Variance-Aware Estimation and Inference for Michaelis--Menten Models with Heteroscedastic Errors and Clustered Measurements

Abstract

Michaelis--Menten analysis is often conducted by nonlinear least squares under a constant-variance assumption, even though enzyme-kinetic data frequently display concentration-dependent heteroscedasticity and often include repeated or clustered measurements. We develop a variance-aware procedure for Michaelis--Menten estimation and inference that is motivated by conditional moment restrictions and implemented through simple conditionally Gaussian working models. For single curves, the method reduces to one-dimensional root finding for $K_m$ followed by closed-form plug-in updates for $V_{\max}$ and a variance scale parameter; the same score logic yields a cluster-level extension through a random-effect-induced working covariance. In simulation, modeling heteroscedasticity improved variance recovery and interval efficiency relative to homoscedastic nonlinear least squares, while cluster-aware semiparametric and NLME fits restored fixed-effect coverage far more effectively than pooled analyses that ignored clustering. In self-driving laboratory and soil exoenzyme data, heteroscedastic models achieved lower information criteria than homoscedastic nonlinear least squares, with the square-root variance function giving the most stable empirical fit among the prespecified working models. We implement the workflow in the companion \texttt{inferMM} package for single-curve, grouped, and clustered Michaelis--Menten analysis. These results show that simple variance-function and covariance modeling can stabilize original-scale Michaelis--Menten inference when variability changes with substrate concentration or measurements are clustered.