# American Institute of Mathematical Sciences

May  2019, 24(5): 2125-2147. doi: 10.3934/dcdsb.2019087

## Distributed delays in Hes1 gene expression model

 Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland

* Corresponding author: Marek Bodnar

Received  December 2017 Revised  January 2019 Published  March 2019

Fund Project: The first author is supported by National Science Centre, Poland, project OPUS no. 2015/17/B/ST1/00693

In the Hes1 gene expression system the protein (present as dimers) bounds to the promoter of its own DNA blocking transcription of its mRNA. This negative feedback leads to an oscillatory behavior, which is observed experimentally. Classical mathematical model of this system consists of two ordinary differential equations with discrete time delay in the term reflecting transcription. However, transcription takes place in the nucleus while translation occurs in the cytoplasm. This means that the delay present in the system is larger than transcription time. Moreover, in reality it is not discrete but distributed around some mean value. In this paper we present the model of the Hes1 gene expression system and discuss similarities and differences between the model with discrete and distributed delays. It turns out that in the case of distributed delays the region of stability of the steady state is larger than in the case of discrete delay. We also derive conditions that guarantee stability of the steady state for particular delay distributions.

Citation: Marek Bodnar. Distributed delays in Hes1 gene expression model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2125-2147. doi: 10.3934/dcdsb.2019087
##### References:
 [1] M. P. Antoch, E.-J. Song, A.-M. Chang, M. H. Vitaterna, Y. Zhao, L. D. Wilsbacher, A. M. Sangoram, D. P. King, L. H. Pinto and J. S. Takahashi, Functional identification of the mouse circadian clock gene by transgenic BAC rescue, Cell, 89 (1997), 655-667. doi: 10.1016/S0092-8674(00)80246-9. Google Scholar [2] J. H. Baek, J. Hatakeyama, S. Sakamoto, T. Ohtsuka and R. Kageyama, Persistent and high levels of Hes1 expression regulate boundary formation in the developing central nervous system, Development, 133 (2006), 2467-2476. doi: 10.1242/dev.02403. Google Scholar [3] A. Bartłomiejczyk, M. Bodnar and M. J. Piotrowska, Analysis of the p53 protein gene expression model, in Proceedings of XIX National Conference on Application of Mathematics in Biology and Medicine, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Regietów, 2015, 15–21.Google Scholar [4] S. Bernard, J. Bélair and M. Mackey, Sufficient conditions for stability of linear differential equations with distributed delay, Discrete Contin. Dyn. Syst. Ser. B., 1 (2001), 233-256. doi: 10.3934/dcdsb.2001.1.233. Google Scholar [5] S. Bernard, B. Cajavec, L. Pujo-Menjouet, M. C. Mackey and H. Herzel, Modelling transcriptional feedback loops: The role of Gro/TLE1 in Hes1 oscillations, Phil. Trans. R. Soc. A, 364 (2006), 1155-1170. doi: 10.1098/rsta.2006.1761. Google Scholar [6] M. Bodnar, Modele reakcji biochemicznych z opóźnionym argumentem: nieujemność rozwiązań i stabilność oscylacji, in Metody Matematyczne w Zastosowaniach, tom 2 (ed. A. Bartłomiejczyk), Centrum Zastosowań Matematyki, Politechnika Gdańska, 2014, chapter 1, 1–20, In polish.Google Scholar [7] M. Bodnar, General model of a cascade of reactions with time delays: global stability analysis, J. Diff. Eqs., 259 (2015), 777–795, arXiv: 1403.5435. doi: 10.1016/j.jde.2015.02.024. Google Scholar [8] M. Bodnar and A. Bartłomiejczyk, Stability