Methods. Research. Dany P. Moualeu-Ngangue 1, Tsu-Wei Chen 1,2 and Hartmut St utzel 1. Summary. Introduction

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1 Research Methods A new method to estimate photosynthetic parameters through net assimilation rate intercellular space CO 2 concentration (A C i ) curve and chlorophyll fluorescence measurements Dany P. Moualeu-Ngangue 1, Tsu-Wei Chen 1,2 and Hartmut St utzel 1 1 Institute for Horticultural Production Systems, Vegetable Systems Modelling Section, Faculty of Natural Sciences, Leibniz Universit at Hannover, Herrenh auser Straße 2, D Hannover, Germany; 2 INRA, UMR759 Laboratoire d Ecophysiologie des Plantes sous Stress Environnementaux, Place Viala, F Montpellier, France Author for correspondence Dany P. Moualeu-Ngangue Tel: moualeu@gem.uni-hannover.de Received: 9 June 2016 Accepted: 8 September 2016 doi: /nph Key words: A Ci curves, chlorophyll fluorescence, fitting method, mesophyll conductance, photosynthetic parameters. Summary Gas exchange (GE) and chlorophyll fluorescence (CF) measurements are widely used to noninvasively study photosynthetic parameters, for example the rates of maximum Rubisco carboxylation (V cmax ), electron transport rate (J), daytime respiration (R d ) and mesophyll conductance (g m ). Existing methods for fitting GE data (net assimilation rate intercellular space CO 2 concentration (A C i ) curve) are based on two assumptions: g m is unvaried with CO 2 concentration in the intercellular space (C i ); and light absorption (a) and the proportion of quanta absorbed by photosystem II (b) are constant in the data set. These may result in significant bias in estimating photosynthetic parameters. To avoid the above-mentioned hypotheses, we present a new method for fitting A C i curves and CF data simultaneously. This method was applied to a data set obtained from cucumber (Cucumis sativus) leaves of various leaf ages and grown under eight different light conditions. The new method had significantly lower root mean square error and a lower rate of failures compared with previously published methods (6.72% versus 24.1%, respectively) and the effect of light conditions on V cmax and J was better observed. Furthermore, the new method allows the estimation of a new parameter, the fraction of incoming irradiance harvested by photosystem II, and the dependence of g m on C i. Introduction Parameters of the Farquhar von Caemmerer Berry (FvCB) photosynthesis model (Farquhar et al., 1980; von Caemmerer & Farquhar, 1981) are often used to characterize the biochemical relationships underlying the net assimilation rate (A) and for modelling the environmental and genetic influences on plant productivity (Gu et al., 2014). The FvCB photosynthesis model was shown to be a useful biochemical approach for photosynthesis in C 3 plants and was cited > 5000 times in the last 35 years (5124 times; Google Scholar, accessed 25 January 2016). Therefore, accurate estimations of the key model parameters maximum ribulose 1,5-bisphosphate carboxylase/oxygenase (Rubisco) carboxylation rate (V cmax ), potential light-saturated electron transport rate (J max ), leaf light respiration rate (R d ) and mesophyll conductance (g m ) are of interest and special importance. Usually, parameters of the model are estimated from leaf-level gas exchange (GE) measurements, namely the CO 2 response curves of net assimilation (hereafter net assimilation rate intercellular space CO 2 concentration (A C i ) curves). To date, several methods of fitting A C i curves to estimate photosynthetic parameters (e.g. R d, V cmax, J or J max and g m ) of the FvCB model have been proposed (Ethier & Livingston, 2004; Dubois et al., 2007; Sharkey et al., 2007; Yin & Struik, 2009; Bellasio et al., 2016; Sharkey, 2016). However, some assumptions associated with g m estimation in current A C i curve-fitting methods introduce bias in fitting other model parameters. If g m is finite, its value has a strong influence on V cmax estimates. For example, Sun et al. (2014a,c) reported that an overestimation of g m might result in a significant underestimation of V cmax, and if g m is infinite, V cmax and J might be underestimated by up to 75% and 60%, respectively. Therefore, accurate g m estimation is a prerequisite for realistic parameterization of the FvCB model. If other methods are available, estimating g m from fitting of GE data is not recommended (Dubois et al., 2007; Pons et al., 2009; Sun et al., 2014a; Sharkey, 2016), because the changes in V cmax and in g m affect the 1543

2 1544 Research New Phytologist net assimilation estimated by the FvCB model similarly. Therefore, it is difficult to distinguish their effects on the observed assimilation rates (Pons et al., 2009; Sun et al., 2014a). Moreover, the high number of parameters simultaneously fitted to an A C i data set increases uncertainties (Ethier et al., 2006; Dubois et al., 2007; Sun et al., 2014a). This might explain why fitting methods are known to overestimate g m in comparison with other methods (Pons et al., 2009; Sun et al., 2014a). For example, Chen et al. (2015) reported that over 90% of the g m values estimated by the curve-fitting method were unrealistically high. Several methods have been developed for g m estimation, including the constant J method (Bongi & Loreto, 1989; Harley et al., 1992), variable J method (Di Marco et al., 1990; Harley et al., 1992), carbon isotope method (Evans et al., 1986, 1994; von Caemmerer & Hubick, 1989; von Caemmerer, 1992; von Caemmerer et al., 2014) and the aforementioned curve-fitting methods (Dubois et al., 2007; Sun et al., 2014a; Sharkey, 2016). The accuracy of g m estimation using the variable J methods strongly depends on the validity of the relation applied to find the electron transport rate (J F ) (Harley et al., 1992; Sun et al., 2014a), including the electron distribution between photosystem I (PSI) and photosystem II (PSII) and PSII electron transport efficiency (Φ PSII ): U PSII ¼ DF =F 0 m ¼ðF 0 m F sþ=f 0 m Eqn 1 where F s is the steady-state fluorescence and F 0 m is the maximum fluorescence during a saturating light pulse (Genty et al., 1989), which can be determined from chlorophyll fluorescence (CF) measurements. The multiphase flash (MPF) method, which was shown to improve the accuracy of Φ PSII, achieves higher robustness of g m estimation by the variable J method (Loriaux et al., 2013). The curve-fitting methods and the constant J method assume that g m is independent of CO 2 concentration. However, g m was reported to vary from five- to nine-fold over the range of C i typically used in A C i curves (Centritto et al., 2003; Flexas et al., 2007; Vrabl et al., 2009; Sun et al., 2014b; Xiong et al., 2015). Therefore, considering g m as constant should be avoided in fitting methods. The constant and variable J methods assume a leaf absorptance (a) and a partitioning factor between PSI and PSII (b = 0.5 in most studies). The product ab, representing the fraction of photosynthetic photon flux density (PPFD) harvested by PSII (Yin et al., 2004), can be calibrated by fitting the CO 2 or light response curves measured under nonphotorespiration conditions (low O 2, Yin et al., 2004, 2009; Bellasio et al., 2016). However, this time-consuming calibration is not conducted in many studies. Even in studies where this calibration is conducted, the value of ab, in many cases, is not calibrated for each individual leaf, but considered to be constant for all treatments in an experiment (e.g. Sun et al., 2014a; Xiong et al., 2015). As a and b are known to vary with species, treatment and light conditions (Laisk & Loreto, 1996), the prevailing current practice still introduces errors in estimating photosynthetic parameters. The goodness of a fitting method depends not only on the assumptions included, but also on the consistency of the model with observation (Miao et al., 2009). The root mean square error (RMSE) of the fit, which quantifies the deviation of the model from observations, is a proper indicator of model accuracy. A higher RMSE indicates a higher deviation between the fitted parameters and the observed values. In addition to RMSE, the value of the chloroplast CO 2 concentration (C c ) at which the transition from Rubisco to RuBP regeneration limitation occurs (C ctr ) can be used as an indicator of the goodness of the estimation when estimating photosynthetic parameters (Sun et al., 2014a). C ctr can be estimated from the intercellular CO 2 concentration at which the transition from Rubisco to RuBP regeneration limitation occurs (C itr ), which was shown to be typically between 100 and 350 ppm (Sharkey et al., 2007). In general, the range of C itr can be roughly estimated from the shape of the Φ PSII C i curve. This information can be used to determine the ranges of C i for fitting V cmax and J values (Sharkey et al., 2007). In recent years, the Dubois method (Dubois et al., 2007) has become a widely used method for fitting the FvCB parameters from A C i curves. This method combines grid search and nonlinear fitting and was shown to be one of the best fitting methods with high accuracy, low sensitivity to sample size, and simplicity in implementation (Miao et al., 2009). Here, we present a new method to fit A C i and Φ PSII C i curves simultaneously to estimate the parameters V cmax, J max, R d, ab and g m in the FvCB model. This method replaces g m with a new parameter, the fraction of incoming photosynthetic photons harvested by PSII (s = ab), to achieve the simultaneous estimation of V cmax and J max, without assuming or fitting a value of g m to convert an A C i curve to an A C c curve. This avoids the errors raised from the assumptions of the current curve-fitting methods. To evaluate our new method, we compared it with the Dubois method using the following criteria: the values of RMSE; the distributions of estimated C ctr and observed ranges derived from Φ PSII C i data sets; g m estimation with a variable J method; and the effect of simultaneously fitting V cmax, J, s or g m, with or without R d and triose phosphate utilization rate (TPU). Materials and Methods Plant material Cucumber seeds (Cucumis sativus L. Aramon ; Rijk Zwaan, De Lier, the Netherlands) were sown in rock-wool cubes ( mm) in a growth chamber at the Institute of Horticultural Production Systems, Leibniz Universit at Hannover, Germany (52.5 N, 9.7 E) on 10 March Seven days after sowing, seedlings were transplanted into larger rock-wool cubes ( cm) for another 7 d until the appearance of the third leaf (leaf length c. 5 cm). Seedlings were then transplanted upon styrofoam floating in a container filled with 25 l of nutrient solution in two growth chambers. There were three plants per container. Each litre of the nutrient solution contained 0.53 g Ca (NO 3 ) 2 and 0.65 g Ferty Basisd unger 1 (Planta GmbH, Regenstauf, Germany). The ph value was adjusted to with 1%

