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We chemically characterize the symmetries underlying the exact solutions of astochastic negatively self-regulating gene. The breaking of symmetry at lowmolecular number causes three effects. Two branches of the solution exist,having high and low switching rates, such that the low switching rate branchapproaches deterministic behavior and the high switching rate branch exhibitssub-Fano behavior. Average protein number differs from the deterministicallyexpected value. Bimodal probability distributions appear as the protein numberbecomes a readout of the ON/OFF state of the gene.

Biological implications of $\mathfrak{so}(2,1)$ symmetry in exact solutions for a self-repressing gene