# What is correct way to calculate the expected overlap of protein sets in two related biological system

I have a question regarding the statistics of protein set differences in two biological samples.

I have performed quantitative proteomics analyses on both WT and KO cells (derived from the WT cells by deleting a gene), both before and after stimulation. I detect the same 100 proteins in all four experiments. I classify the proteins based on their response to the stimulation: some proteins are upregulated (“U”) in the stimulatstaed cells compared to the unstimulated control, some are downregulated (“D”), and some stay constant (“C”). I am interested in what can be gleaned from the sets of proteins with certain response within a given cell type, as well as whether their responses stay the same or differ in the two cell types.

Let’s focus on just U proteins.

The data says there are total of 30 of them in WT cells and 20 in the KO cells, and the sets overlap partially with 10 common proteins. If the correlation of protein classes in the two cell types is zero, the most likely outcome for the overlapping set size is 6 proteins just by chance, where the hypergeometric probability is at its maximum. In R:

dhyper(0:20, 30, 70, 20) ; plot(dhyper(0:20, 30, 70, 20))
sum(dhyper(0:10, 30, 70, 20))


So 6 proteins is expected by chance and the detected 10 is considered a (significant) enrichment. The reason for the low bar of 6 proteins is of course that we are using a fully uncorrelated classes between WT and KO cells as the probability yardstick and any number of proteins above that is considered enrichment (and below, de-enrichment). It just sounds like making too big of a deal from measly 10 protein overlap, which is only 50% of the smaller set.

Is this the correct approach to assign significance to the my data? Does it make a difference that there is cause-effect relationship here, KO cells are derived from the WT cells, so maybe the calculations should be different from the symmetric comparisons where one may not expect any correlation at all?

Your classification into "U," "D," and "C" groups is throwing away a lot of potentially interesting information that addresses the inherent correlation that you rightly expect from the origin of the KO cells from the WT.

Use continuous measures of protein expression instead. Set up a regression model for protein expression with cell type (WT/KO), stimulation (yes/no), and their interaction as predictors. The proteins you care about are those that show a significant interaction coefficient, meaning that the effect of stimulation depends on the cell type.

With 100 proteins you should be able to use the well established tools provided by the Bioconductor limma package. Although developed for microarray gene-expression data, its fundamental basis in regression analysis makes it useful for all sorts of data with multiple dependent variables. See for example this page and its links for how to adapt it to proteomics.