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 & Continuous Dynamical Systems - S, 2019, 12 (7) : 2177-2194. doi: 10.3934/dcdss.2019140
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.   Google Scholar

[2]

J. M. Bean, E. D. Siggia and F. R. Cross, Coherence and timing of cell cycle start examined at single-cell resolution, Mol. Cell., 21 (2006), 3-14.   Google Scholar

[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.   Google Scholar

[4]

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

[5]

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

[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.  Google Scholar

[7]

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.  Google Scholar

[8]

C. R. Brown, C. Mao, F. Elena, M. S. Jurica and H. Boeger, Linking stochastic fluctuations in chromatin structure and gene expression, PLoS Biol., 11 (2013), e1001621.   Google Scholar

[9]

L. Cai, N. Friedman and X. S. Xie, Stochastic protein expression in individual cells at the single molecule level, Nature, 440 (2006), 358-362.   Google Scholar

[10]

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.   Google Scholar

[11]

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.   Google Scholar

[12]

P. J. Choi, L. Cai, K. Frieda and X. S. Xie, A stochastic single molecule event triggers phenotype switching of a bacterial cell, Science, 322 (2008), 442-445.   Google Scholar

[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.   Google Scholar

[14]

M. H. DeGroot and M. J. Schervish, Probability and Statistics, 4th ed. Pearson, 2012. Google Scholar

[15]

J. J. Dennehy and N. I. Wang, Factors influencing lysis time stochasticity in bacteriophage $\lambda$, BMC Microbiol, 11 (2011), 1-12.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[18]

H. B. Fraser, A. E. Hirsh, G. Giaever, J. Kumm and M. B. Eisen, Noise minimization in eukaryotic gene expression, PLoS Biol., 6 (2004), 835-838.   Google Scholar

[19]

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

[20]

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.   Google Scholar

[21]

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

[22]

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

[23]

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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[35]

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

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[40]

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

[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.   Google Scholar

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

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

[44]

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

[45]

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

[46]

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

[47] S. Redner, A Guide to First-Passage Processes, Cambridge University Press, 2001.  doi: 10.1017/CBO9780511606014.  Google Scholar
[48]

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.  Google Scholar

[49] S. K. Ross, Introduction to Probability Models, 10th ed, Academic Press, 2010.   Google Scholar
[50]

F. M. V. Rossi, A. M. Kringstein, A. Spicher, O. M. Guicherit and H. M. Blau, Transcriptional control: Rheostat converted to on/off switch, Mol. Cell, 6 (2000), 723-728.   Google Scholar

[51]

A. Sanchez and J. Kondev, Transcriptional control of noise in gene expression, Proc. Natl. Acad. Sci. U. S. A., 105 (2008), 5081-5086.   Google Scholar

[52]

A. Sanchez, S. Choubey and J. KondevA, Regulation of noise in gene expression, Annu. Rev. Biophys., 42 (2013), 469-491.   Google Scholar

[53]

A. Sanchez, H. G. Garcia, D. Jones and J. Kondev, Effect of promoter architecture on the cell-to-cell variability in gene expression, PLoS Comput. Biol., 7 (2011), e1001100.   Google Scholar

[54]

V. Shahrezaei and P. S. Swain, Analytical distributions for stochastic gene expression, Proc. Natl. Acad. Sci. U. S. A., 105 (2008), 256-261.   Google Scholar

[55]

Y. Shao and I. N. Wang, Bacteriophage adsorption rate and optimal lysis time, Genet., 180 (2008), 471-482.   Google Scholar

[56]

Y. Shao and N. Wang, Effect of late promoter activity on bacteriophage λ fitness, Genet., 181 (2009), 1467-1475.   Google Scholar

[57]

M. Shreshtha, A. Surendran and A. Ghosh, Estimation of mean first passage time for bursty gene expression, Phys. Biol., 13 (2016), 036004.   Google Scholar

