Gene regulatory networks

This example describes the simulation of a gene regulatory network in Siconos. For a complete description of the problem see [Acary.DeJong.Brogliatio2014]

A piecewise linear model is used as a model for the gene regulatory network, in which the variables denote concentrations of gene products.

We consider a two genes system with the following dynamics:

\[\begin{split}\dot x_0 &= -\gamma_0 x_0 + \kappa_0 S^+(x_1, \theta_1^0)S^-(x_0, \theta_0^1) \\ \dot x_1 &= -\gamma_1 x_1 + \kappa_1 S^+(x_0, \theta_0^0)S^-(x_1, \theta_1^1) \\\end{split}\]

with the step functions:

\begin{equation} S^+(x_j, \theta_j^k) = \left\{\begin{array}{cc} 1 & x_j > \theta_j^k \\ \ [0,1] & x_j = \theta_j^k \\ 0 & x_j < \theta_j^k\end{array}\right. \ \ S^-(x_j, \theta_j^k) = \left\{\begin{array}{cc} 0 & x_j > \theta_j^k \\ \ [0,1] & x_j = \theta_j^k \\ 1 & x_j < \theta_j^k\end{array}\right. \end{equation}

\(\theta_j^k\) are thresholds that control inhibition/activation of the expression of a gene.