Abstract
This chapter provides an introduction to the concepts underlying the stochastic modeling of biological systems with Petri Nets. It introduces a timed interpretation of the occurrence of transitions in a net that suites the randomness observed in biochemical reactions occurring in living matter. Thanks to the foundational work of Gillespie in the 70s, this randomness can be easily accounted for by the representative power of Stochastic Petri Nets. The chapter illustrates the Stochastic Petri Net model specification process, the possibilities of analytical and numerical evaluation of model dynamics as well as the basic concepts underlying the simulative approaches, through the application to simple instances of biological systems to help the reader familiarizing with this discrete stochastic modeling formalism. Additional examples of larger scale models are presented, and exercises suggested to consolidate the understanding of the main concepts.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The very useful closure property of statistically independent negative exponential random variables under the minimum operator is easily demonstrated. Suppose X and Y are negative exponential random variables of rate λ X and λ Y , respectively, and let the random variable Z to be defined as Z=MIN{X,Y}. Then, the cumulative density function F Z (t) of Z can be computed as \(F_{Z}(t)=\mathbb{P}[Z\leq t]=1-\mathbb{P}[Z>t]=1-\mathbb{P}[\mathrm{MIN}\{X,Y\}>t]=1- \mathbb{P}[X>t,Y>t]=1-\mathbb{P}[X>t]\mathbb{P}[Y>t]=1-e^{-\lambda_{X}}e^{-\lambda_{Y}}=1-e^{-(\lambda_{X}+\lambda_{Y})}\), which is the same as to say that Z is distributed as a negative exponential random variable of parameter λ X +λ Y .
References
Alexopoulos, C.: Statistical analysis of simulation output: state of the art. In: WSC’07: Proceedings of the 39th conference on Winter simulation, pp. 150–161. IEEE Press, New York (2007)
Arkin, A.J., Ross, J., McAdams, H.H.: Gene Ontology: tool for the unification of biology. Genetics 149(4), 1633–1648 (1998)
Bahi-Jaber, N., Pontier, D.: Modeling transmission of directly transmitted infectious diseases using colored stochastic Petri nets. Math. Biosci. 185(1), 1–13 (2003)
Cao, Y., Li, H., Petzold, L.: Efficient formulation of the stochastic simulation algorithm for chemically reacting systems. J. Chem. Phys. 121(9), 4059–4067 (2004)
Chiola, G., Dutheillet, C., Franceschinis, G., Haddad, S.: Stochastic well-formed colored nets and symmetric modeling applications. IEEE Trans. Comput. 42(11), 1343–1360 (1993)
Chiola, G., Franceschinis, G., Gaeta, R., Ribaudo, M.: Greatspn 1.7: Graphical editor and analyzer for timed and stochastic Petri nets. Perform. Eval. 24, 47–68 (1995)
Ciardo, G., Muppala, J.K., Trivedi, K.S.: Stochastic Petri net package. In: Intern. Workshop on Petri Nets and Performance Models (PNPM’89), pp. 142–151 (1989)
Clark, G., Courtney, T., Daly, D., Deavours, D., Derisavi, S., Doyle, J.M., Sanders, W.H., Webster, P.: The Möbius modeling tool. In: Intern. Workshop on Petri Nets and Performance Models (PNPM’01), pp. 241–250 (2001)
Doob, J.L.: Stochastic Processes. Wiley, New York (1953)
Gibson, M.A., Bruck, J.: Efficient exact stochastic simulation of chemical systems with many species and many channels. J. Phys. Chem. 104(9), 1876–1889 (2000)
Gillespie, D.T.: A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22(4), 403–434 (1976)
Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81(25), 2340–2361 (1977)
Goss, P.J.E., Peccoud, J.: Quantitative modeling of stochastic systems in molecular biology by using stochastic Petri nets. Proc. Natl. Acad. Sci. USA. 95(12), 6750–6755 (1998)