3 New Phytologist Research 1545 sulfuric acid and the nutrient solution was checked twice a week. The growth chamber was heated to maintain 25 C:20 C, day : night temperature. Within and between the growth chambers ( m), there was considerable spatial variation in light intensity ( lmol m 2 s 1 in growth chamber B1; lmol m 2 s 1 in growth chamber B2). Therefore, positioning the measured leaves in different parts of a growth chamber allowed us to create different light exposure intensities in the chamber. The measured leaves were grown under the following light intensities: 143, 188, 297, 376, 481, 601, 661 and 943 lmol m 2 s 1. Furthermore, custom-made leaf holders were used to support the leaves horizontally. The light intensity on the leaf surface was measured weekly and the leaf holders were re-adjusted to maintain a constant light intensity reaching the measured leaves. This experimental set-up was chosen to obtain a broad range of data from different leaf ages and different light environmental conditions. Gas exchange measurements All gas exchange measurements were conducted using a Li-6400XT portable photosynthesis system equipped with a leaf chamber fluorescence head and a B LED light source (Li- Cor Inc., Lincoln, NE, USA) between 08:30 and 13:30 h for growth chamber B1, where the light was switched on at 06:30 h, and between 11:30 and 18:30 h for growth chamber B2, where the lightwasswitchedonat10:30h.toavoiderrors,co 2 leakage was corrected according to Flexas et al. (2007). All types of measurement (light response curves and CO 2 response curves) were conducted on the third leaves of the three plants at each light intensity between day 3 and 42 after leaf appearance. Before starting the CO 2 response curve measurements, the leaf was adapted for 5 20 min to ensure that photosynthesis, stomatal conductance, and fluorescence signal were stable and Rubisco was fully activated. The first point was measured under a reference CO 2 concentration (C a ) of 400 ppm. A C i measurements were conducted at the reference CO 2 concentration of 400 ppm, followed by 300, 200, 100, 50, 150, 250, 350, 600, 900, 1200 and 1500 ppm under saturating light conditions of 1300 lmol m 2 s 1. Together with the GE measurements, the CF measurements were performed using the multiphase flash (MPF) approach (Loriaux et al., 2013). The fluorometer measuring light was turned on and set to measuring light frequency = 10 khz, intensity = 3, filter = 5 and gain= 10. The flash was set to multiphase pulse, target intensity = 9, ramp depth = 30%, measuring frequency = 20 khz and filter = 50 khz. The three phases were 320, 350 and 200 ms long (Chen et al., 2014). Light response curves and respiration rate Following the A C i curves, short light response curves were established based on measurements at 200, 150, 125, 100, 75, 50, 40, 30, 20, 10 and 0 lmol m 2 s 1 PPFD, a reference CO 2 concentration of 400 lmol mol 1, a leaf temperature of c. 25 C, a flow rate of c. 300 lmol s 1 and relative humidity of c. 55%. The short light curve was used to estimate the day respiration rate (R d ; lmol CO 2 m 2 s 1 ) by the Yin method (Yin et al., 2011). Similar values were used under low O 2 for the estimation of the fraction of light harvested by PSII using the Yin calibration (Yin et al., 2004, 2009). Model of photosynthesis In the Rubisco limited phase, the response of net assimilation (A c ; lmol CO 2 m 2 s 1 )toc c is described using the FvCB (Farquhar et al., 1980) equation defined by A c ¼ V cmax ðc c C Þ C c þ K c ð1 þ O=K o Þ R d Eqn 2 where Γ* is the CO 2 compensation point in the absence of photorespiration (lmol mol 1 ), V cmax (lmol CO 2 m 2 s 1 ) is the maximum rate of Rubisco activity at the site of carboxylation, and K C (404 lmol mol 1 )andk O (278 mmol mol 1 ) are the catalytic constants for the carboxylation and oxygenation reactions of Rubisco, respectively; O (210 mmol mol 1 ) is the mole fraction of O 2 at the site of the carboxylation, and C c (lmol mol 1 ) is the mole fraction of CO 2, calculated as: C c ¼ C i A c g m Eqn 3 where g m is the conductance to CO 2 diffusion from the intercellular airspace to the chloroplast, and C i is the intercellular CO 2 concentration. During the RuBP-regeneration limitation, the net assimilation rate is defined by A j ¼ J ðc c C Þ 4C c þ 8C R d Eqn 4 where J (lmol e m 2 s 1 ) is the rate of electron transport. More information about the equations can be found in the literature (Farquhar & von Caemmerer, 1982; Yin et al., 2009). In this paper, parameter J is the electron transport obtained at saturating light, and this can also be called J high or J 1300 according to Buckley & Dıaz-Espejo (2015). During the triose phosphate utilization limitation, the net assimilation rate is defined by A p ¼ 3TPU R d where TPU is the rate of phosphate release in triose phosphate utilization. Estimating mesophyll conductance The actual electron transport rate (J F ; lmol e m 2 s 1 ) was estimated from CF measurements (Genty et al., 1989) using the following equation: J F ¼ a b I inc / PSII Eqn 5

