•Root hairs are known to be important in the uptake of sparingly soluble nutrients by plants, but quantitative understanding of their role in this is weak. This limits, for example, the breeding of more nutrient-efficient crop genotypes.
•We developed a mathematical model of nutrient transport and uptake in the root hair zone of single roots growing in soil or solution culture. Accounting for root hair geometry explicitly, we derived effective equations for the cumulative effect of root hair surfaces on uptake using the method of homogenization.
•Analysis of the model shows that, depending on the morphological and physiological properties of the root hairs, one of three different effective models applies. They describe situations where: (1) a concentration gradient dynamically develops within the root hair zone; (2) the effect of root hair uptake is negligibly small; or (3) phosphate in the root hair zone is taken up instantaneously. Furthermore, we show that the influence of root hairs on rates of phosphate uptake is one order of magnitude greater in soil than solution culture.
•The model provides a basis for quantifying the importance of root hair morphological and physiological properties in overall uptake, in order to design and interpret experiments in different circumstances.
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 x with ε−1 defines a new space variable, y = xε−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.
Table 1. The parameters and variables used in the dimensional model
Root hair and root radius (cm)
Soil buffer power (−)
Nutrient concentration in the solution (μmol ml−1)
Effective nutrient concentration in the root hair zone Va (μmol ml−1)
Nutrient concentration in the domain outside the root hair zone Vb (μmol ml−1)
Initial solution concentration at time t = 0 (μmol ml−1)
Factor that distinguishes between solution culture and soil systems. In the solution culture d = 1; in soil d = 1/(θ + b) (−)
Molecular diffusion coefficient of nutrients in solution (cm2 s−1)
Diffusion coefficient; D = Dl in solution culture, D = Dlθf/(θ + b) in soil (cm2 s−1)
Effective diffusion coefficient taking the impedance caused by root hairs into account (cm2 s−1)
Efflux from root hairs and root (μmol cm−2 s−1)
Impedance factor of soil (−)
Effective root hair uptake (μmol cm−2 s−1)
Net influx into the root hair and root (μmol cm−2 s−1)
Maximal nutrient influx into root hairs and root (μmol cm−2 s−1)
Michaelis–Menten constants for root hairs and root (μmol ml−1)
Distance between two root hairs (cm)
Root length (cm)
Root hair length (cm)
Outer normal vector
Number of root hairs per cm root length (cm−1)
Root hair zone (cm3)
Domain outside the root hair zone (cm3)
The parameter determines which effective model applies (−)
The vector x = (x1,x2,x3) is the macroscopic space variable.
The vector y = (y1,y2,y3) is the microscopic space variable.
Volumetric water content excluding soil particles (−)
Volumetric content of medium excluding root hairs, θh = 1-θhair (−)
Volume fraction of root hairs (−)
Materials and Methods
Dimensional model explicitly considering root hair geometry
We consider a single root with a root hair zone in a homogeneous medium. The medium may be either soil or nutrient solution. In both cases nutrients move to absorbing roots by diffusion and convection. However, in soil there are the additional complications of the tortuosity of the soil pore network, and sorption of the diffusing nutrients on soil surfaces. Following Tinker & Nye (2000), we treat the soil as quasi-homogeneous at the scale of interest and average across micro-scale heterogeneities. In practice the assumption of quasi homogeneity is imperfect. However, a full treatment of micro-scale heterogeneity would unduly complicate the model. We define a composite soil diffusion coefficient that contains an impedance factor for tortuosity and a buffer power for sorption.
Furthermore, we consider Cartesian geometry rather than the more computationally complex cylindrical geometry (Ptashnyk, in press) (Fig. 1b). This is justified as long as the distance between root hairs at the root surface, 2πar/n, is comparable to the distance between the root hair tips, 2π(ar + Lh)/n, where ar is the root radius, Lh is the root hair length and n is the number of root hairs in a root cross-section. In developing the following model we consider a periodic domain containing cylindrical root hairs orthogonal to a planar root surface. In this domain the distance between the root hairs, l, is the characteristic microscopic length scale; the root length scale, L, is the macroscopic length scale; the homogenization technique requires that l << L.
