VIEWPOINT: PART OF A SPECIAL ISSUE ON MATCHING ROOTS TO THEIR ENVIRONMENT A conceptual model of root hair ideotypes for future agricultural environments: what combination of traits should be targeted to cope with limited P availability?

† Background Phosphorus (P) often limits crop production and is frequently applied as fertilizer; however, supplies of quality rock phosphate for fertilizer production are diminishing. Plants have evolved many mechanisms to increase their P acquisition, and an understanding of these traits could result in improved long-term sustain- ability of agriculture. This Viewpoint focuses on the potential beneﬁts of root hairs to sustainable production. † Scope First the various root-related traits that could be deployed to improve agricultural sustainability are cat-alogued, and their potential costs and beneﬁts to the plant are discussed. A novel mathematical model describing the effects of length, density and longevity of root hairs on P acquisition is developed, and the relative beneﬁts of these three root-hair traits to plant P nutrition are calculated. Insights from this model are combined with experi- mental data to assess the relative beneﬁts of a range of root hair ideotypes for sustainability of agriculture. † Conclusions A cost–beneﬁt analysis of root traits suggests that root hairs have the greatest potential for P acquisition relative to their cost of production. The novel modelling of root hair development indicates that the greatest gains in P-uptake efﬁciency are likely to be made through increased length and longevity of root hairs rather than by increasing their density. Synthesizing this information with that from published experiments we formulate six potential ideotypes to improve crop P acquisition. These combine appropriate root hair phenotypes with architectural, anatomical and biochemical traits, such that more root-hair zones are produced in surface soils, where P resources are found, on roots which are metabolically cheap to construct and maintain, and that release more P-mobilizing exudates. These ideotypes could be used to inform breeding programmes to enhance agricultural sustainability.


THE GLOBAL PROBLEM
With the global population set to hit nine billion by 2050 and the resources needed to sustain this population diminishing, unsustainable agronomic practices and environmental change have brought us to the point where a revolution in agricultural production is necessary to ensure future agricultural sustainability and food security. This second green revolution must focus on crops which are tolerant to and productive in lowfertility environments and gains need to be achieved across a range of crops (Lynch, 2007). A new generation of crops adapted to low/reduced input systems will not only enable people in some of the poorest parts of the world to provide themselves with adequate nutritious food and save some products for trade, but will also have a vital part to play in the high-input systems of the developed world by reducing inputs and their associated economic and environmental costs. The key to breeding these crops is the utilization of the large genetic variation in yield that has been identified in low-fertility environments. Of the traits responsible for this yield variation those associated with roots are perceived to have the most potential to deliver crops for the second green revolution and by looking below ground we might make the enormous strides necessary to improve yields to feed nine billion people (Lynch, 2007;Tester and Langridge, 2010;Gregory and George, 2011).
One of the most important inputs to agricultural systems is phosphorus (P) and the issues associated with P bioavailability and acquisition by plants represent problems of global proportions (Stutter et al., 2012). Phosphorus is an essential nutrient required for plant growth and reproduction (White and Brown, 2010), but due to its strong reaction with soil and subsequent lack of mobility it is estimated that 30-40 % of the world's arable land is limited by P bioavailability (Runge-Metzger, 1995). In particular, P bioavailability in the acidic, weathered soils of the tropics and subtropics represents a major limitation to agricultural production (Sanchez et al., 1997). The application of P fertilizers is usually employed to increase soil P bioavailability; however, this is increasingly problematic . While this is a common solution in the intensively managed agricultural systems of the developed world, it is often impossible in subsistence agriculture (Sanchez et al., 1997). Furthermore, in intensive systems up to 80 % of applied P can become immobilized in the soil through the processes of precipitation and adsorption (Jones, 1998), forcing farmers to apply up to five times the P fertilizer required by the crop and resulting in P loading of agricultural soils. In addition, huge quantities of nutrient-rich manure is often spread on soil resulting in soluble Pi (inorganic phosphate) levels which often exceed the crop requirement (Mikkelsen, 2000). Run-off from such land is a primary factor in the eutrophication and hypoxia of lakes and marine estuaries of the developed world (Tiessen, 2008;White and Hammond, 2009). Crucially, the long term sustainability of applying P fertilizers is also highly questionable with world reserves of high-quality rock phosphate, from which they are derived, depleting rapidly Dawson and Hilton, 2011). Rock phosphate is now becoming a strategic material for many countries (Gilbert, 2009). The availability of P fertilizer also has serious implications for subsistence agriculture in the tropics and subtropics where the majority of the earth's population live. Here, lack of fertilizer infrastructure, finance and transport mean that P fertilization is not a viable option for many of these areas (Sanchez et al., 1997). There is much scope for improving the utilization by plants of P that has accumulated in soils, and in the efficiency by which they utilize recently applied fertilizers (Cordell et al., 2009;Stutter et al., 2012;White et al., 2012). The global significance of these issues means that the efficient use of P reserves has become a high priority from a scientific, political and environmental perspective.
We believe it is possible to tackle some of these issues by making better use, in agriculture, of traits exhibited by plants to cope with P deficiency in nature and, thereby, develop crops for reduced-input agricultural systems.

