Introduction

Accurate representation of photosynthetic capacity is critical for modelling the response of terrestrial ecosystems to environmental change1,2. Earth System models use the FvCB biochemical model3 to simulate responses of C3 photosynthesis to environment. The modelled instantaneous carbon-assimilation rate is limited either by Vcmax (μmol m–2 s–1), the maximum rate of carboxylation, or J, the light-dependent electron transport rate, which is asymptotic at high light towards Jmax (μmol m–2 s–1). Both assimilation rates depend on temperature and on the intercellular partial pressure of CO2 (Ci).

Application of the FvCB model3 requires knowledge of three ‘plant-determined’ quantities: Vcmax, Jmax and the ratio of Ci to the ambient partial pressure of CO2 (Ca). This ratio, here called χ, is regulated by stomata. Jmax and Vcmax are closely coordinated4,5. More data are available on Vcmax because it can be inferred from the light-saturated photosynthetic rate, which is commonly measured in the field6. Global models have to contend with the large observed variation (in time and space, and within and between species) of Vcmax. Data analyses have explored its relationship to leaf nutrients7,8,9 and environmental variables10,11. Until recently, however, most models have assigned constant values of Vcmax at standard temperature (conventionally 25 °C: thus Vcmax25) for each of a small number of plant functional types (PFTs), and allowed the temperature-dependent values to follow standard (instantaneous) equations of enzyme kinetics. Models also have to represent the plant-type and environmental dependencies of χ (ref. 12). Most models assign constant per-PFT values of parameters in one of the two widely used models for the response of stomatal conductance to vapour pressure deficit (D). However, these simplifications are not the best possible. Vcmax25 and χ commonly vary at least as much within as between PFTs; while χ has predicted (and observed) relationships to growth temperature (Tg) and to elevation above sea level (z) through its effect on atmospheric pressure, which are neglected in the standard models10.

One strand of recent research has accordingly focused on a search for universal responses to environment, applicable to all (C3) plants. Eco-evolutionary optimality hypotheses12,13,14,15 have been invoked in recent efforts to derive general principles for the prediction of plant traits and productivity10,11,16,17,18. The least-cost hypothesis12,19 proposes that investments in transpiration capacity (maintaining the water transport pathway) and Vcmax are balanced so that photosynthesis is achieved at the lowest total cost in maintenance respiration of leaves and stems. Within this framework, χ varies over a limited range, consistent with tight regulation of the balance between water loss and carbon gain12. The hypothesis predicts that χ should decline with increasing D, decreasing Tg and increasing z. Each of these predictions is quantitatively supported by global compilations of χ values inferred from stable carbon isotope measurements in leaves10,20,21 and wood22. The coordination hypothesis provides a framework to predict Vcmax from physical environmental variables: irradiance (photosynthetic photon flux density, PPFD) and temperature and CO2 (ref. 23). The ‘strong form’24 of this hypothesis states that carboxylation and electron transport are co-limiting under typical daytime growth conditions, so that neither is in excess. Vcmax25 is observed to increase with PPFD, D and z (refs. 10,11,21), and to decline with Tg (refs. 24,25). The coordination hypothesis predicts all these observations. The decline with Tg is predicted because less Rubisco (the key carboxylation enzyme) is required to support photosynthesis in warmer environments24. The increases with D and z are predicted because greater photosynthetic capacity is required to support a given rate of carbon assimilation at lower χ (ref. 26).

Positive relationships between photosynthetic capacities and leaf N (Narea)27,28 and leaf P (Parea)29,30,31,32 are also widely observed. Much leaf N is invested in Rubisco33,34,35,36. Leaf P is required inter alia for cell membranes, nucleic acid synthesis and for ATP and NADPH production9,37. The predictive power of relationships to Narea or Parea is often weak11,38,39,40; however, recent studies8,9 have proposed a framework in which Vcmax25 is constrained by the lesser of two functions, one related to Narea and the other to Parea. Leaf nutrient levels, in turn, may or may not reflect their availability in the soil. Narea can be related to soil pH (or fertility) but is not unambiguously related to soil N availability14, while Parea is related to both soil fertility and total soil P14,41.