The importance of convection relative to diffusion can be estimated by the Péclet number (Roose & Kirk, 2009), Pe = aru/D where u is the water flux and D is the diffusion coefficient. Typically, ar < 10−1 cm, u = 10−7 cm s−1 and the diffusion coefficient, D, is ≤ 10−5 cm2 s−1, so Pe ∼ 10−3. This is true for both solution culture and soil. Thus, we can neglect convection. As a result, the nutrient concentration, c, is described by the diffusion equation:
( Eqn 1)
(where c is the nutrient concentration and D is the effective diffusion coefficient). In solution culture, D = Dl, where Dl is the diffusion coefficient in water, and, in soil, D = Dlθf/(θ + b), where θ is the volumetric water content, f is the diffusion impedance factor and b is the buffer power for sorption on soil surfaces. Initially, the nutrient concentration is assumed to be constant, in other words:
( Eqn 2)
We chose the following boundary conditions:
1External boundaries – far away from the root and root hair zone, as well as at the upper and lower boundaries of the domain; we assume that there is no transfer of nutrients, such that:
( Eqn 3)
This describes the case of an impermeable container.
2Root hair surface – nutrient uptake by root hairs is described by the uptake function, fh, such that:
( Eqn 4)
(where d = 1 in the case of solution culture and d = 1/(θ + b) in the case of soil. In soil, the factor d describes that the root only takes up nutrients from the fluid phase).
3Root surface – nutrient uptake by the root is described by the uptake function, fr, such that:
( Eqn 5)
In Eqns 3–5 the outer normals n are unit vectors that are orthogonal to the respective bounding surfaces and point outwards of the domain.
For uptake by root systems or individual roots under specified conditions of plant growth and nutrient supply, the relationship between influx and concentration at the root surface is usually given by a Michaelis–Menten-type relation (Tinker & Nye, 2000, and references therein). We know of no equivalent information for uptake by root hairs, but assume that a similar sort of relationship applies. Hence, we define:
( Eqn 6)
(F, K and E are the maximal nutrient influx, the Michaelis constant and efflux, respectively; and subscripts h and r denote root hair and root, respectively). Alternative functional forms for fh and fr could also be used without any significant influence on the results presented here as long as the order of magnitudes of fh and fr do not change.
For most roots with root hairs, the ratio between the characteristic root hair scale, l, and root length scale, L, becomes small (i.e. ε = l /L << 1). If the macroscopic length scale, L, is assumed to be 1 cm, ε lies between 6.1·10−3 and 4.3·10−2 for different plant species (see Table 2). We use the method of homogenization to analyse and simplify the model such that we do not explicitly consider every single root hair, but the cumulative effect of all root hair surfaces.
Table 2. Morphological and physiological properties of roots and root hairs. The data were obtained from a literature survey for phosphate uptake by mildly phosphate starved plants
2 The calculated distance between the root hairs approximately the square root of the root surface area that is associated with a single root hair: .
3.2–3.3f, 5.5g, 8h
0.77a, 1.08b, 0.75–0.85c
460a, 560b, 240–380c
1.02a, 1.1b, 1.11–1.49c
In this section, we introduce three effective macroscopic models for nutrient transport and uptake near a root with root hairs. We derive them from the model that explicitly considers the root hair geometry given by Eqns 1–6. The full derivation (see Mathematical Notes S1) involves two steps. First, the single root hair scale model is nondimensionalized and, second, formal multiscale expansion is used to derive the effective equations valid on the root length scale. The method of nondimensionalization involves scaling the variables so that the new variables have no units (Fowler, 1997). In this form the model has fewer parameters and it is easier to analyze, in addition to revealing which processes dominate in any given parameter regime. For different limits of the root hair uptake rate we obtain the three different models. In the following, we present the effective equations in their dimensional forms.
In the homogenized model the domain Va denotes the root hair zone and ca denotes the averaged nutrient concentration within this domain. The surrounding domain outside the root hair zone is denoted as Vb with the corresponding concentration cb. The concentration cb is given by:
( Eqn 7)
(where D is the effective diffusion coefficient; in water D = Dl; and, in the case of soil, D = Dlθf/(θ + b)). On the boundary far away from the root and on the upper and lower boundaries of Vb we apply a no-flux boundary condition:
( Eqn 8)
(where n is the outer normal of the domain). This reflects the typical situation in a container, where nutrients cannot leave the domain.
