Buffering variability in cell regulation motifs close to criticality

Systematic analysis and optimization of early warning signals for critical transitions

“…Eq. 15 displays bistability for a range of values c (the exact range depends on K and k (Proverbio et al, 2022b)) and, in particular, a saddle-node bifurcation between two alternative steady states at a critical value c 0 of the parameter c , such that : where x 0 is the tipping value for the system state. Therefore, system 15 can be used as a paradigmatic example of biological systems, within the saddle-node b-tipping class, to perform optimisation studies that go beyond the local and low-noise-to-signal-ratio approximation provided by normal forms.…”

“…To model combinations of intrinsic and extrinsic noise, we set where α weights the white or multiplicative noise component ( α = 1 corresponds to purely additive Gaussian noise, α = 0 to purely multiplicative); like above, h ( x ) = x or h ( x ) ∝ f ( x ) = x k / (1 + x k ) (Hasty et al, 2000) and dW is a Wiener process with variance σ . Without loss of generality (Proverbio et al, 2022b), we set k = 2.…”

“…As early warning signals are associated with increases of statistical indicators, we need to establish a measure of statistically significant increase, to rule out false positives and false negatives due to random fluctuations in the indicators. To do so, we employ the p-value analysis used in Proverbio et al (2022b) (see STAR Method C.6 for details). It allows to measure at which value of the control parameter c , before c 0 , a significant signal is triggered, thus obtaining a “lead-parameter” depending on noise properties and the considered indicator ℐ (see STAR Method C.6 for details).…”

“…Since we work with distributional data, we propose to employ significance levels on Welch's p-value scores (non-equal variances allowed between the populations), which relate to threshold in confidence intervals and are readily interpreteable (Proverbio et al, 2022b).…”

“…Eq. 15 displays bistability for a range of values c (the exact range depends on K and k (Proverbio et al, 2022b)) and, in particular, a saddle-node bifurcation between two alternative steady states at a critical value c 0 of the parameter c, such that ∂f /∂x| (x,c 0 ) = 0:…”