PLANT STRATEGIES TO OVERCOME P-DEFICIT: BENEFITS AND COSTS
The physiological state of a P-deficient plant is quite specific and the response to P starvation is multigenic. For example, the expression of over 1000 genes changes upon P starvation in arabidopsis (Hammond et al., 2003;Morcuende et al., 2007).
Plants have evolved two broad strategies for P use in nutrient-limiting environments -conservation and active acquisition strategies (White and Hammond, 2008;Veneklaas et al., 2012). Conservation strategies aim to reduce plant P requirement through physiological adjustments which reprioritize internal P-utilization and include reduced growth rates, remobilization of vacuolar P, reduction in nucleic acid pools, the use of P-sparing metabolic pathways; and the replacement of phospholipids with glycolopids and sulpholipids (Vance et al., 2003, Ticconi andAbel, 2004;White and Hammond, 2008;Veneklaas et al., 2012). Conservation processes enable plants to maintain their growth when P bioavailability is low. These strategies are complemented by strategies enabling plants to acquire P more effectively when P bioavailability is low. Under conditions of P starvation, plants exhibit increased root : shoot biomass ratio Hermans et al., 2006), altered root architecture (Lopez-Bucio et al., 2000;Williamson et al., 2001;White et al., 2005), increased lateral rooting and long root hairs (Bates and Lynch, 1996;Hammond et al., 2009). Also high-affinity P transporters are more abundant (Mudge et al., 2002;Smith et al., 2003) and organic acids and phosphatases are synthesized and secreted (Li et al., 2002). The focus of the second green revolution should be P-acquisition strategies, since it is the acquisition of P that will ultimately limit crop production in many cases.
The costs and benefits of the various P-acquisition mechanisms available to plant can be measured both metabolically and ecologically and their optimization is dependent on environmental factors . These are summarized in Table 1, which provides us with a way of rationalizing which of the range of strategies employed by plants should be targeted for development in future crop germplasm.

Modifications to root architecture
Under conditions of P deficiency plants allocate more photosynthate to root production thereby promoting root growth and allowing the root system to explore greater volumes of soil for P . The advantages of this have been demonstrated, for example, in bean (Phaseolus vulgaris) where genotypes with highly branched, actively growing, root systems have been shown to be more P efficient than genotypes lacking such root traits Lynch and Brown, 2001). This has also been confirmed in brassicas  and demonstrated for production of lateral roots (Zhu and Lynch, 2004). However, a number of studies have demonstrated that the increased respiratory burden of a larger root system can be detrimental to plant growth ( Van der Werf et al., 1992;Snapp and Lynch, 1996;Nielsen et al., 1998Nielsen et al., , 2001. Root classes have been shown to differ in their metabolic costs and the development of adventitious roots has been shown to be favoured in P-deficient environments in several plant species . Biomass allocation to the production of adventitious roots allows root exploration of new soil at a reduced carbon cost compared with other root classes due to their greater specific root length and smaller linear construction cost (Zobel, 1992;Miller et al., 2003). Other advantages of adventitious roots include shallow growth, rapid radial dispersion from the shoot, dispersed lateral branching, abundant aerenchyma and proliferation of root hairs. With the greatest concentrations of P typically found in the top few centimetres of a soil profile, P acquisition will be favoured in plants with a large proportion of their roots in the topsoil (Lynch and Brown, 2001). However, plants with a proliferation of surface roots are prone to both drought and waterlogging (Simpson and Pinkerton, 1989;Ho et al., 2004).
The heterogeneous distribution of P in the soil is best exploited by plants exhibiting morphological plasticity of their root system which allows them to proliferate in nutrient-rich patches (Hodge, 2004;Robinson, 2005). Root-system plasticity is an important mechanism to compensate for the large proportion of roots that do not acquire P, provided it does not lead to inter-root competition and increased metabolic cost per unit P acquired (Ge et al., 2000;Rubio et al., 2001).
The formation of cluster roots is considered to be one of the most effective plant adaptations to increase P acquisition (Lamont, 2003). These root structures are made up of zones of tightly packed, short rootlets covered in a dense mat of root hairs which are very effective in mobilizing and taking up P from the soil. A number of P-mobilizing mechanisms are concentrated in cluster roots (Keerthisinghe et al., 1998) into a small volume of soil increasing the surface area of the

