What Does Optimization Theory Actually Predict About Crown Profiles Of

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Plant, Cell and Environment (2013) 36, 1547–1563 doi: 10.1111/pce.12091

What does optimization theory actually predict about crown proﬁles of photosynthetic capacity when models incorporate greater realism?

THOMAS N. BUCKLEY1, ALESSANDRO CESCATTI2 & GRAHAM D. FARQUHAR3

1Department of Biology, Sonoma State University, Rohnert Park, CA, 94928, USA, 2European Commission, Joint Research Centre, Institute for Environment and Sustainability, Ispra, Italy and 3Research School of Biology, The Australian National University, Canberra, Australia

ABSTRACT Amthor 1994). This assumption is popular for two reasons: firstly, it is widely perceived to be identical to optimal N Measured profiles of photosynthetic capacity in plant crowns allocation and, therefore, consistent with natural selection typically do not match those of average irradiance: the ratio (Field 1983a; Hirose & Werger 1987; Sands 1995), and sec- of capacity to irradiance decreases as irradiance increases. ondly, if leaf absorptance is invariant among leaves and if the This differs from optimal profiles inferred from simple time-averaged and instantaneous profiles of irradiance are models. To determine whether this could be explained by equivalent, it renders models of leaf photosynthesis scale- omission of physiological or physical details from such invariant, and therefore applicable at crown scales (Farquhar models, we performed a series of thought experiments using 1989). Most data indeed show that photosynthetic capacity is a new model that included more realism than previous positively correlated with local irradiance within crowns. models. We used ray-tracing to simulate irradiance for 8000 However, the correlation is not linear; instead, it seems to leaves in a horizontally uniform canopy. For a subsample of saturate at high light, such that the photosynthetic capacity 500 leaves, we simultaneously optimized both nitrogen allo- per unit of incident irradiance declines up through the crown cation (among pools representing carboxylation, electron (e.g. Fig. 1) (Evans 1993; Hirose & Werger 1994; Hollinger transport and light capture) and stomatal conductance using 1996; de Pury & Farquhar 1997; Makino et al. 1997; Bond a transdermally explicit photosynthesis model. Few model et al. 1999; Friend 2001; Frak et al. 2002; Kull 2002; Lloyd et al. features caused the capacity/irradiance ratio to vary system- 2010) (but see Gonzales-Real & Baille 2000). Thus, photo- atically with irradiance. However, when leaf absorptance synthetic nitrogen appears to be distributed suboptimally in varied as needed to optimize distribution of light-capture N, relation to irradiance. the capacity/irradiance ratio increased up through the crown Many hypotheses have been offered to explain this appar- – that is, opposite to the observed pattern. This tendency was ent discrepancy. Some are economic: for example, that costs counteracted by constraints on stomatal or mesophyll con- of reallocating N from old, shaded leaves to younger upper- ductance, which caused chloroplastic CO2 concentration to crown leaves override the benefits (Field 1983a), or that the decline systematically with increasing irradiance. Our results ‘goal’ of allocation is not simply crown net carbon gain, but suggest that height-related constraints on stomatal conduct- perhaps something else that favours a different distribution ance can help to reconcile observations with the hypothesis of N (Hollinger 1996; Makino et al. 1997; Ackerly 1999; that photosynthetic N is allocated optimally. Schieving & Poorter 1999; Kull 2002). Other explanations are physiological or structural: for example, that the mechanisms Key-words: canopy; nitrogen; photosynthesis; stomata. underlying allocation (or reallocation) of N cannot achieve optima across the very wide range of irradiances encoun- tered in deep canopies (Evans 1993; Pons & Pearcy 1994; INTRODUCTION Terashima & Hikosaka 1995b), perhaps because of leaf mass Photosynthetic resource allocation in plant crowns has drawn per unit area (LMA) being limited by structural constraints the attention of physiological ecologists for decades. One (Dewar et al. 2012) or by photosynthetic capacity being reason is the importance of allocation in scaling theory: all limited by physiological constraints (Lloyd et al. 2010). models that scale up gas exchange from leaf-level models Another class of explanations has not been thoroughly require assumptions about photosynthetic resource alloca- explored, in our view: namely, that optimization does not, in tion among leaves and over time.A common assumption, and fact, predict a direct proportionality between capacity and which underlies the big-leaf scaling method, is that photosyn- irradiance. Published tests of optimization theory in real thetic capacity is allocated in proportion to local irradiance plant crowns typically calculate optimal distributions using (photosynthetic photon flux density) (Sellers et al. 1992; simplified models, and those models may exclude features that affect the economics of nitrogen use in photosynthesis. Corresponding author: T. N. Buckley. e-mail: tom.buckley@ One such feature that is of central interest in the present sonoma.edu study is the supply of carbon dioxide to the mesophyll. © 2013 John Wiley & Sons Ltd 1547 1548 T. N. Buckley et al.

coordination of photosynthetic capacity with both stomatal (a) 1.0 and mesophyll conductance (Le Roux et al. 2001; Koch et al. 2004; Warren & Adams 2006; Burgess & Dawson 2007; Flexas et al. 2008). Peltoniemi, Duursma & Medlyn (2012) 0.8 recently used a simple model to show that the ratio of optimal photosynthetic capacity to irradiance should be reduced in upper-crown leaves by hydraulic limitations to 0.6 stomatal conductance. Similarly, Warren & Adams (2006) showed that photosynthetic nitrogen and water use efﬁciency are affected by physiological constraints on scaling of meso- 0.4 1:1 phyll conductance with photosynthetic capacity.

capacity (unitless) Other model features may also affect the calculation of

Relative photosynthetic optimal capacity proﬁles. de Pury & Farquhar (1997) showed 0.2 0.34 2 y = 0.94x , r = 0.55 that scale-invariance does not necessarily emerge from opti- 2 y = 0.54x + 0.45, r = 0.49 mization when the key driving variable, local irradiance, 2 y = 0.19ln|x | + 0.90, r = 0.59 varies in complex fashion over the day and with and within 0.0 (b) each canopy layer. Indeed, Bond et al. (1999) suggested that 8 the issue could not be properly addressed without a highly detailed three-dimensional model of crown light penetration. Badeck (1995) and Kull & Kruijt (1998) showed that within- leaf gradients of light and photosynthetic capacity can be 6 important for crown-scale nitrogen allocation. Other studies have shown that photosynthetic nitrogen partitioning among functional pools, including light capture, RuBP carboxylation 4 and regeneration and electron transport, varies with growth irradiance (e.g. Terashima & Inoue 1984; Evans 1987; Average Terashima & Evans 1988; Evans, 1989) (for review, see Terashima & Hikosaka 1995a; Hikosaka & Terashima 1996). 2 Each of these factors may affect computed optimal N pro- ﬁles, yet no analysis to date has brought them together to

to relative irradiance (unitless) re-evaluate the question, how should photosynthetic capacity scale with incident irradiance in plant crowns? As suggested

Ratio of relative photosynthetic capacity 0 0.0 0.2 0.4 0.6 0.8 1.0 by Niinemets (2012) and Buckley et al. (2002), a resolution of the apparent discrepancy between measured and optimal Relative irradiance (unitless) distributions may require the incorporation of ‘more realism’ in the models used to compute optima. Figure 1. Sample observed relationships between photosynthetic The objective of this study was to assess the role of six capacity and irradiance from published data, showing the saturating nature of capacity versus irradiance. Values were model features in calculation of optimal profiles of photosyn- normalized within each dataset and are shown as relative values. thetic capacity: (1) realistic three-dimensional and multi- Symbols in (a) represent different species: closed circles, Tsuga modal crown distributions of irradiance; (2) optimizing N heterophylla; open circles, Pseudotsuga menziesii; closed triangles, allocation to light capture; (3) the co-occurrence of fixed Pinus ponderosa; open triangles, Juglans nigra ¥ regia; closed transdermal (within-leaf) gradients in photosynthetic capac- squares, Nothofagus fusca. In (a), three best-fit relationships are ity and dynamic transdermal gradients in light; (4) optimal shown for all data combined (details given in the legend); the solid regulation of stomatal diffusion; (5) limitations on the ability line is a 1:1 line. In (b), data from (a) are shown with photosynthetic capacity expressed as a ratio with relative of mesophyll conductance to track photosynthetic capacity in irradiance; the horizontal line is the mean of all data shown. a linear and homogeneous fashion; and (6) an upper limit to [Sources: Nothofagus: Hollinger (1996); Tsuga, Pseudotsuga, Pinus: instantaneous leaf transpiration rate, which reflects the need Bond et al. (1999); Juglans: Frak et al. (2002)]. Capacity was given to maintain leaf water potential above the threshold causing as maximum RuBP carboxylation rate for Juglans and as runaway xylem cavitation. We used a numerical model to light-saturated net CO2 assimilation rate for the other data. identify simultaneous optima for allocation of photosynthetic nitrogen among functional pools and among leaves, and for Earlier authors have argued that the economics of N alloca- stomatal conductance over time and among leaves, in a tion cannot be fully understood without also considering sample of 500 leaves from an artificial crown of 8000 leaves the economics of stomatal transpiration (Field, Merino & in which the three-dimensional distribution of irradiance Mooney 1983; Buckley, Miller & Farquhar 2002; Farquhar, was modelled using a ray-tracing technique (Cescatti & Buckley & Miller 2002). Indeed, it is now well known that, Niinemets 2004) and leaf gas exchange was simulated using just as the coordination of photosynthetic capacity with a biochemical and biophysical model (Farquhar, von irradiance can vary systematically within crowns, so can the Caemmerer & Berry 1980, von Caemmerer & Farquhar © 2013 John Wiley & Sons Ltd, Plant, Cell and Environment, 36, 1547–1563 Crown nitrogen and water optimization 1549

