Thermodynamic Constraints Impose a Universal Saturating Response Function Across Biological Scales

Biological systems operate under thermodynamic constraints and information delays. Across scales—from enzymes to neurons to behavior—they exhibit saturating, monotonic, and diminishing-return response profiles. We derive a family of rational response functions from first principles, using the first law of thermodynamics and entropy production bounds, coupled with predictive coding efficiency. These functions, such as $R(t) = A/(1 + kt)$ and $R(t) = At/(B + t)$, outperform exponential and sigmoid models in satisfying energy-limited, information-delayed, and concave dynamics. Empirical examples include Michaelis–Menten kinetics, neural gain modulation, and behavioral delay discounting. A thermodynamic derivation demonstrates that only low-order rational forms allow optimal conversion of finite energy into adaptive output while minimizing prediction error. This framework offers a unifying grammar for dynamic responses in living systems and may extend to artificial adaptive processes under similar constraints.