# American Institute of Mathematical Sciences

## Maintaining gene expression levels by positive feedback in burst size in the presence of infinitesimal delay

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

Received  July 2018 Revised  November 2018 Published  April 2019

Fund Project: The author is supported by the Slovak Research and Development Agency under the contract No. APVV-14-0378 and also by the VEGA grant 1/0347/18

Synthesis of individual molecules in the expression of genes often occurs in bursts of multiple copies. Gene regulatory feedback can affect the frequency with which these bursts occur or their size. Whereas frequency regulation has traditionally received more attention, we focus specifically on the regulation of burst size. It turns out that there are (at least) two alternative formulations of feedback in burst size. In the first, newly produced molecules immediately partake in feedback, even within the same burst. In the second, there is no within-burst regulation due to what we call infinitesimal delay. We describe both alternatives using a minimalistic Markovian drift-jump framework combining discrete and continuous dynamics. We derive detailed analytic results and efficient simulation algorithms for positive non-cooperative autoregulation (whether infinitesimally delayed or not). We show that at steady state both alternatives lead to a gamma distribution of protein level. The steady-state distribution becomes available only after a transcritical bifurcation point is passed. Interestingly, the onset of the bifurcation is postponed by the inclusion of infinitesimal delay.

Citation: Pavol Bokes. Maintaining gene expression levels by positive feedback in burst size in the presence of infinitesimal delay. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019070
##### References:
 [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Washington, D.C., 1972. [2] B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts and P. Walter, Molecular Biology of the Cell, Garland Science New York, 2002. [3] U. Alon, An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman & Hall/CRC, 2007. [4] M. Barrio, K. Burrage, A. Leier and T. Tian, Oscillatory regulation of hes1: Discrete stochastic delay modelling and simulation, PLoS Comput. Biol., 2 (2006), e117. [5] 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. [6] P. Bokes, Y. 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. [7] P. Bokes, J. R. King, A. 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. [8] L. Cai, N. Friedman and X. S. Xie, Stochastic protein expression in individual cells at the single molecule level, Nature, 440 (2006), 358-362. [9] A. Crudu, A. Debussche, A. Muller and O. Radulescu, others, Convergence of stochastic gene networks to hybrid piecewise deterministic processes, Ann. Appl. Probab., 22 (2012), 1822Ƀ1859. doi: 10.1214/11-AAP814. [10] P. Czuppon and P. Pfaffelhuber, Limits of noise for autoregulated gene expression, J. Math. Biol., 77 (2018), 1153-1191. doi: 10.1007/s00285-018-1248-4. [11] R. D. Dar, B. S. Razooky, A. Singh, T. V. Trimeloni, J. M. McCollum, C. D. Cox, M. L. Simpson and L. S. Weinberger, Transcriptional burst frequency and burst size are equally modulated across the human genome, P. Natl. Acad. Sci. USA, 109 (2012), 17454-17459. [12] R. Dessalles, V. Fromion and P. Robert, A stochastic analysis of autoregulation of gene expression, J. Math. Biol., 75 (2017), 1253-1283. doi: 10.1007/s00285-017-1116-7. [13] M. B. Elowitz, A. J. Levine, E. D. Siggia and P. S. Swain, Stochastic gene expression in a single cell, Science, 297 (2002), 1183-1186. [14] 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. [15] M. W. Hirsch, S. Smale and R. L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Academic Press, 2004. [16] G. C. P. Innocentini, M. Forger, O. 