We extend the so-called lower-bound technique for equicontinuous families of Markov operators by introducing the new concept of uniform equicontinuity on balls. Combined with a semi-concentrating condition, it yields a new abstract mathematical result on existence and uniqueness of invariant measures for Markov operators. It allows us to show the tightness of the set of invariant measures for some classes of Markov operators. This, in turn, gives a useful tool for proving a continuous dependence on given parameters for semi–concentrating Markov semigroups. In the second part we formulate an abstract modelling framework that defines a piecewise deterministic Markov process whose transition operator at the times of intervention yields a semi-concentrating Markov operator that is uniformly equicontinuous on balls. We show that this framework applies to a detailed stochastic model for an autoregulated gene in a bacterium that takes random transcription delay into account.
Mots clés : Regulatory network model, Markov operators, invariant measure
@article{AMBP_2016__23_2_171_0,
author = {Hille, Sander and Horbacz, Katarzyna and Szarek, Tomasz},
title = {Existence of a unique invariant measure for a class of equicontinuous {Markov} operators with application to a stochastic model for an autoregulated gene},
journal = {Annales math\'ematiques Blaise Pascal},
pages = {171--217},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {23},
number = {2},
year = {2016},
doi = {10.5802/ambp.360},
language = {en},
url = {http://www.numdam.org/articles/10.5802/ambp.360/}
}
TY - JOUR AU - Hille, Sander AU - Horbacz, Katarzyna AU - Szarek, Tomasz TI - Existence of a unique invariant measure for a class of equicontinuous Markov operators with application to a stochastic model for an autoregulated gene JO - Annales mathématiques Blaise Pascal PY - 2016 SP - 171 EP - 217 VL - 23 IS - 2 PB - Annales mathématiques Blaise Pascal UR - http://www.numdam.org/articles/10.5802/ambp.360/ DO - 10.5802/ambp.360 LA - en ID - AMBP_2016__23_2_171_0 ER -
%0 Journal Article %A Hille, Sander %A Horbacz, Katarzyna %A Szarek, Tomasz %T Existence of a unique invariant measure for a class of equicontinuous Markov operators with application to a stochastic model for an autoregulated gene %J Annales mathématiques Blaise Pascal %D 2016 %P 171-217 %V 23 %N 2 %I Annales mathématiques Blaise Pascal %U http://www.numdam.org/articles/10.5802/ambp.360/ %R 10.5802/ambp.360 %G en %F AMBP_2016__23_2_171_0
Hille, Sander; Horbacz, Katarzyna; Szarek, Tomasz. Existence of a unique invariant measure for a class of equicontinuous Markov operators with application to a stochastic model for an autoregulated gene. Annales mathématiques Blaise Pascal, Tome 23 (2016) no. 2, pp. 171-217. doi : 10.5802/ambp.360. http://www.numdam.org/articles/10.5802/ambp.360/
[1] Stochastic kinetic analysis of developmental pathway bifurcations in phage lambda-infected Escherichia coli cells, Genetics, Volume 149 (1998), pp. 1633-1648
[2] Invariant measures arising from iterated function systems with place dependent probabilities, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 24 (1988), pp. 367-394
[3] Convergence of probability measures, John Wiley & Sons, New York-London-Sydney, 1968, xii+253 pages
[4] Transcriptional pulsing of a developmental gene, Curr. Biol., Volume 16 (2006), pp. 1018-1025 | DOI
[5] Positively invariant regions for systems of nonlinear diffusion equations, Indiana Univ. Math. J., Volume 26 (1977) no. 2, pp. 373-392 | DOI
[6] Large time behavior of solutions of systems of nonlinear reaction-diffusion equations, SIAM J. Appl. Math., Volume 35 (1978) no. 1, pp. 1-16 | DOI
[7] Ergodicity for Infinite Dimensional Systems, London Mathematical Society lecture Note, 229, Cambridge University Press, 1996, xi+339 pages
[8] Qualitative simulation of the initiation of sporulation in Bacillus subtilis, Bull. Math. Biol., Volume 66 (2004), pp. 261-299 | DOI
