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February  2017, 14(1): 249-262. doi: 10.3934/mbe.2017016

## Spatio-temporal models of synthetic genetic oscillators

 School of Mathematics and Statistics, Mathematical Institute, North Haugh, University of St Andrews, St Andrews KY16 9SS, Scotland

* Corresponding author: Mark Chaplain

Received  November 20, 2015 Accepted  April 15, 2016 Published  October 2016

Signal transduction pathways play a major role in many important aspects of cellular function e.g. cell division, apoptosis. One important class of signal transduction pathways is gene regulatory networks (GRNs). In many GRNs, proteins bind to gene sites in the nucleus thereby altering the transcription rate. Such proteins are known as transcription factors. If the binding reduces the transcription rate there is a negative feedback leading to oscillatory behaviour in mRNA and protein levels, both spatially (e.g. by observing fluorescently labelled molecules in single cells) and temporally (e.g. by observing protein/mRNA levels over time). Recent computational modelling has demonstrated that spatial movement of the molecules is a vital component of GRNs and may cause the oscillations. These numerical findings have subsequently been proved rigorously i.e. the diffusion coefficient of the protein/mRNA acts as a bifurcation parameter and gives rise to a Hopf bifurcation. In this paper we first present a model of the canonical GRN (the Hes1 protein) and show the effect of varying the spatial location of gene and protein production sites on the oscillations. We then extend the approach to examine spatio-temporal models of synthetic gene regulatory networks e.g. n-gene repressilators and activator-repressor systems.

