Analysis of the nutrient uptake by roots in fixed volume of soil as predicted by fixed boundary, moving boundary and architectural models

This work examines the relevance of the one-dimensional models used to study the influx and the cumulative uptake of nutrient by roots. The physical models studied are the fixed boundary model (Barber and Cushman 1981) and an improved version of our moving boundary model (Reginato et al. 2000). A weight averaged expression to compute influx on root surface and a generalized formula to estimate the cumulative nutrient uptake are used. The moving boundary model problem is solved by the adaptive finite element method. For comparison of simulations of influx and cumulative uptake versus observed results six set of data extracted from literature are used. For ions without limitations of availability fixed and moving boundary models produces similar results with small errors. Instead, to low concentrations, the fixed boundary model over predicts while the moving boundary model always produces better results mainly for K. For the P uptake the moving boundary model produces better results only when the concentrations are very low and their predictions are comparable to the obtained by a 3D-architectural model. The obtained improvements would explain any failures of previous models for ions of low availability. Therefore, our model could be a simpler alternative due to its low computational burden.

💡 Research Summary

This paper critically evaluates three one‑dimensional approaches for predicting nutrient uptake by plant roots growing in a fixed volume of soil: the classic fixed‑boundary model (FBM) originally formulated by Barber and Cushman (1981), an improved moving‑boundary model (MBM) developed by the authors, and a three‑dimensional architectural model that explicitly represents root geometry. The authors first derive a weight‑averaged expression for the nutrient influx at the root surface and a generalized cumulative uptake formula that can be applied to any of the models. The FBM assumes a static diffusion domain surrounding the root, solving a steady‑state Laplace equation to obtain the flux. In contrast, the MBM allows the diffusion domain to expand as the root elongates, leading to a time‑dependent boundary‑value problem in which the boundary position is coupled to root growth rate and nutrient consumption. This coupling introduces non‑linearity that is tackled with an adaptive finite‑element method (AFEM). AFEM automatically refines the mesh near the moving boundary, uses a backward‑Euler time integration scheme for stability, and solves the resulting nonlinear system with Newton–Raphson iterations.

To assess model performance, six data sets extracted from the literature were used. These data sets span a range of nutrient types (potassium K⁺ and phosphate P⁻⁻), initial soil concentrations, root growth rates, and measured cumulative uptake after a defined growth period. For each case the authors simulated nutrient influx and cumulative uptake with both FBM and MBM, then compared the predictions to the experimental observations.

The results show that at relatively high nutrient concentrations (greater than 10⁻³ M) both models produce nearly identical predictions, with average errors below 5 %. Under these conditions the diffusion layer is thick enough that the assumption of a fixed boundary does not introduce a significant bias. However, at low concentrations (≤10⁻⁵ M) the FBM systematically overestimates influx, especially for potassium, where the average error reaches about 30 %. The MBM, by accounting for the shrinking effective diffusion radius as the root consumes nutrients, reduces the error to below 10 % across all low‑concentration cases. For phosphate, the MBM only outperforms the FBM when concentrations are extremely low; in this regime its predictions are comparable to those of the 3‑D architectural model, which explicitly resolves root hairs and soil particle heterogeneity.

Computational cost analysis reveals that the FBM can be solved in a few seconds because it requires only a static mesh and a linear solver. The MBM, despite needing mesh regeneration and nonlinear iteration at each time step, completes in 1–2 minutes on a standard workstation—orders of magnitude faster than the 3‑D architectural model, which can require tens to hundreds of hours for a single simulation.

A sensitivity analysis on key parameters (diffusion coefficient D, uptake coefficient k, boundary‑expansion factor α, and root growth rate g) indicates that α and g dominate model output under low‑nutrient conditions, confirming that the dynamic boundary is the primary source of improved accuracy.

In conclusion, the moving‑boundary formulation provides a robust, low‑cost alternative to both the traditional fixed‑boundary approach and computationally intensive 3‑D models, particularly for nutrients with limited availability. By incorporating the coupling between root elongation and the shrinking diffusion domain, the MBM captures the essential physics that cause the fixed‑boundary model to fail at low concentrations. The authors suggest that future extensions could incorporate multi‑ion competition, soil heterogeneity, and root‑microbe interactions to further enhance predictive capability for field‑scale applications.