Ross E. McMurtrie, School of Biological, Earth and Environmental Sciences, The University of New South Wales, Sydney, NSW 2052, Australia. Tel: +61-425 306272; Fax: +61-2-9385 1558; E-mail: firstname.lastname@example.org
CO2-enrichment experiments consistently show that rooting depth increases when trees are grown at elevated CO2 (eCO2), leading in some experiments to increased capture of available soil nitrogen (N) from deeper soil. However, the link between N uptake and root distributions remains poorly represented in forest ecosystem and global land-surface models. Here, this link is modeled and analyzed using a new optimization hypothesis (MaxNup) for root foraging in relation to the spatial variability of soil N, according to which a given total root mass is distributed vertically in order to maximize annual N uptake. MaxNup leads to analytical predictions for the optimal vertical profile of root biomass, maximum rooting depth, and N-uptake fraction (i.e., the proportion of plant-available soil N taken up annually by roots). We use these predictions to gain new insight into the behavior of the N-uptake fraction in trees growing at the Oak Ridge National Laboratory free-air CO2-enrichment experiment. We also compare MaxNup with empirical equations previously fitted to root-distribution data from all the world's plant biomes, and find that the empirical equations underestimate the capacity of root systems to take up N.
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 (Nmin) above a specified soil depth (Parton et al. 1988; Comins and McMurtrie 1993; Jackson et al. 1996, 2000), or from Nmin 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 CO2 (eCO2) because the majority of experiments show that root distributions are altered when trees are grown at eCO2, involving in particular an increase in rooting depth (Iversen 2010). For example, at the Oak Ridge National Laboratory (ORNL) forest free-air CO2-enrichment (FACE) experiment, both peak annual root biomass and annual root production approximately doubled at eCO2 (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 eCO2 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 eCO2 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 eCO2 (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).
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.
The MaxNup-optimization hypothesis
According to the MaxNup hypothesis, the annual amount of N exported from the root system to support the growth of nonroot tissues (Unet, g N m−2 land area year−1) is maximized with respect to the vertical distribution of fine-root mass per unit soil volume (R(z), kg DM m−3) and the maximum rooting depth (Dmax, m), subject to the constraint of a given total amount of root biomass per unit land area (Rtot, kg DM m−2). Maximization of Unet for a given amount of root biomass is a reasonable modeling objective at an N-limited site, as any nonoptimal root distribution would result in less N export to aboveground pools. The modeled Unet–Rtot relationship can be viewed as a “return on investment”, the return being annual N gain by nonroot tissues for a given carbon (C) investment in roots. (Note: Under conditions where N is nonlimiting but C is limiting, an analogous hypothesis may apply, namely minimization of annual C investment in roots required to achieve a given annual N uptake. This case will not be considered further here.) We implemented MaxNup using a simple model of annual N uptake per unit soil volume by roots at depth z (Ur(z), g N m−3 year−1) as a saturating function of R(z):
where Uo(z) (potential annual plant N uptake) is the asymptotic N-uptake rate in the limit R(z) →∞ and Ro (kg DM m−3), the root-mass density yielding half the potential N-uptake rate, determines the initial slope of the Ur–R relationship. Uo(z) is assumed to be greatest at the soil surface and to decrease exponentially with depth (cf. Jackson et al. 2000; Jobbágy and Jackson 2001):
where Umax is total potential annual N uptake integrated over all soil depths, and Do is the length scale for exponential decline of available soil N. The assumption that Uo decreases exponentially with depth is supported by measurements of gross mineralization and extractable inorganic N over depths 0–90 cm at the ORNL experiment (Iversen et al. 2011, 2012). (Note: MaxNup may be applied using any given function Uo(z) [eqs. A18 and A19].) Root-mass density R is related to root-length density (Lr, cm root length cm−3): R=πro2ρrLr, where ro (cm) and ρr (kg DM m−3) are the radius and tissue density of roots, respectively. Hence, Lro=Ro/πro2ρr represents root-length density at half potential N-uptake rate.
Our representation of plant N uptake per unit soil volume at depth z in terms of one equation for plant-available N at z (eq. 2) and one for the fraction of available N taken up by roots (eq. 1), with root biomass and its depth distribution held constant over time, is a gross simplification of the complex processes, including root–microbe interactions that operate in plant rhizospheres (Schimel and Bennett 2004; Wardle et al. 2004; Frank and Groffman 2009). The advantage of making these simplifications is that they enable us to derive simple analytic expressions (below) for annual N uptake by spatially distributed root systems. However, as discussed in Appendix A1, a mechanistic basis for equation (1) can be derived from the Barber–Cushman (BC) model (Darrah 1993; Yanai 1994) that describes mass flow and diffusion of soil N toward root surfaces down a concentration gradient, and competition between root N uptake and soil microbial N immobilization. In the BC model, nutrient uptake per unit soil volume depends on root surface area per unit soil volume, which is proportional to Lr and nutrient influx I per unit root surface area, which is a function of solute concentration at the root surface (Yanai 1994). The Ur–R relationship derived from the BC model is well approximated by equation (1) (Fig. A1). The value of Lro fitted to the BC model depends on the capacity of roots to absorb solute N and the rate of solute N transport toward root surfaces relative to the rate of N immobilization by microbial decomposers.
