Maximum entropy models for patterns of gene expression

New experimental methods make it possible to measure the expression levels of many genes, simultaneously, in snapshots from thousands or even millions of individual cells. Current approaches to analyze these experiments involve clustering or low-dimensional projections, and often start with the assumption that distinct cell types exist. Here we use the principle of maximum entropy to obtain a probabilistic description that captures the observed presence or absence of mRNAs from hundreds of genes in cells from the mammalian brain. We construct the Ising model compatible with experimental means and pairwise correlations, and validate it by showing that it gives good predictions for higher-order statistics. We find that the probability distribution of cell states has many local maxima. Grouping cells according to these maxima (or energy minima) gives a classification in good agreement with currently assigned cell types. We show that when assignments disagree our model is dividing cell types into subtypes with clearly distinguishable expression patterns. These results make concrete the intuition that types or classes of cells are emergent behaviors.