# American Institute of Mathematical Sciences

January  2020, 25(1): 241-257. doi: 10.3934/dcdsb.2019180

## Analytical formula and dynamic profile of mRNA distribution

 a. College of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou 450002, China b. Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, China

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

Received  January 2019 Revised  March 2019 Published  January 2020 Early access  July 2019

Fund Project: This work was supported by National Natural Science Foundation of China (11631005, 11871174, 11601491) and the Program for Changjiang Scholars and Innovative Research Team in University (IRT_16R16).

The stochasticity of transcription can be quantified by mRNA distribution $P_m(t)$, the probability that there are $m$ mRNA molecules for the gene at time $t$ in one cell. However, it still lacks of a standard method to calculate $P_m(t)$ in a transparent formula. Here, we employ an infinite series method to express $P_m(t)$ based on the classical two-state model. Intriguingly, we observe that a unimodal distribution of mRNA numbers at steady-state could be transformed from a dynamical bimodal distribution. This indicates that "bet hedging" strategy can be still achieved for the gene that generates phenotypic homogeneity of the cell population. Moreover, the formation and duration of such bimodality are tightly correlated with mRNA synthesis rate, reinforcing the modulation scenario of some inducible genes that manipulates mRNA synthesis rate in response to environmental change. More generally, the method presented here may be implemented to the other stochastic transcription models with constant rates.