1981; Buckley & Farquhar 2004) that explicitly accounts back and forth between sunny and cloudy conditions during for transdermal gradients of irradiance and photosynthetic a single day. psun was arbitrarily varied in different simulations capacity. to represent sunny, moderate or cloudy environments

(psun = 0.9, 0.5 or 0.1, respectively). THE MODEL Optimization method Our simulations used a process-based model of leaf photosynthesis, applied to many leaves in the same individual plant Optimization of nitrogen and water are governed by the crown. In each leaf, the model was constrained by four marginal water and nitrogen costs of fixed carbon: ∂E/∂A and ∂ ∂ parameters that depend on nitrogen investment or transpi- N/ Ad (where Ad is the average of A over the day). When ration rate: Vm25, Jm25, a and gsw(t), which are, respectively, nitrogen and water are optimally allocated, these marginal ∂ ∂ maximum RuBP carboxylation velocity at 25 °C, maximum costs are invariant: N/ Ad is invariant among leaves and potential electron transport rate at 25 °C, leaf absorptance to among photosynthetic N pools, and ∂E/∂A is invariant among photosynthetic irradiance and the diurnal time course of sto- leaves and over time (Cowan & Farquhar 1977; Field 1983b; matal conductance to water vapour. Those four parameters Farquhar 1989; Buckley et al. 2002). The four physiological ∂ ∂ ∂ ∂ were adjusted to satisfy mathematical criteria for optimality. parameters that affect E/ A and N/ Ad and are determined The model and optimization procedures are briefly summa- by resource investment (gsw, Vm25, Jm25 and a) were adjusted rized below, and additional details are given in the Appendix. to achieve arbitrarily chosen target values for the marginal costs: the target value for ∂N/∂Ad was defined as n, and the target value for ∂E/∂A was defined as l.The numerical values Gas exchange model -1 of l and n were arbitrarily chosen (1100 mol H2O mol CO2 -1 We used the photosynthesis and photorespiration model of and 0.22 mol N dmol CO2, respectively) to give reasonable values for leaf gas exchange variables. This optimization was Farquhar et al. (1980) to calculate mesophyll CO2 demand achieved in a series of nested loops.An ‘outer’ loop identified as a function of chloroplastic CO2 concentration (cc), and the leaf photosynthetic nitrogen content (Np) for which expressions for CO2 and H2O diffusion given by von ∂N/∂Ad = n. For each candidate value of Np in the outer loop, Caemmerer & Farquhar (1981) to calculate CO2 supply to the chloroplast and transpiration rate. Our implementation a second loop optimized the partitioning of that Np among of this widely used model was distinct from most previous three pools, each serving a different photosynthetic function: implementations in two respects. Firstly, we specified meso- Nv, RuBP carboxylation; Nj, RuBP regeneration and electron transport; and Nc, light capture. For each candidate vector of phyll conductance to CO2, gm, by one of two alternative pool-wise N allocation fractions, a third loop optimized gsw assumptions (that gm is proportional to Vm25, or that gm is invariant through the crown) and we assessed these alterna- for each time point over the day. Detailed costing (relation- tives in different simulations. Secondly, to account for the ships between resource investment and resource-dependent nitrogen cost of light capture in leaves of differing orienta- photosynthetic parameters) and numerical methods used in tion and irradiance, we used a new model for whole leaf each loop are given in the Appendix. potential electron transport rate, J, that accounts for variable illumination at either leaf surface and variable transdermal Simulations gradients of photosynthetic capacity within leaves (Buckley We performed a range of simulations to assess the effect & Farquhar 2004). In that model, the transdermal capacity of the seven model features outlined in the Introduction profile is a weighted average of exponential profiles from on inferred optimal crown distributions of photosynthetic either leaf surface, the weights being w and w = 1–w for u l u capacity in relation to irradiance. In most cases, we compared the upper and lower surfaces, respectively. This model is a ‘default’ simulation with another in which some feature was described in the Appendix. omitted from, or altered relative to the model. The default The simulated canopy comprised a uniform layer of simulation used the model as described above: that is, pho- random leaves with spherical leaf angle distribution. tosynthetic N was optimized for each functional pool, sto- Direct and diffuse irradiance at each leaf were simulated matal conductance was optimized at each time point and for using a ray-tracing approach (Cescatti & Niinemets 2004), each leaf, mesophyll conductance was assumed proportional described in the Appendix. We used these irradiances to to carboxylation capacity, potential electron transport rate compute two values for all gas exchange variables for each (J) was calculated using the transdermally explicit model of leaf at each point in time: one for sunny conditions and one Buckley & Farquhar (2004) with w (the weighting of the for cloudy conditions. Under perfectly cloudy conditions, u transdermal capacity profile to the upper surface) set equal light is entirely diffuse and thus unimodal, whereas under to the fraction of daily irradiance received at the upper sunny conditions, most light is direct, creating a bi- or multi- surface for each leaf, and the sunshine probability parameter modal light environment. A weighted average of the sunny p was set at 0.5. The other simulations differed from the and cloudy values for leaf net CO assimilation rate was then sun 2 default simulation as follows: computed using the instantaneous probability of sunny conditions (psun) and its complement as the weights. This simu- 1 Cloudy or sunny conditions: To assess the role of realistic lates an environment in which the radiation regime switches three-dimensional and multi-modal crown distributions of © 2013 John Wiley & Sons Ltd, Plant, Cell and Environment, 36, 1547–1563 1550 T. N. Buckley et al.

irradiance, we compared simulations under three scenarios the model of Oren et al. (1999), in which gs = gs,ref – m ln(D/ for the degree of cloudiness of the above-crown light envi- kPa), where D is evaporative demand in kPa and

ronment: psun = 0.1 (cloudy), 0.5 (moderate) or 0.9 (sunny). m = 0.6·gs,ref (Oren et al. 1999,2001;Ewers et al. 2001).This is 2 Variable N investment in leaf absorptance (a):To assess the in a range typical for many tree species (Oren et al. 1999). effects of optimizing N investment in light capture (N ), we c The purpose of these simulations is to investigate the effect compared default simulations with another in which N c of specific model features on optimized profiles of crown (and hence leaf absorptance) was not optimized, but photosynthetic nitrogen. They should each be interpreted as instead was identical for each leaf [such that a was uni- a theoretical device to that end, or ‘thought experiments’ in formly equal to its assimilation-weighted crown average which the model is used to probe relationships among crown from the default simulation (0.826)]. properties in a way that would not be possible using experi- 3 Explicit transdermal gradients in capacity and light:To ments on real crowns in nature. As such, these simulations assess the relevance of transdermal gradients in photosyn- will, by design and by necessity, include features that are not thetic capacity and light (gradients among paradermal empirically realistic. layers), we compared the default simulation with two others. In one, the bias of the transdermal capacity profile RESULTS (wu) was identical for all leaves and equal to its assimilation-weighted crown average from the default Optimal crown photosynthetic resource simulation (0.745); this accounts for transdermal light gra- distributions (default simulation) dients but assumes transdermal capacity profiles cannot Field observations of the ratio of V to I are that it adapt to those gradients within each leaf independently. m25 d decreases systematically up the crown. Figure 1 illustrates The other simulation used the standard model for J (a this trend with published data from several species. Here, we non-rectangular hyperbola), using the same curvature examine the results of simulations (1)–(6) and seek indica- factor (q , Eqn A7,Table 1) as in the default simulation; this j tions of a similar decrease in our model. The fully optimized ignores transdermal gradients entirely, which is equivalent model predicts positive relationships between integrated to assuming transdermal profiles of capacity instantane- daily irradiance (I ) and carboxylation capacity (V ) that ously adjust to match those of irradiance as the light regime d m25 have positive curvature (Fig. 2), such that the ratio of V to shifts over the day (Buckley & Farquhar 2004). m25 I increases systematically up through the crown (Fig. 3); that 4 Variation in chloroplastic CO (c ): We assessed the role of d 2 c is, opposite to observations. Electron transport capacity (J ) optimal stomatal regulation by comparing the default simu- m25 and V are linearly and homogeneously related (Fig. 4), but lation with another in which chloroplastic CO concentra- m25 2 the proportion of photosynthetic N allocated to light capture tion (c ) was identical for each leaf and equal to its c declines as I increases (Fig. 5). Intercellular and chloroplas- assimilation-weighted crown average from the default d tic CO concentrations (c and c ; Fig. 6a) are approximately simulation (225 mmol mol-1),instead of being set by optimi- 2 i c invariant among leaves, except for a steep decrease in rela- zation of stomatal conductance (g ). Because optimization s tion to I in a few leaves at very low I . The decrease occurs of g leads to smaller computed values of ∂A/∂N at any given d d s among leaves with exceptionally low carbon gain, suggesting level of photosynthetic N investment,this simulation used a these leaves would probably not be produced in real canopies smaller value of n (0.152 mol N dmol-1 CO ) to yield the 2 (cf. vertical lines in Fig. 3, indicating values of I above which same whole-crown photosynthetic N content as the default d 90 or 95% of crown carbon gain occurred). simulation. Water, nitrogen and light use efficiencies (WUE, PNUE 5 Constraints on mesophyll conductance (g ):We assessed the m and LUE, respectively) calculated from daily averages of leaf role of constraints on mesophyll conductance by comparing net carbon gain, transpiration rate and irradiance behaved the default simulation with another in which g m quite differently from one another: WUE was nearly invari- was identical among leaves and equal to its assimilation- ant among leaves, whereas LUE and PNUE both increased weighted crown average from the default simulation with I (Fig. 7a). (0.34 mol m-2 s-1), instead of being directly proportional to d carboxylation capacity in each leaf as in the default simula- Effect of complex multi-modal light environment tion.Note that invariant gm is intended as a limiting scenario

