Unifying Weak Independence and Signal Hierarchy Theory: Extended Biological Petri Net Formalism with Application to Vibrio fischeri Quorum Sensing

TL;DR

This paper extends Biological Petri Nets with a unified formalism combining weak independence and signal hierarchy theories, enabling detailed modeling of biochemical and informational interactions, demonstrated through Vibrio fischeri quorum sensing.

Contribution

It introduces a comprehensive 13-tuple formalism for Bio-PNs that integrates two key theories, enhancing the modeling of biochemical reactions and hierarchical signal control.

Findings

Validated weak independence correctness for continuous dynamics

Applied formalism to Vibrio fischeri quorum sensing

Revealed timing of signal saturation as a threshold mechanism

Abstract

Biological Petri Nets (Bio-PNs) require extensions beyond classical formalism to capture biochemical reality: multiple reactions simultaneously affect shared metabolites through convergent production or regulatory coupling, while signal places carry hierarchical control information distinct from material flow. We present a unified 13-tuple Extended Bio-PN formalism integrating two complementary theories: Weak Independence Theory (enabling coupled parallelism despite place-sharing) and Signal Hierarchy Theory (separating information flow from mass transfer). The extended definition adds signal partition (Psi subset P), arc type classification (A), regulatory structure (Sigma), environmental exchange (Theta), dependency taxonomy (Delta), heterogeneous transition types (tau), and biochemical formula tracking (rho). We formalize signal token consumption semantics through two-phase execution (enabling vs. consumption) and prove weak independence correctness for continuous dynamics. Application to Vibrio fischeri quorum sensing demonstrates how energy metabolism (ENERGY signals) orchestrates binary ON/OFF decisions through hierarchical constraint propagation to regulatory signals (LuxR-AHL complex), with 133-fold difference separating states. Analysis reveals signal saturation timing as the orchestrator forcing threshold-crossing, analogous to bacteriophage lambda lysogeny-lysis decisions. This work establishes formal foundations for modeling biological information flow in Petri nets, with implications for systems biology, synthetic circuit design, and parallel biochemical simulation.