Exudation of organic anions
Greater P availability from inorganic sources, increased microbial biomass and altered microbial communities to benefit plant Increased metabolic costs, increased microbial biomass and altered microbial population increasing resource competition with microbes and/or other negative impacts for plants 7,17,19,26,31,34,39,40 Exudation of phosphatases Increases P availability from organic sources, increased microbial biomass and altered microbial communities to benefit plants Increased metabolic costs, increased microbial biomass and altered microbial population increasing resource competition with microbes and/or other negative impacts for plants 8,9,21,34,46 Rhizosphere acidification Greater P availablity from inorganic sources, increased microbial biomass and altered microbial communities to benefit plants Potential increase in metabolic costs, increased microbial biomass and altered microbial population increasing resource competition with microbes and/or other negative impacts for plants 11,12 Associations with AM fungi Greater volume exploited beyond the root, exploitation of narrower soil pores, plasticity of foraging responses, greater P availability through exudation of fungal organic acids and phosphatases, increased microbial biomass and altered microbial communities Large carbon costs for maintenance of the AM-fungal association, altered microbial biomass and communities with negative impacts for plants 16,31,32,38,43 Associations with specific microbes Greater P availability from inorganic and organic sources through P-solubilizing and mineralizing micro-organisms, increased microbial biomass and altered microbial communities to benefit plants Increased microbial biomass and altered microbial population increasing resource competition with microbes and/or other negative impacts for plants 2,28,31,35,41 root system and releasing organic anions and acid phosphatases in an exudative burst . This strategy for P acquisition is employed by only a very few species which tend to inhabit soils with extremely low P availability, which is possibly indicative of the high metabolic input required. Indeed, it is estimated that half the photosynthetic carbohydrate production of a plant might be required for growth, respiration and exudate production by cluster roots .
Another modification to root-system architecture that improves P acquisition is the production of root hairs and this is often concurrent with increases in the root : shoot ratio. Under P-deficient conditions, root hairs can be responsible for up to 90 % of the P acquired by plants, and, in many species, root hairs can contribute almost 70 % of the total surface area of roots (Raghothama, 2005). Root hairs increase the volume of soil explored by roots, access soil pores otherwise inaccessible to the plant (Misra et al., 1988), allow dispersion of root exudates  and are implicit in rhizosheath formation . Modifications to root hair morphology are considered to be the cheapest way by which a plant can increase the surface area of the root system (Bates and Lynch, 2000), but the observed plasticity of this trait could be indicative of a significant metabolic cost or a biological cost in the form of increased susceptibility to pathogens (Hood and Shew, 1997;Zhu et al., 2010).

Modification to root anatomy
A number of anatomical changes can occur in plant roots in response to P deficiency, including the reduction of root diameter or the production of finer roots (Eissenstat et al., 2000;Forde and Lorenzo, 2001). An increase in specific root length, or length of root per mass, enables a plant to produce longer roots per unit of dry matter (Hill et al., 2006), and can be achieved either by reducing root mass density (Fitter, 1985) or by decreasing root diameter. However, greater specific root length has been linked with greater vulnerability to biotic stress (Eissenstat, 1992) and reduced elongation rates and ability to penetrate hard soils.
Another anatomical adaptation to reduce the metabolic cost of roots is the formation of aerenchyma in response to P starvation . The formation of aerenchyma reduces root costs by replacing metabolically active cortical cells with air spaces, thereby increasing the proportion of root mass occupied by non-respiring tissues. The reduced carbon costs associated with replacement of cortical cells by aerenchyma accompanied by the additional P resources made available to the plant through the senescence of cortical cell tissue benefits the P-economy of the plant and contribute to physiological P-utilization efficiency (Lynch and Brown, 1998;Lu et al., 1999;Koide et al., 2000;Fan et al., 2003). The costs of forming aerenchyma are unknown, but might include reduction in niches for colonization by arbuscular mycorrhizal (AM) fungi, reduced radial transport of water and nutrients and reduced resistance to physical stresses Striker et al., 2006Striker et al., , 2007.

