Stable and unstable attractors in Boolean networks

TL;DR

This paper investigates the nature of attractors in Boolean networks, revealing that many observed attractors under deterministic parallel update are artifacts, and identifies a subset of stable attractors with biologically relevant properties.

Contribution

The study demonstrates that most attractors found under synchronous update are artifacts and identifies stable attractors that scale sublinearly with system size, improving biological relevance.

Findings

Most attractors under parallel update are artifacts.

Stable attractors are resistant to response delays.

Number of stable attractors scales sublinearly with system size.

Abstract

Boolean networks at the critical point have been a matter of debate for many years as, e.g., scaling of number of attractor with system size. Recently it was found that this number scales superpolynomially with system size, contrary to a common earlier expectation of sublinear scaling. We here point to the fact that these results are obtained using deterministic parallel update, where a large fraction of attractors in fact are an artifact of the updating scheme. This limits the significance of these results for biological systems where noise is omnipresent. We here take a fresh look at attractors in Boolean networks with the original motivation of simplified models for biological systems in mind. We test stability of attractors w.r.t. infinitesimal deviations from synchronous update and find that most attractors found under parallel update are artifacts arising from the synchronous clocking mode. The remaining fraction of attractors are stable against fluctuating response delays. For this subset of stable attractors we observe sublinear scaling of the number of attractors with system size.