On the need to incorporate sensitivity to CO₂ transfer conductance into the Farquhar–von Caemmerer–Berry leaf photosynthesis model

ABSTRACT

Virtually all current estimates of the maximum carboxylation rate (V_cmax) of ribulose‐1,5‐bisphosphate carboxylase/oxygenase (Rubisco) and the maximum electron transport rate (J_max) for C₃ species implicitly assume an infinite CO₂ transfer conductance (g_i). And yet, most measurements in perennial plant species or in ageing or stressed leaves show that g_i imposes a significant limitation on photosynthesis. Herein, we demonstrate that many current parameterizations of the photosynthesis model of Farquhar, von Caemmerer & Berry (Planta 149, 78–90, 1980) based on the leaf intercellular CO₂ concentration (C_i) are incorrect for leaves where g_i limits photosynthesis. We show how conventional A–C_i curve (net CO₂ assimilation rate of a leaf –A_n– as a function of C_i) fitting methods which rely on a rectangular hyperbola model under the assumption of infinite g_i can significantly underestimate V_cmax for such leaves. Alternative parameterizations of the conventional method based on a single, apparent Michaelis–Menten constant for CO₂ evaluated at C_i[K_m(CO₂)_i] used for all C₃ plants are also not acceptable since the relationship between V_cmax and g_i is not conserved among species. We present an alternative A–C_i curve fitting method that accounts for g_i through a non‐rectangular hyperbola version of the model of Farquhar et al. (1980). Simulated and real examples are used to demonstrate how this new approach eliminates the errors of the conventional A–C_i curve fitting method and provides V_cmax estimates that are virtually insensitive to g_i. Finally, we show how the new A–C_i curve fitting method can be used to estimate the value of the kinetic constants of Rubisco in vivo is presented

Abbreviations

A _c

RuBP‐saturated CO₂ assimilation rate

A _j

RuBP‐limited CO₂ assimilation rate

A _n

net CO₂ assimilation rate

C

gas phase CO₂ concentration in an in vitro assay media

C _c

chloroplastic CO₂ concentration

C _i

intercellular CO₂ concentration

C _i*

intercellular CO₂ photocompensation point

Γ*

chloroplastic CO₂ photocompensation point

Γ

CO₂ compensation point

g _i

CO₂ transfer conductance

g _s

stomatal conductance

I

incident irradiance

J

CO₂‐saturated electron transport rate

J _max

maximum, light‐ and CO₂‐saturated electron transport rate

α

quantum efficiency (number of electrons transferred per incident photon); θ, curvature factor of the non‐rectangular hyperbola describing the light response of J

K _c and K_o

Michaelis–Menten constants for RuBP carboxylation and oxygenation, respectively

K _m(CO₂)_i

apparent Michaelis–Menten constant for CO₂ evaluated at C_i

O

O₂ concentration

R _d

mitochondrial respiration in the light

RuBP

ribulose‐1,5‐bisphosphate

Rubisco

ribulose‐1,5‐bisphosphate carboxylase/oxygenase

S _c/o

Rubisco specificity factor

V _cmax

maximal carboxylation rate

W _c

RuBP‐saturated carboxylation rate.

INTRODUCTION

Over the past two decades, the photosynthetic CO₂ responses of numerous C₃ plant species have been evaluated at the whole‐leaf level in relation to the CO₂ concentration in the intercellular airspace subtending the stomata (C_i) and fitted to the photosynthesis model of Farquhar, von Caemmerer & Berry (1980). These ‘A–C_i curve’ (net CO₂ assimilation rate of a leaf –A_n– as a function of C_i) analyses have been invaluable for elucidating and quantifying in vivo the fundamental biochemical processes underlying the photosynthetic responses of plants to various environmental conditions (von Caemmerer 2000). Several comparative studies document the species variability and the temperature or CO₂ sensitivity of two key model parameters derived from A–C_i curves: V_cmax, the maximum carboxylation rate of ribulose‐1,5‐bisphosphate carboxylase/oxygenase (Rubisco), and J_max, the maximum electron transport rate (e.g. Wullschleger 1993; Leuning 1997, 2002; Wohlfahrt et al. 1999; Medlyn et al. 1999, 2002; Bunce 2000). Such studies are particularly useful to global climate change modellers since the latest generation of surface carbon exchange models incorporate the equations of Farquhar et al. (1980) (Sellers et al. 1997; Pitman 2003). However, these estimates of V_cmax and J_max, and indeed all current canopy or landscape scale carbon exchange models, implicitly assume that once CO₂ has been transported into the leaf's internal airspace subtending the stomata, the diffusional resistance that remains between this point and the carboxylation sites is insignificant and can be ignored. Recent research has shown that this is often not the case. In perennial plants, for example, most reported values of leaf internal CO₂ transfer conductance (g_i) fall below 0.2 mol m⁻² s⁻¹(Table 1), which is sufficiently small, given the photosynthetic capacity of the leaves, to impose a significant limitation on photosynthesis (e.g. Lloyd et al. 1992; Epron et al. 1995; Roupsard, Gross & Dreyer 1996; Miyazawa & Terashima 2001; Warren et al. 2003). Furthermore, Bernacchi et al. (2002) have recently shown that g_i is strongly dependent on leaf temperature (see also Makino, Nakano & Mae 1994) and becomes increasingly limiting to photosynthesis as temperature rises. In such cases, failure to account for the finite g_i when estimating V_cmax from A–C_i curves results in erroneously low values (Epron et al. 1995; Centritto, Loreto & Chartzoulakis 2003; Warren et al. 2003).