[58]

A. Singh and M. Soltani, Quantifying intrinsic and extrinsic variability in stochastic gene expression models, PLoS ONE, 8 (2003), e84301.   Google Scholar

[59]

A. Singh, B. Razooky, C. D. Cox, M. L. Simpson and L. S. Weinbergers, Transcriptional bursting from the hiv-1 promoter is a significant source of stochastic noise in hiv-1 gene expression, Biophys. J., 98 (2010), L32-L34.   Google Scholar

[60]

A. Singh and J. J. Dennehy, Stochastic holin expression can account for lysis time variation in the bacteriophage $\lambda$, J. R. Soc. Interface., 11 (2014), 20140140.   Google Scholar

[61]

M. Soltani, C. A. Vargas-Garcia, D. Antunes and A. Singh, Intercellular variability in protein levels from stochastic expression and noisy cell cycle processes, PLoS Comput. Biol., 12 (2016), e1004972.   Google Scholar

[62] M. R. Spiegel, Theory and problems of probability and statistics, McGraw Hill, 1992.   Google Scholar
[63]

D. G. Spiller, C. D. Wood, D. A. Rand and M. R. White, Measurement of single-cell dynamics, Nature, 465 (2010), 736-745.   Google Scholar

[64]

Q. W. Sun, M. X. Tang and J. S. 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.  Google Scholar

[65]

Q. W. Sun, M. X. Tang and J. S. Yu, Modulation of gene transcription noise by competing transcription factors, J. Math. Biol., 64 (2012), 469-494.  doi: 10.1007/s00285-011-0420-x.  Google Scholar

[66]

D. M. Suter, N. Molina and D. Gatfield, et al., Mammalian genes are transcribed with widely different bursting kinetics, Science, 332 (2011), 472-474. Google Scholar

[67]

M. Thattai and A. V. Oudenaarden, Intrinsic noise in gene regulatory networks, Proc. Natl. Acad. Sci. U. S. A., 98 (2001), 8614-8619.   Google Scholar

[68]

Q. Wang, L. F. Huang, K. W. Wen and J. S. Yu, The mean and noise of stochastic gene transcription with cell division, Math. Biosci. Eng., 15 (2018), 1255-1270.   Google Scholar

[69]

I. N. Wang, Lysis timing and bacteriophage fitness, Genet., 172 (2006), 17-26.   Google Scholar

[70]

I. N. Wang, D. E. Dykhuizen and L. B. Slobodkin, The evolution of phage lysis timing, Evolut. Ecol., 10 (1996), 545-558.   Google Scholar

[71]

R. White, S. Chiba, T. Pang, J. S. Dewey, C. G. Savva, A. Holzenburg, et al., Holin triggering in real time, Proc. Natl. Acad. Sci. U. S. A., 108 (2011), 798-803. Google Scholar

[72]

J. Yu, J. Xiao, X. Ren, K. Lao and X. S. Xie, Probing gene expression in live cells, one protein molecule at a time, Science, 311 (2006), 1600-1603.   Google Scholar

[73]

J. S. Yu and X. J. Liu, Monotonic dynamics of mRNA degradation by two pathways, J. Appl. Anal. Comput., 7 (2017), 1598-1612.   Google Scholar

[74]

J. S. Yu, Q. W. Sun and M. X. 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

[75]

J. J. Zhang, L. N. Cheng and T. S. Zhou, Analytical distribution and tunability of noise in a model of promoter progress, Biophys. J., 102 (2012), 1-11.   Google Scholar

[76]

J. J. Zhang and T. S. Zhou, Promoter-mediated transcriptional dynamics, Biophys. J., 106 (2014), 479-488.   Google Scholar

[77]

T. S. Zhou and J. J. Zhang, Analytical results for a multistate gene model, SIAM J. Appl. Math., 72 (2012), 789-818.  doi: 10.1137/110852887.  Google Scholar