4 1546 Research New Phytologist where a is the fraction of incoming light absorbed by the photosystems, which is related to the chlorophyll content per unit leaf area, b denotes the partitioning fraction of photons between PSI and PSII, which can be assumed to vary between 0.45 and 0.6 (Laisk & Loreto, 1996), I inc is the photosynthetically active photon flux density incident on the leaf (lmol photon m 2 s 1 ), and / PSII (mol e mol 1 photon) is the photochemical yield of PSII estimated from fluorescence measurements (Eqn 1). Values of a was taken between 0.5 and 0.95 (Bauerle et al., 2004; Ritchie, 2014). Because data for chlorophyll content or a are not always available and, moreover, values of b are rarely estimated, we propose to constrain the product s = ab representing the fraction of I inc harvested by PSII between and The boundary interval can be changed if precise values of a or b are available. The aim of the optimization algorithm is to minimize the distance between the minimum of A c and A j, and the observed net assimilation A (lmol m 2 s 1 ) at each C i (lmol mol 1 ) value. In the new fitting method, A, C i, I inc and / PSII were used simultaneously as the input data set for the optimization algorithm and the output was the minimum of A c, A j and eventually A p. From an initial guess value of s combined with input parameters I inc and / PSII, J F was computed from Eqn 5. g m (mol m 2 s 1 ) was then calculated using the equation (Harley et al., 1992; Long & Bernacchi, 2003): AðsI inc U PSII 4ðA þ R d ÞÞ g m ¼ si inc U PSII ðc i C Þ 4ðC i þ 2C ÞðA þ R d Þ Eqn 6 where A, C i, I inc and / PSII are input data. Then, C c at corresponding C i was obtained from Eqn 3. The value of C c was then used in Eqns 2 and 4 with initial guesses of V cmax and J to compute the estimated A (namely ^A). The optimization algorithm checked the sum of square error between the ^A and A, and updated each parameter using the partial derivative of the model with respect to each of the parameters to be estimated. The process was repeated automatically until the convergence criteria were met (the sum of square error was less than the tolerance, in which case the algorithm converged), or other stop conditions of the algorithm were fulfilled (most commonly, the maximum number of iterations was reached, or the step size showed no improvement), in which case the algorithm failed to converge. Furthermore, the C c at which the transition from Rubisco to RuBP regeneration limitation occurs (C ctr ) was determined as: C ctr ¼ K c ð1þo=k o ÞJ 4V cmax 2C 1 J 4V cmax : Eqn 7 Parameters to be estimated Parameters in the photosynthesis model that can be estimated through analyses of GE and CF measurements are V cmax, J, TPU, R d, g m, K O, K C and Γ*. However, only V cmax, J, R d and g m are simultaneously estimated in most curve-fitting methods. As CO 2 concentration influences g m (Flexas et al., 2007, 2008) but is unlikely to affect a and b, g m was replaced by s = ab (Eqn 6) in our method. The advantage of this replacement is that the dependence of g m on C i can be calculated using Eqn 6. With four variables, it is obvious that noisy data sets cause considerable variation of estimations as a result of the increased potential local minimum in the cost function. For this reason, it seems preferable to estimate R d with other methods (e.g. Kok, 1948; Laisk, 1977; Yin et al., 2011) and to fit the other three parameters using Eqns 2 6. Furthermore, including Γ*, K O and K C is possible if the ranges of these parameters can be well defined, in order to avoid convergence to unrealistic local minima. Testing the new method The data set obtained from A C i and light curve measurements was used to evaluate the new method. The new method was compared with a similar fitting method proposed by Dubois et al. (2007; referred to as the Dubois method ) using the same standard convergence conditions of the PROC NLIN in SAS (SAS Institute Inc., Cary, NC, USA). Two comparisons between the new method and the Dubois method were made. In the first comparison, simultaneous fitting of photosynthesis parameters V cmax, J, R d and g m using the Dubois method and simultaneous fitting of V cmax, J, R d and s using the new method were performed. The Levenberg Marquardt algorithm implemented in the PROC NLIN procedure of SAS was used to fit the A C i curve with the standard step size search method as defined in the program. Sample SAS code and a sample EXCEL file are provided in Supporting Information Methods S1 and S2. The results for V cmax, J, the V cmax to J ratio and C ctr were compared. The average g m estimated by the Dubois method was compared with g m at C a = 400 ppm calculated using Eqn 6. Furthermore, R d values obtained using both fitting methods were compared with R d obtained using the Yin method (Yin et al., 2011). To estimate R d by the Yin method, simple linear regressions were performed using the PROC REG procedure of SAS. In the second comparison, R d obtained with the Yin method was used as an input parameter for fitting V cmax, J,andssimulta- neously by the new method, and for fitting V cmax, J, and g m by the Dubois method. The robustness of the fitting methods was compared by the differences between distribution of values estimated by each methods. The Mann Whitney rank sum test (MWW) implemented in SIGMAPLOT 11.0 (Systat Software Inc., San Jose, CA, USA) was applied to different sets of data to compare their distributions. Finally, using R d obtained with the Yin method as input, parameters V cmax, J, TPU and s/g m were simultaneously estimated by the new method and the Dubois method. The results were compared with estimations in previous steps. Sensitivity analysis Sensitivity analysis can be performed to test the influence of the estimated parameter on the model output, and also to quantify the change in convergence rate subject to the number of simultaneously estimated parameters.