Table 1. Literature survey of CO₂ transfer conductance (g_i) measurements in C₃ plant species

Species	g i range^a (mol m^–^2 s^–^1)	Species	g i range^a (mol m^–^2 s^–^1)
Herbaceous annuals		Deciduous, cont’d
Monocots		Populus deltoides×nigra^8	0.50
Oryza sativa^1	0.39–0.50	Prunus persica^3	0.27–0.43
Triticum aestivum^1	0.32–0.53	Quercus ilex^4^,^8	0.10–0.11
Triticum durum^6	0.35–0.60	Quercus petraea^8	0.24
Triticum spp.^4	0.638	Quercus robur^8^,^1^2	0.07–0.27
Dicots		Quercus rubra^2^,^4	0.10–0.18
Xanthium strumarium^4	0.37–0.60	Vitis vinifera^16	0.07–0.21
Beta vulgaris^4	0.34	Evergreens
Cucumis sativus^4	0.45	Arbutus unedo^4	0.16
Glycine max^1^2	0.32	Camellia japonica^1^1	0.07–0.12
Nicotiana tabacum^1^,^5^,^1^2	0.19–0.50	Castanopsis sieboldii^1^1^,^1^5	0.02–0.11
Phaseolus vulgaris^1	0.17–0.29	Cinnamomum camphora^1^1	0.06
Raphanus sativus^1	0.25–0.38	Citrus aurantium^4	0.02
Spinacia oleracea^1^0	0.40	Citrus limon^3	0.15–0.18
Vicia faba^4^,^1^8	0.34–0.46	Citrus paradisi^3	0.15–0.26
Herbaceous perennials		Eucalyptus blakelyi^1	0.16–0.19
Polygonum cuspidatum^1^4	0.08–0.19	Eucalyptus globulus^2^,^4	0.11–0.12
Woody perennials		Hedera helix^4	0.15
Deciduous		Ligustrum lucidum^1^1	0.07–0.11
Acer mono^1^3	0.07–0.15	Macadamia integrifolia^3	0.11–0.13
Alnus japonica^1^3	0.08–0.13	Nerium oleander^4	0.22
Castanea sativa^7^,^9	0.02–0.15	Olea europea^1^9	0.06–0.12
Fagus sylvatica^7	0.10	Pseudotsuga menziesii ^20	0.14–0.20
Juglans nigra×regia^1^7	0.09–0.19	Quercus glauca ^1 ^1	0.07–0.08
Juglans regia^1^7	0.08–0.22	Quercus phillyraeoides ^1 ^1	0.14
Populus maximowiczii^1^3	0.04–0.20	Simmondsia chinensis ^4	0.03

• ¹ von Caemmerer & Evans (1991);
• ² Harley et al. (1992a);
• ³ Lloyd et al. (1992);
• ⁴ Loreto et al. (1992);
• ⁵ Evans et al. (1994);
• ⁶ Loreto et al. (1994);
• ⁷ Epron et al. (1995);
• ⁸ Roupsard et al. (1996);
• ⁹ Lauteri et al. (1997);
• ¹⁰ Delfine et al. (1999);
• ¹¹ Hanba, Miyazawa & Terashima (1999);
• ¹² Gillon & Yakir (2000);
• ¹³ Hanba et al. (2001);
• ¹⁴ Kogami et al. (2001);
• ¹⁵ Miyazawa & Terashima (2001);
• ¹⁶ Flexas et al. (2002);
• ¹⁷ Piel et al. (2002);
• ¹⁸ Terashima & Ono (2002);
• ¹⁹ Centritto et al. (2003);
• ²⁰ Warren et al. (2003).
• ^a Values given are for well‐watered plants, neither salt‐stressed nor senescing. All g_i measurements were done at 25 °C leaf temperature except: ^{8 , 20} (22 °C), ¹² (23–29 °C), ⁶ (25–30 °C), ¹ Oryza sativa and ⁴ Simmondsia chinensis (27 °C), ¹⁶ (30 °C).

As with stomatal conductance (g_s), g_i decreases in response to water or salinity stress (Roupsard et al. 1996; Delfine et al. 1999; Flexas et al. 2002; Centritto et al. 2003; Loreto, Centritto & Chartzoulakis 2003), the response being just as rapid and reversible as for g_s (Delfine et al. 1999; Centritto et al. 2003). As gas exchange studies of stomatal versus non‐stomatal limitations to photosynthesis during drought are typically based on A–C_i curve analyses carried out without considering g_i (e.g. Wilson, Baldocchi & Hanson 2000a and references therein), they overestimate the proportion of non‐stomatal limitation attributed to a drought‐induced loss of biochemical photosynthetic capacity (i.e. they underestimate V_cmax). Indeed, Centritto et al. (2003) recently showed that both V_cmax and J_max in olive (Olea europea) saplings remained unaffected by salt stress, the increased photosynthesis limitation being entirely due to simultaneous decreases in g_i and g_s. Similarly, the age‐related decline of photosynthesis in annual and deciduous perennial species does not appear to be due solely to the reduction of g_s and loss of biochemical photosynthetic capacity. For example, Loreto et al. (1994) found that g_i decreased in parallel to A_n and g_s in maturing, well‐watered wheat plants (Triticum durum) and that the relative contribution of g_i to photosynthetic limitation increased continuously throughout the maturation process. Similar results have also been found in spinach (Delfine et al. 1999), suggesting that g_i is an important factor causing the photosynthetic decline of leaves during ageing (Loreto et al. 1994) and thereby confounds the actual change of biochemical photosynthetic capacity (i.e. V_cmax) throughout this process. This last fact has so far been overlooked in studies concerned with measuring the seasonal variability of V_cmax in annual crops and deciduous forests (e.g. Grossman‐Clarke et al. 1999; Wilson et al. 2000a, Wilson, Baldocchi & Hanson 2000b; Kosugi, Shibata & Kobashi 2003).

This article provides both a review and a critique of the current A–C_i curve analysis method which relies on fixed values for the kinetic constants of Rubisco under the assumption of infinite g_i. Our objectives are: (1) to explain why this method fails to properly estimate V_cmax, given a set of Rubisco kinetic constants, in leaves with low g_i relative to their photosynthetic capacity; and (2) to present a new approach to estimate V_cmax or J_max that accounts for g_i. We start by reviewing the fundamental Michaelis–Menten theory upon which the photosynthesis model of Farquhar et al. (1980) is based to introduce the concepts and equations of our A–C_i curve analysis method. We then illustrate the limitations of the existing approach using examples of incorrect V_cmax determinations and show how these errors are eliminated when using our method. Finally, we demonstrate how our method can be used to estimate all the parameters describing Rubisco‐limited photosynthesis in the model of Farquhar et al. (1980) simultaneously in vivo using the approach of von Caemmerer et al. (1994).

THEORY

Rubisco‐limited photosynthesis

The equations describing the original biochemical model of C₃ leaf photosynthesis of Farquhar et al. (1980) and its subsequent refinements (von Caemmerer & Farquhar 1981; Farquhar & von Caemmerer 1982; Harley & Sharkey 1991) have been presented on numerous occasions (see von Caemmerer 2000 for a detailed review). Here we review the equations describing Rubisco‐limited photosynthesis through a series of theoretical CO₂ response curves starting with the predicted enzyme kinetics of fully activated Rubisco in vitro in O₂‐free media, then successively add the components that lead to the final formulation of whole‐leaf net photosynthesis in relation to C_i. Such an approach emphasizes the applicability of the fundamental Michaelis–Menten equation across all levels (enzyme to leaf) except at C_i. It also highlights the gradual apparent ‘burial’ of initial model parameters following the inclusion of the mitochondrial respiratory flux and the internal diffusion resistance.

