Estimating the frequency of multiplets in single-cell RNA sequencing from cell-mixing experiments

In single-cell RNA-sequencing, it is important to know the frequency at which the sequenced transcriptomes actually derive from multiple cells. A common method to estimate this multiplet frequency is to mix two different types of cells (e.g., human and mouse), and then determine how often the transcriptomes contain transcripts from both cell types. When the two cell types are mixed in equal proportion, the calculation of the multiplet frequency from the frequency of mixed transcriptomes is straightforward. But surprisingly, there are no published descriptions of how to calculate the multiplet frequency in the general case when the cell types are mixed unequally. Here, I derive equations to analytically calculate the multiplet frequency from the numbers of observed pure and mixed transcriptomes when two cell types are mixed in arbitrary proportions, under the assumption that the loading of cells into droplets or wells is Poisson.