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Erschienen in: Software and Systems Modeling 2/2015

01.05.2015 | Special Section Paper

Petri nets in systems biology

verfasst von: Ina Koch

Erschienen in: Software and Systems Modeling | Ausgabe 2/2015

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Abstract

Petri nets are used in many areas. This article discusses the application of Petri nets in systems biology. Using an example from biochemistry, concepts for the automatic decomposition of biochemical systems are introduced. The article focuses on those concepts that fulfill steady-state conditions. Interestingly, all the concepts are based on minimal, semi-positive transition invariants. The article describes, which new definitions for network decomposition can be derived and how they can be interpreted in the context of biology. This is illustrated with the example of the citric acid cycle, for which a new metabolic pathway could be predicted with the help of such an analysis.

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Literatur
1.
Zurück zum Zitat Ackermann, J., Koch, I.: Quantitative analysis. In: Koch, I., Reisig, W., Schreiber, F. (eds.) Modeling in Systems Biology: The Petri Net Approach, pp. 153–178. Springer, Berlin (2011) (Comp. Biol.) Ackermann, J., Koch, I.: Quantitative analysis. In: Koch, I., Reisig, W., Schreiber, F. (eds.) Modeling in Systems Biology: The Petri Net Approach, pp. 153–178. Springer, Berlin (2011) (Comp. Biol.)
2.
Zurück zum Zitat Ackermann, J., et al.: Reduction techniques for network validation in systems biology. J. Theor. Biol. 315, 71–80 (2012)CrossRefMathSciNet Ackermann, J., et al.: Reduction techniques for network validation in systems biology. J. Theor. Biol. 315, 71–80 (2012)CrossRefMathSciNet
3.
Zurück zum Zitat Backhaus, K., et al.: Multivariate Analysis Methods. An Application-Oriented Introduction, 10th edn. Springer, Berlin (2003) (in German) Backhaus, K., et al.: Multivariate Analysis Methods. An Application-Oriented Introduction, 10th edn. Springer, Berlin (2003) (in German)
4.
Zurück zum Zitat Bahi-Jaber, N., Pontier, D.: Modeling transmission of directly transmitted infectious diseases using colored stochastic Petri nets. Math. Biosci. 185, 1–13 (2003)CrossRefMATHMathSciNet Bahi-Jaber, N., Pontier, D.: Modeling transmission of directly transmitted infectious diseases using colored stochastic Petri nets. Math. Biosci. 185, 1–13 (2003)CrossRefMATHMathSciNet
5.
Zurück zum Zitat Berg, J.M., Tymoczko, J.L., Stryer, L.: Biochemistry, 5th edn. W.H. Freeman, New York (2002) Berg, J.M., Tymoczko, J.L., Stryer, L.: Biochemistry, 5th edn. W.H. Freeman, New York (2002)
6.
Zurück zum Zitat Bortfeldt, R.H., Schuster, S., Koch, I.: Exhaustive analysis of the modular structure of the spliceosomal assembly network: a Petri net approach. In Silico Biol. 10, 89–123 (2010). doi:10.3233/ISB-2010-0419 Bortfeldt, R.H., Schuster, S., Koch, I.: Exhaustive analysis of the modular structure of the spliceosomal assembly network: a Petri net approach. In Silico Biol. 10, 89–123 (2010). doi:10.​3233/​ISB-2010-0419
7.
Zurück zum Zitat Chaouiya, C.: Petri net modelling of biochemical systems. Brief. Bioinform. 8.4, 210–219 (2007)CrossRef Chaouiya, C.: Petri net modelling of biochemical systems. Brief. Bioinform. 8.4, 210–219 (2007)CrossRef
8.
Zurück zum Zitat Clarke, B.L.: Complete set of steady states for the general stoichiometric dynamical system. J. Chem. Phys. 75, 4970 (1981)CrossRefMathSciNet Clarke, B.L.: Complete set of steady states for the general stoichiometric dynamical system. J. Chem. Phys. 75, 4970 (1981)CrossRefMathSciNet
9.
