# American Institute of Mathematical Sciences

June  2019, 24(6): 2799-2810. doi: 10.3934/dcdsb.2018275

## Distribution profiles in gene transcription activated by the cross-talking pathway

 1 Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, China 2 School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

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

Received  April 2018 Revised  May 2018 Published  June 2019 Early access  October 2018

Fund Project: This work was supported by Natural Science Foundation of China grants (11631005, 11501138), Guangdong Natural Science Foundation (2016A030313542), Program for Guangzhou Municipal College and University (1201630273), and the Program for Changjiang Scholars and Innovative Research Team in University (IRT_16R16).

Gene transcription is a stochastic process, manifested by the heterogeneous mRNA distribution in an isogenic cell population. Bimodal distribution has been observed in the transcription of stress responsive genes which have evolved to be easily turned on and easily turned off. This is against the conclusion in the classical two-state model that bimodality occurs only when the gene is hardly turned on and hardly turned off. In this paper, we extend the gene activation process in the two-state model by introducing the cross-talking pathway that involves the random selection between a spontaneous weak basal pathway and a stress-induced strong signaling pathway. By deriving exact forms of mRNA distribution at steady-state, we find that the cross-talking pathway is much more likely to trigger the bimodal distribution. Our further analysis reveals an observed transition among the decaying, bimodal and unimodal mRNA distribution for stress gene upon enhanced stimulations. Especially, the bimodality occurs when the stress-induced signalling pathway is more frequently selected, reinforcing the assertion that bimodal transcription is a general feature of stress genes in response to environmental change.

Citation: Feng Jiao, Qiwen Sun, Genghong Lin, Jianshe Yu. Distribution profiles in gene transcription activated by the cross-talking pathway. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2799-2810. doi: 10.3934/dcdsb.2018275
##### References:
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Neuert and A. van Oudenaarden, Using gene expression noise to understand gene regulation, Science, 336 (2012), 183-187.  doi: 10.1126/science.1216379. [17] E. Nadal, G. Ammerer and F. Posas, Controlling gene expression in response to stress, Nat. Rev. Genet., 12 (2011), 833-845.  doi: 10.1038/nrg3055. [18] S. Paliwal, MAPK-mediated bimodal gene expression and adaptive gradient sensing in yeast, Nature, 446 (2007), 46-51. [19] R. B. Paris, A Kummer-type transformation for a 2F2 hypergeometric function, J. Comput. Appl. Math., 173 (2005), 379-382.  doi: 10.1016/j.cam.2004.05.005. [20] J. Peccoud and B. Ycart, Markovian modelling of gene-product synthesis, Theor. Popul. Biol., 48 (1995), 222-234. [21] V. Pelechano, S. Chávez and J.E. Pérez-Ortín, A complete set of nascent transcription rates for yeast genes, Plos One, 5 (2010), e15442. [22] S. Pelet, Transient activation of the HOG MAPK pathway regulates bimodal Gene expression, Science, 332 (2011), 732-735. [23] A. Raj, C.S. Peskin, D. Tranchina, D.Y. Vargas and S. Tyagi, Stochastic mRNA synthesis in mammalian cells, PLoS Biol., 4 (2006), e309. doi: 10.1371/journal.pbio.0040309. [24] J. Ren, F. Jiao, Q. Sun, M. Tang and J. Yu, The dynamics of gene transcription in random environments, AIMS, 23 (2018), 3167-3194.  doi: 10.3934/dcdsb.2018224. [25] A. Sanchez and I. Golding, Genetic determinants and cellular constraints in noisy gene expression, Science, 342 (2013), 1188-1193.  doi: 10.1126/science.1242975. [26] 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. [27] 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. [28] 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. [29] 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. [30] J. Yu and X. Liu, Monotonic dynamics of mRNA degradation by two pathways, J. Appl. Anal. Comput., 7 (2017), 1598-1612. [31] 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. [32] 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. [33] T. Zhou and J. Zhang, Analytical results for a multistate gene model, SIAM J. Appl. Math., 72 (2012), 789-818.  doi: 10.1137/110852887.

