Optimal compression and transcriptional control

The information bottleneck procedure provides a selective compression, using only a limited number of bits to describe the state of a system while preserving as much of the relevant information as possible. We are interested in using these ideas to describe enhancers, the regulatory elements that control transcription in more complex organisms, identifying the state of the enhancer with a compressed description of the concentrations of transcription factors (TF) to which it is responding. In general the bottleneck problem has to be solved numerically, but here we search for limits in which analytic progress is possible. If conditional distributions are Gaussian and effective noise levels are small, we derive a scaling relation onto which all optimal bottleneck curves collapse. We verify this scaling relation on the example of gene expression of the gap genes in early fruit fly development, which even goes slightly above the low noise limit. Mapping the compressed variables of the bottleneck to enhancer states, we explore the connections between noise levels and the cooperativity of responses, and the changing structure of the problem as we include more TF inputs to a single enhancer.