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.   Google Scholar

[2]

J. M. Bean, E. D. Siggia and F. R. Cross, Coherence and timing of cell cycle start examined at single-cell resolution, Mol. Cell., 21 (2006), 3-14.   Google Scholar

[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.   Google Scholar

[4]

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

[5]

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

[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.  Google Scholar

[7]

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.  Google Scholar

[8]

C. R. Brown, C. Mao, F. Elena, M. S. Jurica and H. Boeger, Linking stochastic fluctuations in chromatin structure and gene expression, PLoS Biol., 11 (2013), e1001621.   Google Scholar

[9]

L. Cai, N. Friedman and X. S. Xie, Stochastic protein expression in individual cells at the single molecule level, Nature, 440 (2006), 358-362.   Google Scholar

[10]

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.   Google Scholar

[11]

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.   Google Scholar

[12]

P. J. Choi, L. Cai, K. Frieda and X. S. Xie, A stochastic single molecule event triggers phenotype switching of a bacterial cell, Science, 322 (2008), 442-445.   Google Scholar

[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.   Google Scholar

[14]

M. H. DeGroot and M. J. Schervish, Probability and Statistics, 4th ed. Pearson, 2012. Google Scholar

[15]

J. J. Dennehy and N. I. Wang, Factors influencing lysis time stochasticity in bacteriophage $\lambda$, BMC Microbiol, 11 (2011), 1-12.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[18]

H. B. Fraser, A. E. Hirsh, G. Giaever, J. Kumm and M. B. Eisen, Noise minimization in eukaryotic gene expression, PLoS Biol., 6 (2004), 835-838.   Google Scholar

[19]

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

[20]

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.   Google Scholar

[21]

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

[22]

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

[23]

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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[35]

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

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[40]

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

[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.   Google Scholar

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

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

[44]

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

[45]

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

[46]

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

[47] S. Redner, A Guide to First-Passage Processes, Cambridge University Press, 2001.  doi: 10.1017/CBO9780511606014.  Google Scholar
[48]

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.  Google Scholar

[49] S. K. Ross, Introduction to Probability Models, 10th ed, Academic Press, 2010.   Google Scholar
[50]

F. M. V. Rossi, A. M. Kringstein, A. Spicher, O. M. Guicherit and H. M. Blau, Transcriptional control: Rheostat converted to on/off switch, Mol. Cell, 6 (2000), 723-728.   Google Scholar

[51]

A. Sanchez and J. Kondev, Transcriptional control of noise in gene expression, Proc. Natl. Acad. Sci. U. S. A., 105 (2008), 5081-5086.   Google Scholar

[52]

A. Sanchez, S. Choubey and J. KondevA, Regulation of noise in gene expression, Annu. Rev. Biophys., 42 (2013), 469-491.   Google Scholar

[53]

A. Sanchez, H. G. Garcia, D. Jones and J. Kondev, Effect of promoter architecture on the cell-to-cell variability in gene expression, PLoS Comput. Biol., 7 (2011), e1001100.   Google Scholar

[54]

V. Shahrezaei and P. S. Swain, Analytical distributions for stochastic gene expression, Proc. Natl. Acad. Sci. U. S. A., 105 (2008), 256-261.   Google Scholar

[55]

Y. Shao and I. N. Wang, Bacteriophage adsorption rate and optimal lysis time, Genet., 180 (2008), 471-482.   Google Scholar

[56]

Y. Shao and N. Wang, Effect of late promoter activity on bacteriophage λ fitness, Genet., 181 (2009), 1467-1475.   Google Scholar

[57]

M. Shreshtha, A. Surendran and A. Ghosh, Estimation of mean first passage time for bursty gene expression, Phys. Biol., 13 (2016), 036004.   Google Scholar

[58]

A. Singh and M. Soltani, Quantifying intrinsic and extrinsic variability in stochastic gene expression models, PLoS ONE, 8 (2003), e84301.   Google Scholar