5 New Phytologist Research 1547 The sensitivity index of a parameter p at C i can be ðc iþ for each C i, where ^A is the estimated net assimilation obtained from the model when all other parameters are considered constant. The average sensitivity index of each parameter is computed as the average partial derivative with respect to the parameter using the whole data set. This unitless parameter can help to assess the average rate of change of the estimated photosynthesis rate with respect to the parameter. The sensitivity of A net with respect to each parameter was evaluated using an EXCEL spreadsheet developed for this purpose. To evaluate the robustness of the method and to assess how many parameters can be estimated simultaneously, sensitivities of the fitting methods were tested with different combinations of parameters (see Tables 1 and 2) and the percentage of failures to meet convergence was calculated. Results Comparison of the relationship between V cmax and J The relationships between J and V cmax estimated by the Dubois method (Fig. 1a) and by the new method (Fig. 1b) showed remarkable differences. Clear linear relationships were observed between J and V cmax estimated by the new method for all light intensities combined (R 2 = 0.94; P < ) as well as for individual light intensities (minimum R 2 = 0.91; P < ). A linear relationship was found using the Dubois method but with a lower R 2 value (R 2 = 0. 14; P < ). Moreover, J and V cmax did not increase with light intensities when estimated by the Dubois method, in contrast with values estimated by the new method (except for one point under 376 lmol m 2 s 1 ; Fig. 1b). The new method estimated higher values of J and V cmax than the Dubois method (Fig. 1c,d). Comparison of C ctr values obtained using the new method and the Dubois method C ctr ranges typically between 100 and 350 ppm, as the Rubiscolimited state typically occurs at C i < 200 ppm CO 2 and the RuBP-regeneration-limited state typically occurs at C i > 400 ppm (Sharkey et al., 2007). When the C i Φ PSII curves were observed as suggested by Sharkey et al. (2007), C itr in our data set ranged between 100 and 350 ppm (data not shown), as Φ PSII remained constant when C i > C itr. Because g m is finite and A is positive in that range of values, C ctr is lower than C itr. C ctr estimates showed considerable differences between the two methods. In general, the new method estimated lower C ctr than the Dubois method (Fig. 2). In comparison with the Dubois method, C ctr calculated by the new method fell better into the range of realistic values (Fig. 2). Roughly 10% of C ctr values obtained by the Dubois Table 1 Effects of varying number of parameters to be estimated for 116 net assimilation rate intercellular space CO 2 concentration (A C i ) data sets of observations of cucumber leaves by the new method Parameters % failed V cmax J s R d Γ* K C K O V cmax, J, s 0 Max Min Max SE Min SE V cmax, J, s, R d 1.72 Max Min Max SE Min SE V cmax, J, s, Γ* 2.58 Max Min Max SE Min SE V cmax, J, s, Γ*, K C 3.36 Max Min Max SE Min SE V cmax, J, s, Γ*, K C, K O 2.58 Max Min Max SE Min SE V cmax, J, s, R d, Γ*, K C, K O 6.72 Max Min e Max SE 1.4e 7 9.5e e Min SE V cmax (lmol CO 2 m 2 s 1 ), maximum carboxylation rate; J (lmol e m 2 s 1 ), potential maximum electron transport rate; s, proportion of light harvested by photosystem II; R d, daytime respiration rate; Γ*,CO 2 compensation point in the absence of photorespiration; K C (lmol mol 1 ) and K O (mmol mol 1 ), the catalytic constants for the carboxylation and oxygenation reactions of Rubisco, respectively.

6 1548 Research New Phytologist Table 2 Effects of varying number of parameters to be estimated for 116 net assimilation rate intercellular space CO 2 concentration (A C i ) data sets of observations of cucumber leaves by the Dubois method Parameters % failed V cmax J g m R d Γ* K C K O V cmax, J, g m 1.7 Max Min Max SE Min SE V cmax, J, g m, R d 3.5 Max Min Max SE Min SE V cmax, J, g m, Γ* 7.8 Max Min Max SE Min SE V cmax, J, g m, Γ*, K C 5.2 Max 6e e 10 Min Max SE 2.4e e 5 Min SE V cmax, J, g m, Γ*, K C, K O 19.8 Max Min Max SE Min SE V cmax, J, g m, R d, Γ*, K C, K O 24.1 Max 1.5e e e 13 Min e Max SE 7e e 6 1.6e 24 Min SE V cmax (lmol CO 2 m 2 s 1 ), maximum carboxylation rate; J (lmol e m 2 s 1 ), potential maximum electron transport rate; g m (mol CO 2 m 2 s 1 ), conductance to CO 2 diffusion from the intercellular airspace to the chloroplast; R d, daytime respiration rate; Γ*,CO 2 compensation point in the absence of photorespiration; K C (lmol mol 1 ) and K O (mmol mol 1 ), the catalytic constants for the carboxylation and oxygenation reactions of Rubisco, respectively. method were > 900 lmol mol 1. In these cases, only one to three points were used to fit the A j function. By MWW, a statistically significant difference (P < 0.001) was obtained between C ctr values estimated by the Dubois method and by the new method. Comparing R d values obtained using the new method and the Dubois method About 99% of daytime respiration rate (R d ) values estimated by the Yin method (Yin et al., 2011) were positive. The remaining 1% were less than or equal to zero and were set to zero. The Dubois method and the new method estimated different values of R d in comparison with the Yin method (Fig. S1a), but the Dubois method had more positive values (39%) than the new method (24%). Values obtained using both methods did not correlate well with the values obtained by the Yin method (Fig. S1b). By Kruskal Wallis one-way analysis of variance on ranks, statistically significant differences (P < 0.001) were obtained between R d values estimated by the Dubois method, the new method and the Yin method. Comparison of g m estimations obtained using the new method and the Dubois method In contrast to the Dubois method, which estimates an average g m for the whole curve, the new method allows g m to vary with C i. Over 95% of g m at a reference CO 2 concentration of 400 ppm estimated with the new method and the variable J method ranged between 0.02 and 0.24 mol CO 2 m 2 s 1 and < 5% were negative. Using the Dubois method, 72% of g m reached the maximum boundary of mol CO 2 m 2 s 1. Slightly lower values of mesophyll conductance were obtained using the variable J method (Fig. 3b) in comparison with the new method. No significant difference (P = with the MWW test) between g m values estimated by the variable