Towards a universal model for carbon dioxide uptake by plants

Transcription

1 SUPPLEMENTARY INFORMATION Letters DOI: /s In the format provided by the authors and unedited. Towards a universal model for carbon dioxide uptake by plants Han Wang 1,2,3 *, I. Colin Prentice 1,2,4, Trevor F. Keenan 2,5, Tyler W. Davis 4,6, Ian J. Wright 2, William K. Cornwell 7, Bradley J. Evans 2,8 and Changhui Peng 1,9 * 1 State Key Laboratory of Soil Erosion and Dryland Farming on the Loess Plateau, College of Forestry, Northwest A & F University, Yangling, Shaanxi, China. 2 Department of Biological Sciences, Macquarie University, North Ryde, NSW 2109, Australia. 3 Ecosystems Services and Management Program, International Institute for Applied Systems Analysis, A-2361 Laxenburg, Austria. 4 AXA Chair of Biosphere and Climate Impacts, Department of Life Sciences, Imperial College London, Silwood Park Campus, Buckhurst Road, Ascot SL5 7PY, UK. 5 Climate and Ecosystem Sciences Division, Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA. 6 United States Department of Agriculture-Agricultural Research Service, Robert W. Holley Center for Agriculture and Health, Ithaca, NY 14853, USA. 7 Ecology and Evolution Research Centre, School of Biological, Earth and Environmental Sciences, The University of New South Wales, Randwick NSW 2052, Australia. 8 Faculty of Agriculture and Environment, Department of Environmental Sciences, The University of Sydney, Sydney, NSW 2006, Australia. 9 Department of Biological Sciences, Institute of Environmental Sciences, University of Quebec at Montreal, C.P. 8888, Succ. Centre-Ville, Montréal, Québec H3C 3P8, Canada. * wanghan_sci@yahoo.com; peng.changhui@uqam.ca Nature Plants Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

2 1 2 1 The theoretical framework The standard model of C 3 plant photosynthesis In 1980, Farquhar, von Caemmerer and Berry proposed a model describing the biochemical processes of photosynthesis by C 3 plants which has underpinned research on the environmental regulation of photosynthesis for the past 37 years 1. Its starting point is the observation that the instantaneous rate of photosynthesis is limited either by the capacity of enzyme Rubisco for the carboxylation of RuBP or by the rate of electron transport for the regeneration of RuBP, which depends on the absorbed photosynthetic photon flux density (PPFD). The Rubisco-limited photosynthetic rate A C is given by: 9 A c = V cmax χc a Γ * χc a + K (1) where V cmax is the photosynthetic capacity for carboxylation, c a is the ambient partial pressure of CO 2, χ is the ratio of leaf-internal to ambient CO 2 partial pressures, Γ * is the CO 2 compensation point in the absence of mitochondrial respiration, and K is the effective Michaelis-Menten coefficient of Rubisco. Both Γ * and K are influenced by the partial pressure of oxygen. V cmax, Γ * and K are temperaturedependent, following Arrhenius kinetics. Recent modelling work has relied on the in vivo determinations of their activation energies by Bernacchi et al. 2. The electron-transport limited photosynthetic rate A J is given by: 17 A J = ϕ 0 I abs χc a Γ * χc a + 2Γ * (2) for low PPFD, where I abs is the absorbed PPFD, and φ 0 is the intrinsic quantum efficiency of photosynthesis. With increasing PPFD, I abs must be substituted with a saturating function of the photosynthetic capacity for electron-transport capacity (J max ). Various empirical functions have been used for this purpose The product of χ and c a is the leaf-internal CO 2 partial pressure (c i ). The standard model assumes arbitrarily large mesophyll conductance (g m ), which is the liquid-phase conductance between the intercellular spaces and the chloroplasts. Although this is now known to be an over-simplification 3, most large-scale ecophysiological data analysis and modelling continues to work with the standard model. Thus, for the main analysis we follow common modelling practice in disregarding the finite (and variable) nature of g 4 m. We then extend the analysis to the case of finite g m (Supplementary Methods 2, 4) making the simplifying assumption that the ratio of stomatal conductance (g s ) to g m is independent of environmental factors 5-8. The conclusions from the main analysis are upheld in the finite g m -based analysis (Supplementary Table 2, Supplementary Figure 4). In particular, the theoretical value of 0.5 for the coefficient of ln D 0 (the vapor pressure deficit that would be obtained at sea level under the same H 2 O mole fraction and temperature) lies just outside the 95% confidence intervals of

