doi: 10.3934/dcdsb.2021126

Exact and WKB-approximate distributions in a gene expression model with feedback in burst frequency, burst size, and protein stability

Department of Applied Mathematics and Statistics, Comenius University, Bratislava 842 48, Slovakia

Received  November 2020 Revised  March 2021 Published  April 2021

Fund Project: The author is supported by the Slovak Research and Development Agency under the contract No. APVV-18-0308 and by the VEGA grant 1/0347/18

The expression of individual genes into functional protein molecules is a noisy dynamical process. Here we model the protein concentration as a jump-drift process which combines discrete stochastic production bursts (jumps) with continuous deterministic decay (drift). We allow the drift rate, the jump rate, and the jump size to depend on the protein level to implement feedback in protein stability, burst frequency, and burst size. We specifically focus on positive feedback in burst size, while allowing for arbitrary autoregulation in burst frequency and protein stability. Two versions of feedback in burst size are thereby considered: in the first, newly produced molecules instantly participate in feedback, even within the same burst; in the second, within-burst regulation does not occur due to the so-called infinitesimal delay. Without infinitesimal delay, the model is explicitly solvable; with its inclusion, an exact distribution to the model is unavailable, but we are able to construct a WKB approximation that applies in the asymptotic regime of small but frequent bursts. Comparing the asymptotic behaviour of the two model versions, we report that they yield the same WKB quasi-potential but a different exponential prefactor. We illustrate the difference on the case of a bimodal protein distribution sustained by a sigmoid feedback in burst size: we show that the omission of the infinitesimal delay overestimates the weight of the upper mode of the protein distribution. The analytic results are supported by kinetic Monte-Carlo simulations.

Citation: Pavol Bokes. Exact and WKB-approximate distributions in a gene expression model with feedback in burst frequency, burst size, and protein stability. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021126
References:
[1]

A. A. AlonsoR. BermejoM. Pájaro and C. Vázquez, Numerical analysis of a method for a partial integro-differential equation model in regulatory gene networks, Math. Models Methods Appl. Sci., 28 (2018), 2069-2095.  doi: 10.1142/S0218202518500495.  Google Scholar

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P. BokesJ. R. KingA. T. A. Wood and M. Loose, Multiscale stochastic modelling of gene expression, J. Math. Biol., 65 (2012), 493-520.  doi: 10.1007/s00285-011-0468-7.  Google Scholar

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P. BokesJ. R. KingA. T. A. Wood and M. Loose, Transcriptional bursting diversifies the behaviour of a toggle switch: Hybrid simulation of stochastic gene expression, B. Math. Biol., 75 (2013), 351-371.  doi: 10.1007/s11538-013-9811-z.  Google Scholar

[6]

P. BokesY. T. Lin and A. Singh, High cooperativity in negative feedback can amplify noisy gene expression, B. Math. Biol., 80 (2018), 1871-1899.  doi: 10.1007/s11538-018-0438-y.  Google Scholar

[7]

P. Bokes, Maintaining gene expression levels by positive feedback in burst size in the presence of infinitesimal delay, Discrete Cont. Dyn. Syst. Ser. B, 24 (2019), 5539-5552.  doi: 10.3934/dcdsb.2019070.  Google Scholar

[8]

P. BokesA. BorriP. Palumbo and A. Singh, Mixture distributions in a stochastic gene expression model with delayed feedback: A WKB approximation approach, J. Math. Biol., 81 (2020), 343-367.  doi: 10.1007/s00285-020-01512-y.  Google Scholar

[9]

P. BokesJ. R. KingA. T. A. Wood and M. Loose, Exact and approximate distributions of protein and mRNA levels in the low-copy regime of gene expression, J. Math. Biol., 64 (2012), 829-854.  doi: 10.1007/s00285-011-0433-5.  Google Scholar

[10]

P. Bokes and A. Singh, Protein copy number distributions for a self-regulating gene in the presence of decoy binding sites, PLoS ONE, 10 (2015), e0120555. doi: 10.1371/journal.pone.0120555.  Google Scholar

[11]

P. Bokes and A. Singh, Gene expression noise is affected differentially by feedback in burst frequency and burst size, J. Math. Biol., 74 (2017), 1483-1509.  doi: 10.1007/s00285-016-1059-4.  Google Scholar

