Studying genetic regulatory networks at the molecular level: delayed reaction stochastic models.

Current advances in molecular biology enable us to access the rapidly increasing body of genetic information. It is still challenging to model gene systems at the molecular level. Here, we propose two types of reaction kinetic models for constructing genetic networks. Time delays involved in transcription and translation are explicitly considered to explore the effects of delays, which may be significant in genetic networks featured with feedback loops. One type of model is based on delayed effective reactions, each reaction modeling a biochemical process like transcription without involving intermediate reactions. The other is based on delayed virtual reactions, each reaction being converted from a mathematical function to model a biochemical function like gene inhibition. The latter stochastic models are derived from the corresponding mean-field models. The former ones are composed of single gene expression modules. We thus design a model of gene expression. This model is verified by our simulations using a delayed stochastic simulation algorithm, which accurately reproduces the stochastic kinetics in a recent experimental study. Various simplified versions of the model are given and evaluated. We then use the two methods to study the genetic toggle switch and the repressilator. We define the "on" and "off" states of genes and extract a binary code from the stochastic time series. The binary code can be described by the corresponding Boolean network models in certain conditions. We discuss these conditions, suggesting a method to connect Boolean models, mean-field models, and stochastic chemical models.

[1]  P. R. ten Wolde,et al.  Chemical models of genetic toggle switches. , 2004, The journal of physical chemistry. B.

[2]  Michael A. Gibson,et al.  Efficient Exact Stochastic Simulation of Chemical Systems with Many Species and Many Channels , 2000 .

[3]  A. Arkin,et al.  Simulation of prokaryotic genetic circuits. , 1998, Annual review of biophysics and biomolecular structure.

[4]  H. Riezman,et al.  Transcription and translation initiation frequencies of the Escherichia coli lac operon. , 1977, Journal of molecular biology.

[5]  Ertugrul M. Ozbudak,et al.  Regulation of noise in the expression of a single gene , 2002, Nature Genetics.

[6]  R. Weiss,et al.  Ultrasensitivity and noise propagation in a synthetic transcriptional cascade. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[7]  M. Roussel.  The Use of Delay Differential Equations in Chemical Kinetics , 1996 .

[8]  A. Ninfa,et al.  Development of Genetic Circuitry Exhibiting Toggle Switch or Oscillatory Behavior in Escherichia coli , 2003, Cell.

[9]  N. Friedman,et al.  Stochastic protein expression in individual cells at the single molecule level , 2006, Nature.

[10]  A. van Oudenaarden,et al.  Noise Propagation in Gene Networks , 2005, Science.

[11]  P. Swain,et al.  Gene Regulation at the Single-Cell Level , 2005, Science.

[12]  P. R. ten Wolde,et al.  Enhancement of the stability of genetic switches by overlapping upstream regulatory domains. , 2003, Physical review letters.

[13]  Rui Zhu,et al.  Mesoscopic Description of Chemical Supercritical Hopf bifurcation , 2004, Int. J. Bifurc. Chaos.

[14]  Yiannis N. Kaznessis,et al.  Multi-scale models for gene network engineering , 2006 .

[15]  Julian Lewis.  Autoinhibition with Transcriptional Delay A Simple Mechanism for the Zebrafish Somitogenesis Oscillator , 2003, Current Biology.

[16]  F R Adler,et al.  How to make a biological switch. , 2000, Journal of theoretical biology.

[17]  J. Raser,et al.  Control of Stochasticity in Eukaryotic Gene Expression , 2004, Science.

[18]  K. Aihara,et al.  Stability of genetic regulatory networks with time delay , 2002 .

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

[20]  A. Kierzek,et al.  The Effect of Transcription and Translation Initiation Frequencies on the Stochastic Fluctuations in Prokaryotic Gene Expression* , 2001, The Journal of Biological Chemistry.

[21]  Rui Zhu,et al.  Validation of an algorithm for delay stochastic simulation of transcription and translation in prokaryotic gene expression , 2006, Physical biology.