3 33 34 the regression estimate, but remains close to the fitted central value of The other theoretical values lie within the relevant 95% confidence intervals According to the standard model, c i constrains both the Rubisco-limited and electron-transport limited photosynthesis rates. The actual photosynthetic rate is then given by the lesser of the two rates: 37 A = min {A C, A J } (3) A third potentially limiting rate, the triose phosphate utilization rate, can apply at high c a and/or low temperatures; this is disregarded here as it probably rarely applies under field conditions. 40 The light-use efficiency model The light-use efficiency (LUE) model has its origin in agricultural research, specifically the finding that gross primary production (GPP) of a crop in a given time interval is proportional to the total light (i.e. the total number of moles of photosynthetic photons, or the time-integral of PPFD) absorbed by the crop during that interval 9. A variety of generalized LUE models have been developed for global application, including the Simple Diagnostic Biosphere Model, SDBM 10, the Carnegie Ames Stanford Approach model, CASA 11,12, the Simple Diagnostic Photosynthesis and Respiration Model, SDPRM 13, the MODIS GPP model 14 and the up-scaled GPP product of Beer et al. 15, which is constrained by eddycovariance flux measurements. These models have in common that they are driven by satellite-derived data on green vegetation cover, which together with data on incident PPFD allow the estimation of absorbed PPFD. In the simplest approach adopted in SDBM, the actual value of LUE (the ratio of GPP to absorbed PPFD) is treated as constant. Many subsequent models, including MODIS GPP, have included functions that make LUE dependent on environmental influences and/or vegetation classes. But no general theoretical basis for these functions exists, nor does the LUE concept suggest any obvious route to the inclusion of effects of CO 2 on photosynthesis with the result that the MODIS and Beer et al. products, for example, predict no such effect 16, except to the extent that it is manifested by changes in green vegetation cover that are visible from space The connection between the Farquhar and LUE models is not obvious. Supplementary Equation (2) predicts that the electron-transport limited rate of photosynthesis is proportional to absorbed PPFD but this only applies at relatively low PPFD, and in any case, Supplementary Equation (1) for Rubiscolimited photosynthesis is expected to apply at high PPFD. The conundrum is this: how can GPP (the time-integral of photosynthesis) be proportional to absorbed light, which is the basis of the LUE model, if the response of photosynthesis to PPFD saturates? This question has surfaced occasionally in the literature, but not been fully resolved. Medlyn 17 reviewed some alternative explanations. 64 The co-ordination hypothesis One of the explanations in ref. 17 invokes the co-ordination (or co-limitation) hypothesis, which states that V cmax of leaves at any level in the canopy acclimates spatially and temporally to the prevailing daytime incident PPFD in such a way as to be neither in excess (entailing additional, futile

4 maintenance respiration), nor less than required for full exploitation of the available light. In other words, under typical daytime conditions when most photosynthesis takes place, A J = A c = A. This hypothesis also requires that J max maintain a ratio to V cmax such that strong limitation by J max is avoided. Evidence for the co-ordination hypothesis was presented by Haxeltine and Prentice 18, and Dewar 19, who noted that it can explain many otherwise unexplained responses of C 3 plants to environmental changes: including changes in leaf C:N ratios along environmental gradients, and the widely observed reduction of V cmax under experimentally increased atmospheric CO 2. More recently Maire et al. 20 showed very good agreement between typical daytime values of A J and A C as calculated under the prevailing growth conditions for 31 species (293 data points) based on published studies. A later section (Supplementary Methods 5) shows algebraically how a LUE model, applied at an approximately monthly timescale, can be compatible with the Farquhar model applied at an approximately hourly time scale and therefore, how values of LUE can be predicted consistently with the Farquhar model. 81 The Cowan-Farquhar optimality criterion and the least-cost hypothesis Missing from the Farquhar model is an equation to predict χ, which constrains both the Rubisco- and electron-transport limited rates of carbon fixation and therefore appears in both Supplementary Equations (1) and (2). The value of χ at any moment must be consistent both with the rate of carbon fixation and with the rate of diffusion of CO 2 through the stomata. Although the mechanism of stomatal control is still an active research topic, there is abundant evidence that χ is closely regulated to remain within a narrow range. Current Earth System Models that use the Farquhar model all include an empirical 'closure' that predicts either stomatal conductance or χ. The most commonly used closures are the one-parameter Ball-Berry equation 21 and the two-parameter Leuning equation 22 (or almost equivalently, the Jacobs closure 23 ). The predictions of both equations are formulated in terms of stomatal conductance, and include assimilation rate as a predictor: thereby allowing stomatal conductance to respond (as is observed) to light and temperature, via the assimilation rate. However, these equations can be re-written in terms of χ. It then emerges that χ is independent of the assimilation rate, responding only to humidity: relative humidity in the Ball-Berry model and vapour pressure deficit (vpd) in the Leuning model. And although superficially similar, these equations make predictions that differ in important ways. For example, Ball-Berry allows χ to approach unity as vpd tends to zero, whereas Leuning caps χ at a maximum value. Both equations are usually implemented with different parameter values for different plant functional types, but with no strong basis for the distinctions Cowan and Farquhar 24 proposed a different, theoretical approach based on optimality considerations. They hypothesized that stomatal behaviour maximizes marginal carbon gain (photosynthesis, A) while minimizing marginal water loss (transpiration, E), leading to the criterion E / A = λ where λ is a parameter representing the marginal carbon cost of water. They showed that this hypothesis can account for many observed features of stomatal behaviour. Medlyn et al. 25,26 found that the Cowan- Farquhar criterion can (to a good approximation) predict the existence of an optimal value of χ which is

5 a function of vpd and temperature similar in mathematical form to that proposed in ref. 27. Their prediction of a temperature effect, additional to the effect of temperature on vpd, was further supported by a global dataset of gas exchange measurements The difficulty of defining λ remains a problem in the approach of ref. 24. This was avoided by Prentice et al. 27 through application of the least-cost hypothesis, an idea first proposed explicitly by Wright et al. 29. According to the least-cost hypothesis, plants should minimize the combined costs (per unit of assimilation) of maintaining the required capacities for both carboxylation and transpiration. This criterion is both different from, and more explicit than, the Cowan-Farquhar criterion. If a and b are dimensionless cost factors for the maximum rates of water transport (E) and carbon fixation (V cmax ) respectively (i.e. a and b are maintenance respiration per unit E and V cmax ), the optimality criterion of ref. 27 is: 117 a. (E/A)/ χ + b. (V cmax /A)/ χ = 0 (4) 118 According to Fick s law of diffusion: 119 E = 1.6g s D (5) 120 where D is the vpd, and g s is the canopy stomatal conductance to CO Moreover, 121 A = g s c a (1 χ) (6) 122 Therefore, 123 E/A = 1.6(D/c a )/(1 χ) (7) while the coordination hypothesis (A J = A C = A, Supplementary Methods 1) allows Supplementary Equation (1) to be written as: 126 V cmax /A = (χc a +K)/(χc a Γ * ) (8) Initially neglecting Γ * for simplicity (i.e. assuming χc a >> Γ * ), substituting Supplementary Equations (7) and (8) in (4) and taking derivatives, the optimal value of χ satisfies: (aD/c a )/(1 χ) 2 bk / χ 2 c a = 0 (9) 130 The solution to Supplementary Equation (9) provides the required optimal value (χ o ): 131 χ o = ξ/(ξ+ D) (10) 132 where 133 ξ = (bk/1.6a) (11)

