Qualitative Analysis of Signaling Networks Using Petri Nets and Invariant Computation

Qualitative analysis of biochemical reaction systems reveals fundamental system-level properties that are independent of precise kinetic parameters, often context-dependent, or experimentally inaccessible. By focusing on structural and topological features — such as conservation relations, feedback loops, and pathway interconnections — qualitative analysis identifies invariant behaviors, robustness mechanisms, and potential failure modes inherent to the signaling network. In this study, we use Petri nets as a formal modeling framework to conduct qualitative analysis of the integrated MAPK and PI3K/Akt signaling network. By exploiting structural properties including place invariants, transition invariants, and siphons, the analysis stablishes a direct correspondence between the Petri net structure and biologically meaningful conservation laws, signaling modules, and characteristic dynamic behaviors. The results demonstrate that the proposed model is structurally consistent, biologically plausible, and modular. Minimal semi-positive place invariants confirm mass conservation, indicating that proteins and enzymes circulate within closed molecular pools. Minimal semi-positive transition invariants identify canonical kinase–phosphatase cycles underlying sustained and reversible signaling. Hierarchical decomposition reveals a modular organization reducible to reusable enzymatic motifs, reflecting biological reuse across cascades and supporting scalability. Additionally, the identification of sixteen siphons that are also traps highlights persistent subsystems that ensure continuous regulator availability, confirming the robustness and dynamic sustainability of the integrated network.