# American Institute of Mathematical Sciences

October  2018, 23(8): 3167-3194. doi: 10.3934/dcdsb.2018224

## The dynamics of gene transcription in random environments

 a. Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, China b. Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA c. College of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou 450000, China

* Corresponding author: jsyu@gzhu.edu.cn

Received  August 2017 Revised  November 2017 Published  October 2018 Early access  June 2018

Gene transcription is a stochastic process, as the mRNA copies of the same gene in a population of isogeneic cells are often distributed unevenly. The fluctuation has been attributed to the random transition of system states and random production or degradation of transcripts, as characterized by the prevailing two-state model. In addition, as cells live in heterogeneous environments, noisy signals provide a further source of randomness for transcription activation. In this paper, we study how the coupling of random environmental signals and the core transcription system coordinates transcriptional dynamics and noise by extending the two-state model. One of our major concerns is whether noisy signals activate noisier transcription. We find the exact forms for the steady-states of the mean mRNA level and its noise and clarify their dynamical behavior. Our numerical examples strongly suggest that the randomness of the signals inducing a positive or negative regulation does not make significant impact on transcription. Corresponding to each noisy signal, there is a deterministic signal such that the two signals generate nearly identical temporal profiles for the mean and the noise. When transcription is regulated by pulsatile signals, the mean and the noise exhibit damped but almost synchronized oscillations, indicating that noisy pulsatile signals may even reduce transcription noise at some time intervals. Our further analysis reveals that the transition rates in the core transcription system make more notable impacts on creating transcription noise than what the randomness in external signals may contribute.

Citation: Jian Ren, Feng Jiao, Qiwen Sun, Moxun Tang, Jianshe Yu. The dynamics of gene transcription in random environments. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3167-3194. doi: 10.3934/dcdsb.2018224
##### References:
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Siliciano and L. S. Weinberger, Screening for noise in gene expression indentifies frug synergies, Science, 344 (2014), 1392-1396.   Google Scholar [7] I. Golding, J. Paulsson, S. M. Zawilski and E. C. Cox, Real-time kinetics of gene activity in individual bacteria, Cell, 123 (2005), 1025-1036.  doi: 10.1016/j.cell.2005.09.031.  Google Scholar [8] M. D. Gordon, M. S. Dionne, D. S. Schneider and R. Nusse, WntD is a feedback inhibitor of Dorsal/NF-$κ$B in Drosophila development and immunity, Nature, 437 (2005), 746-749.   Google Scholar [9] S. Hao and D. Baltimore, The stability of mRNA influences the temporal order of the induction of genes encoding inflammatory molecules, Nat. Immunol., 10 (2009), 281-288.  doi: 10.1038/ni.1699.  Google Scholar [10] L. Huang, P. Liu, Z. Yuan, T. Zhou and J. Yu, The free-energy cost of interaction between DNA loops Scientific Reports, 7 (2017), 12610. doi: 10.1038/s41598-017-12765-x.  Google Scholar [11] Q. Li, L. Huang and J. Yu, Modulation of first-passage time for bursty gene expression via random signals, Math. Biosci. Eng., 14 (2017), 1261-1277.  