Invariant non-equilibrium dynamics of transcriptional regulation optimize information flow

Eukaryotic gene regulation is based on stochastic yet controlled promoter switching, during which genes transition between transcriptionally active and inactive states. Despite the molecular complexity of this process, recent studies reveal a surprising invariance of the "switching correlation time" ($T_C$), which characterizes promoter activity fluctuations, across gene expression levels in diverse genes and organisms. A biophysically plausible explanation for this invariance remains missing. Here, we show that this invariance imposes stringent constraints on minimal yet plausible models of transcriptional regulation, requiring at least four system states and non-equilibrium dynamics that break detailed balance. Using Bayesian inference on Drosophila gap gene expression data, we demonstrate that such models (i) accurately reproduce the observed $T_C$-invariance; (ii) remain robust to parameter perturbations; and (iii) maximize information transmission from transcription factor concentration to gene expression. These findings suggest that eukaryotic gene regulation has evolved to balance precision with reaction rate and energy dissipation constraints, favoring non-equilibrium architectures for optimal information transmission.