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, Volume 23 (2016) no. 2, pp. 171-217.

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.

DOI: 10.5802/ambp.360
Classification: 92B05, 37A30, 60J20
Keywords: Regulatory network model, Markov operators, invariant measure
Sander Hille 1; Katarzyna Horbacz 2; Tomasz Szarek 3

1 Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands
2 Department of Mathematics, University of Silesia, Bankowa 14, 40-007 Katowice, Poland
3 Institute of Mathematics, University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland
     author = {Sander Hille and Katarzyna Horbacz and Tomasz Szarek},
     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},
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     year = {2016},
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     url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.360/}
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Sander Hille; Katarzyna Horbacz; Tomasz Szarek. 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, Volume 23 (2016) no. 2, pp. 171-217. doi : 10.5802/ambp.360. https://ambp.centre-mersenne.org/articles/10.5802/ambp.360/

[1] Adam Arkin; John Ross; Harley H. McAdams Stochastic kinetic analysis of developmental pathway bifurcations in phage lambda-infected Escherichia coli cells, Genetics, Volume 149 (1998), pp. 1633-1648

[2] Michael F. Barnsley; Stephen G. Demko; John H. Elton; Jeffrey S. Geronimo Invariant measures arising from iterated function systems with place dependent probabilities, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 24 (1988), pp. 367-394

[3] Patrick Billingsley Convergence of probability measures, John Wiley & Sons, New York-London-Sydney, 1968, xii+253 pages

[4] Jonathan R. Chubb; Tatjana Trcek; Shailesh M. Shenoy; Robert H. Singer Transcriptional pulsing of a developmental gene, Curr. Biol., Volume 16 (2006), pp. 1018-1025 | DOI

[5] Kai-Nai Chueh; Charles C. Conley; Smoller Joel A. Positively invariant regions for systems of nonlinear diffusion equations, Indiana Univ. Math. J., Volume 26 (1977) no. 2, pp. 373-392 | DOI

[6] Edward Conway; David Hoff; Joel Smoller 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] Giuseppe Da Prato; Jerzy Zabczyk Ergodicity for Infinite Dimensional Systems, London Mathematical Society lecture Note, 229, Cambridge University Press, 1996, xi+339 pages

[8] Hidde De Jong; Johannes Geiselmann; Grégrory Batt; Céline Hernandez; Michel Page Qualitative simulation of the initiation of sporulation in Bacillus subtilis, Bull. Math. Biol., Volume 66 (2004), pp. 261-299 | DOI

[9] James A. Dix; A. S. Verkman Crowding effects of diffusion in solutions and cells, Annu. Rev. Biophys., Volume 37 (2008), pp. 247-263 | DOI

[10] Christopher M. Dobson Chemical space and biology, Nature, Volume 432 (2004), pp. 824-828 | DOI

[11] Wolfgang Doeblin 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] Richard M. Dudley Probabilities and Metrics, Lecture Notes Series, 45, Aarhus Universitet, 1976

[13] Johan Elf; Gene-Wei Li; X. Sunney Xie Probing transcription factor dynamics at the single-molecule level in a living cell, Science, Volume 316 (2007), pp. 1191-1194 | DOI

[14] Michael B. Elowitz; Arnold J. Levine; Eric D. Siggia; Peter S. Swain Stochastic gene expression in a single cell, Science, Volume 297 (2002), pp. 1183-1186 | DOI

[15] Anne Farewell; Frederick C. Neidhardt Effect of temperature on in vivo protein synthetic capacity in Escherichia coli, J. Bacteriol., Volume 180 (1998) no. 17, pp. 4704-4710

[16] Alfred L. Goldberg Protein degradation and protection against misfolded or damaged proteins, Nature, Volume 426 (2003), pp. 895-899 | DOI

[17] Ido Golding; Johan Paulsson; Scott M. Zawilski; Edward C. Cox Real-time kinetics of gene activity in individual bacteria, Cell, Volume 123 (2005), pp. 1025-1036 | DOI

[18] Brian C. Goodwin Advances in enzymatic control processes, Adv. Enzyme Regul., Volume 3 (1965), pp. 425-437 | DOI