doi: 10.3934/mbe.2017065.  Google Scholar [12] F. Jiao, M. Tang and J. Yu, Distribution profiles and their dynamic transition in stochastic gene transcription, J. Diff. Eqn., 254 (2013), 3307-3328.  doi: 10.1016/j.jde.2013.01.019.  Google Scholar [13] M. Kærn, T. C. Elston, W. J. Blake and J. J. Collins, Stochasticity in gene expression: From theories to phenotypes, Nature, 6 (2005), 451-464.   Google Scholar [14] A. Kleino, H. Myllymäki, J. Kallio, L. M. Vanha-aho, K. Oksanen, J. Ulvila, D. Hultmark, S. Valanne and M. Rämet, Pirk is a negative regulator of the Drosophila Imd pathway, J. Immunol., 180 (2008), 5413-5422.  doi: 10.4049/jimmunol.180.8.5413.  Google Scholar [15] N. Kumar, T. Platini and R. V. Kulkarni, Exact distributions for stochastic gene expression models with bursting and feedback, Phys. Rev. Lett., 113 (2014), 268105. doi: 10.1103/PhysRevLett.113.268105.  Google Scholar [16] G. Lahav, N. Rosenfeld, A. Sigal, N. Geva-Zatorsky, A. J. Levine, M. B. Elowitz and U. Alon, Dynamics of the p53-Mdm2 feedback loop in individual cells, Nat. genet., 36 (2004), 147-150.  doi: 10.1038/ng1293.  Google Scholar [17] K. Z. Lee and D. Ferrandon, Negative regulation of immune responses on the fly, EMBO J., 30 (2011), 988-990.  doi: 10.1038/emboj.2011.47.  Google Scholar [18] B. Lemaitre and J. Hoffmann, The host defense of Drosophila melanogaster, Annu. Rev. Immunol., 25 (2007), 697-743.  doi: 10.1146/annurev.immunol.25.022106.141615.  Google Scholar [19] J. H. Levine, Y. Lin and M. B. Elowitz, Functional roles of pulsing in genetic circuits, Science, 342 (2013), 1193-1200.  doi: 10.1126/science.1239999.  Google Scholar [20] Y. Li, M. Tang and J. Yu, Transcription dynamics of inducible genes modulated by negative regulations, Math. Med. Biol., 32 (2015), 115-136.  doi: 10.1093/imammb/dqt019.  Google Scholar [21] J. J. Manfredi, The Mdm2-p53 relationship evolves: Mdm2 swings both ways as an oncogene and a tumor suppressor, Genes Dev., 24 (2010), 1580-1589.  doi: 10.1101/gad.1941710.  Google Scholar [22] U. M. Moll and O. Petrenko, The Mdm2-p53 interaction, Mol. Cancer Res., 1 (2003), 1001-1008.   Google Scholar [23] B. Munsky, G. Neuert and A. Oudenaarden, Using gene expression noise to understand gene regulation, Science, 336 (2012), 183-187.  doi: 10.1126/science.1216379.  Google Scholar [24] S. M. O'Donnell, G. H. Holm, J. M. Pierce, B. Tian, M. J. Watson, R. S. Chari, D. W. Ballard, A. R. Brasier and T. S. Dermody, Identification of an NF-kB-dependent gene network in cells infected by mammalian reovirus, J. Virol., (2006), 1077-1086.   Google Scholar [25] A. Raj, C. S. Peskin, D. Tranchina, D. Y. Vargas and S. Tyagi, mRNA synthesis in mammalian cells, PLoS Biol., 4 (2006), 1707-1719.   Google Scholar [26] B. S. Razooky, A. Pai, K. Aull, I. M. Rouzine and L. S. Weinberger, A hardwired HIV latency program, Cell, 160 (2015), 990-1001.  doi: 10.1016/j.cell.2015.02.009.  Google Scholar [27] D. S. Ruelas and W. C. Greene, An integrated overview of HIV-1 latency, Cell, 155 (2013), 519-529.  doi: 10.1016/j.cell.2013.09.044.  Google Scholar [28] D. S. Schneider, How and why does a fly turn its immune system off?, PLoS Biol., 5 (2007), e247. doi: 10.1371/journal.pbio.0050247.  Google Scholar [29] N. Silverman and T. Maniaties, NF-$κ$B signaling pathways in mammalian and insect innate immunity, Genes Dev., 15 (2001), 2321-2342.  doi: 10.1101/gad.909001.  Google Scholar [30] Q. Sun, M. Tang and J. Yu, Modulation of gene transcription noise by competing transcription factors, J. Math. Biol., 64 (2012), 469-494.  