Using combined measurements of gas exchange and chlorophyll fluorescence to estimate parameters of a biochemical C₃ photosynthesis model: a critical appraisal and a new integrated approach applied to leaves in a wheat (Triticum aestivum) canopy

ABSTRACT

We appraised the literature and described an approach to estimate the parameters of the Farquhar, von Caemmerer and Berry model using measured CO₂ assimilation rate (A) and photosystem II (PSII) electron transport efficiency (Φ₂). The approach uses curve fitting to data of A and Φ₂ at various levels of incident irradiance (I_inc), intercellular CO₂ (C_i) and O₂. Estimated parameters include day respiration (R_d), conversion efficiency of I_inc into linear electron transport of PSII under limiting light [κ_2(LL)], electron transport capacity (J_max), curvature factor (θ) for the non‐rectangular hyperbolic response of electron flux to I_inc, ribulose 1·5‐bisphosphate carboxylase/oxygenase (Rubisco) CO₂/O₂ specificity (S_c/o), Rubisco carboxylation capacity (V_cmax), rate of triose phosphate utilization (T_p) and mesophyll conductance (g_m). The method is used to analyse combined gas exchange and chlorophyll fluorescence measurements on leaves of various ages and positions in wheat plants grown at two nitrogen levels. Estimated S_c/o (25 °C) was 3.13 mbar µbar⁻¹; R_d was lower than respiration in the dark; J_max was lower and θ was higher at 2% than at 21% O₂; κ_2(LL), V_cmax, J_max and T_p correlated to leaf nitrogen content; and g_m decreased with increasing C_i and with decreasing I_inc. Based on the parameter estimates, we surmised that there was some alternative electron transport.

INTRODUCTION

To predict productivity of crops such as wheat (Triticum aestivum L.), it is necessary to scale the instantaneous photosynthetic rate at the leaf level up to the daily total at the canopy level, and to integrate this total over the whole growth season. Photosynthetic capacity of a leaf changes with its age and its position in the canopy. These changes in photosynthetic capacity are related to the degradation of chlorophyll and the breakdown of ribulose 1·5‐bisphosphate carboxylase/oxygenase (Rubisco) and other proteins into amino acids that are then exported as a source of nitrogen (N) to growing or storing organs (Chiba et al. 2003). Thus, quantifying the photosynthetic parameters in relation to the N status of plants is an important step to reliably predict crop leaf and canopy photosynthesis.

The model of Farquhar, von Caemmerer & Berry (1980) (hereafter the ‘FvCB model’), later modified by, for example, Sharkey (1985), has been widely used to describe C₃ photosynthesis at the leaf level. This model predicts net photosynthesis rate (A; see Table 1 for the full explanation of all model variables) as the minimum of the Rubisco‐limited rate (A_c), the ribulose 1,5‐bisphosphate (RuBP)‐regeneration or electron (e^‐) transport limited rate (A_j), and the triose phosphate utilization (TPU) limited rate (A_p) of CO₂ assimilation:

Table 1. List of model variables and their units

Variable	Definition	Unit
A	Net photosynthesis rate	µmol CO2 m^−2 s^−1
A c	Rubisco activity limited net photosynthesis rate	µmol CO2 m^−2 s^−1
A H	Net photosynthesis rate under a high O2 condition	µmol CO2 m^−2 s^−1
A j	Electron transport limited net photosynthesis rate	µmol CO2 m^−2 s^−1
A L	Net photosynthesis rate under a low O2 condition	µmol CO2 m^−2 s^−1
A p	Triose phosphate utilization limited net photosynthesis rate	µmol CO2 m^−2 s^−1
b H	Slope of the initial linear part of the A–Ci curve under a high O2 condition	mol m^−2 s^−1 bar^−1
b L	Slope of the initial linear part of the A–Ci curve under a low O2 condition	mol m^−2 s^−1 bar^−1
C a	Ambient air CO2 partial pressure or concentration	µbar or µmol mol^−1
C c	Chloroplast CO2 partial pressure	µbar
C i	Intercellular CO2 partial pressure	µbar
C iH	Intercellular CO2 partial pressure under a high O2 condition	µbar
C iL	Intercellular CO2 partial pressure under a low O2 condition	µbar
C i*	C i‐based CO2 compensation point in the absence of Rd	µbar
C i*H	C i* under a high O2 condition	µbar
C i*L	C i* under a low O2 condition	µbar
f cyc	Fraction of electrons at PSI that follow cyclic transport around PSI	–
f pseudo	Fraction of electrons at PSI that follow pseudocyclic transport	–
f pseudo(b)	Fraction of electrons at PSI that follow the basal pseudocyclic e^‐ flow	–
f Q	Fraction of electrons at reduced plastoquinone that follow the Q cycle	–
g m	Mesophyll diffusion conductance	mol m^−2 s^−1 bar^−1
g mo	Residual mesophyll diffusion conductance in the gm model Eqn 11	mol m^−2 s^−1 bar^−1
h	Number of protons required to produce one ATP	mol mol^−1
I abs	Photon flux density absorbed by leaf photosynthetic pigments	µmol photon m^−2 s^−1
I inc	Photon flux density incident to leaves	µmol photon m^−2 s^−1
J	Linear plus additional pseudocyclic e^‐ transport rate through PSII	µmol e^‐ m^−2 s^−1
J′	Rate of e^‐ transport through PSII, as applied to Eqn 3b	µmol e^‐ m^−2 s^−1
J 2	Rate of all (i.e. linear plus basal and additional pseudocyclic) e^‐ transport through PSII	µmol e^‐ m^−2 s^−1
J c	Rate of e^‐ transport calculated from CO2 uptake measurement	µmol e^‐ m^−2 s^−1
J f	Rate of e^‐ transport calculated from the chlorophyll fluorescence measurement	µmol e^‐ m^−2 s^−1
J max	Maximum value of J under saturated light	µmol e^‐ m^−2 s^−1
J 2max	Maximum value of J2 under saturated light	µmol e^‐ m^−2 s^−1
K mC	Michaelis–Menten constant of Rubisco for CO2	µbar
K mO	Michaelis–Menten constant of Rubisco for O2	mbar
O	Oxygen partial pressure	mbar
O H	High oxygen partial pressure	mbar
O L	Low oxygen partial pressure	mbar
R d	Day respiration (i.e. respiratory CO2 release other than by photorespiration)	µmol CO2 m^−2 s^−1
R dk	Respiratory CO2 release in the dark	µmol CO2 m^−2 s^−1
s	A lumped parameter, = ρ2β[1–fpseudo(b)/(1–fcyc)], see Eqn 7b	–
S c/o	Relative CO2/O2 specificity factor for Rubisco	mbar µbar^−1
T p	Rate of triose phosphate export from the chloroplast	µmol m^−2 s^−1
V cmax	Maximum rate of Rubisco activity‐limited carboxylation	µmol CO2 m^−2 s^−1
Z i	Dummy variables in Eqn 13	Unitless
α 2(LL)	Quantum efficiency of PSII e^‐ transport under strictly limiting light, on the combined PSI‐ and PSII‐absorbed light (i.e. Iabs) basis	mol e^‐ (mol photon)^−1
β	Absorptance by leaf photosynthetic pigments	–
δ	A parameter in the gm model, defining Cc : Ci ratio at saturating light	–
κ 2	Conversion efficiency of incident light into J	mol e^‐ (mol photon)^−1
κ 2(LL)	Value of κ2 at the strictly limiting light	mol e^‐ (mol photon)^−1
θ	Convexity factor for response of J to Iinc (see Eqn 9)	–
θ 2	Convexity factor for response of J2 to Iabs (see Eqn 4b)	–
ρ 2	Proportion of Iabs partitioned to PSII (typically 0.5 but with caution)	–
Φ 1(LL)	Quantum efficiency of PSI e^‐ flow at the strictly limiting light level, on the PSI‐absorbed light basis	mol e^‐ (mol photon)^−1
Φ 2	Quantum efficiency of PSII e^‐ flow on PSII‐absorbed light basis, usually assessed from the chlorophyll fluorescence measurements	mol e^‐ (mol photon)^−1
Φ 2(LL)	Value of Φ2 at the strictly limiting light level	mol e^‐ (mol photon)^−1
Φ CO2	Quantum efficiency of CO2 assimilation on the Iabs basis	mol CO2 (mol photon)^−1
Φ CO2(LL)	Value of ΦCO2 at the strictly limiting light	mol CO2 (mol photon)^−1
Γ	C c‐ or Ci‐based CO2 compensation point in the presence of Rd	µbar
Γ H	Γ under a high O2 condition	µbar
Γ L	Γ under a low O2 condition	µbar
	C c‐based CO2 compensation point in the absence of Rd	µbar
Γ *H	under a high O2 condition	µbar
Γ *L	under a low O2 condition	µbar

• PSI, photosystem I; PSII, photosystem II; Rubisco, ribulose 1·5‐bisphosphate carboxylase/oxygenase.

