# 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  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 & Continuous Dynamical Systems - B, 2020, 25 (1) : 241-257. doi: 10.3934/dcdsb.2019180
##### References:
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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 [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [22] G. Neuert et al., Systematic identification of signal-activated stochastic gene regulation, Science, 339 (2013), 584-587.   Google Scholar [23] J. Peccoud and B. Ycart, Markovian modelling of gene-product synthesis, Theor. Popul. Biol., 48 (1995), 222-234.   Google Scholar [24] S. Pelet et al., Transient activation of the HOG MAPK pathway regulates bimodal Gene expression, Science, 332 (2011), 732-735.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [30] L. J. Slater, Confluent Hypergeometric Functions, Cambridge University Press, Cambridge, England, 1960.   Google Scholar [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.  Google Scholar [32] L. So et al., General properties of the transcriptional timeseries in Escherichia Coli, Nat. Genet., 43 (2011), 554-560.   Google Scholar [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.  Google Scholar [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.  Google Scholar [35] J. Yu and X. Liu, Monotonic dynamics of mRNA degradation by two pathways, J. Appl. Anal. Comput., 7 (2017), 1598-1612.   Google Scholar [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.  Google Scholar [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.   Google Scholar

show all references

##### References:
 [1] G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999.  doi: 10.1017/CBO9781107325937.  Google Scholar [2] T. M. Apostol, Mathematical Analysis, 2$^{nd}$ edition, Addison-Wesley, Boston, USA, 1974.  Google Scholar [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.   Google Scholar [4] L. C. Evans, Partial Differential Equations, 2$^{nd}$ edition, American Math. Society, Providence, USA, 2010. doi: 10.1090/gsm/019.  Google Scholar [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.   Google Scholar [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.  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. W. Hirsch, S. Smale and R. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Elsevier/Academic Press, Amsterdam, 2004.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [22] G. Neuert et al., Systematic identification of signal-activated stochastic gene regulation, Science, 339 (2013), 584-587.   Google Scholar [23] J. Peccoud and B. Ycart, Markovian modelling of gene-product synthesis, Theor. Popul. Biol., 48 (1995), 222-234.   Google Scholar [24] S. Pelet et al., Transient activation of the HOG MAPK pathway regulates bimodal Gene expression, Science, 332 (2011), 732-735.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [30] L. J. Slater, Confluent Hypergeometric Functions, Cambridge University Press, Cambridge, England, 1960.   Google Scholar [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.  Google Scholar [32] L. So et al., General properties of the transcriptional timeseries in Escherichia Coli, Nat. Genet., 43 (2011), 554-560.   Google Scholar [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.  Google Scholar [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.  Google Scholar [35] J. Yu and X. Liu, Monotonic dynamics of mRNA degradation by two pathways, J. Appl. Anal. Comput., 7 (2017), 1598-1612.   Google Scholar [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.  Google Scholar [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.   Google Scholar
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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