J method and the new method was obtained. However, a significant difference (P < 0.001) was found between g m estimations obtained using both methods and the Dubois method. The new method estimated a broad range of g m with C i. For example, 99% of g m values for C i < 100 were negative (Fig. 4). Robustness of the model without simultaneous estimation of R d The method presented by Yin et al. (2011) was used to estimate R d. The results were used as input for fitting the A C i curves with the new and the Dubois methods. The relationship between V cmax and J was linear, similar to Fig. 1(b), using the new method but sigmoidal using the Dubois method (data not shown). Using R d as an input parameter, C ctr estimated by the Dubois method was not affected (Fig. S2a; P = using the MWW test). By contrast, the new method estimated slightly higher C ctr when R d was used as an input parameter (Fig. S2b; P < with the MWW test). The number of parameters to be estimated had

7 New Phytologist Research 1549 (a) (b) Fig. 1 Relationships between the maximum Rubisco carboxylation rate (V cmax ) and the light-saturating electron transport rate (J) obtained using (a) the Dubois method and (b) the new method. Comparisons of (c) J and (d) V cmax values obtained using the Dubois method (y-axes) and the new method (x-axes). Results were outputs of simultaneous estimations of V cmax, J, daytime respiration rate (R d ) and the fraction of incoming photosynthetic photons harvested by photosystem II (s) or mesophyll conductance (g m ) from a data set of 122 net assimilation rate intercellular space CO 2 concentration (A C i ) curve measurements on cucumber leaves. (c) (d) Fig. 2 Comparison between the distributions of the chloroplast CO 2 concentration (C ctr ) at which the transition from Rubisco to RuBPregeneration limitation occurred obtained using the Dubois method (dotted line) and the new method (solid line). Results were calculated from outputs of simultaneous estimations of the maximum Rubisco carboxylation rate (V cmax ), the light-saturating electron transport rate (J), the daytime respiration rate (R d ) and the fraction of incoming photosynthetic photons harvested by photosystem II (s) or mesophyll conductance (g m ) from a data set of 122 net assimilation rate intercellular space CO 2 concentration (A C i ) curves measured on cucumber leaves. greater effects on C ctr estimates in the Dubois method than in the new method (Fig. S2a,b). About 6% of C ctr values obtained by simultaneous fit of g m, V cmax and J were negative and 3% of values were > 800 lmol mol 1. Compared with the Dubois method, the new method showed a higher stability for V cmax and errors in R d had less effect on V cmax (Fig. S3a,b). A significant difference was found between V cmax values fitted with and without R d by the Dubois method (P = using the MWW test). Simultaneous fitting with or without R d showed a smaller variation for V cmax estimated with the new method compared with the Dubois method. No significant difference was found in this case (P = using the MWW test). Values of J estimated by the Dubois method were very stable (Fig. S3c), with a slight deviation from the 1 : 1 line. No significant difference (P = for the Dubois method (Fig. S3c); P = for the new method (Fig. S3d); MWW test) was found between J estimations with and without R d as input. However, there was a difference (P < 0.001) between J estimations obtained using the Dubois method and the new method without joint estimation of R d (similar to Fig. 1d). Using R d as an input parameter affected g m considerably when it was estimated by the Dubois method (Fig. S3e). Including R d values computed by the Yin method led to a better distribution of s (Fig. S3f) with a lower number of s values reaching the maximum bound. The distribution of g m was not statistically different (P = 0.478; MWW test). Effect of simultaneous estimation with TPU Using the method presented by Yin et al. (2011) to estimate R d, and including the TPU limitation in the net assimilation model, the A C i curves with the new and the Dubois methods were evaluated. Compared with the Dubois method, the new method showed a higher stability for J. Significant differences were found between values of J estimated by the Dubois method and the new method (P 0.001; MWW test) and also between values of J estimated with and without TPU limitation by the Dubois method (Fig. S4a). No significant difference was found within V cmax values fitted with or without TPU limitation by the Dubois method (P = 0.069; MWW test) as well as by the new method (P = 0.088;

8 1550 Research New Phytologist (a) (b) Fig. 3 (a) Distribution of mesophyll conductance (g m ) estimated using the new method (solid line), the variable light-saturating electron transport rate (J) method (light absorption, a = 0.875; the proportion of quanta absorbed by photosystem II, b = 0.5; dotted line) and the Dubois method (fraction of incoming photosynthetic photons harvested by photosystem II (s) = ab fitted from 122 net assimilation rate intercellular space CO 2 concentration (A C i ) data sets for cucumber leaves; dashed line) at a reference CO 2 concentration of 400 ppm. An interval of length 0.02 is used on the x-axis, from 0 to 3 mol CO 2 m 2 s 1. (b) Boxplot representing values of g m obtained using the Dubois method, the new method and the variable J method. Different letters indicate differences between root mean square errors (RMSEs) determined using the Mann Whitney rank sum test at significance level P < The bottom and top of the boxes are the first and the third quartiles, respectively, of 122 individual values, the central lines represent the median values and the whiskers represent the 10 th and the 90 th percentiles. difference in s values estimated by the new method (Fig. S4e; P = 0.766) and g m values estimated by the Dubois method (Fig. S4f; P = by the MWW test). Fig. 4 Variation of mesophyll conductance (g m ) with intercellular CO 2 concentration (C i ) for individual leaves under eight different growth irradiances applied to cucumber leaves. The highlighted area represents the reliable area for g m according to Harley et al. (1992). MWW test), although there was a significant difference between V cmax values fitted by the Dubois and the new methods (P < 0.001; MWW test; Fig. S4b). Including TPU in the set of parameters to be estimated had a greater effect on C ctr estimates