## Introduction

Water and nutrients are heterogeneously distributed in soils (Robinson 1996; Hopmans and Bristow 2002; Hodge 2004; Schimel and Bennett 2004). Therefore models of water and nutrient uptake by plants need to consider the spatial distribution of roots. Virtually, all global land-surface and forest ecosystem models do simulate water uptake from multiple soil-depth layers—even models that do not explicitly consider root distributions (Jackson et al. 1996; Woodward and Osborne 2000). Some include equations for water uptake by three-dimensional root distributions (e.g., Somma et al. 1998; Hopmans and Bristow 2002; Simunek and Hopmans 2009), while others have used water-balance modeling to infer optimal root distributions and maximum rooting depths (Kleidon and Heimann 1996; van Wijk and Bouten 2001; Laio et al. 2006; Collins and Bras 2007; Guswa 2008, 2010; Schymanski et al. 2008, 2009).

Modeling of nutrient uptake in global and ecosystem models is rudimentary in comparison to that of water. Most models evaluate nitrogen (N) uptake from simulated bulk soil net N mineralization rate (*N*_{min}) above a specified soil depth (Parton et al. 1988; Comins and McMurtrie 1993; Jackson et al. 1996, 2000), or from *N*_{min} multiplied by a take-up fraction represented by an empirical function of total root mass (Mäkelä et al. 2008; Franklin et al. 2009). Few consider how the spatial distribution of roots affects the efficiency of N capture from soil, although the mechanisms of nutrient transport in soils and uptake by roots have been extensively studied (Hopmans and Bristow 2002).

This shortcoming is a concern in modeling of tree responses to elevated CO_{2} (eCO_{2}) because the majority of experiments show that root distributions are altered when trees are grown at eCO_{2}, involving in particular an increase in rooting depth (Iversen 2010). For example, at the Oak Ridge National Laboratory (ORNL) forest free-air CO_{2}-enrichment (FACE) experiment, both peak annual root biomass and annual root production approximately doubled at eCO_{2} (Iversen et al. 2011), and the greatest increases in root mass occurred at soil depths below 30 cm, leading to enhanced N extraction from deeper in the soil (Iversen 2010; Iversen et al. 2011). At the Duke Forest FACE experiment (Pritchard et al. 2008; Jackson et al. 2009), fine-root biomass increased by 24% in the top 15 cm of soil (Pritchard et al. 2008; Jackson et al. 2009) and there was a shift to deeper rooting (Pritchard et al. 2008; Iversen 2010). Fine root mass also increased at the Rhinelander Forest FACE experiment (Zak et al. 2011).

In all three FACE experiments, increases in annual tree growth at eCO_{2} were associated with increased annual uptake of N by tree roots rather than more efficient use of N taken up (Finzi et al. 2007). It remains uncertain, however, whether the increases in annual N uptake at eCO_{2} were due to increased N availability or more efficient capture of N available in the soil, due, possibly, to deeper rooting (Iversen et al. 2008; Norby et al. 2010; Drake et al. 2011; Hofmockel et al. 2011; Phillips et al. 2011; Zak et al. 2011). Understanding of how the N-uptake fraction, that is, the proportion of plant-available soil N taken up annually by roots, depends on the amount and vertical distribution of root biomass is a key to predicting tree growth responses to eCO_{2} (Iversen et al. 2010), and in turn feedbacks from the terrestrial biosphere to climate (Norby et al. 2010). Moreover, an enhanced understanding of the N-uptake fraction may foster more efficient use of N fertilizers with potential benefits for managed forests, agriculture, and the environment through reduced use of N fertilizers (Tilman et al. 2002).

Current N-uptake models fall well short of providing such an understanding. The above responses of root distributions to eCO_{2} and their consequences for N uptake have not yet been incorporated into land-surface models of the terrestrial biosphere, or into ecosystem models. Instead, coupled land-climate models (Friedlingstein and Prentice 2010) are moving apace to incorporate long-term feedbacks associated with immobilization of N in wood and soils at eCO_{2} (Comins and McMurtrie 1993; McMurtrie and Comins 1996; Luo et al. 2004), although these feedbacks have yet to be verified in forest FACE experiments (Norby et al. 2010; Hofmockel et al. 2011; Zak et al. 2011).

Our objective in this study was to address these short-comings through a new model of the N-uptake fraction that takes account of the vertical distribution of plant-available N in the soil. We define N-uptake fraction as the ratio of the annual rate of plant N uptake to the annual rate at which soil N becomes potentially available to plants. (Potential annual plant-available soil N is the annual rate of supply of bio-available soil N, for which roots and soil microbes compete, sensu Schimel and Bennett 2004). Our model is based on an optimal root-foraging hypothesis (*MaxNup*), according to which “a given total amount of root biomass is distributed vertically in soil in order to maximize annual N supply to aboveground plant organs (i.e., plant N uptake minus the N investment in growing roots)”. N uptake by roots at soil depth *z* is modeled as a saturating function of root-mass density at *z*. Using this function, *MaxNup* predicts the optimal vertical profile of root-mass density, rooting depth, and annual N uptake as functions of total root mass. We use these predictions to evaluate the N-uptake fraction of trees growing at the ORNL FACE experiment, and compare predicted root distributions with empirical equations previously fitted to root-distribution data from the ORNL FACE experiment (Iversen 2010) and global plant datasets (Gale and Grigal 1987; Jackson et al. 1996; Arora and Boer 2003). Because our model is simpler than previous models of N uptake by spatially distributed root systems (Somma et al. 1998; Hopmans and Bristow 2002; Simunek and Hopmans 2009), we are able to derive new simple analytic expressions for optimal root distributions and maximum N uptake. The power of simple models, in which biological mechanisms can be clearly understood, versus complex or computationally intensive simulation models is expounded by May (2004). To the best of our knowledge, this is the first time a model has been used to evaluate the efficiency of N uptake by a spatially distributed root system.