Equation (1) is shown in Figure 1a at four soil depths in a system of roots with Ro= 0.265 kg DM m−3 and Do= 0.3 m. At low root-mass density (R), the bulk of available N is immobilized because the distance between roots is relatively large, which increases the likelihood that solute is immobilized before reaching the root surfaces. As R increases the inter-root distance decreases, so a greater proportion of solute reaches the root surfaces without being immobilized.
The net annual N export to aboveground pools by roots at depth z (Un(z), g N m−3 year−1) is equal to Ur(z) minus the annual N investment in growing roots at depth z:
where Nr (g N kg−1 DM) is the N concentration of roots and τr (year) is root life span. Equation (3) is derived from equation (1) under assumptions that root mass is maintained at a steady state where annual root production equals annual mortality, and that there is no N retranslocation at root senescence (Gordon and Jackson 2000). In contrast to the Ur–R relationship (Fig. 1a), the Un–R relationship has a peak that shifts to lower R values with increasing depth z (Fig. 1b).
We applied MaxNup to the total annual N exported to aboveground pools (Unet, g N m−2 year−1) obtained by integrating equation (3) through the rooting zone from z= 0 (surface) to z=Dmax (maximum rooting depth):
Likewise, annual total N uptake per unit land area (Utot, g N m−2 year−1) is the integral of annual uptake per unit soil volume by roots at depth z (Ur(z), g N m−3 year−1) from the soil surface to Dmax:
where Ur is a saturating function of R(z) (eq. 1; Fig. 1a). Total root mass per unit land area (Rtot, kg DM m−2) is the integral of root-mass density (R(z), kg DM m−3) from the soil surface to Dmax:
Our optimization hypothesis MaxNup is that for a given total root mass Rtot, roots are distributed vertically in order to maximize annual N supply to aboveground pools (Unet). The optimal solution for R(z) and maximum rooting depth Dmax is obtained using the Lagrange multiplier method described in Appendix A2. It turns out that if Nr and τr are constant throughout the root system, the solution for R(z) that maximizes Unet also maximizes Utot.
The model has been parameterized (see Table 1) for plantations of the deciduous tree Liquidambar styraciflua (sweetgum) growing over an 11-year period at CO2 concentrations of 385 (aCO2) and 550 ppm (eCO2) at the ORNL FACE experiment. The CO2 treatment commenced in 1998, 10 years after the plantation was established with two plots operated at eCO2 and three at aCO2. The experiment has been described fully elsewhere (see Norby et al.  and references therein). Methods used for root biomass and N-uptake measurements shown in Figure 2 are described in Iversen et al. (2008) and Norby et al. (2008, 2010). Data from the FACE experiment are publicly available through the Carbon Dioxide Information Analysis Center (http://cdiac.ornl.gov), U.S. Department of Energy, Oak Ridge National Laboratory, Oak Ridge, TN.