and should not be viewed as a representation of empirically Under sunny conditions (psun = 0.9), most relationships pre-

observed patterns of gm. dicted by the model became more scattered than under mod-

6 Constraints on transpiration rate (E):We assessed the role of erate conditions (psun = 0.5), whereas relationships in cloudy

constraints on hydraulic conductance and transpiration rate conditions (psun = 0.1) were less scattered (Figs 2, 3 & 7). The

by comparing the default simulation with another in which ratio Vm25/Id increased with Id at all values of psun. LUE transpiration rate for each leaf (E) was limited to a value, became highly variable under sunny conditions, whereas

Emax, to represent limitations imposed by the need to WUE and PNUE were less variable (Fig. 7). (Note that the

prevent runaway xylem cavitation. When optimized gs scatter introduced by sunﬂecks in these simulations could be

caused E to exceed Emax for a leaf, its gs was reduced so that mistaken for the random variation typically seen in empirical -2 -1 E = Emax. Emax was arbitrarily set at 0.002 mol m s , which data, so we wish to emphasize that these are simulations and -2 -1 corresponds roughly to gs,ref = 0.17 mol m s in terms of not actual data.) © 2013 John Wiley & Sons Ltd, Plant, Cell and Environment, 36, 1547–1563 Crown nitrogen and water optimization 1551

Table 1. Parameters and variables in this paper. Area-based units are per leaf area, except those marked with an asterisk (*), which are per ground area. ‘Irradiance’ means photosynthetic photon ﬂux density

Description Symbol Units Value

-2 -1 Leaf net CO2 assimilation rate A mmol m s – Leaf absorptance to photosynthetic radiation a –– -2 -1 Crown daily average of net CO2 assimilation Ad mol m d *– -2 -1 Demand expression for net CO2 assimilation rate AD mmol m s – - -2 -1 Demand expression (e transport limited) ADj mmol m s – -2 -1 Demand expression (carboxylation limited) ADv mmol m s – -1 -1 Ratio of max photosynthetic rate (Pmax)to[cytf] af mol CO2 s mol cyt f 19.9 -1 -1 Ratio of Pmax to [PSII] ap mol CO2 s mol PSII 18.2 -1 -3 Ratio of [PSI] to [Chl] as mol PSII mol Chl 1.74·10 -2 -1 Supply expression for net CO2 assimilation rate As mmol m s – Solar elevation angle b degrees – Leaf chlorophyll content [Chl] mmol m-2 – -1 Intercellular CO2 mole fraction ci mmol mol – -1 Chloroplastic CO2 mole fraction cc mmol mol – -1 -1 Carboxylation capacity per N cv mmol CO2 s mmol N 4.49 - -1 -1 Electron transport capacity per N cj mmol e s mmol N 9.48 -1 -4 Chlorophyll per electron transport N (Nj) ccj mmol Chl mmol N 4.64·10 -1 -2 Chlorophyll per light capture N (Nc) cc mmol Chl mmol N 3.384·10 -1 Cytochrome f per unit group II N cII mol cyt f mol N 1/9530 -1 PSII per unit group III N cIII mol cyt f mol N 1/5000 -1 PSI per unit group IV N cIV mol cyt f mol N 1/6040 -1 Chl per unit group III N ccIII mol Chl mol N 1/83.3 -1 Chl per unit group IV N ccIV mol Chl mol N 1/32.8 -1 Chl per unit group V N ccV mol Chl mol N 1/26.0 Solar declination d degrees – Time step dt hours – Marginal carbon product of water use ∂E/∂A mol mol-1 – -1 -1 Marginal carbon product of nitrogen use ∂N/∂Ad mol [mol d ] – Leaf transpiration rate E mol m-2 s-1 – Fraction of absorbed photons not used in photosynthesis F e-/photon 0.23

Light-capture fraction of photosynthetic N fc –– Diffuse fraction of daily irradiance fd ––

Electron transport fraction of photosynthetic N fj –– Carboxylation fraction of photosynthetic N fv –– Shortwave radiation F Jm-2 s-1 – -1 Photorespiratory CO2 compensation point G (G 25) mmol mol –, 36.9 * * -2 -1 Boundary layer conductance to CO2 (H2O) gbc (gbw) mol m s – -2 -1 Boundary layer conductance to heat gbh mol m s – -2 -1 Mesophyll conductance to CO2 gm mol m s – -2 -1 Stomatal conductance to CO2 (H2O) gsc (gsw) mol m s – -2 -1 Total conductance to CO2 (H2O) gtc (gtw) mol m s – -2 -1 Beam irradiance at upper (lower) leaf surface Ibu (Ibl) mmol m s – -2 -1 Integrated daily irradiance Id mol m d *– -2 -1 Diffuse irradiance at upper (lower) leaf surface Idu (Idl) mmol m s – -2 -1 Diffuse (beam) irradiance above crown Iod (Iob) mmol m s – -2 -1 Irradiance at upper (lower) leaf surface Iu (Il) mmol m s – -2 -1 Global irradiance (above atmosphere) Iog (Io) mmol m s – Leaf potential e- transport rate J mmol m-2 s-1 –

Day of year Jday - 0 - -2 -1 Light-limited maximum potential e transport rate Ji mmol m s - - -2 -1 Leaf maximum potential e transport rate (at 25 °C) Jm (Jm25) mmol m s - -2 -1 Excess Ji in light-saturated transdermal layers Js mmol m s - -1 Michaelis constant for RuBP carboxylation (at 25 °C) Kc (Kc25) mmol mol -, 404 Michaelis constant for RuBP oxygenation (at 25 °C) Ko (Ko25) kPa -, 24.8 Setpoint for ∂E/∂A l mol mol-1 1100 Crown daily light use efﬁciency LUE mol mol-1 - -1 Setpoint for ∂N/∂Ad n mol d mol 0.22 -2 Light capture nitrogen Nc mmol N m - -2 Electron transport nitrogen Nj mmol N m - -2 Photosynthetic N Np mmol N m - -2 Rubisco nitrogen Nv mmol N m -

Table 1. Continued

Description Symbol Units Value

-2 Group I N (Rubisco) NI mmol N m - -2 Group II N (electron transport, Calvin cycle) NII mmol N m - -2 Group III N (PSII) NIII mmol N m - -2 Group IV N (PSI, LHCI) NIV mmol N m - -2 Group V N (LHCII) NV mmol N m - Oxygen concentration O kPa 21 Crown daily photosynthetic N use efﬁciency PNUE mol d-1 mmol-1 -

Instantaneous sunshine probability psun – 0.5 Curvature factor for photosynthesis qA – 0.99 Curvature factor for electron transport qj – 0.86 -2 -1 Non-photorespiratory CO2 release in the light (at 25 °C) Rd (Rd25) mmol m s -, 0.0089Vm25 Time of day t h - o Air temperature (in Kelvin) Ta (TaK) C, K - o Leaf temperature (in Kelvin) Tl (TlK) C, K - -1 Wind speed (at top of crown) v (vo)ms -,5 -2 -1 Leaf maximum carboxylation rate (at 25 °C) Vm (Vm25) mmol m s - Downwind leaf width w m 0.12 -1 Water vapour mole fraction of the air wa mol mol 1.0 -1 Saturation water vapour mole fraction of the air wsa mol mol - -1 Saturation water vapour mole fraction in leaf wsl mol mol - Bias of transdermal capacity proﬁle to upper leaf surface wu – - Crown daily water use efﬁciency WUE mol kmol-1 -

Effect of optimizing nitrogen allocation to Effect of constraints on leaf transpiration rate light capture When instantaneous leaf transpiration rate was limited to an

When leaf absorptance was held invariant among leaves in imposed maximum value, the relationship between Vm25 and simulations at moderately sunny conditions (psun = 0.5), the Id took on clear negative curvature, such that the ratio Vm25/Id relationship between Vm25 and Id became approximately declined with increasing Id across most of the range of Id linear and homogeneous, such that the ratio Vm25/Id did not (Fig. 10). Assimilation-weighted average ci and cc both increase with Id (Fig. 8). declined with increasing Id in this simulation (Fig. 6c).

