# American Institute of Mathematical Sciences

## Fluctuations of mRNA distributions in multiple pathway activated transcription

 1 School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China 2 Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, China 3 Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

* Corresponding author: Jianshe Yu, Email: jsyu@gzhu.edu.cn

Received  December 2017 Revised  February 2018 Published  June 2018

Fund Project: This work was supported by Natural Science Foundation of China grants (Nos. 11631005 and 11626246) and Program for Changjiang Scholars and Innovative Research Team in University (No. IRT_16R16)

Randomness in gene transcription can result in fluctuations (noise) of messenger RNA (mRNA) levels, leading to phenotypic plasticity in the isogenic populations of cells. Recent experimental studies indicate that multiple pathway activation mechanism plays an important role in the regulation of transcription noise and cell-to-cell variability. Previous theoretical studies on transcription noise have been emphasized on exact solutions and analysis for models with a single pathway or two cross-talking pathways. For stochastic models with more than two pathways, however, exact analytical results for fluctuations of mRNA levels have not been obtained yet. We develop a gene transcription model to examine the impact of multiple pathways on transcription noise for which the exact fluctuations of mRNA distributions are obtained. It is nontrivial to determine the analytical results for transcription fluctuations due to the high dimension of system parameter space. At the heart of our method lies the usage of the model's intrinsic symmetry to simplify the complicated calculations. We show the symmetric relation among system parameters, which allows us to derive the analytical expressions of the dynamical and steady-state fluctuations and to characterize the nature of transcription noise. Our results not only can be reduced to previous ones in limiting cases but also indicate some differences between the three or more pathway model and the single or two pathway one. Our analytical approaches provide new insights into the role of multiple pathways in noise regulation and optimization. A further study for stochastic gene transcription involving multiple pathways may shed light on the relation between transcription fluctuation and genetic network architecture.