Modification to the chemistry and biology of the rhizosphere
The manipulation of rhizosphere biochemistry by plants offers an opportunity to increase P bioavailability . Phosphorus bioavailability can be increased through the acidification of the rhizosphere, the exudation of organic acid anions and the secretion of extracellular phosphatases . Rhizosphere acidification is thought to increase P bioavailability by altering the solubility of inorganic phosphate salts or by affecting P absorption/desorption reactions in the soil (Hinsinger, 2001). While this process can benefit plant P acquisition in alkaline soils, excess acidification of the rhizosphere can result in reducing P availability and can lead to aluminium toxicity in acid soils . It has also been demonstrated that alkalinization and the uptake of calcium ions by plant roots can lead to increased P availability in the rhizosphere (Devau et al., 2010(Devau et al., , 2011. The importance of organic anions for the P nutrition of plants has been reviewed by Ryan et al. (2001) and Lynch and Ho (2005). These compounds are thought to increase P bioavailability by complexing or chelating cations and, thereby, solubilizing inorganic phosphates, by replacing phosphate on sorption sites in the soil and by altering the surface characteristics of soil particles (Bar-Yosef, 1991;Jones and Darrah, 1994;Jones, 1998;Ryan et al., 2001). Organic anions also increase the availability of P from organic compounds by facilitating their mineralization by phosphatases . However, exudation of carbon in the form of organic anions can be metabolically expensive, in some cases representing over half the below-ground carbon allocation Dinkelaker et al., 1989;Johnson et al., 1996a, b;Kirk et al., 1999;Nguyen, 2003). In addition, the short residence time of these compounds in the soil means their effectiveness may be limited (Marschner and Romheld, 1996;Gahoonia and Nielsen, 2003).
The secretion of enzymes into the rhizosphere is another important mechanism by which plants can increase P bioavailability. Organic P (Po) forms a substantial component of soil P (Turner et al., 2002), but before this P becomes available to the plant it must be mineralized by phosphatases (Tarafdar and Jungk, 1987;George et al., 2002;Li et al., 2003). There is significant genotypic variation in exuded phosphatases both between and within plant species (George et al., 2008). However, while this variation is related to the ability of plants to utilize specific organic-P substrates in vitro, similar relationships are not always found when plants are grown in soils (George et al., , 2008. Mycorrhizal associations are considered to be of great importance in increasing the ability of the host plant to increase its P acquisition (Clark and Zeto, 2000;Smith et al., 2003;Morgan et al., 2005;Smith and Read, 2008). The main benefit to the plant is thought to be the increased volume of soil that can be exploited through the fungal mycelium which projects beyond the P-depletion zone of the root (Smith and Read, 2008). The speed that P moves through fungal mycelia is faster than P diffusion through the soil, resulting in rapid movement of P from the soil to the plant (Smith and Read, 2008). Mycorrhizal relationships have also been demonstrated to increase the utilization of soil organic P by plants and to enhance the exploitation of nutrient-rich patches (Feng et al., 2003). A lack of mycorrhizal colonization in species with cluster roots and reduced mycorrhizal colonization in plants with adequate P status, along with strong correlations between plant P acquisition and root traits like root-hair length and root shallowness in the presence of mycorrhizas might indicate that mycorrhizal foraging is costly in comparison with root foraging alone (Neumann and Martinoia, 2002;Brown et al., 2012; L. K. Brown, T. S. George, G. E. Barrett, S. McLaren, S. F. Hubbard and P. J. White, unpubl.). Indeed, mycorrhizal associations incur a substantial metabolic cost to the plant (Koch and Johnston, 1984;Douds et al., 1988;Jakobsen and Rosendahl, 1990;Eissenstat et al., 1993;Peng et al., 1993;Harris and Paul, 1997;Nielsen et al., 1998) which in some cases can be nonbeneficial or even parasitic to the plant Morgan et al., 2005).
Other associations with soil microorganisms can enhance P uptake both directly, by increasing P acquisition through the stimulation of root growth  and indirectly, by providing a source of P which contributes to plant P nutrition (Richardson, 1994). The microbial mass releases phytohormones which can lead to increased root branching and root hair development (Pitts et al., 1998), thereby increasing the surface area of the roots in contact with the soil and the opportunity for roots to take up available P. Microorganisms play an important role in the transfer of P between different organic and inorganic forms through the processes of solubilization and mineralization (Helal and Dressler, 1989;Perrot et al., 1990;Seeling and Zasoski, 1993;Macklon et al., 1997;Rodriguez et al., 1999;Jones et al., 2003;Vance et al., 2003;Hammond et al., 2004). Moreover, microorganisms contribute significantly both to the pool of organic anions and to phosphatase activity in the rhizosphere (Richardson, 1994) and there is evidence that phosphatases derived from fungal sources are more efficient in mineralizing soil organic P than those secreted by plants (Tarafdar et al., 2001). In addition, turnover of the microbial biomass makes P available to the plant indirectly (Oehl et al., 2001) and while microorganisms could be seen to be in competition with plants for available P in the short term (McLaughlin et al., 1988), this subsequent release of P into the rhizosphere provides a direct source of P for plants in the longer term.

POTENTIAL TO MANIPULATE ROOT AND RHIZOSPHERE TRAITS
Mutants with allelic variation and/or altered expression of genes affecting P acquisition through root traits have been generated. For example, transgenic plants that secrete microbial phytases into the rhizosphere have the potential to release P from inositol phosphates and show enhanced growth and P nutrition when inositol hexaphosphate is the major source of P (Richardson, 2001;George et al., 2004George et al., , 2005a. However, when grown in most soils these plants have comparable growth and P nutrition to control plants (George et al., 2004(George et al., , 2005b. Similarly, overexpression of a bacterial citrate synthase gene in tobacco has been reported to increase citrate efflux from roots and to increase the availability of P from calcium phosphate (López-Bucio et al., 2000), but an effect on plant growth and P acquisition is not always observed . The expression of a wheat malate transporter gene (ALMT1) in barley (Hordeum vulgare) has been shown to be effective in increasing P uptake by transgenic plants, but only in severely acidic soil conditions (Delhaize et al., 2009). Mutations altering root morphology also have the potential to enable plants to acquire more P. For example, barley genotypes with long root hairs have higher yields than genotypes with no root hairs on soils with low P availability (Gahoonia and Nielsen, 2004;Brown et al., 2012), and genotypes of bean, maize and brassica with larger root systems have better growth under P-limiting conditions (Rubio et al., 2003;Liu et al., 2004;Hammond et al., 2009). The differential expression of a number of transcription factors and genes have been shown to be critical in greater accumulation of P in plants ( Bari et al., 2006;Chiou et al., 2006). For example, a T-DNA insertional knockout of a specific arabidopsis gene (AtSIZ1) caused exaggerated Pi starvation responses, including increase in root/shoot biomass quotient, cessation of primary root growth and extensive lateral root development and increased root hair production, even though intracellular Pi concentrations in siz1 plants were similar to wild type. Mutations that improve plant growth through better physiological utilization of P when soil P availability is low can also be incorporated in breeding programmes to develop crops for reduced P inputs (Veneklaas et al., 2012). For example, OsPTF1, a bHLH transcription factor from rice, whose expression increases in the roots of P-starved plants, has been shown to enhance tolerance to P starvation (Yi et al., 2005). Recently, a gene for P-use efficiency in rice (Pup-1) has been cloned and shown to affect the proliferation of roots when present (Gamuyao et al., 2012). Thus, we appear to be on the cusp of being able to deploy a number of postgenomic tools to develop crop germplasm that will allow improved P acquisition and greater physiological P utilization (Hammond and White, 2011;Veneklaas et al., 2012).