(1)

The calculation of A_c is straightforward in the FvCB model:

(2)

where is calculated by: .

While calculating A_j, the FvCB model assumes 100% noncyclic e^‐ transport, thus excluding cyclic e^‐ transport around photosystem I (PSI) (CET). A_j is calculated by

((3a))

((3b))

Equation 3a assumes that RuBP regeneration is limited because of insufficient NADPH and implies 100% linear e^‐ transport (LET) – the non‐cyclic e^‐ flux used for carbon reduction and photorespiration. Equation 3b assumes that ATP is insufficient: its two forms result from different assumptions about the operation of the Q cycle and the number of protons (H⁺) required for synthesizing an ATP (von Caemmerer 2000). Equation 3b implies that in addition to LET there is some pseudocyclic e^‐ transport (PET) – the non‐cyclic e^‐ flux not used for carbon reduction and photorespiration (so, the variable J′ in Eqn 3b is different from variable J in Eqn 3a, with J′ > J in both forms of Eqn 3b). PET normally refers to e^‐ transports used for Mehler‐type O₂ reduction. We include other non‐cyclic e^‐ fluxes within the PET category because they all generate a trans‐thylakoid H⁺ gradient permitting ATP synthesis, whereas the reductive process to which these e^‐ fluxes are coupled, needs either none or only small amounts of ATP; so, the ATP generated by these fluxes plays a role in producing the required ATP/NADPH ratio for carboxylation and photorespiration. Many studies (e.g. Harley et al. 1992b; Wullschleger 1993; Warren 2004), in which the FvCB model was applied, calculated J using a saturation light‐response curve, with a constant quantum yield of e^‐ transport that is often corrected empirically for the observed quantum yield of CO₂ assimilation under limiting lights (Φ_CO2(LL)). In fact, Φ_CO2(LL) is a function of excitation partitioning of absorbed light between PSI and photosystem II (PSII), and the e^‐ transfer efficiencies of PSI and PSII. Moreover, any active CET will reduce observed Φ_CO2(LL). Therefore, the FvCB model for A_j was extended for a generalized stoichiometry (Yin, van Oijen & Schapendonk 2004; Yin, Harbinson & Struik 2006):

((4a))

((4b))

((4c))

((4d))

Equation 4b describes a non‐rectangular hyperbolic irradiance response of e^‐ transport rate, according to Farquhar & Wong (1984). Equation 4d sets the required relation for f_cyc, f_pseudo and f_Q if ATP and NADPH produced in the light reactions are to match the requirement by carbon reduction and photorespiration. Equations 3a and 3b are special cases of the extended model (Yin, van Oijen & Schapendonk 2004).

In addition to the limitations set by A_c and A_j, Sharkey (1985) has drawn attention to a third limitation by TPU, which may come into play to set a ceiling value of CO₂ assimilation rate. If the TPU limitation occurs, net photosynthesis rate is simply

(5)

The validity and simplicity of the FvCB model is widely recognized in analysing photosynthesis in response to environmental variables. Relatively less attention has been paid to procedures to estimate its parameters from in vivo measurements. Early reports (e.g. Harley et al. 1992b; Wullschleger 1993) and recent methods (Ethier et al. 2006; Dubois et al. 2007; Sharkey et al. 2007) have estimated only a few parameters (e.g. R_d, V_cmax, J or J_max) of the FvCB model by fitting only to gas exchange (GE) data, using A–C_i curves. However, combining with other types of in vivo measurements can give more insights into photosynthetic regulation and provides opportunities to estimate more parameters. For example, chlorophyll fluorescence (CF) measurements can assess PSII e^‐ transport efficiency (Φ₂) by (where F_s is the steady‐state fluorescence and is the maximum fluorescence during a saturating light pulse) (Genty, Briantais & Baker 1989), although whether is the actual Φ₂ is debatable (e.g. Lavergne & Trissl 1995). Combined GE and CF data have been used to estimate the mesophyll diffusion conductance g_m (Bongi & Loreto 1989; Di Marco et al. 1990; Harley et al. 1992a; Evans & von Caemmerer 1996), which is required to convert C_i into C_c for the use in the FvCB model. Commonly, is properly converted into the flux of e^‐ transport (J_f). Then J_f is compared with J_c– the e^‐ transport rate based on CO₂ exchange data. The difference between J_f and J_c is interpreted as being caused by mesophyll diffusion resistance – the reciprocal of g_m. In early reports (e.g. Loreto et al. 1992) g_m was considered constant for a given leaf at a given temperature. Recent evidence suggests that g_m is variable (Centritto, Loreto & Chartzoulakis 2003; Düring 2003), and the response of g_m to CO₂ and irradiance is similar to that of stomatal conductance g_s (Flexas et al. 2007b).

Combined GE and CF data have also been used to estimate other parameters, for example S_c/o (Peterson 1989), and the proportion of I_abs partitioned to PSII (ρ₂) (Makino, Miyake & Yokota 2002). Laisk & Loreto (1996) and Laisk et al. (2002, 2006) developed a procedure (hereafter the ‘L method’) to estimate many parameters – some of which correspond to the parameters of the FvCB model. The L method consists of the steps:

• 1

  S _c/o is estimated from the response of CO₂ compensation point (Γ) to O₂ levels;

• 2

  R _d is estimated from Γ, S_c/o and C_i‐based initial carboxylation efficiency;

• 3

  R _d and S_c/o are used to calculate J_c, based on Eqn 3a;

• 4

  and the J_c : I_abs ratio at limiting irradiance are used to estimate ρ₂;

• 5

  ρ ₂ is used to convert into J_f;

• 6

  the difference between J_f and J_c is considered to be alternative e^‐ transport;

• 7

  to calculate J_c, g_m is needed, and g_m is derived from data points where a zero (Laisk et al. 2002) or constant (Laisk et al. 2006) alternative e^‐ flux is assumed.

To estimate S_c/o, the L method assumes that within the 1.5–21% O₂ range the variation of carboxylation efficiency is insignificant. According to the theoretical equation for the slope of the relation of Γ versus O₂ (eqn 16.23 of Farquhar & von Caemmerer 1982), the L method will underestimate S_c/o. Furthermore, it employs a circular logic to determine g_m, J_c and alternative e^‐ transport. To avoid this trap, Laisk et al. (2006) estimated g_m iteratively with the constraint that the resulting A–C_c curve has a K_mC closest to its value measured in vitro. This assumes that the kinetics of Rubisco in vitro are similar to those in vivo, which might not always be the case because of the complex biochemical environment of the stroma.

Our objectives are to develop a robust approach to estimate a complete set of parameters of the FvCB model from concurrent in vivo measurements of A and Φ₂ responses to both light and C_i, and to apply this approach to analyse photosynthesis for wheat leaves differing in age and position in canopies of crops grown at different N levels. Attention will be given to: (1) using the extended model, Eqns 4a–d, to elucidate some FvCB model parameters; (2) identifying problems with the existing reports of quantifying photosynthesis; and (3) accommodating both constant and variable g_m modes, given that it is uncertain whether g_m varies with light and C_i levels.

MODEL STEPS

Estimating quantum efficiency of PSII e^‐ transport under limiting light (Φ_2(LL))

From the measurements of Φ₂, J₂ can be expressed as J₂ = Φ₂ρ₂I_abs. Substituting this equation into Eqn 4b and solving for Φ₂ give

(6)

where α_2(LL) is given by Eqn 4c, and ρ₂ = α_2(LL)/Φ_2(LL) (Yin et al. 2006), using the widely held assumption that ρ₂ does not vary with light level. f_cyc and Φ_1(LL) should be known a priori. Using various values for f_cyc and Φ_1(LL) indicates that, unlike J_2max, estimates of θ₂ and Φ_2(LL) are not affected by f_cyc or Φ_1(LL). So, Φ_2(LL) can be obtained using arbitrary values for f_cyc and Φ_1(LL) within a physiologically reasonable range.