Thus, there are two conflicting paradigms to explain worldwide variation in photosynthetic capacity. One emphasizes its predictability from climate, based on optimality principles. The other emphasizes its predictability from leaf nutrients. This second approach has been extended to embrace the assumption that leaf nutrients reflect soil nutrient availability—although this is not universally true42.

To help resolve this contradiction, we assembled a large global dataset of Vcmax25, Narea and Parea data from multiple species and sites. In situ soil measurements (pH, C:N ratio and total P) were available at a subset of the sites. Rather than total soil N, which mainly relates to soil organic content, we used soil C:N as an inverse measure of N availability43. We hypothesized that

  1. (1)

    Photosynthetic capacity is subject to first-order control by climate, as predicted by the coordination and least-cost hypotheses. Vcmax25 increases in proportion to PPFD and increases towards colder and drier environments, due to greater biochemical investment required when χ is low.

  2. (2)

    Photosynthetic capacity is reduced, compared to climate-based predictions, under conditions of low nutrient (N and/or P) availability.

Results

Theoretically predicted values (see ‘Methods’) of the derivatives of ln Vcmax25 against ln PPFD, Tg and ln D are given in Table 1, for comparison with values fitted by statistical models (Table 1, Fig. 1). The value of 1 for the derivative of ln Vcmax25 with respect to ln PPFD implies proportionality, i.e. a 10% increase in PPFD induces a 10% increase in Vcmax25. The value of –0.05 K–1 for the derivative of ln Vcmax25 with respect to Tg implies that a 1 °C increase in growth temperature is predicted to induce a 5% decrease in Vcmax25. Regression coefficients of Vcmax25 against the same climate variables were statistically indistinguishable from theoretically predicted values (Table 1). Analysis of site-mean data explained more variance than a mixed-model analysis of all species (see ‘Methods’), indicating that a greater fraction of variation in photosynthetic capacity can be explained by physical environmental constraints when considering the whole community together, excluding variation within the community. The response of Vcmax25 to D was slightly steeper in the ‘observed’ than the ‘theoretical’ relationship, but the difference was within one standard error. From the random term of the all-species mixed model (see ‘Methods’), species and site identity separately accounted for 22 and 50% of the variation in Vcmax25 that was unexplained by the model’s climate variables (Table S1).

Table 1 Summary statistics for the climatic dependencies of Vcmax25 (μmol m–2 s–1).
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Fig. 1: Partial residual plots for Vcmax25 in relation to climate variables.
figure 1

Partial residual plots for log-transformed Vcmax25: all-species (a, b, c) and site-means (d, e, f). Coefficients and standard errors for the fitted lines are given in Supporting Information Table S4.

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No significant bias was shown for the predicted relationship of Vcmax25 to PPFD, Tg or D (Fig. 2). There was a possible underestimation of Vcmax25 at higher D, but this trend was not significant either in all-species (Fig. 2c; p = 0.12) or site-mean (Fig. 2f; p = 0.09) analyses.

Fig. 2: Partial residual plots for the model bias of theoretically predicted Vcmax25 values in relation to climate variables.
figure 2

Partial residual plots for the model bias of theoretically predicted Vcmax25 values in relation to climate variables: all-species (a, b, c) and site means (d, e, f). Coefficients and standard errors for the fitted lines are given in Supporting Information Table S4.

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Statistical models of photosynthetic capacity (all species and site means) as a function of climate overestimated Vcmax25 in low-P leaves and underestimated Vcmax25 in high-P leaves (Fig. 3b, d). The all-species statistical model also showed a bias in Vcmax25 related to leaf N (Fig. 3a). This relationship was still apparent (p < 0.0001) after removal of three highly influential points. The three species with extremely low Narea values (Turpinia pomifera, Uncaria laevigata and Walsura pinnata) shown in Fig. 3a were sampled in Yunnan, China (21.6°N, 101.5°E). These species possessed very low Vcmax25 (21 μmol m–2 s–1) values, probably a consequence of growth in deep shade. In contrast to the all-species model, the site-mean model showed no bias with respect to Narea (Fig. 3c).

