Distinct characteristics of correlation analysis at the single-cell and the population level

Correlation analysis is widely used in biological studies to infer molecular relationships within biological networks. Recently, single-cell analysis has drawn tremendous interests, for its ability to obtain high-resolution molecular phenotypes. It turns out that there is little overlap of co-expressed genes identified in single-cell level investigations with that of population level investigations. However, the nature of the relationship of correlations between single-cell and population levels remains unclear. In this manuscript, we aimed to unveil the origin of the differences between the correlation coefficients at the single-cell level and that at the population level, and bridge the gap between them. Through developing formulations to link correlations at the single-cell and the population level, we illustrated that aggregated correlations could be stronger, weaker or equal to the corresponding individual correlations, depending on the variations and the correlations within the population. When the correlation within the population is weaker than the individual correlation, the aggregated correlation is stronger than the corresponding individual correlation. Besides, our data indicated that aggregated correlation is more likely to be stronger than the corresponding individual correlation, and it was rare to find gene-pairs exclusively strongly correlated at the single-cell level. Through a bottom-up approach to model interactions between molecules in a signaling cascade or a multi-regulator-controlled gene expression, we surprisingly found that the existence of interaction between two components could not be excluded simply based on their low correlation coefficients, suggesting a reconsideration of connectivity within biological networks which was derived solely from correlation analysis. We also investigated the impact of technical random measurement errors on the correlation coefficients for the single-cell level and the population level. The results indicate that the aggregated correlation is relatively robust and less affected. Because of the heterogeneity among single cells, correlation coefficients calculated based on data of the single-cell level might be different from that of the population level. Depending on the specific question we are asking, proper sampling and normalization procedure should be done before we draw any conclusions.