Expected KL risk quantifies when first-order power-law approximations are sufficient

Abstract

Abstract Biochemical Systems Theory (BST) often replaces nonlinear rate laws by first-order log–Taylor power-law approximations, but deciding when this truncation is adequate remains difficult. We derive a closed-form leading-order expression for the expected conditional Kullback–Leibler (KL) risk incurred by using the first-order model instead of the local second-order log expansion. Under Gaussian log-input fluctuations with covariance $$\Sigma$$ and homoscedastic Gaussian log-output noise with variance $$s^2$$ , the risk reduces to a trace contraction of the local log-curvature Hessian H with $$\Sigma$$ . The criterion is therefore directly estimable from perturbation data or mechanistic models near an operating point. We also identify the leading correction from non-Gaussian inputs through fourth-order cumulants. Toy-model calculations and two biochemical case studies show that the criterion not only matches Monte Carlo estimates, but also identifies operating conditions and perturbation directions for which first-order BST is expected to fail.