Zurück zum Zitat Clarke, B.L.: Stoichiometric network analysis. Cell Biophys. 12, 237–253 (1988)CrossRef Clarke, B.L.: Stoichiometric network analysis. Cell Biophys. 12, 237–253 (1988)CrossRef
10.
Zurück zum Zitat Doi, A., et al.: Simulation based validation of the p53 transcriptional activity with hybrid functional Petri net. In Silico Biol. 6.1–2, 1–13 (2006) Doi, A., et al.: Simulation based validation of the p53 transcriptional activity with hybrid functional Petri net. In Silico Biol. 6.1–2, 1–13 (2006)
11.
Zurück zum Zitat Einloft, J., et al.: MonaLisa—visualization and analysis of functional modules in biochemical networks. Bioinformatics 29, 1469–1470 (2013)CrossRef Einloft, J., et al.: MonaLisa—visualization and analysis of functional modules in biochemical networks. Bioinformatics 29, 1469–1470 (2013)CrossRef
12.
Zurück zum Zitat Esparza, J.: Decidability and complexity of Petri net problems—an introduction. LNCS 1491, 374–428 (1998) Esparza, J.: Decidability and complexity of Petri net problems—an introduction. LNCS 1491, 374–428 (1998)
13.
Zurück zum Zitat Fieber, M.: Design and Implementation of a Generic and Adaptive Tool for Graph Manipulation. Master’s Thesis. Brandenburg University of Technology at Cottbus (2004) (in German) Fieber, M.: Design and Implementation of a Generic and Adaptive Tool for Graph Manipulation. Master’s Thesis. Brandenburg University of Technology at Cottbus (2004) (in German)
14.
Zurück zum Zitat Finney, A., Hucka, M.: Systems biology markup language: level 2 and beyond. Biochem. Soc. Trans. 31, 1472–1473 (2003)CrossRef Finney, A., Hucka, M.: Systems biology markup language: level 2 and beyond. Biochem. Soc. Trans. 31, 1472–1473 (2003)CrossRef
15.
Zurück zum Zitat Fischer, E., Sauer, U.: A novel metabolic cycle catalyzes glucose oxidation and anaplerosis in hungry Escherichia coli. J. Biol. Chem. 278.47, 46446–46451 (2003)CrossRef Fischer, E., Sauer, U.: A novel metabolic cycle catalyzes glucose oxidation and anaplerosis in hungry Escherichia coli. J. Biol. Chem. 278.47, 46446–46451 (2003)CrossRef
16.
Zurück zum Zitat Förster, J., et al.: Genome-scale reconstruction of the Saccharomyces cerevisiae metabolic network. Genome Res. 13, 244–253 (2003)CrossRef Förster, J., et al.: Genome-scale reconstruction of the Saccharomyces cerevisiae metabolic network. Genome Res. 13, 244–253 (2003)CrossRef
17.
Zurück zum Zitat Genrich, H., Küffner, R., Voss, K.: Executable Petri net models for the analysis of metabolic pathways. J. Softw. Tools Technol. Transf. 3.4, 394–404 (2001) Genrich, H., Küffner, R., Voss, K.: Executable Petri net models for the analysis of metabolic pathways. J. Softw. Tools Technol. Transf. 3.4, 394–404 (2001)
18.
Zurück zum Zitat Grafahrend-Belau, E.: Classification of T-Invariants in Biochemical Petri Nets Based on Different Cluster Analysis Techniques. Master’s Thesis. Technical University of Applied Sciences Berlin (2006) (in German) Grafahrend-Belau, E.: Classification of T-Invariants in Biochemical Petri Nets Based on Different Cluster Analysis Techniques. Master’s Thesis. Technical University of Applied Sciences Berlin (2006) (in German)
19.
Zurück zum Zitat Grafahrend-Belau, E., et al.: Modularisation of biochemical networks through hierarchical cluster analysis of T-invariants of biochemical Petri nets. BMC Bioinform. 9, 90 (2008)CrossRef Grafahrend-Belau, E., et al.: Modularisation of biochemical networks through hierarchical cluster analysis of T-invariants of biochemical Petri nets. BMC Bioinform. 9, 90 (2008)CrossRef
20.