Three different models can be derived for the root hair zone. The dimensionless parameter α is given by:
( Eqn 9)
and distinguishes between those models (see Mathematical Notes S1, Eqn 16). The critical parameters determining α are both morphological and physiological parameters (i.e. the inter hair distance l; the factors d = 1 for solution culture and d = 1/(θ + b) for soil; root length L; the effective diffusion coefficient D; maximal nutrient influx into root hairs Fh; and the Michaelis–Menten constant for root hairs Kh). A review of parameter values from the literature for different plant species is presented in Table 2. If α ∼ 1, both uptake and diffusion are important processes; if α < 1, uptake is negligibly small compared with diffusion; and if α > 1, uptake is fast compared with diffusion.
Model 1 (case α ∼ 1) The most interesting situation arises when the rates of uptake and diffusion are comparable (i.e. Fh/Kh ∼ D/L). This is the most common case in soil according to our literature search (see Table 2 and Eqn 9). In this case there is a dynamic development of a depletion zone within the root hair zone. This effectively takes into account the dynamic development of overlapping depletion zones between neighbouring root hairs. In this case the concentration inside the root hair zone, Va, is described by an effective diffusion equation with the sink term describing the uptake by the root hairs:
( Eqn 10)
(where θh is the volume fraction of the homogeneous medium in the root hair zone; d = 1 for solution culture or d = 1/(θ + b) for soil; 2πah/l2 is the root hair surface area density; fa is the root hair uptake described by a surface flux; and is the effective diffusion matrix, in other words, it takes the diffusion impedance caused by the presence of root hairs into account). This is the dimensional form of Eqn 41 in the Mathematical Notes S1. The uptake fa is given by:
( Eqn 11)
which is the dimensional form of Eqn 46 in the Mathematical Notes S1.
Because of the impedance caused by the root hairs, the effective diffusion in the root hair zone is slower than that in the fluid or soil. The effective diffusion matrix is given by:
( Eqn 12)
(where Disotropic is the scaled diffusion matrix, and Dcorrector contains negative diagonal elements reducing the overall diffusion). The corrector is calculated in dependence on the root hair radius, ah, and the inter hair distance, l; see Eqn 41 in the Mathematical Notes S1. If the root hairs are sparse, ah/l << 1, the effective diffusion will be nearly identical to the diffusion coefficient D; whereas if the root hairs are dense, 0 << ah/l < 0.5, the effective diffusion becomes small (see Fig. 1 in the Notes S1). Typical values for ah/l = 4.9 · 10−2 are:
( Eqn 13)
(where D is the effective diffusion coefficient, and θhair = 0.01 is the volume fraction of root hairs given by ). In this specific example the corrector is based on the parameters for wheat plant (see Table 2) (Föhse et al., 1991).
The domain Va is bounded at one side by the root surface. Nutrient flux into the root is described by:
( Eqn 14)
(where n is the outer normal to the root surface; d = 1 for solution culture or d = 1/(θ + b) for soil; and fr describes the flux into the root (see Eqn 5). The other boundary is between the root hair zone, Va, and the domain Vb, where continuity of concentration and continuous flux boundary conditions are applied:
( Eqn 15)
(where n is the outer normal of the root hair domain, Va, pointing into the domain Vb).
In the Mathematical Notes S1 we derived the effective equations using multiscale expansion. A proof that these equations are actually the unique limit of this homogenization problem when ε → 0 can be found in Ptashnyk (in press).
Model 1 describes the root hair uptake with a sink term and in this sense it is similar to the model proposed by Bhat et al. (1976) and Geelhoed et al. (1997). However, the main difference is that the diffusion impedance caused by the presence of root hairs is also explicitly taken into account in our model.
Model 2 (case α > 1) When α > 1, nutrient uptake into the root hairs is vanishingly small. The average concentration, ca, is described by:
( Eqn 16)
The only difference to Model 1 is that Eqn 10 is replaced with Eqn 16.
