A semantics for Boolean networks consistent with regulatory threshold constraints
1Université Paris-Saclay, CNRS, ENS Paris-Saclay, Laboratoire Méthodes Formelles, Gif-sur-Yvette, 91190, France.
Journal of Theoretical Biology
|January 26, 2026
Summary
We introduce threshold semantics for Boolean networks (BNs) to accurately model biological systems by excluding spurious behaviors. This new approach refines dynamic analysis for gene and signaling networks, offering a stricter and more realistic abstraction.
Area of Science:
- Computational Biology
- Systems Biology
- Network Dynamics
Background:
- Boolean networks (BNs) are used for qualitative modeling of biological systems like gene and signaling networks.
- Existing BN semantics (synchronous, asynchronous, most permissive) have limitations in capturing real biological dynamics or introduce spurious behaviors.
- Quantitative models offer more accuracy but are computationally complex.
Purpose of the Study:
- To define and operationalize a new semantics, 'threshold semantics', for Boolean networks.
- To exclude spurious behaviors while maintaining a realistic abstraction of biological processes.
- To clarify relationships between existing and new BN semantics and analyze computational complexity.
Main Methods:
- Defined threshold semantics based on threshold networks, a subclass of multivalued networks.
- Clarified relationships between threshold semantics, linear semantics, and a constrained version of the most permissive semantics.
- Operationalized threshold semantics using symbolic constraints verifiable by a satisfiability solver, defining a trajectory-level operational semantics.
Main Results:
- Threshold semantics excludes spurious behaviors and provides a stricter abstraction than linear semantics and a constrained most permissive semantics.
- The operationalized threshold semantics is proven equivalent to the defined threshold semantics.
- The computational complexity of the new semantics is shown to be PSPACE, matching classical BN semantics.
Conclusions:
- Threshold semantics offers a more accurate and computationally efficient approach to modeling biological system dynamics compared to existing BN semantics.
- This semantics provides a robust qualitative abstraction of biological processes regulated by unknown single thresholds.
- Further research directions include exploring conjectures for future work on threshold semantics.
Keywords:
Boolean networksInfluence graphMost permissive semanticsReachabilitySingle threshold networksThreshold consistencyUpdate semanticsMore Related Videos
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