Typicality in high-dimensional gene regulation out of equilibrium*

Growing evidence shows that cells can use biochemical energy while controlling their genes with networks of proteins. What new regulatory behaviors are unlocked by these energy expenditures? How do these functions grow more sophisticated or constrained as governing networks grow larger or gain interactions? Using graph theory, we explore how multiple transcription factors modulate the expression of genes through their transcription rates, in or out of equilibrium. With additional regulatory proteins, the potential mathematical complexity of these responses—and hence the sensitivity attained by a single transcription factor or the number of times it can change roles from activating to repressing gene expression at different concentrations—can explode rapidly, along with the contrast between nonequilibrium and equilibrium behaviors. Yet biophysical constraints—such as binding site overlaps and shared kinetic features—markedly simplify attainable behaviors. We uncover how many microscopically-distinct biological scenarios can collapse to a few mathematical forms, consistent with the sparse character of transcriptional networks. These collapses follow from how regulatory responses can depend only on coarse-grained structural features of networks. Our findings help identify consequences and signatures of energy expenditure in architectures found in transcription across organisms.