Bounding the equilibrium distribution of Markov population models

SUMMARY We propose a bounding technique for the equilibrium probability distribution of continuous-time Markov chains with population structure and infinite state space. We use Lyapunov functions to determine a finite set of states that contains most of the equilibrium probability mass. Then we apply a refinement scheme based on stochastic complementation to derive lower and upper bounds on the equilibrium probability for each state within that set. To show the usefulness of our approach, we present experimental results for several examples from biology. Copyright © 2011 John Wiley & Sons, Ltd.

[1]  M. Thattai,et al.  Intrinsic noise in gene regulatory networks , 2001, Proceedings of the National Academy of Sciences of the United States of America.

[2]  J. Paulsson.  Summing up the noise in gene networks , 2004, Nature.

[3]  C. J. Burden,et al.  A solver for the stochastic master equation applied to gene regulatory networks , 2007 .

[4]  Pierre Semal,et al.  Bounds for the Positive Eigenvectors of Nonnegative Matrices and for their Approximations by Decomposition , 1984, JACM.

[5]  P J Goss,et al.  Quantitative modeling of stochastic systems in molecular biology by using stochastic Petri nets. , 1998, Proceedings of the National Academy of Sciences of the United States of America.

[6]  S. Banach,et al.  Théorie des opérations linéaires , 1932 .

[7]  T. Kurtz,et al.  Submitted to the Annals of Applied Probability ASYMPTOTIC ANALYSIS OF MULTISCALE APPROXIMATIONS TO REACTION NETWORKS , 2022 .

[8]  J. Schur,et al.  Über lineare Transformationen in der Theorie der unendlichen Reihen. , 1921 .

[9]  John K. Reid,et al.  An Implementation of Tarjan's Algorithm for the Block Triangularization of a Matrix , 1978, TOMS.

[10]  W. Stewart,et al.  Quasi Lumpability, Lower-Bounding Coupling Matrices, and Nearly Completely Decomposable Markov Chains , 1997 .

[11]  Thomas A. Henzinger,et al.  Hybrid numerical solution of the chemical master equation , 2010, CMSB '10.

[12]  Poul E. Heegaard,et al.  Ant-based approach for determining the change of measure in importance sampling , 2007, 2007 Winter Simulation Conference.

[13]  Tsung-Lin Lee,et al.  HOM4PS-2.0: a software package for solving polynomial systems by the polyhedral homotopy continuation method , 2008, Computing.

[14]  James Demmel,et al.  A Supernodal Approach to Sparse Partial Pivoting , 1999, SIAM J. Matrix Anal. Appl..

[15]  Pierre-Jacques Courtois.  Analysis of Large Markovian Models by Parts. Applications to Queueing Network Models , 1985, MMB.

[16]  T. Kurtz.  The Relationship between Stochastic and Deterministic Models for Chemical Reactions , 1972 .

[17]  Alma Riska,et al.  Aggregate matrix-analytic techniques and their applications , 2002 .

[18]  P. Swain,et al.  Intrinsic and extrinsic contributions to stochasticity in gene expression , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[19]  Ofer Biham,et al.  Stochastic simulations of genetic switch systems. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[21]  Carl D. Meyer,et al.  Stochastic Complementation, Uncoupling Markov Chains, and the Theory of Nearly Reducible Systems , 1989, SIAM Rev..

[22]  Stefan Engblom,et al.  Numerical Solution Methods in Stochastic Chemical Kinetics , 2008 .

[23]  Ofer Biham,et al.  Genetic toggle switch without cooperative binding. , 2006, Physical review letters.

[24]  P. Taylor,et al.  Some properties of the rate operators in level dependent quasi-birth-and-death processes with a countable number of phases , 1996 .

[25]  Roger B. Sidje,et al.  Inexact Uniformization Method for Computing Transient Distributions of Markov Chains , 2007, SIAM J. Sci. Comput..

[26]  P. Glynn,et al.  Bounding Stationary Expectations of Markov Processes , 2008 .

[27]  Richard R. Muntz,et al.  Bounding Availability of Repairable Computer Systems , 1989, IEEE Trans. Computers.

[28]  Thomas A. Henzinger,et al.  Formalisms for Specifying Markovian Population Models , 2009, Int. J. Found. Comput. Sci..

[29]  R. Tweedie.  Sufficient conditions for regularity, recurrence and ergodicity of Markov processes , 1975, Mathematical Proceedings of the Cambridge Philosophical Society.

[30]  Thomas A. Henzinger,et al.  Fast Adaptive Uniformization of the Chemical Master Equation , 2009, 2009 International Workshop on High Performance Computational Systems Biology.

[31]  E. Seneta.  Non-negative matrices;: An introduction to theory and applications , 1973 .

[32]  W. Huisinga,et al.  Solving the chemical master equation for monomolecular reaction systems analytically , 2006, Journal of mathematical biology.

[33]  Hamid Bolouri,et al.  Dizzy: Stochastic Simulation of Large-scale Genetic Regulatory Networks , 2005, J. Bioinform. Comput. Biol..

[34]  J. Collins,et al.  Construction of a genetic toggle switch in Escherichia coli , 2000, Nature.