This describes the situation where there is no significant active nutrient uptake within the root hair zone. The concentration gradient is solely created by the flux into the root. The root hairs are impeding the nutrient flux, but they are not influencing the concentration profile. There is no corresponding model in the literature describing this possibility, although a scenario similar to this has been discussed previously by Roose (2000). Our parameter estimation in the section ‘Parameter values’ suggests that this model does not apply for most plant species. However, it can apply in strongly sorbing soils, particularly for root-hair-defective mutants (e.g. of Arabidopsis). Equation 9 enables us to exactly determine parameter regimes where it is possible to neglect root hair uptake.
Model 3 (case α < 1) When α < 1, the nutrient uptake by the root hairs is very large. Thus, nutrient concentration profiles between neighbouring root hairs start overlapping rapidly. This is the most common case in solution culture according to our literature search (see Table 2 and Eqn 9). The average concentration reaches its equilibrium almost instantaneously and thus the dynamic development of a concentration gradient in the root hair zone can be neglected. The concentration, ca, within the root hair zone is constant and given by the solution to fh = 0. For the case of Michaelis–Menten kinetics, it is given by Eqn 6:
( Eqn 17)
It follows that:
( Eqn 18)
in the root hair zone, Va.
The boundary of the domain, Vb, to the root hair zone, Va, is described with a Dirichlet boundary condition:
( Eqn 19)
In this model, root hairs effectively extend the root radius by the root hair length, and thus Model 3 corresponds to the model proposed by Passioura (1963).
We obtained morphological and physiological parameters for phosphate uptake by roots and root hairs from a survey of the literature. The parameters for different plant species are given in Table 2. For a particular plant species, maximum rates of influx into roots and the concentration-dependence of influx (as summarized in the Michaelis–Menten parameters in Eqn 6) vary with the plant's internal nutritional status, growth stage and other conditions (Marschner, 1995; Tinker & Nye, 2000). Hence, in a plant that has been starved of phosphate before measurements of influx are made, root transport systems will be up-regulated compared with phosphate-sufficient plants, and influx will be correspondingly greater. A complete description of the uptake properties of roots of a particular species is therefore complicated and beyond the scope of the current model. Hence, an experiment to paramaterize our model might use plants in nutrient solution under moderate phosphate stress before measurement of short-term influx at different concentrations of phosphate (Narang et al., 2000; Nielsen & Schjørring, 1983). Other methods include using radiolabelled phosphorous in soil for measuring short-term kinetics, uptake measurements between two harvests (Föhse et al., 1991) and the depletion method of Claassen & Barber (1974) (Itoh & Barber, 1983; Krannitz et al., 1991). Values of Fh, Kh and Eh are given in Table 2 for phosphate uptake by different plant species.
The parameter Eh in Eqn 6 is calculated from influx at the minimum concentration at which roots can maintain a net influx of phosphate, Cmin, according to:
( Eqn 20)
(where Cmin is typically in the range of 0.01–0.12 μM for different plant species (Tinker & Nye, 2000)).
The initial concentration of phosphate is taken to be 5 · 10−4μmol cm−3. This is typical for a soil in which phosphate is likely to be limiting (Föhse et al., 1991; Tinker & Nye, 2000). As soil-specific parameters we chose volumetric water content θ = 0.3, impedance factor f = 0.3, and soil buffer power b = 239. These are typical values often found in soils (Tinker & Nye, 2000; Barber, 1995; Roose et al., 2001) and will therefore be used in our simulations.
From the parameter values in Table 2 we found that the dimensionless parameter α is between 0 and 1. Thus, we present simulation results for Models 1 and 3. We implemented the effective models given above using the finite element analysis package Comsol Multiphysics 3.5 (http://www.comsol.com/).
In the following sections we illustrate the application of the three models by considering the effects of root hair properties on phosphate depletion in solution culture and soil. Changes in concentration only occur with distance from and along the root axis. Therefore, it is sufficient to consider axial symmetry.
In the first example we compare Models 1 and 3 with parameters relating to wheat and tomato. We show the one-dimensional concentration profiles of phosphate with respect to the distance from the root axis. The position along the root axis is at the centre of the root hair zone; the root hairs are 0.033 cm long.