Effect of explicit transdermal gradients in capacity and irradiance DISCUSSION The objective of this study was to ask a theoretical question The shape of the relationship between Vm25 and Id was indis- rather than an empirical one: does the apparent discrepancy tinguishable between the default simulation and those in between observed and optimal crown profiles of photosyn- which the bias of the transdermal capacity profile (wu) was thetic capacity disappear when certain detailed features are identical among leaves, or in which transdermal gradients in included in the models used to infer optimal profiles? This capacity and irradiance were ignored altogether (not shown). followed the suggestion by Niinemets (2012) that the discrepancy could not be adequately understood without incorpora- Effect of optimal stomatal regulation tion of greater realism in models. Our analysis took the form

The relationship between Vm25 and Id was broadly similar of a series of ‘thought experiments’: we performed a default whether gs was optimized (and gm was set proportional to simulation using a model that included many detailed fea-

Vm25)orcc was set as a constant, equal to the crown average tures that were meant to impart realism, and then these fea- from the optimized default scenario (Fig. 9). In the latter case tures were individually modiﬁed or excluded in subsequent

– in which gs and gm were irrelevant to carbon gain – the simulations.The purpose of these thought experiments was to relationship between Vm25 and Id exhibited slightly more determine how these features affect calculated optimal scatter at any given Id. crown capacity proﬁles. The main dependent variable of interest in these comparisons was the ratio of carboxylation Effect of constraints on mesophyll conductance capacity to integrated daily irradiance (Vm25/Id). This is because observations typically show the ratio Vm25/Id decreas-

When gm was assumed invariant among leaves, the ratio ing up through the crown (e.g. Fig. 1), whereas most previous

Vm25/Id decreased slightly with Id across most of the range of theoretical studies have concluded that this ratio should be

Id (Fig. 10), and cc systematically declined with Id, whereas ci invariant with Id – that is, in the optimum, Vm25 should be increased (Fig. 6b). linearly and homogeneously related to Id. Therefore, we © 2013 John Wiley & Sons Ltd, Plant, Cell and Environment, 36, 1547–1563 Crown nitrogen and water optimization 1553

400 8 (a) (a)

300 6

200 4 d)

100 –1 2

Sunny (p = 0.9) mol sun –1 Sunny ( p = 0.9) y = 5.31x – 26.4, r 2 = 0.849 sun 0 0 mol s mol 0204060 μ 01530456075 /( d I / 7 ) 250 (b) –1 m25 V s (b)

–2 6 200 5 mol m m

/( 4

m25 150 V 3 100 2

50 1 Moderate ( p = 0.5) sun Moderate ( psun = 0.5) 0 y = 5.01x – 9.0, r 2 = 0.981 0 01020304050 Carboxylation capacity, capacity, Carboxylation 0153045 6 (c) 120 (c) 5

100 irradiance, to daily capacity of carboxylation Ratio 4

80 3

60 2

40 1 Cloudy (psun = 0.1) 20 Cloudy (psun = 0.1) 0

y = 4.85x – 5.5, r 2 = 0.997 0 5 10 15 20 25 30 0 –2 –1 0 5 10 15 20 25 30 Integrated daily irradiance, Id /(mol m d )

Integrated daily irradiance, I /(mol m–2 d–1) d Figure 3. Ratio of carboxylation capacity at 25 °C, Vm25, to daily incident irradiance, Id, shown in relation to Id, under different Figure 2. Optimal relationships between daily irradiance, Id, and sunshine probabilities (psun): (a) psun = 0.9 (sunny); (b) psun = 0.5 carboxylation capacity at 25 °C, Vm25, under different sunshine (moderate, default simulation); and (c) psun = 0.1 (cloudy). The probabilities (psun): (a) psun = 0.9 (sunny); (b) psun = 0.5 (moderate, vertical lines indicate leaves whose daily net carbon gain (Ad)is default simulation); and (c) psun = 0.1 (cloudy). Each point is a such that whole-crown daily net carbon gain would only decline by different leaf in the same crown. Best-ﬁt lines and their equations 5% (solid lines) or 10% (dashed lines) if leaves with equal or and coefﬁcients of determination are shown. lower Ad were excluded.

600 300 (a) 500 280 )

–1 260 ci

s 400 –2

) 240 300 –1 mol m

m c 220 c

/( slope 200 mol m25 psun = 0.9 1.60 m J 200

p = 0.5 ) /( 100 sun 1.56 c Electron transport capacity, c psun = 0.1 1.63 180 Default 0 0 100 200 300 400 160 (b) Carboxylation capacity, V /(mmol m–2 s–1) m25 280 ) or chloroplastic ( Figure 4. Relationship between carboxylation capacity (Vm25) i c 260 and electron transport capacity (Jm25) among leaves in crowns that were optimized under different sunshine probabilities (psun): open circles, psun = 0.1; closed triangles, psun = 0.5 (default simulation); 240 open squares, psun = 0.9. Slopes of lines ﬁtted to relationships for each psun value are given in the legend. 220 asked whether various model details would alter this conclu- 200 sion, predicting instead that Vm25/Id should decrease with increasing Id in the optimum, as is observed in the ﬁeld. 180 Invariant gm mole fraction, intercellular ( Allowing N investment in light capture to vary 2 160 optimally makes the discrepancy worse (c) 280 In our default simulation, the ratio Vm25/Id increased with Id. This is in part because our model allowed leaf absorptance, a, 260

240 0.6

220 0.5 Assimilation-weighted CO Assimilation-weighted 200 0.4 180 /(unitless)

j Limited E f 0.3 , v

f 160 , c f 0.2 0 1020304050 f Carboxylation ( v) –2 –1 Integrated daily irradiance, Id /(mol m d ) f 0.1 Electron transport ( j) Figure 6. Relationships between assimilation weighted CO2 fractions, f Light capture ( c) concentration [intercellular, ci (open triangles); chloroplastic, cc

Photosynthetic nitrogen partitioning Photosynthetic 0.0 (closed circles)] and integrated daily irradiance (Id) for crowns 0 1020304050 optimized with different assumptions about the regulation of CO2 supply to the mesophyll: (a) stomatal conductance (gs) optimized I –2 –1 Integrated daily irradiance, d /(mol m d ) and mesophyll conductance (gm) proportional to carboxylation capacity (default simulation; no arbitrary constraints on CO2 supply); (b) as in (a), but with g invariant among leaves; (c) as in Figure 5. Photosynthetic nitrogen partitioning fractions (fc, light m (a), but with g indirectly constrained by an upper limit (E )on capture; fv, carboxylation; fj, electron transport) in the default s max simulation. Each point is a different leaf in the same crown. transpiration rate (E). © 2013 John Wiley & Sons Ltd, Plant, Cell and Environment, 36, 1547–1563 Crown nitrogen and water optimization 1555

5 0.05 (a) Sunny (psun = 0.9) 6 4 0.04

3 0.03 d) –1 photons) –1

mol 4

2 0.02 –1 mol 2 mol s

1 0.01 m /( d

I a

/ Invariant 2

0 0.00 m25 Default ) V 2 0 15304560 CO

–1 5 0.05 (b) Moderate ( psun = 0.5) 0 O mol

2 4 0.04 010203040 –2 –1 Integrated daily irradiance, Id /(mol m d ) 3 0.03

Figure 8. Relationships between integrated daily irradiance (Id) N), light use efficiency, LUE /(mol CO and the ratio of carboxylation capacity, Vm25, to Id, for simulations 2 0.02 –1 in which leaf absorptance was held invariant among leaves (closed triangles) or was allowed to vary among leaves such that N

mmol investment in light capture was optimized (open circles, default 1 0.01 –1 simulation). These simulations were performed under moderately d 2 sunny conditions (psun = 0.5).