[12]

P. Bokes and A. Singh, Controlling noisy expression through auto regulation of burst frequency and protein stability, in Češka M., Paoletti N. (eds) Hybrid Systems Biology. HSB 2019. Lecture Notes in Computer Science, vol 11705, Springer, Cham, 2019. doi: 10.1007/978-3-030-28042-0_6.  Google Scholar

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A. CruduA. DebusscheA. Muller and O. Radulescu, Convergence of stochastic gene networks to hybrid piecewise deterministic processes, Ann. Appl. Probab., 22 (2012), 1822-1859.  doi: 10.1214/11-AAP814.  Google Scholar

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M. B. ElowitzA. J. LevineE. D. Siggia and P. S. Swain, Stochastic gene expression in a single cell, Science, 297 (2002), 1183-1186.  doi: 10.1126/science.1070919.  Google Scholar

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N. Friedman, L. Cai and X. S. Xie, Linking stochastic dynamics to population distribution: an analytical framework of gene expression, Phys. Rev. Lett., 97 (2006), 168302. doi: 10.1103/PhysRevLett.97.168302.  Google Scholar

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H. Ge, H. Qian and X. S. Xie, Stochastic phenotype transition of a single cell in an intermediate region of gene state switching, Phys. Rev. Lett., 114 (2015), 078101. doi: 10.1103/PhysRevLett.114.078101.  Google Scholar

[21]

L. Ham, R. D. Brackston and M. P. Stumpf, Extrinsic noise and heavy-tailed laws in gene expression, Phys. Rev. Lett., 124 (2020), 108101. doi: 10.1103/PhysRevLett.124.108101.  Google Scholar

[22]

M. A. HernandezB. PatelF. HeyS. GiblettH. Davis and C. Pritchard, Regulation of BRAF protein stability by a negative feedback loop involving the MEK-ERK pathway but not the FBXW7 tumour suppressor, Cell. Signal., 28 (2016), 561-571.  doi: 10.1016/j.cellsig.2016.02.009.  Google Scholar

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E. J. Hinch, Perturbation Methods, Cambridge University Press, 1991. doi: 10.1017/CBO9781139172189.  Google Scholar

[24]

R. Hinch and S. J. Chapman, Exponentially slow transitions on a Markov chain: The frequency of calcium sparks, Eur. J. Appl. Math., 16 (2005), 427-446.  doi: 10.1017/S0956792505006194.  Google Scholar

[25]

J. Holehouse, Z. Cao and R. Grima, Stochastic modeling of auto-regulatory genetic feedback loops: A review and comparative study, Biophys. J. Google Scholar

[26]

P. G. Hufton, Y. T. Lin, T. Galla and A. J. McKane, Intrinsic noise in systems with switching environments, Phys. Rev. E, 93 (2016), 052119. doi: 10.1103/PhysRevE.93.052119.  Google Scholar

[27]

G. C. P. InnocentiniM. ForgerO. Radulescu and F. Antoneli, Protein synthesis driven by dynamical stochastic transcription, B. Math. Biol., 78 (2016), 110-131.  doi: 10.1007/s11538-015-0131-3.  Google Scholar

[28]

J. Jȩdrak, M. Kwiatkowski and A. Ochab-Marcinek, Exactly solvable model of gene expression in a proliferating bacterial cell population with stochastic protein bursts and protein partitioning, Phys. Rev. E, 99 (2019), 042416. doi: 10.1103/PhysRevE.99.042416.  Google Scholar

[29]

J. Jȩdrak and A. Ochab-Marcinek, Influence of gene copy number on self-regulated gene expression, J. Theor. Biol., 408 (2016), 222-236.  doi: 10.1016/j.jtbi.2016.08.018.  Google Scholar

[30]

J. Jȩdrak and A. Ochab-Marcinek, Time-dependent solutions for a stochastic model of gene expression with molecule production in the form of a compound poisson process, Phys. Rev. E, 94 (2016), 032401. doi: 10.1103/PhysRevE.94.032401.  Google Scholar

[31]

