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PhD Mathematical modelling of root systems

Background

World agriculture is facing the challenge of needing to produce more food and feed on a shrinking proportion of arable land in an increasingly erratic environment. While availability of arable land and water resources are squeezed by industrialisation and urbanisation, global capacities for food and feed production will increasingly have to compete with a growing need for energy, chemicals and the production of plants for purposes other than nutrition (for example, clothing, housing, and biofuel). Maintaining the productivity of crops under reduced nutrient and water inputs can be brought about through improved crop management, and plant breeding with the aim to develop improved cultivars and management strategies exhibiting an enhanced ability to capture water and nutrients.

The capacity of roots to extract nutrients and water from a given volume of soil is heavily influenced by the root system architecture (RSA), which is a combination of traits such as root number, angle, elongation and branching, and the ability to penetrate hardpans. Unfortunately, root systems are complex objects to deal with. They consist of large numbers of segments and branches, whose position and topological relationships influence the efficiency of plant resource capture from the environment ^[1]. To account for the variability of root system architecture in terms of mechanisms and to better understand its functional significance, we need to find ways of modelling these processes accurately.

Currently, most plant root architectural models use computer simulations to reproduce the developmental processes of root apical meristems and to construct virtual root architectures: single roots are assembled incrementally through the growth of a set of virtual apical meristems whose activity is determined at each time step of the simulation. The resulting root architecture is a complex data structure describing the geometrical properties of the different roots and the topology of connection they establish with each other. Usually the same morph ogenetic rules are used to prescribe the behaviour of whole sets of meristems, and complex architectures arise as emergent properties of these simple rules. Unfortunately, root architectural models suffer from major limitations. Understanding their behaviour is often difficult because they are merely numerical experiments a root growth processes. They also tend to be difficult to parameterize or used in optimisation techniques due to computational time. Modelling interactions with environmental processes proved also difficult because of the complexity of the architecture of the root networks.

Partial Differential Equation (PDE) models, however, offer compelling alternatives to architectural models ^[2]. Analytical models, exact or approximate solutions to growth equations, can be obtained in the form of mathematical functions. These functions provide insight into the development of the root system as a whole, and can be parameterized Image of continuous models with underlying architectural properties relatively easily from field data. The use of continuous variables to aggregate root morphological properties also facilitates the coupling of growth with environmental and physical processes ^[3], which is useful for a wide range of plant environmental studies. For example, soil mechanics and transport models use partial differential equations to describe processes, and such models would communicate more naturally with continuous descriptions of plant root structure. Finally, representing root systems as continua allows more efficient computational models to be developed.

Objective of the project

The objective of this project is to construct PDE models to describe the growth of root branching structures. The student will also develop numerical techniques to obtain approximate solutions of these equations and to fit the model on experimental data. This project is part of the European project EUROOT and will be supervised by Lionel Dupuy (The James Hutton Institute), Glyn Bengough (University of Dundee), Professor Xavier Draye (Catholic University of Louvain) and the successful candidate will be expected to travel to Belgium for collaborative work. The project is funded for 3.5 years.

As a starting point, PDEs will be based on previous work by the supervisors where root tip dynamics is represented as an advection process in a generalised space E, constituted of the position of the roots and their direction of growth. Three types of root density distribution functions are required to model the growth of root system architectures ^[2]. Root tip density, the number of root tips per unit soil volume, represents the region of the root system where growth occurs. Root length density, the total root length per unit soil volume, correlates with the area of roots in contact with the soil. Finally the branching density, the number of "branching points", defines the connections between roots of two consecutive branching orders.

In a second stage, the PhD student will develop methods to obtain numerical solution of the PDEs. Initially, the project will seek to develop techniques for the discretisation of the PDEs based on the decomposition of the solution into basis functions evolving both in space and time. Under such approximation, PDEs can be transformed into a system of ordinary differential equation which is easier to solve. An example of such technique is the Travelling Wavelet method ^[4,5]. Basis function methods will be preferred because they allow the number of unknown parameters to be reduced and the model to be simulated or fitted more efficiently.

Finally, the last stage of the project will be dedicated to application of the new methods on data obtained by the partners of the EUROOT project. This task will involve fitting models on Root growth data, carrying out model sensitivity analysis and model simulation to identify plant ideotypes. Models will be exploited to identify already existing or idealised genotypes that perform best under the range of stresses studied in the project. Models will be calibrated on various genotypes and will be tested on a range of reference environmental conditions corresponding to experiments carried out throughout the consortium. Simulation of scenarios will identify key biological factors and plant traits that can be targeted by breeding.