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Characterizing Oscillatory and Noisy Periodic Behavior in Markov Population Models

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Quantitative Evaluation of Systems (QEST 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8054))

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Abstract

In systems biology, an interesting problem is to analyze and characterize the oscillatory and periodic behavior of a chemical reaction system. Traditionally, those systems have been treated deterministically and continuously via ordinary differential equations. In case of high molecule counts with respect to the volume this treatment is justified. But otherwise, stochastic fluctuations can have a high influence on the characteristics of a system as has been shown in recent publications.

In this paper we develop an efficient numerical approach for analyzing the oscillatory and periodic character of user-defined observations on Markov population models (MPMs). MPMs are a special kind of continuous-time Markov chains that allow for a discrete representation of unbounded population counts for several population types and transformations between populations. Examples are chemical species and the reactions between them.

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References

  1. Andrei, O., Calder, M.: Trend-based analysis of a population model of the akap scaffold protein. TCS Biology 14 (2012)

    Google Scholar 

  2. Arkin, A., Ross, J., McAdams, H.: Stochastic kinetic analysis of developmental pathway bifurcation in phage λ-infected escherichia coli cells. Genetics 149, 1633–1648 (1998)

    Google Scholar 

  3. Baier, C., Hermanns, H., Haverkort, B., Katoen, J.-P.: Model-checking algorithms for continuous-time Markov chains. IEEE Transactions on Software Engineering 29(6), 524–541 (2003)

    Article  Google Scholar 

  4. Ballarini, P., Guerriero, M.L.: Query-based verification of qualitative trends and oscillations in biochemical systems. TCS 411(20), 2019–2036 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ballarini, P., Mardare, R., Mura, I.: Analysing biochemical oscillation through probabilistic model checking. ENTCS 229(1), 3–19 (2009)

    MathSciNet  Google Scholar 

  6. Barkai, N., Leibler, S.: Biological rhythms: Circadian clocks limited by noise. Nature 403, 267–268 (2000)

    Google Scholar 

  7. Bortolussi, L., Policriti, A.: The importance of being (a little bit) discrete. ENTCS 229(1), 75–92 (2009)

    MathSciNet  Google Scholar 

  8. Dayar, T., Hermanns, H., Spieler, D., Wolf, V.: Bounding the equilibrium distribution of Markov population models. NLAA (2011)

    Google Scholar 

  9. Didier, F., Henzinger, T.A., Mateescu, M., Wolf, V.: Fast adaptive uniformization of the chemical master equation. In: Proc. of HIBI, pp. 118–127. IEEE Computer Society, Washington, DC (2009)

    Google Scholar 

  10. Elowitz, M.B., Leibler, S.: A synthetic oscillatory network of transcriptional regulators. Nature 403(6767), 335–338 (2000)

    Article  Google Scholar 

  11. Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. The Journal of Physical Chemistry 81(25), 2340–2361 (1977)

    Article  Google Scholar 

  12. Gillespie, D.T.: A rigorous derivation of the chemical master equation. Physica A 188, 404–425 (1992)

    Article  Google Scholar 

  13. Henzinger, T.A., Mateescu, M., Wolf, V.: Sliding window abstraction for infinite Markov chains. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 337–352. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  14. Júlvez, J., Kwiatkowska, M., Norman, G., Parker, D.: A systematic approach to evaluate sustained stochastic oscillations. In: Proc. BICoB. ISCA (2011)

    Google Scholar 

  15. Kerr, B., Riley, M.A., Feldman, M.W., Bohannan, B.J.M.: Local dispersal promotes biodiversity in a real-life game of rock-paper-scissors. Nature 418 (2002)

    Google Scholar 

  16. Kirkup, B.C., Riley, M.A.: Antibiotic-mediated antagonism leads to a bacterial game of rock-paper-scissors in vivo. Nature 428 (2004)

    Google Scholar 

  17. Maroto, M., Monk, N.A.M.: Cellular Oscillatory Mechanisms. Advances in Experimental Medicine and Biology, vol. 641. Springer (2009)

    Google Scholar 

  18. Meyer, K., Wiegand, K., Ward, D., Moustakas, A.: Satchmo: A spatial simulation model of growth, competition, and mortality in cycling savanna patches. Ecological Modelling 209, 377–391 (2007)

    Article  Google Scholar 

  19. Perko, L.: Differential Equations and Dynamical Systems. Texts in Applied Mathematics. Springer (2000)

    Google Scholar 

  20. Reichenbach, T., Mobilia, M., Frey, E.: Coexistence versus extinction in the stochastic cyclic lotka-volterra model. Phys. Rev. E 74, 051907 (2006)

    Article  MathSciNet  Google Scholar 

  21. Spieler, D.: Model checking of oscillatory and noisy periodic behavior in Markovian population models. Technical report, Saarland University (2009), Master thesis available at http://mosi.cs.uni-saarland.de/?page_id=93

  22. Stewart, W.J.: Introduction to the numerical solution of Markov chains. Princeton University Press (1994)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Saarland University, Germany

    David Spieler

Authors
  1. David Spieler
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Editor information

Editors and Affiliations

  1. AT&T Labs Research, 180 Park Avenue, Building 103, 07932, Florham Park, NJ, USA

    Kaustubh Joshi

  2. Institut für Technische Informatik, Universität der Bundeswehr München, Werner-Heisenberg Weg 39, 85577, Neubiberg, Germany

    Markus Siegle

  3. Faculty of Electrical Engineering, Mathematics and Computer Science, University of Twente, Drienerlolaan 5, Zilverling Building, 7522 NB, Enschede, The Netherlands

    Mariëlle Stoelinga

  4. Facultad de Matemáticas, Astronomía y Física, Universidad Nacional de Córdoba – CONICET, Medina Allende s/n, X5000HUA, Córdoba, Argentina

    Pedro R. D’Argenio

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Spieler, D. (2013). Characterizing Oscillatory and Noisy Periodic Behavior in Markov Population Models. In: Joshi, K., Siegle, M., Stoelinga, M., D’Argenio, P.R. (eds) Quantitative Evaluation of Systems. QEST 2013. Lecture Notes in Computer Science, vol 8054. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40196-1_8

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  • DOI: https://doi.org/10.1007/978-3-642-40196-1_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-40195-4

  • Online ISBN: 978-3-642-40196-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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