A Mathematical Model of Water and Nutrient Transport in Xylem Vessels of a Wheat Plant

Abstract

At a time of increasing global demand for food, dwindling land and resources, and escalating pressures from climate change, the farming industry is undergoing financial strain, with a need to improve e ciency and crop yields. In order to improve e ciencies in farming, and in fertiliser usage in particular, understanding must be gained of the fertiliser-to-crop-yield pathway. We model one aspect of this pathway; the transport of nutrients within the vascular tissues of a crop plant from roots to leaves. We present a mathematical model of the transport of nutrients within the xylem vessels in response to the evapotranspiration of water. We determine 7 di↵erent classes of flow, including positive unidirectional flow, which is optimal for nutrient transport from the roots to the leaves; and root multidirectional flow, which is similar to the hydraulic lift process observed in plants. We also investigate the e↵ect of di↵usion on nutrient transport and find that di↵usion can be significant at the vessel termini especially if there is an axial e✏ux of nutrient, and at night when transpiration is minimal. Models such as these can then be coupled to whole-plant models to be used for optimisation of nutrient delivery scenarios.

Mathematical Modelling of the Phloem: The Importance of Diffusion on Sugar Transport at Osmotic Equilibrium

“…Note that active loading is similar to the implicit loading of Jensen et al (2012), whilst passive loading is similar to their explicit loading case. Payvandi et al (2014) showed di↵usion to be important in the xylem vessels, and we expect that it may also be of importance in the phloem vessels especially near the ends of the phloem vessels when the fluid velocity falls to zero. The inclusion of di↵usion also allows the modelling of an axial flux of sugar,f (mol m 2 s 1 ), at the vessel termini in the roots (ẑ =ẑ 3 ), which we include for generality.…”

“…We follow a similar approach to the phloem model of Jensen et al (2011Jensen et al ( , 2012) and the xylem model of Payvandi et al (2014), and separate the plant and phloem tissue into three functional zones in theẑ direction; the leaves L, the stem S, and the roots R as shown in Figure 1 in Section 4. The leaf zone is represented by 0 <ẑ <ẑ 1 , the stem zone isẑ 1 <ẑ <ẑ 2 and the root zone isẑ 2 <ẑ <ẑ 3 whereẑ,ẑ 1 ,ẑ 2 andẑ 3 are measured in metres.…”

“…Following the approaches of Thompson & Holbrook (2003a), and Jensen et al (2011Jensen et al ( , 2012, we assume that the loading and unloading of sugar into and out of the phloem vessels, which we represent byF i (mol m 2 s 1 ), is balanced by the flux of sugar in the axial direction. Similar to Payvandi et al (2014) we also allow di↵usion to contribute to the axial flux and allow unloading to also take place in the stem region. Leakage of solutes has been observed along the whole length of the phloem (Minchin & Thorpe, 1987, 1996Van Bel, 2003), and sugar stem reserves are of particular importance to crop grain filling (Rawson & Evans, 1971).…”

“…We follow the approach of Jensen et al (2012); Payvandi et al (2014) to investigate the e↵ect of varying the values of the nondimensional radial and axial loading parameters F , and f . To represent the loading of sugar in the leaf due to photosynthesis, and unloading in the stem and root zones for storage and growth, we prescribe that F L is positive whilst F S and F R are both negative.…”

On the Mathematical Model of the Biomechanics of Green Plants

“…The sieve tube membrane permeability strongly influenced the results of the model. Payvandi et al [23] studied the transport of water and nutrient in xylem vessels of a wheat plant. Solution to the transport of the nutrient was obtained considering convection and diffusion.…”