The qualitative behavior of the dynamical model proposed for the square wave generator is generally dependent on the parameters of the system. Thus, the dynamical response of the system in-vivo would depend on the system parameters. To streamline the design process we need to understand this dependence of the system on the system parameters. The most natural way to do so is to conduct bifurcation analysis against the system parameters. Under such a framework, the system parameters are varied and the qualitative behavior is tracked, and when the system undergoes a change in behavior, we have arrived at a bifurcation point. For instance, varying a certain system parameter might cause the system’s steady state value to change is behavior from stable to unstable, and the system might start to exhibit oscillatory behavior; such a bifurcation is referred to as hopf bifurcation. Further, if the oscillations are stable, we have supercritical hopf bifurcation. For our purpose, we are tracking hopf bifurcation for the square wave generator. There is a two-fold reason for this: we want to identify parameter ranges that lead to oscillations and we want to quantify the extent of ‘squareness’ in these oscillations.
We performed the bifurcation analysis with respect to two system parameters, cooperativity and the the degradation rate. Intuitively, we can predict that higher cooperativity might lead to response with more square like behavior, thus we have included cooperativity in the bifurcation analysis. For the sake of ease, we assume a symmetric system, where all the repressors have the same cooperativities and degradation rates. The results for the bifurcation analysis are shown in Figs. 1, 2 and 3. The hopf branch is clearly visible in Figs. 1, 2. We see that for a give cooperativity, increasing the degradation rate leads to reduction in the amplitude of oscillation and finally the oscillations die out. Frequency has an inverse relationship with amplitude; points close to the hopf branchin in the oscillator region have high frequency, while moving away from the hopf branch, within the oscillatory region, leads to a reduction in frequency.
Fig.1: Colour map representing bifurcation analysis w.r.t. frequency
It is evident from Figs., 1 and 2 that the range of degradation rate over which the system oscillates increasing with increasing cooperativity. Thus, selecting a higher cooperativity would make the system more robust to variation in the degradation rate of the system.
To assess the extent to which the system response is close to a square wave, we define a ‘squareness’ metric. To compute the metric, we calculate the correlation between the system response and an ideal square wave with same period as the system. This correlation value is compared against the extent of squareness of a sinusoidal wave. The final squareness value captures the extent to which the system response is squarer than a sinusoidal signal.
Fig.3: Colour map representing squareness
We see that majority of the oscillatory region shows some square behavior. The maximum squareness is observed for high cooperativity values. Given Figs. 1, 2 and 3, we expect that a point selected in the mid to low degradation ranges for high cooperativity would sustain robust square like oscillations.
