Allocation of carbon (C) between tree components (leaves, fine roots and woody structures) is an important determinant of terrestrial C sequestration. Yet, because the mechanisms underlying C allocation are poorly understood, it is a weak link in current earth-system models. We obtain new theoretical insights into C allocation from the hypothesis (MaxW) that annual wood production is maximized.
MaxW is implemented using a model of tree C and nitrogen (N) balance with a vertically resolved canopy and root system for stands of Norway spruce (Picea abies).
MaxW predicts optimal vertical profiles of leaf N and root biomass, optimal canopy leaf area index and rooting depth, and the associated optimal pattern of C allocation.
Key insights include a predicted optimal C–N functional balance between leaves at the base of the canopy and the deepest roots, according to which the net C export from basal leaves is just sufficient to grow the basal roots required to meet their N requirement. MaxW links the traits of basal leaves and roots to whole-tree C and N uptake, and unifies two previous optimization hypotheses (maximum gross primary production, maximum N uptake) that have been applied independently to canopies and root systems.
Total below-ground carbon (C) allocation by forests can range from 40% of C allocated to wood at fertile sites to 300% at infertile sites (Litton et al., 2007). According to a synthesis of the FLUXNET dataset, the percentage of C allocated to fine roots can vary even more widely, from 10 to 500% of C allocated to above-ground wood (Dybzinski et al., 2011). The percentage of C allocated below ground is also sensitive to elevated CO_{2} (eCO_{2}). Fine root biomass usually increases when trees are grown at eCO_{2} (Luo et al., 2006), with a doubling of fine root mass at eCO_{2} at the Oak Ridge Forest Free-Air CO_{2} Enrichment (FACE) experiment (Norby et al., 2010; Iversen et al., 2011), and a 25% increase at the Duke Forest FACE experiment (Pritchard et al., 2008; Jackson et al., 2009). This large variation in the observed C allocation response to increased fertility and eCO_{2}, coupled with inadequate scientific understanding of the mechanisms underlying C allocation, hampers prediction of future C sequestration by terrestrial ecosystems and hence of feedbacks from the terrestrial biosphere to climate.
Three broad approaches to modelling C allocation may be identified: empirical, mechanistic and goal-seeking (or optimization). In earth-system and forest-ecosystem models, C allocation is usually represented empirically by constant allocation coefficients or simple allocation functions parameterized from tree biomass measurements or tree allometry (Sands & Landsberg, 1996; Sitch et al., 2003; Corbeels et al., 2005; Valentine & Mäkelä, 2012). Empirical models lack generality because measurements of variables that models aim to predict are required to estimate allocation parameters.
In the second approach, simple mechanistic models of C–nitrogen (N) balance are able to predict allocation between roots and shoots (leaves plus stems) (Ågren & Ingestad, 1987), but are less successful when the shoot compartment is separated into leaves and stems. Complex mechanistic models of the small-scale physiology of C allocation (cell enlargement and division, phloem and xylem loading and transport) have been developed (Hölttä et al., 2010), but so far have been unable to explain variation of C allocation with fertility and eCO_{2}. Another mechanistic approach is to distribute C within plants using equations for substrate transport down concentration gradients (Thornley, 1991). This approach produces credible root–shoot responses to fertility and eCO_{2} (Dewar, 1993; Thornley & Cannell, 1996), but resistances to substrate transport are uncertain and not easily measured.
Greater success has come from the third (goal-seeking) approach based on the concept of functional equilibrium between the capture of above- and below-ground resources by shoots and roots, respectively: root C allocation increases when soil resources (nutrients and water) become more limiting, whereas shoot C allocation increases when light becomes more limiting (Brouwer, 1983; Hilbert & Reynolds, 1991). The goal-seeking approach can also be formulated using constrained optimization models of C allocation. These models, which maximize some goal function subject to resource supply constraints, successfully predict responses of C allocation to limiting resources and eCO_{2} (Mäkelä et al., 2008; Dewar et al., 2009; Franklin et al., 2009, 2012; Valentine & Mäkelä, 2012).
Existing goal-seeking models have major shortcomings, however. One is that they fail to consider how resource balance depends on leaf and root traits, such as tissue chemistry, morphology and life span, which are highly plastic and respond to the same drivers as C allocation (Reich, 2002; Brassard et al., 2009; Valentine & Mäkelä, 2012). For example, eCO_{2} consistently decreases specific leaf area, leaf N concentration and stomatal conductance (Ainsworth & Long, 2005) and increases maximum rooting depth (Iversen, 2010), all of which affect resource supply or demand, and hence need to be considered when modelling C allocation at eCO_{2}. A second failing of optimization models of C allocation, discussed by Dybzinski et al. (2011), is that they consider the effect of growing additional roots on total N uptake, but not on the N content and photosynthesis of leaves at different heights in the canopy. In particular, they do not quantify the effect on the C ‘payback’ (i.e. photosynthesis in excess of C required for leaf construction and maintenance; Poorter et al., 2006) of the additional leaves constructed with the extra N taken up.
