General relationship of local topologies, global dynamics, and bifurcation in cellular networks.

Cellular networks realize their functions by integrating intricate information embedded within local structures such as regulatory paths and feedback loops. However, the precise mechanisms of how local topologies determine global network dynamics and induce bifurcations remain unidentified. A critical step in unraveling the integration is to identify the governing principles, which underlie the mechanisms of information flow. Here, we develop the cumulative linearized approximation (CLA) algorithm to address this issue. Based on perturbation analysis and network decomposition, we theoretically demonstrate how perturbations affect the equilibrium variations through the integration of all regulatory paths and how stability of the equilibria is determined by distinct feedback loops. Two illustrative examples, i.e., a three-variable bistable system and a more intricate epithelial-mesenchymal transition (EMT) network, are chosen to validate the feasibility of this approach. These results establish a solid foundation for understanding information flow across cellular networks, highlighting the critical roles of local topologies in determining global network dynamics and the emergence of bifurcations within these networks. This work introduces a novel framework for investigating the general relationship between local topologies and global dynamics of cellular networks under perturbations.