Continuous Petri Nets Faithfully Fluidify Most Permissive Boolean Networks

2025-05-22

AI Summary

Bridging the semantic gap between discrete Boolean models—specifically most permissive Boolean networks (MPBNs)—and continuous dynamical systems (e.g., ODEs) remains a fundamental challenge in multiscale biological modeling. Method: We introduce a fluidization framework based on continuous Petri nets (CPNs), formally proving that CPNs precisely encode the semantics of MPBNs—including their discrete attractor structure—while yielding smooth, differentiable dynamics. Contribution/Results: By leveraging abstract reachability graphs and symbolic dynamics analysis, our approach enables complete behavioral characterization of MPBNs at the continuous scale and supports efficient, lossless attractor identification. Unlike conventional continuous approximations, this method preserves the exact discrete-state semantics and attractor topology of MPBNs, thereby establishing a formally verified, interpretable modeling paradigm for cross-scale analysis of biological pathways.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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.

Problem

• Bridging MPBNs and continuous refinements via CPNs
• Exploring attractors across abstraction scales in MPBNs
• Proving CPN simulation of MP semantics for BNs

Innovation

• Continuous Petri Nets fluidify Most Permissive Boolean Networks
• CPNs link MPBNs to continuous dynamical refinements
• Discrete methods analyze continuous dynamics effectively