Identification of periodic attractors in Boolean networks using a priori information.

Ulrike Münzner,Tomoya Mori,Tatsuya Akutsu,Edda Klipp,Marcus Krantz,Jeffrey J Saucerman

doi:10.1371/journal.pcbi.1009702

Abstract

Boolean networks (BNs) have been developed to describe various biological processes, which requires analysis of attractors, the long-term stable states. While many methods have been proposed to detection and enumeration of attractors, there are no methods which have been demonstrated to be theoretically better than the naive method and be practically used for large biological BNs. Here, we present a novel method to calculate attractors based on a priori information, which works much and verifiably faster than the naive method. We apply the method to two BNs which differ in size, modeling formalism, and biological scope. Despite these differences, the method presented here provides a powerful tool for the analysis of both networks. First, our analysis of a BN studying the effect of the microenvironment during angiogenesis shows that the previously defined microenvironments inducing the specialized phalanx behavior in endothelial cells (ECs) additionally induce stalk behavior. We obtain this result from an extended network version which was previously not analyzed. Second, we were able to heuristically detect attractors in a cell cycle control network formalized as a bipartite Boolean model (bBM) with 3158 nodes. These attractors are directly interpretable in terms of genotype-to-phenotype relationships, allowing network validation equivalent to an in silico mutagenesis screen. Our approach contributes to the development of scalable analysis methods required for whole-cell modeling efforts.

Highlights

Boolean network (BN) analysis is a powerful tool to computationally study biological processes, which include gene regulatory networks [1–3] and neural networks [4]
Boolean networks (BNs) are a convenient way to formalize biological processes, their analysis suffers from the combinatorial complexity with increasing number of nodes n
The long standing O(2n) barrier for detection of periodic attractors in BNs has obstructed the development of large, biological BNs

Summary

Introduction

Boolean network (BN) analysis is a powerful tool to computationally study biological processes, which include gene regulatory networks [1–3] and neural networks [4]. Given an initial state where each node (e.g., each gene) is assigned a value zero (equivalent terms: 0, inactive, false) or one (equivalent terms: 1, active, true), the node values are updated according to the Boolean functions to compute the network state in the time step. Describing biological processes as BNs allows a qualitative, dynamic description, where the attractors can be interpreted as stable states of a cell [5, 6]. Driving biological systems to stable states is important and has been studied using both Boolean models [7] and neural network models [8]. Which attractor is reached often depends on the initial state and testing of all possible initial states may require an enormous computational burden because there exist 2n states in a BN with n nodes

Methods

Results

Discussion

Conclusion