ROOT-HAIR TRAITS FOR THE SECOND GREEN REVOLUTION
An examination of Table 1 reveals one of the most effective and least costly of all the P-acquisition mechanisms to be the production of root hairs and, as such, this trait is a good candidate for manipulation to improve the sustainability of agriculture. Root hair formation, length and longevity have been demonstrated to be regulated by P supply in several crops (e.g. Fohse and Jungk, 1983;Hoffmann and Jungk, 1995;White et al., 2005) and also in Arabidopsis thaliana Lynch, 1996, 2000). In addition, genetic variation in root-hair length has been found in many plant species including wheat, barley, clover, bean, turfgrass and soybean (Caradus, 1979;Green et al., 1991;Yan et al., 1995;Gahoonia and Nielsen, 1997;Wang et al., 2004). Breeding programmes for white clover have been able to select for the root-hair length trait due to its highly heritable nature (Caradus, 1981). Variation in root-hair length within crop species has been shown to correlate with P acquisition (Gahoonia et al., 1999) and yield under P-limited conditions  and experiments using the bald root barley (brb) mutant have provided a general understanding of the importance of root hairs to P acquisition and crop yield (Gahoonia and Nielsen, 2003).
Root-hair trait responses to P deficiency include alterations to the length and density of root hairs and the location and size of the root-hair zone (Foehse and Jungk, 1983;Bates and Lynch, 1996;Ma et al., 2001a, b). Root hairs tend to be sparse in P sufficient plants, but increase in length and density as plants become P deficient (Gahoonia and Nielsen, 1997), with benefits of root hairs for P acquisition reaching a maximum when P-depletion zones around each root hair begin to overlap . Newly formed root hairs become sites for expression of genes encoding phosphate transporters (Mudge et al., 2002) and also release and disperse root exudates throughout the rhizosphere (Hinsinger, 2001;Ryan et al., 2001). The structure of root hairs makes them efficient at foraging for P in cracks or pores where roots themselves are unable to penetrate (Misra et al., 1988). Although the metabolic cost of root hairs might be critical in some situations , and root hair formation is often correlated with greater metabolic costs associated with organic acid exudation and P transporters , the direct cost of carbon allocation to root hairs is generally considered minimal (Bates and Lynch, 2000).
The size of P-depletion zones around plant roots can be manipulated by modifications to root-hair length such that an increase in root-hair length allows the depletion zone to expand around the root providing the plant with access to untapped sources of P. In addition, root hairs are thought to be critical in the formation of rhizosheaths Haling et al., 2010 a, b;Brown et al., 2012) which are physical features of the rhizosphere forming the most intimate interface between the soil and root. The importance of rhizosheath size to P acquisition is derived from the corresponding surface area of the root coming into contact with the rhizosphere.

MODELLING THE RELATIONSHIP BETWEEN COSTS OF ROOT HAIR PHENOTYPES AND P DEPLETION
To test the functionality of root-hair traits, we created a model to represent how root-hair length, root-hair density along the root and the longevity of root hairs determine the amount of P extracted by a unit length of root. The details of the construction of this model can be found in Supplementary Data. In this model, the root is represented as a cylinder of radius R 0 (cm) and length of 1 cm. Root hairs are distributed homogeneously along this cylinder at branching density r (cm 21 ) and form a radius R 1 (cm) around the root (Fig. 1A). We assume root hairs expand radially in straight lines and, therefore, the root-hair-length density r l (cm cm 23 ) is determined as a function of the radial distance from the root centreline R: r l (R) ¼ r/ 2pR. The fraction of soil accessed by root hairs is a function of the root-hair-length density. The greater the root-hair-length density, the larger the fraction of soil is exploited. The fraction of soil f accessed by root hairs in a unit volume of soil is a non-decreasing function of root-hair-length density and tends towards a maximum value of 1. The increase slows down gradually as roots occupy a larger volume of soil (Fig. 1B). The relationship between length/density and soil fraction available to roots has been defined previously as the principle of proportional resource capture (Dupuy and Vignes, 2012). In order to understand this principle, it is useful to consider the region of soil accessed by a portion of root as a cylinder of radius r. If the fraction of soil accessible to roots is f, then the probability of the new portion of the root entering a new region of soil is (1 -f )pr 2 . The increase of the fraction of soil accessible per unit added root length/density can therefore be expressed using a differential equation in which solution f ¼ 1 -exp( -pr 2 r l ); explanations of further simplifications to the model can be found in the Supplementary Data.

