Equilibrium points of a Toggle Switch System

This file contain a simulation of the equilibrium points for a toggle switch circuit

The bi-stability in the protein circuit studied in [1] raises a convergence towards two stable states, called equilibrium points of the system, they are close to their unstable equilibrium point. In some special cases it is undesirable, as, for example; the cultivation of bacteria, proteins or the behavior of biological organisms that affect human life or health. This requires the design of control laws that lead to the stability of the biological circuit towards the desired equilibrium point [1].

This simulation contemplates the study of the genetic circuit of Gardner and Collins, given its bi-stability, the objective is to analytically calculate its equilibrium points using the Routh criterion to ensure that the data produced by the numerical simulation are plausible. Also, the mass action model was used as well as the Hill exponent addressed in [1], to fit the mathematical model to a non-linear second order system.

The numerical simulation allowed to find the equilibria of the Gardner and Collins system, this was possible thanks to the implementation of the numerical integrator ode45 in Matlab, the dissociation constant k has been varied to observe the phase trajectories towards each equilibrium point. It is concluded that the initial states of the system do not alter its convergence towards the calculated equilibrium points.

[1] C. C. Samaniego, N. A. Delateur, G. Giordano y E. Franco, «Biomolecular stabilisation near the unstable equilibrium of a biological system,» de 2019 IEEE 58th Conference on Decision and Control (CDC), 2019.