Fig. 3: Partial residual plots for the model bias of statistically fitted Vcmax25 in relation to leaf nutrients.
figure 3

Partial residual plots for the model bias of statistically fitted Vcmax25 (Table 1) in relation to leaf nutrients, for all-species (a, b) and site-mean (c, d) data. The model bias represents the difference between predicted and observed Vcmax25, where the predicted Vcmax25 was based on the climate-driven regressions fitted from site-mean and all-species data as shown in Table 1. Coefficients and standard errors for the fitted lines are given in Supporting Information Table S4.

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Analysis of the subset of the data with in situ soil measurements indicated that Parea increased with soil C:N ratio, total soil P and soil pH (Figs. 4d–f, S1d–f). Narea increased with soil P (Figs. 4b, S1b) and decreased with soil pH (Figs. 4c, S1c). No relationship was found between leaf N and soil C:N ratio (Figs. 4a, S1a).

Fig. 4: Partial residual plots for leaf nutrients in relation to in situ measured soil properties.
figure 4

Partial residual plots for leaf nutrients (site means) in relation to in situ measured soil properties, for Narea (a, b, c) and Parea (d, e, f) data. Analyses for all species are shown in Fig. S1. Coefficients and standard errors for the fitted lines are given in Supporting Information Table S4.

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Global Vcmax25 could, alternatively, be represented by a minimum function (Eq. 12) of Narea and Parea. This function provided a better fit to the data than linear regression models with Vcmax25 as a function of Narea and Parea and combinations thereof or a model including Narea, Parea and their interaction (Table S2). In site-mean analysis based on the minimum function Parea was shown to be the principal limiting factor (93% of sites). In all-species analysis, Narea was shown to be the principal limiting factor (86% of species; Fig. 5). This contrast agrees with our findings for model bias: Vcmax25 variations within sites are more related to leaf N, while variations between sites (community means) are related to mean leaf P but not to mean leaf N. However, the goodness of fit of these models based on nutrients alone (R2 = 0.05, 0.12 for all species and site mean, respectively) was inferior to that of models based on climate alone (R2 = 0.17, 0.31).

Fig. 5: Visualizing the co-limitation of Vcmax25 by Narea and Parea based on the minimum function model.
figure 5

Visualizing the co-limitation of Vcmax25 by Narea and Parea for global (a) site-mean and (b) all-species analyses, based on the minimum function model. Following Domingues et al.8, blue points represent cases where Parea was the ‘limiting’ nutrient; red points represent cases where Narea was the ‘limiting’ nutrient. The fitted regression line in (a) is Narea = ln (5.96 Parea + 2.01) and in (b) is Narea = ln (158.62 Parea + 0.11).

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Discussion

The optimality framework accounts for the major global patterns of photosynthetic capacity as shown in our dataset. Consistent with hypothesis (1), global patterns of Vcmax25 were found to be predictable to first order from PPFD, growth temperature and vapour pressure deficit. Proportionality to PPFD is consistent with observations on light gradients44, seasonal dynamics45 and the cloud immersion effect, which decreases PPFD and Vcmax25 at mid elevations of tropical mountains39. Vcmax25 was predicted (and found) to be greater in drier environments: consistent with the larger biochemical investment required to achieve optimal photosynthesis when stomata are more closed. We found a somewhat steeper than predicted response to D and thus a slight but non-significant underestimation of Vcmax25 at higher D. This might be because the least-cost hypothesis does not consider the compounding effect of low soil moisture, which often accompanies high D and further decreases stomatal conductance, therefore preventing excessive transpiration but increasing the investment in carboxylation capacity22,46,47. In short-term drying experiments, Vcmax25 typically declines steeply (at different critical pre-dawn water potential values dependent on species48,49), although an increase in leaf-level Vcmax25—which may be accompanied by a reduction in leaf area—can be observed when plants are allowed to acclimate to moderate drought50,51,52,53,54,55. These findings are consistent with the expectation56 that a decrease of Vcmax under drought conditions is linked to a declining hydraulic capacity of the soil–root–xylem system, which can be accommodated over time by leaf shedding. Vcmax25 showed a negative response to growth temperature, which is predicted because greater investment in photosynthetic enzymes is required at lower temperatures to produce the same catalytic activity10,55. Thermal acclimation according to this optimality principle is supported by evidence for a decline of light use efficiency57 and an increase of photosynthetic nitrogen use efficiency58 towards warmer environments, and by increased Vcmax25 at higher elevations21,41. The percentage variance explained by these relationships is modest, however (31% for site-mean data: Table 1), consistent with findings by van der Plas et al.59 on the limits to predictability of ecosystem function from plant traits.

