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.
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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 .
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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
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DOI: https://doi.org/10.1007/978-1-84996-474-6_7
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