Gene-gene cooperativity in small networks.

We show how to construct a reduced description of interacting genes in noisy, small regulatory networks using coupled binary spin variables. Treating both the protein number and gene expression state variables stochastically and on equal footing, we propose a mapping that connects the molecular level description of networks to the binary representation. We construct a phase diagram indicating when genes can be considered to be independent and when the coupling between them cannot be neglected, which can lead to synchrony or correlations. We find that an appropriately mapped Boolean description reproduces the probabilities of gene expression states of the full stochastic system very well, and can be transferred to examples of self-regulatory systems with a larger number of gene copies.