Establishing relationships for estimating a lumped calibration factor and R_d

To use CF measurements properly in relation to GE data, it is necessary to assess whether or not PET occurs. PET‐supporting basal metabolic demands such as nitrite reduction, denoted herein as the basal PET (PET_b), could account for an appreciable fraction of the total e^‐ flux (Noctor & Foyer 1998). When photosynthesis is limited by Rubisco, surplus e^‐ may also follow PET, referred to as the additional PET (PET_a), in support of the Mehler reaction (Asada 1999) or exported to the cytosol via the malate‐oxaloacetate shuttle (Backhausen et al. 2000). To correct for any PET and uncertain factors such as light absorbance by non‐photosynthetic pigments, the relationship between J_c and J_f, or between Φ_CO2 and under non‐photorespiratory (NPR) conditions has commonly been used as a calibration curve. However, the fraction of PET_a is unlikely to be constant across various CO₂ or irradiance levels. To avoid the confounding effect of varying PET_a, we suggest using data from the low irradiance or high CO₂ levels, or both combined, for the calibration procedure. This differs from calibration using the entire light response curves of J_c and J_f (e.g. Cornic & Briantais 1991; Miyake et al. 2004; Fila et al. 2006). Others even used the entire C_i response curve for calibration (e.g. Flexas et al. 2007b). We strongly advise excluding data at the low C_i range of the C_i response curve, because in addition to possible PET_a, the relative rate of photorespiration may be appreciable at low C_i even under low O₂ conditions because of the relatively low C_i : O₂ ratio. That Rubisco may lose activation at low C_i is another complicating factor to consider.

From Eqn 4a, the following relation to calculate e^‐‐transport limited A can be derived:

((7a))

where J₂ in Eqn 4a is replaced by ρ₂βI_incΦ₂. For NPR conditions, Eqn 7a becomes

((7b))

So, using data of the A_j‐limited range under NPR conditions, a linear regression can be performed for the observed A against (I_incΦ₂/4), in whichΦ₂ is based on CF measurements. The slope of the regression will yield the estimate of a lumped parameter s, and the intercept will give an estimate of R_d. The procedure can be used to estimate R_d for photorespiratory (PR) conditions, although it is less certain that the relation between A and (I_incΦ₂/4) will be linear.

Parameter s depends on: (1) β; (2) ρ₂; and (3) the effect of alternative e^‐ pathways in the form of [1 − f_pseudo(b)/(1 − f_cyc)]. Points (1) and (2) are well recognized. Point (3) has been noted occasionally, but its quantitative form is only elucidated here using the extended model; if not recognized (e.g. Miyake et al. 2005), any PET_b has been unknowingly lumped to ρ₂.

Assumptions underlying the linear regression using Eqn 7b are: (1) R_d does not vary much with light level; (2) photosynthesis is hardly limited by TPU, or such a limitation can be effectively mirrored by Φ₂ via feedback on e^‐ transport; and (3) parameter s does not vary with C_i or irradiance levels.

Calculating e^‐ transport parameters κ_2(LL), J_max and θ

It follows, from Eqns 3a, 4a and 7b, that under NPR conditions

((8a))

where J is the sum of LET and PET_a fluxes of PSII whereas J₂ is its total e^‐ fluxes of (LET + PET_b + PET_a). So, the efficiency of converting incident light into J (κ₂) is

((8b))

The value of κ₂ for the strictly limiting light condition is therefore given by

((8c))

By analogy with Eqn 4b, J can be calculated but as a function of I_inc as follows:

(9)

Then, A_j can still be calculated by Eqn 3a, but with J being given by Eqn 9. Parameter κ_2(LL) in Eqn 9 differs fromα_2(LL) in Eqn 4b by κ_2(LL) = α_2(LL)β[1 − f_pseudo(b)/(1 − f_cyc)]. Parameters J_max and θ in Eqn 9 can be estimated by fitting to the calculated value of J (i.e. J = sI_incΦ₂) across a range of light levels, using the κ_2(LL) mentioned earlier as an input.

In some studies (e.g. Piel et al. 2002; Niinemets et al. 2005) in which the CF‐based method was used to estimate g_m, J_f was calculated as , where ρ₂ was fixed at, for example, 0.5. Fixing ρ₂ might be incorrect, as its exact form should be (1 − f_cyc)/[Φ_2(LL)/Φ_1(LL) + (1 − f_cyc)] (Yin et al. 2006). More importantly, this way of calculating J_f implies that all of the PSII e^‐ flux is used for carbon reduction and photorespiration. This can lead to underestimation of g_m if PET_b does exist in vivo. Using our approach (i.e. ) will not have this problem so long as parameter s does not differ significantly between PR and NPR conditions. Again, this condition is usually met, as assumed in existing calibration procedures (e.g. Warren 2004; Flexas et al. 2007b).

Estimating S_c/o

The relationship between A and C_i at the low C_i range (<100 µbar) is linear, and this linearity has been used to estimate S_c/o. In the method of Laisk (1977), the linear A–C_i curves are obtained by using several irradiances, and the intersection of the A–C_i lines will identify R_d and C_i* (e.g. Brooks & Farquhar 1985; von Caemmerer et al. 1994). The Laisk (1977) method does not estimate the C_c‐based S_c/o, because C_i* and differ, as defined by (von Caemmerer et al. 1994; Piel et al. 2002; Niinemets et al. 2005). As g_m remains to be estimated (usually from , i.e. based on the re‐assimilation of photorespiratory CO₂, Laisk & Loreto 1996), the estimate of C_i* is often used as a proxy for (e.g. Flexas et al. 2007b; Warren 2008). In the L method (Laisk et al. 2002), S_c/o was estimated as 0.5(O_H − O_L)/(Γ_H − Γ_L), where Γ_H and Γ_L were obtained by extrapolating the linear part of A–C_i curves at O_H and O_L, respectively. This approach underestimates S_c/o.

We propose a new approach to estimate the real C_c‐based S_c/o, by making use of the fact that Γ_*H − Γ_*L = C_i*H − C_i*L if both g_m at the CO₂ compensation point and R_d are constant across O₂ levels. Let A_H and A_L be the values of A, and C_iH and C_iL be the values of C_i, at O_H and O_L, respectively. A relationship between A_H and A_L for the linear part of A–C_i curves can be expressed as (see Appendix)

(10)

where b_H and b_L are the values of the slope of the linear A–C_i relationships measured in O_H and O_L, respectively. With b_H and b_L estimated via linear regression beforehand and R_d estimated earlier, Eqn 10 is used to estimate S_c/o as the only parameter by fitting A_H as affected by A_L and C_iL–C_iH, or affected by A_L, O_H–O_L and C_iL–C_iH in case of more than two O₂ levels.

Establishing a generic model for g_m and estimating V_cmax and T_p

To be able to deal with both constant and variable g_m modes, we propose a model for g_m by analogy with a phenomenological model of the Leuning (1995)‐type for g_s:

(11)

Parameter δ in this model defines the C_c : C_i ratio at saturating light as . A higher δ gives higher g_m and therefore a higher C_c : C_i ratio.

Combining Eqn 11 with Eqns 2 and 3a, and replacing C_c with (C_i − A_c/g_m) and (C_i − A_j/g_m), respectively, and then solving for A_c and A_j give

(12)

Equation 12 is generic. If δ = 0 and g_mo approaches infinity, Eqn 12 becomes Eqns 2 and 3a, in which C_c is replaced by C_i. If only δ = 0, then g_m = g_mo so Eqn 12 becomes a quadratic model (von Caemmerer & Evans 1991), which was proposed by Ethier & Livingston (2004) as a method to fit g_m when it is assumed to be independent of C_i. In this method, J_max was estimated together with g_m and V_cmax from the A–C_i curves (Ethier et al. 2006), whereas in our method J_max is estimated beforehand from the CF data.

Equation 12, combined with Eqns 1, 2, 3a, 5 and 9, can be used to specifically estimate V_cmax, T_p and g_mo or δ using a least squares regression that minimizes the difference between the measured and estimated values for A. The previously estimated R_d, κ_2(LL), θ, J_max and S_c/o are used as inputs. All the data from the A–C_i and A–I_inc curves will be used in a single fitting, which will identify the transition among A_c, A_j and A_p limitations for each A–C_i and A–I_inc curve. This transition auto‐identifying procedure was occasionally used previously in parameterizing the FvCB model (e.g. Yin et al. 2004). Its advantages over the more commonly used method, in which data points of an A–C_i curve are first assigned to A_c‐, A_j‐ and A_p‐limited ranges and then each section is fitted separately, were discussed recently by Dubois et al. (2007).