Our hypothesis (2) is partially supported by the analysis of bias in the statistically fitted model. Consistent with findings by Maire et al.14, we showed an overestimation of Vcmax25 in leaves with low Parea. These are typical of sites on acid soils and/or low soil P availability, including some wet tropical forests14,39. Many tropical soils are characterized by low total soil P due to long-term weathering60,61, and a dependency of net primary production on P availability has been shown in tropical forests62. Small-scale experimental studies have also suggested that low soil P availability can decrease the light-saturated photosynthetic rate (Asat)63,64,65 and Vcmax66,67. Adaptation strategies to cope with long-term P deficiency include restricting export of triose phosphate to the cytosol68, preventing the phosphorylation of ADP to ATP37,69, phosphate recycling during photorespiration70 and the replacement of phospholipids by galactolipids and sulpholipids71,72, all potentially entailing additional costs to the plant. On the other hand, photosynthesis in tropical forests is typically not limited by N73.

The global relationship between Vcmax25 and Narea74 primarily reflects the large amount of N invested in Rubisco and other photosynthetic enzymes75. Leaves with a high photosynthetic capacity necessarily have a large N content per unit area. Within vegetation canopies, Vcmax25 and Narea both vary greatly, especially along the light gradient from the canopy top to the understory—as shown in many empirical studies23,35,76,77 and further discussed elsewhere78,79. Our data provide no information on the range of light environments within sites and, therefore, our finding of a bias (low-N leaves having lower than predicted Vcmax25) in the all-species analysis is no surprise. However, the relationship disappeared in the site-mean analysis, indicating that Vcmax25 at community level is predictable without the need to consider leaf N. Moreover, we found no support for the hypothesis (assumed in some ecosystem and Earth System models) that leaf N is determined by soil N availability—suggesting that the metabolic component of leaf N is determined by photosynthetic capacity, as proposed by Dong et al.28, rather than vice versa. We did however find that leaf N increases with soil P, which is consistent with the observed effect of soil P on photosynthetic capacity.

A limitation of our analysis is its implicit assumption that mesophyll conductance (gm) is not limiting to photosynthesis. Vcmax as estimated here, therefore, is an ‘apparent’ value and likely to underestimate the true photosynthetic capacity by a variable amount, which cannot be predicted from data currently available at a large scale. However, this simplification reflects the situation in the great majority of ecosystem models, and it has been indicated that ‘greater process knowledge of gm will be required before it can be included [in models]’ (ref. 80, p. 26). A more comprehensive understanding of the relationships between leaf nutrients and photosynthesis will depend on advances in understanding the anatomical and physiological controls of gm (refs. 81,82), and extensions of leaf-level optimality theory to consider these controls.

In conclusion, while the short-term control of photosynthesis is relatively well understood (and modelled), the longer-term control of photosynthetic capacity is different, and subject to conflicting interpretations. Our findings show that the first-order climatic controls of Vcmax25 are relatively strong and predictable, indicating that models must account for them. Our results are not consistent with the model assumption that soil N availability controls leaf N, which in turn controls Vcmax25. They are, however, consistent with previous observational and experimental results indicating the existence of P limitation on leaf P, leaf N and Vcmax25.

