On entropy and information in gene interaction networks.

Modern biological experiments often produce candidate lists of genes presumably related to the studied phenotype. One can ask if the gene list as a whole makes sense in the context of existing knowledge: Are the genes in the list reasonably related to each other or do they look like a random assembly? There are also situations when one wants to know if two or more gene sets are closely related. Gene enrichment tests based on counting the number of genes two sets have in common are adequate if we presume that two genes are related only when they are in fact identical. If by related we mean well connected in the interaction network space, we need a new measure of relatedness for gene sets. We derive entropy, interaction information and mutual information for gene sets on interaction networks, starting from a simple phenomenological model of a living cell. Formally, the model describes a set of interacting linear harmonic oscillators in thermal equilibrium. Because the energy function is a quadratic form of the degrees of freedom, entropy and all other derived information quantities can be calculated exactly. We apply these concepts to estimate the probability that genes from several independent genome-wide association studies are not mutually informative; to estimate the probability that two disjoint canonical metabolic pathways are not mutually informative; and to infer relationships among human diseases based on their gene signatures. We show that the present approach is able to predict observationally validated relationships not detectable by gene enrichment methods. The converse is also true; the two methods are therefore complementary. The functions defined in this paper are available in an R package, gsia, available for download at https://github.com/ucsd-ccbb/gsia.

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