The purpose of this paper is to present new insights into C allocation by trees obtained using an optimization model of forest growth constrained by light and soil N availability that takes into account the plasticity of key leaf and root traits in a unified way. In particular, the model treats leaves and roots in a complementary fashion, by evaluating the C payback of leaves throughout the canopy and, analogously, the N payback of roots throughout the root system. In this way, the model overcomes the shortcomings of previous goal-seeking models described earlier. C payback of leaves at each height within the canopy is evaluated as photosynthesis per unit leaf area over their life span, minus the C costs of leaf respiration and reconstruction. Analogously, root N payback at each depth within the root system is evaluated as root N uptake per unit soil volume over the root life span, minus the N cost of root reconstruction. The model quantifies how trees benefit from allocating more new growth to leaves or roots as a function of their respective C and N paybacks.
In this study, the benefit to trees is evaluated in terms of their annual wood production. Specifically, we use the model to predict optimal C and N allocation from the optimization hypothesis (MaxW) that annual wood production is maximized under constraints of C and N conservation, assuming foliage and fine roots are in a steady state. The objective of maximizing wood production is reasonable for trees, whose survival and reproductive success depend on growing a large woody trunk in order to outcompete neighbours for light. High wood production thus confers a fitness advantage (Brassard et al., 2009; Franklin et al., 2009; Valentine & Mäkelä, 2012). C and N conservation (subject to light and soil N availability) is a universal constraint (Lawton, 1999), and the additional assumption that foliage and fine roots are in a steady state (i.e. production = turnover) is reasonable on the timescale of wood growth.
We derive constraint equations for C and N conservation that take account of the vertical distributions of light incident on leaves within the canopy and plant-available N within the soil. Under these constraints, MaxW predicts optimal leaf N content and leaf C payback as functions of cumulative leaf area index L, and optimal root C density and root N payback as functions of soil depth z. MaxW also predicts the associated optimal C and N allocation to leaves, roots and wood.
Key insights derived from MaxW include generic relationships between the sensitivities of whole-tree resource capture to increased canopy and root-system biomass, leaf traits at the base of the canopy, and root traits at the base of the root system. Most significantly, we establish a relationship – which we call the ‘bottom line’ – between the optimal C payback of basal leaves and the optimal N payback of basal roots. The bottom line also establishes a generic relationship between the optimal total leaf area index (L_{tot}) and maximum rooting depth (D_{max}), which is a key to understanding C allocation and its response to both increased fertility and eCO_{2}.
The structure of the paper is as follows. The next section presents the key elements of the MaxW hypothesis (goal function, optimized traits, constraints; Table 1). In the Results section, the key predictions of MaxW (vertical profiles of leaf N content and fine root C density, L_{tot}, D_{max}, C allocation) are presented and illustrated for stands of Norway spruce (Picea abies L. Karst.) with contrasting fertilities. In the final section, we discuss the implications of these results for our understanding of C allocation responses to changes in resource availability, and how these responses might be better represented in earth-system models; this section also clarifies the relation between MaxW and previously proposed optimization hypotheses. Symbol definitions, parameter values, mathematical details and additional results are given in Table 2 and Supporting Information Table S1, Notes S1–S3.
Table 1. Key elements of the MaxGPP, MaxNup and MaxW optimization hypotheses
MaxGPP
MaxNup
MaxW
Goal function
Annual canopy photosynthesis, A_{tot}
Annual root-system nitrogen (N) uptake, U_{tot}
Annual wood carbon (C) production, C_{w}
Optimized traits
Vertical profile of leaf N content, N_{a}(L); total leaf area index, L_{tot}
Vertical profile of root C density, R(z); maximum rooting depth, D_{max}
Step 1: N_{a}(L), L_{tot}, R(z), D_{max}Step 2: Leaf N : C ratio, n_{f}
Constraints
Fixed canopy N content, N_{tot}
Fixed total root C, R_{tot}
C and N conservation; steady-state foliage and roots
Table 2. Symbol definitions, units and parameter values for Norway spruce (Picea abies)
Symbol
Definition (relevant equations)
Units/value
C, carbon; LAI, leaf area index; N, nitrogen; PPFD, photosynthetic photon flux density.
The value of CUE is 10% lower than that in Dewar et al. (2012) because daytime leaf respiration is incorporated in CUE in (Eqn 3), whereas it was modelled explicitly in Dewar et al. (2012).