C. Jia and R. Grima, Dynamical phase diagram of an auto-regulating gene in fast switching conditions, J. Chem. Phys., 152 (2020), 174110. doi: 10.1063/5.0007221.  Google Scholar

[32]

A. Kozdeba and A. Tomski, Application of the goodwin model to autoregulatory feedback for stochastic gene expression, Math. Biosci., 327 (2020), 108413. doi: 10.1016/j.mbs.2020.108413.  Google Scholar

[33]

P. KurasovA. LückD. Mugnolo and V. Wolf, Stochastic hybrid models of gene regulatory networks - a PDE approach, Math. Biosci., 305 (2018), 170-177.  doi: 10.1016/j.mbs.2018.09.009.  Google Scholar

[34]

D. R. LarsonR. H. Singer and D. Zenklusen, A single molecule view of gene expression, Trends Cell Biol., 19 (2009), 630-637.  doi: 10.1016/j.tcb.2009.08.008.  Google Scholar

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G.-W. Li and X. S. Xie, Central dogma at the single-molecule level in living cells, Nature, 475 (2011), 308-315.  doi: 10.1038/nature10315.  Google Scholar

[36]

Y. T. Lin and C. R. Doering, Gene expression dynamics with stochastic bursts: Construction and exact results for a coarse-grained model, Phys. Rev. E, 93 (2016), 022409. doi: 10.1103/physreve.93.022409.  Google Scholar

[37]

Y. T. Lin and T. Galla, Bursting noise in gene expression dynamics: Linking microscopic and mesoscopic models, J. Roy. Soc. Interface, 13 (2016), 20150772. doi: 10.1098/rsif.2015.0772.  Google Scholar

[38]

T. LipniackiP. PaszekA. Marciniak-CzochraA. R. Brasier and M. Kimmel, Transcriptional stochasticity in gene expression, J. Theor. Biol., 238 (2006), 348-67.  doi: 10.1016/j.jtbi.2005.05.032.  Google Scholar

[39]

M. Masujima, Applied Mathematical Methods in Theoretical Physics, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2009. doi: 10.1002/9783527627745.  Google Scholar

[40]

A. H. Nayfeh, Introduction to Perturbation Techniques, John Wiley & Sons, New York, 1981.  Google Scholar

[41]

J. Newby and J. Chapman, Metastable behavior in Markov processes with internal states, J. Math. Biol., 69 (2014), 941-976.  doi: 10.1007/s00285-013-0723-1.  Google Scholar

[42]

M. PájaroA. A. AlonsoI. Otero-Muras and C. Vázquez, Stochastic modeling and numerical simulation of gene regulatory networks with protein bursting, J. Theor. Biol., 421 (2017), 51-70.  doi: 10.1016/j.jtbi.2017.03.017.  Google Scholar

[43]

J. Rodriguez and D. R. Larson, Transcription in living cells: Molecular mechanisms of bursting, Annu. Rev. Biochem., 89 (2020), 189-212.  doi: 10.1146/annurev-biochem-011520-105250.  Google Scholar

[44]

M. A. Schikora-TamaritC. Toscano-OchoaJ. D. EspinosL. Espinar and L. B. Carey, A synthetic gene circuit for measuring autoregulatory feedback control, Integr. Biol., 8 (2016), 546-555.  doi: 10.1039/C5IB00230C.  Google Scholar

[45]

L. SchuhM. Saint-AntoineE. M. SanfordB. L. EmertA. SinghC. MarrA. Raj and Y. Goyal, Gene networks with transcriptional bursting recapitulate rare transient coordinated high expression states in cancer, Cell Syst., 10 (2020), 363-378.  doi: 10.1016/j.cels.2020.03.004.  Google Scholar

[46]

Z. Schuss, Theory and Applications of Stochastic Processes: An Analytical Approach, Springer Science & Business Media, Berlin/Heidelberg, 2009. Google Scholar

[47]

M. Soltani, P. Bokes, Z. Fox and A. Singh, Nonspecific transcription factor binding can reduce noise in the expression of downstream proteins, Phys. Biol., 12 (2015), 055002. doi: 10.1088/1478-3975/12/5/055002.  Google Scholar

[48]

A. Sundqvist and J. Ericsson, Transcription-dependent degradation controls the stability of the srebp family of transcription factors, P. Natl. Acad. Sci. USA., 100 (2003), 13833-13838. doi: 10.1073/pnas.2335135100.  Google Scholar