Methods

During photosynthesis, Ci declines relative to Ca because C assimilation removes CO2 from the intercellular spaces while the stomata impose a resistance to the diffusion of CO2 into the leaf from the air. The Ci/Ca ratio (χ) is maintained within a limited range (about 0.5–0.9 in C3 plants) that is determined by the growth environment83. According to the least-cost hypothesis12,19, χ is controlled by stomata in such a way as to minimize the sum of the unit costs of the required capacities for transpiration and carboxylation. A consequence of this hypothesis is that for any given set of environmental conditions, there is an optimal value of χ10,12

$${\upchi}_{{\mathrm{opt}}} = \frac{{{\Gamma} \ast }}{{C_a}} + \frac{{( {1-\frac{{{\Gamma} \ast }}{{C_a}}} )\xi }}{{\xi + \sqrt D }},{\mathrm{where}}\;\xi = \sqrt {\left[ {\frac{{\beta \left( {K + {\Gamma} \ast } \right)}}{{1.6\;\eta \ast }}} \right]}$$
(1)

that satisfies the least-cost criterion. Here, Γ* is the photorespiratory compensation point, i.e. the value of Ci at which gross photosynthesis is zero; Κ is the effective Michaelis–Menten coefficient of Rubisco (Pa); D is the leaf-to-air vapour pressure deficit (Pa); η* is the viscosity of water relative to its value at 25 °C and β is the ratio of the unit costs of maintaining carboxylation and transpiration activities at 25 °C, estimated as 146 based on a global compilation of leaf stable carbon isotope measurements10. K is given by

$$K = K_{\mathrm{C}}\left( {1 + O/K_{\mathrm{O}}} \right)$$
(2)

where KC and KO are the Michaelis–Menten coefficients of Rubisco for CO2 and O2, respectively (Pa, reflecting the twin affinities of Rubisco), and O is the partial pressure of O2 (Pa). Γ*, ΚC and KO are functions of temperature, which we apply based on in vivo measurements on tobacco plants84. Γ*, Ca and O also vary with elevation, in direct proportion to atmospheric pressure.

The coordination hypothesis states that under typical daytime growth conditions photosynthesis is co-limited by carboxylation and electron transport. Optimal Vcmax is calculated as

$$V_{{\mathrm{cmax}},\;{\mathrm{opt}}} = \varphi _0I_{{\mathrm{abs}}}[ {( {C_{\mathrm{i}} + K} )/( {C_{\mathrm{i}} + 2{\Gamma}^ \ast } )} ]$$
(3)

where φ0 is the intrinsic quantum efficiency of photosynthesis (mol C mol–1 photons); Iabs is the PPFD absorbed by the leaf (μmol photons m–2 s–1). These values were corrected to 25°C using the Arrhenius equation with activation energies from Bernacchi et al.84,85. Intrinsic quantum efficiency was assumed to follow the temperature dependency of electron transport in light-adapted leaves85

$$\varphi _0 = (0.352 + 0.021\;T_{\mathrm{g}}-3.4 \times 10^{-4}T_g^2)/8$$
(4)

According to Eq. (3) and its derivatives, optimal Vcmax increases in proportion to PPFD. It also increases with Tg. On the other hand, optimal Vcmax25 declines with Tg. This is because the enzyme-kinetic effect, leading to a reduced Vcmax25 requirement at higher temperatures (caused by the temperature dependency of Rubisco activity), is stronger than the photorespiratory effect, leading to an increased Vcmax requirement at higher temperatures (caused by the temperature dependencies of K and Γ*). Experimental manipulations of growth temperature86, repeated measurements on the same plants at different seasons24, global spatial patterns of Vcmax11 and variations of Vcmax25 on a long elevation transect41 are all consistent with the negative temperature dependency of Vcmax25 implied by Eq. (3).

Quantitative predictions of the effect of each climate variable on ln Vcmax25 can be obtained by taking partial derivatives of Eq. (3) with respect to each variable in turn21. Logarithmic transformation is appropriate for magnitude variables described by multiplicative expressions like these87. The theory predicts approximately linear relationships of ln Vcmax25 to ln PPFD, ln D and (without transformation) Tg21. These derivatives were evaluated at the median climate of the dataset (PPFD = 400 μmol m–2 s–1, Tg = 25°C, D = 0.60 kPa) using the deriv package in R (ref. 88) (Table 1).

