Finite state abstraction of a stochastic model of the lactose regulation system of Escherichia coli

This paper focuses on the lactose regulation system in Escherichia coli bacteria, one of the most extensively studied examples of positive feedback in a naturally occurring gene network. State-of-the-art nonlinear dynamical system models predict a bi-stability phenomenon that is confirmed in experiments. However, such deterministic models fail to explain experimental observations of spontaneous transition between the two stable states in the system and the simultaneous occurrence of both steady states in a population of cells. In this paper, we propose a stochastic model that explains this phenomenon. Furthermore, we also extract a coarser two-state continuous-time Markov chain as a higher level abstraction of this model, and show that macroscopic properties are retained in the abstraction

[1]  S Pestka,et al.  Anti-mRNA: specific inhibition of translation of single mRNA molecules. , 1984, Proceedings of the National Academy of Sciences of the United States of America.

[2]  John C. Doyle,et al.  Surviving heat shock: control strategies for robustness and performance. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[3]  Natasha A. Neogi,et al.  Dynamic Partitioning of Large Discrete Event Biological Systems for Hybrid Simulation and Analysis , 2004, HSCC.

[4]  D. Gillespie.  A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions , 1976 .

[5]  D. Gillespie,et al.  Avoiding negative populations in explicit Poisson tau-leaping. , 2005, The Journal of chemical physics.

[6]  Christos G. Cassandras,et al.  Introduction to Discrete Event Systems , 1999, The Kluwer International Series on Discrete Event Dynamic Systems.

[7]  Claire J. Tomlin,et al.  Lateral Inhibition through Delta-Notch Signaling: A Piecewise Affine Hybrid Model , 2001, HSCC.

[8]  C. Tomlin,et al.  Mathematical Modeling of Planar Cell Polarity to Understand Domineering Nonautonomy , 2005, Science.

[9]  J. Monod,et al.  [The kinetics of the biosynthesis of beta-galactosidase in Escherichia coli as a function of growth]. , 1952, Biochimica et biophysica acta.

[10]  P. Dhurjati,et al.  Oscillatory behavior of beta-galactosidase enzyme activity in Escherichia coli during perturbed batch experiments. , 1987, Biotechnology and bioengineering.

[11]  K. Burrage,et al.  Stochastic models for regulatory networks of the genetic toggle switch. , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[12]  Jeffrey W. Smith,et al.  Stochastic Gene Expression in a Single Cell , .

[13]  A. Novick,et al.  ENZYME INDUCTION AS AN ALL-OR-NONE PHENOMENON. , 1957, Proceedings of the National Academy of Sciences of the United States of America.

[14]  T. Kepler,et al.  Stochasticity in transcriptional regulation: origins, consequences, and mathematical representations. , 2001, Biophysical journal.

[15]  A. Agung Julius,et al.  Approximate Abstraction of Stochastic Hybrid Automata , 2006, HSCC.

[16]  E. D. Sontagc,et al.  Nonmonotone systems decomposable into monotone systems with negative feedback , 2005 .

[17]  J. Hespanha,et al.  Models for Multi-Specie Chemical Reactions Using Polynomial Stochastic Hybrid Systems , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[18]  S. Leibler,et al.  Bacterial Persistence as a Phenotypic Switch , 2004, Science.

[19]  David Angeli,et al.  Multistability in monotone I/O systems , 2004 .

[20]  D. Gillespie.  Approximate accelerated stochastic simulation of chemically reacting systems , 2001 .

[21]  M. Mackey,et al.  Feedback regulation in the lactose operon: a mathematical modeling study and comparison with experimental data. , 2003, Biophysical journal.

[22]  R. Cox,et al.  Quantitative relationships for specific growth rates and macromolecular compositions of Mycobacterium tuberculosis, Streptomyces coelicolor A3(2) and Escherichia coli B/r: an integrative theoretical approach. , 2004, Microbiology.

[23]  Ádám M. Halász,et al.  Understanding the Bacterial Stringent Response Using Reachability Analysis of Hybrid Systems , 2004, HSCC.

[24]  Spring Berman,et al.  Algorithms for the Analysis and Synthesis of a Bio-inspired Swarm Robotic System , 2006, Swarm Robotics.

[25]  D. Zipser,et al.  The lactose operon , 1970 .

[26]  Jose M. G. Vilar,et al.  Modeling network dynamics: the lac operon, a case study , 2004 .

[27]  Ertugrul M. Ozbudak,et al.  Predicting stochastic gene expression dynamics in single cells. , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[28]  Eduardo Sontag,et al.  Untangling the wires: A strategy to trace functional interactions in signaling and gene networks , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[29]  W A Knorre,et al.  Oscillations of the rate of synthesis of beta-galactosidase in Escherichia coli ML 30 and ML 308. , 1968, Biochemical and biophysical research communications.

[30]  Dominique Schneider,et al.  Qualitative Analysis and Verification of Hybrid Models of Genetic Regulatory Networks: Nutritional Stress Response in , 2005, HSCC.

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

[32]  Linda R. Petzold,et al.  Improved leap-size selection for accelerated stochastic simulation , 2003 .

[34]  Eduardo D. Sontag,et al.  Nonmonotone systems decomposable into monotone systems with negative feedback , 2006 .

[35]  Ertugrul M. Ozbudak,et al.  Multistability in the lactose utilization network of Escherichia coli , 2004, Nature.

[36]  J. Mahaffy,et al.  Stability analysis for a mathematical model of the lac operon , 1999 .

[37]  S. Leibler,et al.  Phenotypic Diversity, Population Growth, and Information in Fluctuating Environments , 2005, Science.