[49]

D. M. SuterN. MolinaD. GatfieldK. SchneiderU. Schibler and F. Naef, Mammalian genes are transcribed with widely different bursting kinetics, Science, 332 (2011), 472-474.  doi: 10.1126/science.1198817.  Google Scholar

[50]

Z. Vahdat, K. Nienałtowski, Z. Farooq, M. Komorowski and A. Singh, Information processing in unregulated and autoregulated gene expression, in 2020 European Control Conference (ECC), IEEE, 2020,258-263. doi: 10.23919/ECC51009.2020.9143689.  Google Scholar

[51]

N. G. van Kampen, Stochastic Processes in Physics and Chemistry, Lecture Notes in Mathematics, 888. North-Holland Publishing Co., Amsterdam-New York, 1981.  Google Scholar

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B. XuH.-W. Kang and A. Jilkine, Comparison of deterministic and stochastic regime in a model for Cdc42 oscillations in fission yeast, B. Math. Biol., 81 (2019), 1268-1302.  doi: 10.1007/s11538-019-00573-5.  Google Scholar

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show all references

References:
[1]

A. A. AlonsoR. BermejoM. Pájaro and C. Vázquez, Numerical analysis of a method for a partial integro-differential equation model in regulatory gene networks, Math. Models Methods Appl. Sci., 28 (2018), 2069-2095.  doi: 10.1142/S0218202518500495.  Google Scholar

[2]

A. Andreychenko, L. Bortolussi, R. Grima, P. Thomas and V. Wolf, Distribution approximations for the chemical master equation: Comparison of the method of moments and the system size expansion, in Modeling Cellular Systems, Springer, 2017, 39-66.  Google Scholar

[3]

M. Assaf and B. Meerson, WKB theory of large deviations in stochastic populations, J. Phys. A: Math. Theor., 50 (2017), 263001. doi: 10.1088/1751-8121/aa669a.  Google Scholar

[4]

P. BokesJ. R. KingA. T. A. Wood and M. Loose, Multiscale stochastic modelling of gene expression, J. Math. Biol., 65 (2012), 493-520.  doi: 10.1007/s00285-011-0468-7.  Google Scholar

[5]

P. BokesJ. R. KingA. T. A. Wood and M. Loose, Transcriptional bursting diversifies the behaviour of a toggle switch: Hybrid simulation of stochastic gene expression, B. Math. Biol., 75 (2013), 351-371.  doi: 10.1007/s11538-013-9811-z.  Google Scholar

[6]

P. BokesY. T. Lin and A. Singh, High cooperativity in negative feedback can amplify noisy gene expression, B. Math. Biol., 80 (2018), 1871-1899.  doi: 10.1007/s11538-018-0438-y.  Google Scholar

[7]

P. Bokes, Maintaining gene expression levels by positive feedback in burst size in the presence of infinitesimal delay, Discrete Cont. Dyn. Syst. Ser. B, 24 (2019), 5539-5552.  doi: 10.3934/dcdsb.2019070.  Google Scholar

[8]

P. BokesA. BorriP. Palumbo and A. Singh, Mixture distributions in a stochastic gene expression model with delayed feedback: A WKB approximation approach, J. Math. Biol., 81 (2020), 343-367.  doi: 10.1007/s00285-020-01512-y.  Google Scholar

[9]

P. BokesJ. R. KingA. T. A. Wood and M. Loose, Exact and approximate distributions of protein and mRNA levels in the low-copy regime of gene expression, J. Math. Biol., 64 (2012), 829-854.  doi: 10.1007/s00285-011-0433-5.  Google Scholar

[10]

P. Bokes and A. Singh, Protein copy number distributions for a self-regulating gene in the presence of decoy binding sites, PLoS ONE, 10 (2015), e0120555. doi: 10.1371/journal.pone.0120555.  Google Scholar

[11]

P. Bokes and A. Singh, Gene expression noise is affected differentially by feedback in burst frequency and burst size, J. Math. Biol., 74 (2017), 1483-1509.  doi: 10.1007/s00285-016-1059-4.  Google Scholar

[12]