of delay induced oscillations in gene expression of Hes1 protein model, Non. Anal. - Real., 13 (2012), 2227-2239. doi: 10.1016/j.nonrwa.2012.01.017. Google Scholar [9] M. Bodnar, U. Foryś and J. Poleszczuk, Analysis of biochemical reactions models with delays, J. Math. Anal. Appl., 376 (2011), 74-83. doi: 10.1016/j.jmaa.2010.10.038. Google Scholar [10] M. Bodnar and M. J. Piotrowska, Stability analysis of the family of tumour angiogenesis models with distributed time delays, Communications in Nonlinear Science and Numerical Simulation, 31 (2016), 124-142. doi: 10.1016/j.cnsns.2015.08.002. Google Scholar [11] K. L. Cooke and P. van den Driessche, On zeroes of some transcendental equations, Funkcj. Ekvacioj, 29 (1986), 77-90. Google Scholar [12] G. Fu, Z. Wang, J. Li and R. Wu, A mathematical framework for functional mapping of complex phenotypes usingdelay differential equations, J. Theor. Biol., 289 (2011), 206-216. doi: 10.1016/j.jtbi.2011.08.002. Google Scholar [13] Y. Haupt, R. Maya, A. Kazaz and M. Oren, Mdm2 promotes the rapid degradation of p53, Nature, 387 (1997), 296-299. doi: 10.1038/387296a0. Google Scholar [14] Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, vol. 1473 of Lecture Notes in Mathematics, Springer-Verlag, New York, 1991. doi: 10.1007/BFb0084432. Google Scholar [15] H. Hirata, S. Yoshiura, T. Ohtsuka, Y. Bessho, T. Harada, K. Yoshikawa and R. Kageyama, Oscillatory expression of the bHLH factor Hes1 regulated by a negative feedback loop, Science, 298 (2002), 840-843. doi: 10.1126/science.1074560. Google Scholar [16] M. Jensen, K. Sneppen and G. Tiana, Sustained oscillations and time delays in gene expression of protein Hes1, FEBS Lett., 541 (2003), 176-177. doi: 10.1016/S0014-5793(03)00279-5. Google Scholar [17] F. Li and J. Sun, Stability analysis of a reduced model of the lac operon under impulsive andswitching control, Nonlinear Anal.-Real, 12 (2011), 1264-1277. doi: 10.1016/j.nonrwa.2010.09.022. Google Scholar [18] J. Miⱸkisz, J. Poleszczuk, M. Bodnar and U. Foryś, Stochastic models of gene expression with delayed degradation, Bull. Math. Biol., 73 (2011), 2231-2247. doi: 10.1007/s11538-010-9622-4. Google Scholar [19] N. A. Monk, Oscillatory expression of Hes1, p53, and NF-$\kappa$B driven by transcriptional time delays, Curr. Biol., 13 (2003), 1409-1413. Google Scholar [20] B. Novak and J. J. Tyson, Quantitative analysis of a molecular model of mitotic control in fission yeast, J. Theor. Biol., 173 (1995), 283-305. doi: 10.1006/jtbi.1995.0063. Google Scholar [21] W. Pan, Z. Wang, H. Gao and X. Liu, Monostability and multistability of genetic regulatory networks with different types of regulation functions, Nonlinear Anal.-Real, 11 (2010), 3170-3185. doi: 10.1016/j.nonrwa.2009.11.011. Google Scholar [22] M. Sturrock, A. J. Terry, D. P. Xirodimas, A. M. Thompson and M. A. Chaplain, Spatio-temporal modelling of the Hes1 and p53-Mdm2 intracellular signalling pathways, J. Theoret. Biol., 273 (2011), 15-31. doi: 10.1016/j.jtbi.2010.12.016. Google Scholar [23] I. Yamaguchi, Y. Ogawa, Y. Jimbo, H. Nakao and K. Kotani, Reduction theories elucidate the origins of complex biological rhythmsgenerated by interacting delay-induced oscillations, PLoS ONE, 6 (2011), e26497.Google Scholar [24] S. Zeiser, H. Liebscher, H. Tiedemann, I. Rubio-Aliaga, G. Przemeck, M. de Angelis and G. Winkler, Number of active transcription factor binding sites is essential for the Hes7 oscillator, Theor. Biol. Med. Model., 3 (2006), 11-16. doi: 10.1186/1742-4682-3-11. Google Scholar

