The Hill coefficient as Fisher information: An information-theoretic identity for allosteric cooperativity

The Hill coefficient n_H has quantified cooperative binding in biology for over a century, yet its information-theoretic content is rarely made explicit. We prove a sharply local algebraic identity: for a Hill-type dose-response curve, the Fisher information with respect to the log-dose parameterθ=ln(c/K_d), evaluated at the midpointp=1/2 (equivalently c=K_d), equals I_θ=n_H²/4. The result is parameterisation-dependent: away from the midpoint it generalises to I(θ)=n_H²p(1-p), recovering n_H²/4 only at half-saturation. The identity itself follows from the standard Fisher information of a Bernoulli outcome under a logistic parameterisation; our contribution is its biological reading as a quantitative midpoint decision-precision measure for cooperative biological switches. For hemoglobin (n_H≈2.8), I_θ=1.96. For the bacterial flagellar motor (n_H≈10.3), I_θ≈26.5. We examine five biological systems spanning molecular, cellular, organismal, and quantum-biological scales, then connect the identity to Frank's theorem (J. Evol. Biol. 2009) that natural selection maximises Fisher information. We separate direct applications, where a single cooperative-binding curve carries a well-defined Hill coefficient and the identity applies pointwise (hemoglobin, the bacterial flagellar motor), from conceptual extensions, where a binary biological decision need not be underwritten by a single cooperative-binding Hill coefficient (gene-expression thresholds, chemotactic adaptation, the avian compass). Four falsifiable predictions are stated.