obtained using the Dubois method than using the new method (Fig. S4c). Significant differences were found between C ctr values estimated with and without simultaneous TPU estimation by both the Dubois method and the new method (P < 0.001; MWW test; Fig. S4c). However, TPU estimations obtained using the Dubois method and the new method were within the same range (P = 0.371; MWW test; Fig. S4d). Including TPU limitation did not lead to a significant Root mean square errors in fitting methods Using the fitted V cmax, J, R d, TPU, g m and s, A C i curves of each leaf were calculated. RMSE was calculated for each curve using measured data. The distribution of RMSE of 122 A C i curves showed that the RMSE of the Dubois method was higher than that of the new method (Fig. 5; P by one-way analysis of variance on ranks test). For the new method, including R d and TPU as input parameters did not lead to a significant difference in RMSE. Lower RMSE obtained using the new method showed its higher accuracy in comparison with the Dubois method. The new method was also more stable because using R d computed by the Yin method did not lead to a significant difference in RMSE (P > 0.05; MWW test). Using R d computed by the Yin method, no significant difference was found between RMSE estimated by the new method, with and without including TPU limitation (P > 0.05; MWW test). Model sensitivity to estimated parameters It was observed that the Dubois method resulted in a lower sensitivity index to changes in K C and K O than the new method (Fig. 6). V cmax and J had almost the same sensitivity index when estimated by the new method. When using the Dubois method, both parameters had completely different sensitivities. The new method was more robust than the Dubois method regarding the number of parameters to be estimated (Tables 1, 2). The new method was able to estimate six parameters of FvCB,

9 New Phytologist Research 1551 Fig. 5 Comparing the distributions of root mean square errors (RMSEs) obtained using the Dubois method and the new method with and without simultaneous estimation of daytime respiration rate (R d ) and triose phosphate utilization rate (TPU). Dub I, the Dubois method with simultaneous estimation of R d and without TPU limitation; Dub II, the Dubois method with R d as input data estimated by the Yin method and without TPU limitation; Dub III, the Dubois method with TPU limitation and with R d as input data estimated by the Yin method; New I, the new method with simultaneous estimation of R d and without TPU limitation; New II, the new method without TPU limitation and with R d as input data estimated by the Yin method; New III, the new method with simultaneous estimation of TPU, but with R d as input data estimated by the Yin method. Different letters indicate differences between RMSEs determined using the Mann Whitney rank sum test at significance level P < The bottom and top of the boxes are the first and the third quartiles, respectively, of 122 individual values measured on cucumber leaves, the central lines represent the median values and the whiskers represent the 10 th and the 90 th percentiles. with a failure percentage of 6.7% versus 24.1% for the Dubois method. Including Γ* in the estimated parameter set gave various values from 22 to 120 lmol CO 2 m 2 s 1 by the new method, while negative values were found when using the Dubois method (Table 2). Coefficient of variation The new method had a larger coefficient of variation for J and V cmax than the Dubois method (Fig. S5a,b). Eighty-eight per cent of the coefficients of variation of V cmax and J estimated by the new method were between 0 and 40%, and 1.5% did not exist (not a number). This happened usually when, despite convergence of the optimization algorithm, the approximate hessian was singular. In this case, the skewness of the parameter was also not a number and the approximate SE was not a number. The algorithm gave a warning message that there was a possible error in the estimation. For the Dubois method, 26.2% of standard errors did not exist, and the rest were between 0 and 40%. Discussion Estimating photosynthetic parameters is fundamental for studying the ecophysiological responses and acclimation of leaves to the environment. We have presented a new method combining Φ PSII C i and A C i curves to fit the parameters in the FvCB model of C 3 photosynthesis. The main features of the new method are that it requires no assumption regarding the dependence of g m on C i and ab for calculating the fraction of incoming photosynthetically active radiation harvested by PSII. Fig. 6 Average sensitivity index (logarithmic scale) of the estimated photosynthesis rate of cucumber leaves with respect to each photosynthetic parameter. V cmax (lmol CO 2 m 2 s 1 ), maximum carboxylation rate; J (lmol e m 2 s 1 ), potential maximum electron transport rate; s, proportion of light harvested by photosystem II; g m (mol CO 2 m 2 s 1 ), conductance to CO 2 diffusion from the intercellular airspace to the chloroplast; R d, daytime respiration rate; Γ*,CO 2 compensation point in the absence of photorespiration; K C (lmol mol 1 ) and K O (mmol mol 1 ), the catalytic constants for the carboxylation and oxygenation reactions of Rubisco, respectively. Roles of mesophyll conductance in curve fitting For decades, plant ecophysiologists have used the parameters V cmax and J in the FvCB model to study plant photosynthesis in response to environmental factors (e.g. Wullschleger, 1993; Centritto et al., 2003; Chen et al., 2015) and to investigate leaf nitrogen economics using the ratio between V cmax and J (e.g. Kattge & Knorr, 2007). Accurate estimations of V cmax and J require statistical methods for fitting A C c curves. If there is no method available for direct measurement of C c (Tholen et al., 2012), g m has to be estimated to transform A C i curves into A C c curves using Eqn 3. This means that g m plays an essential role in parameter estimation. Therefore, many studies reported that inaccurate estimation of g m leads to underestimation of V cmax and J (Flexas et al., 2008; Warren, 2008; Niinemets et al., 2009; Sun et al., 2014b,c). g m values obtained from published curve-fitting methods represent an average value of g m during A C i curve measurements. However, if there are rapid responses of g m to CO 2 (e.g. Xiong et al., 2015), g m obtained from curve-fitting methods might be underestimated for C i ranging between c. 200 and 400 ppm and overestimated for C i outside this range. This probably explains why the variable J method gave lower g m than curvefitting methods (Pons et al., 2009; Fig. 3). Furthermore, large coefficients of variation of g m obtained using the Dubois method (Fig. S5c) indicate unreliable g m estimates according to the goodness criterion of de Kauwe et al. (2015). Most V cmax values estimated by the Dubois method were < 200 lmol CO 2 m 2 s 1, while the new method gave values approaching 300 lmol CO 2 m 2 s 1 (Fig. S4b). The fact that overestimated g m may strongly decrease V cmax and slightly decrease J explains the discrepancies in V cmax (Fig. 1c) and J (Fig. 1d) between the Dubois method and our approach. In our

