## Introduction

More than 90% of all terrestrial plants form mycorrhizas, mutualistic symbiotic associations between plant roots and soil fungi (Strack *et al.*, 2003). Mycorrhizas may offer several benefits to the host plant, including faster growth, improved nutrition, greater drought resistance, and protection from pathogens. The fungus benefits from the mycorrhizal symbiosis by receiving photosynthesis products from the plant. The two most common types of mycorrhizas are the ectomycorrhizas and the endomycorrhizas, also known as arbuscular mycorrhizas. Arbuscular mycorrhizas differ from ectomycorrhizas in that their fungal mycelium penetrates into the cortical cells of plant roots. Many experimental studies have focused on arbuscular mycorrhizal fungi and their contribution to plant phosphorus uptake (Smith *et al.*, 2003). To understand the mechanisms that control and influence the mycorrhizal pathway of the uptake of nutrients into plants, mathematical modelling can be very useful. Many plant nutrient uptake models exist (Darrah *et al.*, 2005; Tinker & Nye, 2000; Roose *et al.*, 2001). It has been pointed out that it is necessary to include mycorrhizal fungi in such models of plant nutrient uptake (Tinker & Nye, 2000). It is thought that the external fungal hyphae should give mycorrhizal plants an enormous spatial advantage to access low-mobility ions such as phosphorus. However, to date no such model has been available.

### Objectives

In this paper, we develop a spatially explicit and dynamic model for the uptake of low-mobility nutrients, such as phosphorus, by mycorrhizal roots with a growing hyphal network (mycelium). We assume that single root models are well characterized, and aim to include in these models the uptake by fungal mycelium in the form of a sink term. For the derivation of this sink term, we consider two spatial scales: the scale of a single root surrounded by a growing mycelium and the scale of a single cylindrical hypha. We model nutrient transport towards the plant root in the soil as well as within the hyphal network. Whilst the model is applicable to general mycelial fungi and solutes, we present specific numerical simulations for phosphate uptake by the fungal species *Scutellospora calospora* (Nicol. & Gerd.) (Jakobsen *et al.*, 1992).

Based on literature estimates of the necessary model parameters, we quantitatively assess the contribution of external hyphae to plant mineral nutrition and find those processes and parameter values that should be measured more accurately to improve the predictive power of this model.

For analysis and solution of mathematical models, we apply the technique of nondimensionalization (Fowler, 1997), a process of changing variables by scaling so that the new variables have no units. This procedure leads to a simpler form of equations with fewer parameters that are all dimensionless. The dimensionless parameters are useful in their own right as they describe the relative importance of different processes included in the model compared with each other. One well-known dimensionless parameter is the Péclet number, which shows the importance of convection in comparison to dispersion. In order to make the paper more accessible to a broader readership, we describe the mathematics involved in nondimensionalizing the models we derive in this paper in appendices. The resulting simplified models are presented in the main text.