American Institute of Mathematical Sciences

Advanced
November  2019, 12(7): 2177-2194. doi: 10.3934/dcdss.2019140

The mean and noise of FPT modulated by promoter architecture in gene networks

Kunwen Wen a,b, , Lifang Huang c, , Qiuying Li d, , Qi Wang a, and  Jianshe Yu a,,

a. 

Center for Applied Mathematics, Guangzhou University, Guangzhou 510006, China
b. 

School of Mathematics, Jiaying University, Meizhou 514015, China
c. 

School of Statistics and Mathematics, Guangdong University of Finance and Economics, Guangzhou 510320, China
d. 

College of Mathematics and Statistics, Huanghuai University, Zhumadian 463000, China

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

Received  February 2018 Revised  May 2018 Published  December 2018

Increasing experimental evidences suggest that cell phenotypic variation often depends on the accumulation of some special proteins. Recently, a lot of studies have shown that the complexity of promoter architecture plays a major role in regulating transcription and controlling expression dynamics and further phenotype. One unanswered question is why the organism chooses such a complex promoter architecture and how the promoter architecture affects the timing of proteins amount up to a given threshold. To address this issue, we study the effect of promoter architecture on the first-passage time (FPT) by formulating a multi-state gene model, that may reflect the complexity of promoter architecture. We derive analytical formulae for FPT moments in each case of irreversible promoter and reversible promoter regulation, which is the first time to give these analytical results in the existing literature. We show that the mean and noise of FPT increase with the state number of promoter architecture if the mean residence time at $ off$ states is not fixed. Inversely, if the mean residence time at $ off$ states is fixed, then complex promoter architecture will not vary the mean of FPT but will tend to decrease the noise of FPT. Our results show that, in the same inactive promoter states, the noise of FPT with promoters in irreversible case is always less than that in reversible case. In conclusion, our results reveal the effect of the promoter architecture on FPT and enhance understanding of the regulation mechanism of gene expression.

Keywords: Mean and noise, FPT, promoter architecture, gene networks.
Mathematics Subject Classification: Primary: 34A30, 60E05, 60J10; Secondary: 65D30, 65F05.
Citation: Kunwen Wen, Lifang Huang, Qiuying Li, Qi Wang, Jianshe Yu. The mean and noise of FPT modulated by promoter architecture in gene networks. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2177-2194. doi: 10.3934/dcdss.2019140
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show all references

References:
[1]

A. Amir, O. Kobiler, A. Rokney, A. B. Oppenheim and J. Stavans, Noise in timing and precision of gene activities in a genetic cascade, Molec. Syst. Biol., 3 (2007), 71. 

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[3]

A. Becskei, B. B. Kaufmann and A. V. Oudenaarden, Contributions of low molecule number and chromosomal positioning to stochastic gene expression, Nat. Gen., 37 (2005), 937-944. 

[4]

W. J. Blake, M. Kærn, C. R. Cantor and J. J. Collins, Noise in eukaryotic gene expression, Nature, 422 (2003), 633-637. 

[5]

W. J. Blake, G. Balazsi and M. A. Kohanski, et al., Phenotypic consequences of promotermediated transcriptional noise, Mol. Cell, 24 (2006), 853-865.

[6]

J. A. Bonachela and S. A. Levin, Evolutionary comparison between viral lysis rate and latent period, J. Theor. Biol., 345 (2014), 32-42.  doi: 10.1016/j.jtbi.2013.12.006.

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P. C. Bressloff, Stochastic switching in biology: from genotype to phenotype, J. Phys. A: Math. Theor., 50 (2017), 133001,136pp.  doi: 10.1088/1751-8121/aa5db4.

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L. B. Carey, D. V. Dijk, P. M. A. Sloot, J. A. Kaandorp and E. Segal, Promoter sequence determines the relationship between expression level and noise, PLoS Biol., 11 (2013), e1001528. 

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L. Chantranupong and R. H. Heineman, A common, non-optimal phenotypic endpoint in experimental adaptations of bacteriophage lysis time, BMC Evolut. Biol., 12 (2012), 37. 

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[13]

A. Coulon, O. Gandrillon and G. Beslon, On the spontaneous stochastic dynamics of a single gene: Complexity of the molecular interplay at the promoter, BMC Syst. Biol., 4 (2010), 2. 

