Characterizing Oscillatory and Noisy Periodic Behavior in Markov Population Models
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.
Unable to display preview. Download preview PDF.
- 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
- 6.Barkai, N., Leibler, S.: Biological rhythms: Circadian clocks limited by noise. Nature 403, 267–268 (2000)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
- 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
- 19.Perko, L.: Differential Equations and Dynamical Systems. Texts in Applied Mathematics. Springer (2000)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