Goss, P.J.E., Peccoud, J.: Analysis of the stabilizing effect of Rom on the genetic network controlling ColE1 plasmid replication. Proc. Pac. Symp. Biocomput. 4, 65–76 (1999)
Gross, D., Miller, D.R.: The randomization technique as a modeling tool and solution procedure for transient Markov processes. Oper. Res. 32(2), 343–361 (1984)
Hardy, S., Robillard, P.N.: Modeling and simulation of molecular biology systems using Petri nets: modeling goals of various approaches. J. Bioinform. Comput. Biol. 2(4), 619–637 (2004)
Heiner, M., Gilbert, D., Donaldson, R.: Petri nets for systems and synthetic biology. In: Formal Methods for Computational Systems Biology. LNCS, vol. 5016, pp. 215–264. Springer, Berlin (2008)
Heiner, M., Lehrack, S., Gilbert, D., Marwan, W.: Extended stochastic Petri nets for modelbased design of wetlab experiments. In: Computational Models for Cell Processes. Transactions on Computational Systems Biology XI. LNCS, vol. 5750, pp. 138–163. Springer, Berlin (2009)
Hitchcock, S.E.: Extinction probabilities in predator-prey models. J. Appl. Probab. 23(1), 1–13 (1986)
Lotka, A.J.: Undamped oscillations derived from the laws of mass action. J. Am. Chem. Soc. 42, 1595–1599 (1920)
Marsan, A.M., Conte, G., Balbo, G.: A class of generalized stochastic Petri nets for the performance evaluation of multiprocessor systems. ACM Trans. Comput. Syst. 2(2), 93–122 (1984)
Marwan, W., Sujatha, A., Starostzik, C.: Reconstructing the regulatory network controlling commitment and sporulation in physarum polycephalum based on hierarchical petri net modelling and simulation. J. Theor. Biol. 236(4), 349–365 (2005)
McColluma, J.M., Peterson, G.D., Cox, C.D., Simpson, M.L., Samatova, N.F.: The sorting direct method for stochastic simulation of biochemical systems with varying reaction execution behavior. Comput. Biol. Chem. 30(1), 29–39 (2006)
Meyer, J.F., Movaghar, A., Sanders, W.H.: Stochastic Activity Networks: Structure, Behavior, and Application. In: International Workshop on Timed Petri Nets, pp. 106–115. IEEE Comput. Soc., Los Alamitos (1985)
Molloy, M.K.: Performance analysis using stochastic Petri nets. IEEE Trans. Comput. 31(9), 913–917 (1982)
Muppala, J.K., Ciardo, G., Trivedi, K.S.: Stochastic reward nets for reliability prediction. In: Communications in Reliability, Maintainability and Serviceability, pp. 9–20 (1994)
Mura, I., Csiksz-Nagy, A.: Stochastic Petri Net extension of a yeast cell cycle model. J. Theor. Biol. 254(4), 850–860 (2008)
Peleg, M., Rubin, D., Altman, R.B.: Using Petri net tools to study properties and dynamics of biological systems. J. Am. Med. Inform. Assoc. 12(2), 181–199 (2005)
Srivastava, R., Peterson, M.S., Bentley, W.E.: Stochastic kinetic analysis of the Escherichia coli stress circuit using σ−32 targeted antisense. Biotechnol. Bioeng. 75(1), 120–129 (2001)
Tsavachidou, D., Liebman, M.N.: Modeling and simulation of pathways in menopause. J. Am. Med. Inform. Assoc. 9(5), 461–471 (2002)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag London Limited
About this chapter
Cite this chapter
Mura, I. (2011). Stochastic Modeling. In: Koch, I., Reisig, W., Schreiber, F. (eds) Modeling in Systems Biology. Computational Biology, vol 16. Springer, London. https://doi.org/10.1007/978-1-84996-474-6_7
Download citation
DOI: https://doi.org/10.1007/978-1-84996-474-6_7
Publisher Name: Springer, London
Print ISBN: 978-1-84996-473-9
Online ISBN: 978-1-84996-474-6
eBook Packages: Computer ScienceComputer Science (R0)