Continuous Petri Nets Faithfully Fluidify Most Permissive Boolean Networks

TL;DR

This paper introduces a novel approach that uses Continuous Petri Nets to faithfully represent Most Permissive Boolean Networks, bridging discrete and continuous models for biological network analysis.

Contribution

It proves that Continuous Petri Nets can simulate the semantics of Most Permissive Boolean Networks, enabling continuous analysis while preserving discrete behavior.

Findings

CPNs create a formal link between MPBNs and continuous models.

The methodology extends discrete analysis techniques to continuous systems.

Proof of simulation of MP semantics by CPNs enhances biological network modeling.

Abstract

The analysis of biological networks has benefited from the richness of Boolean networks (BNs) and the associated theory. These results have been further fortified in recent years by the emergence of Most Permissive (MP) semantics, combining efficient analysis methods with a greater capacity of explaining pathways to states hitherto thought unreachable, owing to limitations of the classical update modes. While MPBNs are understood to capture any behaviours that can be observed at a lower level of abstraction, all the way down to continuous refinements, the specifics and potential of the models and analysis, especially attractors, across the abstraction scale remain unexplored. Here, we fluidify MPBNs by means of Continuous Petri nets (CPNs), a model of (uncountably infinite) dynamic systems that has been successfully explored for modelling and theoretical purposes. CPNs create a formal link between MPBNs and their continuous dynamical refinements such as ODE models. The benefits of CPNs extend beyond the model refinement, and constitute well established theory and analysis methods, recently augmented by abstract and symbolic reachability graphs. These structures are shown to compact the possible behaviours of the system with focus on events which drive the choice of long-term behaviour in which the system eventually stabilises. The current paper brings an important keystone to this novel methodology for biological networks, namely the proof that extant PN encoding of BNs instantiated as a CPN simulates the MP semantics. In spite of the underlying dynamics being continuous, the analysis remains in the realm of discrete methods, constituting an extension of all previous work.