Fig. 4.

For an unstable protein, positive feedback provides the lowest noise in event timing for a fixed mean FPT. (Top) Noise in timing ($C{V}_T^2$) as a function of the feedback strength $c$ for different control strategies. The value of $k_{\max }$ is changed in Eq. 14/Eq. 15 so as to keep $⟨T⟩=40\text{mins}$ fixed. The performance of the negative feedback worsens with increasing feedback strength. In contrast, positive feedback with an optimal value of $c$ provides the highest precision in event timing. Other parameters used are $\gamma =0.05{\text{min}}^{-1}$, $X=500$ molecules, $H=1$, $b=2$, and for positive feedback $r=0.05$. (Bottom) The minimum value of $C{V}_T^2$ obtained via positive feedback increases monotonically with the protein degradation rate. A smaller basal promoter strength $r=0.01$ in Eq. 15 gives better noise suppression than a larger value $r=0.05$. For comparison purposes, $C{V}_T^2$ obtained without any feedback ($c=0$), and a linear feedback with $c_1$ and $c_2$ in Eq. 13 chosen to minimize $C{V}_T^2$ for a given $⟨T⟩=40\text{mins}$ are also shown. The parameter values used are $X=500$ molecules, $H=1$, and $b=2$.