0 0.00 01020304050

Water use efficiency , WUEWater /(kmol H 5 0.05 (c) Cloudy (psun = 0.1)

4 0.04 6

3 0.03 d) –1 WUE 4 2 LUE 0.02 mol PNUE –1 Nitrogen use efficiency, PNUE /(mol CO 1 0.01 mol s μ /( d I / 2

0 0.00 m25 V 0 5 10 15 20 25 30 Invariant cc –2 –1 Default Integrated daily irradiance, Id /(mol m d )

Figure 7. Predicted crown profiles in photosynthetic resource 0 efficiency (ratios of daily net carbon gain to resource use): water 0 10203040 use efficiency, WUE (left axis, closed circles), photosynthetic Integrated daily irradiance, I /(mol m–2 d–1) nitrogen use efficiency, PNUE (right axis, closed squares) and light d use efficiency, LUE (right axis, open triangles) for a crown optimized under (a) sunny conditions (psun = 0.9) (default Figure 9. Relationships between integrated daily irradiance (Id) simulation), (b), moderate conditions (psun = 0.5) or (c) cloudy and the ratio of carboxylation capacity, Vm25, to Id, when stomatal conditions (psun = 0.1). conductance was optimized (default simulation, open circles) or chloroplastic CO2 concentration, cc, was invariant among leaves and equal to the assimilation-weighted average from the default simulation. © 2013 John Wiley & Sons Ltd, Plant, Cell and Environment, 36, 1547–1563 1556 T. N. Buckley et al.

was identical in all transdermal layers, and that Vm25 and Jm25 were directly proportional to leaf Chl content among leaves. 6 Those assumptions ensured that partitioning of photosynthetic N among light capture, carboxylation and electron transport could not vary with irradiance.As a result,the return

from N investment in light capture declined as Id increased d)

–1 in that model. This, in turn, required reduced investment in 4 photosynthetic capacity relative to irradiance at high light (in mol ∂ ∂

–1 order to keep the marginal N cost of carbon, N/ Ad,invariant among leaves as required for optimization), so predictions

mol s from that model are more in line with observations. By con- μ

/( trast, our simulations used the model of Buckley & Farquhar d I / 2 (2004), which assumes the transdermal profile of capacity is a Default m25 weighted average of exponential profiles that decline from V Invariant gm either leaf surface.We also allowed N partitioning to vary with irradiance by optimising N allocation to light capture, car- Limited E boxylation and regeneration separately.This captures the ten- 0 dency for photosynthetic partitioning to acclimate to local 0 10203040 environmental conditions, both within and among leaves. In short, the contrast between our results and those of Badeck Integrated daily irradiance, I /(mol m–2 d–1) d (1995) results from greater flexibility of photosynthetic partitioning in our model, so if partitioning is in fact somehow Figure 10. Relationships between integrated daily irradiance (I ) d constrained in real crowns,that could help explain the discrep- and the ratio of carboxylation capacity, Vm25,toId, for two alternative simulations that included ad hoc constraints on ancy between observed and optimal capacity profiles. mesophyll CO2 supply (closed triangles and open squares) not present in the default simulation (open circles). Closed triangles Constraints on CO2 supply to the mesophyll (‘invariant gm’), mesophyll conductance, gm, invariant among leaves. Open squares (‘limited E’), stomatal conductance indirectly help resolve the discrepancy limited by an upper bound (E ) on transpiration rate (E). Best-fit max Most previous analyses of optimal crown N allocation have lines for values above I = 10 mol m-2 d-1 are shown for each case. d either assumed a constant value for intercellular or chloro-

plastic CO2 concentration (ci or cc, respectively), either to vary among leaves as needed to optimize N investment implicitly or explicitly (Hirose & Werger 1987; Gutschick & in light capture. The result of this optimization was that a Wiegel 1988; Evans 1993; Hikosaka & Terashima 1995; Sands increased with Id, which caused absorbed irradiance to increase 1995; Ackerly 1999; Bond et al. 1999), or have speciﬁed sto- non-linearly and with positive curvature in relation to incident matal conductance (gsw) using empirical models that assume irradiance. Because absorbed irradiance, not incident irradi- gsw is linearly proportional to photosynthetic rate (Field ance per se, determines the return on N investment in carboxy- 1983a; Hollinger 1996; Haefner, Buckley & Mott 1997). Our lation and electron transport capacities (Vm25 and Jm25), these results suggest that those assumptions are reasonable proxies capacities therefore also increased non-linearly with Id, and for optimal stomatal behaviour because predictions from our therefore the ratio Vm25/Id also increased with Id (e.g. Fig. 3). model were qualitatively similar whether gsw was optimized Previous authors have pointed out that leaf absorptance should over time and among leaves, or if instead the general char- be greater in high light than in low light (e.g. Evans 1989; acter of stomatal control was represented indirectly, by

Hikosaka & Terashima 1995), yet few analyses of optimal N holding cc constant. In particular, optimization of gsw did not allocation in crowns have allowed absorptance to vary. Our give rise to any systematic trends in ci or cc in relation to simulations suggest, in any event, that including this feature irradiance in the default simulation. does not resolve the discrepancy between actual and optimal However, when the model incorporated features that capacity proﬁles, but instead it makes it worse. caused CO2 supply to the mesophyll to decline systematically

in relation to increasing irradiance, the ratio Vm25/Id also Accounting for transdermal gradients in light declined with increasing Id across most of the range of Id.This and capacity has little impact result held whether CO2 supply was restricted by limitations on mesophyll or stomatal conductance, although the effect We found that explicitly accounting for transdermal gradients was far greater in the latter case (e.g. Fig. 10).The reason that of irradiance and photosynthetic capacity had no qualitative lower photosynthetic capacity is optimal if cc is lower is effect on predicted optimal proﬁles of photosynthetic capac- simply that the marginal carbon product of N, ∂Ad/∂N,is ity. This contrasted with the conclusions of Badeck (1995), positively related to cc and negatively to photosynthetic N who also used a transdermally explicit model. However, that (Buckley et al. 2002); thus, to maintain invariance among model made two assumptions that differed from our own: it leaves in ∂Ad/∂N as required for optimality, a decrease in cc assumed that photosynthetic capacity per unit chlorophyll necessitates a decrease in photosynthetic N investment. © 2013 John Wiley & Sons Ltd, Plant, Cell and Environment, 36, 1547–1563 Crown nitrogen and water optimization 1557

Hydraulic constraints in relation to height are well known contain N) – leaves and wood devoid of N would sufﬁce if and widely studied (Yoder et al. 1994; Hubbard, Bond & their sole function were to cast shade on competitors, so these Ryan 1999; Ryan et al. 2000; Phillips et al. 2002; Delzon et al. considerations do not bear on the optimal distribution of N 2004; Franks 2004; Ryan, Phillips & Bond 2006), so such among leaves. Kull (2002) suggested that the proper goal constraints may help explain the apparent discrepancy function is actually a compromise between maximizing between observed and optimal photosynthetic capacity – at carbon gain and maximizing PNUE, and that the costs of leaf least in tall trees, where height is likely to substantially limit construction must be included somehow; Hollinger (1996) hydraulic conductance. In this respect, our results echo those made a similar suggestion. Yet, maximal PNUE must depend of Peltoniemi et al. (2012), who used a simple two-leaf model, on the total supply of nitrogen, which leads to the question, in which stomatal conductance, was directly proportional to how much carbon should be invested in roots? Thus, it seems hydraulic conductance, to show that optimal photosynthetic again that crown carbon gain should nevertheless be maxi- N investment should be reduced in upper crown leaves expe- mized because carbon can be used to capture more N. riencing hydraulic limitation. However, this explanation is It has also been argued that lower-crown leaves that ini- unlikely to explain the discrepancy in very short canopies, tially developed in higher-light conditions cannot fully remo- such as crops like wheat (e.g. de Pury & Farquhar 1997). bilize photosynthetic nitrogen when they begin to become

Evidence also suggests that gm can only track photosyn- shaded (Hikosaka & Terashima 1995), or that the costs of thetic capacity up to a point, beyond which it saturates remobilization skew optimal N distributions (Field et al. (Niinemets et al. 2005, 2006; Warren & Adams 2006), prob- 1983). However, the cost of remobilizing N would not affect ably due to limits to leaf structural plasticity (Flexas et al. comparisons of ∂N/∂Ad among leaves, assuming that cost is

