Fig. 6: Effect of degradation rates on stability.
a The system is stable if and only if the maximal real part of the eigenvalues of the interaction matrix \({\lambda }_{M,{r}_{\max }}\le 1\), regardless of the value of ω which increases with mRNA degradation rates (Eq. (24)). The scaled eigenvalues \(\tilde{\lambda }\to {\lambda }_{M}-1\) in the limit of fast mRNA degradation rate ω → ∞ (Eq. (23)). b Eigenvalue spectrum for different degradation rates τ. When mRNA and protein degradation rates are comparable, all eigenvalues fall within a circular region (red). On the other hand, when τm ≪ τp, the eigenvalue spectrum approximately resembles two circular regions, one corresponding to the dynamics of mRNAs and one for that of proteins. In this limit, increasing mRNA degradation rate only shifts the eigenvalues for the mRNA sector to more negative values, leaving the maximal real part of the eigenvalues approximately unchanged, ω = 0.5 (green) vs ω = 0.4 (blue).
