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 O(1.19(n)) time for K = 2, which is much faster than the naive O(2(n)) time algorithm, where n is the number of genes and K 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.
Highlights
The advent of DNA microarrays and oligonucleotide chips has significantly sped up the systematic study of gene interactions [1,2,3,4]
Since the number of singleton attractors is small in most cases, it is expected that the algorithm does not examine many partial Gene Activity Profile (GAP) with large m
After reordering all genes according to their outdegrees from largest to smallest, the average case time complexity of the algorithm for K = 1 to K = 10 is given in the second row of Table 2
Summary
The advent of DNA microarrays and oligonucleotide chips has significantly sped up the systematic study of gene interactions [1,2,3,4]. Mochizuki introduced a general model of genetic networks based on nonlinear differential equations [21] He analyzed the number of steady states in that model, where steady states are again closely related to singleton attractors in Boolean networks. Devloo et al proposed algorithms for finding steady states of various biological networks using constraint programming [20], which can be applied to identification of singleton attractors in Boolean networks. Though we do not have strong evidence that small attractors are more important than those with long periods, it seems that cell cycles correspond to small attractors and large attractors are not so common (with the exception of circadian rhythms) in real biological networks As a minimum, these extensions show that application of the proposed techniques is not limited to the singleton attractor problem.
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