[59]

A. Singh, B. Razooky, C. D. Cox, M. L. Simpson and L. S. Weinbergers, Transcriptional bursting from the hiv-1 promoter is a significant source of stochastic noise in hiv-1 gene expression, Biophys. J., 98 (2010), L32-L34.   Google Scholar

[60]

A. Singh and J. J. Dennehy, Stochastic holin expression can account for lysis time variation in the bacteriophage $\lambda$, J. R. Soc. Interface., 11 (2014), 20140140.   Google Scholar

[61]

M. Soltani, C. A. Vargas-Garcia, D. Antunes and A. Singh, Intercellular variability in protein levels from stochastic expression and noisy cell cycle processes, PLoS Comput. Biol., 12 (2016), e1004972.   Google Scholar

[62] M. R. Spiegel, Theory and problems of probability and statistics, McGraw Hill, 1992.   Google Scholar
[63]

D. G. Spiller, C. D. Wood, D. A. Rand and M. R. White, Measurement of single-cell dynamics, Nature, 465 (2010), 736-745.   Google Scholar

[64]

Q. W. Sun, M. X. Tang and J. S. 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.  Google Scholar

[65]

Q. W. Sun, M. X. Tang and J. S. Yu, Modulation of gene transcription noise by competing transcription factors, J. Math. Biol., 64 (2012), 469-494.  doi: 10.1007/s00285-011-0420-x.  Google Scholar

[66]

D. M. Suter, N. Molina and D. Gatfield, et al., Mammalian genes are transcribed with widely different bursting kinetics, Science, 332 (2011), 472-474. Google Scholar

[67]

M. Thattai and A. V. Oudenaarden, Intrinsic noise in gene regulatory networks, Proc. Natl. Acad. Sci. U. S. A., 98 (2001), 8614-8619.   Google Scholar

[68]

Q. Wang, L. F. Huang, K. W. Wen and J. S. Yu, The mean and noise of stochastic gene transcription with cell division, Math. Biosci. Eng., 15 (2018), 1255-1270.   Google Scholar

[69]

I. N. Wang, Lysis timing and bacteriophage fitness, Genet., 172 (2006), 17-26.   Google Scholar

[70]

I. N. Wang, D. E. Dykhuizen and L. B. Slobodkin, The evolution of phage lysis timing, Evolut. Ecol., 10 (1996), 545-558.   Google Scholar

[71]

R. White, S. Chiba, T. Pang, J. S. Dewey, C. G. Savva, A. Holzenburg, et al., Holin triggering in real time, Proc. Natl. Acad. Sci. U. S. A., 108 (2011), 798-803. Google Scholar

[72]

J. Yu, J. Xiao, X. Ren, K. Lao and X. S. Xie, Probing gene expression in live cells, one protein molecule at a time, Science, 311 (2006), 1600-1603.   Google Scholar

[73]

J. S. Yu and X. J. Liu, Monotonic dynamics of mRNA degradation by two pathways, J. Appl. Anal. Comput., 7 (2017), 1598-1612.   Google Scholar

[74]

J. S. Yu, Q. W. Sun and M. X. 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

[75]

J. J. Zhang, L. N. Cheng and T. S. Zhou, Analytical distribution and tunability of noise in a model of promoter progress, Biophys. J., 102 (2012), 1-11.   Google Scholar

[76]

J. J. Zhang and T. S. Zhou, Promoter-mediated transcriptional dynamics, Biophys. J., 106 (2014), 479-488.   Google Scholar

[77]

T. S. Zhou and J. J. Zhang, Analytical results for a multistate gene model, SIAM J. Appl. Math., 72 (2012), 789-818.  doi: 10.1137/110852887.  Google Scholar

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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Markov chain model for calculating inter arrival time distribution
Figure Options