doi: 10.1007/s00285-011-0420-x.  Google Scholar [31] Q. Sun, M. Tang and J. Yu, Temporal profile of gene transcription noise modulated by cross-talking signal transduction pathways, Bull. Math. Biol., 74 (2012), 375-398.  doi: 10.1007/s11538-011-9683-z.  Google Scholar [32] M. Tang, The mean and noise of stochastic gene transcription, J. Theor. Biol., 253 (2008), 271-280.  doi: 10.1016/j.jtbi.2008.03.023.  Google Scholar [33] M. Tang, The mean frequency of transcriptional bursting and its variation in single cells, J. Math. Biol., 60 (2010), 27-58.  doi: 10.1007/s00285-009-0258-7.  Google Scholar [34] Y. Taniguchi, P. J. Choi, G. W. Li, H. Chen, M. Babu, J. Hearn, A. Emili and X. S. Xie, Quantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells, Science, 329 (2010), 533-538.   Google Scholar [35] S. Tay, J. J. Hughev, T. K. Lee, T. Lipniacki, S. R. Quake and M. W. Covert, Single-cell NF-$κ$B dynamics reveal digital activation and analog information processing, Nature, 466 (2010), 267-271.   Google Scholar [36] M. Thattai and A. van Oudenaarden, Intrinsic noise in gene regulatory networks, Proc. Natl. Acad. Sci., 98 (2001), 8614-8619.  doi: 10.1073/pnas.151588598.  Google Scholar [37] M. Turcotte, J. Garcia-Ojalvo and G. M. Süel, A genetic timer through noise-induced stabilization of an unstable state, Proc. Natl. Acad. Sci., 105 (2008), 15732-15737.  doi: 10.1073/pnas.0806349105.  Google Scholar [38] Q. Wang, L. Huang, K. Wen and J. Yu, The mean and noise of stochastic gene transcription with cell division, Math. Biosci. Eng., 15 (2018), 1255-1270.   Google Scholar [39] J. Yu and X. Liu, Monotonic dynamics of mRNA degradation by two pathways, J. Appl. Anal. Comput., 7 (2017), 1598-1612.   Google Scholar [40] J. Yu, J. Xiao, X. Ren, K. Lao and X. S. Xie, Probing gene expression in live cells, one protein molecule at a time, Science, 311 (2006), 1600-1603.  doi: 10.1126/science.1119623.  Google Scholar [41] D. Zenklusen, D. R. Larson and R. H. Singer, Single-RNA counting reveals alternative modes of gene expression in yeast, Nat. Struct. Mol. Biol., 15 (2008), 1263-1271.   Google Scholar [42] J. Zhang and T. Zhou, Promoter-mediated transcriptional dynamics, Biophys. J., 106 (2014), 479-488.  doi: 10.1016/j.bpj.2013.12.011.  Google Scholar

show all references

##### References:
 [1] K Aggarwal and N. Silverman, Positive and negative regulation of the Drosophila immune response, BMB Rep., 41 (2008), 267-277.  doi: 10.5483/BMBRep.2008.41.4.267.  Google Scholar [2] D. Alarcon-Vargas and Z. Ronai, p53-Mdm2-the affair that never ends, Carcinogenesis, 23 (2002), 541-547.  doi: 10.1093/carcin/23.4.541.  Google Scholar [3] L. Ashall, C. A. Horton, D. E. Nelson, P. Paszek, C. V. Harper and K. Sillitoe, Pulsatile stimulation determines timing and specificity of NF-kB-dependent transcription, Science, 324 (2009), 242-246.   Google Scholar [4] L. Cai, C. K. Dalal and M. B. Elowitz, Frequency-modulated nuclear localization bursts coordinate gene regulation, Nature, 455 (2008), 485-491.  doi: 10.1038/nature07292.  Google Scholar [5] C. K. Dalal, L. Cai, Y. Lin, K. Rahbar and M. B. Elowitz, Pulsatile dynamics in the yeast proteome, Curr. Bio., 24 (2014), 2189-2194.  doi: 10.1016/j.cub.2014.07.076.  Google Scholar [6] R. D. Dar, N. N. Hosmane, M. R. Arkin, R. F. Siliciano and L. S. Weinberger, Screening for noise in gene expression indentifies frug synergies, Science, 344 (2014), 1392-1396.   