Input parameters
a_{N}
Slope of linear relationship between A_{sat} and N_{a}a (S1.10)
Annual gross primary production (= A_{tot}) (3, 5)
kg C m^{−2} land yr^{−1}
I(L)
Incident PPFD at cumulative LAI, L (S1.11)
mol m^{−2} land s^{−1}
L_{crit}
Critical value of L, below which N_{a}(L) is constant (S1.1)
m^{2} leaf m^{−2} land
L
Cumulative or shading one-sided LAI (4)
m^{2} leaf m^{−2} land
L_{tot}
Total canopy LAI (4)
m^{2} leaf m^{−2} land
m_{a}(L)
Leaf mass per unit area at cumulative LAI, L (11)
kg DW m^{−2} leaf
N_{a}*
Leaf N content at the canopy base (= n_{f} Ω m_{a}*) (S1.1)
kg N m^{−2}
N_{a}(L)
Leaf N content per unit leaf area at cumulative LAI, L (4)
kg N m^{−2} leaf
n_{f}
Leaf N : C ratio (3)
kg N kg^{−1} C
N_{tot}
Total canopy N content (3, 4)
kg N m^{−2} land
NPP
Annual net primary production (1)
kg C m^{−2} land yr^{−1}
R(z)
Root C per unit soil volume, or root C density at depth z (6)
kg C m^{−3} soil
R_{tot}
Total root C per unit land area (2, 6)
kg C m^{−2} land
U_{o}(z)
Potential annual N uptake per unit soil volume at soil depth z (S2.2)
kg N m^{−3} yr^{−1}
U_{r}(z)
Annual N uptake per unit soil volume at soil depth z (9)
kg N m^{−3} yr^{−1}
U_{tot}
Total annual N uptake rate (7, 9)
kg N m^{−2} land yr^{−1}
X_{C}(L)
Leaf C payback at cumulative LAI L (11)
kg C m^{−2} leaf area
X_{N}(z)
Root N payback at soil depth z (12)
kg N m^{−3} soil
z
Soil depth (6)
m
ε_{f}(L)
Leaf C payback per unit leaf N investment (13)
kg C kg^{−1} N
ε_{r}(z)
Root N payback per unit root C investment (14)
kg N kg^{−1} C
λ, λ_{1}, λ_{2}
Lagrange multipliers (S3.1, S1.2, S2.3)
kg C kg^{−1} N yr^{−1}, kg C kg^{−1} N yr^{−1}, kg N kg^{−1} C yr^{−1}
σ
Parameter combination CUE ξm_{c} (S3.11)
kg C mol^{−1} CO_{2} (s yr^{−1})
ϕ
Annual N uptake as fraction of U_{max} (S2.7)
–
Ψ, Ψ_{1}
Goal functions (S3.1, S1.2)
kg C m^{−2} yr^{−1}
Description
The MaxW optimization hypothesis
According to the MaxW hypothesis, annual wood carbon production (C_{w}, kg C m^{−2} land area yr^{−1}) is maximized with respect to five plant traits: the vertical profiles of leaf N content (N_{a}(L), kg N m^{−2} leaf area; L = shading or cumulative leaf area index with L =0 at the top of the canopy) and fine root C density (R(z), kg C m^{−3} soil volume; z = soil depth, m), canopy leaf area index (L_{tot}, m^{2} leaf area m^{−2} land area), maximum rooting depth (D_{max}, m), and foliar N : C ratio n_{f} (kg N kg^{−1} C). The maximization is subject to the constraints of C and N conservation, in which foliage and fine root biomass are assumed to be in a steady state. Table 1 summarizes the key elements of MaxW and two other optimization hypotheses (MaxGPP and MaxNup), which we will show are closely related to MaxW.
