A general, robust framework for determining the key species that forewarns sudden transitions in biological circuits

The Cdc2-cyclin B/Wee1 kinase system exhibits bistability between alternative steady states, which emerges due to the mutual inhibition between Cdc2-cyclin B and Wee1 kinases. Alternative steady states are {\em M phase-like} state and {\em G2 arrest} state, which have implications in the cell cycle progression at the G2 phase in eukaryotic cells. A slight alteration in the feedback strength can drive sudden transitions between these contrasting alternative states upon crossing a critical threshold or a tipping point. The phenomenon of {\em critical slowing down} (CSD) has been widely used to identify the proximity to a tipping point. However, determining the key variable or species that best signals CSD is a challenging task and holds significance in complex biochemical processes. Here, we determine the key variable or observation direction (OD) from the direction of CSD to best detect an upcoming transition in the Cdc2-cyclin B/Wee1 model system. We find that with increasing feedback strength, the Cdc2-cyclin B is the OD, as it produces a stronger signal than that of Wee1. With decreasing feedback strength, both Cdc2-cyclin B and Wee1 produce similar signals and can be used as OD. Further, the noise-sensitive direction highlights the effect of stochasticity in Cdc2-cyclin B and Wee1 for increasing and decreasing feedback strength, respectively. We also perform sensitivity analyses that reveal the robustness of the OD. Finally, we compare the efficacy of OD with principal component analysis while detecting a tipping point, and also validate its general applicability to epithelial-mesenchymal transition for cancer progression.