2012). Because relationships between gm and Vm25 vary directly proportional to N content: in that case, the remobi- widely and cannot be generalized (Flexas et al. 2012), we lization cost would merely add a constant to ∂N/∂Ad.Another used two limiting cases to bound the range of constraints on possibility is that leaves store nitrogen in photosynthetically gm – either gm tracked Vm25 perfectly, or it was held invariant active pools such as Rubisco and electron transport chain among leaves. When the crown was optimized under invari- components in order to hedge against future deficits, or as a ant gm, Vm25/Id declined slightly across most of the range of reservoir for N acquired during vigorous early season growth irradiance. Thus, constraints on gm can, in theory at least, when water and nitrogen are plentiful.The economic benefits contribute to reconciling observed and inferred optimal of storage are difficult to express and have not been included crown N profiles. However, that simulation also made an in any analyses of crown N allocation, to our knowledge. unrealistic prediction, namely that ci increases up through the However, delayed costs and benefits of water use have pre- crown. This occurred because the gm constraint reduced N viously been addressed in the context of optimal stomatal investment in photosynthetic capacity at high irradiance control (Cowan 1977, 1986; Cowan & Farquhar 1977; Comins (which is, of course, the very phenomenon that we set out to & Farquhar 1982), so the problem may be tractable. Our explain), but this, in turn, reduced total CO2 demand, leading model also excludes temporal buffering of the light environ- to the increase in ci. Our results thus suggest that while gm ment by non-steady-state photosynthesis (enabled by large constraints may contribute to reconciling the discrepancy potential pool sizes for photosynthetic intermediates) which between optimal and observed capacity profiles, they must be can allow an intermediate photosynthetic rate to persist combined with another factor that independently reduces across sun- and shade-flecks (Pearcy 1990). Future analyses ci. As we found much stronger reductions in Vm25/Id with should consider these additional details. increasing Id when gsw was constrained than when gm was Others have posited upper or lower limits on the proper- constrained, we conclude that hydraulic constraints on gsw ties of leaves within crowns in order to explain the tendency remain the strongest hypothesis to resolve the discrepancy. for profiles of capacity to be ‘less steep’ than those of irradiance. For example, Lloyd et al. (2010) suggested that photo- Other explanations for the discrepancy between synthetic capacity has an upper physiological limit, and observed and optimal profiles Dewar et al. (2012) suggested that LMA has a lower limit. Both studies concluded that, for a given total capacity, crown A variety of other arguments have been advanced to explain photosynthesis is greatest if capacity profiles are less steep the apparent discrepancy between actual and optimal pro- than irradiance profiles, once upper limits to capacity or files of photosynthetic capacity. One class of arguments lower limits to LMA are accounted for. However, both centres on the idea that crown net carbon gain is not the studies also assumed that the profile of capacity must follow correct goal function to understand the adaptive significance an exponential curve. While such profiles may be empirically of photosynthetic nitrogen allocation. For example, Ackerly observed, there is no a priori reason to suppose they are (1999) suggested that leaf production and height growth optimal, in the sense that no possible redistribution of pho- are more important because of competition for light, and tosynthetic nitrogen among leaves would increase total Schieving & Poorter (1999) showed how light competition carbon gain. The latter condition requires invariance of the between species could skew vertical profiles of photosyn- marginal C product of N, ∂A/∂N (Field 1983a; Farquhar thetic capacity and reduce crown carbon gain. However, leaf 1989), which, in turn, requires close proportionality between construction and height growth represent investments of irradiance and capacity, unless, as we have shown here, reduced carbon, not nitrogen (although leaves and wood do another determinant of net photosynthesis, such as CO2 © 2013 John Wiley & Sons Ltd, Plant, Cell and Environment, 36, 1547–1563 1558 T. N. 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Sellers P.J., Berry J.A., Collatz G.J., Field C.B. & Hall F.G. (1992) Canopy limitation, which presents a challenge for numerical optimi- reﬂectance, photosynthesis and transpiration. III: a reanalysis using zation. We addressed this by calculating AD as the hyperbolic improved leaf models and a new canopy integration scheme. Remote Sensing of the Environment 42, 187–216. minimum (Eqn A3) of the two limiting rates, which ‘smooths’ Terashima I. & Evans J.R. (1988) Effects of light and nitrogen nutrition on the the transition: organization of the photosynthetic apparatus in spinach. Plant and Cell Physiology 29, 143–155. AAADDvDjA= minh {},,θ , (A3) Terashima I. & Hikosaka K. (1995a) Comparative ecophysiology of leaf and canopy photosynthesis. Plant, Cell & Environment 18, 1111–1128. 2 Terashima I. & Hikosaka K. (1995b) Comparative ecophysiology of leaf and where minh{x,y,q} is deﬁned as the lesser root z of qz – canopy photosynthesis. Plant, Cell & Environment 18, 1111–1128. (x + y)z + xy = 0 and q < 1. [Note that Eqns A1–A3 are iden- Terashima I. & Inoue Y. (1984) Comparative photosynthetic properties of tical to the original model of Farquhar et al. (1980) when cc is palisade tissue chloroplasts and spongy tissue chloroplasts of Camellia japonica L.: functional adjustment of photosynthetic apparatus to light envi- above the CO2 compensation point, but not below. This dis- ronment within a leaf. Plant and Cell Physiology 25, 555–563. tinction is immaterial in our model because it only applies Warren C.R. (2007) Stand aside stomata, another actor deserves centre stage: when A < 0, a situation that cannot occur in our model the forgotten role of the internal conductance to CO2 transfer. Journal of because it would cause ∂A/∂E to become negative. Assimila- Experimental Botany 59, 1475–1487. Warren C.R. & Adams M.A. (2006) Internal conductance does not scale with tion is also limited by CO2 diffusion from the atmosphere to photosynthetic capacity: implications for carbon isotope discrimination and the sites of carboxylation (As): the economics of water and nitrogen use in photosynthesis. Plant, Cell & Environment 29, 192–201. ccac− Yoder B.J., Ryan M.G., Waring R.H., Schoettle A.W. & Kaufmann M.R. (1994) A = , (A4) s −−−111++ Evidence of reduced photosynthetic rates in old trees. Forest Science 40, gggbc sc m 513–527. where ca is the ambient CO2 concentration, gbc is the bound- Received 5 November 2012; accepted for publication 1 March 2013 ary layer conductance to CO2 (discussed below under the

section Leaf temperature), gsc is the stomatal conductance to

APPENDIX CO2 (gsc = 1.6·gsw, where gsw is the stomatal conductance to H O (discussed below under the section Numerical optimi- We used a process-based model of leaf gas exchange, in which 2 zation procedures)) and g is mesophyll conductance to CO key resource-dependent physiological parameters are deter- m 2 (discussed below under the section Mesophyll conductance). mined for each leaf by numerical optimization.These param- The actual value of A is then the intersection of A and A : eters are maximum RuBP carboxylation velocity at 25 °C D s

(Vm25), maximum potential electron transport rate at 25 °C AA=∩sD A. (A5) (Jm25), leaf absorptance to photosynthetically active light (a) and the diurnal time course of stomatal conductance to water Leaf transpiration rate, E, was computed as vapour [gsw(t)] (variables and symbols are deﬁned in Table 1). ()()() Egw=−tw sl w a105−+. w sl w a (A6) Leaf gas exchange model (von Caemmerer & Farquhar 1981), where gtw is the total In the biochemical photosynthesis model of Farquhar et al. -1 conductance to water vapour, wsl (mol mol ) is the saturation (1980), the net assimilation rate (A) at any given chloroplas- water vapour mole fraction at the leaf temperature Tl, and wa tic CO2 concentration (cc) is the lesser of two rates: a RuBP is the water vapour mole fraction of the air. gtw = gsw·gbw/ carboxylation-limited rate (ADv) and a RuBP regeneration- (gsw + gbw), where gsw and gbw are stomatal and boundary layer limited rate (ADj): conductances to water vapour, respectively. gsw was numeri- cally optimized (see the section Numerical optimization pro- Vcmc( − Γ ) A = * − R (A1) cedures); computation of gbw is discussed under the section Dv ++()d cKcc1 OK o Leaf temperature. and Modelling potential electron transport rate and − Γ J (cc ) light capture A = * − R , Dj + Γ d (A2) 42(cc *) Potential electron transport rate, J, has traditionally been modelled as a saturating function of total irradiance, I, and where Vm is the maximum RuBP carboxylation velocity, G* is maximum potential electron transport rate, Jm. A common the photorespiratory CO2 compensation point, Kc is the form for this relationship is the hyperbolic minimum of Jm and

Michaelis constant for RuBP carboxylation, O is the concen- Ji: J = minh{Jm, Ji, qj}, where minh{x, y, q} is deﬁned in the text tration of O2, Ko is the Michaelis constant for RuBP oxygena- below Eqn A3, Ji = aI(1 - F)/2, a is the leaf absorptance, F tion, Rd is the rate of non-photorespiratory CO2 release in (0.23; Farquhar & Wong 1984) is the fraction of photons that the light and J is the potential electron transport rate. are absorbed but do not lead to photo-oxidation of water, the

We assumed Rd25 = 0.0089·Vm25 (de Pury & Farquhar 1997). 1/2 factor accounts for the two photosystems and qj < 1isa