Annual wood C production determined from C conservation
Conservation of C implies that annual net primary production (NPP, kg C m^{−2} land area yr^{−1}) is equal to total C allocated to the production of leaves, fine roots and wood (which includes tree stems, branches and coarse roots). Hence annual wood C production (the goal function, Table 1) is
Cw=NPPâˆ’Cfâˆ’Cr(Eqn 1)
where C_{f}, C_{r} and C_{w} (kg C m^{−2} land area yr^{−1}) are annual foliage, fine root and wood production, respectively. For simplicity, we do not model maintenance and growth respiration of each compartment explicitly, but instead assume NPP = CUE × A_{tot}, where CUE is plant carbon-use efficiency, and A_{tot} (kg C m^{−2} land area yr^{−1}) is annual gross canopy photosynthesis (i.e. gross primary production, GPP); this assumption does not affect the key insights derived here. Steady-state foliage and root C (kg C m^{−2} land area) are
Ftot=CfÏ„fandRtot=CrÏ„r,(Eqn 2)
respectively, where τ_{f} and τ_{r} (yr) are leaf and root life spans, respectively. (Eqn 1) can then be expressed as
Cw=CUEAtotâˆ’NtotnfÏ„fâˆ’RtotÏ„r,(Eqn 3)
where n_{f} is the foliar N : C ratio and N_{tot} (kg N m^{−2} land area) = n_{f}F_{tot} is the steady-state canopy N content. N_{tot} and A_{tot} can be expressed as canopy integrals of leaf N content (N_{a}(L)) and gross leaf photosynthesis (A_{a}(L), kg C m^{−2} leaf area yr^{−1}), respectively, over cumulative leaf area indices from L =0 (top of canopy) to L = L_{tot} (total canopy leaf area index):
Ntot=âˆ«0LtotNa(L)dL,(Eqn 4)
Atot=âˆ«0LtotAa(Na(L),L)dL,(Eqn 5)
where gross leaf photosynthesis A_{a} is a function of leaf N content N_{a}(L) and incident light, which decreases exponentially with increasing L (Eqn S1.9). Incident light is assumed constant over time. Similarly, R_{tot} can be expressed as an integral of root C density (R(z)) over the rooting zone from soil depth z =0 (soil surface) to z = D_{max} (maximum rooting depth):
Conservation of N implies that total N uptake (U_{tot}, kg N m^{−2} land area yr^{−1}) is equal to total N allocated to leaves, plus roots, plus wood:
Utot=Cfnf(1âˆ’r)+Crnr+Cwnw(Eqn 7)
=Ntot(1âˆ’r)Ï„f+RtotnrÏ„r+Cwnw,(Eqn 8)
where r is the fraction of leaf N retranslocated at leaf senescence, and n_{r} and n_{w} are root and wood N : C ratios, respectively. We assume there is no retranslocation of N from old to new root and wood tissues (Gordon & Jackson, 2000). N : C ratios (n_{f}, n_{r} and n_{w}) are assumed to be the same for all leaves, roots and wood, respectively; whereas n_{r} and n_{w} are fixed parameters, n_{f} is one of the traits optimized under MaxW (Table 1). U_{tot} can be expressed as an integral of N uptake per unit soil volume by roots at soil depth z (U_{r}(z), kg N m^{−3} yr^{−1}) from z =0 to z = D_{max}:
Following McMurtrie et al. (2012), U_{r}(z) is assumed to be a saturating function of R(z) that approaches an upper limit U_{o}(z) (potential annual plant N uptake) asymptotically in the limit R(z) → ∞. The shape of this function is a consequence of competition between plant roots and soil microbes for available soil N, which moves by diffusion and mass flow to the root surface, where plant uptake occurs (Schimel & Bennett, 2004; Kuzyakov & Xu, 2013). At low R(z), roots are far apart and soil N is mostly immobilized by microbes before reaching the root surface. According to our N uptake model (McMurtrie et al., 2012), as R(z) increases, roots are closer together and solute is less likely to be immobilized before reaching the root surface. Hence, as R(z) increases, the fraction of available N that is immobilized decreases and the fraction taken up by roots increases. U_{o}(z) is greatest at the soil surface and decreases exponentially with soil depth (Jackson et al., 2000; Jobbágy & Jackson, 2001). We assume there is no horizontal variation of N availability. Total potential annual N uptake integrated over all soil depths (U_{max}, kg N m^{−2} yr^{−1}) is constant over time.
MaxW as a constrained optimization problem
According to the MaxW optimization hypothesis, annual wood production, defined by (Eqn 1), is maximized subject to conservation of C and N. From (Eqn 3) and (Eqn 8), this is equivalent to the mathematical problem of maximizing C_{w} given by (Eqn 3) subject to the C-N conservation constraint:
This constrained optimization problem can be solved using the Lagrange multiplier method (see Notes S3). Leaf and root life spans (τ_{f} and τ_{r}), root and wood N : C ratios (n_{r} and n_{w}) and parameters r and CUE are assumed constant (Table 2). Other variables in Eqns 1-10 are determined by optimization (Table 1): the vertical profiles of leaf N (N_{a}(L)) and root C density (R(z)), total leaf area index (L_{tot}), maximum rooting depth (D_{max}), and foliar N : C ratio (n_{f}). Optimal solutions are presented as general equations in terms of N_{a}(L), A_{a}(L), R(z) and U_{r}(z). However, solutions are illustrated using specific functions for leaf photosynthesis (A_{a}(L), Dewar et al. (2012)) and N uptake (U_{r}(z), McMurtrie et al. (2012)) parameterized for Norway spruce.
Results
In this section we present new insights into the optimal vertical profiles of leaf N within the canopy and root C in soil, the ‘bottom line’ and C allocation, derived from the MaxW optimization hypothesis.
