arxiv: 2605.13168 · v1 · submitted 2026-05-13 · 📊 stat.ME

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

Mijeong Kim , Minkyoung Cha , Ah Young Jeong

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Pith reviewed 2026-05-14 18:26 UTC · model grok-4.3

classification 📊 stat.ME

keywords Michaelis-Mentenheteroscedasticityclustered datavariance modelingnonlinear regressionenzyme kineticsinference

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The pith

A variance-aware procedure using simple working models stabilizes Michaelis-Menten estimates of Km and Vmax when errors vary with concentration or data are clustered.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops an estimation method for Michaelis-Menten models that replaces the usual constant-variance assumption with explicit working variance functions and random-effect covariances. For single curves the procedure reduces to one-dimensional root finding for Km followed by closed-form updates for Vmax and a scale parameter; the same score equations extend directly to clustered data. Simulations show that the heteroscedastic and cluster-aware fits recover variances more accurately and produce intervals with better efficiency and coverage than standard nonlinear least squares. Real enzyme-kinetic and soil data confirm that prespecified variance functions, especially the square-root form, yield lower information criteria than the homoscedastic baseline. The approach is implemented in an accompanying package for routine single-curve, grouped, and clustered analyses.

Core claim

A variance-aware procedure for Michaelis-Menten estimation and inference 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 Km followed by closed-form plug-in updates for Vmax 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.

What carries the argument

Variance-aware procedure based on conditional moment restrictions and implemented via conditionally Gaussian working models with prespecified variance functions

If this is right

Modeling heteroscedasticity improves variance recovery and interval efficiency relative to homoscedastic nonlinear least squares.
Cluster-aware fits restore fixed-effect coverage more effectively than pooled analyses that ignore clustering.
The square-root variance function gives the most stable empirical fit among the prespecified working models.
The workflow applies uniformly to single-curve, grouped, and clustered Michaelis-Menten data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same conditional-moment approach could be applied to other nonlinear kinetic models that currently rely on constant-variance assumptions.
If the working models prove robust, experimental designs in enzyme assays could deliberately vary substrate concentrations to exploit the stabilized inference.
Routine adoption would reduce the need for data transformations that alter the original-scale interpretation of Km and Vmax.

Load-bearing premise

The prespecified working variance functions and conditionally Gaussian models sufficiently approximate the true error distribution and clustering structure without biasing the estimates of Km and Vmax.

What would settle it

A dataset in which the square-root or other working variance function produces Km and Vmax point estimates that differ materially from homoscedastic nonlinear least squares while known simulation truth shows no bias would falsify the claim of practical stabilization.

Figures

Figures reproduced from arXiv: 2605.13168 by Ah Young Jeong, Mijeong Kim, Minkyoung Cha.

**Figure 2.** Figure 2: Fitted Michaelis–Menten curves for the best-fitting model [ [PITH_FULL_IMAGE:figures/full_fig_p026_2.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives a clean, implementable way to handle heteroscedasticity and clustering in Michaelis-Menten fits while keeping the mean-parameter estimates consistent.

read the letter

The main takeaway is that the authors use conditional moment restrictions to build estimating equations for Km and Vmax that remain unbiased even when the working variance model is misspecified. For single curves the procedure collapses to a one-dimensional root find for Km followed by closed-form updates for Vmax and the scale parameter; the clustered version adds a random-effect working covariance that preserves the same unbiasedness property. That is the concrete novelty relative to standard nonlinear least squares or off-the-shelf NLME fits. They also ship an R package, inferMM, which lowers the barrier for anyone who wants to try it. In the simulations the heteroscedastic versions recover variances better and produce more efficient intervals than homoscedastic NLS, while the cluster-aware fits restore proper coverage that pooled analyses lose. The real-data examples (self-driving lab and soil exoenzyme) show the square-root variance function giving the lowest information criteria, which is useful practical evidence. The derivations look straightforward and avoid circularity; the moment conditions deliver consistency for the mean parameters under correct specification of the Michaelis-Menten curve itself. The soft spots are modest. Efficiency still depends on how well one of their prespecified variance functions matches the data, so users will need to compare a few options via information criteria as the authors do. The simulation designs cover the main regimes but do not stress very small numbers of clusters or extreme imbalance, which could be checked in revision. Overall the work is focused and the empirical gains are clear. It is aimed at statisticians and biochemists who routinely analyze enzyme-kinetic curves with concentration-dependent noise or repeated measures. A reader who needs reliable original-scale inference in that setting will find the method and code immediately usable. The paper is solid enough on both the theory and the evidence to deserve a serious referee rather than a desk rejection.

Circularity Check

0 steps flagged

Derivation is self-contained via standard conditional moment restrictions

full rationale

The paper derives its estimators from conditional moment restrictions on the Michaelis-Menten mean function, yielding unbiased estimating equations for Km and Vmax that remain valid under variance misspecification. Single-curve estimation reduces to explicit one-dimensional root finding for Km followed by closed-form plug-in updates for Vmax and scale; the clustered extension uses a random-effect working covariance that preserves the same unbiasedness property. No step reduces by construction to a fitted input, no self-citation supplies a load-bearing uniqueness theorem, and no ansatz is smuggled in; the procedure is a direct quasi-likelihood application whose consistency and efficiency gains are independently verified by simulation and real-data information criteria.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The method relies on conditional moment restrictions as motivation and working models for heteroscedasticity and clustering; the variance scale is estimated rather than fixed a priori.

free parameters (1)

variance scale parameter
Estimated in closed-form plug-in after determining Km via root finding.

axioms (1)

domain assumption Conditionally Gaussian working models adequately represent the data-generating process for inference purposes.
Invoked to implement the variance-aware procedure through simple models.

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discussion (0)

Reference graph

Works this paper leans on

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