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##### References:
 [1] W. Chu and W. Zhang, Transformations of Kummer-type for 2F2-series and their q-analogues, J. Comput. Appl. Math., 216 (2008), 467-473.  doi: 10.1016/j.cam.2007.05.024. [2] P. L. Felmer, Random dynamics of gene transcription activation in single cells, J. Differential Equations, 247 (2009), 1796-1816.  doi: 10.1016/j.jde.2009.06.006. [3] 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. [4] S. Iyer-Biswas, F. Hayot and C. Jayaprakash, Stochasticity of gene products from transcriptional pulsing, Phys. Rev. E, 79 (2009), 031911. doi: 10.1103/PhysRevE.79.031911. [5] F. Jiao, Q. Sun, M. Tang, J. Yu and B. Zheng, Distribution modes and their corresponding parameter regions in stochastic gene transcription, SIAM J. Appl. Math., 75 (2015), 2396-2420.  doi: 10.1137/151005567. [6] 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. [7] 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. [8] M. Kaern, T. C. Elston, W. J. Blake and J. J. Collins, Stochasticity in gene expression: From theories to phenotypes, Nat. Rev. Genet., 6 (2005), 451-464.  doi: 10.1038/nrg1615. [9] J. Kuang, M. Tang and J. Yu, The mean and noise of protein numbers in stochastic gene expression, J. Math. Biol., 67 (2013), 261-291.  doi: 10.1007/s00285-012-0551-8. [10] D. R. Larson, C. Fritzsch, L. Sun, X. Meng, D. S. Lawrence and R. H. Singer, Direct observation of frequency modulated transcription in single cells using light activation, eLife, 2 (2013), e00750. doi: 10.7554/eLife.00750. [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. [12] 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. [13] G. Lin, J. Yu, Z. Zhou, Q. Sun and F. Jiao, Fluctuations of mRNA distributions in multiple pathway activated transcription, Discrete Contin. Dyn. Syst. B, (2018), in press. doi: 10.3934/dcdsb.2018219. [14] J. Macia, S. Regot, T. Peeters, N. Conde, R. Sol$\acute{e}$ and F. Posas, Dynamic signaling in the Hog1 MAPK pathway relies on high basal signal transduction, Sci. Signal, 2 (2009), ra13. doi: 10.1126/scisignal.2000056. [15] C. Miller et al., Dynamic transcriptome analysis measures rates of mRNA synthesis and decay in yeast, Mol. Syst. Biol., 7 (2011), 458. [16] B. Munsky, G. Neuert and A. van Oudenaarden, Using gene expression noise to understand gene regulation, Science, 336 (2012), 183-187.  doi: 10.1126/science.1216379. [17] E. Nadal, G. Ammerer and F. Posas, Controlling gene expression in response to stress, Nat. Rev. Genet., 12 (2011), 833-845.  doi: 10.1038/nrg3055. [18] S. Paliwal, MAPK-mediated bimodal gene expression and adaptive gradient sensing in yeast, Nature, 446 (2007), 46-51. [19] R. B. Paris, A Kummer-type transformation for a 2F2 hypergeometric function, J. Comput. Appl. Math., 173 (2005), 379-382.  doi: 10.1016/j.cam.2004.05.005. [20] J. Peccoud and B. Ycart, Markovian modelling of gene-product synthesis, Theor. Popul. Biol., 48 (1995), 222-234. [21] V. Pelechano, S. Chávez and J.E. Pérez-Ortín, A complete set of nascent transcription rates for yeast genes, Plos One, 5 (2010), e15442. [22] S. Pelet, Transient activation of the HOG MAPK pathway regulates bimodal Gene expression, Science, 332 (2011), 732-735. [23] A. Raj, C.S. Peskin, D. Tranchina, D.Y. Vargas and S. Tyagi, Stochastic mRNA synthesis in mammalian cells, PLoS Biol., 4 (2006), e309. doi: 10.1371/journal.pbio.0040309. [24] J. Ren, F. Jiao, Q. Sun, M. Tang and J. Yu, The dynamics of gene transcription in random environments, AIMS, 23 (2018), 3167-3194.  doi: 10.3934/dcdsb.2018224. [25] A. Sanchez and I. Golding, Genetic determinants and cellular constraints in noisy gene expression, Science, 342 (2013), 1188-1193.  doi: 10.1126/science.1242975. [26] 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. [27] 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. [28] 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. [29] 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. [30] J. Yu and X. Liu, Monotonic dynamics of mRNA degradation by two pathways, J. Appl. Anal. Comput., 7 (2017), 1598-1612. [31] 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. [32] 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. [33] T. Zhou and J. Zhang, Analytical results for a multistate gene model, SIAM J. Appl. Math., 72 (2012), 789-818.  doi: 10.1137/110852887.
Three modes of mRNA distribution generated by the two-state modal. (a) The decaying distribution for which $P_m$ deceases in $m$ for $m = 0, 1, 2, \cdots$. (b) The unimodal distribution for which $P_m$ takes exactly one peak at some $m>0$. (c) The bimodal distribution for which $P_m$ takes exactly two peaks with the first one at $m = 0$, and the other one at some $m > 0$. The parameter sets $(k_{on}, k_{off}, k_b, k_d)$ in (a), (b), and (c) are chosen as $(0.5, 1.5, 20, 1), (2, 2, 10, 1)$, and $(0.1, 0.2, 15, 1)$, respectively
Gene transcription modulated by the cross-talking pathway [31]
Cross-talking pathway can generate bimodal distribution when the gene is (a) easily turned on (${\rm T}_{{\rm off}}<1$) and easily turned off (${\rm T}_{{\rm on}}<1$); (b) easily turned on (${\rm T}_{{\rm off}}<1$) but hardly turned off (${\rm T}_{{\rm on}}>1$); and (c) hardly turned on (${\rm T}_{{\rm off}}<1$) and hardly turned off (${\rm T}_{{\rm on}}>1$). mRNA synthesis rate $v = 30$ and the other parameters are set to be $(\lambda_1, \lambda_2, \gamma, q_1) = (0.6, 6, 2, 0.2)$ in (a), $(\lambda_1, \lambda_2, \gamma, q_1) = (0.2, 4, 0.2, 0.1)$ in (b), and $(\lambda_1, \lambda_2, \gamma, q_1) = (0.2, 4, 0.2, 0.2)$ in (c)
The transcriptional transition pattern of osmostress-responsive genes in yeast [22]. (a) At low salt concentrations (below 0.05 M NaCl solution), the weak basal pathway is more frequently selected, and therefore most cells are off, giving rise to the decaying distribution. (b) At intermediate salt concentrations, the stress-induced strong signaling pathway is more likely to be activated, leading to the bimodal distribution with two distinct cell identities representing silent and highly expressed cells. (c) At the highest salt concentrations (above 0.15 M NaCl solution), the strong pathway dominates the gene activation, giving rise to the most on cells and the unimodal distribution.
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