Citation: Genghong Lin, Jianshe Yu, Zhan Zhou, Qiwen Sun, Feng Jiao. Fluctuations of mRNA distributions in multiple pathway activated transcription. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2018219
##### References:
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Levine, Enhancer control of transcriptional bursting, Cell, 166 (2016), 358-368. doi: 10.1016/j.cell.2016.05.025. [12] F. Geng and B. C. Laurent, Roles of SWI/SNF and HATs throughout the dynamic transcription of a yeast glucose-repressible gene, EMBO J., 23 (2004), 127-137. doi: 10.1038/sj.emboj.7600035. [13] I. Golding, J. Paulsson and S. M. Zawilski, et al., Real-time kinetics of gene activity in individual bacteria, Cell, 123 (2005), 1025-1036. doi: 10.1016/j.cell.2005.09.031. [14] T. Gregor, E. F. Wieschaus and A. P. McGregor, et al., Stability and nuclear dynamics of the bicoid morphogen gradient, Cell, 130 (2007), 141-152. doi: 10.1016/j.cell.2007.05.026. [15] N. E. Ilott, J. A. Heward and B. Roux, et al., Long non-coding RNAs and enhancer RNAs regulate the lipopolysaccharide-induced inflammatory response in human monocytes, Nat. Commun. 5 (2014), 3979. doi: 10.1038/ncomms7814. [16] F. Jiao, Q. Sun and M. Tang, et al., Distribution modes and their corresponding parameter regions in stochastic gene transcription, SIAM J. Appl. Math., 75 (2015), 2396-2420. doi: 10.1137/151005567. [17] F. Jiao, M. Tang and J. Yu, Distribution profiles and their dynamic transition in stochastic gene transcription, J. Differential Equations, 254 (2013), 3307-3328. doi: 10.1016/j.jde.2013.01.019. [18] D. L. Jones, R. C. Brewster and R. Phillips, Promoter architecture dictates cell-to-cell variability in gene expression, Science, 346 (2014), 1533-1536. doi: 10.1126/science.1255301. [19] 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. [20] D. R. Larson, C. Fritzsch and L. Sun, et al., Direct observation of frequency modulated transcription in single cells using light activation, eLife 2 (2013), e00750. doi: 10.7554/eLife.00750. [21] 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. [22] T. L. Lenstra, J. Rodriguez and H. Chen, et al., Transcription dynamics in living cells, Annu. Rev. Biophys., 45 (2016), 25-47. doi: 10.1146/annurev-biophys-062215-010838. [23] 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. [24] N. Molinaa, D. M. Suter and C. Rosamaria, et al., Stimulus-induced modulation of transcriptional bursting in a single mammalian gene, Proc. Natl. Acad. Sci. USA, 110 (2013), 20563-20568. doi: 10.1073/pnas.1312310110. [25] B. Munsky and A. V. Oudenaarden, Using gene expression noise to understand gene regulation, Science, 336 (2012), 183-187. doi: 10.1126/science.1216379. [26] J. Peccoud and B. Ycart, Markovian modeling of gene-product synthesis, Theor. Popul. Biol., 48 (1995), 222-234. doi: 10.1006/tpbi.1995.1027. [27] A. Raj, S. A. Rifkin and E. Andersen, et al., Variability in gene expression underlies incomplete penetrance, Nature, 463 (2010), 913-918. doi: 10.1038/nature08781. [28] A. Raj, C. S. Peskin and D. Tranchina, et al., Stochastic mRNA synthesis in mammalian cells, PLoS Biol. 4 (2006), e309. doi: 10.1371/journal.pbio.0040309. [29] J. M. Raser and E. K. O'Shea, Noise in Gene Expression: Origins, Consequences, and Control, Science, 309 (2005), 2010-2013. doi: 10.1126/science.1105891. [30] J. Ren, F. Jiao, Q. Sun, M. Tang and J. Yu, The dynamics of gene transcription in random environments, Discrete Contin. Dyn. Syst. B, 23 (2018), 3167-3194. doi: 10.3934/dcdsb.2018224. [31] A. Sánchez and J. Kondev, Transcriptional control of noise in gene expression, Proc. Natl. Acad. Sci. USA, 105 (2008), 5081-5086. [32] J. Schott, S. Reitter and J. Philipp, et al., Translational regulation of specific mRNAs controls feedback inhibition and survival during macrophage activation, PLoS Genet. 10 (2014), e1004368. doi: 10.1371/journal.pgen.1004368. [33] L. H. So, A, Ghosh and C. Zong, et al., General properties of transcriptional time series in Escherichia coli, Nat. Genet., 43 (2011), 554-560. doi: 10.1038/ng.821. [34] 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. [35] 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. [36] D. M. Suter, N. Molina and D. Gatfield, et al., Mammalian genes are transcribed with widely different bursting kinetics, Science, 332 (2011), 472-474. doi: 10.1126/science.1198817. [37] 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. [38] 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. [39] M. Vera, J. Biswas and A. Senecal, et al., Single-cell and single-molecule analysis of gene expression regulation, Annu. Rev. Genet., 50 (2016), 267-291. doi: 10.1146/annurev-genet-120215-034854. [40] 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. [41] X. Yang, Y. Wu and Z. Yuan, Characteristics of mrna dynamics in a multi-on model of stochastic transcription with regulation, Chinese J. Phys., 55 (2017), 508-518. doi: 10.1016/j.cjph.2016.12.006. [42] J. Yu, Q. Sun and M. Tang, The nonlinear dynamics and fluctuations of mRNA levels in cross-talking pathway activated transcription, J. Theor. Biol., 363 (2014), 223-234. doi: 10.1016/j.jtbi.2014.08.024. [43] J. Yu and X. Liu, Monotonic dynamics of mRNA degradation by two pathways, J. Appl. Anal. Comput., 7 (2017), 1598-1612. [44] 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.