Curve 1 of Fig. 1 illustrates the CO₂ response of a fully activated Rubisco preparation assayed in vitro under saturating concentrations of ribulose‐1,5‐bisphosphate (RuBP) and in O₂‐free media. The curve follows the classical Michaelis–Menten equation

Figure 1

Hypothetical CO₂ response curves from purified Rubisco preparations and corresponding C₃ plant source leaves. Curves 1 and 2 illustrate the CO₂ response of the RuBP‐saturated carboxylation rate (W_c) of fully activated Rubisco enzymes assayed in vitro in the absence or presence of O₂, respectively; curve 3 represents the whole‐leaf CO₂ response of W_c minus photorespiration [W_c(1 − Γ*/C_c)] evaluated at C_c; and curves 4 and 5 are the corresponding CO₂ response of the net RuBP‐saturated CO₂ assimilation rate (A_c) of the leaf evaluated at C_c and C_i, respectively. The upper panel (a) highlights the gradual change of the CO₂ concentration required to half‐saturate the CO₂ response according to Michaelis–Menten theory while the lower panel (b) shows the corresponding change of the CO₂ concentration at which the gross RuBP‐saturated CO₂ assimilation rate (W_cin vitro; W_c (1 − Γ*/C_c) or A_c + R_din vivo) equals zero. Broken lines sections indicate where CO₂ assimilation would normally no longer be Rubisco‐limited in wild‐type plants.

(1)

which describes the kinetic relationship between an enzyme and its substrate (x) as a rectangular hyperbola with asymptotes at V_xmax (ordinate) and –K_x (abscissa). Following the notation of Farquhar et al. (1980), W_c denotes the RuBP‐saturated carboxylation rate, V_cmax is the maximal carboxylation rate, and K_c is the Michaelis–Menten constant for CO₂; here C denotes the gas phase CO₂ concentration in the media. The Michaelis–Menten relation is fundamentally a function of diminishing returns. Its derivative

(2)

decreases continuously with increasing C, thereby describing the monotonic loss of free enzyme catalytic sites available to take advantage of further additions of substrate. That is, Eqn 2 describes the monotonic loss of carboxylation efficiency by Rubisco with increasing C. As long as the activity of the enzyme is unchanged, the rate of carboxylation efficiency loss is predictable and is given by

(3)

Thus, for a given V_cmax, the curvature of the CO₂ response function is entirely determined by K_c– the CO₂ concentration required to achieve a catalytic rate equal to 0.5V_cmax.

Owing to the inherent bifunctionality of Rubisco with respect to CO₂ and O₂ (Andrews & Lorimer 1987), introduction of O₂ to the in vitro assay media inevitably diverts a portion of the enzyme's catalytic activity towards the oxygenation of RuBP (Bowes, Ogren & Hageman 1971; Bowes & Ogren 1972; Andrews, Lorimaer & Tolbert 1973) and thereby decreases W_c at subsaturating C (Fig. 1, curve 2). In the presence of the competitive inhibitor O₂, the Michaelis–Menten equation describing the CO₂ response of W_c becomes (Laing, Ogren & Hageman 1974)

(4)

which shows that the effective CO₂ concentration required to half‐saturate W_c– the effective Michaelis–Menten constant for CO₂ (von Caemmerer 2000) – is linearly dependent on the O₂ concentration in the media (O) and equals K_c(1 + O/K_o) (Fig. 1a), where K_o is the Michaelis–Menten constant for O₂ in the competitive oxygenation reaction. Graphically, the loss of carboxylation efficiency by Rubisco in the presence of O₂ is represented by the shift of the abscissa asymptote from –K_c to –K_c(1 + O/K_o), which reduces the curvature of the rectangular hyperbola accordingly.

In intact, fully illuminated leaves from which Rubisco was presumably purified to perform the above‐mentioned in vitro assays, the photorespiratory release of 0.5 mole of CO₂ which follows the oxygenation of each mole of RuBP (Tolbert 1971; Keys 1986) further depresses the gross yield of the photosynthetic process (Fig. 1, curve 3) by an amount equal to W_c(Γ*/C_c) (Farquhar & von Caemmerer 1982), where Γ* is the chloroplastic CO₂ photocompensation point (Laisk 1977), that is the chloroplastic CO₂ concentration (C_c) at which W_c equilibrates with photorespiration (Fig. 1b). The dependence of Γ* on O is related to Rubisco's relative specificity for CO₂ as opposed to O₂ (S_c/o); following Laing et al. (1974)

(5)

Due to the photorespiratory flux, the CO₂ concentration required to half‐saturate W_c(1 − Γ*/C_c) inside the chloroplast increases to K_c(1 + O/K_o) + 2Γ* (Fig. 1a), as can be deduced from the denominator of the equation (written in Michaelis–Menten form) describing the CO₂ response of the net carboxylation rate following photorespiration:

(6)

The new effective Michaelis–Menten constant for the combined carboxylation–photorespiration reaction is now equal to K_c(1 + O/K_o) + Γ*. Once again, we note that the curvature of the rectangular hyperbola is reduced accordingly.