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. [17] G. C. P. Innocentini, S. Guiziou, J. Bonnet and O. Radulescu, Analytic framework for a stochastic binary biological switch, Phys. Rev. E, 94 (2016), 062413. [18] S. Intep and D. J. Higham, Zero, one and two-switch models of gene regulation, Discrete Cont. Dyn-B., 14 (2010), 495-513. doi: 10.3934/dcdsb.2010.14.495. [19] J. Jedrak 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. [20] C. Jia, H. Qian, M. Chen and M. Q. Zhang, Relaxation rates of gene expression kinetics reveal the feedback signs of autoregulatory gene networks, J. Chem. Phys., 148 (2018), 095102. [21] N. Kumar, T. Platini and R. V. Kulkarni, Exact distributions for stochastic gene expression models with bursting and feedback, Phys. Rev. Lett., 113 (2014), 268105. [22] D. R. Larson, R. H. Singer and D. Zenklusen, A single molecule view of gene expression, Trends Cell Biol., 19 (2009), 630-637. [23] G. Lin, J. Yu, Z. Zhou, Q. Sun and F. Jiao, Fluctuations of mrna distributions in multiple pathway activated transcription, Discrete Cont. Dyn-B., 2018. [24] Y. T. Lin and N. E. Buchler, Efficient analysis of stochastic gene dynamics in the non-adiabatic regime using piecewise deterministic markov processes, J. Roy. Soc. Interface, 15 (2018), 20170804. [25] Y. T. Lin, P. G. Hufton, E. J. Lee and D. A. Potoyan, A stochastic and dynamical view of pluripotency in mouse embryonic stem cells, PloS Comput. Biol., 14 (2018), e1006000. [26] 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, 10pp. doi: 10.1103/physreve.93.022409. [27] M. Masujima, Applied Mathematical Methods in Theoretical Physics, John Wiley & Sons, 2009. doi: 10.1002/9783527627745. [28] N. A. M. Monk, Oscillatory expression of hes1, p53, and nf-$\kappa$b driven by transcriptional time delays, Curr. Biol., 13 (2003), 1409-1413. [29] M. Pájaro, A. A. Alonso, I. 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. [30] J. Peccoud and B. Ycart, Markovian modeling of gene-product synthesis, Theor. Popul. Biol., 48 (1995), 222-234. [31] A. Raj, C. S. Peskin, D. Tranchina, D. Y. Vargas and S. Tyagi, Stochastic mRNA synthesis in mammalian cells, PLoS Biol., 4 (2006), e309. [32] J. Ren, F. Jiao, Q. Sun, M. Tang and J. Yu, The dynamics of gene transcription in random environments, Discrete Cont. Dyn-B., 23 (2018), 3167-3194. doi: 10.3934/dcdsb.2018224. [33] R. Rudnicki and M. Tyran-Kamińska, Piecewise Deterministic Processes in Biological Models, Springer, 2017. doi: 10.1007/978-3-319-61295-9. [34] M. A. Schikora-Tamarit, C. Toscano-Ochoa, J. D. Espinos, L. Espinar and L. B. Carey, A synthetic gene circuit for measuring autoregulatory feedback control, Integr. Biol., 8 (2016), 546-555. [35] F. Veerman, C. Marr and N. Popović, Time-dependent propagators for stochastic models of gene expression: an analytical method, J. Math. Biol., 77 (2018), 261-312. doi: 10.1007/s00285-017-1196-4. [36] H. Wang, P. Liu, Q. Li and T. Zhou, Entangled signal pathways can both control expression stability and induce stochastic focusing, FEBS Lett., 592 (2018), 1135-1149. [37] S. Winkelmann and C. Schütte, Hybrid models for chemical reaction networks: Multiscale theory and application to gene regulatory systems, J. Chem. Phys., 147 (2017), 114115.