show all references

##### References:
 [1] M. P. Antoch, E.-J. Song, A.-M. Chang, M. H. Vitaterna, Y. Zhao, L. D. Wilsbacher, A. M. Sangoram, D. P. King, L. H. Pinto and J. S. Takahashi, Functional identification of the mouse circadian clock gene by transgenic BAC rescue, Cell, 89 (1997), 655-667. doi: 10.1016/S0092-8674(00)80246-9. Google Scholar [2] J. H. Baek, J. Hatakeyama, S. Sakamoto, T. Ohtsuka and R. Kageyama, Persistent and high levels of Hes1 expression regulate boundary formation in the developing central nervous system, Development, 133 (2006), 2467-2476. doi: 10.1242/dev.02403. Google Scholar [3] A. Bartłomiejczyk, M. Bodnar and M. J. Piotrowska, Analysis of the p53 protein gene expression model, in Proceedings of XIX National Conference on Application of Mathematics in Biology and Medicine, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Regietów, 2015, 15–21.Google Scholar [4] S. Bernard, J. Bélair and M. Mackey, Sufficient conditions for stability of linear differential equations with distributed delay, Discrete Contin. Dyn. Syst. Ser. B., 1 (2001), 233-256. doi: 10.3934/dcdsb.2001.1.233. Google Scholar [5] S. Bernard, B. Cajavec, L. Pujo-Menjouet, M. C. Mackey and H. Herzel, Modelling transcriptional feedback loops: The role of Gro/TLE1 in Hes1 oscillations, Phil. Trans. R. Soc. A, 364 (2006), 1155-1170. doi: 10.1098/rsta.2006.1761. Google Scholar [6] M. Bodnar, Modele reakcji biochemicznych z opóźnionym argumentem: nieujemność rozwiązań i stabilność oscylacji, in Metody Matematyczne w Zastosowaniach, tom 2 (ed. A. Bartłomiejczyk), Centrum Zastosowań Matematyki, Politechnika Gdańska, 2014, chapter 1, 1–20, In polish.Google Scholar [7] M. Bodnar, General model of a cascade of reactions with time delays: global stability analysis, J. Diff. Eqs., 259 (2015), 777–795, arXiv: 1403.5435. doi: 10.1016/j.jde.2015.02.024. Google Scholar [8] M. Bodnar and A. Bartłomiejczyk, Stability of delay induced oscillations in gene expression of Hes1 protein model, Non. Anal. - Real., 13 (2012), 2227-2239. doi: 10.1016/j.nonrwa.2012.01.017. Google Scholar [9] M. Bodnar, U. Foryś and J. Poleszczuk, Analysis of biochemical reactions models with delays, J. Math. Anal. Appl., 376 (2011), 74-83. doi: 10.1016/j.jmaa.2010.10.038. Google Scholar [10] M. Bodnar and M. J. Piotrowska, Stability analysis of the family of tumour angiogenesis models with distributed time delays, Communications in Nonlinear Science and Numerical Simulation, 31 (2016), 124-142. doi: 10.1016/j.cnsns.2015.08.002. Google Scholar [11] K. L. Cooke and P. van den Driessche, On zeroes of some transcendental equations, Funkcj. Ekvacioj, 29 (1986), 77-90. Google Scholar [12] G. Fu, Z. Wang, J. Li and R. Wu, A mathematical framework for functional mapping of complex phenotypes usingdelay differential equations, J. Theor. Biol., 289 (2011), 206-216. doi: 10.1016/j.jtbi.2011.08.002. Google Scholar [13] Y. Haupt, R. Maya, A. Kazaz and M. Oren, Mdm2 promotes the rapid degradation of p53, Nature, 387 (1997), 296-299. doi: 10.1038/387296a0. Google Scholar [14] Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, vol. 1473 of Lecture Notes in Mathematics, Springer-Verlag, New York, 1991. doi: 10.1007/BFb0084432. Google Scholar [15] H. Hirata, S. Yoshiura, T. Ohtsuka, Y. Bessho, T. Harada, K. Yoshikawa and R. Kageyama, Oscillatory expression of the bHLH factor Hes1 regulated by a negative feedback loop, Science, 298 (2002), 840-843. doi: 10.1126/science.1074560. Google Scholar [16] M. Jensen, K. Sneppen and G. Tiana, Sustained oscillations and time delays in gene expression of protein Hes1, FEBS Lett., 541 (2003), 176-177. doi: 10.1016/S0014-5793(03)00279-5. Google Scholar [17] F. Li and J. Sun, Stability analysis of a reduced model of the lac operon under impulsive andswitching control, Nonlinear Anal.