6 134 The parameter ξ expresses the sensitivity of χ o to D If the simplifying assumption χc a >> Γ * is omitted, the derivations in Supplementary Equation (4) are slightly changed, and the following more exact form (equation (8) in ref. 27) emerges: 137 χ o = Γ * /c a + (1 Γ * /c a ) ξ/(ξ+ D), where ξ = [b(k+γ * )/1.6a] (12) As only the ratio b/a (not the individual terms b and a) affects χ ο by either formula we will later use the composite parameter β to denote the value of b/a at 25 C. 140 Incorporating finite g m into the least-cost framework The total conductance (g) for CO 2 diffusing from the ambient atmosphere to the carboxylation reaction sites can be estimated from the two most important limiting conductances, g s and g m, in series: 143 g = g s g m / (g s + g m ) (13) Note that g m affects CO 2 diffusion for carboxylation, but not H 2 O diffusion during transpiration. Replacing stomatal with total conductance for carboxylation, Supplementary Equation (6) therefore becomes: 147 A = c a (1 χ c ) g s g m / (g s + g m ) (14) 148 and Supplementary Equations (7) and (8) become: 149 E/A = 1.6 (D/c a ) (g s + g m ) / [(1 χ c ) g m ] (15) 150 V cmax /A = (χ c c a + K)/(χ c c a Γ * ) (16) 151 where χ c is the ratio of the chloroplastic to ambient CO Similarly, by applying the optimality criterion: 153 a. (E/A c )/ χ c + b. (V cmax /A c )/ χ c = 0, (17) to Supplementary Equations (15) and (16), the optimal ratio of chloroplast to ambient CO 2 (χ co ) is given by (assuming χ c c a >> Γ * ): 156 χ co = ξ c /(ξ c + D), where ξ c = {bk/[1.6a(1+ g s /g m )]} = ξ / (1+ g s /g m ), (18) 157 and in the full version (relaxing the assumption χ c c a >> Γ * ): 158 χ co = Γ * /c a + (1 Γ * /c a ) ξ c /(ξ c + D), where ξ c = [b(k+γ * )/1.6a/(1+ g s /g m )] (19) Therefore, χ co is not influenced by g s and g m separately, but by their ratio. The form of the model for χ co resembles that for χ o, but the sensitivity parameter ξ is adjusted by a factor [1/(1+ g s /g m )].

7 Effects of temperature, humidity and elevation on the optimal ratio of leaf-internal to ambient CO 2 partial pressures (χ o ) Supplementary Equation (12) is expected to be more accurate than Supplementary Equations (10)-(11) at low c i. However, for analytical simplicity, we use Supplementary Equations (10)-(11) to derive the effects of temperature, vpd and elevation on χ o under field conditions The parameter a is expected to be influenced by the (temperature-dependent) viscosity of water 27 ; while b, the ratio of mitochondrial respiration to carboxylation capacity, is generally taken to be constant 1. Equation (11) of ref. 27 suggests that a also depends on plant properties, including plant height, the density and permeability of conducting tissues, the ratio of conducting tissue cross-sectional area to the leaf area, and the maximum water potential difference between soil and leaf which in turn depends on both soil moisture and plant strategy (e.g. isohydric versus anisohydric). We neglect these complexities in the current derivation for global-scale analysis, given the limited data availability on those associated plant traits and the still limited information on the controls of the variation in plant hydraulic traits In the model for χ co the ratio of g s to g m is assumed to be independent of environment. Even though both g s and g m vary with environmental conditions, including light, moisture and temperature, their covariation under a wide range of conditions supports this assumption at least as a first approximation 5. Moreover, data indicate that the value of g s /g m is quite conservative, with a median of about 1.4 (I.J. Wright, unublished data). The derivation of the environmental dependencies of χ co then follows the same logical steps as that for χ. To further refine the model of χ co (i.e. the sensitivity of χ co to environmental changes), a deep understanding of the mechanistic regulations of g s and g m is required. Moreover, particular attention should be paid to their relative changes along different environmental gradients. Nevertheless, our proposed model of χ co provides a framework for incorporating the effects of g m into land surface models via a theory-based way. 185 The effective Michaelis-Menten coefficient of Rubisco (K) is given by: 186 K = K c (1 + P o /K o ), (20) where K c and K o are the Michaelis-Menten coefficients of Rubisco for carboxylation and oxygenation, respectively, in partial pressure units, and P o is the partial pressure of O Applying logit transformation, logit (χ o ) = ln [χ o /(1 χ o )], to Supplementary Equations (10)-(11) yields a linearized model as follows: 191 ln [χ o /(1 χ o )] = ½ ln b ½ ln a + ½ ln K ½ ln D ½ ln 1.6 (21) The dependencies of a (through the viscosity η) and K (through K c and K o ) on temperature (T), and the dependency of K (through P o ) and D on elevation (z), are denoted by f 1 (T), f 2 (T), g 1 (z) and g 2 (z). Supplementary Equation (21) is then equivalent to:

8 195 ln [χ o /(1 χ o )] = ½ ln f 1 (T) + ½ ln f 2 (T) + ½ ln g 1 (z) ½ ln D 0 ½ ln g 2 (z) + C, (22) 196 where C = ½ (ln b ln a ref + ln K ref ln 1.6) = ½ (ln β + ln K ref ln 1.6) (23) and a ref and K ref are the values of a and K under standard conditions (T = 298 K, z = 0). β denotes the ratio of b to a ref. D 0 (the vpd that would be obtained at sea level under the same H 2 O mole fraction and temperature) in Supplementary Equation (22) is distinguished from D in Supplementary Equation (21) because the latter includes the effect due to atmospheric pressure, g 2 (z). Elevation increases the effective vpd because a parcel of air with a given water content, raised to a higher elevation, has a reduced vapour pressure while the saturation vapour pressure remains constant. Many other environmental factors commonly change with elevation, but the pressure effects need to be considered when estimating the partial effect of elevation on χ o, i.e. when other factors are held constant. Effects of elevational changes in temperature and/or vpd will be superimposed on the pressure effects The same reasoning applies in the model for χ o except that the term ξ is multiplied by [1/(1+ g s /g m )], which then leads to a different expression for the constant term C c : 208 ln [χ co /(1 χ co )] = ½ ln f 1 (T) + ½ ln f 2 (T) + ½ ln g 1 (z) ½ ln D 0 ½ ln g 2 (z) + C c, (24) 209 where C c = ½ [ln b ln a ref + ln K ref ln 1.6 ln(g s /g m + 1)] = ½ (ln β c + ln K ref ln 1.6) (25) Therefore, β c in Supplementary Equation (25) is the product of β in Supplementary Equation (23) and (g s /g m + 1) Moisture effect of D Based on Supplementary Equation (22), the predicted coefficient of ln D 0 is 0.5. Wang et al. 31, using the evapotranspiration deficit (ΔE, the difference between actual and potential evapotranspiration) as a proxy for D 0, indicated a coefficient of 0.25 instead of 0.5; however, we found that this apparent deviation from theory is due to saturation of the relationship between ΔE and D 0 at high values of D 0 (Supplementary Figure 8). 218 Temperature dependency of a The parameter a is directly proportional to η, according to equation (11) in ref. 27. The temperature dependency of η can be described by the Vogel equation: 221 η = 10 3 exp [A + B/(C + T)] (26) with the following fitted parameters A = 3.719, B = 580 and C = Thus, the sensitivity of η to temperature is given by: 224 (1/η) η/ T = ln η/ T = B/(C + T) 2 (27)

9 allowing the response of η to T, within the physiologically relevant range, to be well approximated by an exponential response to ΔΤ = T 298 K relative to a reference value at T = 298 K (η ref ): 227 η η ref exp [ B/(C + T) 2 ΔΤ] (28) 228 Therefore, f 1 (T) = exp [ B/(C + T) 2 ΔΤ ] (29) 229 At standard temperature (T = 298 K), B/(C + T) 2 = and 230 f 1 (T) = exp ( ΔΤ) (30) 231 Temperature dependency of K The Arrhenius relationship describing the response of a biochemical rate parameter (x) to temperature can be expressed as: 234 ln x/ T = (ΔH/R).(1/T 2 ) (31) where R = J mol 1 K 1 and the activation energies ΔH are kj mol 1 for K c and kj mol 1 for K o, denoted as ΔH c and ΔH o, respectively, from in vivo determinations 33. Therefore, the partial derivative of K with respect to temperature from Supplementary Equation (20), and the sensitivity of K to temperature, are given by: 239 K/ T = (K c /K o )[(ΔH c /R)(1/T 2 )(P o + K o ) (ΔH o /R)(1/T 2 )P o ] (32) 240 and 241 (1/K) K/ T = [(ΔH c /R)(1/T 2 ) (P o + K o ) (ΔH o /R)(1/T 2 ) P o ]/(P o + K o ) (33) 242 leading to: 243 f 2 (T) = exp([(δh c /R)(1/T 2 ) (P o + K o ) (ΔH o /R)(1/T 2 ) P o ]/(P o + K o ) ΔΤ) ( At T = 298 K and z = 0 the sensitivity of K c to temperature is given by (ΔH c /R)(1/T 2 ) = and of K o by (ΔH o /R)(1/T 2 ) = , with P o = Pa, K c = Pa and K o = Pa, yielding: 246 f 2 (T) = exp ( ΔΤ). (35) 247 Elevation dependency of K Using a standard approximation for the decline in atmospheric pressure with elevation 34, the partial pressure of O 2 can be expressed as a simple function of elevation: 250 P o = exp ( z) (36) 251 with z in km. The sensitivity of K to elevation can then be derived:

10 252 ln K/ z = ( ln P o / z)( ln K/ ln P o ) 253 = (P o /K)( ln P o / z)( K/ P o ) 254 = P o /(P o + K o ) (37) 255 Therefore, g 1 (z) = exp[ P o /(P o + K o )z] (38) 256 and at T = 298 K and z = 0, 257 g 1 (z) = exp( z). (39) 258 Elevation dependency of D 259 D can similarly be expressed as a function of elevation: 260 D = e s e a0 exp ( z) (40) where e s is the saturation vapour pressure and e a0 is the actual vapour pressure that would be obtained at sea level under the same H 2 O mole fraction and temperature. The dependency of D on elevation is then: 264 ln D/ z = e a0 exp ( 0.114z) / [e s e a0 exp ( 0.114z)] (41) Here exp ( 0.114z) can be taken as equal to unity, to a good approximation, within the relevant range of z, yielding the approximation: 267 ln D/ z = e a0 /D 0 = R 0 /(1 R 0 ), (42) 268 hence 269 g 2 (z) = exp {0.114 [ R 0 /(1 R 0 )] z} (43) where R 0 = e a0 /e s (relative humidity). Evaluating Supplementary Equation (43) at sea level and R 0 = 50% yields g 2 (z) exp (0.114z) (44) Note that this theoretically derived elevation effect on D varies strongly with R 0, approaching infinity as R 0 tends to Linearized expressions for χ o in terms of environmental predictors Substituting f 1 (T), f 2 (T), g 1 (z) and g s (z) in Supplementary Equation (22) with the derived relationships (Supplementary Equations (30), (35), (39) and (44)), we obtain:

11 278 ln [χ o /(1 χ o )] = ½ ( ) ΔΤ ½ ( ) z ½ ln D 0 + C 279 = ΔΤ z 0.5 ln D 0 + C (45) A value for the constant C (1.189) was obtained based on the global carbon isotope dataset (Supplementary Methods 3) by fitting a linear regression model with the coefficients of the predictors fixed according to Supplementary Equation (52), giving: 283 ln [χ o /(1 χ o )] = ΔΤ z 0.5 D (46) Supplementary Equation (46) predicts χ o = 0.77 for standard environmental conditions, and allows us to estimate β 240 from Supplementary Equation (23). When the statistical model was fitted separately for gymnosperms and angiosperms, the intercept fitted in Supplementary Equation (46) was for angiosperms and for gymnosperms. We thus obtained β 198 for gymnosperms and 243 for angiosperms, implying that the ratio of the cost factors for transpiration to carboxylation (1/β) is about 20% larger for gymnosperms. However, according to Supplementary Equations (22) and (23), this difference results in a value of χ o for gymnosperms that is only about 0.02 less than that for angiosperms Based on the environmental response of ln [χ o /(1 χ o )] in Supplementary Equation (46) and applying the chain rule, the approximate responses of χ o to temperature, elevation and vpd can be derived for standard conditions: 295 χ o / ΔΤ = χ o (1 χ o ) 0.01 K 1 (47) 296 χ o / z = χ o (1 χ o ) 0.01 km 1 (48) 297 χ o / D 0 = 0.5 χ o (1 χ o ) 0.1 kpa 1 (49) 298 providing rules of thumb for the general magnitudes of each variable s effect on χ o In the model for χ co, a value of was obtained for the constant C c based on observational χ c estimated from the global carbon isotope dataset (Supplementary Methods 4) using the same logic as described above. This allows us to estimate β c 200 from Supplementary Equation (25). Therefore, by using the observationally based mean value of g m /g s = 1.4, and deducting the term of (g s /g m + 1) 1 from β c, the ratio of cost factor b to a at reference temperature (i.e. β in Supplementary Equation (23)) is estimated as Full expressions for the optimal leaf-internal partial pressure of CO Supplementary Equations (10)-(11) yield an expression for the optimal leaf-internal partial pressure of CO 2i :

12 308 c i = ξc a ξ + D,ξ = βk 1.6η * (50) Here η * is the viscosity of water relative to its value at 25 C, representing the effect of changing viscosity on the value of a. The more exact expression for χ o (Supplementary Equation (12)) yields: 311 * ( ) c i = ξc + a Γ* D ξ + D, ξ = β K + Γ 1.6η * (51) Supplementary Equation (51) is used in the derivation of the GPP model (Supplementary Methods 5). Supplementary Equations (50) and (51) can also be applied using c c instead of c i to take account of finite g m. However, the term β then needs to be replaced by β c, being the product of two components: the ratio of cost factors b to a at reference temperature (i.e. β defined in c i model) and (g s /g m +1) 1. 3 Estimating χ from stable carbon isotope data Vascular-plant leaf stable carbon isotope (δ 13 C) data were compiled from published and unpublished sources 35. This global dataset of 2833 species includes 3985 observations from 594 sites located on seven continents. Taxonomies were resolved based on The Plant List (v1.1). Species with uncertain identifications were included in the dataset if a higher-level identity was known. Measurements on the upper, most exposed leaves were used as the value for a species in the dataset if both shade and sun leaves were measured. Carbon isotope discrimination (Δ) values of 3651 leaf samples from C 3 plants were either directly obtained from the researchers original reports, or estimated taking into account the isotopic signature of the air, adjusting for both the year and latitude of collection relative to a 1992 reference 36. Soil ph for the top 30 cm layer was estimated based on the Harmonised World Soil Database at a resolution of 30 arc-seconds 37. Observationally based values of χ were derived from Δ with the standard equation 38 : 328 χ = Δ a' b' a' (52) where a and b have standard values 4.4 and 27, representing the diffusional and biochemical components of carbon isotope discrimination, respectively. Studies have suggested that a is conservative while b shows some variation among species. More complex expressions have been developed 26 but their full application requires more information than is usually available for multispecies data sets. Supplementary Equation (52) performs well in comparison to c i :c a ratios determined independently from gas-exchange measurements; this is probably because the standard value for b subsumes several potential corrections to the true discrimination by Rubisco 39. We excluded 102 samples where Supplementary Equation (52) yielded χ > 1, which can arise due to the re-fixation of respired, 13 C-depleted CO 2, resulting in a final sample size of 3549 observations.