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##### References:
 [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Washington, D.C., 1972. [2] B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts and P. Walter, Molecular Biology of the Cell, Garland Science New York, 2002. [3] U. Alon, An Introduction to Systems Biology: Design Principles of Biological Circuits, Chapman & Hall/CRC, 2007. [4] M. Barrio, K. Burrage, A. Leier and T. Tian, Oscillatory regulation of hes1: Discrete stochastic delay modelling and simulation, PLoS Comput. Biol., 2 (2006), e117. [5] 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. [6] P. Bokes, Y. 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. [7] P. Bokes, J. R. King, A. 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. [8] L. Cai, N. Friedman and X. S. Xie, Stochastic protein expression in individual cells at the single molecule level, Nature, 440 (2006), 358-362. [9] A. Crudu, A. Debussche, A. Muller and O. Radulescu, others, Convergence of stochastic gene networks to hybrid piecewise deterministic processes, Ann. Appl. Probab., 22 (2012), 1822Ƀ1859. doi: 10.1214/11-AAP814. [10] P. Czuppon and P. Pfaffelhuber, Limits of noise for autoregulated gene expression, J. Math. Biol., 77 (2018), 1153-1191. doi: 10.1007/s00285-018-1248-4. [11] R. D. Dar, B. S. Razooky, A. Singh, T. V. Trimeloni, J. M. McCollum, C. D. Cox, M. L. Simpson and L. S. Weinberger, Transcriptional burst frequency and burst size are equally modulated across the human genome, P. Natl. Acad. Sci. USA, 109 (2012), 17454-17459. [12] R. Dessalles, V. Fromion and P. Robert, A stochastic analysis of autoregulation of gene expression, J. Math. Biol., 75 (2017), 1253-1283. doi: 10.1007/s00285-017-1116-7. [13] M. B. Elowitz, A. J. Levine, E. D. Siggia and P. S. Swain, Stochastic gene expression in a single cell, Science, 297 (2002), 1183-1186. [14] 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. [15] M. W. Hirsch, S. Smale and R. L. Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Academic Press, 2004. [16] G. C. P. Innocentini, M. Forger, O. 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. [17] G. C. P. Innocentini, S. Guiziou, J. Bonnet and O. Radulescu, Analytic framework for a stochastic binary biological switch, Phys. Rev. E, 94 (2016), 062413. [18] S. Intep and D. J. Higham, Zero, one and two-switch models of gene regulation, Discrete Cont. Dyn-B., 14 (2010), 495-513. doi: 10.3934/dcdsb.2010.14.495. [19] J. Jedrak 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. [20] C. Jia, H. Qian, M. Chen and M. Q. Zhang, Relaxation rates of gene expression kinetics reveal the feedback signs of autoregulatory gene networks, J. Chem. Phys., 148 (2018), 095102. [21] N. Kumar, T. Platini and R. V. Kulkarni, Exact distributions for stochastic gene expression models with bursting and feedback, Phys. Rev. Lett., 113 (2014), 268105. [22] D. R. Larson, R. H. Singer and D. Zenklusen, A single molecule view of gene expression, Trends Cell Biol., 19 (2009), 630-637. [23] G. Lin, J. Yu, Z. Zhou, Q. Sun and F. Jiao, Fluctuations of mrna distributions in multiple pathway activated transcription, Discrete Cont. Dyn-B., 2018. [24] Y. T. Lin and N. E. Buchler, Efficient analysis of stochastic gene dynamics in the non-adiabatic regime using piecewise deterministic markov processes, J. Roy. Soc. Interface, 15 (2018), 20170804. [25] Y. T. Lin, P. G. Hufton, E. J. Lee and D. A. Potoyan, A stochastic and dynamical view of pluripotency in mouse embryonic stem cells, PloS Comput. Biol., 14 (2018), e1006000. [26] 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, 10pp. doi: 10.1103/physreve.93.022409. [27] M. Masujima, Applied Mathematical Methods in Theoretical Physics, John Wiley & Sons, 2009. doi: 10.1002/9783527627745. [28] N. A. M. Monk, Oscillatory expression of hes1, p53, and nf-$\kappa$b driven by transcriptional time delays, Curr. Biol., 13 (2003), 1409-1413. [29] M. Pájaro, A. A. Alonso, I. 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. [30] J. Peccoud and B. Ycart, Markovian modeling of gene-product synthesis, Theor. Popul. Biol., 48 (1995), 222-234. [31] A. Raj, C. S. Peskin, D. Tranchina, D. Y. Vargas and S. Tyagi, Stochastic mRNA synthesis in mammalian cells, PLoS Biol., 4 (2006), e309. [32] J. Ren, F. Jiao, Q. Sun, M. Tang and J. Yu, The dynamics of gene transcription in random environments, Discrete Cont. Dyn-B., 23 (2018), 3167-3194. doi: 10.3934/dcdsb.2018224. [33] R. Rudnicki and M. Tyran-Kamińska, Piecewise Deterministic Processes in Biological Models, Springer, 2017. doi: 10.1007/978-3-319-61295-9. [34] M. A. Schikora-Tamarit, C. Toscano-Ochoa, J. D. Espinos, L. Espinar and L. B. Carey, A synthetic gene circuit for measuring autoregulatory feedback control, Integr. Biol., 8 (2016), 546-555. [35] F. Veerman, C. Marr and N. Popović, Time-dependent propagators for stochastic models of gene expression: an analytical method, J. Math. Biol., 77 (2018), 261-312. doi: 10.1007/s00285-017-1196-4. [36] H. Wang, P. Liu, Q. Li and T. Zhou, Entangled signal pathways can both control expression stability and induce stochastic focusing, FEBS Lett., 592 (2018), 1135-1149. [37] S. Winkelmann and C. Schütte, Hybrid models for chemical reaction networks: Multiscale theory and application to gene regulatory systems, J. Chem. Phys., 147 (2017), 114115.
A: Random telegraph model for gene expression. Protein concentration increases if gene is On and decreases if gene is Off. Short On periods accompanied by fast production lead to burst-like gene expression. B: The burst dynamics can be deciphered by looking on the fast timescale (the scaling factor is set to $\varepsilon = 0.05$ here). C: Protein dynamics in the reduced drift-jump model. Between bursts (vertical dotted lines) the protein level decays exponentially (solid lines; one such period is demarcated by the braces). D: Positive feedback response function of the Michaelis-Menten type (11) as function of protein level $x$ for selected values of the threshold level $\kappa$. The dotted black lines ending in coloured markers indicate that the response is half-maximal at $x = \kappa$
Transcritical bifurcation of steady-state protein mean in the undelayed and infinitesimally delayed models. Analytical results (13) and (23) (solid lines) are cross-validated with kinetic Monte Carlo estimates (34) (discrete markers). Each individual marker gives the average value based on one hundred independent estimates (34)
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