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]

Qi Wang, Lifang Huang, Kunwen Wen, Jianshe Yu. The mean and noise of stochastic gene transcription with cell division. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1255-1270. doi: 10.3934/mbe.2018058
[2]

Ö. Uğur, G. W. Weber. Optimization and dynamics of gene-environment networks with intervals. Journal of Industrial & Management Optimization, 2007, 3 (2) : 357-379. doi: 10.3934/jimo.2007.3.357
[3]

Erik Kropat, Gerhard-Wilhelm Weber, Erfan Babaee Tirkolaee. Foundations of semialgebraic gene-environment networks. Journal of Dynamics & Games, 2020  doi: 10.3934/jdg.2020018
[4]

Roberto Serra, Marco Villani, Alex Graudenzi, Annamaria Colacci, Stuart A. Kauffman. The simulation of gene knock-out in scale-free random Boolean models of genetic networks. Networks & Heterogeneous Media, 2008, 3 (2) : 333-343. doi: 10.3934/nhm.2008.3.333
[5]

Yves Achdou, Manh-Khang Dao, Olivier Ley, Nicoletta Tchou. A class of infinite horizon mean field games on networks. Networks & Heterogeneous Media, 2019, 14 (3) : 537-566. doi: 10.3934/nhm.2019021
[6]

Fabio Camilli, Elisabetta Carlini, Claudio Marchi. A model problem for Mean Field Games on networks. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4173-4192. doi: 10.3934/dcds.2015.35.4173
[7]

Zhen Li, Jicheng Liu. Synchronization for stochastic differential equations with nonlinear multiplicative noise in the mean square sense. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5709-5736. doi: 10.3934/dcdsb.2019103
[8]

Liliana Borcea, Fernando Guevara Vasquez, Alexander V. Mamonov. Study of noise effects in electrical impedance tomography with resistor networks. Inverse Problems & Imaging, 2013, 7 (2) : 417-443. doi: 10.3934/ipi.2013.7.417
[9]

Ziheng Chen, Siqing Gan, Xiaojie Wang. Mean-square approximations of Lévy noise driven SDEs with super-linearly growing diffusion and jump coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4513-4545. doi: 10.3934/dcdsb.2019154
[10]

Alain Bensoussan, Xinwei Feng, Jianhui Huang. Linear-quadratic-Gaussian mean-field-game with partial observation and common noise. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020025
[11]

Stamatios Katsikas, Vassilli Kolokoltsov. Evolutionary, mean-field and pressure-resistance game modelling of networks security. Journal of Dynamics & Games, 2019, 6 (4) : 315-335. doi: 10.3934/jdg.2019021
[12]

Jian Ren, Feng Jiao, Qiwen Sun, Moxun Tang, Jianshe Yu. The dynamics of gene transcription in random environments. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3167-3194. doi: 10.3934/dcdsb.2018224
[13]

Quan Hai, Shutang Liu. Mean-square delay-distribution-dependent exponential synchronization of chaotic neural networks with mixed random time-varying delays and restricted disturbances. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020221
[14]

Shunfu Jin, Haixing Wu, Wuyi Yue, Yutaka Takahashi. Performance evaluation and Nash equilibrium of a cloud architecture with a sleeping mechanism and an enrollment service. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2407-2424. doi: 10.3934/jimo.2019060
[15]

Ying Hao, Fanwen Meng. A new method on gene selection for tissue classification. Journal of Industrial & Management Optimization, 2007, 3 (4) : 739-748. doi: 10.3934/jimo.2007.3.739
[16]

Jian-Wu Xue, Xiao-Kun Xu, Feng Zhang. Big data dynamic compressive sensing system architecture and optimization algorithm for internet of things. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1401-1414. doi: 10.3934/dcdss.2015.8.1401
[17]

Gokhan Calis, O. Ozan Koyluoglu. Architecture-aware coding for distributed storage: Repairable block failure resilient codes. Advances in Mathematics of Communications, 2018, 12 (3) : 465-503. doi: 10.3934/amc.2018028
[18]

Zizi Wang, Zhiming Guo, Huaqin Peng. Dynamical behavior of a new oncolytic virotherapy model based on gene variation. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1079-1093. doi: 10.3934/dcdss.2017058
[19]

Baltazar D. Aguda, Ricardo C.H. del Rosario, Michael W.Y. Chan. Oncogene-tumor suppressor gene feedback interactions and their control. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1277-1288. doi: 10.3934/mbe.2015.12.1277
[20]

Marek Bodnar. Distributed delays in Hes1 gene expression model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2125-2147. doi: 10.3934/dcdsb.2019087

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (102)
  • HTML views (548)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences