Team:Hong Kong HKUST/Modelling Stability Analysis

Home

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 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 goes back or forth and returns to the original point finally; 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, a stably performed system is critical to investigate the 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.

For the ODE system

Suppose $\overrightarrow{p_e}$ denotes an equilibrium concentration of the repressor proteins. Then $\overrightarrow{p_e}$ is stable or unstable as follows:

  1. If the eigenvalues of $\overrightarrow{f^'}$($\overrightarrow{p_e}$) all have negative real part, then $\overrightarrow{y_e}$ is asymptoticallystable;
  2. 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:

Fig. 1. A graphical representation of the approximation of Hill’s function by piecewise-linear function. (Polynikis et al., 2009)


Then, equation (14)-(16) are reduced into the following form:





Using linear algebra, we solved the approximated analytical solutions:

We then apply the analytical solutions into a Jacobian Matrix of the system of ODEs to assess the stability:

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 $a\geq1+\frac2n$.