Aggregation for Computing Multi-Modal Stationary Distributions in 1-D Gene Regulatory Networks

This paper proposes aggregation-based, three-stage algorithms to overcome the numerical problems encountered in computing stationary distributions and mean first passage times for multi-modal birth-death processes of large state space sizes. The considered birth-death processes which are defined by Chemical Master Equations are used in modeling stochastic behavior of gene regulatory networks. Computing stationary probabilities for a multi-modal distribution from Chemical Master Equations is subject to have numerical problems due to the probability values running out of the representation range of the standard programming languages with the increasing size of the state space. The aggregation is shown to provide a solution to this problem by analyzing first reduced size subsystems in isolation and then considering the transitions between these subsystems. The proposed algorithms are applied to study the bimodal behavior of the lac operon of E. coli described with a one-dimensional birth-death model. Thus, the determination of the entire parameter range of bimodality for the stochastic model of lac operon is achieved.

[1]  Daniel T Gillespie,et al.  Stochastic simulation of chemical kinetics. , 2007, Annual review of physical chemistry.

[2]  Jeremy T. Bradley,et al.  Fluid computation of passage-time distributions in large Markov models , 2012, Theor. Comput. Sci..

[3]  Ferhan Pekergin,et al.  Determining the bistability parameter ranges of artificially induced lac operon using the root locus method , 2015, Comput. Biol. Medicine.

[4]  Glenn Vinnicombe,et al.  Noise in Gene Regulatory Networks , 2008, IEEE Transactions on Automatic Control.

[5]  M. Khammash,et al.  The finite state projection algorithm for the solution of the chemical master equation. , 2006, The Journal of chemical physics.

[6]  William J. Stewart,et al.  Introduction to the numerical solution of Markov Chains , 1994 .

[7]  Thomas A. Henzinger,et al.  Solving the chemical master equation using sliding windows , 2010, BMC Systems Biology.

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

[9]  Ling Tang,et al.  Efficient and Reliable Computation of Birth-Death Process Performance Measures , 2012, INFORMS J. Comput..

[10]  David K. Smith.  Calculation of steady-state probabilities of M/M queues: further approaches , 2002, Adv. Decis. Sci..

[11]  Ferhan Pekergin,et al.  Numerically Efficient Analysis of a One-Dimensional Stochastic Lac Operon Model , 2015, ISCIS.

[12]  L. Allen.  An introduction to stochastic processes with applications to biology , 2003 .

[13]  C. Guzelis,et al.  Coexistence of Deterministic and Stochastic Bistability in a 1-D Birth-Death Process with Hill Type Nonlinear Birth Rates , 2015 .

[14]  Pierre Semal,et al.  Computable Bounds for Conditional Steady-State Probabilities in Large Markov Chains and Queueing Models , 1986, IEEE J. Sel. Areas Commun..

[15]  Ward Whitt,et al.  Engineering Solution of a Basic Call-Center Model , 2005, Manag. Sci..

[16]  Holger Hermanns,et al.  Bounding the equilibrium distribution of Markov population models , 2010, Numer. Linear Algebra Appl..

[17]  L. Cavaş,et al.  Discriminant-based bistability analysis of a TMG-induced lac operon model supported with boundedness and local stability results , 2016 .

[18]  M. Hegland.  Solving the Chemical Master Equation with the Aggregation-Disaggregation Method , 2009 .

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

[20]  Parosh Aziz Abdulla,et al.  Fast Adaptive Uniformization of the Chemical Master Equation , 2010 .