Google Scholar [7] I. Golding, J. Paulsson, S. M. Zawilski and E. C. Cox, Real-time kinetics of gene activity in individual bacteria, Cell, 123 (2005), 1025-1036.  doi: 10.1016/j.cell.2005.09.031.  Google Scholar [8] M. D. Gordon, M. S. Dionne, D. S. Schneider and R. Nusse, WntD is a feedback inhibitor of Dorsal/NF-$κ$B in Drosophila development and immunity, Nature, 437 (2005), 746-749.   Google Scholar [9] S. Hao and D. Baltimore, The stability of mRNA influences the temporal order of the induction of genes encoding inflammatory molecules, Nat. Immunol., 10 (2009), 281-288.  doi: 10.1038/ni.1699.  Google Scholar [10] L. Huang, P. Liu, Z. Yuan, T. Zhou and J. Yu, The free-energy cost of interaction between DNA loops Scientific Reports, 7 (2017), 12610. doi: 10.1038/s41598-017-12765-x.  Google Scholar [11] Q. Li, L. Huang and J. Yu, Modulation of first-passage time for bursty gene expression via random signals, Math. Biosci. Eng., 14 (2017), 1261-1277.  doi: 10.3934/mbe.2017065.  Google Scholar [12] F. Jiao, M. Tang and J. Yu, Distribution profiles and their dynamic transition in stochastic gene transcription, J. Diff. Eqn., 254 (2013), 3307-3328.  doi: 10.1016/j.jde.2013.01.019.  Google Scholar [13] M. Kærn, T. C. Elston, W. J. Blake and J. J. Collins, Stochasticity in gene expression: From theories to phenotypes, Nature, 6 (2005), 451-464.   Google Scholar [14] A. Kleino, H. Myllymäki, J. Kallio, L. M. Vanha-aho, K. Oksanen, J. Ulvila, D. Hultmark, S. Valanne and M. Rämet, Pirk is a negative regulator of the Drosophila Imd pathway, J. Immunol., 180 (2008), 5413-5422.  doi: 10.4049/jimmunol.180.8.5413.  Google Scholar [15] N. Kumar, T. Platini and R. V. Kulkarni, Exact distributions for stochastic gene expression models with bursting and feedback, Phys. Rev. Lett., 113 (2014), 268105. doi: 10.1103/PhysRevLett.113.268105.  Google Scholar [16] G. Lahav, N. Rosenfeld, A. Sigal, N. Geva-Zatorsky, A. J. Levine, M. B. Elowitz and U. Alon, Dynamics of the p53-Mdm2 feedback loop in individual cells, Nat. genet., 36 (2004), 147-150.  doi: 10.1038/ng1293.  Google Scholar [17] K. Z. Lee and D. Ferrandon, Negative regulation of immune responses on the fly, EMBO J., 30 (2011), 988-990.  doi: 10.1038/emboj.2011.47.  Google Scholar [18] B. Lemaitre and J. Hoffmann, The host defense of Drosophila melanogaster, Annu. Rev. Immunol., 25 (2007), 697-743.  doi: 10.1146/annurev.immunol.25.022106.141615.  Google Scholar [19] J. H. Levine, Y. Lin and M. B. Elowitz, Functional roles of pulsing in genetic circuits, Science, 342 (2013), 1193-1200.  doi: 10.1126/science.1239999.  Google Scholar [20] Y. Li, M. Tang and J. Yu, Transcription dynamics of inducible genes modulated by negative regulations, Math. Med. Biol., 32 (2015), 115-136.  doi: 10.1093/imammb/dqt019.  Google Scholar [21] J. J. Manfredi, The Mdm2-p53 relationship evolves: Mdm2 swings both ways as an oncogene and a tumor suppressor, Genes Dev., 24 (2010), 1580-1589.  doi: 10.1101/gad.1941710.  Google Scholar [22] U. M. Moll and O. Petrenko, The Mdm2-p53 interaction, Mol. Cancer Res., 1 (2003), 1001-1008.   Google Scholar [23] B. Munsky, G. Neuert and A. Oudenaarden, Using gene expression noise to understand gene regulation, Science, 336 (2012), 183-187.  doi: 10.1126/science.1216379.  Google Scholar [24] S. M. O'Donnell, G. H. Holm, J. M. Pierce, B. Tian, M. J. Watson, R. S. Chari, D. W. Ballard, A. R. Brasier and T. S. Dermody, Identification of an NF-kB-dependent gene network in cells infected by mammalian reovirus, J. Virol., (2006), 1077-1086.   