Distributed delays in Hes1 gene expression model

Marek Bodnar

doi:10.3934/dcdsb.2019087

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2019, Volume 24, Issue 5: 2125-2147. Doi: 10.3934/dcdsb.2019087

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Distributed delays in Hes1 gene expression model

Marek Bodnar^,

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

^* Corresponding author: Marek Bodnar
^* Corresponding author: Marek Bodnar

Received: November 30, 2017

Revised: December 31, 2018

Early access: March 2019

Published: March 2019

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 34K13, 37L15; Secondary: 37N25.

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Figure 1. A sketch of negative feedback loop for the Hes1 system

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Figure 2. 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 $

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Figure 3. 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

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Figure 4. 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]

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Figure 5. 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 $

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Figure 6. 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 $

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Figure 7. 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 $

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Figure 8. 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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