Photosynthetic data

The leaf-trait dataset comprised measurements at 266 sites for a total of 1637 species and 5000 individuals, and soil measurements for 39% of sites (Fig. S2). The dataset consists of field measurements made in natural (unfertilized) vegetation, from several published data sources7,8,14,20,28,73,89,90,91,92,93,94. The numbers of species recorded within each PFT (ref. 95) are provided in Table S3. Vcmax values were derived either from CO2 response (A–Ci) curves (94% of the dataset) or the one-point method6 from single measurements of light-saturated net photosynthesis (Asat) (6% of the dataset). The one-point method provides a way to estimate Vcmax knowing only Asat, day respiration (Rd), temperature and atmospheric pressure

$$V_{{\mathrm{cmax}}}\left[ {{\mathrm{est}}} \right] \approx \left( {A_{{\mathrm{sat}}} + R_{\mathrm{d}}} \right)\left( {C_{\mathrm{i}} + K} \right)/( {C_{\mathrm{i}}-{\Gamma}^ \ast } ).$$
(5)

If no respiration measurement was available, the following approximation was used instead

$$V_{{\mathrm{cmax}}}\left[ {{\mathrm{est}}} \right] \approx A_{{\mathrm{sat}}}/[ {( {C_{\mathrm{i}}-{\Gamma}^ \ast } )/( {C_{\mathrm{i}} + K} )-0.015} ]$$
(6)

where Rd is assumed to be 1.5% of Vcmax6,40,96. Rogers et al.97 indicated that the one-point method could result in a twofold underestimation of photosynthetic capacity in the Arctic region. Burnett et al.98 however estimated errors in photosynthetic capacity at around 20% at most, suggesting that Vcmax data obtained in this way (which, in any case, constitute only a small fraction of the dataset) can be justified in the context of a global survey. If measurements were made at a temperature other than 25 °C, reported Vcmax and Jmax values were standardized to 25°C using activation energies provided by Bernacchi et al.84,85.

Climate data

Monthly average values of mean daily maximum (Tmax, °C) and minimum (Tmin, °C) temperatures were extracted at the 0.5° grid location of each site from Climate Research Unit data (CRU TS 4.01)99, either for the measurement year or for the period 1991–2010 at sites not reporting measurement year. These data were three-dimensionally interpolated to actual site locations (longitude, latitude, elevation) using Geographically Weighted Regression (GWR) in ArcGIS. Mean daytime air temperature (Tg) was estimated for each month by assuming the diurnal temperature cycle to follow a sine curve, with daylight hours determined by latitude and month

$$T_g = T_{{\mathrm{max}}}\left\{ {1/2 + \left( {1-x^2} \right)^{1/2}/2\;\cos ^{-1}x} \right\} + T_{\min }\left\{ {1/2-\left( {1-x^2} \right)^{1/2}/2\;\cos ^{-1}x} \right\},\;x = -\tan \lambda \tan \;\delta$$
(7)

where λ is latitude and δ is the monthly average solar declination100. Monthly values of Tg were averaged over the thermal growing season, i.e. months with mean daily temperature > 0 °C.

Incident solar radiation data were derived from WATCH Forcing Data ERA-Interim101 at the same period and resolution, and also interpolated by GWR. Solar radiation (W m–2) was converted to PPFD by multiplication by the energy-to-flux conversion factor 2.04 (μmol J–1)102. PPFD was averaged across the thermal growing season. Mean atmospheric pressures (Patm) were derived using the barometric formula102,103. D (kPa) was estimated using the Magnus–Tetens formula46

$$D = e_{\mathrm{s}}-e_{\mathrm{a}},$$
(8)

with

$$e_{\mathrm{s}} = 0.611\;\exp \;[17.27\;T/(T + 237.3)],\;{\mathrm{where}}\;T = \left( {T_{\min } + T_{\max }} \right)/2$$
(9)

and

$$e_{\mathrm{a}} = \left[ {P_{{\mathrm{atm}}}W_{{\mathrm{air}}}R_{\mathrm{v}}} \right]/\left[ {R_{\mathrm{d}} + W_{{\mathrm{air}}}R_{\mathrm{v}}} \right]$$
(10)

where Wair is the mass mixing ratio of water vapour to dry air; Wair = qair / (1 – qair), where qair is the specific humidity (kg/kg) derived from WATCH Forcing Data ERA-Interim101, Rd and Rv are the specific gas constants of dry air and water vapour, Rd = R/Md and Rv = R/Mv, where R is the universal gas constant (8.314 J–1 K–1), Md is the molar mass of dry air (28.963 g mol–1) and Mv is the molar mass of water vapour (18.02 g mol–1).

