Team:Hong Kong HKUST/Modelling Stability Analysis
Stability Analysis
In our tri-stable system, stability plays an indispensable role in determining the robustness of the system. What we learnt from the bi-stable switch is the variation in biological parts, e.g. promoters and ribosome binding sites (RBS), would cause the system to be unstable. Therefore, we performed stability analysis to assess the conditions and constraints that the system gives three stable states.
Here we focus on examining the equilibrium points, at which all species-of-interest i.e. repressor proteins reach equilibrium and that their concentrations do not change with time (i.e. $\frac{dx}{dt}=\frac{dy}{dt}=\frac{dz}{dt}=0$). Then, we can determine whether the equilibrium points are stable or unstable. A stable point is defined as: given a slight perturbation to an equilibrium point, the system automatically returns to the original point subsequently; while an unstable point is an equilibrium point where the system changes dramatically upon any perturbation. Moreover, parameters, such as relative efficiency of the promoter and binding affinity of the repressors to DNA, are known to have great influence on the stability of the equilibrium points. Therefore, in order to achieve a stable system, it is crucial to investigate the suitable numerical range of the parameters and accordingly, this will provide insights on the selection of proper combinations of parts in the circuits.
(I) Poincare-Perron Theorem
Poincare-Perron Theorem was employed for the stability analysis.
Suppose $\overrightarrow{p_e}$ denotes an equilibrium concentration of the repressor proteins. Then $\overrightarrow{p_e}$ is stable or unstable as follows:
- If the eigenvalues of $\overrightarrow{f'}$($\overrightarrow{p_e}$) all have negative real part, then $\overrightarrow{y_e}$ is asymptoticallystable;
- If any of the eigenvalues of $\overrightarrow{f'}$($\overrightarrow{p_e}$) has positive real part, then $\overrightarrow{y_e}$ is unstable.
In order to obtain the equilibrium concentrations of the three repressor proteins, we have to solve $\frac{dx}{dt}=\frac{dy}{dt}=\frac{dz}{dt}=0$, such that:
Where $x_s$, $y_s$ and $z_s$ denote the equilibrium concentration of the repressor proteins. However, solving a nonlinear ODE system analytically is hard and the solution is expected to be too lengthy and complex. We therefore employed piecewise-linear functions (Jutta,Gebert & Nicole,Radde & Gerhard-Wilhelm,Weber, 2007; Polynikis, Hogan, & Di Bernardo, 2009) to approximate the Hill’s term in the equation as follow:
After computation, we obtained the following results:
| Eigen-value \ Analytical solution | $\begin{pmatrix}x_s\\y_s\\z_s\end{pmatrix}=\begin{pmatrix}a\\a\\0\end{pmatrix}$ | $\begin{pmatrix}x_s\\y_s\\z_s\end{pmatrix}=\begin{pmatrix}a\\0\\a\end{pmatrix}$ | $\begin{pmatrix}x_s\\y_s\\z_s\end{pmatrix}=\begin{pmatrix}0\\a\\a\end{pmatrix}$ |
|---|---|---|---|
| $\lambda_1$ | < 0 | < 0 | < 0 |
| $\lambda_2$ | < 0 | < 0 | < 0 |
| $\lambda_3$ | < 0 | < 0 | < 0 |
The result shows that all eigenvalues of the 3 solutions are smaller than 0, according to the criterion of Poincare-Perron Theorem, these three solutions are all stable equilibrium points only if $n\geq2$ and $a\geq1+\frac2n$.
(II) Phase Portrait
Another method can be adopted to analyse the system's stability is phase portrait. (P, 2009) In order to give a clear picture, we begin with the illustration of the phase portrait of a simpler yet similar system, bistable switch.
The arrows on the plot represent the vector fields of the system, which indicate the direction that the system tends to evolve to. Specifically speaking, a direction to which vector fields point tells where the system tends to approach; conversely, when vector field points away from certain region, that region is where the system does not tend to stay. On the other hand, as shown in figure 2, there are two functions namely nullclines which are obtained by setting $\frac{dx}{dt}=0$ and $\frac{dy}{dt}=0$, have 3 intersections. The intersections, as defined above, represent the equilibrium points of the system. Nonetheless, according to the phase portrait, not all the points are stable; in fact only 2 points are stable to which the arrows points, while one of the intersections is unstable from which the arrows point away, while at the unstable point, all the repressor proteins’ concentrations are of equal concentration. And upon slight perturbation at this point, the system switches to either one of the stable points. This property is shown pictorially that when it is perturbed slightly away from the unstable point, the arrows point to one of the stable points. And the other intersections are considered stable due to their recovery ability under perturbation.
As expected, phase portrait can be applied to tri-stable toggle switch, which provides us another way to analyze the system stability pictorially.
We notice there are three intersections on the phase plane which corresponds to the three stable points identified by Poincare-Perron Theorem in the previous section: $\begin{pmatrix}x_s\\y_s\\z_s\end{pmatrix}=\begin{pmatrix}a\\a\\0\end{pmatrix}$ , $\begin{pmatrix}x_s\\y_s\\z_s\end{pmatrix}=\begin{pmatrix}a\\0\\a\end{pmatrix}$ and $\begin{pmatrix}x_s\\y_s\\z_s\end{pmatrix}=\begin{pmatrix}0\\a\\a\end{pmatrix}$.
Accordingly, it shows that the system has three steady states, which conforms to our original design.
Nonetheless, when Hill’s coefficient (n) is adjusted to a value smaller than 2 and ‘a’ is set to be smaller than 1, the three stable points disappear and reduce to a single intersection, which indicates that under the above condition, the system no longer exhibits tristability. We therefore conclude that for the Tristable Switch to attain tristability, the following constraints must be satisfied:
