Strong modules and asynchronous attractors of Boolean networks

TL;DR

This paper provides an algebraic framework for analyzing asynchronous Boolean networks by decomposing them into strong modules, enabling efficient attractor computation under certain conditions.

Contribution

It introduces a dependent sum construction for asynchronous attractors and shows polynomial-time computation when modules are small and networks are sparse or functions are simple.

Findings

Asynchronous attractors are products of attractors of strong modules.

Polynomial-time attractor computation is possible under specific size and sparsity conditions.

Application demonstrated on a published Boolean model.

Abstract

We consider Boolean networks with interaction graphs partitioned into strongly connected components, which we call strong modules. This type of network decomposition has been considered in the literature, primarily from the perspective of attractor detection algorithms. In this paper, we aim to provide an algebraic basis for this line of research in the case of asynchronous Boolean networks. We prove that the asynchronous attractors of a network can be described as a dependent sum construction: as products of attractors of its controlled strong modules. We then show that a representation of all attractors can be computed in polynomial time under two conditions: the strong modules are small, and either the network is sparse or its defining functions have small size circuits (in particular when they are nested canalizing). We illustrate these results on a published Boolean model.