10 1552 Research New Phytologist approach, where the variable J method (Eqn 6) was embedded into the Dubois method, the possible biases resulting from g m are avoided. Using Eqn 6, s was estimated and then g m was calculated. Therefore, slight differences between g m values obtained using the variable J method and the new curve-fitting method (Fig. 3) resulted from their differences in s. Using the variable J method, a and b are usually assumed to be constant in a data set because their calibration is laborious. Our new method does not require this assumption. Furthermore, if a is measured, the new method allows estimation of b. Using data for 15 A C i and short light curve measurements under ambient O 2 and 2% O 2, respectively, for the estimation of s by both the new method and the Yin calibration (Bellasio et al., 2016), no statistically significant difference was found between the two estimations (t-test: t = 1.690; P = 0.102; Fig. S6). However, further work is required to confirm this approach. Another disadvantage of the new method is that it requires CF data during the GE measurements. Therefore, possible errors from CF measurements (e.g. Loriaux et al., 2013) affect the fitted results. The choice of the device for measurement should take into account accuracies of GE measurements (Flexas et al., 2007; Yin et al., 2011) and CF measurements. Model sensitivity A sensitivity analysis allowed us to quantify the dependence of the model output on individual parameters (Fig. 6) and the effect of combining different parameter sets on the failure rate of the estimation (Tables 1, 2). The unrealistic range of C ctr obtained by the Dubois method can be explained by the lower sensitivity to V cmax in comparison with J (Fig. 6). Because the sensitivity index does not include the dependence between different parameters in the model, simultaneous estimations of parameters might be impossible although a high sensitivity index was found. This is particularly the case when parameters to be estimated are multiplied or divided by each other in the model. For that reason, we recommend avoiding the estimation of parameters that are mathematically related to each other through product and divisions. Mathematical dependence between two parameters plays a key role in parameter fitting. In fact, updating the set of parameters to be fitted depends on the sensitivity index which is computed from the partial derivative of the cost function (least square function). When two parameters of different magnitude are dependent (in the mathematical sense), the optimization algorithm will tend to sacrifice the parameter that affects the cost function less. That might be the case when g m and V cmax are simultaneously estimated by the Dubois method. In the new method, this dependence is more constrained and g m is computed using a variant of the variable J method. Effect of growing light conditions on photosynthetic parameters By the new method, estimated V cmax and J in leaves grown at higher light intensity were higher, as suggested in the literature (Murchie & Horton, 1997; Murchie et al., 2005). By contrast, this trend was less clear in the results obtained with the Dubois method. This might be attributable to the errors of the Dubois method in g m estimation. Results obtained using the Yin method showed that R d increased with growth light intensity (data not shown), which is consistent with previous studies (Noguchi et al., 2005). However, neither the new nor the Dubois method could deliver reasonable estimates of R d. Without simultaneous estimation of R d, fewer g m and s values reached the boundary of the search interval (Fig. S2e,f). Therefore, we suggest separate estimation of R d. Other considerations In this work, Γ* was taken from the literature and assumed to be constant for each species, although it has been reported to vary (Evans & Loreto, 2000; Pons & Westbeek, 2004). As the model showed a high sensitivity to Γ* (Fig. 5; Table 1), co-estimation using the new method can be considered. Conclusions The new fitting method avoids erroneous assumptions for mesophyll conductance included in the existing methods. This is a major step forward because it captures the variations of mesophyll conductance with intercellular CO 2 concentration, and it estimates the fraction of photosynthetic photon flux density harvested by PSII, in a range comparable with the Yin calibration (Yin et al., 2004, 2009). We further demonstrated that the new method performed better than the Dubois method because it had lower RMSE; estimated C ctr in the range obtained from the analysis of Φ PSII C i curves; returned similar values of g m to g m estimated by the variable J method; allowed less influence of simultaneous estimation of R d and TPU on RMSE; captured the influences of growth light on the V cmax to J relationship; allowed differentiation of the effects of growing irradiance on photosynthetic parameters; and returned similar ranges of photosynthesis parameters independently of simultaneous estimation of TPU. Acknowledgements We thank Yi-Chen Pao for her help during the measurements, Ilona Napp for her help setting up the experiment, Dr Christian Blume for the material he provided for low O 2 measurements and all three anonymous referees for their comments and suggestions, which significantly contributed to improving the quality of the publication. Author contributions D.P.M-N. and T-W.C. planned and designed the experiment and performed the measurements. D.P.M-N. developed the method and analyzed the data. D.P.M-N., T-W.C. and H.S. discussed the data and wrote the manuscript.

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