A zero-order approach for estimating nutrient uptake length in streams: A Michaelis-Menten-based theoretical analysis

Abstract

Nutrient uptake in streams is commonly quantified by uptake length (S_w). Unlabeled nutrient addition experiments (steady plateau or pulse) are widely used because they are cost-effective, yet the standard analysis—linear regression of log-transformed, dilution-corrected concentrations—assumes first-order kinetics. That assumption breaks down under nutrient-saturated conditions and leads to biased estimates of S_w. Using a one-dimensional advection framework coupled with Michaelis–Menten (M–M) uptake, we demonstrate analytically that the classical first-order approach can overestimate S_w, particularly for pulse additions in which saturation is pronounced. We derive and evaluate an alternative, zero-order method that approximates M–M behavior when the substrate is non-limiting. Under nutrient-saturated conditions, the zero-order method yields S_w estimates that are substantially closer to the M–M benchmark than the first-order method. Therefore, the zero-order approach should be adopted to estimate S_w when the nutrient is not limiting. Furthermore, the new approach expands applicability to low-dispersion streams and improves inter-study comparability of stream nutrient retention metrics.