As K_mC and K_mO are believed to be conservative among C₃ species, in principle they could be treated as constants in the curve‐fitting process. However, existing in vivo working values of K_mC and K_mO depend on whether g_m is assumed to be infinite (e.g. Bernacchi et al. 2001) or constant (e.g. von Caemmerer et al. 1994; Bernacchi et al. 2002), and are, therefore, not suitable if g_m is shown to vary. Thus, we will also try to fit K_mC and K_mO, which will require measurements at a minimum of two O₂ levels, with at least one set of data obtained under PR conditions to have a good estimate of K_mO. Data under PR conditions are also required because the estimation of g_m using the A_j part of the model works only in the presence of photorespiration.

Often, A–C_i curves are obtained from plants of the same genotype under a range of conditions (e.g. measured on leaves of different ages). K_mC and K_mO, like S_c/o, are constant for a specific genotype, although S_c/o varies with species (Jordan & Ogren 1981) and within a species (Pettigrew & Turley 1998). To maintain a single value for K_mC or K_mO at the same time allowing multiple values for other parameters (e.g. V_cmax) to be estimated from a single fitting to multiple A–C_i curves, we will use dummy variables Z_i where

(13)

where i = 1, 2, . . . , n for the case that n A–C_i curves are to be fitted, and where each curve will have its own value for V_cmax (i.e. V_cmax,i). The values of dummy variables Z_i (where i = 1, 2, . . . , n) are set to 1 for the i‐th curve, and 0 for other curves.

Once A is calculated from Eqn 12, g_m can be calculated using the equation obtained by replacing C_c in Eqn 11 with (C_i − A/g_m) and then solving for g_m:

((14a))

If g_mo = 0, Eqn 14a is simplified as

((14b))

MATERIALS AND METHODS

Plant growth conditions, treatments and experimental design

Eight containers (66.6 × 88.8 × 35.0 cm) were placed in a walk‐in growth chamber in a randomized block design. Seeds of wheat (cv. Minaret) were sown and then thinned to 508 seedlings m⁻². The soil per container had 3.47 g N, and was mixed with 28 g P₂O₅ and 8.4 g K₂O. In four containers, no extra N (low N treatment) was applied. In the other four (high N treatment), N was added twice: 5.13 g before sowing and 2.16 g at stem elongation per container. The chamber, illuminated by lamps, had an irradiance of 550–650 µmol m⁻² s⁻¹ at canopy level. The photoperiod was 15 h d⁻¹, the relative humidity was 75% and temperature was 16/11 °C (light/dark period) until the leaf 3 stage and 21/16 °C thereafter. The plants were watered daily.

GE and CF measurements

We selected leaf 4 and the flag leaf on the main stems. Measurements were performed at different stages (during elongating, when full‐grown and when senescing) on four leaves (one leaf from one replicate). Simultaneous GE and CF measurements at both 21% and 2% O₂ levels were made at flowering and 2 weeks after flowering for full‐grown flag leaves. Only GE data at 21% O₂ was collected for other stages of the flag leaf and for leaf 4.

We used an open GE system (Li‐Cor 6400; Li‐Cor Inc, Lincoln, NE, USA) and an integrated fluorescence chamber head (LI‐6400‐40). A nearby main‐stem leaf of the same position was positioned with the pre‐labeled leaf in order to cover the full area (2 cm²) of the chamber. All measurements were made at a leaf temperature of 25 °C and a leaf‐to‐air vapour pressure difference (VPD) of 1.0–1.6 kPa. For C_i response curves, C_a was increased stepwise: 50, 100, 150, 200, 250, 350, 500, 650, 1000 and 1500 µmol mol⁻¹, while keeping I_inc at 1000 µmol m⁻² s⁻¹. For the I_inc response curves, photon flux densities were in an increasing series: 0, 20, 50, 100, 150, 200, 500, 1000, 1500 and 2000 µmol m⁻² s⁻¹, at the same time keeping C_a at 350 µmol mol⁻¹ for measurements at 21% O₂, and keeping C_a at 1000 µmol mol⁻¹ for measurements at 2% O₂ to ensure an NPR condition. Light and CO₂ responses for the two O₂ levels were measured on the same leaf pairs. For the measurements at 2% O₂, a gas cylinder containing a mixture of 2% O₂ and 98% of N₂ was used. Gas from the cylinder was humidified and supplied to the Li‐Cor 6400 where CO₂ was blended with the gas. The effect of varying O₂ on the CO₂ and H₂O sensitivity of the Li‐6400 was corrected. All CO₂ exchange data were also corrected for: (1) leakage of CO₂ into and out of the leaf cuvette, using heat‐killed leaves according to Flexas et al. (2007a); and (2) diffusion of CO₂ respired by leaf tissue under the gasket (Pons & Welschen 2002).

The value of R_dk was measured 5 min after leaves had been placed in the darkness. Then F_v/F_m (the maximum quantum yield for PSII e^‐ transport for dark‐adapted leaves) was measured: the beam intensity was 0.1 µmol m⁻² s⁻¹, the saturating light pulse was >8500 µmol m⁻² s⁻¹ for 0.8 s. For measurements at each irradiance or CO₂ step, A was allowed to reach steady‐state, after which F_s was recorded from the leaf, and then a saturating light‐pulse (>8500 µmol m⁻² s⁻¹ for 0.8 s) was applied to determine .

Leaf N content measurements

The portion of the leaf pairs used for above measurements was cut. Its area was measured with a Li‐Cor area meter (Li‐Cor Inc). The leaf material was then weighed after drying at 70 °C to constant weight, and its total N content was analysed using an element analyser based on the micro‐Dumas combustion method.

Curve‐fitting methods

Simple linear regressions were performed using Microsoft Excel. Non‐linear fitting was carried out using the GAUSS method in PROC NLIN of SAS (SAS Institute Inc, Cary, NC, USA). The SAS codes can be obtained upon request.

RESULTS

Estimated photochemical efficiency of photosystem II under limiting light

The estimated Φ_2(LL) using Eqn 6, in which was used in place of Φ₂, did not vary with f_cyc, nor with Φ_1(LL) or β. So arbitrary values for f_cyc (0.0), Φ_1(LL) (0.95) and β (0.84) were used. Equation 6 described light responses of well with R² > 0.99. The estimated Φ_2(LL) differed slightly between stages, between N levels, and between PR and NPR conditions, and was slightly lower than F_v/F_m (Table 2). The latter suggests that either F_v/F_m differs from Φ_2(LL) or the response of Φ₂ to low light is more complex than that generated by a non‐rectangular hyperbola of e^‐ transport rate. The concurrently fitted θ₂ and J_2max are not given as they will not be used further.

Table 2. Values (standard error of estimate in brackets if applicable) of photosynthetic parameters estimated for flag leaves of wheat plants at flowering and 2 weeks after flowering grown at low nitrogen (N) and high N levels. Estimates may be made separately for photorespiratory (PR) and non‐photorespiratory (NPR) conditions if necessary. Parameter definitions and units are given in Table 1 or in the text