Google Scholar [25] A. Raj, C. S. Peskin, D. Tranchina, D. Y. Vargas and S. Tyagi, mRNA synthesis in mammalian cells, PLoS Biol., 4 (2006), 1707-1719.   Google Scholar [26] B. S. Razooky, A. Pai, K. Aull, I. M. Rouzine and L. S. Weinberger, A hardwired HIV latency program, Cell, 160 (2015), 990-1001.  doi: 10.1016/j.cell.2015.02.009.  Google Scholar [27] D. S. Ruelas and W. C. Greene, An integrated overview of HIV-1 latency, Cell, 155 (2013), 519-529.  doi: 10.1016/j.cell.2013.09.044.  Google Scholar [28] D. S. Schneider, How and why does a fly turn its immune system off?, PLoS Biol., 5 (2007), e247. doi: 10.1371/journal.pbio.0050247.  Google Scholar [29] N. Silverman and T. Maniaties, NF-$κ$B signaling pathways in mammalian and insect innate immunity, Genes Dev., 15 (2001), 2321-2342.  doi: 10.1101/gad.909001.  Google Scholar [30] Q. Sun, M. Tang and J. Yu, Modulation of gene transcription noise by competing transcription factors, J. Math. Biol., 64 (2012), 469-494.  doi: 10.1007/s00285-011-0420-x.  Google Scholar [31] Q. Sun, M. Tang and J. Yu, Temporal profile of gene transcription noise modulated by cross-talking signal transduction pathways, Bull. Math. Biol., 74 (2012), 375-398.  doi: 10.1007/s11538-011-9683-z.  Google Scholar [32] M. Tang, The mean and noise of stochastic gene transcription, J. Theor. Biol., 253 (2008), 271-280.  doi: 10.1016/j.jtbi.2008.03.023.  Google Scholar [33] M. Tang, The mean frequency of transcriptional bursting and its variation in single cells, J. Math. Biol., 60 (2010), 27-58.  doi: 10.1007/s00285-009-0258-7.  Google Scholar [34] Y. Taniguchi, P. J. Choi, G. W. Li, H. Chen, M. Babu, J. Hearn, A. Emili and X. S. Xie, Quantifying E. coli proteome and transcriptome with single-molecule sensitivity in single cells, Science, 329 (2010), 533-538.   Google Scholar [35] S. Tay, J. J. Hughev, T. K. Lee, T. Lipniacki, S. R. Quake and M. W. Covert, Single-cell NF-$κ$B dynamics reveal digital activation and analog information processing, Nature, 466 (2010), 267-271.   Google Scholar [36] M. Thattai and A. van Oudenaarden, Intrinsic noise in gene regulatory networks, Proc. Natl. Acad. Sci., 98 (2001), 8614-8619.  doi: 10.1073/pnas.151588598.  Google Scholar [37] M. Turcotte, J. Garcia-Ojalvo and G. M. Süel, A genetic timer through noise-induced stabilization of an unstable state, Proc. Natl. Acad. Sci., 105 (2008), 15732-15737.  doi: 10.1073/pnas.0806349105.  Google Scholar [38] Q. Wang, L. Huang, K. Wen and J. Yu, The mean and noise of stochastic gene transcription with cell division, Math. Biosci. Eng., 15 (2018), 1255-1270.   Google Scholar [39] J. Yu and X. Liu, Monotonic dynamics of mRNA degradation by two pathways, J. Appl. Anal. Comput., 7 (2017), 1598-1612.   Google Scholar [40] J. Yu, J. Xiao, X. Ren, K. Lao and X. S. Xie, Probing gene expression in live cells, one protein molecule at a time, Science, 311 (2006), 1600-1603.  doi: 10.1126/science.1119623.  Google Scholar [41] D. Zenklusen, D. R. Larson and R. H. Singer, Single-RNA counting reveals alternative modes of gene expression in yeast, Nat. Struct. Mol. Biol., 15 (2008), 1263-1271.   Google Scholar [42] J. Zhang and T. Zhou, Promoter-mediated transcriptional dynamics, Biophys. J., 106 (2014), 479-488.  doi: 10.1016/j.bpj.2013.12.011.  Google Scholar
The two-state transcription model. The transcription system randomly switches between the inactive (gene OFF) state and the active (gene ON) state with rates $\lambda>0$ and $\gamma>0$. When the gene is active, mRNA molecules are produced with a rate $\upsilon>0$. Independent of system states, mRNA molecules are degraded with a rate $\delta>0$
