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We sort different populations of blood cells using a number of fluorescent flow cytometry markers and then sequence RNA. We want to see what the transcriptome tells us about the similarity and relation between these cells. In my experience on bulk RNA-seq data, there is a very good agreement between flow cytometry and mRNA expression for the markers. It's good to remember we sort using a few markers only, while there are hundreds if not thousands of cell surface proteins.

Should we exclude genes of these sorting (CD) marker proteins when we perform PCA or other types of clustering?

The argument is that after sorting, these genes may dominate clustering results, even if the rest of the transcriptome would tell otherwise, and thus falsely confirm similarity relations inferred from sorting.

Should we at least check the influence of these genes on clustering results?

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I do not think there is a simple "yes" or "no" answer here.

A good starting point would be, as you suggest, use all the genes and assess the results in the light of the marker genes and expected results. This could both serve as as good quality control as well as give you overview of all the processes happening in the cells.

Depending on the marker genes effect and biological question you may then want to remove the marker genes with potential ordering effect, or even other genes, e.g. by GO terms or pathways.

You most likely want to account for the cell cycle phases as well. And check for other technical factors. I recommend Bioconductor pipeline to get inspired https://www.bioconductor.org/help/workflows/simpleSingleCell/ when it comes to scRNA-seq analyses.

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