Profiles of leaf N and root C density
In Notes S3, we show that, if the MaxW optimization hypothesis holds, two other optimal hypotheses also hold:
The total amount of canopy N (N_{tot}) is distributed vertically within the canopy so as to maximize GPP (A_{tot}). This hypothesis, which we call MaxGPP (Table 1, Notes S1), has been the subject of much past theoretical and experiment work (Field, 1983; Hirose & Werger, 1987; Anten et al., 1995; Franklin & Ågren, 2002; McMurtrie & Dewar, 2011; Dewar et al., 2012; Niinemets, 2012). However, it has not been shown previously that maximization of GPP is implied by maximization of wood production.
The total amount of root C (R_{tot}) is distributed vertically within the soil so as to maximize annual N uptake (U_{tot}). This is the MaxNup hypothesis proposed by McMurtrie et al. (2012) (Table 1, Notes S2). The result that MaxNup is implied by MaxW has similarly not been shown previously.
Results (1) and (2) display a complementarity between above- and below-ground function, with leaf N distributed to maximize canopy C uptake (MaxGPP), and root C distributed to maximize root-system N uptake (MaxNup). These two results are surprising because in MaxGPP and MaxNup, respectively, the values of N_{tot} and R_{tot} are constrained to be fixed, whereas MaxW allows N_{tot} and R_{tot} to vary within the less restrictive constraint imposed by C and N conservation ((Eqn 10)). The explanation is that the vertical profiles of leaf N and root C density predicted by MaxW coincide with those that would be predicted by MaxGPP and MaxNup if N_{tot} and R_{tot} were fixed at their optimal values predicted by MaxW.
We emphasize that these two results (i.e. MaxW implies MaxGPP and MaxNup) are completely general, and do not depend on detailed assumptions about leaf photosynthesis and root N uptake. However, for illustration purposes, the solution of MaxGPP for optimal N_{a}(L) and L_{tot} is presented in Notes S1 under the following assumptions applicable to Norway spruce (Dewar et al., 2012): that gross leaf photosynthesis (A_{a}) follows a rectangular-hyperbolic light response, that light-saturated photosynthetic rate is linearly related to leaf N content, that leaf N : C ratio is independent of canopy depth, and that leaf mass per unit area (m_{a}(L), kg DW m^{−2}) may vary with canopy depth but has a minimum value (m_{a}*) determined by the need for structural strength to withstand herbivory, wind and other hazards. Similarly, the solution of MaxNup for optimal R(z) and D_{max} is presented in Notes S2 for a specific choice of root N uptake model (McMurtrie et al., 2012).
Optimal vertical profiles of leaf N content, N_{a}(L), and root C density, R(z), derived from MaxW are illustrated in Figs 1(a) and 2(a). Leaf N content decreases with increasing L in the upper canopy, but is constant in the lower canopy. The explanation for constant leaf N content in the lower canopy is that N_{a}(L) is proportional to leaf mass per unit area (N_{a}(L) = n_{f }Ω m_{a}(L), where Ω (kg C kg^{−1} DW) is the carbon content of biomass) and hence has a minimum value n_{f }Ω m_{a}*, which applies in the lower canopy. R(z) decreases with increasing soil depth z and is zero at z = D_{max} ((Eqn 7) of McMurtrie et al., 2012).
Fig. 1(b) shows the optimal vertical profile for leaf C payback (X_{C}(L), kg C m^{−2} leaf area, L = cumulative leaf area index), defined as photosynthesis over the leaf life span minus the fraction respired and the C cost of constructing new leaf biomass:
Similarly, Fig. 2(b) shows the optimal vertical profile for root N payback (X_{N}(z), kg N m^{−3} soil volume, z = soil depth), defined as N uptake over the root life span minus the N cost of constructing new roots:
XN(z)=Ur(z)Ï„râˆ’nrR(z).(Eqn 12)
As Fig. 1(a) shows, GPP is maximized by preferentially allocating N to leaves in the upper canopy where N is utilized more efficiently to produce C. A measure of that efficiency is leaf C payback per unit leaf N investment (Falster et al., 2012),
Îµf(L)=XC(L)Na(L)(1âˆ’r),(Eqn 13)
which decreases with increasing L (Fig. 1c). Similarly, total N uptake by the root system is maximized by preferentially growing roots at soil depths where N-uptake efficiency is high. Analogous to (Eqn 13), the efficiency of roots to take up N is defined as root N payback per unit root C investment (McMurtrie et al., 2012),
Îµr(z)=XN(z)R(z),(Eqn 14)
which is largest near the soil surface and decreases with depth z (Fig. 2c).
