Comparison between numeric and approximate analytic solutions for the prediction of soil metal uptake by roots. Example of cadmium

How exogenous ligand enhances the efficiency of cadmium phytoextraction from soils?

Many experiments showed that exogenous ligands could enhance cadmium (Cd) phytoextraction efficiency in soils. Previous studies suggested that the dissociation and the apoplastic uptake of Cd complex could not fully explain the increase of root Cd uptake. Two hypotheses are evaluated to explain enhanced Cd uptake in the presence of ligand: i) enhanced apoplastic uptake of complex due to reduced apoplastic resistance and ii) complex internalization by membrane transporters. RESULTS: show that the ligand affinity for Cd is a key characteristic determining the potential mechanism for enhanced Cd uptake. When low molecular weight organic acids are applied, the complex dissociation could generally be fast (> 10-3.3 s-1) and result in the increased Cd uptake. When hydrophilic aminopolycarboxylic acids (APCAs) are applied in experiments without water or temperature stresses to the plant, the root water uptake flux could very likely be high (> 10-7.8 dm s-1), and the strong apoplastic complex uptake could enhance the root Cd uptake. When lipophilic APCAs are applied, the strong internalization of the complex by membrane transporters could result in the increased Cd uptake if the maximum internalization rate is high (> 10-12 mol dm-2 s-1). However, the complex internalization by membrane transporters must be experimentally confirmed.

Analytical solution of Nye-Tinker-Barber model by Laplace transform

The Nye-Tinker-Barber model is a classical convection-diffusion model for nutrient uptake by plant roots in cylindrical coordinates and has one nonlinear left Robin boundary condition with Michaelis-Menten function of concentration. First the Michaelis-Menten function is fitted into a function of time by numerical concentration at root surface from difference scheme, and then the Laplace and numerical inverse Laplace transforms - Zakian inversion method are taken to obtain the approximate analytical solution. Compared with other solutions made by difference scheme, Stehfest inversion method and previous analytical methods, it is found that the analytical solution obtained by Laplace and Zakian inversion transforms has higher accuracy and computation efficiency. This analytical method can be extended to other nutrient uptake models with Michaelis-Menten function.

Approximate nutrient flux and concentration solutions of the Nye-Tinker-Barber model by the perturbation expansion method

The Nye-Tinker-Barber model is a basic and representative one for single-ion nutrient uptake by plant root from the soil and we aim to derive its approximate analytical solutions of flux and concentration. We divide the rhizosphere into the inner and the outer fields, match the inner and the outer solutions near the root surface, and then obtain the approximate analytical solutions of nutrient uptake flux at the root surface and global nutrient concentration of the diffusion or the convection-diffusion Nye-Tinker-Barber model. The analytical and numerical fluxes of K⁺ and [Formula: see text] decay quickly to 0 in less than 3 days while [Formula: see text] and Cd²⁺ gradually decrease in more than 15 days; the depletion profile spread of [Formula: see text] is apparently narrower than [Formula: see text] and K⁺ in 24 days. The different flux and concentration patterns of 4 nutrients result from their mobility and solubility in the rhizosphere. In comparison with the numerical simulations and the previous analytical results, we find that the analytical flux will overestimate the numerical flux of [Formula: see text] and Cd²⁺ while the analytical concentration can accurately predict the numerical concentration; the flux and the concentration solutions of the convection-diffusion Nye-Tinker-Barber model can be simplified to the diffusion versions by the Péclet number, and they can more widely describe the transport of nutrients of different attributes in soils of different textures with different levels of saturation, conductivity and permeability.