[9] Crowding effects of diffusion in solutions and cells, Annu. Rev. Biophys., Volume 37 (2008), pp. 247-263 | DOI
[10] Chemical space and biology, Nature, Volume 432 (2004), pp. 824-828 | DOI
[11] Sur les propriétés asymptotiques de mouvement régis par certains types de chaines simples, Bull. Math. Soc. Roum. Sci., Volume 39 (1937) no. 1, pp. 57-115
[12] Probabilities and Metrics, Lecture Notes Series, 45, Aarhus Universitet, 1976
[13] Probing transcription factor dynamics at the single-molecule level in a living cell, Science, Volume 316 (2007), pp. 1191-1194 | DOI
[14] Stochastic gene expression in a single cell, Science, Volume 297 (2002), pp. 1183-1186 | DOI
[15] Effect of temperature on in vivo protein synthetic capacity in Escherichia coli, J. Bacteriol., Volume 180 (1998) no. 17, pp. 4704-4710
[16] Protein degradation and protection against misfolded or damaged proteins, Nature, Volume 426 (2003), pp. 895-899 | DOI
[17] Real-time kinetics of gene activity in individual bacteria, Cell, Volume 123 (2005), pp. 1025-1036 | DOI
[18] Advances in enzymatic control processes, Adv. Enzyme Regul., Volume 3 (1965), pp. 425-437 | DOI
[19] On a unique ergodicity of some Markov processes, Potential Anal., Volume 36 (2012), pp. 589-606 | DOI
[20] A schematic method of deriving the rate laws for enzyme-catalyzed reactions, J. Phys. Chem., Volume 60 (1956) no. 10, pp. 1375-1378 | DOI
[21] On ergodicity of some Markov processes, Ann. Probab., Volume 38 (2010) no. 4, pp. 1401-1443 | DOI
[22] The mean and noise of protein numbers in stochastic gene expression, J. Math. Biol., Volume 67 (2013) no. 2, pp. 261-291 | DOI
[23] Cell division and the stability of cellular population, J. Math. Biol., Volume 38 (1999) no. 3, pp. 241-261 | DOI
[24] On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., Volume 186 (1973), pp. 481-488 | DOI
[25] Lower bound technique for Markov operators and iterated function systems, Random Comput. Dynam., Volume 2 (1994) no. 1, pp. 41-77
[26] Molecular distributions in gene regulatory dynamics, J. Theor. Biol., Volume 247 (2011), pp. 84-96 | DOI
[27] Stochastic mechansisms in gene expression, Proc. Natl. Acad. Sci., Volume 94 (1997) no. 3, pp. 814-819 | DOI
[28] Macromolecule diffusion and confinement in prokarytic cells, Curr. Op. Biotech., Volume 22 (2011) no. 1, pp. 117-126 | DOI
[29] Bimodal gene expression in noncooperative regulatory systems, Proc. Natl. Acad. Sci., Volume 107 (2010) no. 51, pp. 22096-22101 | DOI
[30] Semigroups of Linear Operators and Applications to Partial Dfferential Equations, Applied Mathematical Sciences, 41, Springer-Verlag, Berlin, 1983, viii+279 pages
[31] Regulation of intracellular proteolysis in Escherichia coli, J. Bacteriol., Volume 115 (1973) no. 1, pp. 107-116
[32] Noise in gene expression: origins, consequences, and control, Science, Volume 309 (2005), pp. 2010-2013 | DOI
[33] Dynamical consequences of bandpass feedback loops in a bacterial phosphorelay, PLoS One, Volume 6 (2011) no. 9, e25102 pages | DOI
[34] Backtracking by single RNA polymerase molecules observed at near-base-pair resolution, Nature, Volume 426 (2003), pp. 685-687 | DOI
[35] A note on a theorem of Karlin, Statist. Probab. Lett., Volume 54 (2001) no. 2, pp. 183-187 | DOI
[36] Mammalian genes are transcribed with widely different bursting kinetics, Science, Volume 332 (2011), pp. 472-474 | DOI
[37] Noise in genetic and neural networks, Chaos, Volume 16 (2006), 026101 pages | DOI
[38] Invariant measures for nonexpansive Markov operators on Polish spaces, Diss. Math., Volume 415 (2003), 64 pages
[39] Feller processes on nonlocally compact spaces, Ann. Probab., Volume 34 (2006) no. 5, pp. 1849-1863 | DOI
[40] The mean frequency of transcriptional bursting and its variation in single cells, J. Math. Biol., Volume 60 (2010) no. 1, pp. 27-58 | DOI
[41] Genetical feedback repression: I. Single locus models, J. Theor. Biol., Volume 45 (1974) no. 2, pp. 311-326 | DOI
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