Model parameterization and validation
We used root-hair density values (r ¼ 500 cm 21 ) typical of those reported in the literature (Tinker and Nye, 2000;Ma et al., 2001b). Data for root diameter R 0 (0 . 5 mm) and roothair length R 1 -R 0 (0 . 7 mm) were derived from experimental work by Brown et al. 9 cm 2 s 21 ) typical of that used in previous work by Ma et al. (2001a, b). We assumed a root hair longevity T of 5 d.  Ö zacar, 2003). Based on these assumptions, model predictions were compared with the average plant P content per unit root length measured in barley cultivars grown in the glasshouse   White, unpubl.), plants were grown for 7 d and had, on average, a total root length of 170 cm. The shoot dry weight was 0 . 04 g. Root dry weight was not derived, but for the estimation of total plant P content, we assumed a root dry weight of 0 . 01 g. The plant P concentration equalled 3 mg P g 21 dry matter. Using these values, we calculated the specific P uptake (total accumulated P per unit root length) per unit root length of 8 . 8 10 24 mg P cm 21 root.
The model underestimates the observed total P content per unit root length and predicts a total P content per unit root length in the order of 10 25 mM cm 21 (Fig. 2). It is likely that differences between the observed and modelled total P content per unit root length might be explained by the rate at which P is replenished in the rhizosphere, due to processes such as root exudation and the release of phosphatases.
Model predictions show that due to competition with other root hairs, root-hair density has less effect on P acquisition than any other root-hair trait (Fig. 2). Unless root-hair length is increased drastically at the same time, less is to be gained by increased root-hair density. Root-hair length is the trait that has the largest effect on P uptake (Fig. 2). The cost of increasing root-hair traits, however, might be prohibitive, particularly due to the physical constraints to growth and maintenance of long root-hair cells. Longevity has less impact on P uptake than root-hair length (Fig. 2), but the benefit is in the same order of magnitude. It is likely also that longevity has low physiological cost and holds more promise to improve P acquisition than other traits. However, this is only true in the case where the limiting factor is the P-replenishment rate. In the case where the limiting factor is total soil P, longevity will have a similar effect on P acquisition as root-hair density.
Costs for root-hair length, density and longevity used exponential empirical functions with constant growth rates, b 1 , b 2 and b 3 , and varying amplitudes a 1 , a 2 and a 3 for root-hair density, length and longevity, respectively. In order to infer the value of the parameters of the cost functions, we made the hypothesis that current genotypes are close to an optimal P uptake. Results show that cost functions follow the same order as the efficiency of each of the root traits: root-hair length (a2 ¼ 0 . 13) has the highest improvement cost, root-hair longevity has an intermediate cost (a3 ¼ 0 . 096) and root-hair density has the lowest improvement cost (a1 ¼ 0 . 0018) (Fig. 3). To summarize, root-hair density has a low cost but is quite inefficient at generating increased P uptake, and root-hair length is efficient at improving P uptake but the associated costs are the highest.
Optimal root-hair traits for low P soils can be derived from this analysis. For example, if P availability is reduced by 15 %, the new optimal root-hair traits will require 3 . 5 % increase in root-hair density, 6 . 9 % increase in root-hair length, and root-hair longevity would have to be increased by 6 . 2 %. In this new configuration, specific P uptake would be reduced by 0 . 7 %. This analysis provides insights into the type of modifications that would allow crops to maintain yield under reduced P availability. Further research should now concentrate on the quantification of the physiological costs associated with root-hair traits. These cost functions will be critical to predict accurately the performance of ideotypes in field conditions.  2. Specific P uptake and calculated cost associated with root-hair (RH) traits. (A) Influence of root-hair density, root-hair length and root-hair longevity on specific P uptake. The x-axis indicates the fraction of the change in density, length and longevity with respect to wild type (WT). The y-axis indicates the corresponding changes in specific P uptake. (B) Estimated cost function for root-hair density, root-hair length and root-hair longevity. The x-axis indicates the fraction of the change in density, length and longevity with respect to wild type (WT). The y-axis is the cost of building these traits.

CONCEPTUAL MODEL OF BARLEY ROOT-HAIR FUNCTION IN P-LIMITED ENVIRONMENTS
By combining insights from studies in the literature, the model described above and our recently published experiments we have produced a conceptual model of barley roothair function in P-limited environments which identifies targets for future breeding programmes for improved P acquisition (Fig. 4). Our recent results have established that the presence of root hairs is implicit to the sustainable yield of barley in P-limiting conditions; however, root-hair length is not important in maintaining yield under these conditions . Further investigations into the possible compensatory role of AM fungi in P acquisition indicated that significant AM colonization is only found in barley plants lacking root hairs, and that AM associations did not provide an effective compensatory mechanism for P acquisition in the absence of root hairs (L. K. Brown, T. S. George, G. E. Barrett, S. McLaren, S. F. Hubbard and P. J. White, unpubl.). Therefore, genotypes with root hairs of an optimum length and density, producing a contiguous depletion zone, with no mycorrhizal infection appear to perform best (Figs 2 and 4). It should be noted that due to the large genotypic variation in traits such as root-hair length among different barley cultivars (Gahoonia and Nielsen, 2004), this conceptual model might not hold true for all cultivars, and this is important to bear in mind when breeding for optimum P acquisition, i.e. genotypes with shorter root hairs may benefit from increased root-hair length. Our results and modelling suggest that the ideal root-hair zone would be comprised of contiguous root hair depletion zones with variable longevity which would allow soil exploration to be maximized but would not waste valuable energy as a result of the competition between root hairs.