Zurück zum Zitat Grunwald, S., et al.: Petri net modelling of gene regulation of the Duchenne muscular dystrophy. Biosystems 92, 189–205 (2008)CrossRef Grunwald, S., et al.: Petri net modelling of gene regulation of the Duchenne muscular dystrophy. Biosystems 92, 189–205 (2008)CrossRef
21.
Zurück zum Zitat Heinrich, R., Rapoport, T.A.: A linear steady-state treatment of enzymatic chains. General properties, control and effector strength. Eur. J. Biochem. 42, 89–95 (1974)CrossRef Heinrich, R., Rapoport, T.A.: A linear steady-state treatment of enzymatic chains. General properties, control and effector strength. Eur. J. Biochem. 42, 89–95 (1974)CrossRef
22.
Zurück zum Zitat Hoops, S., et al.: COPASI—a COmplex PAthway SImulator. Bioinformatics 22.24, 3067–3074 (2006)CrossRef Hoops, S., et al.: COPASI—a COmplex PAthway SImulator. Bioinformatics 22.24, 3067–3074 (2006)CrossRef
23.
Zurück zum Zitat Kielbassa, J., et al.: Modeling of the U1 snRNP assembly pathway in alternative splicing in human cells using Petri nets. Comput. Biol. Chem. 33, 46–61 (2009)CrossRefMATHMathSciNet Kielbassa, J., et al.: Modeling of the U1 snRNP assembly pathway in alternative splicing in human cells using Petri nets. Comput. Biol. Chem. 33, 46–61 (2009)CrossRefMATHMathSciNet
24.
Zurück zum Zitat Klamt, S., Gilles, E.D.: Minimal cut sets in biochemical reaction networks. Bioinformatics 20.2, 226–234 (2004)CrossRef Klamt, S., Gilles, E.D.: Minimal cut sets in biochemical reaction networks. Bioinformatics 20.2, 226–234 (2004)CrossRef
25.
Zurück zum Zitat Koch, I.: Petri Nets and GRN models. In: Das, S., et al. (eds.) Handbook of Research on Computational Methodologies in Gene Regulatory Networks, pp. 604–637. IGI Global, Hershey, NY (2010) Koch, I.: Petri Nets and GRN models. In: Das, S., et al. (eds.) Handbook of Research on Computational Methodologies in Gene Regulatory Networks, pp. 604–637. IGI Global, Hershey, NY (2010)
26.
Zurück zum Zitat Koch, I., Junker, B.H., Heiner, M.: Application of Petri net theory for modelling and validation of the sucrose breakdown pathway in the potato tuber. Bioinformatics 21, 1219–1226 (2005)CrossRef Koch, I., Junker, B.H., Heiner, M.: Application of Petri net theory for modelling and validation of the sucrose breakdown pathway in the potato tuber. Bioinformatics 21, 1219–1226 (2005)CrossRef
27.
Zurück zum Zitat Koch, I., Reisig, W., Schreiber, F.: Modeling in Systems Biology: The Petri Net Approach. Springer, Berlin (2011). (Comp. Biol.)CrossRef Koch, I., Reisig, W., Schreiber, F.: Modeling in Systems Biology: The Petri Net Approach. Springer, Berlin (2011). (Comp. Biol.)CrossRef
28.
Zurück zum Zitat Larhlimi, A., Bockmayr, A.: A new constraint-based description of the steady-state flux cone of metabolic networks. Disc. Appl. Math. 157, 2257–2266 (2009)CrossRefMATHMathSciNet Larhlimi, A., Bockmayr, A.: A new constraint-based description of the steady-state flux cone of metabolic networks. Disc. Appl. Math. 157, 2257–2266 (2009)CrossRefMATHMathSciNet
29.
Zurück zum Zitat Lautenbach, K.: Exact conditions of liveness for a class of Petri nets. Berichte der GMD 82. Sankt Augustin: Gesellschaft für Mathematik und Datenverarbeitung (1973) (in German) Lautenbach, K.: Exact conditions of liveness for a class of Petri nets. Berichte der GMD 82. Sankt Augustin: Gesellschaft für Mathematik und Datenverarbeitung (1973) (in German)
30.
