Distribution and enumeration of attractors in probabilistic boolean networks

Many mathematical models for gene regulatory networks have been proposed. In this study, the authors study attractors in probabilistic Boolean networks (PBNs). They study the expected number of singleton attractors in a PBN and show that it is (2 - (1/2)^L-1)ⁿ, where n is the number of nodes in a PBN and L is the number of Boolean functions assigned to each node. In the case of L=2, this number is simplified into 1.5n. It is an interesting result because it is known that the expected number of singleton attractors in a Boolean network (BN) is 1. Then, we present algorithms for identifying singleton and small attractors and perform both theoretical and computational analyses on their average case time complexities. For example, the average case time complexities for identifying singleton attractors of a PBN with L=2 and L=3 are O(1.601n) and O(1.763n), respectively. The results of computational experiments suggest that these algorithms are much more efficient than the naive algorithm that examines all possible 2n states.