Parameter		Flowering		Two weeks after flowering
		Low N	High N	Low N	High N
F v/Fm	PR	0.777	0.789	0.731	0.765
	NPR	0.759	0.773	0.708	0.754
Φ 2(LL)	PR	0.748 (0.006)	0.748 (0.006)	0.713 (0.007)	0.734 (0.007)
	NPR	0.734 (0.007)	0.725 (0.008)	0.688 (0.007)	0.725 (0.008)
R d‐CF^a	PR	0.939	1.317	0.940	0.986
	NPR	1.375	1.573	0.778	0.949
R d‐Kok^b	PR	0.821	1.188	0.784	0.794
	NPR	1.169	1.380	0.488	0.614
R dk	PR	1.369	1.736	1.464	1.752
	NPR	1.510	1.656	1.346	1.606
s		0.380 (0.005)	0.403 (0.004)	0.336 (0.006)	0.386 (0.007)
κ 2(LL)	PR	0.284	0.301	0.239	0.283
	NPR	0.279	0.292	0.231	0.280
θ	PR	0.780 (0.046)	0.725 (0.049)	0.890 (0.049)	0.689 (0.067)
	NPR	0.812 (0.053)	0.869 (0.036)	0.879 (0.059)	0.745 (0.073)
J max ^c	PR	198.2 (6.3)	251.5 (8.9)	90.9 (2.9)	163.9 (5.9)
	NPR	156.8 (5.3)	181.1 (5.3)	92.4 (3.3)	142.4 (5.6)
b H		0.123	0.129	0.078	0.129
b L		0.171	0.193	0.094	0.153
V cmax ^d		58.5 (3.3)	65.8 (3.7)	34.7 (2.0)	51.3 (2.9)
T p ^d		10.9 (0.14)	13.2 (0.23)	6.8 (0.15)	9.5 (0.14)
J max ^e	PR	204.7 (7.8)	252.4 (9.1)	110.6 (6.4)	158.0 (4.0)
	NPR	152.9 (2.9)	185.7 (11.4)	87.7 (1.9)	131.5 (2.4)
V cmax ^e		58.5 (0.8)	65.8 (0.8)	34.7 (0.7)	51.8 (0.8)
T p ^e		11.1 (0.19)	12.9 (0.13)	7.4 (0.25)	10.1 (0.71)

• ^a R _d estimated from the A–(I_incΦ₂/4) relation using chlorophyll fluorescence (CF) information.
• ^b R _d estimated from the Kok method.
• ^c J _max estimated from CF measurements (see the text).
• ^d V _cmax and T_p estimated from the previously estimated J_max as input (see the text).
• ^e J _max, V_cmax and T_p estimated from gas exchange data only based on the variable g_m mode (see the text).

Estimated R_d

Using combined GE and CF data, R_d was estimated as the intercept of the linear regression of A against (I_incΦ₂/4), based on Eqns 7a and 7b, where is used for Φ₂. A sensitivity analysis with a range of values (0.7–1.3) for the ratio of to Φ₂ showed that the estimated R_d is totally insensitive to this ratio. With available data we compared this new method with the Kok method, which extrapolates the linear section of the light response of A to a zero light intensity to identify R_d (Villar, Held & Merino 1994).

An example using the two methods based on data of I_inc between 20 and 200 µmol m⁻² s⁻¹ is shown in Fig. 1. For NPR conditions, Eqn 7b described the data slightly, but consistently, better than the Kok method (with a higher r²) for the four cases (two flag‐leaf stages × two N treatments). The estimate of R_d obtained using the Kok method was consistently lower (by a factor of 0.78 on the average) than the estimate obtained using the A–(I_incΦ₂/4) relation, that is, Eqn 7b, for the NPR conditions (Table 2).

Figure 1

The estimation of day respiration (R_d) by linear regression based on two methods: (a) the A − (I_incΦ₂/4) relation according to Eqn 7a and 7b; and (b) the Kok method, for the flag leaves at 2 weeks after the flowering of plants grown under low nitrogen environments. Measurements were conducted under both photorespiratory (open circles and the dotted line) and non‐photorespiratory (filled circles and the full line) conditions. Data points represent measurements for individual replicate leaves. The negative intercept of the linear equation is the estimated R_d.

Linearity between A and (I_incΦ₂/4) was also evident for the PR condition (Fig. 1). Again, the A–(I_incΦ₂/4) relation, that is, Eqn 7a, generally gave slightly better fit than the Kok method, and the estimate of R_d by the Kok method was consistently lower; on the average it was 0.86 times the estimate given by Eqn 7a for PR conditions (Table 2).

Regardless of the method used, the estimated R_d did not differ significantly (P > 0.10) between PR and NPR conditions, and was consistently lower than R_dk under both conditions (Table 2). Both R_d and R_dk were higher for high N than low N leaves.

Estimated values for parameters s, κ_2(LL), θ and J_max

To minimize the chance of any influence by PET_a on the estimate of parameter s, we excluded data points from the A–I_inc curves that were obtained in high light (≥500 µmol m⁻² s⁻¹) at the same time including only those data measured at high C_i (>500 µbar) in the A–C_i curves at 2% O₂ (Fig. 2). The points from high C_i lay on the same locus as those measured at low irradiance, suggesting the same underlying relationship. This means that either the limitation at high C_i by TPU did not occur or that the limitation occurred but acted via a feedback effect on e^‐ transport such that A was limited by e^‐ transport. Data from high light intensities were mostly below the line (Fig. 2), suggesting the occurrence of PET_a at these light levels. With R_d estimated a priori as mentioned earlier, the estimated parameter s differed between low and high N, and between the two stages (Table 2).

Figure 2

Net CO₂ assimilation rate (A), measured under a non‐photorespiratory condition, of the flag leaves at flowering (FLW) and 2 weeks after flowering (2WA) for wheat plants grown under low nitrogen (N) and high N conditions as a function of I_incΦ₂/4. Data points for linear regression (lines) are from low light levels of the A–I_inc curve (open circles) and from high CO₂ level of A–C_i curve (open squares). The slope of each regression line is the estimated value for parameter s in Eqn 7b. Data points shown in filled triangles are from high light levels (i.e. 500, 1000, 1500 and 2000 µmol m⁻² s⁻¹) of the A–I_inc curve. The black square in each panel represents average respiration in the darkness (R_dk) of replicate leaves. Other data points represent measurements for individual replicate leaves.

As expected, the estimate of parameter s varied proportionally with an uncertain factor, the ΔF/F′_m : Φ₂ ratio. However, this uncertainty does not have any impact on the calculation of J and κ_2(LL) because a change in parameter s is cancelled out by a proportional change in Φ₂ or Φ_2(LL) (see Eqns 8a and 8c). The κ_2(LL) calculated from Eqn 8c ranged from 0.231 to 0.301 mol e^‐ (mol photon)⁻¹ (Table 2) and correlated with leaf N content (Fig. 3). The small differences in Φ_2(LL) between PR and NPR conditions result in similarly small differences in κ_2(LL).

Figure 3

Estimated conversion efficiency of limiting incident light into linear electron transport of photosystem II [κ_2(LL)] under the photorespiratory condition in relation to leaf nitrogen content (N). Points are representing estimates for the full‐grown flag leaves at flowering and 2 weeks after flowering for wheat plants grown under low nitrogen and high nitrogen conditions.

Using the estimated parameter s, the potential e^‐ flux J can be calculated as sI_incΔF/F′_m. Equation 9 was then fitted to the calculated J across the range of light levels, to estimate parameters θ and J_max. The estimated θ were generally above the common value 0.7, and were higher for NPR than for PR conditions (Table 2). Not surprisingly, the estimated J_max were higher in high N than in low N leaves, and higher at flowering than 2 weeks later. Except for the low N leaves at 2 weeks after flowering, J_max were higher under PR than NPR conditions (Table 2). Figure 4 illustrates the reduced J_max and the higherθ (the latter meaning that J reaches saturation at lower light levels) under NPR conditions.

Figure 4

The response of potential e^‐ transport rate J (i.e. LET plus PET_a) to incident irradiance, under photorespiratory (open symbols and the dotted curve) and non‐photorespiratory (filled symbols and the full curve) conditions, for the flag leaves at flowering of wheat plants grown with high nitrogen supply. Data are shown as means of three to four replicate leaves (±standard errors). The curves are drawn from Eqn 9 using parameter values fitted to the data points.

Estimated S_c/o

The initial, linear part of the A–C_i curves is clearly seen in our data. The slope of this linear phase (i.e. the initial carboxylation efficiency) was consistently higher at low than at high O₂ levels, that is, b_L > b_H (Table 2). The present Rubisco kinetics theory attributes the difference between b_L and b_H to the ‘residual carboxylation resistance’ rather than to the diffusion resistance 1/g_m (see Appendix). As shown earlier, R_d did not differ significantly between 2% and 21% O₂ levels, so Eqn 10 can be used to estimate S_c/o. As S_c/o is expected to be constant for a specific genotype, data from the four situations (two N treatments × two flag‐leaf stages) were pooled for this curve fitting, using the previously estimated values of R_d, b_L and b_H (Table 2) as inputs. The estimated S_c/o (at 25 °C) was 3.13(0.18) mbar  µbar⁻¹, yielding 34 µbar for at the atmospheric O₂ level. The estimated S_c/o using the L method was 2.93 mbar µbar⁻¹; so, the L method underestimated S_c/o by about 7% for our data.