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

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

[24]  Teresa Head-Gordon,et al.  Computational challenges in structural and functional genomics , 2001, IBM Syst. J..

[25]  Kevin Burrage,et al.  Stochastic approaches for modelling in vivo reactions , 2004, Comput. Biol. Chem..

[26]  S. Kauffman.  Metabolic stability and epigenesis in randomly constructed genetic nets. , 1969, Journal of theoretical biology.

[27]  A. Arkin,et al.  Stochastic mechanisms in gene expression. , 1997, Proceedings of the National Academy of Sciences of the United States of America.

[28]  Stefan Bornholdt,et al.  Stable and unstable attractors in Boolean networks. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  D. Kennell,et al.  Evidence for variable rates of ribosome movement in Escherichia coli. , 1976, Journal of molecular biology.

[30]  X. Xie,et al.  Probing Gene Expression in Live Cells, One Protein Molecule at a Time , 2006, Science.

[31]  M. Ptashne,et al.  Transcriptional activation by recruitment , 1997, Nature.

[32]  Edward R. Dougherty,et al.  Probabilistic Boolean networks: a rule-based uncertainty model for gene regulatory networks , 2002, Bioinform..

[33]  Siren R. Veflingstad,et al.  Effect of time delay on pattern formation : Competition between homogenisation and patterning , 2005 .

[34]  Yiannis N Kaznessis,et al.  Model-driven designs of an oscillating gene network. , 2005, Biophysical journal.

[35]  D. Volfson,et al.  Delay-induced stochastic oscillations in gene regulation. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[36]  S. Kauffman.  A proposal for using the ensemble approach to understand genetic regulatory networks. , 2004, Journal of theoretical biology.

[37]  I. Cowell,et al.  Repression versus activation in the control of gene transcription. , 1994, Trends in biochemical sciences.

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

[39]  Farren J. Isaacs,et al.  Signal Processing in Single Cells , 2005, Science.

[40]  M. Elowitz,et al.  A synthetic oscillatory network of transcriptional regulators , 2000, Nature.

[41]  E. A. Gaffneya,et al.  Gene Expression Time Delays and Turing Pattern Formation Systems , 2006 .

[42]  W. McClure,et al.  Rate-limiting steps in RNA chain initiation. , 1980, Proceedings of the National Academy of Sciences of the United States of America.

[43]  F. Neidhart.  Escherichia coli and Salmonella. , 1996 .

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

[45]  Mads Kærn,et al.  Noise in eukaryotic gene expression , 2003, Nature.

[46]  Stuart A. Kauffman,et al.  The origins of order , 1993 .

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

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

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

[50]  Nicholas J. Guido,et al.  A bottom-up approach to gene regulation , 2006, Nature.

[51]  M. Fussenegger,et al.  An engineered epigenetic transgene switch in mammalian cells , 2004, Nature Biotechnology.

[52]  J. Raser,et al.  Noise in Gene Expression: Origins, Consequences, and Control , 2005, Science.

[53]  M. Dreyfus,et al.  Interdependence of translation, transcription and mRNA degradation in the lacZ gene. , 1992, Journal of molecular biology.

[54]  P. V. Hippel,et al.  An Integrated Model of the Transcription Complex in Elongation, Termination, and Editing , 1998 .

[55]  T. Elston,et al.  Stochasticity in gene expression: from theories to phenotypes , 2005, Nature Reviews Genetics.

[56]  Zhonghuai Hou,et al.  Internal noise stochastic resonance in a circadian clock system , 2003 .

[57]  Rui Zhu,et al.  A General Modeling Strategy for Gene Regulatory Networks with Stochastic Dynamics , 2006, J. Comput. Biol..

[58]  N. Monk.  Oscillatory Expression of Hes1, p53, and NF-κB Driven by Transcriptional Time Delays , 2003, Current Biology.