ROOT HAIR IDEOTYPES FOR INCREASED P ACQUISITION IN BARLEY
Root architectural ideotype: increasing the surface area of root hairs in P-replete soil Increasing the length and/or density of root hairs increases the potential of a plant to acquire P. One architectural ideotype for improving P acquisition would include an increase in the numbers of root hairs in surface soils where the majority of P is found. This could be achieved in two ways: (1) by increasing the density of root hairs within existing root-hair zones or (2) by modifying root architecture to increase the number of root-hair zones by producing more roots.
There is an optimum root-hair length and root-hair density for the most efficient P acquisition which is determined when P-depletion zones around each root hair begin to overlap (see Fig. 4; Ma et al., 2001a). This concept of competition between root hairs within the root-hair zone is an important consideration when looking at potential ways of increasing the number of root hairs.
Increasing the number of roots and root branches will increase the number of root-hair zones. The potential to couple this with the ability of the plant to respond to P deficiency by proliferating their roots in the P-rich surface layers of soil (Lynch and Brown, 2001) and/or nutrient-rich patches (Hodge, 2004) will have the effect of concentrating more roothair zones in the areas where most P is available. It should be noted, however, that the metabolic costs of soil exploration by root systems are substantial and can exceed 50 % of daily photosynthesis (Lambers et al., 2002).

Root anatomical ideotype: reducing the cost of root hairs
Another proposed component of the root-hair ideotype is based on the idea that P acquisition by root hairs can be enhanced by increasing the longevity of the individual root hairs. The typical active life of a functional root hair for nutrient acquisition is only a few days (McElgunn and Harrison, 1969). In contrast the cell walls of root hairs often remain for many days, long after the root hair cells have ceased to function (Clarkson, 1996). Because of the short life of individual root hairs, the root-hair zone 'moves' through the soil by continuously changing its location following the growth of the root apex, and the period of time for contact between root hairs and soil at a specific location is determined by the lifetime of the root hair. The data of Claassen et al. (1981) suggests that the soil surrounding root-hair zones becomes depleted of available P within a few days and an increase in the longevity of root hairs to greater than a few days would, therefore, not increase P acquisition (Jungk, 2001). However, the time taken to exhaust the soil of available P is likely to vary greatly depending on soil type, being much shorter in extremely P-deficient soils than in the less-deficient soils which are likely to be typical of future low-input agricultural systems. However, it is not known whether all genotypes have root hairs whose longevity is sufficient to take advantage of the available P and we could hypothesize that increased plant available P could be accessed by increasing the functional life of a root hair, thereby keeping high affinity P transporters operational for longer. While the P concentration of the soil decreases The z-axis represents total cost and colourmap indicates variations from optimal P acquisition. The x-axis indicates root-hair length, the y-axis indicates root-hair longevity and the z-axis indicates the cost of making these traits. The vertical line indicates the optimal set of root-hair traits under a 15 % reduction in P availability. The new optimal root-hair traits will require a 3 . 5 % increase in roothair density, a 6 . 9 % increase in root-hair length, and root-hair longevity would have to be increased by 6 . 2 %. In this new configuration, specific P uptake would be reduced by 0 . 7 %.
quickly around newly formed root hairs, this is slowly replenished through the processes of diffusion and mineralization. As such, root hairs which remain functional for longer might be able to acquire more P. Increasing the longevity might also increase the amount of P-liberating compounds that can be exuded by the root hair, thereby having greater potential for liberating more plant-available P. The allocation of root-hair zones to less metabolically demanding root classes is another way of achieving more roothair zones at a minimum cost. Miller et al. (2003) have shown that adventitious roots acquire P at reduced metabolic cost compared with basal or tap roots and the relative biomass of adventitious roots increases under P-deficit conditions. Targeting root-hair zones to roots with aerenchyma could greatly improve the efficiency with which roots take up P. Indeed, there may be potential for the 'saving' associated with this root formation to 'pay' for the additional cost of more root-hair zones. In addition, increased root turnover could potentially improve P acquisition by increasing soil exploration and replacing older roots with younger ones which are more active in P uptake (Steingrobe et al., 2001). The roothair zones associated with the new roots will enhance P acquisition while the senescence of the older roots will allow the remobilization of resources, thereby, reducing the metabolic cost of maintenance respiration.
Biochemical modifications to rhizosphere ideotype: extracting more from root-hair zones The production and secretion of phosphatases and the exudation of organic anions by roots, particularly in the root-hair zone where the potential for P uptake is greatest, could  Modification 1 represents an increase in root-hair numbers and length and the inset demonstrates the resulting overlap of P-depletion zones which are defined by red dotted lines. Modification 2 represents the optimum status, which in the case of the Optic cultivar, means no change. Here the inset demonstrates a zone of contiguous root-hair zones with the extent of the P-depletion zone defined by the red dotted lines. Modification 3 represents a scenario where no root hairs are present and the roots are colonized by AM fungi which produce a P-depletion zone identified by the pink highlighted area surrounding the roots in the inset. The associated yield for each modification is represented by the green barley heads, the greater the number the greater the yield. Potential target traits or ideotypes for improving P acquisition are divided into the three categories.
(1) Architectural -(a) intensive: increasing the total root-hair surface area (more red root-hair zones on more roots); (b) extensive: increasing the number of red root-hair zones in the surface area of soil where the greatest amount of P is typically found.
(2) Anatomical -(a) intensive: longer-living root hairs represented by green root-hair zones; (b) extensive: red root-hair zones on low-cost green roots. (3) Biochemical -(a) intensive: root-hair zones delivering exudates represented by yellow dots; (b) extensive: root-hair zones recruiting beneficial microbes represented by blue circles.
increase P acquisition by plants. Significant genotypic variation in the secretion of organic acids and phosphatases has been observed both between and within plant species (Asmar, 1997;Wouterlood et al., 2004;Richardson et al., 2005;George et al., 2008). The ultimate ideotype to target with regards root hairs and exudation is similar to that of cluster roots formed by a number of species including Lupinus albus (Neumann et al., 1999). An ideotype which targets root-hair zones with an enhanced ability to exude organic anions, followed temporally by phosphatases, is likely to maximize the potential to liberate plant-available P in the zone surrounding root hairs where it can be readily taken up . The order in which these compounds are released could be crucial in increasing plant-available P whereby the release of organic acids would prime soil conditions with soluble Po prior to increased phosphatase activity which would enhance mineralization. With exudates representing 5 -25 % of photosynthate production in species producing cluster roots (Dinkelaker et al., 1995), the targeting of enhanced exudation in the roothair zones, where the high affinity phosphate transporters are found, could help reduce the substantial metabolic investment involved in exudate production.
Root-hair zones might also deliver compounds to select specific microbial communities with a greater ability to increase P availability to plants. This could be through selection of microorganisms which produce phytohormones to enhance root growth, those which produce P-active exudates and those which turnover rapidly releasing P for plant consumption. In order to screen for this trait the functionality of individual microbial species would need to be understood and the identity of appropriate compounds secreted by plants established.
The lack of a role for AM fungi None of the ideotypes suggested in this conceptual model for barley include AM fungi. While associations with AM fungi have been demonstrated to improve P nutrition of many plant species (Smith et al., 2003) White, unpubl.) suggest that barley breeding programmes should focus on maintaining or improving other root traits such as increased root-hair length, density and longevity or greater exudation of P-liberating compounds.

