Using Penicillium chrysogenum. Biotechnol Bioeng 51: 558-572
14. Nielsen J, J?rgensen HS (1996) A Kinetic model for the penicillin biosynthetic pathwway in Penicillium Chrysogenum. Control Eng 4: 765-771
15. Varner J, Ramkrishna D (1999) The non-linear analysis of cybernetic models. Guidelines for model formulation. J Biotechnol 71: 67-104
16. Straight JV, Ramkrishna D (1990) Regulation of complex growth dynamics-substitutable and complementary process, In: Abstracts of Papers of the American Chemical Society 200,4-BIOT
1. Straight JV, Ramkrishna D (1994) Cybernetic modeling and regulation of metabolic pathways growth on complementary nutrients. Biotechnol Prog 10:574-587
2. Straight JV, Ramkrishna D (1990) Regulation of complex growth dynamics-substitutable and complementary process, In: Abstracts of Papers of the American Chemical Society 200, 4-BIOT
3. Ramakrishna R, Ramkrishna D, Konopka AE (1996) Cybernetic modeling of growth in mixed, substitutable environments: Preferential and simulation utilization. Biotechnol Bioeng 52, 141-151.
4. Liu YH, Yuan JQ (2008) A cybernetic approach based dynamic model for myeloma cell cultivation. Appl Math Comput 205: 84-97
5. Guardia MJ, Gambhir A, Europa AF (1986) Cybernetic modeling and regulation of metabolic pathways in multiple steady states of hybridoma cells. Biotechnol Bioeng 28: 927-937
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