The model of transcription in a random environment. The production and degradation of mRNAs follow the same mechanism as in the two-state model. The transcription is initiated by parallel pathways $O_1$ and $O_2$, along which the durations for initiation are independent and exponentially distributed with rates $\lambda_1>0$ and $\lambda_2>\lambda_1$, respectively. The pathway selection probabilities are determined by the random processes $Q_1(t)$ and $Q_2(t) = 1-Q_1(t)$
The temporal profiles of the expected pathway selection probability $q_2(t)$ in pulsatile environments. The probability $q_2(t)$ given in Example 1 exhibits distinct dynamics, with zero, one, two or three critical points as shown in the panels a, b, c and d where we set $a_- = 0$, $a^+ = 1$. The vector ($\kappa_1$, $\kappa_2$, $\kappa_3$, $\kappa^+$, $\kappa_-$) takes values a: (0.2, 0.3, 0.25, 0.3, 0.25), b: (5, 0.3, 0.25, 0.3, 0.25), c: (5, 0.8, 0.25, 0.3, 0.25), d: (10, 1, 0.45, 0.3, 0.25)
Minimal impact of noisy signals in positively regulated genes. In each panel, the solid curve represents the transcription activated by a stable signal with $Q_2(t)\equiv 0.6$, and the three dashed-curves represent the transcription activated by random signals with an increasing $Q_2(t)$ that approaches $0.6$ as $t\to \infty$. These curves are generated by applying the parameter values in (56), $\lambda_2 = 5\lambda_1$, and the data specified within each panel to Eqs. (28)-(35). In a-d, the temporal profiles of the mean mRNA level, the variance, the noise strength $\phi(t)$, and the noise $\eta^2(t)$ are shown, respectively. Panels e and f show the dependance of $\phi(t)$ and $\eta^2(t)$ on the mean level. As the dashed-curves do not deviate from the solid curve considerably, noisy signals exhibit only a minimal impact on the mean mRNA level and the transcription noise in positively regulated genes
Weak impact of random signals in negative regulations. In all panels, the solid curves represent the transcription profiles for the system activated by deterministic signals, and the dash-dot curves are for the system with random signals. The selection probability $Q_2(t)$ decreases with $a_i = 0.6-0.1(i-1)$ for $i = 1, 2, \cdots, 6$, and $a_i = 0.1$ for $i>6$. The waiting time $T_i-T_{i-1}$ follows an exponential distribution with a rate $\kappa_i>0$ for random signals, and equals $1/\kappa_i$ for deterministic signals. The activation strengths $\kappa_i = \kappa_2$ for $i>2$, and $(\kappa_1, \kappa_2) = (1, 2)$, $(0.5, 1)$, and $(0.25, 1)$, respectively. In a-c, the temporal profiles of the mean mRNA level, the noise and the noise strength are shown, respectively. Panel d shows the dependance of $\eta^2(t)$ on the mean level. The six curves in each panel are apparently clustered in three groups, indicating that noisy signals make only a weak impact on transcription
, the mean transcription level oscillates around the mean level of the constant system. In Fig. 6b and c, the noise strength $\phi(t)$ and the noise $\eta^2(t)$ also oscillates with some reduced magnitudes. The two curves for the noise against the mean level in Fig. 6d are almost identical, indicating that the oscillations of the mean transcription level and the noise are almost synchronized">Figure 6.  