Zurück zum Zitat Liao, J., Hou, S.-Y., Chao, Y.-P.: Pathway analysis, engineering, and physiological considerations for redirecting central metabolism. Biotechnol. Bioeng. 52, 129–140 (1996)CrossRef Liao, J., Hou, S.-Y., Chao, Y.-P.: Pathway analysis, engineering, and physiological considerations for redirecting central metabolism. Biotechnol. Bioeng. 52, 129–140 (1996)CrossRef
31.
Zurück zum Zitat Matsuno, H., et al.: Hybrid Petri net representation of gene regulatory network. Proc. Pac. Symp. Biocomput. 5, 338–349 (2000) Matsuno, H., et al.: Hybrid Petri net representation of gene regulatory network. Proc. Pac. Symp. Biocomput. 5, 338–349 (2000)
33.
Zurück zum Zitat Mura, I.: Stochastic modeling. In: Koch, I., Reisig, W., Schreiber, F. (eds.) Modeling in Systems Biology: The Petri Net Approach, pp. 121–152. Springer, Berlin (2011) (Comp. Biol) Mura, I.: Stochastic modeling. In: Koch, I., Reisig, W., Schreiber, F. (eds.) Modeling in Systems Biology: The Petri Net Approach, pp. 121–152. Springer, Berlin (2011) (Comp. Biol)
34.
Zurück zum Zitat Nagasaki, N., et al.: Cell illustrator 4.0: a computational platform for systems biology. Stud. Health Technol. Inform. 162, 160–181 (2011) Nagasaki, N., et al.: Cell illustrator 4.0: a computational platform for systems biology. Stud. Health Technol. Inform. 162, 160–181 (2011)
36.
Zurück zum Zitat Peleg, M., Rubin, D., Altman, R.B.: Using Petri net tools to study properties and dynamics of biological systems. J. Am. Med. Inf. Assoc. 12.2, 369–371 (2005) Peleg, M., Rubin, D., Altman, R.B.: Using Petri net tools to study properties and dynamics of biological systems. J. Am. Med. Inf. Assoc. 12.2, 369–371 (2005)
37.
Zurück zum Zitat Pèrés, S., et al.: ACoM: a classification method for elementary flux modes based on motif finding. Biosystems 103(3), 410–419 (2011)CrossRef Pèrés, S., et al.: ACoM: a classification method for elementary flux modes based on motif finding. Biosystems 103(3), 410–419 (2011)CrossRef
38.
Zurück zum Zitat Pfeiffer, T., et al.: METATOOL: for studying metabolic networks. Bioinformatics 15.3, 251–257 (1999)CrossRef Pfeiffer, T., et al.: METATOOL: for studying metabolic networks. Bioinformatics 15.3, 251–257 (1999)CrossRef
39.
Zurück zum Zitat Popova-Zeugmann, L., Heiner, M., Koch, I.: Time Petri nets for modeling and analysis of biochemical networks. Fundam. Inform. 67, 149–162 (2005)MATHMathSciNet Popova-Zeugmann, L., Heiner, M., Koch, I.: Time Petri nets for modeling and analysis of biochemical networks. Fundam. Inform. 67, 149–162 (2005)MATHMathSciNet
40.
Zurück zum Zitat Priami, C., et al.: Application of a stochastic name-passing calculus to representation and simulation of molecular processes. Inf. Proc. Lett. 80, 25–31 (2001)CrossRefMATHMathSciNet Priami, C., et al.: Application of a stochastic name-passing calculus to representation and simulation of molecular processes. Inf. Proc. Lett. 80, 25–31 (2001)CrossRefMATHMathSciNet
41.
Zurück zum Zitat Reddy, V.N.: Modeling Biological Pathways: A Discrete Event Systems Approach. Master’s Thesis. University of Maryland, USA (1994) Reddy, V.N.: Modeling Biological Pathways: A Discrete Event Systems Approach. Master’s Thesis. University of Maryland, USA (1994)
42.