As Woodrow & Berry (1988) remarked, one has to make a leap of faith to presume that the photosynthetic response of intact leaves can be interpreted in terms of the kinetic behaviour of Rubisco in vitro. The question is not so much the validity of the fundamental Michaelis–Menten model in vivo, but more the accuracy of the in vitro system and our ability to establish in vitro conditions that will not greatly affect the original in vivo Rubisco affinities for CO₂ and O₂– otherwise the constants K_c, K_o, and S_c/o will be affected (Kane et al. 1994).

Curve 4 of Fig. 1 represents the CO₂ response of the overall net RuBP‐saturated CO₂ assimilation rate of a leaf (A_c) evaluated at C_c. The curve is identical to curve 3 in all respects except that it is shifted down on the ordinate due to constant CO₂ input from the mitochondrial respiration of the illuminated cells (R_d) (Fig. 1b). Consequently, the chloroplastic concentration at which all the respiratory processes of the leaf are in equilibrium with W_c no longer occurs at Γ*, but further down the abscissa at Γ– the overall CO₂ compensation point of the leaf (Fig. 1b). It follows that Γ* is now found where A_c equals –R_d (Fig. 1b).

Following Eqn 6, the Michaelis–Menten form of the equation describing the CO₂ response of A_c evaluated at C_c is given as

(7)

Although the effective Michaelis–Menten constant for the overall net photosynthetic process is now increased to K_c(1 + O/K_o) + Γ (Fig. 1a), the curvature of the rectangular hyperbola describing curve 4 remains unchanged relative to curve 3 (Eqn 6) due to the lowering of the ordinate asymptote from V_cmax to ‘V_cmax − R_d’ (Eqn 7). Equation 7 is the Michaelis–Menten equivalent of the commonly used equation of Farquhar et al. (1980)

(8)

which of course is Eqn 6 minus R_d. Since there is a continuum of V_cmax and R_d combinations which will produce the ordinate asymptote ‘V_cmax − R_d’, Eqn 8 requires that either Γ* or R_d be known a priori to estimate V_cmax and ‘K_c(1 + O/K_o)’ accurately from a non‐linear least‐squares fit to the CO₂ response curve (knowledge of g_i is also required to express the A–C_i curve in terms of C_c). Equation 7, on the other hand, requires no a priori information (other than g_i) to properly estimate Γ, ‘V_cmax − R_d’ and ‘K_c(1 + O/K_o)’ using non‐linear regression methods. In both cases, the validity of the least‐squares fit strongly depends on how well the curvature of the CO₂ response function follows the rectangular hyperbola model.

Under steady‐state conditions

A_c = g_i(C_i − C_c)

(9)

Solving Eqn 9 for C_c and substituting in Eqns 7 or 8 gives a quadratic equation (von Caemmerer & Evans 1991) whose solution is the positive root

(10)

Equation 10 represents the CO₂ response of A_c evaluated at C_i (Fig. 1, curve 5). The A_c versus C_i curve now follows a non‐rectangular hyperbola. It shares the same ordinate asymptote (V_cmax − R_d) and x‐intercept (Γ) as the A_c versus C_c curve (Fig. 1, curve 4), but its curvature is decreased relative to the rectangular hyperbola model to a degree determined by the magnitude of g_i. Again, since there is a continuum of K_c(1 + O/K_o) and g_i combinations which will produce that curvature, Eqn 10, when derived from Eqns 7 and 9, requires that either K_c(1 + O/K_o) or g_i be known a priori to properly estimate the remaining parameters from a non‐linear least‐squares fit to the A_c versus C_i curve. As mentioned above, a priori knowledge of either Γ* or R_d is also required if the alternate form of Eqn 10 (based on Eqns 8 and 9) is used.

Equation 9 indicates that whenever there is a steady‐state CO₂ flux through the leaf, there has to be a concentration gradient between C_i and C_c. Below Γ the flux is negative, so C_i is smaller than C_c, hence the appearance of an intercellular CO₂ photocompensation point (C_i*) smaller than Γ* at –R_d (Fig. 1b). The difference between Γ* and C_i* is rarely appreciated since g_i is usually assumed to be infinite in A–C_i curve analyses. Moreover, the CO₂ flux at –R_d is usually very small so it is not expected to generate a significant drawdown between Γ* and C_i*. However, Warren et al. (2003) estimated that this drawdown can be as high as 15 µbar in Douglas‐fir (Pseudotsuga menziesii) and were able to use the measured differences between Γ* and C_i* to estimate g_i in this species using Eqn 9.