[14]

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[15]

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[16]

V. Elgart, T. Jia, A. T. Fenley and R. Kulkarni, Connecting protein and mrna burst distributions for stochastic models of gene expression, Phys. Biol., 8 (2011), 046001. 

[17]

P. L. Felmer, A. Quaas, M. X. Tang and J. S. Yu, Random dynamics of gene transcription activation in single cells, J. Differ. Equ., 247 (2009), 1796-1816.  doi: 10.1016/j.jde.2009.06.006.

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K. R. Ghusinga, C. A. Vargas-Garcia and A. Singh, A mechanistic stochastic framework for regulating bacterial cell division, Sci. Rep., 6 (2016), 30229. 

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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. 

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R. Heineman and J. Bull, Testing optimality with experimental evolution: Lysis time in a bacteriophage, Evolution, 61 (2007), 1695-1709. 

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G. Hornung, R. Bar-Ziv and D. Rosin, et al., Noise-mean relationship in mutated promoters, Genome. Res., 22 (2012), 2409-2417.

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L. F. Huang, J. J. Zhang, P. J. Liu and T. S. Zhou, Effects of promoter leakage on dynamics of gene expression, BMC Syst. Biol., 9 (2015), 16. 

[24]

L. F. Huang, Z. J. Yuan, J. S. Yu and T. S. Zhou, Fundamental principles of energy consumption for gene expression, Chaos, 25 (2015), 123101, 10pp..  doi: 10.1063/1.4936670.

[25]

L. F. Huang, P. J. Liu, Z. J. Yuan, T. S. Zhou and J. S. Yu, The free-energy cost of interaction between DNA loops, Sci. Rep., 7 (2017), 12610. 

[26]

F. Jiao, M. X. Tang and J. S. Yu, Distribution profiles and their dynamic transition in stochastic gene transcription, J. Diff. Equat., 254 (2013), 3307-3328.  doi: 10.1016/j.jde.2013.01.019.

[27]

M. Kærn, 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. 

[28]

O. Kobiler, A. Rokney and N. Friedman, et al., Quantitative kinetic analysis of the bacteriophage λ genetic network, Proc. Natl. Acad. Sci. U. S. A., 102 (2005), 4470-4475.

[29]

J. H. Kuang, M. X. Tang and J. S. 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.

[30]

N. Kumar, A. Singh and R. V. Kulkarni, Transcriptional Bursting in Gene Expression: Analytical Results for General Stochastic Models, PLoS Comput. Biol., 11 (2015), e1004292. 

[31]

W. Li, D. Notani and M.G. Rosenfeld, Enhancers as non-coding RNA transcription units: Recent insights and future perspectives, Nat. Rev. Genet., 17 (2016), 207-223. 

[32]

Q. Y. Li, L. F. Huang and J. S. 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.

[33]

Y. Y. Li, M. X. Tang and J. S. Yu, Transcription dynamics of inducible genes modulated by negative regulations, Math. Med. Biol., 32 (2015), 115-136.  doi: 10.1093/imammb/dqt019.

[34]

G. H. Lin, J. S. Yu, Z. Zhou, Q. W. Sun and F. Jiao, Fluctuations of mRNA distributions in multiple pathway activated transcription, Discrete Contin. Dyn. Syst. Ser. B, (2018), Accepted for publication, 26 pp.

[35]

H. H. McAdams and A. Arkin, Stochastic mechanisms in gene expression, Proc. Natl. Acad. Sci. U. S. A., 94 (1997), 814-819. 

[36]

B. Munsky, G. Neuert and A. V. Oudenaarden, Using gene expression noise to understand gene regulation, Science, 336 (2012), 183-187.  doi: 10.1126/science.1216379.

[37]

K. F. Murphy, G. Balazsi and J. J. Collins, Combinatorial promoter design for engineering noisy gene expression, Proc. Natl. Acad. Sci. U. S. A., 104 (2007), 12726-12731. 

[38]

R. Murugan and G. Kreiman, On the minimization of fluctuations in the response times of autoregulatory gene networks, Biophys. J., 101 (2011), 1297-1306. 

[39]

A. Ochab-Marcinek and M. Tabaka, Bimodal gene expression in noncooperative regulatory systems, Proc. Natl. Acad. Sci. U. S. A., 107 (2010), 22096-22101. 

[40]

M. Osella and M. C. Lagomarsino, Growthrate-dependent dynamics of a bacterial genetic oscillator, Phys. Rev. E., 87 (2013), 012726. 

