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 $\beta_{1(3)}$.

It is obvious that such a difference can influence the attractor of this system when we draw the time response diagram and phase trajectory diagram for the three proteins ($x_1, x_2, x_3$) (Figure xxx). We present two potential attractors: a fixed point and a limit cycle. If the attractor is a fixed point, the gene expression is easy to converge to a steady state which means the oscillations die out. If the attractor is a limit cycle, the oscillations are stable. We simulate the oscillation by scanning different initial protein numbers to further illuminate the difference between these two attractors. The oscillation behaviours is sensitive to the initial state when the attractor is a fixed point. So we cannot make sure that we will observe the oscillations in this situation. But the oscillation behaviours is robust when the attractor is a limit cycle.

Another question arises: Whether the attractor results from the relative promoter strength or the absolute promoter strength. To find out the answer, we scan a range of promoter strength and construct a index named relative amplitude $A_r$, \begin{equation} A_r=\frac{2\cdot(max(x_1)-min(x_1))}{max(x_1)+min(x_1)}. \end{equation}

This index is calculated when the simulation time approaches the infinity ($t=120000\ min$). So if the relative amplitude approaches 0, we conclude that the system is attracted to a fixed point.

The simulation results reveal that there is a critical surface between these two limiting states (Figure xxx). An attracting limit cycle emerges as the absolute promoter strengths increase, which is consistent with the previous result (Strelkowa and Barahona, 2011). They also assumed that the promoter strengths for all genes are identical and constructed a lumped parameter $c$. The parameter $c$ is the strength of the repressive coupling, which is calculated by \begin{equation} c=\frac{\beta_1\cdot K_{tl}}{K_R\cdot K_P}. \end{equation}

Figure 3: 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.

They found that there is a only fixed point $p_m$ when $c$ is small enough and the uniform solution $p_m$ becomes unstable via a Hopf bifurcation as $c$ increases resulting in a stable limit cycle.

Then how the promoter strength of every gene influences the final result? We try to use average promoter strength to simplify this question. Comparing four different mean values, we find that the harmonic mean of the promoter strengths exhibits good competence in distinguishing these two cases. If the harmonic mean of these three genes' promoter strengths is large enough, a stable oscillation can be observed easily in theory. While any one of the promoter strengths is too weak, the harmonic mean will be too small to lead to a stable oscillation, because harmonic mean is easily affected by the minimal value.

The model for genetic toggle switch, by contrast, is more simple. It has been acknowledged that the relative promoter strength is more important to the steady states. If the promoter strengths are approximately equal, or, in other words, if they are in the same order, the topological structure of toggle switch creates two stable and one unstable steady states. If one promoter strength is considerably larger than the another one, only one stable steady state will be produced.

Figure 4: Results of parameter scanning. The left diagram and the right diagram are the same results from different point of views. The colorbar indicates the relative amplitudes of different simulation result. The blank regions reveal that there is a fixed point attractor in the parameter space. The colored regions imply that there is a limit cycle attractor in the parameter space.

Figure 5: Distributions of average promoter strength for two cases. The red areas represent the average promoter strength distribution for those systems with a fixed point attractor. The yellow areas represent the average promoter strength distribution for those systems with a limit cycle attractor. The orange areas represent the overlapping areas of these two distributions