IMPLICATIONS FOR FUTURE BREEDING PROGRAMMES
Improved understanding of root-hair traits involved in P acquisition will help new cultivar breeding programmes. Screening of populations for phenotypes associated with root traits is a useful, though time-consuming, approach (Lynch, 2007). Such breeding programmes should exploit the large genotypic variation in root-hair traits associated with enhanced P acquisition. Importantly, it has been observed that many traits present in traditional landraces and older cultivars may have been bred out of modern-day genotypes which are adapted to high-input, intensive agricultural systems (Bingham et al., 2012). It is traits like these that may now be of value in the current context of finite resources and the need to increase the productivity of poorer soils through the reintroduction of genes regulating such adaptive traits. The impact of the environment on genotype should also be taken into consideration and screens for new crops adapted to problem or marginal soils should be carried out under low-input conditions in the presence of biotic and abiotic stresses. Where soils are less problematic it is possible that modifications to only a single root-hair trait may be sufficient to enhance access to P. When considering which traits to target for improved P acquisition it is important to keep any costs to the plant to a minimum. The benefit of modifying root-hair traits for improved P acquisition depends on the relative metabolic and ecological costs associated with the trait compared with the benefits of the improved P acquisition. Root costs have been found to be particularly important in plants under P stress .
In addition to the need to balance the potential costs and benefits to the plant of the various ideotypes, it is clear that a combination of these traits would provide optimum P acquisition. Under severe P deficiency the combined benefits of increased number and longevity of root hairs are likely to increase P acquisition. This can be complimented by increased exudation of P-liberating compounds. Breeding programmes which involve the integration of P absorption, solubilization and mineralization traits are likely to produce the greatest benefits to P acquisition. Furthermore, superior fitness will be achieved by genotypes that can acquire limited soil resources at reduced metabolic cost, as they can allocate more metabolic resources to defence, growth and reproduction.
Breeding programmes should also consider the implications of phenotypic plasticity ) that helps to maximize P acquisition without wasting valuable carbon resources. Plasticity in the expression of selected root traits to improve P acquisition is preferable to constitutively expressed traits with the consequential metabolic costs of these modifications only being incurred in P-limiting conditions. While the conceptual model presented here has focused on one species (barley) the principles and concepts behind its creation could be easily adapted to other species, given an understanding of the specific adaptive traits and mechanisms that they employ, e.g. the optimum root-hair-length density, the ability of the plant to liberate plant-available P through modifications to rhizosphere biochemistry and its relationships with other microorganisms such as AM fungi. The use of such a model to improve our understanding of the relative importance of the various root-hair traits involved in improving P acquisition for different species could allow breeders to tailor their programmes to produce optimum genotypes for each crop species.
Ideotypes with root-hair traits or a combination of root-hair traits, e.g. increased numbers of root hairs/root-hair zones, increased longevity of root hairs and increased exudation of P-liberating compounds have the potential to improve the P efficiency of crop plants. It is traits such as these that should be targeted through breeding programmes to increase crop production while reducing the environmental impact and improving the long-term sustainability of agriculture. The fact that the genetic control of some of these traits is already known, as highlighted previously, means that their deployment in current germplasm could be relatively rapid. Developments which improve the efficiencies by which humans utilize finite resources such as phosphate have the potential to make a contribution on a global scale in a future of population growth and environmental change.

SUPPLEMENTARY DATA
Supplementary data are available online at www.aob.oxfordjournals.org and consist of details of the mathematical model representing how root-hair length, root-hair density along the root and the longevity of root hairs determine the amount of P extracted by a unit length of root.