The bottom line
In Notes S3, a third key result is derived from MaxW, namely that leaf C payback per unit N investment at the base of the canopy and root N payback per unit root C investment at the base of the rooting zone are related by the remarkably simple equation:
Îµf(Ltot)Îµr(Dmax)=1.(Eqn 15)
(Eqn 15) – which we call the ‘bottom line’ – is a relationship between optimal traits of leaves at the base of the canopy and roots at the base of the root system. It has the following simple biological interpretation. Suppose that the C payback from leaves at the canopy base (between L = L_{tot} – δL and L = L_{tot}) is entirely allocated to grow roots at the base of the root zone (between z = D_{max} – δD and z = D_{max}). Since leaves and roots have lifespans τ_{f} and τ_{r}, respectively, then for small δL and δD we have the rate balance equation
XC(Ltot)Î´LÏ„f=R(Dmax)Î´DÏ„r(Eqn 16)
Similarly, suppose that the N payback from roots at z = D_{max} is entirely allocated to leaves at the base of the canopy, so that
which, from Eqns 13 and 14, is equivalent to the bottom line (Eqn 15). Thus the bottom line can be understood as a quid pro quo between basal leaves and basal roots, where C export by basal leaves over their life span is just sufficient to grow the basal roots required to meet their N requirement. Likewise, N export from basal roots over their life span is just sufficient to grow the basal leaves required to meet their C requirement. In this marginal situation, growing more leaves or roots confers no further benefit to wood production, which is therefore at a maximum. However, this interpretation of the bottom line should not be viewed literally as a transfer of C exported by basal leaves to basal roots, or of N in the opposite direction. The model makes no assumption about the source of structural C and N incorporated into specific leaves, roots and woody tissue.
As shown in Fig. 1(c), the optimal values of leaf C payback per unit leaf N investment at the canopy base, that is, ε_{f}(L_{tot}) given by (Eqn 13), are 0.45 and 0.81 kg C g^{−1} N when the soil N supply (U_{max}) is 0.012 and 0.008 kg N m^{−2} yr^{−1}, respectively. The corresponding optimal values of root N payback per unit root C investment at the base of the root zone, that is, ε_{r}(D_{max}) given by (Eqn 14), are 2.24 and 1.23 g N kg^{−1} C, respectively (Fig. 2c). Notice that the bottom line (Eqn 15) is satisfied in each case, but when soil N supply is reduced, ε_{r}(D_{max}) is smaller, while ε_{f}(L_{tot}) is larger.
Carbon allocation
(Eqn 15), the bottom line, is a condition for optimal functional balance between basal leaves and basal roots. In this section we explain how that condition, when coupled with the C-N conservation constraint ((Eqn 10)), can be used to determine optimal annual C allocation to leaves (C_{f}), fine roots (C_{r}) and wood (C_{w}).
The bottom line ((Eqn 15)) relates L_{tot} and D_{max}. A second equation relating L_{tot} and D_{max} can be derived from the C-N conservation constraint ((Eqn 10)) by substituting expressions for A_{tot} and N_{tot} as functions of L_{tot} (obtained from MaxGPP in Notes S1), and expressions for R_{tot} and U_{tot} as functions of D_{max} (obtained from MaxNup in Notes S2). The simultaneous solution of these two equations for L_{tot} and D_{max} can then be used to determine N_{tot}, R_{tot}, C_{f}, C_{f} and C_{w} (see Notes S3). These optimal traits are functions of leaf and root N : C ratios (n_{f} and n_{r}) and, for reasons explained by Dewar et al. (2012), the value of C_{w} has an optimum with respect to n_{f}; the final step in MaxW is then to optimize n_{f} (see Notes S3).
The optimal value of n_{f} decreases when N supply declines (n_{f} = 0.033 and 0.028 when U_{max} = 0.012 and 0.008 kg N m^{−2} yr^{−1}, respectively). Other modelled responses to reduced N supply are listed in Table S1. Predicted values of C_{f}, C_{r} and C_{w} all decrease with decreasing N supply, consistent with recent meta-analyses of C allocation patterns in forests (Litton et al., 2007; Dybzinski et al., 2011). However, the predicted fraction of NPP allocated to foliage is largely insensitive to increased N supply (0.07 at both U_{max} = 0.008 and 0.012 kg N m^{−2} yr^{−1}), the main allocation response being a tradeoff between decreased root allocation (0.19 and 0.15, respectively) and increased wood allocation (0.74 and 0.77), consistent with the general pattern in forest data (Litton et al., 2007; Dybzinski et al., 2011).
