  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 which 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 leading 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 transfered to examples of self-regulatory systems with a larger
number of gene copies.
