Using the principle of entropy maximization to infer genetic interaction networks from gene expression patterns

Abstract

We describe a method based on the principle of entropy maximization to identify the gene interaction network with the highest probability of giving rise to experimentally observed transcript profiles. In its simplest form, the method yields the pairwise gene interaction network, but it can also be extended to deduce higher-order interactions. Analysis of microarray data from genes in Saccharomyces cerevisiae chemostat cultures exhibiting energy metabolic oscillations identifies a gene interaction network that reflects the intracellular communication pathways that adjust cellular metabolic activity and cell division to the limiting nutrient conditions that trigger metabolic oscillations. The success of the present approach in extracting meaningful genetic connections suggests that the maximum entropy principle is a useful concept for understanding living systems, as it is for other complex, nonequilibrium systems.

Fig. 1.

A plot of the rank-ordered standard deviations (σ) of raw expression profiles for the 4,670 genes in the short-period data set. Arrows indicate cutoff values for 582 genes and 1,008 genes. (Inset) The Spearman correlation between the calculated interactions and those that result when noise of a fixed amplitude is added to the raw profiles. Random noise from a Gaussian distribution of width δ is added to each of the raw expression profiles, and the network for the noise-enhanced data was calculated and compared with the network from the raw data by using the Spearman correlation. When the 500 genes with largest profile amplitudes are retained (solid line), noise of δ = 52.2, ≈10 times the estimated background level, does not significantly change the network. The network is more sensitive to noise when 1,000 genes are retained (dashed line), but the correlation is still 0.9 between the original network and that with noise added at six times the estimated background.

Fig. 2.

The distribution of the relative strengths of pairwise and three-gene interactions among the 582 genes in the short-period data set showing the largest fluctuations during metabolic oscillations. Each curve is normalized to unit area.

Fig. 3.

The network of the strongest 110 pairwise interactions inferred by entropy maximization using the 582 genes showing the most marked fluctuations in transcript levels in the data set from yeast chemostat cultures showing 40-min metabolic oscillations (26). Nodes are identified by gene names and color-coded to indicate the cell process in which they participate (there is some ambiguity in assigning genes to categories). The solid blue lines denote positive couplings, and the dashed red lines denote negative couplings. The identity of the hubs circled in red is discussed in the text.

Fig. 4.

A diagrammatic representation of the cellular processes identified by the network hubs among the 582 (level 1) and the 1,008–2,000 (level 2) genes exhibiting the most marked fluctuations in transcript levels during 40-min metabolic oscillations. PKA and TOR represent the PKA and TOR nutrient signaling pathways; other ovals contain the designations of hub genes identified as described in the text and color-coded by cell process as in Fig. 3 (see SI Text for details and references on the hub genes).