Exponent in the empirical relationship for cumulative root proportion2,3, (15)
0.914 (tundra), 0.972 (aCO2), 0.984 (eCO2)
Diffusion coefficient of nutrient in soil1, (A3)
0.052 cm2 day−1
Gross N-uptake fraction (=Utot/Umax) and net N-uptake fraction (=Unet/Umax), (12), (13)
Peak values of φN and φnet, (A28), (A27)
Lagrange multiplier ∂Un/∂R, (A10)
g N kg−1 DM year−1
Rate of solute loss from the rhizosphere through immobilisation by soil microbes12, (A3)
Root-tissue density8, (1)
380 kg DM m−3
Expression , (13)
Root lifespan8, (3)
Goal function, (A10)
g N m−2 year−1
Expression , (A25)
Optimal vertical profiles of root-mass density and N uptake predicted by MaxNup
The maximum possible Unet would be achieved if root-mass density (R(z)) were held at the peak of the Un–R relationship shown in Figure 1b at all depths z. However, if the total amount of root biomass (Rtot) is too low, then it is not possible to operate at the peak of the Un–R relationship at all depths unless maximum rooting depth (Dmax) is small; a higher value of Unet might be achieved by operating to the left of the peak of the Un–R relationship with deeper roots (larger Dmax). As shown in Appendix A2, for a given total root mass (Rtot), Unet is maximized when the increase in N-uptake rate Ur(z) associated with a small local increase in root-mass density R(z) (i.e., the marginal gain in N uptake, ∂Ur/∂R) has the same value throughout the rooting zone, which is given by its value Uo(Dmax)/Ro at the maximum rooting depth (Dmax), where the optimal R= 0 (eq. A17). Provided Nr, τr, and Ro (i.e., Lro, ro, and ρr) are independent of z, we obtain the vertical profile of optimal R(z):
which is illustrated in Figure 1c. The length scale for the decrease of R(z) in equation (7) (2Do) is twice the length scale for the exponential decrease of available soil N (Uo(z), eq. 2). The optimal profile of R(z) (Fig. 1c) is thus less steep than the profile of Uo(z) (Fig. 1d). (Note: This conclusion might not hold if Ro were a decreasing function of depth.) Total root mass (eq. 6) is
An analogous equation for the optimal profile of N supply to aboveground pools Un(z) is given by equation (A20). Total annual N uptake Utot and annual N supply to aboveground pools Unet are obtained from equations (5) and (4), respectively:
where Umax is total potential N uptake, and
The solid circles in Figure 1a and b indicate the optimal values of Ur(z), Un(z), and R(z) at four soil depths for a total root mass of Rtot= 0.19 kg DM m−2. Figure 1c and d shows the corresponding complete vertical profiles of optimal R(z) (eq. 7) and Ur(z) (eq. 9) as well as the optimal profiles for a root system with twice the total root mass (Rtot= 0.38 kg DM m−2). With this doubling of Rtot, the optimal root-mass density R(z) increases throughout the rooting zone, and the maximum rooting depth Dmax increases from 0.74 to 0.97 m (Figs. 1c and 2a). Total N uptake Utot, evaluated from equation (10), increases from 6.8 to 8.7 g N m−2 land area year−1. In contrast, the total potential N uptake over the rooting zone () is 13.6 g N m−2 land area year−1, which greatly exceeds the predicted values of Utot, indicating that much available soil N remains untapped by these root systems. In Figure 1d, the optimal profiles of Ur(z) are compared with the vertical profile of annual potential N uptake Uo(z). The difference between Ur(z) and Uo(z) is considerable. For these two values of Rtot, respectively, 50% and 36% of potentially available soil N throughout the rooting zone are not taken up; in fact, as the results below will show, in terms of Unet, it would be uneconomic to do so due to the additional N cost of growing more root mass.
Relationships between optimal rooting depth, N-uptake fraction, and total root mass predicted by MaxNup
The optimization hypothesis MaxNup predicts the optimal values of maximum rooting depth (Dmax), total annual N uptake (Utot), and annual N export to aboveground pools (Unet) as functions of total root mass (Rtot) (eqs. 8, 10, and 11, respectively). Dmax can be eliminated from equations (8) and (10) to obtain a relationship between the fraction of available N taken up annually (φN=Utot/Umax) and total root mass Rtot:
An analogous relationship between the fraction of available N distributed to aboveground pools (φnet=Unet/Umax) and Rtot is obtained by combining equations (11) and (12):
where . Equation (13) was fitted to annual measurements of peak annual root mass to a soil depth of 60 cm (Rtot) and annual N supply to aboveground pools (Unet) from the ORNL FACE experiment with Rtot as independent variable and Unet as dependent variable. Values of Umax and the product RoDo were estimated with τr= 1 year (Iversen et al. 2008) and Nr= the average of measured root N concentration over all plots and years (6.8 g N kg−1 DM, Norby et al. 2008), yielding Umax= 13.6 g N m−2 year−1 and Ro= 0.265 kg DM m−3 with Do= 0.3 m. The corresponding value of root-length density at half potential N-uptake rate is Lro= 0.77 cm−2. Modeled relationships between gross N-uptake fraction (φN) and Rtot (eq. 12) and net N-uptake fraction (φnet) and Rtot (eq. 13) are shown in Figure 2b and c, respectively, along with data from the ORNL experiment. The modeled relationship between maximum rooting depth (Dmax) and Rtot is illustrated in Figure 2a. Initially, both Dmax and φN increase rapidly with Rtot, but, because N uptake is a saturating function of root-mass density (eq. 1), their rates of increase slow as Rtot increases further, with φN approaching 1 asymptotically as Rtot→∞. Compared to φN, the net N-uptake fraction φnet increases less rapidly with Rtot due to the increasing N cost of growing more root mass. φnet reaches a peak when Rtot= 0.38 kg DM m−2 (indicated by the open circles in Figs. 1a and b and 2) and then decreases at higher Rtot (cf. Franklin et al. 2009). When φnet is at its peak, Un(z) is maximized with respect to R(z) throughout the rooting zone (i.e., ∂Un/∂R) = 0, open circles Fig. 1b).
