9-Dimensional Genetic Model

We consider a genetic model which is adapted from the one described in [1]. The dynamics is defined by the following ODE:

$\left\{ \begin{array}{lcl} \dot{x}_1 & = & 50 x_3 - 0.1 x_1 x_6 \\ \dot{x}_2 & = & 100 x_4 - x_1 x_2 \\ \dot{x}_3 & = & 0.1 x_1 x_6 - 50 x_3 \\ \dot{x}_4 & = & x_2 x_6 - 100 x_4 \\ \dot{x}_5 & = & 5 x_3 + 0.5 x_1 - 10 x_5 \\ \dot{x}_6 & = & 50 x_5 + 50 x_3 + 100 x_4 - x_6\cdot (0.1 x_1 + x_2 + 2 x_8 + 1) \\ \dot{x}_7 & = & 50 x_4 + 0.01 x_2 - 0.5 x_7 \\ \dot{x}_8 & = & 0.5 x_7 - 2 x_6 x_8 + x_9 - 0.2 x_8 \\ \dot{x}_9 & = & 2 x_6 x_8 - x_9 \end{array} \right.$

We consider the initial set

$x1 \in [0.98,1.02], x2 \in [1.28,1.32], x3 \in [0.08,0.12], x4 \in [0.08,0.12],$
$x5 \in [0.08,0.12], x6 \in [1.28,1.32], x7 \in [2.48,2.52], x8 \in [0.58,0.62],$
$x9 \in [1.28,1.32]$

and the time horizon [0,3]. The following figures shows the projections of the interval overapproximations of the flowpipes computed by Flow* 2.1.0. The total time cost is about 100 seconds.

References

[1] J. M. G. Vilar, H. Y. Kueh, N. Barkai, and S. Leibler.
Mechanisms of noise-resistance in genetic oscillators.
Proc. of the National Academy of Sciences of the United States of America, 99(9):5988–5992, 2002.

[2] X. Chen and S. Sankaranarayanan.
Decomposed Reachability Analysis for Nonlinear Systems.
In IEEE Real Time Systems Symposium (RTSS), pp. 13-24, 2016.