Statistical analysis

The climate data were used to make theoretical predictions of relationships between photosynthetic capacity and climate variables based on the optimality framework, and independently, to derive statistical relationships by multiple regression (Tables S2 and S4). Separate statistical analyses were carried out for individual species, and for site-averaged measurements. In the analyses of individual species (i), each data-point represents the average of one or more measurements on a particular species at a site (n = 2513). In the analyses of site-averaged measurements (ii), each data-point represents an average for a site (across all individual and species; n = 266) (Table 1). Analyses of type (i) (‘all species’) data were carried out by means of a linear mixed effects model using the nlme package in R88. Climate variables (Tg, D, PPFD) were included as fixed terms, with site and species as random intercepts. A crossed rather than a fully nested random design was used because some species occurred at more than one site. Ordinary least squares multiple linear regression, using the lm function in R88, was used for analyses of type (ii) (‘site mean’) data. Regression relationships were visualized using partial residual plots, obtained with the visreg package in R88. Partial residual plots display the relationship between values of the response variable versus each predictor variable, after those responses have been adjusted to hold all other predictors constant at their median values in the dataset. Photosynthetic capacities, PPFD and D were natural log-transformed before analysis so that the resulting regression coefficients can be directly compared with theoretical predictions (Table 1).

Model data comparisons

Model bias (B, %) in Vcmax25 was calculated as follows:

$$B = 100\left( {V_{{\mathrm{cmax}}25}\left[ {{\mathrm{pred}}} \right]-V_{{\mathrm{cmax}}25}\left[ {{\mathrm{obs}}} \right]} \right)/V_{{\mathrm{cmax}}25}\left[ {{\mathrm{obs}}} \right]$$
(11)

where Vcmax25[pred] is a predicted value and Vcmax25[obs] an observed value. Using theoretically predicted values, we explored whether B was significantly related to the climate variables. If so, this would indicate that the true responses of Vcmax25 to climate variables were different from the predicted ones—pointing to something missing (or wrong) in the theory. Then, we explored whether bias in the values predicted by the statistical models (both all-species and site-mean models) was significantly related to leaf Narea and Parea. If found, such bias would indicate effects of leaf nutrients, additional to the effects of the climate variables considered.

Alternative models for the response to leaf nutrients

An alternative statistical model for photosynthetic capacity is a ‘minimum function’ of Narea or Parea8. The following differentiable equation is almost exactly equivalent to a minimum function (Fig. S3):

$$Z = -\left( {1/k} \right)\;\ln \;\left[ {e^{-kx} + e^{-ky}} \right]$$
(12)

where Z is the response variable (Vcmax25), x and y are the predictor variables (Narea, Parea) and k ≫ 1. Equation (12) is the ‘log-sum-exp’ formula, which provides a continuous approximation to the minimum function—allowing its use in regression, and comparison of goodness-of-fit statistics with ordinary linear regression (Table S2). The larger the value of k, the closer the approximation to the minimum function. A simple sensitivity analysis showed that large values of k (≥10) gave best performance (Table S5), indicating that the minimum function fitted the data better than a smooth transition between N and P limitation. Equation (12) was fitted to both all-species and site-mean data (Fig. 5). The equation was plotted using an iterative least squares procedure using the akima, stats and grDevices packages in R88.

Statistics and reproducibility

Data collection, formulae and statistical analyses are described in ‘Methods’. All statistical analyses used R software (ref. 88), applying ordinary linear regression for site-mean analysis and a mixed effects model for all-species analysis. All R packages applied are referenced in ‘Methods’. The relevant statistics for the main analyses are presented in Supplementary Information.

Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.