P. Bokes and A. Singh, Controlling noisy expression through auto regulation of burst frequency and protein stability, in Češka M., Paoletti N. (eds) Hybrid Systems Biology. HSB 2019. Lecture Notes in Computer Science, vol 11705, Springer, Cham, 2019. doi: 10.1007/978-3-030-28042-0_6.  Google Scholar

[13]

P. C. Bressloff and J. M. Newby, Metastability in a stochastic neural network modeled as a velocity jump Markov process, SIAM J. Appl. Dyn. Syst., 12 (2013), 1394-1435. doi: 10.1137/120898978.  Google Scholar

[14]

J. A. CañizoJ. A. Carrillo and M. Pájaro, Exponential equilibration of genetic circuits using entropy methods, J. Math. Biol., 78 (2019), 373-411.  doi: 10.1007/s00285-018-1277-z.  Google Scholar

[15]

D. R. Cox and D. Oakes, Analysis of Survival Data, Chapman & Hall/CRC, 1984.  Google Scholar

[16]

A. CruduA. DebusscheA. Muller and O. Radulescu, Convergence of stochastic gene networks to hybrid piecewise deterministic processes, Ann. Appl. Probab., 22 (2012), 1822-1859.  doi: 10.1214/11-AAP814.  Google Scholar

[17]

J. Dattani and M. Barahona, Stochastic models of gene transcription with upstream drives: exact solution and sample path characterization, J. Roy. Soc. Interface, 14 (2017), 20160833. doi: 10.1098/rsif.2016.0833.  Google Scholar

[18]

M. B. ElowitzA. J. LevineE. D. Siggia and P. S. Swain, Stochastic gene expression in a single cell, Science, 297 (2002), 1183-1186.  doi: 10.1126/science.1070919.  Google Scholar

[19]

N. Friedman, L. Cai and X. S. Xie, Linking stochastic dynamics to population distribution: an analytical framework of gene expression, Phys. Rev. Lett., 97 (2006), 168302. doi: 10.1103/PhysRevLett.97.168302.  Google Scholar

[20]

H. Ge, H. Qian and X. S. Xie, Stochastic phenotype transition of a single cell in an intermediate region of gene state switching, Phys. Rev. Lett., 114 (2015), 078101. doi: 10.1103/PhysRevLett.114.078101.  Google Scholar

[21]

L. Ham, R. D. Brackston and M. P. Stumpf, Extrinsic noise and heavy-tailed laws in gene expression, Phys. Rev. Lett., 124 (2020), 108101. doi: 10.1103/PhysRevLett.124.108101.  Google Scholar

[22]

M. A. HernandezB. PatelF. HeyS. GiblettH. Davis and C. Pritchard, Regulation of BRAF protein stability by a negative feedback loop involving the MEK-ERK pathway but not the FBXW7 tumour suppressor, Cell. Signal., 28 (2016), 561-571.  doi: 10.1016/j.cellsig.2016.02.009.  Google Scholar

[23]

E. J. Hinch, Perturbation Methods, Cambridge University Press, 1991. doi: 10.1017/CBO9781139172189.  Google Scholar

[24]

R. Hinch and S. J. Chapman, Exponentially slow transitions on a Markov chain: The frequency of calcium sparks, Eur. J. Appl. Math., 16 (2005), 427-446.  doi: 10.1017/S0956792505006194.  Google Scholar

[25]

J. Holehouse, Z. Cao and R. Grima, Stochastic modeling of auto-regulatory genetic feedback loops: A review and comparative study, Biophys. J. Google Scholar

[26]

P. G. Hufton, Y. T. Lin, T. Galla and A. J. McKane, Intrinsic noise in systems with switching environments, Phys. Rev. E, 93 (2016), 052119. doi: 10.1103/PhysRevE.93.052119.  Google Scholar

[27]

G. C. P. InnocentiniM. ForgerO. Radulescu and F. Antoneli, Protein synthesis driven by dynamical stochastic transcription, B. Math. Biol., 78 (2016), 110-131.  doi: 10.1007/s11538-015-0131-3.  Google Scholar

[28]

