A Metacell Model of Single Cell RNA-seq Counts Yields a Gaussian Mixture Model in PCA Space

Single cell RNA sequencing (scRNAseq) has revolutionized cellular biology over the past decade. At its most basic, a scRNAseq dataset is a cell by gene matrix with entries giving the transcription count of a particular gene in a particular cell. A common workflow applies log-normalization and z-scoring to the count matrix and then embeds the cells in a lower dimensional space using principal component analysis (PCA). A large number of computational tools use the PCA embedding as a starting point, but statistical models for the distribution of cells in the PCA space are lacking. As a result, many tools depend on ad-hoc assumptions that likely lead to signal loss. In this work, we address this gap using the metacell model, a statistical model of count matrices introduced by previous authors. We show that application of the workflow to count matrices generated under a metacell model yields PCA embeddings distributed as a Gaussian mixture model (GMM). To our knowledge, this is the first result connecting a statistical model of counts to the distribution of cells in PCA space. Our derivation depends on methods from random matrix theory (RMT) and we extend and establish connections between RMT and scRNAseq count matrices. We then use the predicted GMM to test the metacell model on 11 datasets. The predictions are accurate for counts generated by the metacell model but underestimate the variance of real cells for most of our datasets. We show that this error largely arises from gene correlations within metacells, which the metacell model does not capture. Some of our datasets are fit relatively well by the metacell model, suggesting that the fit of the metacell model to true datasets depends on sufficiently dense sampling. Finally, we show that the strength of noise in the PCA space varies significantly across metacells, suggesting that statistical modeling of PCA embeddings can improve existing downstream workflows. Our results provide a framework for deriving and testing statistical models of PCA embeddings from statistical models of counts and identify a limitation of the metacell description for scRNAseq datasets.