Stochastic reachability analysis in complex biological networks

The impact of noise on cellular networks and its interplay with their rich dynamics are increasingly being characterized as important phenomena that must be thoroughly investigated for a useful understanding of biological dynamics. At the same time, the mathematical modeling and analysis of these networks in a stochastic setting presents a number of challenges, such as the need for a large number of computationally expensive stochastic simulations to collect statistics about the occurrence of important events or correlate their occurrence with the noise intensity. In this paper, we demonstrate the use of new techniques of stochastic reachability analysis to address these problems. Specifically, we study the problem of computing bounds on the probability of a biological stochastic process to reach certain parts of the state space in a finite time. The techniques presented are based on the algorithmic construction of barrier certificates using convex optimization, and are illustrated through the use of a biologically important system: the bacteriophage lambda genetic switch

[1]  D. Gillespie A rigorous derivation of the chemical master equation , 1992 .

[2]  Linda R. Petzold,et al.  Stochastic modelling of gene regulatory networks , 2005 .

[3]  L. Collatz,et al.  Differential equations: An introduction with applications , 1986 .

[4]  W. Ebeling Stochastic Processes in Physics and Chemistry , 1995 .

[5]  M. Ptashne A genetic switch : phage λ and higher organisms , 1992 .

[6]  C. Rao,et al.  Stochastic chemical kinetics and the quasi-steady-state assumption: Application to the Gillespie algorithm , 2003 .

[7]  W H Lamers,et al.  The expression of liver-specific genes within rat embryonic hepatocytes is a discontinuous process. , 1994, Differentiation; research in biological diversity.

[8]  F. Bunz,et al.  Cell death and cancer therapy. , 2001, Current opinion in pharmacology.

[9]  D. Bennett,et al.  Differentiation in mouse melanoma cells: Initial reversibility and an on-off stochastic model , 1983, Cell.

[10]  N. Kampen,et al.  Stochastic processes in physics and chemistry , 1981 .

[11]  Linda R. Petzold,et al.  Stochastic Modeling of Gene Regulatory Networks y , 2005 .

[12]  D. Gillespie The chemical Langevin equation , 2000 .

[13]  Peter J Seiler,et al.  SOSTOOLS: Sum of squares optimization toolbox for MATLAB , 2002 .

[14]  D. Gillespie,et al.  Stochastic Modeling of Gene Regulatory Networks † , 2005 .

[15]  George J. Pappas,et al.  Stochastic safety verification using barrier certificates , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[16]  Nir Friedman,et al.  Quantitative kinetic analysis of the bacteriophage λ genetic network , 2005 .

[17]  A. Arkin,et al.  Stochastic kinetic analysis of developmental pathway bifurcation in phage lambda-infected Escherichia coli cells. , 1998, Genetics.

[18]  Gerald A. Edgar,et al.  Stopping times and directed processes , 1992 .

[19]  D. Gillespie Exact Stochastic Simulation of Coupled Chemical Reactions , 1977 .

[20]  G. TEMPLE,et al.  Stability and Control , 1953, Nature.

[21]  J. Hasty,et al.  Noise-based switches and amplifiers for gene expression. , 2000, Proceedings of the National Academy of Sciences of the United States of America.