The validity of the non‐rectangular hyperbola model (Eqn 10) rests on the assumption that g_i is purely diffusional and is therefore not affected by the changes in CO₂ concentration inside the leaf necessary to develop an A–C_i curve. This has been verified to some extent by both the isotopic (von Caemmerer & Evans 1991) and the chlorophyll fluorescence (Harley et al. 1992a; Loreto et al. 1992) methods commonly used to estimate g_i. The recent evidence indicating a close association between carbonic anhydrase (Gillon & Yakir 2000) and aquaporin membrane channels (Terashima & Ono 2002) and g_i does not change this view since both proteins only act in facilitating the passive CO₂ diffusion process. Although Mächler, Müller & Dubach (1990) presented data suggesting the presence of an active CO₂ pump at the chloroplast envelope, we believe that their conclusion is due to calculation artefacts arising from using the rectangular hyperbola model to describe photosynthesis in relation to both C_c and C_i. Indeed, under this assumption, g_i will necessarily appear to scale in direct proportion to the carboxylation efficiency (dA_c/dC_c) which, as suggested in Eqns 2 and 3, rises increasingly sharply as C_c approaches Γ* (substitute the appropriate terms from Eqn 6 into Eqns 2 and 3), thereby simulating, in the corresponding g_i case, an active transport system. No chloroplastic CO₂ concentrating mechanism has yet been identified in C₃ plants (Badger & Price 1994).

RuBP‐limited photosynthesis

The half‐asymptotic values of W_c(1 − Γ*/C_c) and A_c on curves 3–5 (Fig. 1a) are shown on broken line portions of the Rubisco‐limited CO₂ response curves to indicate that they are never reached under steady‐state conditions in wild‐type plants since the regeneration rate of the RuBP pool of the leaf falls behind the potential rate of RuBP carboxylation/oxygenation by Rubisco and begins limiting photosynthesis at a lower C_c (Laisk & Oja 1974; Lilley & Walker 1975). Like A_c, the net RuBP‐limited CO₂ assimilation rate (A_j) of a leaf can also be expressed in terms of an enzymatic Michaelis–Menten process whose overall efficiency diminishes with increasing C_c (rectangular hyperbola model):

(11)

where J is the CO₂‐saturated electron transport rate of the thylakoids reactions which ultimately supply the necessary energy in the form of ATP and NADPH for the regeneration of RuBP. Detailed treatments of the biochemical reactions involved in the process and of stoichiometric alternatives to Eqn 11 are given in Farquhar & von Caemmerer (1982) and von Caemmerer (2000). Equation 11 is given here in a form equivalent to Eqn 8, but it is understood that it can undergo the same derivations as outlined in Eqns 6–10 simply by replacing V_cmax by J/4 and K_c(1 + O/K_o) by 2Γ*. Following Farquhar & Wong (1984), the dependence of J on irradiance is given as

θJ ² − J(αI + J_max) + αIJ_max = 0

(12)

where I is the incident irradiance, α is the quantum efficiency (number of electrons transfered per incident photon), and θ is the convexity (curvature factor) of the non‐rectangular hyperbola.

CONVENTIONAL A–C_i CURVE FITTING METHOD

Disregarding the possible limitation imposed on photosynthesis at high CO₂ concentrations by the rate of triose phosphates utilization (Harley & Sharkey 1991), the net CO₂ assimilation rate of a leaf is given as

A _n = min{A_c,A_j}

(13)

where the appropriate equations describing A_c and A_j on the basis of C_i or C_c are given above. The conventional A–C_i curve fitting method commonly uses Eqns 8 and 11 to describe A_c and A_j, respectively, and assumes g_i to be infinite; thus

(14)

and

(15)

Equations 14 and 15 require that a total of six parameters be estimated from non‐linear regression techniques, which, for reasons explained earlier, is beyond the iterative power of such techniques. Here, however, the leap of faith mentioned by Woodrow & Berry (1988) is taken and values for the kinetic constants of Rubisco (S_c/o or Γ*, K_c, K_o) are chosen a priori from previously published estimates to constrain the least‐squares fits to Eqns 14 and 15.

There are many S_c/o, K_c, and K_o estimates to choose from in the literature (see among others the multispecies comparative studies of Yeoh, Badger & Watson 1980, 1981; Jordan & Ogren 1981, 1983; Bird, Cornelius & Keys 1982; Makino, Mae & Ohira 1985; Parry et al. 1989; Kane et al. 1994; Laisk & Loreto 1996), but as far as the parameterization of the model of Farquhar et al. (1980) is concerned, these have largely been restricted to values found in spinach and tobacco. Table 2 lists commonly used choices for Γ* (often taken as C_i*), K_c, and K_o and for the activation energy of the Arrhenius function describing their respective temperature response (e.g. Harley et al. 1992b; Wullschleger 1993; de Pury & Farquhar 1997; Walcroft et al. 1997; Wang & Leuning 1998; Medlyn et al. 1999, 2002; Dreyer et al. 2001; Wilson, Baldocchi & Hanson 2001; Kosugi et al. 2003). The rationale behind this approach is that the kinetic properties of Rubisco are so fundamental to the efficiency of the photosynthetic process that they should be largely conserved among C₃ plant species (von Caemmerer 2000).

