Weak Independence and Coupled Parallelism in Biological Petri Nets

Motivation: Biological Petri Nets (Bio-PNs) model biochemical pathways where multiple reactions simultaneously affect shared metabolites through convergent production or regulatory coupling. However, classical Petri net independence theory requires transitions to share no places -- a constraint that fails to capture biological reality. This mismatch prevents parallel simulation and incorrectly flags biologically valid models as structurally problematic. Results: To resolve this fundamental limitation, we introduce weak independence -- a novel formalization distinguishing resource conflicts from biological coupling. Building on this theory, we extend the Bio-PN definition from a classical 5-tuple to a 12-tuple by adding regulatory structure, environmental exchange classification, dependency taxonomy, heterogeneous transition types, and biochemical formula tracking. This extended formalism enables systematic classification of three place-sharing modes: competitive (conflict), convergent (superposition), and regulatory (read-only). Validating our approach on 100 diverse BioModels (1,775 species, 2,234 reactions across metabolism, signaling, and gene regulation), we find that 96.93% of transition pairs exhibit weak independence -- confirming that biological networks inherently favor cooperation over competition. Our SHYpn implementation demonstrates the practical impact, achieving up to 2.6x speedup on 30% of evaluated models. Availability and Implementation: Open-source at https://github.com/simao-eugenio/shypn (MIT License).