Citation: Cicely K. Macnamara, Mark A. J. Chaplain. Spatio-temporal models of synthetic genetic oscillators. Mathematical Biosciences & Engineering, 2017, 14 (1) : 249-262. doi: 10.3934/mbe.2017016
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Plots showing (a) the hes1 mRNA concentration and (b) the Hes1 protein concentration varying in space and time. The hes1 mRNA gene-site where transcription occurs is located at $x_{m}=0.0$ and the Hes1 protein production sites where translation occurs are located at $x_{p}=\pm0.5$. The plots show sustained oscillations of both hes1 mRNA and Hes1 protein. Baseline parameter set $\mathbb{P}$
Plots showing (a) the total hes1 mRNA concentration and (b) the total Hes1 protein concentration over time. The hes1 mRNA gene-site where transcription occurs is located at $x_{m}=0.0$ and the Hes1 protein production sites where translation occurs are located at $x_{p}=\{\pm0.1, \pm0.3, \pm0.5, \pm0.7, \pm0.9\}$ (see legend). The plots show that if the Hes1 protein production sites are either too close or too far away from the hes1 mRNA gene-site, then oscillations are lost. Baseline parameter set $\mathbb{P}$
Plots showing (a) the total hes1 mRNA concentration and (b) the total Hes1 protein concentration over time. The hes1 mRNA gene-site where transcription occurs is located at $x_{m}=0.0$ and the Hes1 protein production sites where translation occurs are located at $x_{p}=\{\pm0.1, \pm0.3, \pm0.5, \pm0.7, \pm0.9\}$ (see legend). The plots show that if the Hes1 protein production sites are either too close or too far away from the hes1 mRNA gene-site, then oscillations are lost. Parameters are as for the baseline parameter set $\mathbb{P}$, except $\epsilon$ which is taken to be $0.25$
Plots showing (a) the total hes1 mRNA concentration and (b) the total Hes1 protein concentration over time. The hes1 mRNA gene-site where transcription occurs is located at $x_{m}=0.0$. Translation occurs uniformly in the cytoplasm as per equations (3) and (4), with the boundaries beyond which protein production occurs located at $x_{p}=\{0.1, 0.3, 0.5, 0.7, 0.9\}$ (see legend). The plots show that if Hes1 protein production is either too close or too far away from the hes1 mRNA gene-site, then oscillations are lost. Parameters are as for the baseline parameter set $\mathbb{P}$
Plots showing (a) the mRNA concentration and (b) the protein concentration varying in space and time for the three-gene repressilator system. $m_1, p_1$ top row; $m_2, p_2$ middle row; $m_3, p_3$ bottom row. The mRNA gene-sites are in different locations, i.e. $x_{m1}=0.0$, $x_{m2}=\pm0.2$, $x_{m3}=\pm0.4$, and the protein production sites are in different locations, i.e. $x_{p1}=\pm0.5$, $x_{p2}=\pm0.7$ and $x_{p3}=\pm0.9$. The plots show sustained oscillations of each of the three mRNA and protein species. Baseline parameter set $\mathbb{P}$
Plots showing (a) the total mRNA concentration for species 1 and (b) the total protein concentration for species 1 over time for the three-gene repressilator. Case (a) -solid black curve: the gene sites are $x_{m1}=x_{m2}=x_{m3}=0.0$ and protein production sites are $x_{p1}=x_{p2}=x_{p3}=\pm0.5$. Case (b) -blue dotted curve: the gene sites are $x_{m1}=0.0$, $x_{m2}=\pm0.2$, $x_{m2}=\pm0.4$, and protein production sites are $x_{p1}=\pm0.5$, $x_{p2}=\pm0.7$ and $x_{p3}=\pm0.9$. Case (c) -red dashed curve: the gene sites are $x_{m1}=x_{m2}=x_{m3}=0.0$ and protein production sites are $x_{p1}=x_{p2}=x_{p3}=\pm0.9$. Case (d) -cyan dot-dashed curve: the gene sites are $x_{m1}=0.0$, $x_{m2}=\pm0.05$, $x_{m3}=\pm0.1$, and protein production sites are $x_{p1}=\pm0.9$, $x_{m2}=\pm0.95$ and $x_{p3}=\pm1.0$. Baseline parameter set $\mathbb{P}$
Plots showing (a) the total mRNA concentration for species 1 and (b) the total protein concentration for species 1 over time for the three-gene repressilator. For all cases protein production (translation) is constant throughout the cytoplasm $|x|>0.5$. Case (a) -solid black curve: the gene sites are $x_{m1}=x_{m2}=x_{m3}=0.0$. Case (b) -blue dotted curve: the gene sites are $x_{m1}=x_{m2}=x_{m2}=\pm0.2$. Case (c) -red dashed curve: the gene sites are $x_{m1}=0.0$, $x_{m2}=\pm0.2$ and $x_{m3}=\pm0.4$. Case (d) -cyan dot-dashed curve: the gene sites are $x_{m1}=\pm0.1$, $x_{m2}=\pm0.3$ and $x_{m3}=\pm0.5$. Baseline parameter set $\mathbb{P}$
Plots showing (a) the mRNA concentration and (b) the protein concentration varying in space and time for a two-gene activator-repressor system. $m_1, p_1$ top row; $m_2, p_2$ bottom row. The mRNA gene-sites where transcription occurs are located at $x_{m_i}=0.0, i=1, 2$ and the protein production sites where translation occurs are located at $x_{p_i}= \pm0.9, i=1, 2$. The plots show sustained oscillations of mRNA and protein concentrations for both species. Baseline parameter set $\mathbb{P}$, $\beta_m=10.0$
Plots showing (a) the total mRNA concentration for species 1 and (b) the total protein concentration for species 1 over time for the two gene activator-repressor system. The mRNA gene-sites where transcription occurs are located at $x_{m_i}=0.0, i=1, 2$ and the protein production sites where translation occurs are located at $x_{p_i}= \pm0.8, \pm0.88, \pm0.94, \pm1.0, i=1, 2$ [(a) solid line, black, (b) dotted line, blue, (c) dashed line, red, (d) dot-dashed line, cyan, respectively]. The plots show that if the protein production sites are either too close or too far away from the mRNA gene-sites, then sustained oscillations are lost. Baseline parameter set $\mathbb{P}$, $\beta_m=10.0$
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