J. Jȩdrak, M. Kwiatkowski and A. Ochab-Marcinek, Exactly solvable model of gene expression in a proliferating bacterial cell population with stochastic protein bursts and protein partitioning, Phys. Rev. E, 99 (2019), 042416. doi: 10.1103/PhysRevE.99.042416.  Google Scholar

[29]

J. Jȩdrak and A. Ochab-Marcinek, Influence of gene copy number on self-regulated gene expression, J. Theor. Biol., 408 (2016), 222-236.  doi: 10.1016/j.jtbi.2016.08.018.  Google Scholar

[30]

J. Jȩdrak and A. Ochab-Marcinek, Time-dependent solutions for a stochastic model of gene expression with molecule production in the form of a compound poisson process, Phys. Rev. E, 94 (2016), 032401. doi: 10.1103/PhysRevE.94.032401.  Google Scholar

[31]

C. Jia and R. Grima, Dynamical phase diagram of an auto-regulating gene in fast switching conditions, J. Chem. Phys., 152 (2020), 174110. doi: 10.1063/5.0007221.  Google Scholar

[32]

A. Kozdeba and A. Tomski, Application of the goodwin model to autoregulatory feedback for stochastic gene expression, Math. Biosci., 327 (2020), 108413. doi: 10.1016/j.mbs.2020.108413.  Google Scholar

[33]

P. KurasovA. LückD. Mugnolo and V. Wolf, Stochastic hybrid models of gene regulatory networks - a PDE approach, Math. Biosci., 305 (2018), 170-177.  doi: 10.1016/j.mbs.2018.09.009.  Google Scholar

[34]

D. R. LarsonR. H. Singer and D. Zenklusen, A single molecule view of gene expression, Trends Cell Biol., 19 (2009), 630-637.  doi: 10.1016/j.tcb.2009.08.008.  Google Scholar

[35]

G.-W. Li and X. S. Xie, Central dogma at the single-molecule level in living cells, Nature, 475 (2011), 308-315.  doi: 10.1038/nature10315.  Google Scholar

[36]

Y. T. Lin and C. R. Doering, Gene expression dynamics with stochastic bursts: Construction and exact results for a coarse-grained model, Phys. Rev. E, 93 (2016), 022409. doi: 10.1103/physreve.93.022409.  Google Scholar

[37]

Y. T. Lin and T. Galla, Bursting noise in gene expression dynamics: Linking microscopic and mesoscopic models, J. Roy. Soc. Interface, 13 (2016), 20150772. doi: 10.1098/rsif.2015.0772.  Google Scholar

[38]

T. LipniackiP. PaszekA. Marciniak-CzochraA. R. Brasier and M. Kimmel, Transcriptional stochasticity in gene expression, J. Theor. Biol., 238 (2006), 348-67.  doi: 10.1016/j.jtbi.2005.05.032.  Google Scholar

[39]

M. Masujima, Applied Mathematical Methods in Theoretical Physics, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2009. doi: 10.1002/9783527627745.  Google Scholar

[40]

A. H. Nayfeh, Introduction to Perturbation Techniques, John Wiley & Sons, New York, 1981.  Google Scholar

[41]

J. Newby and J. Chapman, Metastable behavior in Markov processes with internal states, J. Math. Biol., 69 (2014), 941-976.  doi: 10.1007/s00285-013-0723-1.  Google Scholar

[42]

M. PájaroA. A. AlonsoI. Otero-Muras and C. Vázquez, Stochastic modeling and numerical simulation of gene regulatory networks with protein bursting, J. Theor. Biol., 421 (2017), 51-70.  doi: 10.1016/j.jtbi.2017.03.017.  Google Scholar

[43]

J. Rodriguez and D. R. Larson, Transcription in living cells: Molecular mechanisms of bursting, Annu. Rev. Biochem., 89 (2020), 189-212.  doi: 10.1146/annurev-biochem-011520-105250.  Google Scholar

[44]

M. A. Schikora-TamaritC. Toscano-OchoaJ. D. EspinosL. Espinar and L. B. Carey, A synthetic gene circuit for measuring autoregulatory feedback control, Integr. Biol., 8 (2016), 546-555.  doi: 10.1039/C5IB00230C.  Google Scholar

[45]

