Inference of Boolean networks under constraint on bidirectional gene relationships

The coefficient of determination (CoD) has been used to infer Boolean networks (BNs) from steady-state data, in particular, to estimate the constituent BNs for a probabilistic BN. The advantage of the CoD method over design methods that emphasise graph topology or attractor structure is that the CoD produces a network based on strong predictive relationships between target genes and their predictor (parent) genes. The disadvantage is that spurious attractor cycles appear in the inferred network, so that there is poor inference relative to the attractor structure, that is, relative to the steady-state behaviour of the network. Given steady-state data, there should not be a significant amount of steady-state probability mass in the inferred network lying outside the mass of the data distribution; however, the existence of spurious attractor cycles creates a significant amount of steady-state probability mass not accounted for by the data. Using steady-state data hampers design because the lack of temporal data causes CoD design to suffer from a lack of directionality with regard to prediction. This results in spurious bidirectional relationships among genes in which two genes are among the predictors for each other, when actually only one of them should be a predictor of the other, thereby creating a spurious attractor cycle. This paper characterises the manner in which bidirectional relationships affect the attractor structure of a BN. Given this characterisation, the authors propose a constrained CoD inference algorithm that outperforms unconstrained CoD inference in avoiding the creation of spurious non-singleton attractor. Algorithm performances are compared using a melanoma-based network.