The relationships shown in Figure 2b and c indicate that there is a diminishing N return from increased C investment in roots. Under a doubling of Rtot from 0.19 to 0.38 kg DM m−2 (respectively, large solid and open circles in Fig. 2b and c), φN increases by only 28% (0.50–0.64) while φnet increases by only 11% (0.41–0.45). For both φN and φnet, the diminishing returns from increased Rtot are associated with competition between N uptake by roots and N immobilization by soil microbes, which led to the saturating Ur–R relationship in equation (1). For φnet, the more severe negative return from increased Rtot beyond the peak reflects the additional N cost of growing more root mass, which may be offset somewhat if the extra roots have lower N concentrations (Nr), or longer life spans (τr), or are thicker (higher ro) (cf. Iversen et al. 2008) (see Appendices A2 and A3). Thus, if Rtot were to exceed its value (0.38 kg DM m−2) for peak φnet, then although total N uptake would increase (cf. Fig. 1d), N supply to aboveground plant organs would decrease, which would seem to be detrimental. However, it may be advantageous to an individual plant for its Rtot to increase beyond the peak if it allows that plant to lock up nutrients at the expense of its competitors, and this strategy can be evolutionarily stable (King 1993; Hodge 2009; Dybzinski et al. 2011; Franklin et al. 2012).
Comparison with previous modeling of root-depth distributions
where F(z) represents the fraction of root-mass density at depth z and Z0 is the length scale of the exponential decline of root-mass density with depth (Arora and Boer 2003), and
where CRP(z) represents the cumulative root proportion between the soil surface and depth z (Gale and Grigal 1987; Jackson et al. 1996; Arora and Boer 2003). Equations (14) and (15) are equivalent if . Estimated values of the empirical parameter β for the ambient and elevated CO2 treatments at the ORNL FACE experiment are 0.972 and 0.984, respectively (Iversen 2010), and range over the world's biomes from 0.914 for tundra vegetation, which tends to be shallow rooted, to 0.976 for temperate coniferous forests (Jackson et al. 1996). A fundamental problem with equation (15) is that it is inconsistent with data on how the CRP–z relationship actually changes with increasing total root mass Rtot, namely, the maximum rooting depth increases and the entire CRP–z relationship shifts downward (Arora and Boer 2003). In contrast, root-depth distributions derived from MaxNup (eqs. 7 and A22) are consistent with data in this respect (Fig. 1c), while they are also consistent with equations (14) and (15) near the soil surface. MaxNup therefore provides a more robust model of root-depth distribution than do equations (14) and (15).
Figure 3 compares the net N-uptake fraction (φnet) predicted by MaxNup with that obtained from equation (14) as functions of Rtot over a representative range of β values. (The net N-uptake fraction obtained from equation (14) was evaluated from the expression , where Un was obtained by substituting eq. 14 into eqs. 1 and 3.) As expected, at any value of Rtot, φnet predicted by MaxNup exceeds the net N-uptake fraction obtained from the empirical distribution (Fig. 3), because Unet, and hence φnet, is reduced by any departure from the optimal root profile represented by equation (7), or any variation in maximum rooting depth from the optimum. Thus, equation (14) underestimates the capacity of root systems to take up N. The underestimation is especially large for the shallow empirical distribution (β= 0.914), whereas the empirical distribution with β= 0.972 only slightly underestimates Unet. However, the empirical distribution, which has roots extending to infinite depth, differs qualitatively from that predicted by MaxNup, for which maximum rooting depth (Dmax) varies from 0 to 1.3 m as Rtot increases from 0 to 1 kg DM m−2 (Fig. 2a).