Fig. 3(a,b) shows curves of A_{tot} vs N_{tot} and U_{tot} vs R_{tot}, derived, respectively, from MaxGPP and MaxNup. These curves are generated by varying N_{tot} (Fig. 3a) or R_{tot} (Fig. 3b) with the leaf N : C ratio fixed at the value predicted by MaxW at high and low soil fertilities (respectively, red and blue curves); the red and blue circles indicate the corresponding optimal values of N_{tot} (Fig. 3a) or R_{tot} (Fig. 3b) predicted by MaxW at high and low soil fertilities. With an increase in soil fertility, the U_{tot}–R_{tot} relationship is raised (Fig. 3b), but the A_{tot}–N_{tot} relationship changes only minimally (Fig. 3a). Hence, with increasing fertility, the optimal solution of MaxW moves up the A_{tot}–N_{tot} curve, while it moves from one U_{tot}–R_{tot} curve to another. It is shown in Notes S1 and S2 that, at the optimal solution, the slopes of the A_{tot}–N_{tot} and U_{tot}–R_{tot} curves satisfy dA_{tot}/dN_{tot} = A_{a}(L_{tot})/N_{a}(L_{tot}) and dU_{tot}/dR_{tot} = U_{r}(D_{max})/R(D_{max}). These two results imply that the sensitivities of whole-tree resource capture to increased canopy N content and root-system C are intimately related to the optimal traits of basal leaves and roots. Furthermore, the slopes of the two curves at the optimum are inversely related through the bottom line ((Eqn 15)).
Discussion
Spatial separation in acquisition of above- and below-ground resources is a characteristic of terrestrial plants that is pivotal in determining their structural growth (Cheeseman, 1993). Our model predicts above- and below-ground structure in relation to vertical gradients of light in the canopy and plant-available N in soil. Leaves and roots perform complementary roles in resource acquisition and utilization – leaves take up C and export a portion of it to other plant organs, including roots, but they require N from roots to do so; roots take up N and export a portion of it to above-ground organs, but they require C from leaves to do so.
The complementary role of leaves and roots lies at the heart of our model and is reflected in the following three key insights derived from the MaxW–optimization hypothesis, which amounts to an evolutionary guiding principle:
MaxGPP – N is distributed within the canopy so as to maximize annual GPP, leading to a relationship between the sensitivity of GPP to increased canopy N content and the traits of basal leaves: dA_{tot}/dN_{tot} = A_{a}(L_{tot})/N_{a}(L_{tot}) (Eqn S1.24);
MaxNup – roots are distributed so as to maximize annual N uptake, leading to a relationship between the sensitivity of annual N uptake to increased root-system biomass and the traits of basal roots: dU_{tot}/dR_{tot} = U_{r}(D_{max})/R(D_{max}) (Eqn S2.8);
The bottom line – leaf C payback per unit leaf N investment at the canopy base, multiplied by root N payback per unit root C investment at the base of the root system, is equal to unity (ε_{f}(L_{tot}) × ε_{r}(D_{max}) = 1, (Eqn 15)), a result relating the traits of basal leaves and roots, as well as total leaf area index (L_{tot}) to maximum rooting depth (D_{max}).
Collectively, these three results and the C–N conservation constraint ((Eqn 10)) provide a link from the functional traits of leaves and roots to stand-scale variables (L_{tot}, D_{max}, N_{tot}, R_{tot}, A_{tot}, U_{tot} and C_{w}) that should be useful for upscaling.
The first key result (MaxGPP) has been applied widely (Niinemets, 2012). It has been shown that the vertical profile of leaf N content predicted by MaxGPP is proportional to the light profile (Field, 1983; Anten et al., 1995; Sands, 1995; McMurtrie & Dewar, 2011; Niinemets, 2012). However, this prediction does not match data, which show that the canopy extinction coefficient for leaf N is less than the light extinction coefficient (Hirose & Werger, 1987; Anten et al., 1995; Meir et al., 2002; Dewar et al., 2012; Niinemets, 2012; Peltoniemi et al., 2012). This discrepancy between MaxGPP and data was resolved for Norway spruce by Dewar et al. (2012) through the additional constraint that there is a lower limit to leaf mass per unit area. The prediction from MaxGPP shown in Fig. 1(a) (Eqns S1.1 and S1.14) is that leaf N content is proportional to light in the upper canopy (above L = L_{crit}) and constant in the lower canopy (below L = L_{crit}). The transition at L = L_{crit} is a consequence of a lower limit to leaf mass per unit area, as discussed earlier. However, the sharpness of the transition also depends on our assumption of constant incident light. If incident light were variable (McMurtrie & Dewar, 2011), the transition from upper to lower canopy would be more gradual. The model of Dewar et al. (2012) correctly predicts measurements of canopy N content, leaf area index and leaf N concentration of Norway spruce. The A_{tot}–N_{tot} relationship predicted by Dewar et al.'s model is essentially the same as the relationship shown in Fig. 3(a).