-Real, 12 (2011), 1264-1277. doi: 10.1016/j.nonrwa.2010.09.022. Google Scholar [18] J. Miⱸkisz, J. Poleszczuk, M. Bodnar and U. Foryś, Stochastic models of gene expression with delayed degradation, Bull. Math. Biol., 73 (2011), 2231-2247. doi: 10.1007/s11538-010-9622-4. Google Scholar [19] N. A. Monk, Oscillatory expression of Hes1, p53, and NF-$\kappa$B driven by transcriptional time delays, Curr. Biol., 13 (2003), 1409-1413. Google Scholar [20] B. Novak and J. J. Tyson, Quantitative analysis of a molecular model of mitotic control in fission yeast, J. Theor. Biol., 173 (1995), 283-305. doi: 10.1006/jtbi.1995.0063. Google Scholar [21] W. Pan, Z. Wang, H. Gao and X. Liu, Monostability and multistability of genetic regulatory networks with different types of regulation functions, Nonlinear Anal.-Real, 11 (2010), 3170-3185. doi: 10.1016/j.nonrwa.2009.11.011. Google Scholar [22] M. Sturrock, A. J. Terry, D. P. Xirodimas, A. M. Thompson and M. A. Chaplain, Spatio-temporal modelling of the Hes1 and p53-Mdm2 intracellular signalling pathways, J. Theoret. Biol., 273 (2011), 15-31. doi: 10.1016/j.jtbi.2010.12.016. Google Scholar [23] I. Yamaguchi, Y. Ogawa, Y. Jimbo, H. Nakao and K. Kotani, Reduction theories elucidate the origins of complex biological rhythmsgenerated by interacting delay-induced oscillations, PLoS ONE, 6 (2011), e26497.Google Scholar [24] S. Zeiser, H. Liebscher, H. Tiedemann, I. Rubio-Aliaga, G. Przemeck, M. de Angelis and G. Winkler, Number of active transcription factor binding sites is essential for the Hes7 oscillator, Theor. Biol. Med. Model., 3 (2006), 11-16. doi: 10.1186/1742-4682-3-11. Google Scholar
A sketch of negative feedback loop for the Hes1 system
The comparison of the condition proved in Theorem 2.3 (the red solid line) and $\tau_ {\rm cr}$ for the case of discrete delays (the blue dashed line) for three different values of $d_1$
The dependence of the critical average delay value on the Hill coefficient. Time delay value is given before rescaling, in minutes. All parameters are as proposed by Monk [19]. The lines indicate critical average delay for different delay distributions: the dotted blue line for Erlang distribution with $m = 1$; the solid red line for Erlang distribution with $m = 2$; the dashed green line for discrete delay. The stability region is to the left of the curves
The dependence of the critical average delay value on the Hill coefficient for the uniform distribution (solid red line) and the triangular distribution (dotted blue line). The lines indicate critical average delay. The dashed green line for critical discrete delay was plotted for comparison. The stability region is to the left of the curves. Time delay value is given before rescaling, in minutes. All parameters are as proposed by Monk [19]
The stability region in $(\mu,d_1)$-plane for different values of the shape parameter and various values of $\tau_m$. The solid vertical red line indicates the set of parameters $(\mu,d_1)$ for the parameters proposed by Monk [19] and with the Hill coefficient varying from $1.2$ to $15$. The average delay is set to $1.71$
The stability region in $(\mu,d_1)$-plane for different values of the shape parameter and various values of $\tau_m$. The solid vertical red line indicates the set of parameters $(\mu,d_1)$ for the parameters proposed by Monk [19] and with the Hill coefficient varying from $1.2$ to $15$. The average delay is set to $1.71$
The stability region in $(\mu,d_1)$-plane for the uniform and triangular distributions and different values of $\delta$. The solid vertical red line indicates the set of parameters $(\mu,d_1)$ for the parameters proposed by Monk [19] and with the Hill coefficient varying from $1.2$ to $15$. The average delay is set to $1.71$
The dependence of the critical value of the first derivative of the function $f$ for which destabilization occurs on the variance of the distribution for various distributions. The shaded region in the left-hand side panel is zoomed out in the right-hand panel
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