[19] Rafał Kapica; Tomasz Szarek; Maciej Śleçzka On a unique ergodicity of some Markov processes, Potential Anal., Volume 36 (2012), pp. 589-606 | DOI

[20] Edward L. King; Cart Altman 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] Tomasz Komorowski; Szymon Peszat; Tomasz Szarek On ergodicity of some Markov processes, Ann. Probab., Volume 38 (2010) no. 4, pp. 1401-1443 | DOI

[22] Juhong Kuang; Moxun Tang; Jianshe Yu The mean and noise of protein numbers in stochastic gene expression, J. Math. Biol., Volume 67 (2013) no. 2, pp. 261-291 | DOI

[23] Andrzej Lasota; Michael C. Mackey Cell division and the stability of cellular population, J. Math. Biol., Volume 38 (1999) no. 3, pp. 241-261 | DOI

[24] Andrzej Lasota; James Yorke On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., Volume 186 (1973), pp. 481-488 | DOI

[25] Andrzej Lasota; James Yorke Lower bound technique for Markov operators and iterated function systems, Random Comput. Dynam., Volume 2 (1994) no. 1, pp. 41-77

[26] Michael C. Mackey; Marta Tyran-Kamińska; Romain Yvinec Molecular distributions in gene regulatory dynamics, J. Theor. Biol., Volume 247 (2011), pp. 84-96 | DOI

[27] Harley H. McAdams; Adam Arkin Stochastic mechansisms in gene expression, Proc. Natl. Acad. Sci., Volume 94 (1997) no. 3, pp. 814-819 | DOI

[28] Jacek T. Mika; Bert Poolman Macromolecule diffusion and confinement in prokarytic cells, Curr. Op. Biotech., Volume 22 (2011) no. 1, pp. 117-126 | DOI

[29] A. Ochab-Marcinek; M. Tabaka Bimodal gene expression in noncooperative regulatory systems, Proc. Natl. Acad. Sci., Volume 107 (2010) no. 51, pp. 22096-22101 | DOI

[30] Amnon Pazy Semigroups of Linear Operators and Applications to Partial Dfferential Equations, Applied Mathematical Sciences, 41, Springer-Verlag, Berlin, 1983, viii+279 pages

[31] Martin J. Pine Regulation of intracellular proteolysis in Escherichia coli, J. Bacteriol., Volume 115 (1973) no. 1, pp. 107-116

[32] Jonathan M. Raser; Erin K. O’Shea Noise in gene expression: origins, consequences, and control, Science, Volume 309 (2005), pp. 2010-2013 | DOI

[33] Shaunak Sen; Jordi Garcia-Ojalvo; Michael B. Elowitz Dynamical consequences of bandpass feedback loops in a bacterial phosphorelay, PLoS One, Volume 6 (2011) no. 9, e25102 pages | DOI

[34] Joshua W. Shaevitz; Elio A. Abbondanzieri; Robert Landick; Steven M. Block Backtracking by single RNA polymerase molecules observed at near-base-pair resolution, Nature, Volume 426 (2003), pp. 685-687 | DOI

[35] Örjan Stenflo A note on a theorem of Karlin, Statist. Probab. Lett., Volume 54 (2001) no. 2, pp. 183-187 | DOI

[36] David M. Suter; Nacho Molina; David Gatfield; Kim Schneider; Ueli Schibler; Felix Naef Mammalian genes are transcribed with widely different bursting kinetics, Science, Volume 332 (2011), pp. 472-474 | DOI

[37] Peter S. Swain; André Longtin Noise in genetic and neural networks, Chaos, Volume 16 (2006), 026101 pages | DOI

[38] Tomasz Szarek Invariant measures for nonexpansive Markov operators on Polish spaces, Diss. Math., Volume 415 (2003), 64 pages

[39] Tomasz Szarek Feller processes on nonlocally compact spaces, Ann. Probab., Volume 34 (2006) no. 5, pp. 1849-1863 | DOI

[40] Moxun Tang 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] Jawahar Tiwari; Alex Fraser; Richard Beckman Genetical feedback repression: I. Single locus models, J. Theor. Biol., Volume 45 (1974) no. 2, pp. 311-326 | DOI

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