Reducing Boolean networks with backward equivalence

Georgios A. Argyris, Alberto Lluch Lafuente, Mirco Tribastone, Max Tschaikowski, Andrea Vandin*

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

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Background: Boolean Networks (BNs) are a popular dynamical model in biology where the state of each component is represented by a variable taking binary values that express, for instance, activation/deactivation or high/low concentrations. Unfortunately, these models suffer from the state space explosion, i.e., there are exponentially many states in the number of BN variables, which hampers their analysis. Results: We present Boolean Backward Equivalence (BBE), a novel reduction technique for BNs which collapses system variables that, if initialized with same value, maintain matching values in all states. A large-scale validation on 86 models from two online model repositories reveals that BBE is effective, since it is able to reduce more than 90% of the models. Furthermore, on such models we also show that BBE brings notable analysis speed-ups, both in terms of state space generation and steady-state analysis. In several cases, BBE allowed the analysis of models that were originally intractable due to the complexity. On two selected case studies, we show how one can tune the reduction power of BBE using model-specific information to preserve all dynamics of interest, and selectively exclude behavior that does not have biological relevance. Conclusions: BBE complements existing reduction methods, preserving properties that other reduction methods fail to reproduce, and vice versa. BBE drops all and only the dynamics, including attractors, originating from states where BBE-equivalent variables have been initialized with different activation values The remaining part of the dynamics is preserved exactly, including the length of the preserved attractors, and their reachability from given initial conditions, without adding any spurious behaviours. Given that BBE is a model-to-model reduction technique, it can be combined with further reduction methods for BNs.

Original languageEnglish
Article number212
JournalBMC Bioinformatics
Publication statusPublished - 23 May 2023

Bibliographical note

Funding Information:
We thank reviewers of the original CMSB 2021 submission and of this submission for their help in improving the paper. We also thank the CMSB 2021 attendants for their comments and suggestions on the work. We thank Laure Bally-Cuif, author of [46], for her fruitful information on the modelling approach helping us in the case study in Additional file 3.

Funding Information:
Partially supported by the DFF project REDUCTO 9040-00224B, the Poul Due Jensen Grant 883901, the Villum Investigator Grant S4OS, and the PRIN project SEDUCE 2017TWRCNB. The funders had no direct role in the design of the study, collection, analysis and interpretation of the data, or writing of the manuscript. This publication was produced with the co-funding European Union - Next Generation EU, in the context of The National Recovery and Resilience Plan, Investment 1.5 Ecosystems of Innovation, Project Tuscany Health Ecosystem (THE), CUP: B83C22003920001.

Publisher Copyright:
© 2023, The Author(s).


  • Attractors analysis
  • Boolean network
  • Model reduction
  • Partition refinement
  • State-space generation


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