Parameterization of the g_m model and estimated V_cmax and T_p

The calculated g_m, using the variable J method, varied with C_i and light levels (Fig. 5 for the 21% O₂ level), in line with the report of Flexas et al. (2007b). The g_m obtained at 2% O₂ had a similar trend, but not surprisingly were more scattered. Variations of g_m with C_i or I_inc mean that parameter δ in Eqn 11 is not equal to zero. The value for g_m at low light was close to zero, meaning that g_mo should be close to zero.

Figure 5

The calculated mesophyll diffusion conductance (g_m) using the variable J method (Harley et al. 1992a) by the equation: (filled circles) based on the average of A, C_i and of three to four replicate leaves obtained at 21% O₂, and g_m calculated according to Eqn 14b using parameters fitted to the model presented in this study (curves) in which δ = 2.539. Panels (a), (c), (e) and (g) are for the responses to C_i, and panels (b), (d), (f) and (h) are for the responses to incident irradiance level. The estimates were made for the flag leaves of low nitrogen (N) supply at flowering (a,b), those of high N supply at flowering (c,d), those of low N supply 2 weeks later (e,f), and those of high N supply 2 weeks later (g,h).

Because there are uncertainties related to K_mC and K_mO and any TPU limitation was mirrored by e^‐ transport, we firstly used the A_j part of Eqn 12 to fit δ and g_mo to the A–C_i and A–I_inc curves, using the previously estimated R_d, κ_2(LL), θ, J_max and S_c/o as inputs. We excluded the data of 500, 1000 and 1500 µmol m⁻² s⁻¹ in the A–I_inc curves, as for these I_inc, A was not limited by e^‐ transport under NPR conditions (Fig. 2). The estimated g_mo did not differ significantly from zero (P > 0.05), and the estimated δ did not differ significantly between the two O₂ levels (P > 0.05) and varied slightly between the four situations (two N treatments × two leaf stages), ranging from 1.01 (0.04) to 1.98 (0.25). The fact that δ is not equal to zero confirms the variable g_m mode.

Next, the full model in which g_mo was fixed at zero, combined with other required equations, was fitted to all data. In total eight A–C_i curves and eight A–I_inc curves (i.e. two N treatments × two leaf stages × two O₂ levels) were used for the curve fitting applying the dummy variable approach. The A_p limitation was also included in this step as A–C_i curves go very flat at high C_i (Fig. 6) indicating that TPU limitation occurs. The estimated δ did not differ significantly among the four situations (i.e. two N treatments × two leaf stages), and the overall estimate forδ was 2.539 (0.423). A higher δ with use of the full model probably indicates that PET_a occurred at low C_i values, which would not be revealed if only the A_j‐part of the model is used. The estimated K_mC was 168 (17) µbar and K_mO was 473 (48) mbar. Our estimated K_mC is just outside the lower range, and K_mO is at the upper side of the range, of in vitro measurements (reviewed by von Caemmerer et al. 1994). Our K_mC is also lower than, and K_mO is higher than, their in vivo estimates based on the constant g_m mode (von Caemmerer et al. 1994; Bernacchi et al. 2002). Partly because of our low K_mC and high K_mO, the estimated V_cmax was low (Table 2). The estimated T_p ranged from 6.8 to 13.2 µmol m⁻² s⁻¹. The overall model fit was good (R² = 0.963), although the model may not describe individual curves exactly (Fig. 6).

Figure 6

Net CO₂ assimilation rates (A) of the flag leaves of low nitrogen (N) supply at flowering (a,b), of high N supply at flowering (c,d), of low N supply 2 weeks later (e,f), and of high N supply 2 weeks later (g,h), under 2% O₂ (filled symbols and continuous curve) and 21% O₂ (open symbols and dotted curve) conditions. Data are shown as means of three to four replicate leaves (±standard errors). Panels (a), (c), (e) and (g) are for the responses of A to C_i, and panels (b), (d), (f) and (h) are for the responses to incident irradiance level. The curves are drawn from the model using fitted parameter values. Note that in panels (b), (d), (f) and (h), responses to irradiance under 2% O₂ condition were made using a high C_a (1000 µmol mol⁻¹) whereas those under 21% O₂ condition were measured using a C_a of 350 µmol mol⁻¹ (see the text); so the difference between the two should attribute to the loss in A caused by photorespiration. The difference between the two curves in panels (a), (c), (e) and (g) should not be considered as the difference in A between photorespiratory and non‐photorespiratory conditions because photorespiration also occurs at low C_i even under the 2% O₂ condition, and it is suppressed to a large extent at high C_i even under the 21% O₂ condition.

When assuming C_c = C_i (i.e. infinite g_mo and δ = 0), the model gave a significantly worse fit (P < 0.001), providing statistical support for the need to discard this restricted model. The estimated K_mC and K_mO values obtained with the infinite g_m mode were 213 (17) µbar and 409 (37) mbar, respectively. The estimated V_cmax was reduced by ca. 5% and virtually no change was obtained for T_p, compared with their estimates from the variable g_m mode.

Our estimates for K_mC and K_mO were obtained using data of only two O₂ levels, and thus had relatively high standard errors and should be considered as tentative. As according to the variable g_m mode, g_m is higher for the A_c‐ than for the A_j‐limited range, it is thought that K_mC and K_mO values obtained using this mode lie between their lower values based on the constant g_m mode (e.g. 272 µbar and 166 mbar, Bernacchi et al. 2002) and higher values based on the infinite g_m mode (e.g. 405 µbar and 278 mbar, Bernacchi et al. 2001). Sensitivity analysis showed that the model fit was little affected by K_mC or K_mO within the ranges, because many other parameters had already been determined. A change in K_mC from 272 to 405 µbar in six steps resulted in little change in parameter δ or K_mO, but not surprisingly had a large effect on V_cmax (an increase of ca. 33%). A similar change in K_mO from 166 to 278 mbar resulted in little change in δ, a ca. 30% increase in K_mC and a ca. 8% increase in V_cmax. A combined parallel increase in K_mC and K_mO within these ranges resulted in little change in δ and ca. 20% increase in V_cmax. Because of sensitivities of V_cmax to K_mC and K_mO (and especially to K_mC), both these parameters would need to be estimated with data obtained at more O₂ levels if the variable g_m mode were to be proven to be a general phenomenon.

Values of J_max, V_cmax and T_p estimated from the GE data only

We used our estimated R_d, κ_2(LL), θ, S_c/o, K_mC, K_mO, g_mo and δ as inputs, to estimate J_max, V_cmax and T_p from A–C_i and A–I_inc curves. As J_max estimated from CF data differed between PR and NPR conditions, a dummy variable approach was employed to allow the same estimate for V_cmax but different values of J_max. The estimated J_max for the NPR condition was about 20% less than for the PR condition (Table 2). The estimated J_max values from the GE data agreed with those from CF. Similarly, the estimated V_cmax and T_p from the GE data agreed with those earlier estimates (Table 2). Assuming an infinite g_m resulted in a significantly poorer fit (P < 0.001) in all the four cases. If K_mC and K_mO were adjusted, an infinite g_m mode reduced V_cmax by ca. 5%, reduced J_max by ca. 2 and 10% for NPR and PR conditions, respectively, and resulted in virtually no change in T_p.

Parameter values estimated for other leaf stages and positions

For the fourth leaf and for the young and old stages of the flag leaf we did not conduct CF and NPR measurements, so we were unable to estimate their κ_2(LL), θ and R_d. However, using the GE measurements of the flag leaf at flowering and 2 weeks after, the J_max, V_cmax and T_p estimated from only the data of 21% O₂ were found to agree well with the estimates made from the combined data of both 21 and 2% O₂ levels (results not shown). Thus we estimated J_max, V_cmax and T_p for other leaf stages and positions using the GE data at 21% O₂ and an estimate of κ_2(LL)[ranging from 0.217 to 0.302 mol e^‐ (mol photon)⁻¹] calculated from the linear relation of Fig. 3. Further, we set θ at 0.773 (the average in Table 2 for the PR condition), and used the estimated R_d from the Kok method but with the correction factor 0.86 (R_d calculated with the Kok method was on average 0.86 times R_d from the A–(I_incΦ₂/4) relation for the PR condition, Table 2). S_c/o, K_mC, K_mO, g_mo and δ were taken as the earlier estimates.