[41]

E. M. Ozbudak, M. Thattai, I. Kurtser, A. D. Grossman and A. Oudenaarden, Regulation of noise in the expression of a single gene, Nat. Gen., 31 (2002), 69-73. 

[42] A. Papoulis and S. U. Pillai, Probability, random variables and stochastic processes, 4th ed, McGraw Hill, McGraw Hill. 
[43]

J. Paulsson, Models of stochastic gene expression, Phys. Lif. Rev., 2 (2005), 157-175. 

[44]

J. M. Pedraza and J. Paulsson, Effects of Molecular Memory and Bursting on Fluctuations in Gene Expression, Science, 319 (2008), 1144331. 

[45]

J. M. Raser and E. K. O Shea, Noise in gene expression: Origins, consequences, and control, Science, 309 (2005), 2010-2013. 

[46]

J. M. Raser and E. K. O Shea, Control of stochasticity in eukaryotic gene expression, Science, 304 (2004), 1811-1814. 

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J. Ren, F. Jiao, Q. W. Sun, M. X. Tang and J. S. Yu, The dynamics of gene transcription in random environments, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 3167-3194.  doi: 10.3934/dcdsb.2018224.

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Figure 1.  Simplified sketch of complex promoter architecture for a gene expression model. Where the gene activity proceeds sequentially through $on$ state and several $off$ states and then returns to $on$ state, with the indicated rates: $\lambda_{0}$ is the rate of gene inactivation; $\lambda_{L}$ is the rate of gene activation; $\lambda_{k}$ is the transition rate from the $k^{th}$ $off$ state to the $(k+1)^{th}$ $off$ state with $(k = 1,2,\cdots,L-1)$; $\lambda_{k}^{'}$ is the transition rate from the $(k+1)^{th}$ $off$ state to the $k^{th}$ $off$ state with $(k = 1,2,\cdots,L-1)$; $k_{m}$ is the rate of protein synthesis. To investigate the effects of promoter architecture on FPT, we divide promoters into two categories: A: Irreversible promoter structure; B: Reversible promoter structure
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Figure 2.  Markov chain model for calculating inter arrival time distribution
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Figure 3.  Effects of the amount of promoter state on FPT moments. $(A)$ The relationship between the mean of FPT and the amount of promoter state. $(B)$ The dependence of the noise intensity of FPT on the amount of promoter state. It confirms that the mean and noise intensity is always monotonically increasing with the quantity of promoter state; the mean and noise intensity by the irreversible promoter modulation is always less than those regulated by reversible promoter. Here, $X = 1500, b = 3, \lambda_{0} = 1/50, k_{m} = 10, \lambda_{i} = 1/10 (i = 1,2, ..., 5), \lambda_{j}^{'} = 1/10 (j = 1, ..., 4)$ (see [12])
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Figure 4.  Effects of promoter architecture on the noise of FPT. The dependence of the noise intensity on the promoter architecture in the case where the residence time at $off$ state is immobilized. $(A)$ The noise intensity of FPT regulated by irreversible promoter; $(B)$ The noise intensity of FPT modulated by reversible promoter. Showing the noise intensity is always a monotonically decreasing function of promoter state for both irreversible promoter regulation and reversible promoter regulation. Here, $X = 1500, b = 3, \lambda_{0} = 1/50, k_{m} = 10$ (see [12]), we fix the average residence time at $off$ state, and take a 5 state model as an example, the residence time at $off$ state is 30 min, the transition rates among inactive promoters in the irreversible case are $\lambda_{1} = 1/6, \lambda_{2} = 1/9, \lambda_{3} = 1/7, \lambda_{4} = 1/8$; the transition rate among inactive promoters in the reversible case are $\lambda_{1} = 1/3, \lambda_{1}^{'} = 1/5, \lambda_{2} = 1, \lambda_{2}^{'} = 1/3, \lambda_{3} = 1/2, \lambda_{3}^{'} = 1/4, \lambda_{4} = 53/670$
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Figure 5.  Comparison of the noise intensity of FPT of irreversible promoter modulation (solid green line) and reversible promoter modulation (solid red line) at the same promoter state. $(A), (B), (C), (D)$ represent the promoter state of 3, 4, 5, 6 respectively. The noise intensity regulated by irreversible promoter is always smaller than that regulated by reversible promoter. The parameter values are the same as those for Figure 4
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