Linear cuts in Boolean networks

TL;DR

This paper introduces linear cuts in Boolean networks, a structural feature that resolves regulatory conflicts and simplifies analysis of attractors, providing a new interpretation aligned with multi-valued models.

Contribution

It defines and studies networks with linear cuts, linking structural properties to attractor analysis and offering a novel interpretation of Boolean semantics.

Findings

Linear cuts eliminate regulatory conflicts in Boolean networks.

Attractors correspond to minimal trap spaces under linear cuts.

Reachability of attractors becomes easier to characterize.

Abstract

Boolean networks are popular tools for the exploration of qualitative dynamical properties of biological systems. Several dynamical interpretations have been proposed based on the same logical structure that captures the interactions between Boolean components. They reproduce, in different degrees, the behaviours emerging in more quantitative models. In particular, regulatory conflicts can prevent the standard asynchronous dynamics from reproducing some trajectories that might be expected upon inspection of more detailed models. We introduce and study the class of networks with linear cuts, where linear components -- intermediates with a single regulator and a single target -- eliminate the aforementioned regulatory conflicts. The interaction graph of a Boolean network admits a linear cut when a linear component occurs in each cycle and in each path from components with multiple targets to components with multiple regulators. Under this structural condition the attractors are in one-to-one correspondence with the minimal trap spaces, and the reachability of attractors can also be easily characterized. Linear cuts provide the base for a new interpretation of the Boolean semantics that captures all behaviours of multi-valued refinements with regulatory thresholds that are uniquely defined for each interaction, and contribute a new approach for the investigation of behaviour of logical models.