The estimates of J_max, V_cmax and T_p depended on N treatments, leaf positions and stages, and linearly correlated to leaf N content with J_max = 5.75 + 99.38N, r² = 0.70, V_cmax = 4.36 + 30.40N, r² = 0.68 and T_p = 0.93 + 5.37N, r² = 0.73; n = 14. This indicates that the temporal and spatial variation in the photosynthetic capacity within the wheat canopies grown at different N levels can be largely attributed to the variation in their leaf N content. The estimated J_max, V_cmax and T_p were highly correlated (r > 0.96). The mean J_max : V_cmax ratio was 3.1 mol e^‐ (mol CO₂)⁻¹. This ratio is higher than the commonly reported ratio (ca 1.5) using K_mC and K_mO of von Caemmerer et al. (1994) for the constant g_m mode, partly because of the uncertainty in our K_mC and K_mO estimates, and partly because of the variable g_m mode. Adjusting to an infinite g_m with the C_i based K_mC (213) and K_mO (409) estimates did not change the ratio significantly.

DISCUSSION

Estimating the parameters of the FvCB model is pivotal to predicting the rates of and analysing the physiological limitations to C₃ photosynthesis under various environmental conditions. Previous reports have mostly estimated a limited number of parameters of this model, mainly using the GE data. We present an integrated approach, combined with the dummy variable method, to estimate eight parameters [R_d, κ_2(LL), θ, J_max, S_c/o, V_cmax, T_p and g_m] from combined GE and CF measurements. This approach was applied to analyse photosynthetic rates for leaves differing in age and position in the canopies of wheat grown at two N levels.

Day respiration

Previously R_d has been estimated with the L method, the Laisk (1977) method, or the Kok method. Both the L and Laisk (1977) methods estimate R_d using the linear A–C_i relation obtained at very low C_i values. Our proposed method based on Eqns 7a and 7b, like the Kok method, explores the linear part of an A–I_inc curve but uses both GE and CF data. Equations 7a and 7b assume that over a certain range of values the relationship between PSII e^‐ transport and CO₂ fixation is linear. Use of CF as in Eqns 7a and 7b to estimate R_d may be subject to criticism, as the linearity between and Φ_CO2 has been reported to break at low light levels (e.g. Seaton & Walker 1990). However, the reported non‐linearity between and Φ_CO2 at low light may be, at least partly, caused by the uncertainty in estimating R_d (Edwards & Baker 1993), as R_d accounts for a large portion of the flux at low light. An advantage of Eqns 7a and 7b over the Kok method is that Eqns 7a and 7b use the information of CF (in addition to GE) to compensate for any non‐linearity between A and I_inc. This non‐linearity is caused by the loss of Φ₂ that develops as the irradiance increases above the light‐limited region. Equations 7a and 7b can be applied to a wider range of I_inc than the Kok method, for which the maximum irradiance is around 100 µmol m⁻² s⁻¹ (Björkman & Demmig 1987; Long, Postl & Bolhár‐Nordenkampf 1993). Moreover, the leaf‐to‐leaf variation in photosynthetic rates can be reflected by ΔF/F′_m; as a result, our method can give a slightly better fit than the Kok method when data of replicate leaves are analysed (Fig. 1). Further tests would be needed to examine the general reliability of our method for estimating R_d.

Our estimated R_d did not differ significantly (P > 0.10) between PR and NPR conditions (Table 2). This agrees with other reports (Brooks & Farquhar 1985; Kirschbaum & Farquhar 1987), although both increased CO₂ (Villar et al. 1994) and lowered O₂ (Sharp, Matthews & Boyer 1984) were reported to inhibit R_d to a certain extent.

The estimated R_d was lower than R_dk (Table 2), agreeing with others using the Laisk (1977) method (Sharp et al. 1984; Brooks & Farquhar 1985; Villar et al. 1994; Villar, Held & Merino 1995; Piel et al. 2002) or the L method (Laisk & Loreto 1996; Laisk et al. 2002). Whether this difference is caused by a real inhibition or an artifact has often been debated when direct measurements of R_d have been performed (Loreto, Delfine & Marco 1999). An in vivo metabolic study (Tcherkez et al. 2005) did support the inhibition of R_d by light, and indicated that the main inhibited steps were the entrance of hexose molecules into the glycolytic pathway and the Krebs cycle. Another complicating factor is that the extent to which the intensity of irradiance inhibits R_d is uncertain; in some cases (e.g. Haupt‐Herting, Klug & Fock 2001) inhibition appears not to be intensity dependent, whereas in other cases it does (Brooks & Farquhar 1985; Kirschbaum & Farquhar 1987; Laisk & Loreto 1996; Laisk et al. 2002) although how it responds to light intensity was hard to quantify. As in the FvCB model we assume that R_d is independent of light intensity.

Electron transport parameters

Parameter κ_2(LL)– the conversion efficiency of incident irradiance into linear e^‐ transport – has been commonly used, but only approximately determined, in existing applications of the FvCB model. We have presented a formal procedure for estimating this parameter. NPR measurements were included so that κ_2(LL) could be estimated less ambiguously, and the extended model places this parameter into a more realistic physiological context. Separation of PET_a from the basal components of alternative e^‐ transports was proposed, and care was taken to avoid the effect of photorespiration and light‐ and CO₂‐dependent PET_a on the estimate of κ_2(LL) by not using data of regions of A–C_i and A–I_inc curves where increases in PET_a and photorespiration are possible. This procedure obviates the need for knowing underlying parameters β, ρ₂, f_cyc and f_pseudo(b). The assessment of ρ₂, f_cyc and f_pseudo(b) would need detailed measurements such as data of Laisk et al. (2007), using the principle described by Yin et al. (2006) on the basis of the assumption that β can be represented by leaf absorptance. However, any existence of non‐photosynthetic pigments in leaves would mean that β is smaller than the measured leaf absorptance.

The estimated κ_2(LL) (Table 2) are generally higher than the values of 0.24 (Harley et al. 1992b; Warren 2004) and 0.18 (Wullschleger 1993) used in some studies, where this parameter was fixed as constant across stages and growth CO₂ levels (Harley et al. 1992b), across N treatments (Warren 2004) or across species (Wullschleger 1993). Our results for wheat showed that like V_cmax, J_max and T_p, κ_2(LL) is related to leaf N (Fig. 3). This correlation is possibly because some of the components of κ_2(LL) vary with leaf N: the parameter β depends on leaf chlorophyll content (Evans 1993) and this is highly dependent on leaf N, and we found that Φ_2(LL) also varied with leaf N. If leaf absorptance is measured, our procedure can still be applied as long as leaf absorbed irradiance is used in Eqns 7 to 9 in places of I_inc, and parameter β should then be interpreted as the fraction of the leaf‐absorbed irradiance attributable to photosynthetic pigment absorptance.

Little sensitivity of A to C_i over the high C_i range of A–C_i curves (Fig. 6) suggests that TPU limitation occurred. The fact that in constructing calibration curves the points at high C_i lay on the same locus as those measured at low irradiance (Fig. 2) confirms the assertion (e.g. Sharkey, Berry & Sage 1988; Sharkey et al. 2007) that e^‐ transport rate is regulated by the TPU limitation. In relation to this, our estimated θ were higher and J_max were lower for NPR than for PR conditions (Table 2, Fig. 4). The J_max estimate was in line with the data of Laisk et al. (2006) that light‐saturated e^‐ transport rate was higher at normal than low O₂ levels. A reduced J_max and the faster irradiance saturation under NPR conditions (Fig. 4) could be because of the feedback effect on e^‐ transport when sink activity is incapable of accepting the increased CO₂ fixation when photorespiration is eliminated (e.g. Sharkey et al. 1988; Stitt 1991; Harbinson 1994). Further physiological studies are needed to understand this difference.

Rubisco parameters and mesophyll diffusion conductance

Given the aforementioned weakness of the L and Laisk (1977) methods in estimating S_c/o, we have described a new method for estimating this parameter using curve fitting based on Eqn 10. The assumption about R_d and g_m for the validity of Eqn 10 can be generally met (see Appendix); if not, we suggest S_c/o as an additional parameter to be estimated in the subsequent step, based on Eqn 12. Our curve‐fitting exercises show that these two approaches yield virtually the same estimate of S_c/o. Equation 10 allows the separate estimation of S_c/o that reduces the likelihood of over‐fitting to Eqn 12 in the next step. The estimated S_c/o (at 25 °C) was 3.13 mbar µbar⁻¹, which is within the range of in vitro values reported for various C₃ species (von Caemmerer et al. 1994; von Caemmerer 2000) and the same as an in vitro value 3.13 mbar µbar⁻¹ (after converting the units) reported by Makino, Mae & Ohira (1988) for wheat at 25 °C.