The simple minimum of ADv and ADj is not differentiable curvature factor. Because this model does not distinguish at the transition between carboxylation and regeneration between photons received at the upper and lower surfaces of © 2013 John Wiley & Sons Ltd, Plant, Cell and Environment, 36, 1547–1563 Crown nitrogen and water optimization 1561 broad leaves, nor between leaves with different transdermal calculated separately for each side of each leaf. This was profiles of photosynthetic capacity, it precludes rigorous repeated at different solar angles (in 3 degree intervals). The assessment of the roles of light capture and varying transder- actual irradiance at each leaf surface were calculated as the mal irradiance regime in N optimization. We therefore used product of the above crown beam and diffuse incoming irra- the J model of Buckley & Farquhar (2004), which treats upper diance and the relative beam and diffuse leaf irradiance. and lower surface irradiances separately and explicitly simu- The photosynthesis model used total upper and lower lates the transdermal profile of electron transport capacity as surface irradiances (Iu = Idu + Ibu, Il = Idl + Ibl), where d and b a weighted average of exponential profiles declining from denote diffuse and beam irradiances, respectively, and u and either leaf surface.In this study,we assumed that the weighting l denote upper and lower leaf surfaces, respectively. These for the upper surface (wu) is equal to the time-averaged irradiances were computed as fractions of above-crown fraction of total leaf irradiance received at that surface (the values (Iod, Iob) by ray-tracing, as described earlier. The weight for the lower surface is the complement of wu,1- wu). above-crown diffuse and beam irradiances were computed as

This captures the tendency for the transdermal capacity Iod = fdIog and Iob = Iog – Iod, respectively,where Iog is the global proﬁle to adapt to irradiance regime (Ögren & Evans 1993). irradiance (photosynthetic photon ﬂux density) and fd is the

The model is diffuse fraction of daily above-crown global irradiance. fd was computed following Roderick (1999): JJJJJ=−+minh {}ismsj,,,θ (A7) ⎧ YKX≤ ⎪ 00 where Js is related to the amount of excess light absorbed by fd = ⎨AAKifXKX01+<≤ 0 1 (A9) light-saturated transdermal layers when some other layers are ⎪ ⎩ YKX11> light-limited, and is given by Js = max{0, Jsu, Jsl}, where 2 where X0 = 0.26, X1 = 0.80–0.0017|lat| + 0.000044·lat (lat is α − ⎛ α − ⎞2 latitude in degrees; lat =-30° for the simulations shown here), = FIuum w J− τ FI uum w J Jsu ()⎜1 ⎟ () Y0 = 0.96, Y1 = 0.05, A0 = Y1 - A1X1, A1 = (Y1 – Y0)/(X1 – X0), 1−τ ⎝ αFIlum−−1 w J ⎠ (A8) and K is the ratio of daily global irradiance to that above the α −−−()⎛ α −−()⎞2 = FIl 1 wJum− τ FI l1 wJ um atmosphere, given by Jsl ()⎜1 ⎟ 1−τ ⎝ αFIuum− w J ⎠ Kp=+023.. 050 sun (A10) and I and I are the irradiances at the upper and lower leaf u l (Roderick 1999). At each point in time, we computed gas surfaces, respectively, t is leaf transmissivity to non-reflected exchange under two conditions – sunny and cloudy – and light and a is the leaf absorptance. a is calculated from leaf weighted the resulting values by p (the ratio of sunshine chlorophyll content (Eqn A17), which, in turn, depends sun hours to total daytime hours, or the instantaneous probability on N allocation, as discussed below. t is calculated as of sunshine) and its complement, respectively. Under cloudy exp(-3.9·[Chl]), where [Chl] is the leaf chlorophyll content conditions,I = 0.23I and under sunny conditions I = 0.73I , (mmol m-2) (Eqn A18 below). Full derivation of this model is og o og o where I is the global irradiance (photosynthetic photon given in Buckley & Farquhar (2004). o flux density) above the atmosphere. Io equals Io,maxsin b, where I (mmol m-2 s-1) = 2413[1 + 0.033 cos(2pJ /365)] Modelling the crown light environment o,max day (Leuning et al. 1995), Jday is day of year (January 1 = 0; Jday = 0 The light regime of an artificial, horizontally uniform canopy for the simulations shown here) and b is the solar elevation composed of flat circular leaves was estimated using ray- angle. tracing. This allows direct numerical simulation of irradiance Because crown light penetration was computed at equal at the leaf scale,by accounting for the spatial-angular arrange- intervals of b, time (t, hours) was computed from b ment and optical properties of individual leaves. The canopy rather than the reverse: t(b) = 12(1 – arccos((sinb - a)/b)), was built with 8000 randomly located Lambertian leaves, where a = sin(lat p/180)·sin d, b = cos(lat p/180)cos d and characterized by a spherical angular distribution. The crown d =-23.4(p/180)cos[2p(Jday + 10)/365] (Leuning et al. 1995; de height was 5 m, the leaf radius was 6 cm and the LAI was Pury & Farquhar 1997). Time step size (dt, hours) was esti- 4m2 m-2. In the computation of the leaf beam irradiance, the mated as the product of the solar angle step size (db = p/60) condition of penumbra generated by the partial obstruction and the derivative of t(b) with respect to b: of the solar disk was simulated according to Cescatti & Niinemets (2004). One thousand rays were fired from each δβtba=−−cos5 2 () sin β2 . (A11) leaf in the direction of the sun disk and the fraction of the sun visible from the leaf surface was computed. Un-intercepted Daily totals for irradiance and gas exchange variables were diffuse light irradiance was estimated by firing 1000 rays from then computed by summing the products of their respective the sky hemisphere to each leaf, assuming a standard overcast instantaneous values and dt at each value of b. sky distribution of the incoming radiation. Fluxes of radiation scattered by phytoelements and soil were computed assuming Leaf temperature a Lambertian behaviour of the surfaces. Finally, the relative The parameters Vm, Jm, G*, Kc, Ko and Rd were calculated beam and diffuse irradiance (including scattered fluxes) were from their values at 25 °C (indicated by the subscript ‘25’) © 2013 John Wiley & Sons Ltd, Plant, Cell and Environment, 36, 1547–1563 1562 T. N. Buckley et al.

using temperature responses given by June, Evans & 1.08gbh, and boundary layer conductance to CO2, gbc, was

Farquhar (2004) for Jm and by Bernacchi et al. (2001) for the 1.37gbw. other parameters. Leaf temperature (Tl) was determined using the following approximation, which arises by expand- Mesophyll conductance 4 4 ing Tl – Ta in terms of DT = (Tl – Ta) and ignoring second- order or higher terms in DT, and by approximating the The conductance to CO2 transfer from the intercellular saturation vapour pressure versus temperature curve with a spaces to the sites of carboxylation, termed mesophyll con- line segment (e.g. Peisker 1973; Jones 1992): ductance (gm), varies greatly among species and in relation to photosynthetic capacity (e.g. Warren 2007). Because the

physical nature of gm remains poorly understood, it is not yet Φεσ+−fTlgD()1 4 − TT=+ ir a aK tw air . (A12) possible to calculate gm from known biophysical parameters, la σ 3 ++ 42fTir a cglgs p bh tw so we speciﬁed gm by using two alternative assumptions. The

default assumption was that gm is proportional to Vm25, that is, that it scales linearly and homogeneously with photosyn- In Eqn A12, F is the shortwave radiation, fir is the fraction -2 -1 -2 -1 of infrared radiation from the sky visible at this point in thetic capacity: gm/(mol m s ) = 0.004Vm25/(mmol m s ). The slope of 0.004 was taken from the relationship among the crown, ea is the atmospheric IR emissivity, s (5.67 ¥ -8 -2 -1 -4 estimates of gm and Vm25 by Warren & Adams (2006). The 10 Jm s K ) is the Stefan–Boltzmann constant, TaK is the air temperature in Kelvins, l is the latent heat of vaporization, alternative assumption was that gm is constant and is inde- pendent of photosynthetic capacity; in that scenario, gm was Dair is the vapour mole fraction deﬁcit of the air,cp is the molar set at 0.34 mol m-2 s-1, the assimilation-weighted mean from heat capacity of air, gbh is boundary layer conductance to heat the default simulation. These two alternatives should be and s is the slope of the saturation H2O vapour mole fraction versus temperature curve. F (J m-2 s-1) was computed as viewed as limiting cases.

