Algorithms for Finding Small Attractors in Boolean Networks

Abstract

A Boolean network is a model used to study the interactions between different genes in genetic regulatory networks. In this paper, we present several algorithms using gene ordering and feedback vertex sets to identify singleton attractors and small attractors in Boolean networks. We analyze the average case time complexities of some of the proposed algorithms. For instance, it is shown that the outdegree-based ordering algorithm for finding singleton attractors works in time for , which is much faster than the naive time algorithm, where is the number of genes and is the maximum indegree. We performed extensive computational experiments on these algorithms, which resulted in good agreement with theoretical results. In contrast, we give a simple and complete proof for showing that finding an attractor with the shortest period is NP-hard.

“Immunetworks”, intersecting circuits and dynamics

“…When 1 ≤ c ≤ 2, the number of possible limit cycles has recently been proven to be in general super-polynomial with respect to n [68,69,70,71,72].…”

Integer Programming-Based Approach to Attractor Detection and Control of Boolean Networks

“…Recently, several methods have been developed for efficiently finding and/or enumerating attractors in BNs [10]- [12], [14], [17], [32], whereas it is known that finding a singleton attractor (i.e., a steady state) is NP-hard [1]. Devloo et al developed a method using transformation to a constraint satisfaction problem [11].…”

“…However, theoretical analysis of the average case or worst case complexity was not performed in these studies. Zhang et al developed algorithms for enumerating singleton attractors and small attractors and analyzed the average case time complexities of these algorithms [32]. Akutsu et al, Melkman et al, and Tamura et al developed algorithms with guaranteed worst case time complexities for detection of singleton attractors for BNs with restricted Boolean functions [4], [27], [31].…”

An Improved Algorithm for Detecting a Singleton Attractor in a Boolean Network Consisting of AND/OR Nodes

“…Let v i (t) represent the state of v i at time t. The overall expression level of all the genes in the network at time t is given by the vector Fig. 1 In this paper, we treat Boolean functions which can be represented by either If a BN is acyclic and does not have self-loops, there is a polynomial time algorithm for detecting an attractor [1,23]. In such a case, the number of attractors is only one and it is a singleton attractor.…”

“…Milano and Roli independently proposed a similar reduction [16]. Zhang et al developed algorithms with guaranteed average case time complexity [23]. For example, it is shown that in the average case, one of the algorithms identifies all singleton attractors in O(1.19 n ) time for a random BN with maximum indegree two.…”