Abstract
This chapter describes basic principles for modeling genetic regulatory networks, using three different classes of formalisms: discrete, hybrid, and continuous differential systems. A short review of the mathematical tools for each formalism is presented. Based on several simple examples, which are worked out in detail, this chapter illustrates the study and analysis of the networks’ dynamics, their temporal evolution and asymptotic behaviors.
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Notes
- 1.
Provided that \(({\kappa }_{i} +{\sum\limits}_{j\in \Omega }{\kappa }_{ji}/{\gamma }_{i})\not\in {\Theta }_{i}\) for all \(\Omega \subseteq {G}_{i}\).
- 2.
For the interaction graph of Fig. 2.9, we have b 1 ∈ { 1, 2}, t 11, t 12 ∈ { 1, b 1} and \({b}_{2} = {\theta }_{21} = 1\). If b 1 = 1, then \({t}_{11} = {t}_{12} = 1\), and there are 18 possible instantiations of the parameters K, which lead to a set of 18 different asynchronous state graphs. If b 1 = 2 there are two cases. First, if \({t}_{11} = 1 < {t}_{21} = 2\), there are 60 possible instantiations of the parameters K, which lead to a set \(\mathcal{S}\) of 42 different asynchronous state graphs. Second, if \({t}_{11} = 2 > {t}_{21} = 1\), there are 60 possible instantiations of the parameters K, which lead also to a set of 42 different asynchronous state graphs, but 12 of them are contained in \(\mathcal{S}\). Hence, the total number of asynchronous state graphs is \(18 + 42 + 42 - 12 = 90\).
- 3.
An infinite path of S is an infinite sequence of states \({x}^{0}{x}^{1}{x}^{2},\ldots \) such that, for all \(k \in \mathbb{N}\): if x k has a successor in S, then x k → x k + 1 is an arc of S, and \({x}^{k} = {x}^{k+1}\) otherwise.
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Acknowledgements
It is a pleasure for GB, JPC and AR to thank the biologist Janine Guespin-Michel, who has actively participated to the definition of our formal logic methodology in such a way that our techniques from computer science and the SMBioNet software become truly useful for biologists. She has also been at the origin of the Pseudomonas aeruginosa hypothesis. The authors would also like to thank F. Cazals for his remarks and careful reading of the chapter.
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Bernot, G., Comet, JP., Richard, A., Chaves, M., Gouzé, JL., Dayan, F. (2013). Modeling and Analysis of Gene Regulatory Networks. In: Cazals, F., Kornprobst, P. (eds) Modeling in Computational Biology and Biomedicine. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31208-3_2
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