## Introduction

Root hairs are lateral extensions of epidermal cells, and these root hairs increase the effective surface area of the root system available for water and nutrient uptake. They are particularly important for nutrients that are sparingly soluble in the soil, such as phosphate (Marschner, 1995). The widths of phosphorus-depletion zones around nonmycorrhizal roots are closely related to root hair length, and plants grown under phosphate-limiting conditions form longer root hairs (Bates & Lynch, 1996; Zhang *et al.*, 2003). Conversely, mutant plants with impaired root hair growth–such as root-hair-defective *Arabidopsis* mutants *rhd2* and *rhd6*, which are involved in hair initiation and elongation respectively–have a reduced capacity for phosphate uptake under phosphate-limiting conditions (Bates & Lynch, 2000a). A better understanding of how root hairs mediate phosphate uptake will enhance the development of more phosphate-efficient crops. This can help to minimize fertilizer use and pollution risk (Narang *et al.*, 2000; Wissuwa, 2003). Given the complexity of root hair–soil interactions, and the difficulty of measuring these interactions experimentally, development of mathematical models is necessary. Mathematical models will enable comparisons to be made between different root hair properties, such as their geometry and rates of nutrient uptake.

Previous approaches to modelling root hair in nutrient-uptake models fall in three categories. First, the effective root radius is extended by the length of the hairs, and any concentration gradients along the hair length are not allowed for (Passioura, 1963). Second, the continuity equation for nutrient transport to the root surface is modified with a separate sink term describing nutrient influx into the hairs (Bhat *et al.*, 1976). Third, the nutrient transport equation is solved in a three-dimensional model that takes into account the geometry of root hairs explicitly (Geelhoed *et al.*, 1997).

Modelling such multiscale problems in three dimensions is computationally challenging and generally beyond the scope of standard numerical methods (such as Comsol Multiphysics, PHREEQC, Orchestra, MIN3P, etc.) used in rhizosphere research. While three-dimensional numerical simulations could be utilized to address single root scale phenomena, they are usually very costly and the translation of such results from single root scale to root system scale and field scale is seriously challenging. For such multiscale problems, such as root hair nutrient uptake, the homogenization method (Pavliotis & Stuart, 2008) provides a possible solution. With this method, spatial heterogeneities at different scales can be transformed into a tractable homogeneous description. Equations that are valid on a macroscale are derived by transparently incorporating the relevant information about the microscale geometry and model properties.

The method of homogenization is particularly suitable for domains with a periodic microstructure. The microstructure of root hairs is illustrated in Fig. 1(a). On the left of Fig. 1(a) the different properties of root hairs and the surrounding soil solution are illustrated by periodic changes of dark and light regions. The microscopic length scale is given by the inter-root hair distance, *l*. If the ratio between the single root hair scale, *l*, and root length scale, *L*, is small (i.e. *ε* = *l*/*L* << 1), it is possible to derive an effective macroscopic model describing the root hair zone function. In the graph on the left of Fig. 1(a) the heterogeneities can be distinguished. However, viewed from a distance, as in the graph shown on the right of the figure, the heterogeneities average out. This is, in essence, what the homogenization technique does: it describes how the root hair functions blend into the model viewed on the coarser root length scale. The space variable, ** x**, reveals the properties of the system on the scale of the whole root hair zone. Scaling

**with**

*x**ε*

^{−1}defines a new space variable,

**=**

*y*

*x**ε*

^{−1}, which reflects the microscopic properties on the scale of the single root hair. One of the fundamental assumptions of homogenization is that the two variables

*x*and

*y*can be treated as independent of each other when

*ε*becomes small (Pavliotis & Stuart, 2008). A well-known example is the macroscopic Darcy law derived from the Stokes equations (Hornung, 1997), whereby the role of exact particle shape on hydraulic permeability can be explained.

In this work, we use the method of homogenization to develop an effective model of nutrient transport in the root hair zone of a single root that contains the relevant information about the root hair geometry implicitly. We consider a root with root hairs in a homogeneous medium. In the case of soil, this is a major simplification because the root hair size can be comparable to the soil particle size. We will address this issue in a follow-up paper, but for now we assume that the soil around each root hair is homogeneous (Bhat *et al.*, 1976; Geelhoed *et al.*, 1997).

We analyse the development of nutrient-depletion zones around a root with root hairs for different root morphologies and uptake properties and thereby obtain three different effective models. The notation used is given in Table 1.

Symbol | Description |
---|---|

a_{h},a_{r} | Root hair and root radius (cm) |

b | Soil buffer power (−) |

c | Nutrient concentration in the solution (μmol ml^{−1}) |

c_{a} | Effective nutrient concentration in the root hair zone V_{a} (μmol ml^{−1}) |

c_{b} | Nutrient concentration in the domain outside the root hair zone V_{b} (μmol ml^{−1}) |

c_{0} | Initial solution concentration at time t = 0 (μmol ml^{−1}) |

d | Factor that distinguishes between solution culture and soil systems. In the solution culture d = 1; in soil d = 1/(θ + b) (−) |

D_{l} | Molecular diffusion coefficient of nutrients in solution (cm^{2} s^{−1}) |

D | Diffusion coefficient; D = D_{l} in solution culture, D = D_{l}θf/(θ + b) in soil (cm^{2} s^{−1}) |

Effective diffusion coefficient taking the impedance caused by root hairs into account (cm^{2} s^{−1}) | |

E_{h},E_{r} | Efflux from root hairs and root (μmol cm^{−2} s^{−1}) |

f | Impedance factor of soil (−) |

f_{a} | Effective root hair uptake (μmol cm^{−2} s^{−1}) |

f_{h}, f_{r} | Net influx into the root hair and root (μmol cm^{−2} s^{−1}) |

F_{h},F_{r} | Maximal nutrient influx into root hairs and root (μmol cm^{−2} s^{−1}) |

K_{h},K_{r} | Michaelis–Menten constants for root hairs and root (μmol ml^{−1}) |

l | Distance between two root hairs (cm) |

L | Root length (cm) |

L_{h} | Root hair length (cm) |

n | Outer normal vector |

N | Number of root hairs per cm root length (cm^{−1}) |

t | Time (s) |

V_{a} | Root hair zone (cm^{3}) |

V_{b} | Domain outside the root hair zone (cm^{3}) |

α | The parameter determines which effective model applies (−) |

x | The vector = (xx_{1},x_{2},x_{3}) is the macroscopic space variable. |

y | The vector = (yy_{1},y_{2},y_{3}) is the microscopic space variable. |

θ | Volumetric water content excluding soil particles (−) |

θ_{h} | Volumetric content of medium excluding root hairs, θ_{h} = 1-θ_{hair} (−) |

θ_{hair} | Volume fraction of root hairs (−) |