Citation: Feng Jiao, Jian Ren, Jianshe Yu. Analytical formula and dynamic profile of mRNA distribution. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 241-257. doi: 10.3934/dcdsb.2019180
##### References:
 [1] G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999.  doi: 10.1017/CBO9781107325937. [2] T. M. Apostol, Mathematical Analysis, 2$^{nd}$ edition, Addison-Wesley, Boston, USA, 1974. [3] R. D. Dar et al., Transcriptional burst frequency and burst size are equally modulated across the human genome, Proc. Natl. Acad. Sci. U.S.A., 109 (2012), 17454-17459. [4] L. C. Evans, Partial Differential Equations, 2$^{nd}$ edition, American Math. Society, Providence, USA, 2010. doi: 10.1090/gsm/019. [5] S. Fiering et al., Single cell assay of a transcription factor reveals a threshold in transcription activated by signals emanating from the T-cell antigen receptor, Genes Dev., 4 (1990), 1823-1834. [6] D.T. Gillespie, Stochastic simulation of chemical kinetics, Annu. Rev. Phys. Chem., 58 (2007), 35-55.  doi: 10.1146/annurev.physchem.58.032806.104637. [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. [8] M. W. Hirsch, S. Smale and R. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Elsevier/Academic Press, Amsterdam, 2004. [9] 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. [10] F. Jiao, Q. Sun, G. Lin and J. Yu, Distribution profiles in gene transcription activated by the cross-talking pathway, Discrete Contin. Dyn. Syst. B, 24 (2019), 2799-2810.  doi: 10.3934/dcdsb.2018275. [11] 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. [12] 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. [13] 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. [14] 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. [15] D. R. Larson, What do expression dynamics tell us about the mechanism of transcription, Curr. Opin. Genet. Dev., 21 (2011), 591-599.  doi: 10.1016/j.gde.2011.07.010. [16] 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. [17] 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. [18] 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, 24 (2019), 1543-1568.  doi: 10.3934/dcdsb.2018219. [19] N. Molina et al., Stimulus-induced modulation of transcriptional bursting in a single mammalian gene, Proc. Natl. Acad. Sci. U.S.A., 110 (2013), 20563-20568. [20] A. Mugler, A. M. Walczak and C. H. Wiggins, Spectral solutions to stochastic models of gene expression with bursts and regulation, Phys. Rev. E, 80 (2009), 041921. doi: 10.1103/PhysRevE.80.041921. [21] 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. [22] G. Neuert et al., Systematic identification of signal-activated stochastic gene regulation, Science, 339 (2013), 584-587. [23] J. Peccoud and B. Ycart, Markovian modelling of gene-product synthesis, Theor. Popul. Biol., 48 (1995), 222-234. [24] S. Pelet et al., Transient activation of the HOG MAPK pathway regulates bimodal Gene expression, Science, 332 (2011), 732-735. [25] 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. [26] 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. [27] A. Sanchez and I. Golding, Genetic determinants and cellular constraints in noisy gene expression, Science, 342 (2013), 1188-1193.  doi: 10.1126/science.1242975. [28] V. Shahrezaei and P. S. Swain, Analytical distributions for stochastic gene expression, Proc. Natl. Acad. Sci. USA, 105 (2008), 17256-17261.  doi: 10.1073/pnas.0803850105. [29] S. O. Skinner et al., Measuring mRNA copy number in individual Escherichia coli cells using single-molecule fluorescent in situ hybridization, Nat. Protoc., 8 (2013), 1100-1113. [30] L. J. Slater, Confluent Hypergeometric Functions, Cambridge University Press, Cambridge, England, 1960. [31] A. R. Stinchcombe, C. S. Peskin and D. Tranchina, Population density approach for discrete mRNA distributions in generalized switching models for stochastic gene expression, Phys. Rev. E, 85 (2012), 061919. doi: 10.1103/PhysRevE.85.061919. [32] L. So et al., General properties of the transcriptional timeseries in Escherichia Coli, Nat. Genet., 43 (2011), 554-560. [33] M. Tabaka and R. Hołyst, Binary and graded evolution in time in a simple model of gene induction, Phys. Rev. E, 82 (2010), 052902. doi: 10.1103/PhysRevE.82.052902. [34] 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. [35] J. Yu and X. Liu, Monotonic dynamics of mRNA degradation by two pathways, J. Appl. Anal. Comput., 7 (2017), 1598-1612. [36] 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.  doi: 10.3934/mbe.2018058. [37] 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] G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999.  doi: 10.1017/CBO9781107325937. [2] T. M. Apostol, Mathematical Analysis, 2$^{nd}$ edition, Addison-Wesley, Boston, USA, 1974. [3] R. D. Dar et al., Transcriptional burst frequency and burst size are equally modulated across the human genome, Proc. Natl. Acad. Sci. U.S.A., 109 (2012), 17454-17459. [4] L. C. Evans, Partial Differential Equations, 2$^{nd}$ edition, American Math. Society, Providence, USA, 2010. doi: 10.1090/gsm/019. [5] S. Fiering et al., Single cell assay of a transcription factor reveals a threshold in transcription activated by signals emanating from the T-cell antigen receptor, Genes Dev., 4 (1990), 1823-1834. [6] D.T. Gillespie, Stochastic simulation of chemical kinetics, Annu. Rev. Phys. Chem., 58 (2007), 35-55.  