In Model 1 (case α ∼ 1), root hairs take up phosphate according to the model described in Eqns 7, 8 and 10–15. In Model 3 (case α < 1), the uptake of phosphate by root hairs is very fast, leading to effectively instantaneous uptake in comparison with diffusion through the solution. Thus, the root radius is essentially extended by the root hair length, and gradients within the root hair zone can be neglected. The model describing this situation is given in Eqns 7, 8, 18 and 19. In Fig. 2 we show the gradient caused by depletion, for solution culture and soil. After 1 d the depletion zone in solution culture extends up to 3 cm away from the root surface (Fig. 2a). In comparison, this zone is only 0.12 cm wide in soil (Fig. 2b). The dynamic development of the gradient can be seen in the close-up of the root hair zone. In solution culture, the depletion in the root hair zone is in the order of seconds (Fig. 2c), whereas, in soil, it is in the order of hours (Fig. 2d).
In the second example we analyse the effect of root growth on phosphate depletion and uptake in Model 1. The root hair zone is situated 0.5 cm behind the root tip and is 1 cm in length along the root; the root hairs are 0.033 cm long. The root tip is assumed to grow at a rate of 2 cm d−1 (Watt et al., 2006). Changes with respect to the root surface and along the root axis are considered. Fig. 3 shows the depletion zones around a root after 2 d, for both solution culture and soil, according to three different scenarios. In scenario 1 it is assumed that both root and root hairs are taking up phosphate. In scenario 2 root uptake is neglected, and in scenario 3 only root uptake is considered. In this way we show the effect of the root hair zone on the depletion of phosphate. In the case of solution culture (Fig. 3a), the root hairs cause a strong depletion in scenarios 1 and 2, while the depletion caused by the root only in scenario 3 is negligible because of fast replenishment in the absence of sorption. In soil, as a result of strong sorption, the width of the depletion zone is maintained at the width of the root hair zone even after the root hair zone, has grown further downwards (Fig. 3b). The root on its own causes a typical depletion profile, which is steeper, but more narrow, than in solution culture. We present the dynamic development of the depletion zones in the three scenarios in Supporting Information Videos S1–S6.
The corresponding cumulative phosphate uptake is shown in Fig. 4. After 2 d, the overall uptake in solution culture in the two scenarios where root hairs are absorbing phosphate is larger than in the scenario where only the root is absorbing. The uptake is 2.7 and 1.7 times larger for scenarios 1 and 2, respectively. The contributions of root and root hairs to overall uptake in scenario 1 is represented by the dashed lines in Fig. 4a. After 2 d, roots contributed to 36% of the overall uptake and root hairs contributed to 64% of the overall uptake. This situation is different in soil. When root hairs are taking up phosphate the depletion is so strong that the root uptake becomes negligibly small. Therefore, total phosphate uptake of scenarios 1 and 2 are similiar (Fig. 4b). Root uptake is only significant when no earlier depletion, by root hairs, occurs. Root and root hair uptake is 28 times larger than root uptake alone. The root hair zone contributes 96% to the overall uptake and the root contributes 4% to the overall uptake.
We presented a root hair scale model for nutrient uptake by root hairs and derived three different effective models, which were dependent on the morphological and physiological properties of root hairs. The first model describes the effect of root hairs with a sink term in the diffusion equation. This results in the development of a concentration gradient within the root hair zone. The second model describes a situation where root hair uptake is negligibly small. In the third model all the nutrients inside the root hair zone are taken up instantaneously, and thus the root hairs effectively increase the root radius by the root hair length. Based on our model analysis using published morphological and physiological parameters we have shown that Model 1 (case α ∼ 1) and Model 3 (case α < 1) are primarily applicable in real systems. Model 2 is suitable for strongly sorbing soils, particularly for root hair-defective mutants. Whether we are in a solution culture or homogeneous soil system only changes the characteristic timescale but not the models.