Increased N uptake by forests growing at elevated CO2
The MaxNup hypothesis provides equations predicting how increasing root mass (Rtot) affects total annual N uptake (Utot, Fig. 2b) and N supply to aboveground pools (Unet, Fig. 2c). These equations are consistent with data from the ORNL FACE experiment (Fig. 2b and c), suggesting that the observed increase in N uptake at eCO2 in that experiment may be a consequence of the measured increase in root growth at eCO2. Mechanisms for increased N uptake under eCO2 at the ORNL experiment were evaluated by Johnson et al. (2004), who used measurements of in situ soil incubations to estimate the size of the mineralizable N pool, which was reasoned to be large enough to have supplied the additional N taken up at eCO2. It was speculated that plant roots are somehow able to outcompete microbes at eCO2 (cf. Mosier et al. 2002; Schimel and Bennett 2004). Our modeling supports that view and proposes a mechanism in terms of increased root growth at eCO2 leading to increased gross N-uptake fraction (φN). However, the issue of why fine-root biomass increases at eCO2 at the ORNL FACE and other experiments (e.g., Luo et al. 2006) cannot be resolved by MaxNup itself. Resolution of that issue might be achieved by linking MaxNup to a compatible model of forest productivity that predicts optimal belowground-C allocation (e.g., Mäkelä et al. 2008; Franklin et al. 2009).
MaxNup offers insight into the puzzling observation from the ORNL FACE experiment that although total annual N uptake increased greatly at eCO2, the bulk of the increase in N uptake at eCO2 was used to grow more roots (Iversen et al. 2008; Norby et al. 2010), and there was no difference in annual N supply to aboveground pools between ambient CO2 (aCO2) and eCO2 treatments over 11 years of experimentation. The contrast is illustrated by measured annual values of total N uptake (Utot) and N supply to aboveground pools (Unet) shown in Norby et al. (2006, 2010). According to the MaxNup hypothesis, φnet varies much less than φN with increasing Rtot and even decreases if Rtot is large enough (Fig. 2b and c). This prediction may help to explain why measured annual N supply to aboveground pools is insensitive to eCO2.
Validation of equations (7)–(11) derived from MaxNup would provide robust predictors of root-depth distributions and N uptake that are suitable for incorporation into forest-ecosystem models and coupled models of land-biogeochemistry and climate. In these larger scale models, daily N uptake could be calculated (see Appendix A4) by multiplying daily soil-N supply derived from an established decomposition model such as CENTURY (Parton et al. 1988) by the gross N-uptake fraction φN derived from MaxNup; only two parameters (Ro and Do) would be required to calculate φN (eq. 12). The large-scale models would then be better placed to simulate the effects of global change on rooting depth, root and soil C, plant N uptake, and terrestrial C sequestration. Incorporation of these effects into ecosystem models is crucial so models will have a capacity to simulate the depth distribution of available soil N, and its variation over time, in contrast to our unrealistic assumptions above that the vertical profiles of available soil N (eq. 2) and root biomass are constant over time. According to MaxNup, the depth distribution of plant-available soil N determines the depth distribution of roots. However, in natural ecosystems, the depth distribution of available soil N is dependent on the depth distribution of C and N inputs to soil from decomposing roots. Therefore, the depth distribution of roots feeds back on the distribution of available soil N, so that the two distributions may vary in concert during stand development.
Another potential application of MaxNup is to quantify the effect of N fertilization on the efficiency of N capture by roots. Experiments in forests (Miller et al. 1976), peatland (Iversen et al. 2011) and agricultural systems (Tilman et al. 2002; Chen et al. 2011) have shown that N-uptake efficiency decreases with N addition. However, there is a dearth of models that relate the fraction of fertilizer taken up to the spatial distributions of available soil N and roots. If an improved understanding of N-uptake fraction based on MaxNup leads to more efficient utilization of N fertilizers, then it may be possible to optimize yield with reduced N fertilizer inputs, with benefits to food security and the environment (Tilman et al. 2002; Parry and Hawkesford 2010). Fertilizer practices conducive to improved N-uptake efficiency may include optimal spatial placement of fertilizer, timing of fertilizer applications, use of organic rather than inorganic N fertilizers (which would have different diffusion rates in soil and hence different Ro, as well as different root absorbing capacities), and mixed cropping (Tilman et al. 2002; Chen et al. 2011; Good and Beatty 2011).