doi: 10.1146/annurev.physchem.58.032806.104637. [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. [8] M. W. Hirsch, S. Smale and R. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Elsevier/Academic Press, Amsterdam, 2004. [9] 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. [10] F. Jiao, Q. Sun, G. Lin and J. Yu, Distribution profiles in gene transcription activated by the cross-talking pathway, Discrete Contin. Dyn. Syst. B, 24 (2019), 2799-2810.  doi: 10.3934/dcdsb.2018275. [11] 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. [12] 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. [13] 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. [14] 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. [15] D. R. Larson, What do expression dynamics tell us about the mechanism of transcription, Curr. Opin. Genet. Dev., 21 (2011), 591-599.  doi: 10.1016/j.gde.2011.07.010. [16] 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. [17] 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. [18] 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, 24 (2019), 1543-1568.  doi: 10.3934/dcdsb.2018219. [19] N. Molina et al., Stimulus-induced modulation of transcriptional bursting in a single mammalian gene, Proc. Natl. Acad. Sci. U.S.A., 110 (2013), 20563-20568. [20] A. Mugler, A. M. Walczak and C. H. Wiggins, Spectral solutions to stochastic models of gene expression with bursts and regulation, Phys. Rev. E, 80 (2009), 041921. doi: 10.1103/PhysRevE.80.041921. [21] 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. [22] G. Neuert et al., Systematic identification of signal-activated stochastic gene regulation, Science, 339 (2013), 584-587. [23] J. Peccoud and B. Ycart, Markovian modelling of gene-product synthesis, Theor. Popul. Biol., 48 (1995), 222-234. [24] S. Pelet et al., Transient activation of the HOG MAPK pathway regulates bimodal Gene expression, Science, 332 (2011), 732-735. [25] 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. [26] 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. [27] A. Sanchez and I. Golding, Genetic determinants and cellular constraints in noisy gene expression, Science, 342 (2013), 1188-1193.  doi: 10.1126/science.1242975. [28] V. Shahrezaei and P. S. Swain, Analytical distributions for stochastic gene expression, Proc. Natl. Acad. Sci. USA, 105 (2008), 17256-17261.  doi: 10.1073/pnas.0803850105. [29] S. O. Skinner et al., Measuring mRNA copy number in individual Escherichia coli cells using single-molecule fluorescent in situ hybridization, Nat. Protoc., 8 (2013), 1100-1113. [30] L. J. Slater, Confluent Hypergeometric Functions, Cambridge University Press, Cambridge, England, 1960. [31] A. R. Stinchcombe, C. S. Peskin and D. Tranchina, Population density approach for discrete mRNA distributions in generalized switching models for stochastic gene expression, Phys. Rev. E, 85 (2012), 061919. doi: 10.1103/PhysRevE.85.061919. [32] L. So et al., General properties of the transcriptional timeseries in Escherichia Coli, Nat. Genet., 43 (2011), 554-560. [33] M. Tabaka and R. Hołyst, Binary and graded evolution in time in a simple model of gene induction, Phys. Rev. E, 82 (2010), 052902. doi: 10.1103/PhysRevE.82.052902. [34] 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. [35] J. Yu and X. Liu, Monotonic dynamics of mRNA degradation by two pathways, J. Appl. Anal. Comput., 7 (2017), 1598-1612. [36] 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.  doi: 10.3934/mbe.2018058. [37] 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.
Increasing mRNA synthesis rate $v$ improves the formation of the intermediate bimodal distribution. (a) The mRNA distribution does not form bimodality when $v$ is relatively small. (b) The intermediate bimodality appears when $v$ increases across the threshold value. (c) As $v$ increases further, the duration of the bimodality prolongs with its second peak moving to the right.
Dynamic transitions among three mRNA distribution modes. (a, b) Pattern (Ⅰ): If $P_m(t)$ takes a unimodal distribution at steady-state, then for sufficient large synthesis rate $v = 10 \delta$, its profile develops from the original decaying to the intermediate bimodality, and finally switches to the unimodality. However, when $v$ decreases below the threshold value $v = 5 \delta$, the intermediate bimodality disappears. (c) Pattern (Ⅱ): If $P_m(t)$ takes a bimodal distribution at steady-state, then its profile transits from the decaying to the bimodality at some time points, and maintains the bimodality in a long run. (d) Pattern (Ⅲ): If $P_m(t)$ takes a decaying distribution at steady-state, then it maintains the same distribution mode within the whole time regime. (Inset) Fano factor versus time for the three patterns.
The three modes of the steady-state mRNA distribution. (a) When $\bar{v} = v/ \delta$ is fixed, the $\lambda$-$\gamma$ plane can be divided into three connected regions, and the values of $( \lambda, \gamma)$ in each region generate a corresponding steady-state distribution mode [11]: (b) The decaying distribution that $P_m$ deceases in $m$ for $m = 0, 1, 2, \cdots$; (c) The unimodal distribution that $P_m$ takes exactly one peak at some $m>0$; (d) The bimodal distribution that $P_m$ takes exactly two peaks with the first one at $m = 0$, and the other one at some $m>0$
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