A model for water uptake by plant roots.

We present a model for water uptake by plant roots from unsaturated soil. The model includes the simultaneous flow of water inside the root network and in the soil. It is constructed by considering first the water uptake by a single root, and then using the parameterized results thereby obtained to build a model for water uptake by the developing root network. We focus our model on annual plants, in particular the model will be applicable to commercial monocultures like maize, wheat, etc. The model is solved numerically, and the results are compared with approximate analytic solutions. The model predicts that as a result of water uptake by plant roots, dry and wet zones will develop in the soil. The wet zone is located near the surface of the soil and the depth of it is determined by a balance between rainfall and the rate of water uptake. The dry zone develops directly beneath the wet zone because the influence of the rainfall at the soil surface does not reach this region, due to the nonlinear nature of the water flow in the partially saturated soil. We develop approximate analytic expressions for the depth of the wet zone and discuss briefly its ecological significance for the plant. Using this model we also address the question of where water uptake sites are concentrated in the root system. The model indicates that the regions near the base of the root system (i.e. close to the ground surface) and near the root tips will take up more water than the middle region of the root system, again due to the highly nonlinear nature of water flow in the soil

Homogenization of two fluid flow in porous media

The macroscopic behavior of air and water in porous media is often approximated using Richards’ equation for the fluid saturation and pressure. This equation is parametrized by the hydraulic conductivity and water release curve. In this paper, we use homogenization to derive a general model for saturation and pressure in porous media based on an underlying periodic porous structure. Under an appropriate set of assumptions, i.e., constant gas pressure, this model is shown to reduce to the simpler form of Richards’ equation. The starting point for this derivation is the Cahn-Hilliard phase field equation coupled with Stokes equations for fluid flow. This approach allows us, for the first time, to rigorously derive the water release curve and hydraulic conductivities through a series of cell problems. The method captures the hysteresis in the water release curve and ties the macroscopic properties of the porous media to the underlying geometrical and material properties

A mathematical model of plant nutrient uptake

The classical model of plant root nutrient uptake due to Nye. Tinker and Barber is developed and extended. We provide an explicit closed formula for the uptake by a single cylindrical root for all cases of practical interest by solving the absorption-diffusion equation for the soil nutrient concentration asymptotically in the limit of large time. We then use this single root model as a building block to construct a model which allows for root size distribution in a more realistic plant root system, and we include the effects of root branching and growth. The results are compared with previous theoretical and experimental studies

Mathematical models of avascular cancer

This review will outline a number of illustrative mathematical models describing the growth of avascular tumours. The aim of the review is to provide a relatively comprehensive list of existing models in this area and discuss several representative models in greater detail. In the latter part of the review, some possible future avenues of mathematical modelling of avascular tumour development are outlined together with a list of key questions

Mathematical models of avascular cancer

This review will outline a number of illustrative mathematical models describing the growth of avascular tumours. The aim of the review is to provide a relatively comprehensive list of existing models in this area and discuss several representative models in greater detail. In the latter part of the review, some possible future avenues of mathematical modelling of avascular tumour development are outlined together with a list of key questions

Mathematical model of plant nutrient uptake

This thesis deals with the mathematical modelling of nutrient uptake by plant roots. It starts with the Nye-Tinker-Barber model for nutrient uptake by a single bare cylindrical root. The model is treated using matched asymptotic expansion and an analytic formula for the rate of nutrient uptake is derived for the first time. The basic model is then extended to include root hairs and mycorrhizae, which have been found experimentally to be very important for the uptake of immobile nutrients. Again, analytic expressions for nutrient uptake are derived. The simplicity and clarity of the analytical formulae for the solution of the single root models allows the extension of these models to more realistic branched roots. These models clearly show that the `volume averaging of branching structure' technique commonly used to extend the Nye-Tinker-Barber with experiments can lead to large errors. The same models also indicate that in the absence of large-scale water movement, due to rainfall, fertiliser fails to penetrate into the soil. This motivates us to build a model for water movement and uptake by branched root structures. This model considers the simultaneous flow of water in the soil, uptake by the roots, and flow within the root branching network to the stems of the plant. The water uptake model shows that the water saturation can develop pseudo-steady-state wet and dry zones in the rooting region of the soil. The dry zone is shown to stop the movement of nutrient from the top of the soil to the groundwater. Finally we present a model for the simultaneous movement and uptake of both nutrients and water. This is discussed as a new tool for interpreting available experimental results and designing future experiments. The parallels between evolution and mathematical optimisation are also discussed