The second key result (MaxNup) has been applied to Liquidambar styraciflua (sweetgum) growing at the Oak Ridge Forest FACE experiment; the U_{tot}–R_{tot} relationship predicted by MaxNup (Eqn S2.7) gives a good fit to measurements of U_{tot} and R_{tot} for sweetgum (McMurtrie et al., 2012).
The third key result (the bottom line) provides key insights into C allocation between leaves and roots, because it implies that increased fertility or eCO_{2} must lead to reciprocal changes in ε_{f}(L_{tot}) and ε_{r}(D_{max}) to preserve the bottom line. To see this more explicitly, we can use Eqns 11-14 to express the bottom line (Eqn 15) in the form
where the first and second factors on the left-hand side are ε_{f}(L_{tot}) and ε_{r}(D_{max}), respectively. Increasing soil N supply (U_{max}) will increase specific root uptake, U_{r}(D_{max})/R(D_{max}) (Eqn S2.4), and hence ε_{r}(D_{max}); in order to preserve the bottom line, possible responses include increasing D_{max} to counter the increase in ε_{r}(D_{max}), or increasing L_{tot} to decrease ε_{f}(L_{tot}) through reduced light at the base of the canopy, as predicted for the Norway spruce stand (see Table S1). Elevated CO_{2} will increase photosynthetic N-use efficiency, A_{a}(L_{tot})/N_{a}(L_{tot}), and hence ε_{f}(L_{tot}); possible responses then include increasing L_{tot} and D_{max}, and decreasing n_{f}, all of which are consistent with FACE experiments (Ainsworth & Long, 2005; Norby et al., 2005; Iversen, 2010; Norby & Zak, 2011).
The MaxW optimization hypothesis explains above- vs below-ground C allocation as an emergent property of plant adaptation in which the vertical profiles of leaf photosynthesis, root N uptake and the C and N payback from leaves and roots conspire to maximize wood production. MaxW avoids the need for a specific C-allocation mechanism (cf. Cheeseman, 1993). Moreover, we have represented photosynthesis and root N uptake as quite general functions of leaf N content (A_{a}(N_{a},L)) and root C density (U_{r}(R,z)), respectively. Crucially, therefore, the key insights we obtained from MaxW (i.e. MaxGPP, MaxNup and the bottom line) are not specific to a particular photosynthesis or N-uptake model. Our optimization model of C allocation is therefore flexible and suitable for adoption in earth-system models. In particular, our approach should make it possible to reduce the number of input parameters required to run current earth-system models, because some parameters (e.g. allocation fractions) are determined using optimization rather than empirical or mechanistic submodels (Dewar et al., 2009).
Our results also shed light on the question of what plants maximize, which is a contentious issue in plant optimization modelling (Dewar et al., 2009; Anten & During, 2011; Franklin et al., 2012). Optimization criteria considered previously include maximization of GPP, GPP minus respiration, GPP minus respiration and leaf production, GPP minus respiration and leaf and root production, wood production, tree height growth, and entropy production (Franklin, 2007; Mäkelä et al., 2008; Dewar et al., 2009; Franklin et al., 2009, 2012; Dewar, 2010; Valentine & Mäkelä, 2012). Our conclusion that the MaxGPP and MaxNup optimization hypotheses are nested within MaxW lends support to the nested optimization approach adopted by Franklin (2007) and Franklin et al. (2009). MaxW is also consistent with maximum entropy production when the latter is applied to trees under the same assumption of steady-state foliage and fine roots that we adopted here (see Franklin et al., 2012, appendix B, case λ = 1).
Further research is needed to test the model's predictions for other tree species and under different environmental conditions, and to extend the approach to nonsteady-state conditions, to plants that obey optimization criteria other than MaxW such as nonwoody plants, to plants limited by soil resources other than nitrogen, and to individual plants in competition with neighbours. For nonwoody plants, the appropriate optimization criterion may be maximization of growth of reproductive structures, which would be mathematically analogous to MaxW but with variables for wood production (C_{w}) and N : C ratio (n_{w}) replaced by equivalent variables for reproductive structures. Insights into the appropriate choice of optimization criterion here might also be gained from maximum entropy production, by extending this thermodynamic criterion from woody (Franklin et al., 2012; appendix B) to nonwoody plants. Further work such as this may reveal whether the three key results derived here from MaxW apply to terrestrial plants more generally (Lawton, 1999; Harte, 2002; May, 2004).
Acknowledgements
We are grateful to Lasse Tarvainen and Göran Wallin for providing parameters for Norway spruce for the model described in Dewar et al. (2012), Torgny Näsholm and Remko Duursma for comments on the manuscript and Enid Harrison for inspiration.