Statistical physics models of transcriptional state

The state of a cell can be defined in part by the expression levels of all its genes. New experiments combine super-resolution microscopy with combinatorial probes to count each RNA molecule transcribed from each of many genes, giving a snapshot of the cell state in its high dimensional space. We use ideas from statistical physics to describe the distribution of ~100 genes from ~80,000 human cells. If we treat each gene as on or off (expressed above or below its mean level), we can build maximum entropy models for the resulting binary variables, matching their means and pairwise correlations. These are Ising models, and provide surprisingly good predictions of higher-order correlations in the network. However, if we try to match the mean and pairwise correlations of the full range of molecular counts, the resulting models fail completely, since the observed (co)variances fall outside the bounds of what these models can achieve. This leads to the problem of building maximum entropy models that match higher moments, or even full marginal distributions, along with pairwise correlations. In a different direction, we ask how the distribution of expression levels evolves as we coarse grain our description of the system, removing modes that make the smallest contribution to the total variance. Distributions appear to approach a fixed non-Gaussian form with fewer modes remaining, pointing toward simpler but still high dimensional dynamics.