Signal hierarchical petri nets: Formal semantics of hierarchical regulatory control of biological systems.

Cellular commitment thresholds-such as metabolite concentrations triggering irreversible developmental transitions-cannot be derived as structural properties of network topology in classical computational formalisms. Existing numerical approaches (e.g., ODE bifurcation analysis) predict thresholds only after full parameterization; the threshold is an output of fitting, not of topology. The fundamental reason: classical models treat metabolites either as passive substrates (metabolic networks) or as Boolean switches (gene networks), lacking unified semantics for molecules functioning simultaneously as metabolic currencies and regulatory signals. We present signal hierarchy theory establishing formal foundations for hierarchical biological control. The theory proves two structural theorems: signal flow graphs must be acyclic, and lower-layer signal depletion structurally preempts higher-layer transitions-formalizing that resource exhaustion overrides regulatory programs regardless of controller abundance. We implement this theory as signal hierarchical Petri nets (SHPN), extending classical Bio-PN from 5-tuple to 13-tuple formalism with signal flow arcs enabling consumptive information propagation. Modelers designate any metabolite as signal place (ATP, GTP, NADH, cAMP, Ca2+) when biological evidence supports regulatory gating. Unlike test arcs (non-consuming catalysis), signal flow arcs create basin boundaries with commitment threshold Mcommit=θ+Ws computable directly from arc parameters-no simulation, no fitting. The formalism unifies metabolic-regulatory modeling: normal arcs for horizontal mass transfer, signal flow arcs for vertical information propagation. Applied to Bacillus subtilis sporulation, the formalism captures the Fujita and Losick (2005) observation: abrupt induction of Spo0A* yields only ≈5% sporulation while gradual KinA phosphorelay accumulation yields ≈52%. The asymmetry emerges structurally from the σH commitment separatrix (≈1.60μM): gradual accumulation allows the σH positive-feedback loop to cross threshold stochastically; an abrupt pulse fires before [σH] builds. Stochastic sweep simulation (25 conditions, 50 replicates) reproduces the 52% vs. ≈5% contrast; all arc parameters (Γ, phosphorelay rates, σH feedback) derive from published biological measurements-none was optimized to match this ratio.