Zurück zum Zitat Reddy, V.N., Liebman, M.N., Mavrovouniotis, M.L.: Qualitative analysis of biochemical reaction systems. Comput. Biol. Med. 26.2, 9–24 (1996)CrossRef Reddy, V.N., Liebman, M.N., Mavrovouniotis, M.L.: Qualitative analysis of biochemical reaction systems. Comput. Biol. Med. 26.2, 9–24 (1996)CrossRef
43.
Zurück zum Zitat Reddy, V.N., Mavrovouniotis, M.L., Liebman, M.N.: Petri net representations in metabolic pathways. In: Hunter, L., Searls, D., Shavlik, J. (eds.) Proceedings of the First International Conference on Intelligent Systems for Molecular Biology, vol. 1, pp. 328–336. AAAI Press, Menlo Park, CA, USA (1993) Reddy, V.N., Mavrovouniotis, M.L., Liebman, M.N.: Petri net representations in metabolic pathways. In: Hunter, L., Searls, D., Shavlik, J. (eds.) Proceedings of the First International Conference on Intelligent Systems for Molecular Biology, vol. 1, pp. 328–336. AAAI Press, Menlo Park, CA, USA (1993)
44.
Zurück zum Zitat Regev, A., Silverman, W., Shapiro, E.: Representation and simulation of biochemical processes using the pi-calculus process algebra. Proc. Pac. Symp. Biocomput. 6, 459–470 (2001) Regev, A., Silverman, W., Shapiro, E.: Representation and simulation of biochemical processes using the pi-calculus process algebra. Proc. Pac. Symp. Biocomput. 6, 459–470 (2001)
45.
Zurück zum Zitat Sackmann, A.: Modelling and Simulation of Signaltransduction Pathways of Saccharomyces cerevisiae Based on Petri Net Theory. Diploma Thesis. Ernst Moritz Arndt-University, Greifswald (2005) (in German) Sackmann, A.: Modelling and Simulation of Signaltransduction Pathways of Saccharomyces cerevisiae Based on Petri Net Theory. Diploma Thesis. Ernst Moritz Arndt-University, Greifswald (2005) (in German)
46.
Zurück zum Zitat Sackmann, A., Heiner, M., Koch, I.: Application of Petri net based analysis techniques to signal transduction pathways. BMC Bioinform. 7, 482 (2006)CrossRef Sackmann, A., Heiner, M., Koch, I.: Application of Petri net based analysis techniques to signal transduction pathways. BMC Bioinform. 7, 482 (2006)CrossRef
47.
Zurück zum Zitat Sackmann, A., et al.: An analysis of the Petri net based model of the human body iron homeostasis process. Comput. Biol. Chem. 31, 1–10 (2007)CrossRefMATH Sackmann, A., et al.: An analysis of the Petri net based model of the human body iron homeostasis process. Comput. Biol. Chem. 31, 1–10 (2007)CrossRefMATH
48.
Zurück zum Zitat Schrijver, A.: Theory of Linear and Integer Programming. Wiley-VCH, Weinheim (1998)MATH Schrijver, A.: Theory of Linear and Integer Programming. Wiley-VCH, Weinheim (1998)MATH
49.
Zurück zum Zitat Schuster, S., Dandekar, T., Fell, D.A.: Detection of elementary flux modes in biochemical networks: a promising tool for pathway analysis and metabolic engineering. Trends Biotechnol. 17.2, 53–60 (1999)CrossRef Schuster, S., Dandekar, T., Fell, D.A.: Detection of elementary flux modes in biochemical networks: a promising tool for pathway analysis and metabolic engineering. Trends Biotechnol. 17.2, 53–60 (1999)CrossRef
50.
Zurück zum Zitat Schuster, S., Hilgetag, C.: On elementary flux modes in biochemical reaction systems at steady state. J. Biol. Syst. 2, 165–182 (1994)CrossRef Schuster, S., Hilgetag, C.: On elementary flux modes in biochemical reaction systems at steady state. J. Biol. Syst. 2, 165–182 (1994)CrossRef
51.