0.5666(Iu + Il), based on the ratio of extraterrestrial quantum flux (2413 mmol m-2 s-1) to energy flux (1367 J m-2 s-1) (de Numerical optimization procedures Pury & Farquhar 1997). To compute fir, we assumed that IR The outer N loop and the stomatal conductance optimization penetrated the crown with the same extinction coefficient loop both used the golden search method, initiated by triplet as diffuse visible irradiance, so fir (unitless) = (Idu + Idl)/Iod. ea bracketing of the solution (Press et al. 1992), to identify the 5 5 was computed as (10 wa/TaK) (Leuning et al. 1995); 10 con- values of leaf photosynthetic N or gsw for which ∂N/∂Ad and -1 verts units of wa (mol mol ) to Pa, assuming atmospheric ∂E/∂A equalled the imposed ‘target’ values (n and l, respec- -1 pressure of 100 kPa. l (J mol ) = 18.01·(2501 - Tl·0.002378). tively). The inner N loop, which partitioned total N among -1 Dair (mol mol ) = wsa – wa, where wsa is the saturation mole functional pools, used a Nelder–Mead downhill simplex to -1 -1 fraction at air temperature Ta. cp (J mol K ) = 29.25. The identify the optimal partitioning vector (fv, fj, fc, the fractions -1 -1 slope of ws versus T, s (mol mol K )atTa is wsa·0.0173·237.3/ of N invested in carboxylation, electron transport and light 2 -2 -1 (Ta + 237.3) . gbh (mol m s ) was computed as the sum of capture, respectively; see below for more details). Imple- terms representing forced (gbhw) and free convection (gbhf). menting a simplex algorithm in the finite space defined by The former is complementary allocation fractions (in this example, the space is the triangular plane fragment formed by bounding gvwbhw = 0., 123 (A13) the plane 1 = fv + fj + fc between 0 and 1 for each allocation fraction), is difficult because the simplex can jump out of this space. Thus, in our simulations, the simplex searched an infi- -1 where w is downwind leaf width (m) and v (m s ) is the wind nite two-dimensional space defined by log ratio transforma- speed, which we assumed declines exponentially in the crown tion of the allocation fractions; thus, x = ln(fv/fj) and y = ln(fv/ in the same fashion as diffuse irradiance, so v = fir·vo, where vo fc). x and y coordinates of simplex vertices in this space were is the wind speed above the crown. Conductance for free y y−x transformed back to fv, fj and fc as follows: fc = 1/(1 + e + e ), convection, gbhf,is y−x y fj = fce and fv = fce .

= ()κ ⋅×()83 − 025. gcwTTwbhf05.. p 16 10 l a , (A14) Computing photosynthetic parameters from N pools where k (J s-1 m-1 K-1) is the thermal conductivity of air Our treatment of functional nitrogen economy was derived -5 [0.026 + 5.6 ¥ 10 (Ta – 20)], and the quantity raised to the from that presented by Hikosaka & Terashima (1995), who 0.25 power at right is the Grashof number (dimensionless). considered ﬁve N pools or ‘groups’. Group I is N in Rubisco;

Because Tl depends on gbhf (cf. Eqns A12 and A14), we com- group II is N in the electron transport chain [excluding pho- puted an initial estimate of Tl using gbhf = 0, applied this to tosystems I and II (PSI and PSII, respectively)], coupling

Eqn A14 to compute gbhf and recalculated Tl with Eqn A12; factor and Calvin cycle enzymes other than Rubisco; group we then repeated this cycle one more time. Subsequent itera- III is N in the PSII core; group IV is N in the PSI core, tions produced negligible changes in leaf temperature.Bound- combined with light-harvesting complex I (LHCI); and group ary layer conductance to water vapour, gbw, was computed as V is N in light-harvesting complex II (LHCII). © 2013 John Wiley & Sons Ltd, Plant, Cell and Environment, 36, 1547–1563 Crown nitrogen and water optimization 1563

We re-organized these groups into three functional pools: which is solved for cj: pool ‘V’ (denoted as Nv) is simply group I from Hikosaka & χχ χ = 4aaf p II III Terashima (1995) and represents investment in carboxylation j . (A21) 056. ()aafχχ II+ p III capacity. Pool ‘J’ (denoted as Nj) is groups II and III combined, and represents investment in electron transport capac- From Hikosaka & Terashima (1995), cII and cIII are 9530 mol ity. Pool ‘C’ (denoted as Nc) is groups IV and V combined, N mol-1 cyt f and 5000 mol N mol-1 PSII, respectively, and represents investment in light capture. These pools are -1 -1 -1 and af and as are 19.9 mol s mol and 0.00174 mol mol , related to photosynthetic parameters as follows: - -1 -1 respectively, giving cj = 0.009478 mol e s mol Norcj = 9.478 mmol e- s-1 mmol-1 N. VNmvv25 = χ , (A15) (c) cjc and cc: Chlorophyll is associated with Hikosaka & Terashima’s (1995) groups III, IV and IV such that JNmjj25 = χ , (A16) [] Chl =++χχχcIIINNN III cIV IV cV V, (A22) and where ccIII, ccIV and ccV are the molar ratios of N/binding Chl α = [][]Chl() Chl+ 0., 076 (A17) in froups III, IV and V, respectively. In our treatment, Nc = NIV + NV. From above, NIII = NII(afcII/apcIII) and where Nj = NII(1 + afcII/apcIII). Thus, NIII = NjafcII/(afcII + apcIII), which applied to Eqn A22 gives [] Chl =+χχcjNN j c c. (A18) ⎛ χχ ⎞ [] af II cIII Chl = ⎜ ⎟NNj++χχ cIV IV cV N V. (A23) In Eqns A17 and A18, [Chl] is the leaf chlorophyll content ⎝ aafχχ II+ p III ⎠ and the c terms are nitrogen costing coefficients, as discussed below. The expression for leaf absorptance (a) was given by The first term in parentheses is thus ccj, from Eqn A18. From Evans (1998). The expression for [Chl] reflects the fact that Hikosaka & Terashima (1995), ccIII is 1/83.3, giving ccj = 4.64 ¥ 10-4 mol Chl mol-1 N. Inspection of Eqns A18 and A23 shows group III, which is part of Nj in our treatment, includes chlorophyll binding proteins that are covalently bound to the () χχχcIVVNN+ =+ cIVIVcVV N N, (A24) PSII core. Thus, allocation to electron transport capacity and light capture are not fully separable. We estimated the nitro- which can be rearranged to give cc as gen costing coefficients (cv, cj, cjc, cc) as follows. χχ+ () -1 cIV cVNN V IV (a) cv: There are 6290 mol N mol Rubisco molecules χ = . c + () (A25) (Hikosaka & Terashima 1995) and 8 active sites per mol- 1 NNVIV ecule. Assuming a turnover rate of 3.53 s-1 (von Caemmerer Thus, cc is a function of NV/NIV. This equation can be solved -1 -1 et al. 1994), this gives (3.53 mol CO2 s site ) (8 mol sites- for the ratio NV/NIV: mol-1 Rubisco)/(6290 mol N mol-1 Rubisco) = 0.004490 mol -1 -1 -1 -1 [] CO2 s mol N, or cv = 4.49 mmol CO2 s mmol N. NV 1 ⎛ Chl Nj ⎞ = ⎜ −−χχcj cIV ⎟. (A26) (b) cj: In our treatment, Nj = NII + NIII, Hikosaka & NNIVχ cV⎝ IV NIV ⎠ Terashima (1995) gave functional dependences of maximum To simplify further, we adopt Hikosaka & Terashima’s (1995) photosynthetic rate (Pmax) on the concentrations of cytochrome f ([cyt f]) and PSII ([PSII]), which they took as rep- assumption that [Chl] is proportional to [PSI] (and thus to NIV). This leads to [PSII] = as[Chl], where as is a proportion- resentative of Groups II and III: Pmax = af[cyt f] = ap[PSII]. This implies a stoichiometric constraint between Groups II ality constant. By definition [PSII] = cIVNIV,socIVNIV = as[Chl], allowing [Chl]/NIV in Eqn A26 to be replaced with and III: [PSII] = (af/ap)[cyt f]. Denoting the N costs of groups cIV/as. Likewise, NIV in the denominator of the second term in II and III as cII and cIII, respectively, this yields (cIIINIII) = parentheses in Eqn A26 can be replaced with as[Chl]/cIV, and (af/ap)(cIINII)or(NIII/NII) = (af/ap)(cII/cIII). Thus, NII + NIII = Nj can be replaced with Jm/cj. This gives NII(1 + (af/ap)(cIII/cII)), or in terms of Nj, N 1 ⎛ χ ⎛ χ J ⎞ ⎞ NN=+ N = N()1 + aχχ a . (A19) V =−IV 1 cj m − χ . j II III II f II p III ⎜ ⎜ []⎟ cIV ⎟ (A27) NaIVχ cV ⎝ s ⎝ χ j Chl ⎠ ⎠

In Hikosaka & Terashima (1995), Pmax equalled 0.56 times the This shows that Nv/NIV depends on the ratio of Jm to leaf Chl rate of O evolution at saturating CO and light. The latter 2 2 content. Evans (1989) found that this ratio varied from 0.264 rate is J /4, so P = 0.56J /4. From above, P = a [cyt m max m max f to 0.356 mol mol-1, averaging 0.305. When this range for f] = afcIINII. Combining these expressions for Pmax and solving NV/NIV is applied to Eqn A25, using values for the other for N gives N = (0.56/4)J /(a c ). Replacing J with c N II II m f II m j j parameters in Eqn A27 given by Hikosaka & Terashima and applying this to Eqn A19 gives (1995) or calculated above, the resulting estimates for cc range from 0.03337 to 0.03416 and average 0.03384 mol Chl ⎛ χ ⎞ 056. afII mol-1 N.The extremes differ from the average by only Ϯ 2%, NNjjj=+χ ⎜1 ⎟, (A20) χ ⎝ χ ⎠ -1 4afII ap III so we simply assumed cc = 0.03384 mol Chl mol N.