13 Estimating the ratio of mesophyll to ambient CO 2 partial pressure (χ c ) from stable carbon isotope data According to the comprehensive equation in Ubierna & Farquhar 40 : 341 Δ = 1 1 t c a a c s b c a + a s c s c i c a + 1+ t c a i c c m 1 t c a + b c c c a α b R e d c c Γ * α b f Γ* α e A + R d c a α f c a (53) Here, c a, c s, c i and c c are the CO 2 partial pressures in the ambient air, leaf surface, leaf intercellular spaces and chloroplast, respectively. a b, a s, a m, b, e and f are the fractionations associated with diffusion through the boundary layer (2.8 ), in air (4.4 ), in water (1.8 ), by Rubisco carboxylation (27 to 30 ), during respiration (0 to 5 ) and photorespiration (8 to 16 ), respectively. The terms α b, α e and α f are 1 + b, 1 + e, 1 + f, respectively. R d and A are the rates of dark respiration and assimilation, respectively. t represents the ternary effect, which is related to transpiration rate, the conductance to diffusion of CO 2 in air and the fractionation for the isotopologues of CO 2 diffusing in air. Following the first three simplications listed in Figure 1 by Ubierna & Farquhar 40 : (1) the ternary effect is negligible (i.e. t = 0 ); (2) α b = α e = α f = 1; (3) infinite boundary-layer conductance (c s = c a ), and assuming R d << A, so that R d /(R d + A) R d /A, the equation can be rewritten more simply as: Δ = a s (1 χ) + a m (χ χ c ) + bχ c eb 0 (χ c + κ) fγ (54) where b 0 = R d /V cmax = , κ = Κ/c a and γ = Γ * /c a. (55) Furthermore, given that the CO 2 flux from the outside to the intercellular spaces must be the same as that from the intercellular spaces to the chloroplast, denoting the ratio of g m to g s as θ, we have: (1 χ) g s = (χ χ c ) θ g s (56) Therefore: 1 χ = θ (1 χ c )/(1 + θ) (57) and χ χ c = (1 χ c )/(1 + θ) (58) Substituting these expressions into Supplementary Equation (54) and solving for χ c gives: χ c = Δ θa s + a m 1+θ b θa s + a m 1+θ + eb 0 κ + fγ eb 0 (59) We assumed a constant value of θ = 1.4, based on data compiled by IJW, and consistent with values in the literature 41.

14 Given the uncertainties in parameters b, e and f, we chose the values (b = 30, e = 0, f = 16) that can produce the best fit (R 2 = ) in the regression of χ c against temperature, ln vpd and elevation (Supplementary Table 2). 5 A light-use efficiency model for GPP General principle and a first-order model A first-order global light-use efficiency (LUE) model for GPP was proposed by Wang et al. 31. This model assumed that the electron-transport and Rubisco-limited rates of photosynthesis (A J, A C ) are colimiting under typical daytime conditions 20. In other words, A = A J = A C, where: 373 A J = ϕ 0 I abs c i Γ * c i + 2Γ * (60) 374 A C = V cmax c i Γ * c i + K (61) The assumption of co-limitation allows Supplementary Equation (60) to predict GPP as a fraction (LUE) of I abs. Using m to denote the CO 2 limitation term (c i Γ * )/(c i +2Γ * ) and with c i described by Supplementary Equation (51), the model can be re-written as follows, completely driven by temperature, moisture, elevation and atmospheric CO 2 partial pressure: 379 A = ϕ 0 I abs m, (62) 380 where 381 m = c a Γ c a + 2Γ + 3Γ 1.6Dη β K + Γ ( ) (63) 382 Accounting for the optimal J max :V cmax ratio Supplementary Equation (60) is accurate only if the response of GPP to increasing I abs remains linear at least up to the coordination point. In other words, the model of ref. 31 implicitly assumed that J max is arbitrarily large. However, in reality the limitation by J max may be significant, conferring significant curvature on the light response before the coordination point is reached. By considering a non- 42 rectangular hyperbola relationship between A and I abs, we allow for a limitation of finite Jmax on photosynthesis:

15 389 A = ϕ 0 I abs m ϕ 0I abs J max 2 (64) The dependence of A on I abs in Supplementary Equation (64) is not linear. However, the non-linear light response implied by this biochemical model and the linear light response in the empirical LUE model can be reconciled, provided that plants adjust their photosynthetic capacities (J max and V max ) over time in such a way as to acclimate to typical daytime conditions. We implement this acclimation by assuming that (a) an optimal J max exists that maximizes the differences between the benefit and the cost of maintaining this J max, which is assumed to include the maintenance of light-harvesting complexes and various proteins involved in the electron transport chain; (b) the benefit is the assimilation rate A, while the cost is the product of J max and a parameter c (defined as the unit cost of maintaining J max ); and (c) V cmax and J max vary with environment on a month-by-month basis, whereas the unit costs b and c of maintaining V cmax and J max respectively remain constant; and (d) V cmax and J max are related via the coordination hypothesis (A c = A J = A), an assumption that is supported by independent meta-analysis 20. Mathematically: A/ J max = c (65) Treating c as a constant, we obtain from Supplementary Equations (64) and (65): 404 c = A J max = 4 ϕ 0 I abs m( ϕ 0 I abs ) 3 ( ) 2 + J max (66) The square-root term in Supplementary Equation (64) can now be obtained from Supplementary Equation (66): ϕ I 0 abs J max 2 = 1 4c m (67) 408 Substituting the result into Supplementary Equation (64), the revised biochemical model becomes: 409 A = ϕ 0 I abs m 1 4c m 2 3 (68) 410 Note that this expression for A is proportional to I abs. 411 The coordination hypothesis (A c = A J = A) links Supplementary Equation (61) with Supplementary

16 412 Equation (64) generating the following equation: 413 V cmax c i Γ * c i + K = ϕ 0 I abs m ϕ I 0 abs J max 2 (69) which can be re-written as: ϕ 0 I abs ϕ 0 I abs ( ) 2 + J max 4 ( ) = 4V c cmax i Γ* 2 J max ( c i + K)m (70) Substituting Supplementary Equation (70) into Supplementary Equation (66) and expanding the CO 2 limitation term m, we can express Supplementary Equation (66) as: 418 c = A = 16( c i + 2Γ * ) 2 ( c i Γ * ) J max J max ( c i + K) V cmax 3 (71) Taking typical values of J max /V cmax = and χ = , we estimate c = for standard conditions (T = 25 C, z = 0 km, c a = 400 ppm). Therefore, the final LUE model is: 421 A = ϕ 0 I abs m 1 c* m 2 3 (72) 422 where 423 c * = 4c, (73) 424 m = c a Γ c a + 2Γ + 3Γ 1.6Dη β K + Γ ( ) (74) 425 The term 4c in Supplementary Equation (68) is now replaced by c * = Again, this model can be applied using c c instead of c i except that the term β must be replaced by β c. The c c -based LUE model is then: 428 A c = ϕ 0 I abs m c 1 c* m c 2 3 (75)