L. SchuhM. Saint-AntoineE. M. SanfordB. L. EmertA. SinghC. MarrA. Raj and Y. Goyal, Gene networks with transcriptional bursting recapitulate rare transient coordinated high expression states in cancer, Cell Syst., 10 (2020), 363-378.  doi: 10.1016/j.cels.2020.03.004.  Google Scholar

[46]

Z. Schuss, Theory and Applications of Stochastic Processes: An Analytical Approach, Springer Science & Business Media, Berlin/Heidelberg, 2009. Google Scholar

[47]

M. Soltani, P. Bokes, Z. Fox and A. Singh, Nonspecific transcription factor binding can reduce noise in the expression of downstream proteins, Phys. Biol., 12 (2015), 055002. doi: 10.1088/1478-3975/12/5/055002.  Google Scholar

[48]

A. Sundqvist and J. Ericsson, Transcription-dependent degradation controls the stability of the srebp family of transcription factors, P. Natl. Acad. Sci. USA., 100 (2003), 13833-13838. doi: 10.1073/pnas.2335135100.  Google Scholar

[49]

D. M. SuterN. MolinaD. GatfieldK. SchneiderU. Schibler and F. Naef, Mammalian genes are transcribed with widely different bursting kinetics, Science, 332 (2011), 472-474.  doi: 10.1126/science.1198817.  Google Scholar

[50]

Z. Vahdat, K. Nienałtowski, Z. Farooq, M. Komorowski and A. Singh, Information processing in unregulated and autoregulated gene expression, in 2020 European Control Conference (ECC), IEEE, 2020,258-263. doi: 10.23919/ECC51009.2020.9143689.  Google Scholar

[51]

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Figure 1.  Upper Left: The model includes bursty protein production and continuous protein decay, and allows for feedback in burst frequency, burst size, and protein stability, as quantified by functions $ \alpha(x) $, $ \beta(x) $, and $ \gamma(x) $, respectively. Bottom Left: Bursts lead to an instantaneous increases in protein concentration; between bursts protein concentration decays continuously. Bottom Right: In the deterministic limit $ \varepsilon\to0 $, the concentration changes per unit time by the difference of the production rate $ \alpha(x)\beta(x) $ (solid line) and the decay rate $ \gamma(x) $ (dotted line). The intersections of the two are the fixed points (FPTs) of the deterministic model. Upper Right: The distribution potential (16) is a Lyapunov function of the deterministic model: it possesses minima/maxima where the deterministic model exhibits stable/unstable points. The depicted example pertains to feedback in burst size, with $ \gamma(x) = x $, $ \alpha(x)\equiv 1 $, $ \beta(x) = 0.4 + 1.6x^4/(1 + x^4) $
Figure 2.  Left: For small values of the noise parameter $ \varepsilon $, the sample paths of the stochastic model (coloured curves) are close to the solutions of the deterministic model (black curves). The stable/unstable fixed points of the deterministic model (14) are shown as dashed/dotted horizontal lines. Right: Transitions between the basins of attractions of the stable steady states occur on an extremely slow timescale. Parameter values: Feedback is in burst size, with $ \gamma(x) = x $, $ \alpha(x)\equiv 1 $, $ \beta(x) = 0.4 + 1.6x^4/(1 + x^4) $. The noise parameter is varied in the left panel and fixed to $ \varepsilon = 0.05 $ in the right panel
Figure 3.  Upper Panels: Simulation-based time-dependent distributions (coloured curves) approach, as simulation time increases, the WKB stationary distribution (dashed black curve). This is markedly different from the exact stationary distribution of the model without infinitesimal delay (dotted black curve). The initial condition is $ x_0 = 0 $ (left panel) and $ x_0 = 3 $ (right panel). Bottom Panels: Large-time simulation-based distributions (solid curve) are compared to the WKB approximation (dashed curve). Parameter values: Feedback is in burst size, with $ \gamma(x) = x $, $ \alpha(x)\equiv 1 $, $ \beta(x) = 0.4 + 1.6x^4/(1 + x^4) $, except the bottom right panel, where $ \beta(x) = 0.4 + 2.6x^4/(1 + x^4) $. The noise parameter is fixed to $ \varepsilon = 0.05 $ in the upper panels; in the bottom panels, it assumes values that are specified in the inset
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