Fisher Information Decay in a Circulant Genetic Channel: Eigenvalue Stability, Entropy Approximation, and a Critical Threshold at p(1 − p) = 1/9
Year: 2026
Type: preprint
Abstract: We model genetic transmission from a maternal to an offspring population as a noisy binary information channel governed by a symmetric circulant transition matrix C. The output Fisher information is derived exactly as Iout(p) = λ₂²/[2pq(p² + q²)], where λ₂ = 2p − 1 is the nontrivial eigenvalue of C and q = 1 − p, providing a spectral account of how allele-frequency contrast is lost to depolarization each generation. A key structural result is that the Fisher information loss per generation due to depolarization is a constant 4, independent of allele frequency. We introduce a heterozygosity-based entropy measure G(p) = √p(1 − p) + 2p(1 − p), originally derived in Glasenapp et al. [1], and confirm that it approximates Shannon entropy with maximum absolute error 0.039 bits. A structural proof shows that the conditional channel entropy satisfies H(Y | X) = HS (p) exactly, so the approximation H(Y | X) ≈ G(p) follows directly with the same error bound. A critical allele-frequency threshold at p∗(1 − p∗) = 1/9 (p∗ ≈ 0.873) is identified and supported by four independent characterizations: geometric (Fisher information volume det g∗ = 9), algebraic (λ₂² = 5/9 at threshold), structural (√p∗(1 − p∗) = 1/3) and statistical (fluctuation ratio Ψ(p∗) = 3). We further draw explicit structural analogies between this discrete channel framework and Villani’s theory of optimal transport and entropy production, identifying the Fokker–Planck drift–diffusion decomposition as the continuous counterpart of the selection–mutation balance in the circulant model.
Author: Jacqueline, Siqueira Glasenapp
Language: English
Subfield: Genetics
Domain: Life Sciences
Open Access status: green
OpenAlex ID: W7140128299