Finally, we note three generalizations of the model presented here. First, MaxNup predicts that the marginal gain in nutrient uptake ∂Un/∂R is constant throughout the rooting zone. This prediction was applied here to an idealized soil in which N availability decays exponentially with soil depth (eq. 2), but should hold more generally (see eqs. A12 and A19), including to three-dimensional and patchy distributions of available soil N (Robinson 1996; Hopmans and Bristow 2002; Hodge 2004; Schimel and Bennett 2004). Furthermore, it should be possible to apply MaxNup to soil nutrients other than N, whose availability decreases with soil depth (Jackson et al. 2000; Jobbágy and Jackson 2001); it is advantageous that the Barber–Cushman model has been parameterized for a wide range of nutrients (e.g., Darrah 1993; Yanai 1994). Second, root distributions are presumably optimized relative to the supply of water as well as nutrients (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). Analogous to MaxNup would be the prediction of root distributions that maximize water uptake for a given total root mass (MaxWup). However, important differences between MaxWup and MaxNup would be that in dry conditions available soil water may increase with soil depth, in contrast to our assumption (eq. 2) that N availability decreases with depth, and that plants may sometimes need to be conservative in their water use, or may need to maintain some deep tap roots in order to survive drought periods. Third, extension of the MaxNup hypothesis, as presented above, to systems limited by two (or more) resources (water or different chemical forms of N or different nutrients) could be accomplished by modifying the optimization hypothesis to consider maximization of multiple objective functions (annual uptake of each resource) under the constraint of fixed annual C investment in roots. That hypothesis would lead to an equation, relating annual C investment in roots to uptake fractions of each resource; C investment in roots could include C costs of N acquisition (Fisher et al. 2010) and root-respiratory costs integrated over the root life span (cf. the leaf-life span integrals in McMurtrie and Dewar (2011)).
Funding was provided by the Australian Research Council (Discovery Project DP0881765) and the U.S. Department of Energy, Office of Science, Biological and Environmental Research Program. We thank S. Schymanski and anonymous referees for comments on the manuscript.
Conflict of Interest
The authors declare no conflict of interest.
Appendix A1. Relationship between N uptake Ur and root-length density Lr
In our root-optimization model, annual N uptake per unit soil volume at depth z (Ur(z), g N m−3 year−1) is a rectangular hyperbolic function of root-mass density (R(z), kg DM m−3) (eq. 1), where Uo(z) is the upper limit to N-uptake rate by roots at depth z (potential annual N uptake) and Ro is the root-mass density at half potential N uptake. Equation (1) can also be expressed as a rectangular hyperbolic function of root-length density Lr(z) =R(z)/(πro2ρr) (cm−2), in which the root-length density at half potential N uptake is Lro=Ro/(πro2ρr).
The purpose of this section is to clarify the mechanistic basis of the Ur–Lr relationship by deriving it from the Barber–Cushman (BC) model (Darrah 1993; Yanai 1994), an established mechanistic model of solute transport in soil and uptake by roots. In particular, we show how the parameter Lro=Ro/πro2ρr is related to root and soil properties, and justify the assumption implicit in equation (1) that N uptake Ur is linearly related to potential N-uptake rate Uo.
In the BC model, nutrient uptake rate per unit soil volume (U, mol cm−3 day−1) is proportional to root surface area per unit soil volume:
where ro (cm) is root radius, I is the rate of active nutrient uptake per unit root surface area (mol cm−2 day−1), which is often expressed as a Michaelis–Menten function of solute concentration at the root surface (Co, mol cm−3). Following Yanai (1994), we consider the case I=αCo, where α (cm day−1) is root absorbing capacity, which is assumed to be constant. Solute moves to the root surface by mass flow and diffusion down a concentration gradient generated by uptake at the root surface; the concentration gradient may be reversed at high rates of mass flow. In order to calculate Co, we need to determine the concentration of solute in solution (Cs(r), mol cm−3) at radial distances r extending from the root surface ro to a distance rx, the half-distance to nearest neighboring root (assuming radial symmetry). For a regular array of parallel roots, the value of rx is inversely related to root-length density Lr:
Let Uo (mol cm−3 day−1) and µ (day−1) represent, respectively, rates of supply of plant-available N per unit soil volume and solute loss, for example, through immobilization by microbial decomposers, which compete with plant roots for available solute (Darrah 1993; Mosier et al. 2002; Schimel and Bennett 2004; Frank and Groffman 2009) and are assumed here to be uniformly distributed, or through abiotic or chemical immobilization (e.g., associated with decomposition of lignified litter; Corbeels et al. 2005a, b). We assume that the inter-root distance is much smaller than the length scale for vertical variation of Uo (rx≪Do), so that the depth-dependence of Uo can be neglected in solving for Cs(r) at a particular soil depth. The BC model, as formulated by Yanai (1994), determines Cs(r) by considering the steady-state balance between nutrient inputs to and outputs from a unit-length cylinder of radius r surrounding the root surface with ro≤r≤rx. Nutrient inputs (mol N cm−1 root length day−1) to the cylinder occur through the supply rate Uo (which is spatially uniform within the cylinder), and mass flow and diffusion toward the root surface:
where E (cm3 water cm−3 soil volume day−1) is the rate of extraction of water by roots from unit soil volume, Δ (cm2 day−1) is the effective diffusion coefficient of nutrient in soil, and b is soil buffering power (cf. Darrah 1993; Yanai 1994). We assume that the rate of water extraction E by roots is determined by plant evaporative demand, and that water is supplied at the same rate E uniformly distributed within unit soil volume. Nutrient outputs from the cylinder are through uptake at the root surface (at r=ro), and immobilization within the cylinder, which is proportional to the local solute concentration (Corbeels et al. 2005a, b):
Differentiating the N-balance equation (N input = N output) with respect to r leads to the second-order differential equation:
The two boundary conditions are that solute flux is (1) zero at r=rx, and (2) equal to uptake per unit root surface area at .
