Abstract
In this manuscript, we propose a mathematical framework to couple transcription and translation in which mRNA production is described by a set of master equations, while the dynamics of protein density is governed by a random differential equation. The coupling between the two processes is given by a stochastic perturbation whose statistics satisfies the master equations. In this approach, from the knowledge of the analytical time-dependent distribution of mRNA number, we are able to calculate the dynamics of the probability density of the protein population.
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References
Abramowitz M, Stegun IA (1964) Handbook of mathematical functions with formulas, graphs and mathematical tables. Government Printing Office, U.S
Arnold L (1998) Random dynamical systems. Springer, Berlin
Blake WJ, Kaern M, Cantor CR, Collins JJ (2003) Noise in eukaryotic gene expression. Nature 422:633–637
Cai L, Friedman N, Xie X (2006) Stochastic protein expression in individual cells at the single molecule level. Nature 440(7082):358–62. doi:10.1038/nature04599
Cogburn R, Torrez WC (1981) Birth and death processes with random environments in continuous time. J Appl Probab 18(1):19–30
Crudu A, Debussche A, Muller A, Radulescu O (2012) Convergence of stochastic gene networks to hybrid piecewise deterministic processes. Ann Appl Probab 22(5):1822–1859
Crudu A, Debussche A, Radulescu O (2009) Hybrid stochastic simplifications for multiscale gene networks. BMC Syst Biol 3(1):89
Delbrück M (1940) Statistical fluctuations in autocatalytic reactions. J Chem Phys 8:120–124
Elowitz MB, Levine AJ, Siggia ED, Swain PS (2002) Stochastic gene expression in a single cell. Science 297(5584):1183–1186. doi:10.1126/science.1070919
Ferguson M, Le Coq D, Jules M, Aymerich S, Radulescu O, Declerck N, Royer C (2012) Reconciling molecular regulatory mechanisms with noise patterns of bacterial metabolic promoters in induced and repressed states. Proc Natl Acad Sci USA 109(1):155–160
Ferreira RC, Bosco FAR, Briones MRS (2009) Scaling properties of transcription profiles in gene networks. Int J Bioinform Res Appl 5(2):178–186
Ferreira RC, Briones MRS, Antoneli F. A model of gene expression based on random dynamical systems reveals modularity properties of gene regulatory networks. arXiv:1309.0765 (2013)
Friedman N, Cai L, Xie XS (2006) Linking stochastic dynamics to population distribution: an analytical framework of gene expression. Phys Rev Lett 97(16):168,302
Golding I, Paulsson J, Zawilski S, Cox E (2005) Real-time kinetics of gene activity in individual bacteria. Cell 123(6):1025–36. doi:10.1016/j.cell.2005.09.031
Hornos JEM, Schultz D, Innocentini GCP, Wang J, Walczak AM, Onuchic JN, Wolynes PG (2005) Self-regulating gene: an exact solution. Phys Rev E 72(5):e051,907. doi:10.1103/PhysRevE.72.051907
Innocentini GCP, Forger M, Ramos A, Radulescu O, Hornos JEM (2013) Multimodality and flexibility in stochastic gene expression. Bull Math Biol 75:2600–2630
Innocentini GCP, Hornos JEM (2007) Modeling stochastic gene expression under repression. J Math Biol 55(3):413–431. doi:10.1007/s00285-007-0090-x
Iyer-Biswas S, Hayot F, Jayaprakash C (2009) Stochasticity of gene products from transcriptional pulsing. Phys Rev E 79:031,911
Kepler TB, Elston TC (2001) Stochasticity in transcriptional regulation: origins, consequences, and mathematical representations. Biophys J 81(6):3116–3136. doi:10.1016/S0006-3495(01)75949-8
Lipniacki T, Paszek P, Marciniak-Czochra A, Brasier AR, Kimmel M (2006) Transcriptional stochasticity in gene expression. J Theor Biol 238:348–367. doi:10.1016/j.jtbi.2005.05.032
Ozbudak EM, Thattai M, Kurtser I, Grossman AD, van Oudenaarden A (2002) Regulation of noise in the expression of a single gene. Nat Genet 31:69–73
Paulsson J (2005) Models of stochastic gene expression. Phys Life Rev 2:157–175
Peccoud J, Ycart B (1995) Markovian modelling of gene product synthesis. Theor Popul Biol 48:222–234
Pirone J, Elston T (2004) Fluctuations in transcription factor binding can explain the graded and binary responses observed in inducible gene expression. J Theor Biol 226:111–121
Ramos AF, Innocentini GCP, Hornos JEM (2011) Exact time-dependent solutions for a self-regulating gene. Phys Rev E 83(6):e062,902. doi:10.1103/PhysRevE.83.062902
Raser JM, O’Shea EK (2004) Control of stochasticity in eukaryotic gene expression. Science 304(5678):1811–1814. doi:10.1126/science.1098641
Shahrezaei V, Swain PS (2008) Analytical distributions for stochastic gene expression. Proc Natl Acad Sci USA 105(45):17256–17261. doi:10.1073/pnas.0803850105
van Kampen NG (2007) Stochastic processes in physics and chemistry, 3rd edn. Elsevier, Amsterdam
Yu J, Xiao J, Ren X, Lao K, Xie XS (2006) Probing gene expression in live cells, one protein molecule at a time. Science 311(5767):1600–1603. doi:10.1126/science.1119623
Acknowledgments
We would like to thank the referees for their insights. Work supported by FAPESP, SP, Brazil (G. I., contract 2012/04723-4) and CNPq, Brazil (G. I., contract 202238/2014-8; M. F., contract 307238/2011-3; F. A., contract 306362/2012-0). O. R. thanks CNRS and LABEX Epigenmed for support.
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Innocentini, G.C.P., Forger, M., Radulescu, O. et al. Protein Synthesis Driven by Dynamical Stochastic Transcription. Bull Math Biol 78, 110–131 (2016). https://doi.org/10.1007/s11538-015-0131-3
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DOI: https://doi.org/10.1007/s11538-015-0131-3