Zurück zum Zitat Schuster, S., Hilgetag, C., Schuster, R.: Determining elementary modes of functioning in biochemical reaction networks at steady state. In: Ghista, D.N. (eds.) Biomed. and Life Phys. Vieweg Wiesbaden, pp. 101–114 (1996) Schuster, S., Hilgetag, C., Schuster, R.: Determining elementary modes of functioning in biochemical reaction networks at steady state. In: Ghista, D.N. (eds.) Biomed. and Life Phys. Vieweg Wiesbaden, pp. 101–114 (1996)
52.
Zurück zum Zitat Schuster, S., et al.: Exploring the pathway structure of metabolism: decomposition into subnetworks and application to Mycoplasma pneumoniae. Bioinformatics 18, 352–361 (2002) Schuster, S., et al.: Exploring the pathway structure of metabolism: decomposition into subnetworks and application to Mycoplasma pneumoniae. Bioinformatics 18, 352–361 (2002)
53.
Zurück zum Zitat Srivastava, R., Peterson, M.S., Nentley, W.E.: Stochastic kinetic analysis of the Escherichia coli stress circuit using \(\sigma \)-32 targeted antisense. Biotechol. Bioeng. 231.1, 120–129 (2001)CrossRef Srivastava, R., Peterson, M.S., Nentley, W.E.: Stochastic kinetic analysis of the Escherichia coli stress circuit using \(\sigma \)-32 targeted antisense. Biotechol. Bioeng. 231.1, 120–129 (2001)CrossRef
54.
Zurück zum Zitat Steinhausen, D., Langer, K.: Cluster Analysis. An Introduction to Methods for Automatic Classification. de Gruyter, Berlin (1977) (in German) Steinhausen, D., Langer, K.: Cluster Analysis. An Introduction to Methods for Automatic Classification. de Gruyter, Berlin (1977) (in German)
55.
Zurück zum Zitat Voss, K., Heiner, M., Koch, I.: Steady state analysis of metabolic pathways using Petri nets. In Silico Biol. 3.3, 367–387 (2003) Voss, K., Heiner, M., Koch, I.: Steady state analysis of metabolic pathways using Petri nets. In Silico Biol. 3.3, 367–387 (2003)
56.
Zurück zum Zitat Wang, L., Li, P.: Microfluidic DNA microarray analysis: a review. Anal. Chim. Acta 687.1, 12–27 (2011) Wang, L., Li, P.: Microfluidic DNA microarray analysis: a review. Anal. Chim. Acta 687.1, 12–27 (2011)
57.
Zurück zum Zitat Wick, L.M., Quadroni, M., Egli, T.: Short- and long-term changes in proteome composition and kinetic properties in a culture of Escherichia coli during transition from glucose-excess to glucose limited growth conditions in continuous culture and vice versa. Environ. Microbiol. 3, 588–599 (2001)CrossRef Wick, L.M., Quadroni, M., Egli, T.: Short- and long-term changes in proteome composition and kinetic properties in a culture of Escherichia coli during transition from glucose-excess to glucose limited growth conditions in continuous culture and vice versa. Environ. Microbiol. 3, 588–599 (2001)CrossRef
58.
Zurück zum Zitat Windhager, L., Erhard, F., Zimmer, R.: Fuzzy Modeling. In: Koch, I., Reisig, W., Schreiber, F. (eds.) Modeling in Systems Biology: The Petri Net Approach. Comp. Biol., pp. 179–205. Springer, Berlin (2011) Windhager, L., Erhard, F., Zimmer, R.: Fuzzy Modeling. In: Koch, I., Reisig, W., Schreiber, F. (eds.) Modeling in Systems Biology: The Petri Net Approach. Comp. Biol., pp. 179–205. Springer, Berlin (2011)
Metadaten
Titel
Petri nets in systems biology
verfasst von
Ina Koch
Publikationsdatum
01.05.2015
Verlag
Springer Berlin Heidelberg
Erschienen in
Software and Systems Modeling / Ausgabe 2/2015
Print ISSN: 1619-1366
Elektronische ISSN: 1619-1374
DOI
https://doi.org/10.1007/s10270-014-0421-5

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