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Piecewise-Deterministic Markov Processes as Limits of Markov Jump Processes

Part of: Markov processes Limit theorems Physiological, cellular and medical topics

Published online by Cambridge University Press:  04 January 2016

Uwe Franz ,
Volkmar Liebscher  and
Stefan Zeiser
Uwe Franz*
Affiliation:
University of Franche-Comté
Volkmar Liebscher*
Affiliation:
Ernst Moritz Arndt University Greifswald
Stefan Zeiser*
Affiliation:
Kinesis Pharma BV
*
∗ Postal address: Faculty of Mathematics of Besançon, University of Franche-Comté, Route de Gray 16, 25 030 Besançon cedex, France. Email address: [email protected]
∗∗ Postal address: Faculty of Mathematics and Sciences, Ernst Moritz Arndt University Greifswald, Walther-Rathenau-Straβe 47, 17487 Greifswald, Germany. Email address: [email protected]
∗∗∗ Postal address: Kinesis Pharma BV, Lage Mosten 29, 4822 NK Breda, The Netherlands. Email address: [email protected]
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Abstract

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A classical result about Markov jump processes states that a certain class of dynamical systems given by ordinary differential equations are obtained as the limit of a sequence of scaled Markov jump processes. This approach fails if the scaling cannot be carried out equally across all entities. In the present paper we present a convergence theorem for such an unequal scaling. In contrast to an equal scaling the limit process is not purely deterministic but still possesses randomness. We show that these processes constitute a rich subclass of piecewise-deterministic processes. Such processes apply in molecular biology where entities often occur in different scales of numbers.

Keywords

Piecewise-deterministic Markov processstochastic model of gene regulationlimit theoremSkorokhod space

MSC classification

Primary: 60F15: Strong theorems
Secondary: 60J28: Applications of continuous-time Markov processes on discrete state spaces 92C40: Biochemistry, molecular biology
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 44 , Issue 3 , September 2012 , pp. 729 - 748
Copyright
© Applied Probability Trust 

References

Ball, K., Kurtz, T. G., Popovic, L. and Rempala, G. (2006). Asymptotic analysis of multiscale approximations to reaction networks. Ann. Appl. Prob. 16, 1925–1925.CrossRefGoogle Scholar
Becskei, A. and Serrano, L. (2000). Engineering stability in gene networks by autoregulation. Nature 405, 590–593.CrossRefGoogle ScholarPubMed
Blake, W. J., Kærn, M., Cantor, C. R. and Collins, J. J. (2003). Noise in eukaryotic gene expression. Nature 422, 633–637.CrossRefGoogle ScholarPubMed
Bundschuh, R., Hayot, F. and Jayaprakash, C. (2003). Fluctuations and slow variables in genetic networks. Biophys. J. 84, 1606–1615.CrossRefGoogle ScholarPubMed
Davis, M. H. A. (1984). Piecewise-deterministic Markov processes: a general class of nondiffusion stochastic models. J. R. Statist. Soc. B 46, 353–388.Google Scholar
Doob, J. L. (1953). Stochastic Processes. John Wiley, New York.Google Scholar
Gillespie, D. T. (1976). A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22, 403–434.CrossRefGoogle Scholar
Gillespie, D. T. (1977). Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81, 2340–2361.Google Scholar
Hume, D. A. (2000). Probability in transcriptional regulation and its implications for leukocyte differentiation and inducible gene expression. Blood 96, 2323–2328.CrossRefGoogle ScholarPubMed
Kurtz, T. G. (1970). Solutions of ordinary differential equations as limits of pure Jump Markov processes. J. Appl. Prob. 7, 49–58.CrossRefGoogle Scholar
Kurtz, T. G. (1971). Limit theorems for sequences of Jump Markov processes approximating ordinary differential processes. J. Appl. Prob. 8, 344–356.CrossRefGoogle Scholar
Kurtz, T. G. (1980). Relationships between stochastic and deterministic population models. In Biological Growth and Spread (Proc. Conf. Heidelberg, 1979; Lecture Notes Biomath. 38), Springer, Berlin, pp. 449–467.Google Scholar
Maamar, H. and Dubnau, D. (2005). Bistability in the Bacillus subtilis K-state (competence) system requires a positive feedback loop. Mol. Microbiol. 56, 615–624.Google Scholar
Skorohod, A. V. (1956). Limit theorems for stochastic processes. Theory Prob. Appl. 1, 261–290.Google Scholar
Zeiser, S., Franz, U. and Liebscher, V. (2009). Autocatalytic genetic networks modeled by piecewise-deterministic Markov processes. J. Math. Biol. 60, 207–246.Google Scholar
Zeiser, S., Franz, U., Müller, J. and Liebscher, V. (2009). Hybrid modeling of noise reduction by a negatively autoregulated system. Bull. Math. Biol. 71, 1006–1024.CrossRefGoogle ScholarPubMed
Zeiser, S., Franz, U., Wittich, O. and Liebscher, V. (2008). Simulation of genetic networks modelled by piecewise deterministic Markov processes. IET Systems Biol. 2, 113–135.CrossRefGoogle ScholarPubMed