It is useful to express equation (A3) in terms of dimensionless variables. If I=αCo (Yanai 1994) and transport by mass-flow is ignored (E= 0), which is a reasonable approximation under many circumstances (Nye 1977; Robinson 1986), the dimensionless variables are
subject to the boundary conditions at and at , where . Uptake as a proportion of nutrient supply is (from eq. A1)
Numerical solution is obtained by writing equation (A7) as a pair of first-order differential equations for rates of change of Cs′ and , and then finding a solution that satisfies boundary conditions at both and . The solution shows that solute concentration is depleted adjacent to the root (cf. Yanai 1994). The modeled relationship between U/Uo and derived from equation (A8) is shown in Figure A1. U/Uo increases with Lr and is asymptotic to 1 in the limit of high Lr, which is consistent with equation (1). The qualitative explanation for this pattern is that at low Lr (i.e., high rx), available soil N (Uo) is mostly immobilized by soil microbes before reaching the root surface. As Lr increases, rx decreases so that more solute reaches the root surface instead of being immobilized. N uptake by the root U approaches Uo asymptotically in the limit Lr→∞ (i.e., rx→ro). The rectangular-hyperbolic function
is a good approximation to the U–Lr relationship derived from the numerical solution of equation (A7) (Fig. A1). Values of Lro′ that best fit the relationship between U/Uo and Lr′ shown in Figure A1 are , 1.02, and 5.05 when α′= 1000, 100, and 10, respectively. As an aside, the dependence of Lro on µ/(Δb) (eq. A6) implies that the root-length density required for uptake of 50% of available N is inversely related to ion mobility (Δ) (cf. Raven et al. 1992; Cambui et al. 2011).
Simulations including mass flow are shown in Figure A2. We assume water extraction rate E= 0.01 cm3 water cm−3 soil volume day−1, which is a credible value, noting that the volumetric water content of most soils is in the range 0.1–0.4 cm3 water cm−3 soil volume at field capacity. Simulated daily N uptake U is enhanced when mass flow is included (Fig. A2), and the mass-flow effect can be large at low root-length densities. Further research is required to determine how the shape of the U–Lr relationship changes when E is large or when uptake kinetics of roots and microbes are modeled as Michaelis–Menten functions of Co, and to determine using MaxNup how the relationship between total N uptake and total root mass is affected. Note that the numerical solution for the U–Lr relationship derived from equation (A3) could be substituted directly into equation (A14) (below) to obtain the optimum solution for Un(z) throughout the root zone, thus obviating the need for an empirical approximation to the Ur–R relationship such as equation (1).
There is a discrepancy in timescales between the BC model, which simulates short-term solute transport processes, and equation (1), which evaluates annual N uptake. However, uptake is proportional to Uo in both equations, so that comparison across timescales is reasonable provided Lr and Lro are constant over an annual timescale. The issue of extra-polating from daily to annual timescales is discussed further in Appendix A4 below.
Appendix A2. Maximizing net supply to aboveground pools: solution by the Lagrange multiplier method
Our hypothesis of optimal root function is that annual net N supply to aboveground pools per unit land area (Unet, g N m−2 year−1, eq. 4) is maximized with respect to the vertical profile of root mass per unit soil volume (R(z)) and maximum rooting depth (Dmax, m) under the constraint that total root biomass per unit land area (Rtot, kg DM m−2, eq. 6) is fixed. This constrained optimization problem may be solved by the Lagrange multiplier method. We introduce a Lagrange multiplier λ for the constraint of fixed Rtot and maximize the goal function:
independently with respect to R(z) and Dmax, where the function Un(R(z),z) is given by equation (3). The explicit z-dependence of Un indicated here arises because potential N uptake (Uo, g N m−3 year−1), and N concentration (Nr, g N kg−1 DM) and longevity (τr, year) of roots are functions of z (cf. Pregitzer et al. 1998; Iversen et al. 2008, 2011; Iversen 2010). The modeled relationship between Un and R, obtained when Uo decreases exponentially with z (eq. 2) and Nr and τr do not vary with z, is shown in Figure 1b.
