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<p>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 <InlineEquation ID="IEq1"><EquationSource Format="TEX">\(\Sigma\)</EquationSource></InlineEquation> and homoscedastic Gaussian log-output noise with variance <InlineEquation ID="IEq2"><EquationSource Format="TEX">\(s^2\)</EquationSource></InlineEquation>, the risk reduces to a trace contraction of the local log-curvature Hessian <i>H</i> with <InlineEquation ID="IEq3"><EquationSource Format="TEX">\(\Sigma\)</EquationSource></InlineEquation>. 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.</p>

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

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  • 发表日期：

    2026年5月27日

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  • Chikoo Oosawa

摘要

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.