An updated model for nitrate uptake modelling in plants. II. Assessment of active root involvement in nitrate uptake based on integrated root system age: measured versus modelled outputs

A mechanistic mathematical model for describing and predicting the dynamics of high-affinity nitrate intake into roots of maize and other plant species

A fully mechanistic dynamical model for plant nitrate uptake is presented. Based on physiological and regulatory pathways and based on physical laws, we form a dynamic system mathematically described by seven differential equations. The model evidences the presence of a short-term positive feedback on the high-affinity nitrate uptake, triggered by the presence of nitrate around the roots, which induces its intaking. In the long run, this positive feedback is overridden by two long-term negative feedback loops which drastically reduces the nitrate uptake capacity. These two negative feedbacks are due to the generation of ammonium and amino acids, respectively, and inhibit the synthesis and the activity of high-affinity nitrate transporters. This model faithfully predicts the typical spiking behavior of the nitrate uptake, in which an initial strong increase of nitrate absorption capacity is followed by a drop, which regulates the absorption down to the initial value. The model outcome was compared with experimental data and they fit quite nicely. The model predicts that after the initial exposure of the roots with nitrate, the absorption of the anion strongly increases and that, on the contrary, the intensity of the absorption is limited in presence of ammonium around the roots.

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.

The Thermodynamic Flow-Force Interpretation of Root Nutrient Uptake Kinetics: A Powerful Formalism for Agronomic and Phytoplanktonic Models

The ion influx isotherms obtained by measuring unidirectional influx across root membranes with radioactive or stable tracers are mostly interpreted by enzyme-substrate-like modeling. However, recent analyses from ion transporter mutants clearly demonstrate the inadequacy of the conventional interpretation of ion isotherms. Many genetically distinct carriers are involved in the root catalytic function. Parameters Vmax and Km deduced from this interpretation cannot therefore be regarded as microscopic parameters of a single transporter, but are instead macroscopic parameters (Vmapp and Kmapp, apparent maximum velocity and affinity constant) that depend on weighted activities of multiple transporters along the root. The flow-force interpretation based on the thermodynamic principle of irreversible processes is an alternative macroscopic modeling approach for ion influx isotherms in which macroscopic parameters L_j (overall conductance of the root system for the substrate j) and π_j (thermodynamic parameter when J_j = 0) have a straightforward meaning with respect to the biological sample studied. They characterize the efficiency of the entire root catalytic structure without deducing molecular characteristics. Here we present the basic principles of this theory and how its use can be tested and improved by changing root pre- and post-wash procedures before influx measurements in order to come as close as possible to equilibrium conditions. In addition, the constant values of Vm and Km in the Michaelis-Menten (MM) formalism of enzyme-substrate interpretation do not reflect variations in response to temperature, nutrient status or nutrient regimes. The linear formalism of the flow-force approach, which integrates temperature effect on nutrient uptake, could usefully replace MM formalism in the 1-3-dimension models of plants and phytoplankton. This formalism offers a simplification of parametrization to help find more realistic analytical expressions and numerical solution for root nutrient uptake.

Breaking conceptual locks in modelling root absorption of nutrients: reopening the thermodynamic viewpoint of ion transport across the root

BACKGROUND

The top-down analysis of nitrate influx isotherms through the Enzyme-Substrate interpretation has not withstood recent molecular and histochemical analyses of nitrate transporters. Indeed, at least four families of nitrate transporters operating at both high and/or low external nitrate concentrations, and which are located in series and/or parallel in the different cellular layers of the mature root, are involved in nitrate uptake. Accordingly, the top-down analysis of the root catalytic structure for ion transport from the Enzyme-Substrate interpretation of nitrate influx isotherms is inadequate. Moreover, the use of the Enzyme-Substrate velocity equation as a single reference in agronomic models is not suitable in its formalism to account for variations in N uptake under fluctuating environmental conditions. Therefore, a conceptual paradigm shift is required to improve the mechanistic modelling of N uptake in agronomic models.

SCOPE

An alternative formalism, the Flow-Force theory, was proposed in the 1970s to describe ion isotherms based upon biophysical 'flows and forces' relationships of non-equilibrium thermodynamics. This interpretation describes, with macroscopic parameters, the patterns of N uptake provided by a biological system such as roots. In contrast to the Enzyme-Substrate interpretation, this approach does not claim to represent molecular characteristics. Here it is shown that it is possible to combine the Flow-Force formalism with polynomial responses of nitrate influx rate induced by climatic and in planta factors in relation to nitrate availability.

CONCLUSIONS

Application of the Flow-Force formalism allows nitrate uptake to be modelled in a more realistic manner, and allows scaling-up in time and space of the regulation of nitrate uptake across the plant growth cycle.

An updated model for nitrate uptake modelling in plants. I. Functional component: cross-combination of flow-force interpretation of nitrate uptake isotherms, and environmental and in planta regulation of nitrate influx

BACKGROUND AND AIMS

In spite of major breakthroughs in the last three decades in the identification of root nitrate uptake transporters in plants and the associated regulation of nitrate transport activities, a simplified and operational modelling approach for nitrate uptake is still lacking. This is due mainly to the difficulty in linking the various regulations of nitrate transport that act at different levels of time and on different spatial scales.

METHODS

A cross-combination of a Flow-Force approach applied to nitrate influx isotherms and experimentally determined environmental and in planta regulation is used to model nitrate in oilseed rape, Brassica napus. In contrast to 'Enzyme-Substrate' interpretations, a Flow-Force modelling approach considers the root as a single catalytic structure and does not infer hypothetical cellular processes among nitrate transporter activities across cellular layers in the mature roots. In addition, this approach accounts for the driving force on ion transport based on the gradient of electrochemical potential, which is more appropriate from a thermodynamic viewpoint.

KEY RESULTS AND CONCLUSIONS

Use of a Flow-Force formalism on nitrate influx isotherms leads to the development of a new conceptual mechanistic basis to model more accurately N uptake by a winter oilseed rape crop under field conditions during the whole growth cycle. This forms the functional component of a proposed new structure-function mechanistic model of N uptake.