Fig. 2

Information-theoretic single-cell analysis. Recall that I(g) measures the heterogeneity of a cellular population with respect to the expression of g: \(I(g)=0\) when transcripts are expressed uniformly and increases as transcripts are expressed preferentially in a subset of cells, reaching a maximum \(I(g)=\log (N)\), where N is the number of cells sequenced, when only one cell expresses the gene. a–e Plots of expression heterogeneity, I(g) (normalised by the theoretical maximum, \(\log (N)\)) against log mean expression for the bench-marking sc-Seq data sets described in the main text. In each panel, each point represents a gene profiled. The number of genes associated with large values of I(g) increases with the number of cell types present in the population profiled, showing I(g) as a valid measure of cell type diversity. Panel a shows data from a technical control [42] (number of cell types, \(C = 1\)), b a mixture of three cancerous cell lines [46] (\(C = 3\)), c FACS sorted immune cells [52] (\(C = 4\)), d a sample of mouse bone marrow [41] (\(C = 14\)), and e a multi-organ mouse cell atlas [44] (\(C = 56\)). f–h Biologically meaningful cell annotations are associated with high inter-cluster heterogeneity. Established cell annotations for the f Tian, g Zheng and h Stumpf data are associated with higher inter-cluster heterogeneity than expected by chance (i.e., in randomly permuted clusters; significance is assessed using a one-sided exact test with \(10^4\) permutations; y axes show \(\log _{10}(p + 1)\)). In all panels the red line shows \(p < 0.05\), false discovery rate corrected for 500 trials [2, 8]. Genes below this threshold are significantly different gene expression patterns across the set of identified cell types. i Summary statistics for the total inter-cluster heterogeneity \(H_S = \sum _g H_S(g)\) based on established empirical and randomly permuted cell annotations (\(10^4\) random permutations in each case). These statistics show the strong association of high \(H_S\) with biologically meaningful groupings of cells. j A Uniform Manifold Approximation and Projection (UMAP) [29] plot of the top 500 genes by I(g) for the Stumpf data set; each point is a cell, coloured by its scEC cluster. This shows that I(g) is able to capture the continuous variation of developing cell types