Download full-size image

Download as PowerPoint slide

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])
Figure Options

Download full-size image

Download as PowerPoint slide

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 $\lambda_{1} = 1/3, \lambda_{1}^{'} = 1/5, \lambda_{2} = 1, \lambda_{2}^{'} = 1/3, \lambda_{3} = 1/2, \lambda_{3}^{'} = 1/4, \lambda_{4} = 53/670$
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Alain Bensoussan, Xinwei Feng, Jianhui Huang. Linear-quadratic-Gaussian mean-field-game with partial observation and common noise. Mathematical Control & Related Fields, 2021, 11 (1) : 23-46. doi: 10.3934/mcrf.2020025
[2]

Xiaoxian Tang, Jie Wang. Bistability of sequestration networks. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1337-1357. doi: 10.3934/dcdsb.2020165
[3]

Jie Li, Xiangdong Ye, Tao Yu. Mean equicontinuity, complexity and applications. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 359-393. doi: 10.3934/dcds.2020167
[4]

D. R. Michiel Renger, Johannes Zimmer. Orthogonality of fluxes in general nonlinear reaction networks. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 205-217. doi: 10.3934/dcdss.2020346
[5]

Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344
[6]

Lars Grüne. Computing Lyapunov functions using deep neural networks. Journal of Computational Dynamics, 2020  doi: 10.3934/jcd.2021006
[7]

Pedro Aceves-Sanchez, Benjamin Aymard, Diane Peurichard, Pol Kennel, Anne Lorsignol, Franck Plouraboué, Louis Casteilla, Pierre Degond. A new model for the emergence of blood capillary networks. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2021001
[8]

Leslaw Skrzypek, Yuncheng You. Feedback synchronization of FHN cellular neural networks. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021001
[9]

Mingchao Zhao, You-Wei Wen, Michael Ng, Hongwei Li. A nonlocal low rank model for poisson noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021003
[10]

Hongfei Yang, Xiaofeng Ding, Raymond Chan, Hui Hu, Yaxin Peng, Tieyong Zeng. A new initialization method based on normed statistical spaces in deep networks. Inverse Problems & Imaging, 2021, 15 (1) : 147-158. doi: 10.3934/ipi.2020045
[11]

Charlotte Rodriguez. Networks of geometrically exact beams: Well-posedness and stabilization. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021002
[12]

Illés Horváth, Kristóf Attila Horváth, Péter Kovács, Miklós Telek. Mean-field analysis of a scaling MAC radio protocol. Journal of Industrial & Management Optimization, 2021, 17 (1) : 279-297. doi: 10.3934/jimo.2019111
[13]

Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2021, 17 (1) : 185-204. doi: 10.3934/jimo.2019106
[14]

Laura Aquilanti, Simone Cacace, Fabio Camilli, Raul De Maio. A Mean Field Games model for finite mixtures of Bernoulli and categorical distributions. Journal of Dynamics & Games, 2020  doi: 10.3934/jdg.2020033
[15]

Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324
[16]

Petr Pauš, Shigetoshi Yazaki. Segmentation of color images using mean curvature flow and parametric curves. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1123-1132. doi: 10.3934/dcdss.2020389
[17]

Yu Zhou, Xinfeng Dong, Yongzhuang Wei, Fengrong Zhang. A note on the Signal-to-noise ratio of $ (n, m) $-functions. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020117
[18]

Jie Zhang, Yuping Duan, Yue Lu, Michael K. Ng, Huibin Chang. Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020071
[19]

Jérôme Lohéac, Chaouki N. E. Boultifat, Philippe Chevrel, Mohamed Yagoubi. Exact noise cancellation for 1d-acoustic propagation systems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020055
[20]

Shang Wu, Pengfei Xu, Jianhua Huang, Wei Yan. Ergodicity of stochastic damped Ostrovsky equation driven by white noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1615-1626. doi: 10.3934/dcdsb.2020175

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (112)
  • HTML views (550)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences