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Infinite level-dependent QBD processes and matrix-analytic solutions for stochastic chemical kinetics

Part of: Physiological, cellular and medical topics Markov processes

Published online by Cambridge University Press:  01 July 2016

Tuğrul Dayar ,
Werner Sandmann ,
David Spieler  and
Verena Wolf
Tuğrul Dayar*
Affiliation:
Bilkent University
Werner Sandmann*
Affiliation:
Clausthal University of Technology
David Spieler*
Affiliation:
Saarland University
Verena Wolf*
Affiliation:
Saarland University
*
∗ Postal address: Department of Computer Engineering, Bilkent University, TR-06800 Bilkent, Ankara, Turkey.
∗∗ Postal address: Department of Applied Stochastics and Operations Research, Clausthal University of Technology, Erzstr. 1, D-38678 Clausthal-Zellerfeld, Germany. Email address: werner.sandmann@tu-clausthal.de
∗∗∗ Postal address: Faculty of Computer Science, Saarland University, D-66123 Saarbrücken, Germany.
∗∗∗ Postal address: Faculty of Computer Science, Saarland University, D-66123 Saarbrücken, Germany.
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Abstract

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Systems of stochastic chemical kinetics are modeled as infinite level-dependent quasi-birth-and-death (LDQBD) processes. For these systems, in contrast to many other applications, levels have an increasing number of states as the level number increases and the probability mass may reside arbitrarily far away from lower levels. Ideas from Lyapunov theory are combined with existing matrix-analytic formulations to obtain accurate approximations to the stationary probability distribution when the infinite LDQBD process is ergodic. Results of numerical experiments on a set of problems are provided.

Keywords

Stochastic chemical kineticslevel-dependent quasi-birth-and-death processstate space truncationLyapunov boundmatrix-analytic solution

MSC classification

Primary: 60J22: Computational methods in Markov chains
Secondary: 60J28: Applications of continuous-time Markov processes on discrete state spaces 92C45: Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.) 92D25: Population dynamics (general)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 43 , Issue 4 , December 2011 , pp. 1005 - 1026
Copyright
Copyright © Applied Probability Trust 2011 

References

Baumann, H. and Sandmann, W. (2010). Numerical solution of level dependent quasi-birth-and-death processes. Procedia Comput. Sci. 1, 1561–1569.CrossRefGoogle Scholar
Berman, A. and Plemmons, R. J. (1994). Nonnegative Matrices in the Mathematical Sciences. Society for Industrial and Applied Mathematics, Philadelphia, PA.CrossRefGoogle Scholar
Bharucha-Reid, A. T. (1960). Elements of the Theory of Markov Processes and Their Applications. McGraw-Hill, New York.Google Scholar
Bright, L. and Taylor, P. G. (1995). Calculating the equilibrium distribution in level dependent quasi-birth-and-death processes. Commun. Statist. Stoch. Models 11, 497–525.CrossRefGoogle Scholar
Dayar, T. and Stewart, W. J. (1996). On the effects of using the Grassmann–Taksar–Heyman method in iterative aggregation–disaggregation. SIAM J. Sci. Comput. 17, 287–303.Google Scholar
Dayar, T., Hermanns, H., Spieler, D. and Wolf, V. (2011). Bounding the equilibrium distribution of Markov population models. To appear in Numer. Linear Algebra Appl.Google Scholar
Engblom, S. (2006). Computing the moments of high dimensional solutions of the master equation. Appl. Math. Comput. 180, 498–515.Google Scholar
Gardner, T. S., Cantor, C. R. and Collins, J. J. (2000). Construction of a genetic toggle switch in Escherichia coli. Nature 403, 339–342.CrossRefGoogle ScholarPubMed
Gaver, D. P., Jacobs, P. A. and Latouche, G. (1984). Finite birth-and-death models in randomly changing environments. Adv. Appl. Prob. 16, 715–731.Google Scholar
Gibson, S. and Seneta, E. (1987). Augmented truncation of infinite stochastic matrices. J. Appl. Prob. 24, 600–608.Google Scholar
Glynn, P. W. and Zeevi, A. (2008). Bounding stationary expectations of Markov processes. In Markov Processes and Related Topics: A Festschrift for Thomas G. Kurtz (Inst. Math. Statist. Collect. 4), Institute of Mathematical Statistics, Beachwood, OH, pp. 195–214.Google Scholar
Golub, G. H. and Van Loan, C. F. (1996). Matrix Computations. Johns Hopkins University Press, Baltimore, MD.Google Scholar
Grassmann, W. K., Taksar, M. I. and Heyman, D. P. (1985). Regenerative analysis and steady state distributions for Markov chains. Operat. Res. 33, 1107–1116.Google Scholar
Hanschke, T. (1999). A matrix continued fraction algorithm for the multiserver repeated order queue. Math. Comput. Modelling 30, 159–170.Google Scholar
Kurtz, T. G. (1972). The relationship between stochastic and deterministic models for chemical reactions. J. Chemical Phys. 57, 2976–2978.Google Scholar
Latouche, G. and Ramaswami, V. (1999). Introduction to Matrix Analytic Methods in Stochastic Modeling. Society for Industrial and Applied Mathematics, Philadelphia, PA.Google Scholar
Latouche, G. and Taylor, P. (2002). Truncation and augmentation of level-independent QBD processes. Stoch. Process. Appl. 99, 53–80.Google Scholar
Loinger, A., Lipshtat, A., Balaban, N. Q. and Biham, O. (2007). Stochastic simulations of genetic switch systems. Phys. Rev. E 75, 021904.Google Scholar
McQuarrie, D. A. (1967). Stochastic approach to chemical kinetics. J. Appl. Prob. 4, 413–478.Google Scholar
Neuts, M. F. (1981). Matrix-Geometric Solutions in Stochastic Models. Johns Hopkins University Press, Baltimore, MD.Google Scholar
Oppenheim, I., Shuler, K. E. and Weiss, G. H. (1969). Stochastic and deterministic formulation of chemical rate equations. J. Chemical Phys. 50, 460–466.Google Scholar
Ramaswami, V. and Taylor, P. G. (1996). Some properties of the rate operators in level dependent quasi-birth-and-death processes with a countable number of phases. Commun. Statist. Stoch. Models 12, 143–164.Google Scholar
Seneta, E. (1981). Nonnegative Matrices and Markov Chains, 2nd edn. Springer, New York.Google Scholar
Singer, K. (1953). Application of the theory of stochastic processes to the study of irreproducible chemical reactions and nucleation processes. J. R. Statist. Soc. B 15, 92–106.Google Scholar
Staff, P. J. (1970). A stochastic development of the reversible Michaelis-Menten mechanism. J. Theoret. Biol. 27, 221–232.Google Scholar
Stewart, W. J. (1994). Introduction to the Numerical Solution of Markov Chains. Princeton University Press.Google Scholar
Thattai, M. and van Oudenaarden, A. (2001). Intrinsic noise in gene regulatory networks. Proc. Nat. Acad. Sci. USA 98, 8614–8619.Google Scholar
Thorne, J. (1997). An investigation of algorithms for calculating the fundamental matrices in level dependent quasi birth death processes. , The University of Adelaide.Google Scholar
Tweedie, R. L. (1975). Sufficient conditions for regularity, recurrence and ergodicity of Markov processes. Math. Proc. Camb. Phil. Soc. 78, 125–136.Google Scholar
Tweedie, R. L. (1998). Truncation approximation of invariant measures for Markov chains. J. Appl. Prob. 35, 517–536.Google Scholar
Van Kampen, N. G. (1992). Stochastic Processes in Physics and Chemistry. North-Holland.Google Scholar
Zhao, Y. Q. and Liu, D. (1996). The censored Markov chain and the best augmentation. J. Appl. Prob. 33, 623–629.Google Scholar