Figure 1. Illustration of decoherence and ‘Correlation by Individual Level Product’ (CILP).

(a) Co-expression networks in two different environments. Environment one represent normal (or control/baseline) conditions where the expression levels of gene 1 and gene 2 are highly correlated and co-regulated. (b) Environment 2 represents a stressful or unhealthy condition leading to lower correlation between the expression levels of gene 1 and gene 2. At the network level, this translates into a lower network degree (i.e., lower average correlation across genes or fewer connected nodes). We call this change in the correlation structure 'decoherence'. This is what we could expect if stressful conditions lower transcriptional robustness and lead to dysregulation of gene expression. (c) Differences in correlation between the expression levels of gene 1 and gene 2 in cases versus controls, or between genotypes, which translates into an average difference in product between these groups.

Figure 1—figure supplement 1. Simulations reveal power to detect correlation QTLs across a range of effect sizes.

(A) Power to detect correlation QTLs as a function of the simulated effect size and the significance cutoff. Each point represents the results from 10,000 pairs of simulated traits, across 1,000 individuals; effect sizes represent the degree to which samples originating from different genotypic classes exhibit different levels of correlation for each trait pair (higher effect sizes = larger differences in correlation between each genotypic class). For each simulated set of 10,000 pairs, we tested for correlation QTL using the approach described in the main text. (B) Comparison of simulation results: 1) when a multivariate normal distribution was used to simulate pairs of continuous trait values; 2) when a negative binomial distribution was used to simulate pairs of count data; and 3) when data were simulated as in 1, but mean gene expression levels for each gene in a given pair were included as covariates in linear models testing for correlation QTL.

Figure 1—figure supplement 2. Simulated batch effects on transcriptional correlation structure do not produce false positive correlation QTLs.

500 genes were simulated in 100 individuals over two batches, with no covariance between genes in batch 1, and a covariance of 0.3 among all pairs of genes in batch 2. Panel A presents a PCA on the simulated gene expression data, with individuals colored by batch. To assess the false positive rate under the null, a genotype was simulated with no effect, and CILP was applied to estimate the genetic effect on the degree of correlation between each gene pair. Panel B shows that the observed p-values from CILP both when adjusting for batch (by including it as a covariate in a linear model) as well as ignoring batch are not greater than p-values observed under the expected uniform distribution, suggesting that batch effects do not cause inflation or affect the estimation of the genetic effect.