Belief propagation in genotype-phenotype networks.

Graphical models have proven to be a valuable tool for connecting genotypes and phenotypes. Structural learning of phenotype-genotype networks has received considerable attention in the post-genome era. In recent years, a dozen different methods have emerged for network inference, which leverage natural variation that arises in certain genetic populations. The structure of the network itself can be used to form hypotheses based on the inferred direct and indirect network relationships, but represents a premature endpoint to the graphical analyses. In this work, we extend this endpoint. We examine the unexplored problem of perturbing a given network structure, and quantifying the system-wide effects on the network in a node-wise manner. The perturbation is achieved through the setting of values of phenotype node(s), which may reflect an inhibition or activation, and propagating this information through the entire network. We leverage belief propagation methods in Conditional Gaussian Bayesian Networks (CG-BNs), in order to absorb and propagate phenotypic evidence through the network. We show that the modeling assumptions adopted for genotype-phenotype networks represent an important sub-class of CG-BNs, which possess properties that ensure exact inference in the propagation scheme. The system-wide effects of the perturbation are quantified in a node-wise manner through the comparison of perturbed and unperturbed marginal distributions using a symmetric Kullback-Leibler divergence. Applications to kidney and skin cancer expression quantitative trait loci (eQTL) data from different mus musculus populations are presented. System-wide effects in the network were predicted and visualized across a spectrum of evidence. Sub-pathways and regions of the network responded in concert, suggesting co-regulation and coordination throughout the network in response to phenotypic changes. We demonstrate how these predicted system-wide effects can be examined in connection with estimated class probabilities for covariates of interest, e.g. cancer status. Despite the uncertainty in the network structure, we demonstrate the system-wide predictions are stable across an ensemble of highly likely networks. A software package, geneNetBP, which implements our approach, was developed in the R programming language.