Pulsatile signals induce oscillatory dynamics and noises. The transcriptional dynamics of a constantly activated system (the solid curves) and a system activated by pulsatile signals (the dash-dot curves) are shown. In the pulsatile system, $Q_2(t)$ oscillates between $a_{2i-1} = 0$ and $a_{2i} = 1$ for $i\ge 1$. The inter-arrival time $T_i-T_{i-1}$ follows independent and exponential distribution with rate $\kappa_i>0$, $(\kappa_1, \cdots, \kappa_7) = (20, 1, 0.06, 0.09, 0.007, 0.015, 0.0011)$, $\kappa_{8} = \kappa_{10} = \cdots = \kappa^+ = 0.001$, and $\kappa_{9} = \kappa_{11} = \cdots = \kappa_- = 0.0003$. In the constant system, the selection probability for the signal pathway is the limit of the pulsatile system $\kappa_-/(\kappa_-+\kappa^+) = 3/13$. As shown in Fig. 6a, the mean transcription level oscillates around the mean level of the constant system. In Fig. 6b and c, the noise strength $\phi(t)$ and the noise $\eta^2(t)$ also oscillates with some reduced magnitudes. The two curves for the noise against the mean level in Fig. 6d are almost identical, indicating that the oscillations of the mean transcription level and the noise are almost synchronized
-6. $\lambda_1/\gamma = 0.1$, and $\gamma = 0.94\delta$ or $1.2\delta$ in a and c, and $\lambda_1/\gamma = 2$, and $\gamma = 0.94\delta$ or $1.2\delta$ in b and d. The systems are regulated negatively with $Q_2$ and the inter-arrival time $T_i-T_{i-1}$ defined as in Fig. 5 with $\kappa_1 = 0.5$ and $\kappa_2 = \kappa_3 = \cdots = 1$. In a and b, the mean mRNA levels are clustered and increase rapidly in the first 20 hours to reach the unique peaks at about the same time, and then decay to the same limit. The maximum mRNA level increases about 7-fold from a to b, in response to the 20-fold increase in the activation strength. In Panels c and d, the temporal profiles for the noise strengthes are clearly clustered in two groups. As the inactivation rate $\gamma$ increases from $0.94\delta$ to $1.2\delta$, the noise strengthes shifted down by 6.5 in Panel c and by 1.7 in Panel d at each time $t$. Corresponding to the 20-fold increase in the activation strength from Panels c to d, the noise strength has about 5-fold reduction">Figure 7.  The dominant roles of the transition rates. The temporal profiles for the transcription activated by deterministic signals (solid curves) vs. that activated by noise signals (dash-dot curves) with varying activation and inactivation rates are shown in Panels a-d. We fix $\delta = 0.173$, $\upsilon = 118\delta$, and $\lambda_2 = 5\lambda_1$ as in Fig. 4-6. $\lambda_1/\gamma = 0.1$, and $\gamma = 0.94\delta$ or $1.2\delta$ in a and c, and $\lambda_1/\gamma = 2$, and $\gamma = 0.94\delta$ or $1.2\delta$ in b and d. The systems are regulated negatively with $Q_2$ and the inter-arrival time $T_i-T_{i-1}$ defined as in Fig. 5 with $\kappa_1 = 0.5$ and $\kappa_2 = \kappa_3 = \cdots = 1$. In a and b, the mean mRNA levels are clustered and increase rapidly in the first 20 hours to reach the unique peaks at about the same time, and then decay to the same limit. The maximum mRNA level increases about 7-fold from a to b, in response to the 20-fold increase in the activation strength. In Panels c and d, the temporal profiles for the noise strengthes are clearly clustered in two groups. As the inactivation rate $\gamma$ increases from $0.94\delta$ to $1.2\delta$, the noise strengthes shifted down by 6.5 in Panel c and by 1.7 in Panel d at each time $t$. Corresponding to the 20-fold increase in the activation strength from Panels c to d, the noise strength has about 5-fold reduction
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