An open problem: Why are motif-avoidant attractors so rare in asynchronous Boolean networks?

TL;DR

This paper investigates the rarity and fragility of motif-avoidant attractors in asynchronous Boolean networks, combining large-scale computational analysis with theoretical insights to understand their behavior and impact.

Contribution

It provides the first large-scale quantitative analysis of motif-avoidant attractors, revealing their scarcity in biological models and their sensitivity to network modifications.

Findings

MAAs are extremely rare in biological models.

Network reduction can introduce MAAs into dynamics.

Linear extensions of edges can disrupt MAAs even in sparse networks.

Abstract

Asynchronous Boolean networks are a type of discrete dynamical system in which each variable can take one of two states, and a single variable state is updated in each time step according to pre-selected rules. Boolean networks are popular in systems biology due to their ability to model long-term biological phenotypes within a qualitative, predictive framework. Boolean networks model phenotypes as attractors, which are closely linked to minimal trap spaces (inescapable hypercubes in the system's state space). In biological applications, attractors and minimal trap spaces are typically in one-to-one correspondence. However, this correspondence is not guaranteed: motif-avoidant attractors (MAAs) that lie outside minimal trap spaces are possible. MAAs are rare and (despite recent efforts) poorly understood. Here we summarize the current state of knowledge regarding MAAs and present several novel observations regarding their response to node deletion reductions and linear extensions of edges. We conduct large-scale computational studies on an ensemble of 14,000 models derived from published Boolean models of biological systems, and more than 100 million Random Boolean Networks. Our findings quantify the rarity of MAAs (in particular, we found no MAAs in the biological models), but highlight the role of network reduction in introducing MAAs into the dynamics. We also show that MAAs are fragile to linear extensions: in sparse networks, even a single linear node can disrupt virtually all MAAs. Motivated by this observation, we improve the upper bound on the number of delays needed to disrupt a motif-avoidant attractor.