Team:HZAU-China/Analysis

After describing the biological processes and choosing a set of empirical parameters, we want to simulate our designed processing modules. Before the simulation, we characterized some promoters to estimate the promoter strength, which will sometimes influence the cell's state directly. To take the intrinsic noise into consideration, we simulate the stochastic time course trajectories of the state of a chemical reaction network using Gillespie algorithm (Gillespie, 2001). We will analysis our two designs respectively.

Design 1

4.1.1 The effect of promoter strength

It is widely believed that the repressilator will exhibit a stable oscillation. But it is not always May. Most of the models about the repressilator retain some assumptions made by Elowitz and Leibler (Elowitz and Leibler, 2000), including \begin{equation} \begin{aligned} \beta_{1(i)}&=\beta_{1}, K_{tl(i)}=K_{tl},\\ K_{R(i)}&=K_{R}, K_{P(i)}=K_{P}. \end{aligned} \end{equation}

However the three genes are not identical. Our characterization of the promoters showed that the transcription rates for these three genes are different. For this reason, we treat them differently. We use $i=1$ to indexes gene cI, $i=2$ to indexes gene tetR, $i=3$ to indexes gene lacI. The promoter strength of placI which drives cI is represented by $\beta_{1(1)}$; the promoter strength of pcI which drives tetR in repressilator and drives lacI in toggle switch is represented by $\beta_{1(2)}$; the promoter strength of ptet is represented by $\beta_{1(3)}$. \begin{figure}[!htb] \centering \includegraphics[width=\textwidth]{fig123456.pdf} \caption{ Dynamics and phase diagrams of repressilator model in different cases. Parameters we used are $\beta_0=0.03, K_{tl}=6.93, K_{R}=0.347, K_{P}=0.0173$. (A) Dynamics of repressilator model with $\beta_1=(30,30,30)$; (B) Dynamics of repressilator model with $\beta_1=(30,15,20)$; (C) Phase diagram of repressilator model with $\beta_1=(30,30,30)$. The trajectory forms a limit cycle. (D) Phase diagram of repressilator model with $\beta_1=(30,15,20)$. The trajectory converge to a fixed point. (E) Phase diagram of repressilator model with $\beta_1=(30,30,30)$ and different initial protein numbers. The behaviours are robust. (F) Phase diagram of repressilator model with $\beta_1=(30,15,20)$ and different initial protein numbers. The behaviours are sensitive.} \label{fig4} \end{figure}