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

TL;DR

This paper explores the $rak{so}(2,1)$ symmetry in exact solutions of a stochastic self-repressing gene, revealing how symmetry breaking affects gene expression dynamics and distribution patterns.

Contribution

It characterizes the symmetry underlying exact stochastic solutions of a self-repressing gene and analyzes the effects of symmetry breaking on gene expression behavior.

Findings

Existence of two solution branches with different switching rates

Low switching rate branch approaches deterministic behavior

High switching rate branch shows sub-Fano noise characteristics

Abstract

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