Bridging Genetics and Information Theory: Fisher Information Limits on Genetic Stability, Diversity, and Regulation

I introduce a Fisher Information (FI) based framework for quantifying the fidelity of genetic information transmission across generations. Genetic inheritance is modeled as a noisy information channel defined by a symmetric circulant transition matrix, allowing closed-form expressions for Fisher Information at both the source population and the transmitted signal. This formulation complements entropy-based approaches by directly characterizing estimation precision and the loss of predictability in genetic systems. I show that information decay is governed entirely by the square of the nontrivial channel eigenvalue, providing a natural and biologically interpretable measure of information retention under mutation and recombination. Within this framework, I identify a sharp loss of predictability in gene regulatory networks (GRNs), detectable both as a collapse of classical Fisher Information, and as an entanglement-collapse threshold in the formally equivalent quantum channel representation. Remarkably, the functional instability limit observed in GRNs coincides with the quantum coherence limit within a narrow parameter range, revealing a shared information-theoretic boundary. This convergence provides a principled explanation for the sharp, non-monotonic boundaries reported in GRN inference from single-cell RNA-seq data. More broadly, the results suggest that biological regulatory stability is constrained by fundamental limits on information transmission, linking population genetics, information geometry, and quantum-inspired models of gene regulation.