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=Example 1. Synthetic Oscillator=

Introduction
The synthetic oscillatory network designed by Elowitz and Leibler is ...........

Mathematical Formulation
The activities of a gene are regulated by other genes through the interactions between them, i.e., the transcription and translation factors. Here, we assume that this system follows Hill kinetic law.

$$\begin{align} \frac{dm_{i}}{dt} &=-a_{i}m_{i}+\sum\limits_{j}b_{ij}\frac{p_{j}^{H_{ij}}}{K_{ij}+p_{j}^{H_{ij}}}+l_{i}, \\ \frac{dp_{i}}{dt} &=-c_{i}p_{i}+d_{i}m_{i}, (i=1,2,...,n) \end{align}\,\!$$

where $$m_{i}(t), p_{i}(t)\in {\mathbb{R}}$$ are concentrations of mRNA and protein of the $$i$$th node at time $$t$$, respectively, $$a_{i}$$ and $$c_{i}$$ are the degradation rates of the mRNA and protein, $$d_{i}$$ is the translation rate. Term (1) describes the transcription process and term (2) describes the translation process. Negative and positive signs of $$b_{ij}$$ indicates the mutual interaction relationship that could be attributed to negative or positive feedback. The values describe the strength of promoters which is tunable by inserting different promoters in gene circuits. $$H_{ij}$$ is Hill coefficient describing cooperativity. $$K_{ij}$$ is the apparent dissociation constant derived from the law of mass action (equilibrium constant for dissociation). We can write $$K_{ij}=\left( \hat{K}_{ij}\right) ^{n}$$ where $$\hat{K}$$ is ligand concentration producing half occupation (ligand concentration occupying half of the binding sites), that is also the microscopic dissociation constant.

A Tunable Oscillator
The original three repressors model is described as follows:%

$$\begin{align} \frac{dm_{1}}{dt} &=-am_{1}+b\frac{p_{3}^{H_{13}}}{K+p_{3}^{H_{1}}}, \\ \frac{dm_{2}}{dt} &=-am_{2}+b\frac{p_{1}^{H_{21}}}{K+p_{1}^{H_{2}}}, \\ \frac{dm_{3}}{dt} &=-am_{3}+b\frac{p_{2}^{H_{32}}}{K+p_{2}^{H_{32}}}, \\ \frac{dp_{1}}{dt} &=-cp_{1}+dm_{1}, \\ \frac{dp_{2}}{dt} &=-cp_{2}+dm_{2}, \\ \frac{dp_{3}}{dt} &=-cp_{3}+dm_{3},\text{ } \end{align}\,\!$$

where $$a, b,$$ $$c,$$ $$d,$$ $$H_{1},$$ $$H_{2},$$ $$H_{3},$$ $$K$$ are tunable parameters that could change wave amplitude and frequency. For simplicity, we assume that $$H_{13}=H_{21}=H_{32}=2,$$ meaning that the system contains only positively cooperative reaction that once one ligand molecule is bound to the enzyme, its affinity for other ligand molecules increases.

$$\begin{align} figure\text{ 1}\text{: wave amplitude} && \\ figure\text{ 2}\text{: wave frequency} && \\ figure\text{ 3}\text{: sensitivity analysis} && \end{align}\,\!$$



An Alternative Topology That Leads to Oscillation
The original three repressors model is described as follows:%

$$\begin{align} \frac{dm_{1}}{dt} &= -a_{1}x_{1}+\frac{b_{1}}{K_{1}+p_{2}^{H_{12}}}, \\ \frac{dm_{2}}{dt} &= -a_{2}x_{2}+\frac{b_{2}p_{3}^{H_{23}}}{% K_{2}+p_{1}^{H_{21}}+p_{3}^{H_{23}}}, \\ \frac{dm_{3}}{dt} &= -a_{3}x_{3}+\frac{b_{3}}{K_{3}+p_{2}^{H_{32}}} \\ \frac{dp_{1}}{dt} &= -c_{1}p_{1}+d_{1}m_{1}, \\ \frac{dp_{2}}{dt} &= -c_{2}p_{2}+d_{2}m_{2}, \\ \frac{dp_{3}}{dt} &= -c_{3}p_{3}+d_{3}m_{3}, \end{align}\,\!$$