Théorèmes limites pour des processus de Markov à sauts. Synthèse des résultats et applications en biologie moleculaire

Les processus de Markov a sauts permettent la modelisation des phenomenes stochastiques en biologie moleculaire. Neanmoins, il y a peu de resultats mathematiques sur la dynamique de ces processus. De plus, leur simulation sur ordinateur rencontre des difficultes dues au temps d'execution. Nous presentons des resultats permettant de reduire la complexite de la dynamique stochastique. Ces methodes utilisent des theoremes limites probabilistes.

[1]  J. Rawlings,et al.  Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics , 2002 .

[2]  U Alon,et al.  Generation of oscillations by the p53-Mdm2 feedback loop: a theoretical and experimental study. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[3]  Iosif Ilitch Gikhman,et al.  Introduction to the theory of random processes , 1969 .

[4]  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.

[5]  J. Elf,et al.  Spontaneous separation of bi-stable biochemical systems into spatial domains of opposite phases. , 2004, Systems biology.

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

[7]  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.

[8]  K. Burrage,et al.  Bistability and switching in the lysis/lysogeny genetic regulatory network of bacteriophage lambda. , 2004, Journal of theoretical biology.

[9]  Sergey Plyasunov,et al.  ON HYBRID SIMULATION SCHEMES FOR STOCHASTIC REACTION DYNAM ICS , 2005, math/0504477.

[10]  R. W. R. Darling.  Fluid Limits of Pure Jump Markov Processes: a Practical Guide , 2002 .

[11]  N. Berglund,et al.  Geometric singular perturbation theory for stochastic differential equations , 2002, math/0204008.

[12]  C. Rao,et al.  Control, exploitation and tolerance of intracellular noise , 2002, Nature.

[13]  Tianhai Tian,et al.  A multi-scaled approach for simulating chemical reaction systems. , 2004, Progress in biophysics and molecular biology.

[14]  F. Dunstan,et al.  STOCHASTIC APPROACH TO CHEMICAL REACTION KINETICS , 1981 .

[15]  Alexander N. Gorban,et al.  Reduced description in the reaction kinetics , 2000 .

[16]  T. Kurtz.  Limit theorems for sequences of jump Markov processes approximating ordinary differential processes , 1971, Journal of Applied Probability.

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

[18]  R.W.R. Darling,et al.  Structure of large random hypergraphs , 2005 .

[19]  Uri Alon,et al.  Dynamics of the p53-Mdm2 feedback loop in individual cells , 2004, Nature Genetics.

[20]  George Yin.  On Limit Results for a Class of Singularly Perturbed Switching Diffusions , 2001 .

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

[22]  J. Banavar,et al.  Computer Simulation of Liquids , 1988 .

[23]  Balding,et al.  Diffusion-controlled reactions in one dimension: Exact solutions and deterministic approximations. , 1989, Physical review. A, General physics.

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

[25]  Kunihiko Kaneko,et al.  Discreteness-induced stochastic steady state in reaction diffusion systems : self-consistent analysis and stochastic simulations , 2004, physics/0409027.

[26]  Katherine C. Chen,et al.  Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell. , 2003, Current opinion in cell biology.

[27]  M. Delbrück.  Statistical Fluctuations in Autocatalytic Reactions , 1940 .

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

[29]  T. Kurtz.  Solutions of ordinary differential equations as limits of pure jump markov processes , 1970, Journal of Applied Probability.

[30]  Onno Boxma,et al.  ON/OFF STORAGE SYSTEMS WITH STATE-DEPENDENT INPUT, OUTPUT, AND SWITCHING RATES , 2005, Probability in the Engineering and Informational Sciences.

[31]  Wilhelm Huisinga,et al.  ADAPTIVE SIMULATION OF HYBRID STOCHASTIC AND DETERMINISTIC MODELS FOR BIOCHEMICAL SYSTEMS , 2005 .

[32]  M. Elowitz,et al.  Protein Mobility in the Cytoplasm ofEscherichia coli , 1999, Journal of bacteriology.

[33]  M. Nowak,et al.  Stochastic Tunnels in Evolutionary Dynamics , 2004, Genetics.

[34]  Masaru Tomita,et al.  Space in systems biology of signaling pathways – towards intracellular molecular crowding in silico , 2005, FEBS letters.

[35]  Richard L. Tweedie,et al.  Markov Chains and Stochastic Stability , 1993, Communications and Control Engineering Series.

[36]  A. Swishchuk,et al.  Semi-Markov Random Evolutions , 1994 .

[37]  A. Skorokhod.  Asymptotic Methods in the Theory of Stochastic Differential Equations , 2008 .

[38]  C. Cercignani,et al.  Many-Particle Dynamics And Kinetic Equations , 1997 .

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

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

[41]  Mark H. Davis.  Markov Models and Optimization , 1995 .

[42]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

[43]  Sangyoub Lee,et al.  EXCLUDED VOLUME EFFECT ON THE DIFFUSION-INFLUENCED BIMOLECULAR REACTIONS , 1997 .

[44]  I. Bose,et al.  Graded and binary responses in stochastic gene expression , 2004, Physical biology.

[45]  D. Aldous.  Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists , 1999 .

[46]  S. Tapscott,et al.  Modeling stochastic gene expression: implications for haploinsufficiency. , 1998, Proceedings of the National Academy of Sciences of the United States of America.

[47]  S. Solomon,et al.  The importance of being discrete: life always wins on the surface. , 1999, Proceedings of the National Academy of Sciences of the United States of America.

[48]  David H. Sharp,et al.  Model for cooperative control of positional information in Drosophila by bicoid and maternal hunchback. , 1995, The Journal of experimental zoology.

[49]  M Ander,et al.  SmartCell, a framework to simulate cellular processes that combines stochastic approximation with diffusion and localisation: analysis of simple networks. , 2004, Systems biology.