On a stochastic gene expression with pre-mRNA, mRNA and protein contribution. (English) Zbl 1343.92171
Summary: In this paper we develop a model of stochastic gene expression, which is an extension of the model investigated in the paper [T. Lipniacki et al., “Transcriptional stochasticity in gene expression”, ibid. 238, No. 2, 348–367 (2006; doi:10.1016/j.jtbi.2005.05.032)] In our model, stochastic effects still originate from random fluctuations in gene activity status, but we precede mRNA production by the formation of pre-mRNA, which enriches classical transcription phase. We obtain a stochastically regulated system of ordinary differential equations (ODEs) describing evolution of pre-mRNA, mRNA and protein levels. We perform mathematical analysis of a long-time behavior of this stochastic process, identified as a piece-wise deterministic Markov process (PDMP). We check exact results using numerical simulations for the distributions of all three types of particles. Moreover, we investigate the deterministic (adiabatic) limit state of the process, when depending on parameters
it can exhibit two specific types of behavior: bistability and the existence of the limit cycle. The latter one is not present when only two kinds of gene expression products are considered.
MSC:
| 92C40 | Biochemistry, molecular biology |
| 60J20 | Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) |
Keywords:
stochastic gene expression; pre-mRNA; piece-wise deterministic Markov process; invariant densityReferences:
| [1] | Bakhtin, Y.; Hurth, T., Invariant densities for dynamical systems with random switching, Nonlinearity, 25, 2937-2952 (2012) · Zbl 1251.93132 |
| [2] | Benaïm, M.; le Borgne, S.; Malrieu, F.; Zitt, P. A., Quantitative ergodicity for some switched dynamical systems, Electron. Commun. Probab., 17, 56, 1-14 (2012) · Zbl 1347.60118 |
| [3] | Benaïm, M.; le Borgne, S.; Malrieu, F.; Zitt, P. A., Qualitative properties of certain piecewise deterministic Markov processes, Ann. Inst. H. Poincaré Probab. Stat., 51, 3, 1040-1075 (2015) · Zbl 1325.60123 |
| [4] | Blake, W.; Kaern, M.; Cantor, C.; Collins, J., Noise in eucaryotic gene expression, Nature, 422, 633-637 (2003) |
| [5] | Bobrowski, A., Degenerate convergence of semigroups related to a model of stochastic gene expression, J. Math. Anal. Appl., 73, 3, 345-366 (2006) · Zbl 1114.47068 |
| [6] | Bobrowski, A.; Pichór, K.; Lipniacki, T.; Rudnicki, R., Asymptotic behavior of distribution of mRNA and protein levels in a model of stochastic gene expression, J. Math. Anal. Appl., 333, 753-769 (2007) · Zbl 1115.92018 |
| [8] | Crudu, A.; Debussche, A.; Muller, A.; Radulescu, O., Convergence of stochastic gene networks to hybrid piecewise deterministic processes, Ann. Appl. Probab., 22, 5, 1822-1859 (2012) · Zbl 1261.60073 |
| [9] | Cui, P.; Zhang, S.; Ding, F.; Ali, S.; Xiong, L., Dynamic regulation of genome-wide pre-mRNA splicing and stress tolerance by the Sm-like protein LSm5 in Arabidopsis, Genome Biol., 15, R1 (2014) |
| [10] | Davis, M., Piece-wise deterministic Markov processesa general class of non-diffusion stochastic processes, J. R. Stat. Soc. B, 46, 353-388 (1984) · Zbl 0565.60070 |
| [11] | Friedman, N.; Cai, L.; Xie, X. S., Linking stochastic dynamics to population distributionan analytical framework of gene expression, Phys. Rev. Lett., 97, 168302 (2006) |
| [12] | Gillespie, D., Exact stochastic simulation of coupled chemical reactions, J. Phys. Chem., 81, 25, 2340-2361 (1977) |
| [13] | Goodwin, B. C., Oscillatory behavior in enzymatic control processes, Adv. Enzyme Regul., 3, 425-438 (1965) |
| [14] | Jaruszewicz, J.; Zuk, P.; Lipniacki, T., Type of noise defines global attractors in bistable molecular regulatory systems, J. Theor. Biol., 317, 140-151 (2013) · Zbl 1368.92053 |
| [15] | Kepler, T.; Elston, T., Stochasticity in transcriptional regulationorigins, consequences, and mathematical representations, Biophys. J., 81, 3116-3136 (2001) |
| [16] | Kim, J.; Marioni, J., Inferring the kinetics of stochastic gene expression from single-cell RNA-sequencing data, Genome Biol., 14, R7 (2013) |
| [17] | Klamka, J., Controllability of dynamical systems - a survey, Arch. Control Sci., 2, 281-307 (1993) |
| [18] | Komorowski, T.; Tyrcha, J., Asymptotic properties of some Markov operators, Bull. Pol. Acad. Sci. Math., 43, 221-228 (1989) · Zbl 0767.47012 |
| [19] | Lasota, A.; Mackey, M., Chaos, Fractals and Noise, Stochastic Aspects of Dynamics (1994), Springer: Springer New York · Zbl 0784.58005 |
| [20] | Lipniacki, T.; Paszek, P.; Marciniak-Czochra, A.; Brasier, A. R.; Kimmel, M., Transcriptional stochasticity in gene expression, J. Theor. Biol., 238, 348-367 (2006) · Zbl 1445.92113 |
| [21] | Lipniacki, T.; Pruszynski, K.; Paszek, P.; Brasier, A. R.; Kimmel, M., Single TNF \(\alpha\) trimers mediating NF-\( \kappa B\) activation: stochastic robustness of NF-\( \kappa\) signalling, BMC Bioinform., 8, 376 (2007) |
| [24] | Maniatis, T.; Reed, R., An extensive network of coupling among gene expression machines, Nature, 416, 499-506 (2002) |
| [26] | Peccoud, J.; Ycart, B., Markovian modeling of gene-product synthesis, Theor. Popul. Biol., 48, 2, 222-234 (1995) · Zbl 0865.92006 |
| [27] | Pedraza, J.; Paulsson, J., Effects of molecular memory and bursting on fluctuations in gene expression, Science, 319, 339-343 (2008) |
| [28] | Pichór, K.; Rudnicki, R., Continuous Markov semigroups and stability of transport equations, J. Math. Anal. Appl., 249, 668-685 (2000) · Zbl 0965.47026 |
| [29] | Rudnicki, R., On asymptotic stability and sweeping for Markov operators, Bull. Pol. Acad. Sci. Math., 43, 245-262 (1995) · Zbl 0838.47040 |
| [30] | Rudnicki, R., Markov operatorsapplications to diffusion processes and population dynamics, Appl. Math., 27, 1, 67-238 (2000) · Zbl 1001.47027 |
| [33] | Walter, W., Differential and Integral Inequalities (1970), Springer: Springer New York · Zbl 0252.35005 |
| [34] | Wang, Y.; Hori, Y.; Hara, S.; Doyle III, F. J., Collective oscillation period of inter-coupled biological negative cyclic feedback oscillators, IEEE Trans. Autom. Control, 60, 5, 1392-1397 (2015) · Zbl 1360.92016 |
| [35] | Watson, J.; Baker, T.; Bell, S.; Gann, A.; Levine, M.; Losick, R., Molecular Biology of the Gene (2013), Benjamin Cummings: Benjamin Cummings San Francisco |
| [36] | Yap, K.; Makeyev, E., Regulation of gene expression in mammalian nervous system through alternative pre-mRNA splicing coupled with RNA quality control mechanisms, Mol. Cell. Neurosci., 56, 420-428 (2013) |
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