17 where m c = and β c = c a Γ * c a + 2Γ * + 3Γ * 1.6Dη * β 1+ g s g m β c (K + Γ * ) (76) (77) 433 Testing the predicted temperature dependency of J max25 /V cmax The derivation above implies the existence of an optimal ratio of J max to V cmax. Solving for the ratio J max /V cmax in Supplementary Equation (71), we obtain the following value at the growth temperature: 436 J max V cmax = 4 c i + K ( ) 3 ( c i Γ * )( c i + 2Γ * ) 2 c * (78) Together with Supplementary Equation (51) for c i, Supplementary Equation (78) predicts this ratio as a function of temperature, vpd, elevation and atmospheric CO 2 partial pressure. Following a modified Arrhenius function with temperature acclimation 43, the dependencies of J max and V cmax on temperature can be described as: 441 k T = k 25 e Ha( T g T ref ) RT ref T g 1+ e 1+ e T ref ( a ΔS +b ΔS T g ) H d RT ref T g ( a ΔS +b ΔS T g ) H d RT g (79) where k T and k 25 denote V cmax (or J max ) at the growth temperature (T g ) and at the reference temperature (T ref ), respectively. H a and H d are the activation and deactivation energies and a ΔS and b ΔS are parameters describing the acclimation of the entropy factor, ΔS. Thus, combing Supplementary Equations (78) and (79), our prediction of J max /V cmax at the growth temperature also generates a prediction of this ratio at 298 K, denoted J max25 /V cmax25. We generated simulations of J max25 /V cmax25 at each grid cell in the CRU CL gridded climatology 44, using CRU grid cell elevations, and bioclimatic variables calculated from the STASH model 45,46 as before. The results were regressed against growth temperature (Supplementary Figure 5), indicating a strong relationship (r 2 = 0.93): J max25 /V cmax25 = T g (80)

18 Data indicate a closely similar response of this ratio to growth temperature 43, with an intercept of 2.59 ± 0.17, a slope of ± and r 2 = Flux data based monthly GPP observations Monthly GPP values were derived from eddy-covariance measurements of net ecosystem CO 2 exchange (NEE) and photosynthetic photon flux density (PPFD) in the Free and Fair Use subset of the global FLUXNET archive. Altogether 2452 months of GPP data (between 2002 and 2006) were obtained, including data from 146 sites across 27 countries. These data are archived and freely available (Data link: For each site and month, paired instantaneous NEE and PPFD measurements were collected and non-linear regression was used to fit a rectangular hyperbola or, in months showing non-significant curvature, linear regression was used. Outliers were removed following an implementation of Peirce s criterion 47,48. Ecosystem respiration was assumed to be given by the y-intercept of the regression. (This approximation is not necessarily less accurate than the common use of a dependency on air temperature over the diurnal cycle, as leaf mitochondrial respiration is inhibited in the daytime 49.) In this way we generated a minimal data-based statistical model, with either one or two parameters, to obtain GPP as a function of instantaneous PPFD for each site and month 50. In initial tests, alternative flux partitioning approaches (including the complete omission of NEE measured at zero or low PPFD) were found to have only minor effects on estimated GPP. Instantaneous PPFD measurements at the sites were gapfilled by down-scaling analytically computed instantaneous top-of-the-atmosphere solar radiation by the ratio of observed daily solar surface irradiance to daily top-of-the-atmosphere irradiance. The gapfilled instantaneous PPFD values were then used to estimate GPP based on the empirical model and integrated to provide monthly totals. Site-months without regression parameters (due to a paucity of measurements, or poor model fit) were not processed. 7 Comparing model predictability with other LUE models Yuan et al. 51 compared daily GPP outputs from seven LUE models, including VPRM, VPM, MODIS, EC-LUE, CFlux, CFix and CASA, with observations from the LaThuile flux data set. The coefficient of determination (R 2 ) and root-mean-squared error of prediction (RMSE) were calculated for each model and each flux site. Median, upper and lower quartile, and the maximum and minimum values were reported with bar plots. We used the WebPlotDigitizer tool ( to extract median values from Figure 1 in ref. 51. Results were compared to statistical fits calculated for our LUE model based on the comparison with our flux-derived GPP dataset. RMSE values of our model were divided by 30 to convert from g C m 2 month 1 to g C m 2 day 1. We compare our model R 2 and RMSE values with the means and ranges of reported median R 2 and RMSE from all seven other LUE models reported by Yuan et al. (Supplementary Table 3). 8 Comparing the model-predicted effect of elevated CO 2 on photosynthesis with a FACE experiment meta-analysis

19 Ainsworth and Long conducted a meta-analysis on the response of photosynthesis to enhanced CO 2 based on free-air CO 2 enrichment (FACE) experiments 52. Twelve FACE studies were included. The responses of photosynthetic traits to CO 2 enhancement (by ~ 200 ppm on average across studies) are reported in their Appendix 2, with confidence intervals provided. Traits considered included apparent quantum yield (equivalent to LUE), stomatal conductance (g s ), instantaneous χ, V cmax /J max, instantaneous transpiration efficiency (ITE, defined as the ratio of assimilation rate to g s ), and lightsaturated CO 2 uptake (A sat ). Based on the coordinates of those FACE sites, we extracted the CRU grid cell elevations, and the growing season mean values of vpd and temperature calculated from CRU CL gridded climatology 44. χ was then estimated for each site with equation (1) with A = A sat. The responses of LUE, g s, χ, V cmax /J max and ITE to CO 2 enhancement were predicted for each site. LUE for the ambient and elevated CO 2 was calculated with equations (2) and (3) based on the extracted values of environmental variables. The response of V cmax /J max was calculated by evaluating the partial derivative of V cmax /J max with respect to c A (from Supplementary Equation (78)) at the FACE experiment locations. The simpler form of the model for χ (Supplementary Equations (10)-(11)) predicts no response to c a, while the more exact prediction (Supplementary Equation (12)) shows a slight reduction in χ. The meta-analysis suggests it is reasonable to ignore the response of χ to c a, therefore, the predicted response in g s is proportional to the effects on A sat and c a (Supplementary Equation (6)) note this equation is general, applying to A at any light level including saturation while the changes in ITE are expected to be the same as the changes in c a (Supplementary Equation (7)). Implementing the reported changes in A sat (31%) and c a (55%) from the meta-analysis into Supplementary Equations (6) and (7), the values of the predicted response of ITE and g s were obtained (we used the observed values of A sat here because prediction of A sat would require information on average light conditions, which was missing). The mean values of the responses from all sites were compared with the results from meta-analysis.