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##### References:
 [1] A. Aguirre, M. E. Rubio and V. Gallo, Notch and EGFR pathway interaction regulates neural stem cell number and self-renewal, Nature, 467 (2010), 323-327. doi: 10.1038/nature09347. [2] A. Arkin, J. Ross and H. H. Mcadams, Stochastic kinetic analysis of developmental pathway bifurcation in phage λ-infected escherichia coli cells, Genetics, 149 (1998), 1633-1648. [3] C. R. Bartman, S. C. Hsu and C. C. Hsiung, et al., Enhancer regulation of transcriptional bursting parameters revealed by forced chromatin looping, Mol. Cell, 62 (2016), 237-247. [4] W. J. Blake, M. KAErn and C. R. Cantor, et al., Noise in eukaryotic gene expression, Nature, 422 (2003), 633-637. doi: 10.1038/nature01546. [5] P. Bokes and A. Singh, Gene expression noise is affected differentially by feedback in burst frequency and burst size, J. Math. Biol., 74 (2016), 1-27. doi: 10.1007/s00285-016-1059-4. [6] L. Cai, N. Friedman and X. S. Xie, Stochastic protein expression in individual cells at the single molecule level, Nature, 440 (2006), 358-362. doi: 10.1038/nature04599. [7] R. I. Clark, K. J. Woodcock and F. R. Geissmann, et al., Multiple TGF-β superfamily signals modulate the adult Drosophila immune response, Curr. Biol., 21 (2011), 1672-1677. [8] R. D. Dar, N. N. Hosmane and M. R. Arkin, et al., Screening for noise in gene expression identifies drug synergies, Science, 344 (2014), 1392-1396. doi: 10.1126/science.1250220. [9] M. B. Elowitz, A. J. Levine and E. D. Siggia, et al., Stochastic gene expression in a single cell, Science, 297 (2002), 1183-1186. doi: 10.1126/science.1070919. [10] P. L. Felmer, A. Quaas and M. Tang, et al., Random dynamics of gene transcription activation in single cells, J. Differential Equations, 247 (2009), 1796-1816. doi: 10.1016/j.jde.2009.06.006. [11] T. Fukaya, B. Lim and M. Levine, Enhancer control of transcriptional bursting, Cell, 166 (2016), 358-368. doi: 10.1016/j.cell.2016.05.025. [12] F. Geng and B. C. Laurent, Roles of SWI/SNF and HATs throughout the dynamic transcription of a yeast glucose-repressible gene, EMBO J., 23 (2004), 127-137. doi: 10.1038/sj.emboj.7600035. [13] I. Golding, J. Paulsson and S. M. Zawilski, et al., Real-time kinetics of gene activity in individual bacteria, Cell, 123 (2005), 1025-1036. doi: 10.1016/j.cell.2005.09.031. [14] T. Gregor, E. F. Wieschaus and A. P. McGregor, et al., Stability and nuclear dynamics of the bicoid morphogen gradient, Cell, 130 (2007), 141-152. doi: 10.1016/j.cell.2007.05.026. [15] N. E. Ilott, J. A. Heward and B. Roux, et al., Long non-coding RNAs and enhancer RNAs regulate the lipopolysaccharide-induced inflammatory response in human monocytes, Nat. Commun. 5 (2014), 3979. doi: 10.1038/ncomms7814. [16] F. Jiao, Q. Sun and M. Tang, et al., Distribution modes and their corresponding parameter regions in stochastic gene transcription, SIAM J. Appl. Math., 75 (2015), 2396-2420. doi: 10.1137/151005567. [17] F. Jiao, M. Tang and J. Yu, Distribution profiles and their dynamic transition in stochastic gene transcription, J. Differential Equations, 254 (2013), 3307-3328. doi: 10.1016/j.jde.2013.01.019. [18] D. L. Jones, R. C. Brewster and R. Phillips, Promoter architecture dictates cell-to-cell variability in gene expression, Science, 346 (2014), 1533-1536. doi: 10.1126/science.1255301. [19] 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. [20] D. R. Larson, C. Fritzsch and L. Sun, et al., Direct observation of frequency modulated transcription in single cells using light activation, eLife 2 (2013), e00750. doi: 10.7554/eLife.00750. [21] 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. [22] T. L. Lenstra, J. Rodriguez and H. Chen, et al., Transcription dynamics in living cells, Annu. Rev. Biophys., 45 (2016), 25-47. doi: 10.1146/annurev-biophys-062215-010838. [23] 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. [24] N. Molinaa, D. M. Suter and C. Rosamaria, et al., Stimulus-induced modulation of transcriptional bursting in a single mammalian gene, Proc. Natl. Acad. Sci. USA, 110 (2013), 20563-20568. doi: 10.1073/pnas.1312310110. [25] B. Munsky and A. V. Oudenaarden, Using gene expression noise to understand gene regulation, Science, 336 (2012), 183-187. doi: 10.1126/science.1216379. [26] J. Peccoud and B. Ycart, Markovian modeling of gene-product synthesis, Theor. Popul. Biol., 48 (1995), 222-234. doi: 10.1006/tpbi.1995.1027. [27] A. Raj, S. A. Rifkin and E. Andersen, et al., Variability in gene expression underlies incomplete penetrance, Nature, 463 (2010), 913-918. doi: 10.1038/nature08781. [28] A. Raj, C. S. Peskin and