Under small variations δR(z) and δDmax, the change in Ψ is
so that root-scale N-uptake efficiency, defined as N supply to aboveground pools either per unit root mass (Un/R), or per unit root N (Un/(R Nr)), is maximized at the base of the rooting zone. Furthermore it follows from equation (1) that if R > 0, and hence that root-scale N-uptake efficiency is not maximized elsewhere in the root system. It also follows from equations (1) and (A15) that R(Dmax) = 0 and . Equation (A14) then becomes
If the N concentration and life span of roots do not vary with depth, then Utot is maximized when Unet is maximized, and at the optimum we have
The optimal solutions for Ur(z) versus R(z) and Un(z) versus R(z) are shown in Figure 1a and b, respectively, for total root mass Rtot= 0.19 kg DM m−2 (closed circles, ∂Ur/∂R= 14.5 and ∂Un/∂R= 7.7 g N kg−1 DM year−1) and Rtot= 0.38 kg DM m−2 (open circles, ∂Ur/∂R= 6.8 g N kg−1 DM year−1 and ∂Un/∂R= 0). Thus, for Rtot= 0.38 kg DM m−2, Un(z) is maximized with respect to R(z) throughout the rooting zone.
Appendix A3. Solving equation (A16) for the optimal vertical profiles of root-mass density and N uptake as functions of maximum rooting depth Dmax
For given functions Uo(z), Ro(z), Nr(z), and τr(z), equation (A19) can be solved to determine the optimal vertical root distribution R(z). Assuming that Nr, τr, and Ro (i.e., Lro, ro, and ρr) are independent of z and that available soil N declines exponentially with soil depth (eq. 2), we obtain solutions for the optimal vertical profiles of root-mass density (R(z), eq. 7) and annual total N uptake (Ur(z), eq. 9), total root mass (Rtot, eq. 8), total annual N uptake (Utot, eq. 10), annual N supply to aboveground pools (Unet, eq. 11), gross N-uptake fraction (φN=Utot/Umax, eq. 12), and net N-uptake fraction (φnet=Unet/Umax, eq. 13). The optimal vertical profile of annual N supply to aboveground pools Un(z) is obtained by substituting equations (7) and (9) into equation (3):
Average root-mass density over the rooting zone is Rav=Rtot/Dmax. Root systems with a given total root mass (Rtot) can range from shallow rooted (small Dmax) with large average root-mass density (large Rav) to deep rooted (large Dmax) with small Rav. From equation (8), the optimal value of Rav satisfies the equation:
It follows from equation (A21) that for a given Rtot, optimal Rav is an increasing function of Ro and a decreasing function of Do. Hence, for a given total root mass, the optimal root system will tend to be shallow with high Rav if Ro is large and Do is small, and deep with low Rav if Ro is small and Do is large.
The cumulative root proportion (CRP) between the soil surface and depth z can be obtained from equation (7):
Equations (8), (10), and (11) can be expressed in terms of dimensionless variables , , φN=Utot/Umax (gross N-uptake fraction), and φnet=Unet/Umax (net N-uptake fraction) to obtain
where the dimensionless quantity . The peak value of net N-uptake fraction φnet, shown in Figure 2c, occurs when Dmax=–Do ln(ζ), and
The peak value of ϕnet is
The value of ϕN when ϕnet is maximized is
Using parameter values for sweetgum at the ORNL FACE site (Table 1), we obtain ζ= 0.0398, so that at the peak Dmax= 0.97 m, Rtot= 0.38 kg DM m−2, = 0.64 and = 0.45.
Appendix A4. Extension of root-optimization model to accommodate variation of N availability over time
According to equation (1), the upper limit to N uptake by roots at depth z (Uo(z)) is an annual rate. However, many soil N-cycling models allow N supply to vary on a daily time-step, for example, as a function of daily soil moisture and temperature (Parton et al. 1988; Comins and McMurtrie 1993; Corbeels et al. 2005a, b). This variation can be accommodated within the MaxNup-optimization hypothesis by defining uoi(z) (g N m−3 day−1) as potential daily N uptake per unit soil volume at depth z on day i. Then N uptake per unit soil volume at depth z on day i is
If root-mass density at z R(z) and root-N concentration Nr(z) are constant over the year, then annual N supply to aboveground pools is
Assuming that the length scale for the exponential decrease of available soil N with depth (Do) is constant over the year, we have
where umaxi is potential N uptake per unit land area on day i integrated over all soil depths. Note that potential N uptake varies from day to day, but its distribution with depth does not. Equation (A30) can be written as
Equations in Appendices A2 and A3 above are retrieved if Umax in equation (2) is replaced by . This extension indicates how the predictions of MaxNup could be incorporated into daily time-step ecosystem and land-surface models.