In Model 1, a sink term for root hair uptake in the conservation equation results in phosphate depletion in the root hair zone. Such sink terms have been developed before for soil systems (de Willigen & van Noordwijk, 1994; Geelhoed et al., 1997; Bhat et al., 1976; Baldwin et al., 1973); in the following we discuss the differences between our model and those sink terms. First previous work derived a macroscopic equation using a sink term for root hair uptake, which is based on a concentration averaged around the root hairs. We essentially follow this approach but derive the sink term in a mathematically rigorous way using the method of homogenization. Furthermore, we can predict, from root morphological and physiological parameters, when this specific form of sink term is valid. Further still, in all previous models the impedance of diffusion caused by the root hairs has been neglected. Previous models consider impedance to be caused by soil particles but not by root hairs. Finally, the sink terms are based on steady-state approximations of linear (Bhat et al., 1976; Baldwin et al., 1973) or zero sink (Geelhoed et al., 1997; de Willigen & van Noordwijk, 1994) boundary conditions at the root hair surface. In contrast to previous models, our model uses nonlinear Michaelis–Menten boundary conditions. In Model 3, root hairs effectively extend the root radius, and phosphate inside the root hair zone is taken up instantaneously. Such an ‘equivalent cylinder’ has already been discussed by Passioura (1963). We have been able to determine parameter regimes where each model is valid. When the characteristic velocity scale of uptake is in the same order as that of diffusion, Fh/Kh : D/L ∼ 1, Model 1 applies. When Fh/Kh : D/L ∼ ε−1, Model 3 is suitable. For typical values D = 10−5 cm2 s−1, l = 0.01 cm, and L = 1 cm, the characteristic velocity scale of uptake needs to be in the order of 10−5 cm s−1 for Model 1 and larger than 10−3 cm s−1 for Model 3.
The models presented here provide an explanation of phosphate uptake by various plants with different root hair architectures. Previous studies have demonstrated that root hairs are important in phosphate uptake and that the root hair length is influenced by the external phosphate concentration (Bates & Lynch, 1996). The fact that the width of the phosphate-depletion zones around nonmycorrhizal roots are closely related to root hair length (Marschner, 1995) is demonstrated in Fig. 3. This provides a mechanism for the plant to maximize phosphate uptake under limiting conditions.
The validity of our model could be tested by measuring phosphate-uptake dynamics in root hair mutants. Root hair mutants are available in Arabidopsis (Schiefelbein & Somerville, 1990; Grierson et al., 2001), rice (Ma et al., 2001), barley (Gahoonia et al., 2001), tomato (Hochmuth et al., 1985) and Zea mays (Wen & Schnable, 1994). The Arabidopsisrhd2 and rhd6 mutants have altered hair initiation and elongation, respectively, and both have reduced shoot biomass compared with the wild type when grown under phosphate-limiting conditions (Bates & Lynch, 2000b). This was interpreted as reflecting a reduced ability by the rhd2 and rhd6 mutants to take up phosphate, or may reflect an inability to respond to phosphate-limiting conditions. Our model can be utilized to explain alterations in phosphate uptake by these mutants. In particular, we can predict how quickly the concentrations in the root hair zone will drop to zero. This indicates the timescale of measurements required to observe the dynamics of the depletion zone development. The parameter α ranges between 0.37 and 0.54 for the morphological and physiological parameters of Arabidopsis given in Table 2. Thus, they are at the border between Model 1 (case α ∼ 1) and Model 3 (case α < 1). This means that the root hairs deplete the root hair zone very quickly, but Model 3 might still overestimate uptake. We suggest using the most inclusive model in this case, in other words, Model 1, which includes the sink term for phosphate uptake in the root hair zone. In this case, the timescale of depletion is given by
For the Arabidopsis species given in Table 2, the timescale of depletion ranges from 0.79 to 8.61 s in solution culture and from 3.16 to 34.36 min in soil. In a hydrogel with the same impedance factor as soil, the development of the depletion profile would be slowed down in comparison to nutrient solution culture and thus be more readily observable. If the morphological and physiological parameters for rhd6 and rhd2 are known, our model can predict the α-regime and timescale of depletion compared with wild-type species. This can help to optimize the design of a specific validation experiment.
This work was supported by the Vienna Science and Technology Fund (WWTF, grant no.: MA07-008) and by the Austrian Science Fund (FWF, grant no.: T341-N13). Daniel Leitner was funded by BBSRC (grant ref. BB/C518014/1). Andrea Schnepf is a Hertha-Firnberg Research Fellow. Tiina Roose is a Royal Society University Research Fellow.