D. Tranchina, et al., Stochastic mRNA synthesis in mammalian cells, PLoS Biol. 4 (2006), e309. doi: 10.1371/journal.pbio.0040309. [29] J. M. Raser and E. K. O'Shea, Noise in Gene Expression: Origins, Consequences, and Control, Science, 309 (2005), 2010-2013. doi: 10.1126/science.1105891. [30] J. Ren, F. Jiao, Q. Sun, M. Tang and J. Yu, The dynamics of gene transcription in random environments, Discrete Contin. Dyn. Syst. B, 23 (2018), 3167-3194. doi: 10.3934/dcdsb.2018224. [31] A. Sánchez and J. Kondev, Transcriptional control of noise in gene expression, Proc. Natl. Acad. Sci. USA, 105 (2008), 5081-5086. [32] J. Schott, S. Reitter and J. Philipp, et al., Translational regulation of specific mRNAs controls feedback inhibition and survival during macrophage activation, PLoS Genet. 10 (2014), e1004368. doi: 10.1371/journal.pgen.1004368. [33] L. H. So, A, Ghosh and C. Zong, et al., General properties of transcriptional time series in Escherichia coli, Nat. Genet., 43 (2011), 554-560. doi: 10.1038/ng.821. [34] 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. [35] 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. [36] D. M. Suter, N. Molina and D. Gatfield, et al., Mammalian genes are transcribed with widely different bursting kinetics, Science, 332 (2011), 472-474. doi: 10.1126/science.1198817. [37] 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. [38] 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. [39] M. Vera, J. Biswas and A. Senecal, et al., Single-cell and single-molecule analysis of gene expression regulation, Annu. Rev. Genet., 50 (2016), 267-291. doi: 10.1146/annurev-genet-120215-034854. [40] 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. [41] X. Yang, Y. Wu and Z. Yuan, Characteristics of mrna dynamics in a multi-on model of stochastic transcription with regulation, Chinese J. Phys., 55 (2017), 508-518. doi: 10.1016/j.cjph.2016.12.006. [42] J. Yu, Q. Sun and M. Tang, The nonlinear dynamics and fluctuations of mRNA levels in cross-talking pathway activated transcription, J. Theor. Biol., 363 (2014), 223-234. doi: 10.1016/j.jtbi.2014.08.024. [43] J. Yu and X. Liu, Monotonic dynamics of mRNA degradation by two pathways, J. Appl. Anal. Comput., 7 (2017), 1598-1612. [44] 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.
Schematic representation of gene transcription activated by multiple signaling pathways. (a) Entangling pathways $S_1$, $S_2$, $\cdots$, $S_n$ converge to the promoter of a target gene to activate gene transcription. (b) The pathway $S_i$ has a probability $q_i$ to activate the transcription with constant kinetic rates $\lambda_i$, $\gamma$, $\nu$, and $\delta$
Different dynamical behaviors of the noise strength $\Phi^{*}$ on $\lambda_3$. The three curves are generated by the analytical form (47) with $n = 4$, $q_1 = 0.3$, $q_2 = 0.25$, $q_3 = 0.15$, $q_4 = 0.3$, $\lambda_1 = 0.15$, $\lambda_2 = 0.26$, $\lambda_4 = 10.18$, $\nu = 2.1$, $\delta = 1$, and $\gamma$ respectively equals $10.65$, $0.03$, and $0.56$ in (a), (b), and (c)
Nonlinear dependance of the noise strength $\Phi^{*}$ on $P_E^*$. The up and down dependance curve of $\Phi^{*}$ for $0.005<P_E^*<0.997$ is generated by varying $\gamma$ from $0.0001$ to $5.5$, where $\lambda_1 = 0.015$, $\lambda_2 = 1.53$, $\nu = 7.5$, $\delta = 0.1$, and $q_1 = q_2 = 0.5$
Distinct dynamical behaviors of the noise strength $\Phi^{*}$ on $\lambda_2$ with or without the constraint on the mean $m^*$. The two curves are generated by the analytical form (47) with $n = 2$, $\gamma = 0.3$, $\nu = 15.4$, $\delta = 1$, and $q_1 = q_2 = 0.5$. (a) The mean transcriptional level is fixed as $m^* = 4.5$. (b) No constraint on the mean $m^*$ and $\lambda_1 = 0.0856$
The initial states at time $t$ and transition probabilities toward the terminal state $(E,m)$ at time $t+h$. If the gene is ON with $m$ copies of the mRNA molecules at $t+h$, then all of the initial states at time $t$, listed in (a), (b), (c), and (d), can reach $(E,m)$ with a transition probability of zero or first order of $h$
 Initial State Terminal State Transition Probability (a) $(E,m)$ $(E,m)$ $P_{E}(m,t)\cdot(1-\nu h)(1-\gamma h)(1-m \delta h)$ (b) $(E,m+1)$ $(E,m)$ $P_{E}(m+1,t)\cdot(m+1)\delta h$ (c) $(E,m-1)$ $(E,m)$ $P_{E}(m-1,t)\cdot\nu h$ (d) $(O_i,m)$ $(E,m)$ $P_{i}(m,t)\cdot\lambda_i h$
 Initial State Terminal State Transition Probability (a) $(E,m)$ $(E,m)$ $P_{E}(m,t)\cdot(1-\nu h)(1-\gamma h)(1-m \delta h)$ (b) $(E,m+1)$ $(E,m)$ $P_{E}(m+1,t)\cdot(m+1)\delta h$ (c) $(E,m-1)$ $(E,m)$ $P_{E}(m-1,t)\cdot\nu h$ (d) $(O_i,m)$ $(E,m)$ $P_{i}(m,t)\cdot\lambda_i h$
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