We have presented a model, Eqn 11, which accommodates either a constant or a variable g_m mode. Our results for wheat (Fig. 5), using both the variable J method and curve fitting, suggest that g_m is variable, in line with of results of Flexas et al. (2007b) for six other species but in contrast with the earlier assumption that g_m is independent of CO₂ and light levels. This highlights an important uncertainty in our understanding of gaseous diffusion processes in leaves. Further studies would be needed to elucidate whether this uncertainty is because of our incomplete understanding of the physiological mechanism for g_m or any faulty assumption the methods for estimating g_m rely on. Nonetheless, the results we obtained when fitting the variable g_m model with a constant δ in Eqn 14b supports the proposition that g_m correlates with leaf photosynthetic capacity (Loreto et al. 1992, 1994; Epron et al. 1995; Evans & von Caemmerer 1996; Piel et al. 2002; Singsaas, Ort & Delucia 2003; Ethier et al. 2006) although this correlation has not always been observed (Niinemets et al. 2005).

If the variable g_m mode turns out to be the general case, our study highlights the need to re‐estimate in vivo the true K_mC and K_mO values, as existing values were based on an infinite g_m or a constant g_m mode. When we assumed an infinite g_m, our estimated V_cmax was reduced by ca. 5%, disagreeing with reports (Piel et al. 2002; Ethier & Livingston 2004; Warren 2004) that V_cmax was underestimated by 40–50% if g_m was assumed infinite. However, these earlier reports assumed that K_mC and K_mO were constant regardless of the scenarios set for g_m. Furthermore, these reports assumed that g_m was constant, whereas our results suggest that g_m is variable and has a relatively high value at low C_i or high light levels (Fig. 5) where A is Rubisco activity limited. A higher g_m would yield an estimate of V_cmax closer to that obtained from an infinite g_m mode. This reasoning is supported by our observation that when using the same K_mC and K_mO, the C_i‐based V_cmax was ca. 45% of the estimate using a constant g_m.

Whereas Flexas et al. (2007b) and our results (Fig. 5) show that g_m and g_s respond in parallel to changes in CO₂ and light levels, a recent review of Flexas et al. (2008) showed that g_m and g_s vary in parallel to a number of abiotic factors such as water stress, salt stress and temperature. It is unknown whether g_m also resembles g_s in responding to VPD. Warren (2008) showed a strong response of g_s, but little response of g_m, to VPD, whereas an earlier report (Bongi & Loreto 1989) showed that g_m decreased at increased VPD. If the latter phenomenon is general, our model could be extended by altering parameter δ in Eqn 11 to become a function of VPD. This again merits more experimental and modelling studies.

Assessment of alternative electron transports

It has been reported that g_m obtained by the variable J method was similar to the value by other methods (e.g. Loreto et al. 1992; Bernacchi et al. 2002), for example, the constant J method that is applicable within the high C_i range (Harley et al. 1992a). We reason that g_m estimated by the variable J method, which is often recommended as suitable in the low C_i range, may have been confounded by PET_a occurring at low C_i. Any PET_a, if not properly accounted for, will lead to an underestimation of g_m; so, a high g_m at the low C_i could have been cancelled out by an underestimation because of the existence of PET_a. This may explain why the calculated g_m sometimes declined at C_i close to the CO₂ compensation point (Fig. 5a,g). This decline at low C_i was also clearly shown by Flexas et al. (2007b). Our model‐based approach, which implicitly accounts for PET_a, produces a continuously declining g_m with increasing CO₂ and with decreasing light intensity (Fig. 5). However, leaf‐to‐leaf variations and uncertainties associated with K_mC and K_mO make it difficult to obtain the exact fraction of potential e^‐ flux J that was attributable to PET_a.

Besides PET_a, alternative e^‐ transport also contains basal components in the form of PET_b or CET. The fractions of these two components in the total e^‐ flux of PSI are lumped in parameter s. Our parameter s is mathematically equivalent to the ratio of 4(A + R_d) to (I_incΦ₂) calculated by Miyake et al. (2004, 2005) who set R_da priori. The estimated values of s (Table 2) are lower than 0.44, the estimate of Miyake et al. (2004) for tobacco (Nicotiana tabacum L.). This could be because of the differences in species, leaf age and N levels between the two studies. However, Miyake et al. (2004) assessed A by means of O₂ evolution, so their estimate of 0.44 probably refers only to ρ₂β. The lower s in our study could be because of the significant PET_b, which results in O₂ evolution but not in CO₂ fixation (cf. Yin et al. 2006). Therefore, applying the value 0.44 to the analysis of Miyake et al. (2005) – where A was measured in CO₂ uptake – may be inappropriate.

Neither PET_b nor CET can be exactly determined from our data. As a result, the Q‐cycle activity required for the demands of metabolism supported by LET cannot be assessed using Eqn 4d. However, the extended model can explore the range of variation for uncertain parameters (Yin et al. 2006). If we assume no CET nor PET_b and an absolute efficiency for Φ_1(LL) (Trissl & Wilhelm 1993) and use (1 − f_cyc)/[Φ_2(LL)/Φ_1(LL) + (1 − f_cyc)] as the theoretical value for ρ₂ (Yin et al. 2006), the estimated β in Eqn 8a varied from 0.54 to 0.68, depending on N treatment and leaf stages. These are highly unlikely low values, indicating that CET or PET_b or both must have existed in leaves if Φ₂ can be accurately represented by ΔF/F′_m and Φ_1(LL) approaches to 1.

ACKNOWLEDGMENTS

We thank Dr A. Schapendonk and Mr S. Pot of Plant Dynamics B.V., and Dr T. Pons and Mr R. Welschen of Utrecht University for their support. The work was partially financed by the European research project WatNitMed (INCO‐CT‐2004‐509107). The stay of P. Romero at the Wageningen University was partially financed by Instituto Nacional de Investigación y Tecnología Agraria y Alimentaria (INIA) with the project RTA2005‐00103‐00‐00.

Appendix

Deriving Eqn 10 and the partitioning of initial total mesophyll resistance

The linear part of the A–C_i curves at O_H and O_L (see Table 1 for all variables) can be expressed by definition, respectively as

((A1))

((A2))

Equations A1 and A2 can be rewritten, respectively, as

((A3))

((A4))

Subtracting Eqn A3 from Eqn A4 gives

((A5))

Note that C_i* and differ as follows: . However, if both g_m at the CO₂ compensation point and R_d are independent of the O₂ levels, C_i*H–C_i*L equals Γ_*H–Γ_*L. This assumption for R_d, which is supported by our estimates (see Results), is also used in the FvCB model and the L method. The assumption about the common g_m at C_i*H and C_i*L can be supported theoretically (see further discussion). Thus, the term C_i*H–C_i*L in Eqn A5 can be expressed as 0.5(O_H–O_L)/S_c/o, and Eqn A5 can be rewritten as

((A6))

Solving A_H from Eqn A6 gives Eqn 10.

The initial slope of an A–C_i curve can be theoretically derived from the special case of the A_c part of Eqn 12, where g_m = g_mo and δ = 0. The inverse of this slope, called the total mesophyll resistance (Laisk & Loreto 1996; Laisk et al. 2002, 2006), can be expressed as

((A7))

The second term on the right side of Eqn A7 has been called ‘residual carboxylation resistance’ (von Caemmerer et al. 1994), and depends on the O₂ level. So, the difference between b_L and b_H can be considered to be caused by the second term, rather than by the first term for the diffusion component 1/g_m in the initial phase of the A–C_i curve.

It is difficult to experimentally confirm the independence of g_m at C_i* on O₂. At the ambient CO₂ level, the independence has been supported by some experimental estimates (e.g. Loreto et al. 1992) or been assumed for the analysis to estimate g_m (e.g. Laisk et al. 2006), although Evans et al. (1994) reported a larger g_m in 2% O₂ compared with 21% O₂ when an ambient CO₂ was at 350 µbar.