Table 2. Kinetic constants of Rubisco commonly used to parameterize the photosynthesis model of Farquhar et al. (1980)

Reference	Species	Parameter	Value at 25 °C	Units	E (kJ K^−1 mol^−1)
Farquhar et al. (1980)^a^,^c	Spinacia oleracea (15–35 °C)	K c	460	µbar	59.413
		K o	330	mbar	35.982
Jordan & Ogren (1984)^a^,^d	S. oleracea (7–35 °C)	Γ*	45.146	µmol mol^−1	29.213
		K c	274.22	µbar	70.372
		K o	418.29	mbar	14.351
Brooks & Farquhar (1985)^b	S. oleracea (15–35 °C)	C i*	42.382	µmol mol^−1	30.037
von Caemmerer et al. (1994)^b	Nicotiana tabacum (25 °C)	C i*	36.9	µbar	–
		K c	404	µbar	–
		K o	248	mbar	–
Bernacchi et al. (2001)^b	N. tabacum (10–40 °C)	C i*	42.893	µmol mol^−1	37.83
		K c	406.07	µmol mol^−1	79.43
		K o	276.9	mmol mol^−1	36.38

• The temperature response of the parameters was described by fitting the original data to the Arrhenius function normalized relative to 25 °C: Parameter(T) =Parameter(25 °C) exp [(T − 298.15)/E 298.15 RT]. R is the gas constant (8.314 J K⁻¹ mol⁻¹); T is the absolute temperature (T °C + 273.15)°K; E is the energy of activation; 1 cal = 4.184 J.
• ^a Kinetic constants determined in vitro.
• ^b Kinetic constants determined in vivo assuming g_i = ∞.
• ^c Activation energies (E) for K_c and K_o are from Badger & Collatz (1977).
• ^d After Harley et al. (1992b).

Given a set of Rubisco kinetic constants, Eqn 13 can easily be fitted to the overall A–C_i curve once the respective domain of the A_c and A_j functions is defined. Von Caemmerer & Farquhar (1981) noted on the basis of C_i that the transition from A_c to A_j consistently occurred between 200 and 250 µbar in Phaseolus vulgaris when measurements were performed at ambient O₂ concentration and high irradiance. Furthermore, this transition zone appeared conserved over a range of leaf temperatures and nitrogen nutrition. Since then, it has become established practice to use the 200–250 µbar transition zone of P. vulgaris as the cut‐off point for fitting A_c to the lowest portion of A–C_i curves performed under standard conditions in any species (e.g. Wullschleger 1993; Wohlfahrt et al. 1999; Bunce 2000). This approach is usually justified by the fact that the theoretical transition point between A_c and A_j is determined by the balance between J_max and V_cmax (von Caemmerer 2000) which have been found to correlate conservatively among species (Wullschleger 1993; Leuning 2002; Medlyn et al. 2002).

Our studies on conifers revealed that the above‐mentioned method repeatedly failed to properly fit the A–C_i curves we produced (n = 60) from shoots of 50‐year‐old Douglas‐fir trees for which g_i was estimated to range from 0.14 to 0.20 mol m⁻² s⁻¹ at 22 °C (Warren et al. 2003). Neither the Rubisco kinetic constants shown in Table 2 nor other recommended values (e.g. Collatz et al. 1991; Harley & Tenhunen 1991; Bernacchi et al. 2002) produced acceptable results – the curvature of the fitted A_c function being always too pronounced to satisfy Eqn 13(Fig. 2). Since our measurements were done using an integrating sphere at high diffuse irradiance (see Warren et al. 2003), we considered it likely that the transition between A_c and A_j occurred at an appreciably higher C_i than 200–250 µbar (e.g. 300–400 µbar; Makino, Mae & Ohira 1988, Makino et al. 1994). However, we found that increasing the domain of the A_c function would solve the problem only if it were more than doubled and exclusively for sets of Rubisco kinetic constants that gave K_c(1 + O/K_o) values greater than 700 µbar.

