Stability Analysis: Hysteresis and Oscillations

The first two synthetic gene circuits, which appeared in Nature in January 2000, are referred to as the Toggle Switch and the Repressilator. As the name suggests, the former switches from high to low fluorescence (and vice versa) under the action of two different input signals. The latter, in contrast, shows oscillations in fluorescence expression. However, both circuits work in these ways only if certain conditions about kinetic parameter values and initial species concentrations are met. Stability analysis, whose coverage started in Chap. 5, permits to find out the requisites under which a synthetic gene circuit settles into a steady state that represents a properly-working configuration. In this Chapter we will analyze the model of these two synthetic gene circuits with the open-source software XPPAUT. First, we will show how to design a hysteresis curve for the Toggle Switch and determine the requirements for the circuit to lie into two possible stable steady states (multistationarity). Then, we will obtain a graphical representation for a so-called Hopf bifurcation that, in the Repressilator case, signals the presence of limit cycles that guarantee sustained oscillations.