Understanding Behavior of Biological Networks with Invariant Computation.

This chapter presents a qualitative, invariant-based analysis of the integrated MAPK and PI3K/Akt signaling network using PNs as a formal modeling framework. By exploiting structural properties such as P-invariants, T-invariants, and siphons, the study establishes a direct correspondence between the mathematical structure of the PN and biologically meaningful conservation laws, signaling modules, and dynamic behaviors. Minimal semi-positive P-invariants are shown to represent conserved molecular pools corresponding to distinct proteins and enzymes across their biochemical states, while minimal T-invariants capture canonical kinase-phosphatase modification cycles underlying sustained signaling activity. The hierarchical and modular organization of the network is revealed through systematic decomposition into reusable enzymatic motifs. Furthermore, the identification of siphons that are simultaneously traps highlights structurally persistent subnetworks associated with key regulatory components. Overall, the analysis demonstrates that the integrated MAPK and PI3K/Akt network is mass-preserving, dynamically sustained, and modular, providing a consistent qualitative foundation for understanding signaling robustness and pathway crosstalk independently of kinetic assumptions.