Adsorption and desorption dynamics of citric acid anions in soil

The functional role of organic acid anions (e.g. citrate, oxalate, malonate, etc) in soil has been intensively investigated with special focus either on (i) microbial respiration and soil carbon dynamics, (ii) nutrient solubilization, or (iii) metal detoxification. Considering the potential impact of sorption processes on the functional significance of these effects, comparatively little is known about the adsorption and desorption dynamics of organic acid anions in soils. The aim of this study therefore was to experimentally characterize the adsorption and desorption dynamics of organic acid anions in different soils using citrate as a model carboxylate. Results showed that both adsorption and desorption processes were fast, reaching a steady state equilibrium solution concentration within approximately 1 hour. However, for a given total soil citrate concentration(ctot) the steady state value obtained was critically dependent on the starting conditions of the experiment (i.e. whether most of the citrate was initially present in solution (cl) or held on the solid phase (cs)). Specifically, desorption-led processes resulted in significantly lower equilibrium solution concentrations than adsorption led processes indicating time-dependent sorption hysteresis. As it is not possible to experimentally distinguish between different sorption pools in soil (i.e. fast, slow, irreversible adsorption/desorption), a new dynamic hysteresis model was developed that relies only on measured soil solution concentrations. The model satisfactorily explained experimental data and was able to predict dynamic adsorption and desorption behaviour. To demonstrate its use we applied the model to two relevant scenarios (exudation and microbial degradation), where the dynamic sorption behaviour of citrate occurs. Overall, this study highlights the complex nature of citrate sorption in soil and concludes that existing models need to incorporate both a temporal and sorption hysteresis component to realistically describe the role and fate of organic acids in soil processes

Challenges in imaging and predictive modeling of rhizosphere processes

Background Plant-soil interaction is central to human food production and ecosystem function. Thus, it is essential to not only understand, but also to develop predictive mathematical models which can be used to assess how climate and soil management practices will affect these interactions. Scope In this paper we review the current developments in structural and chemical imaging of rhizosphere processes within the context of multiscale mathematical image based modeling. We outline areas that need more research and areas which would benefit from more detailed understanding. Conclusions We conclude that the combination of structural and chemical imaging with modeling is an incredibly powerful tool which is fundamental for understanding how plant roots interact with soil. We emphasize the need for more researchers to be attracted to this area that is so fertile for future discoveries. Finally, model building must go hand in hand with experiments. In particular, there is a real need to integrate rhizosphere structural and chemical imaging with modeling for better understanding of the rhizosphere processes leading to models which explicitly account for pore scale processes

Can VEGFC form turing patterns in the Zebrafish embryo?

This paper is concerned with a late stage of lymphangiogenesis in the trunk of the zebrafish embryo. At 48 hours post-fertilisation (HPF), a pool of parachordal lymphangioblasts (PLs) lies in the horizontal myoseptum. Between 48 and 168 HPF, the PLs spread from the horizontal myoseptum to form the thoracic duct, dorsal longitudinal lymphatic vessel, and parachordal lymphatic vessel. This paper deals with the potential of vascular endothelial growth factor C (VEGFC) to guide the differentiation of PLs into the mature lymphatic endothelial cells that form the vessels. We built a mathematical model to describe the biochemical interactions between VEGFC, collagen I, and matrix metalloproteinase 2 (MMP2). We also carried out a linear stability analysis of the model and computer simulations of VEGFC patterning. The results suggest that VEGFC can form Turing patterns due to its relations with MMP2 and collagen I, but the zebrafish embryo needs a separate control mechanism to create the right physiological conditions. Furthermore, this control mechanism must ensure that the VEGFC patterns are useful for lymphangiogenesis: stationary, steep gradients, and reasonably fast forming. Generally, the combination of a patterning species, a matrix protein, and a remodelling species is a new patterning mechanism