20 Supplementary Table 1 Regression summaries for the sensitivity parameter ξ in Supplementary Equation (10) based on gas-exchange measurements. Values of ξ for 101 species were derived from an independent global dataset of measured instantaneous CO 2 and water vapour exchanges 28. Lin et al. fitted the stomatal conductance model: g s = 1.6 A (1 + ξ / D) / c a, where g s, A, D and c a are stomatal conductance to CO 2, net photosynthesis rate, and vapour pressure deficit and CO 2 concentration in the cuvette, as measured with standard portable gas exchange instruments. Data were fitted with a nonlinear regression model for each species separately. We tested the relationship of ln ξ to ΔT g, ln D and z using multiple linear regression. Environmental predictors were derived from CRU CL1.0 monthly climatology at 0.5 resolution 44. No significant relationship of ξ to elevation was detected, probably due to the limited elevation range of the measurements (all sites were at 1600 m above sea level). Predictor Coefficient Standard Error t-value P-value Conditional R 2 ΔT g ln D z intercept

21 Supplementary Table 2 Alternative regression summary on the ratio of mesophyll to ambient CO 2 partial pressure (χ c ). χ c is estimated by the simplified standard equation using best-fitting parameters (Supplementary Methods 4). Predictor Theoretical value Fitted coefficient Confidence intervals 2.5% 97.5% Multiple R 2 ΔT g ln D z intercept

22 Supplementary Table 3 Comparison of R 2 and RMSE (g C m -2 day -1 ) between our model and the means and ranges of reported median values from seven other LUE models, as reported by Yuan et R 2 RMSE This study Yuan et al. This study Yuan et al. All ecosystems ± ± Shrubland ± ± Deciduous broadleaf forest ± ± Evergreen broadleaf forest ± ± Evergreen needleleaf forest ± ± Grassland ± ± Mixed forest ± ± al. 51

23 Supplementary Figure 1 Distribution of leaf stable carbon isotope sample sites used in the analysis. Biome types for each site were assigned based on BIOME4 53 except for wetlands and alpine vegetation types, where the assignments given in the original publications were followed Tropical forest Warm temperate forest Temperate forest Boreal forest/tundra Savanna and woodland Grassland and shrubland Desert Wetland Alpine

24 Supplementary Figure 2 Predicted global pattern of environmental controls on the ratio of leafinternal to ambient CO 2 partial pressure (χ). (a) growing-season mean temperature (T g ) and (b) vapour pressure deficit from CRU data (D 0 ); (c) elevation (z) from the CRU grid (z); (d) χ o as predicted by equation (1).

25 Supplementary Figure 3 Global pattern of predicted temperature and moisture influences on the ratio of leaf-internal to ambient CO 2 partial pressure (χ o ). The observed highest and lowest values of growing-season mean temperature and vpd were used as endpoints to derive the colour scale. χ o was calculated for the temperature and moisture conditions at each CRU grid cell, and compared to these endpoints. Elevation effects were ignored here as they are slight in a global map perspective. Moisture influence Temperature influence

26 Supplementary Figure 4 Observations and predictions of the ratio of chloroplast to ambient CO 2 partial pressure (χ c ) with means and standard deviations marked for (a) different plant functional types and (b) different biome types.

27 Supplementary Figure 5 Temperature dependency of the ratio of maximum electron transport capacity to maximum carboxylation capacity at 25 C (J max25 /V cmax25 ). Predictions (grey dots) were made with Supplementary Equations (79) and (80), using 0.5 gridded CRU climatology 44 and c a = 380 ppm. The fitted regression line is shown.

28 Supplementary Figure 6 Predicted global pattern of GPP by C3 plants controlled by (a) intrinsic quantum yield and absorbed light, and modified by (b) CO2 limitation and (c) maximum electron transport capacity y y (a) GPP = φ0 Iabs 8000 x y yy (b) GPP = φ0 Iabs m xx x y 6000 yy x yy x (c) 0 GPP = φ0 Iabs m [1 (c*/m)2/3] GPP (gc m-2 a-1)

29 Supplementary Figure 7 Observations and predictions of the ratio of leaf-internal to ambient CO 2 partial pressure (χ) with means and standard deviations shown for different plant functional types. Predicted χ Evergreen broadleaf tree Deciduous broadleaf tree Evergreen needleleaf tree Deciduous needleleaf tree Savanna tree Shrub C3 grass Observed χ r = RMSE = 0.086

30 Supplementary Figure 8 Relationship between evapotranspiration deficit and vapour pressure deficit. Evapotranspiration deficit is defined as the difference between equilibrium and actual evapotranspiration as computed by the STASH model 34,35. Vapour pressure deficit is estimated with standard methods from relative humidity and maximum and minimum temperatures. Both calculations are driven by the 0.5 gridded CRU climatology 44. The solid line is the 1:1 line.