Figure 2

Typical CO₂ response of the net CO₂ assimilation rate (A_n) of a 1‐year‐old Douglas‐fir shoot evaluated at C_i. Measurements were made at 22 °C and at saturating diffuse irradiance (1600 µmol m⁻² s⁻¹) in an integrating sphere. Least‐squares regression fits to the initial A–C_i curve portion (filled circles) were performed according to the standard A–C_i curve analysis method described in the text. The number beside each curve indicates the maximal carboxylation rate (V_cmax) obtained assuming infinite CO₂ transfer conductance (g_i) (Eqn 14), using popular Rubisco kinetic constant values (see Table 2 for Jordan & Ogren (1984) and Bernacchi et al. (2001); for Bernacchi et al. (2002) Γ* = 33.86 µmol mol⁻¹, K_c = 195.41 µmol mol⁻¹, and K_o = 150.46 mmol mol⁻¹ at 22 °C).

Surprisingly, after over 20 years of use, to our knowledge there are no reports of similar difficulties with the standard A–C_i curve fitting method. For example, Wullschleger (1993) did a retrospective analysis of previously published A–C_i curves from 109 different C₃ plant species following the above‐mentioned method, using the Rubisco kinetic constant values of Jordan & Ogren (1984) shown in Table 2, and reported no anomalies with the procedure. However, there are many cases in his report where V_cmax values were derived from invalid curve fits. Out of the 84 A–C_i curves that we analysed, V_cmax < 65 µmol m⁻² s⁻¹, as estimated by Wullschleger (1993), 15 could not be properly assessed due to insufficient C_i range or low light intensity, but one‐third of the remaining 69 gave wrong V_cmax estimates based on invalid curve fits. Two such examples (Sun & Ehleringer 1986; Siebke et al. 1990) are reproduced in Fig. 3. The V_cmax values we obtained by following the procedure outlined above were within 1 µmol m⁻² s⁻¹ of those reported by Wullschleger (1993), that is 32 and 29 µmol m⁻² s⁻¹ for Sun & Ehleringer (1986) and Siebke et al. (1990), respectively, but it is clear from Fig. 3 that these were derived from invalid curve fits that underestimate the true V_cmax values which, according to the Rubisco kinetic constants of Jordan & Ogren (1984), would in both cases be about 56 µmol m⁻² s⁻¹ after accounting for g_i via Eqn 10 (see method outlined below).

Figure 3

Least‐squares regression fits to the initial portion (filled circles) of A–C_i curves reproduced from (a) Sun & Ehleringer (1986) and (b) Siebke et al. (1990). Curve fits of Eqn 14 assume infinite CO₂ transfer conductance (g_i) and were performed according to Wullschleger (1993). The maximal carboxylation rate (V_cmax) values obtained using the Rubisco kinetic constants of Jordan & Ogren (1984) (see Table 2) were (a) 33 and (b) 30 µmol m⁻² s⁻¹. Curves fits of Eqn 10 were performed on the same data points using the same Rubisco kinetic constants. The values obtained were (a) 57 and (b) 56 µmol  m⁻² s⁻¹ for V_cmax and (a) 0.1 and (b) 0.15 mol m⁻² s⁻¹ for g_i.

NEW A–C_i CURVE FITTING METHOD

According to the non‐rectangular hyperbola model (Eqn 10), A_c reaches its half‐asymptotic value of 0.5(V_cmax − R_d) when C_i equals K_c(1 + O/K_o) + 2Γ + 0.5(V_cmax − R_d)/g_i (Eqns 7 and 9) (Fig. 1a, curve 5). Under the assumption of infinite g_i, the least‐squares rectangular hyperbola approximation to Eqn 10 that will produce the correct value for V_cmax is given by

(16)

The value K_m(CO₂)_i is the apparent Michaelis–Menten constant for CO₂ evaluated at C_i. Since the value of C_i representing the half‐asymptotic value of A_c in the initial non‐rectangular hyperbola increases as g_i decreases, so will K_m(CO₂)_i. That is, the correct rectangular hyperbola approximation to Eqn 10 is one that keeps the ordinate asymptote at V_cmax − R_d and increases the value of the abscissa asymptote in inverse proportion to g_i. The conventional A–C_i curve fitting method does the reverse; it fixes the abscissa asymptote at –K_c(1 + O/K_o) − Γ according to the Rubisco kinetic constants chosen a priori, and decreases the ordinate asymptote in inverse proportion to g_i. This leads to a significant underestimation of V_cmax for leaves with low g_i relative to their photosynthetic capacity.

Figure 4a gives a graphical example of the extent of the V_cmax underestimation as g_i decreases and of the K_m(CO₂)_i value required to restore V_cmax to its initial true value. The results shown in the figure were generated by fitting Eqn 14 (hollow circles) or Eqn 16 (filled circles) to a series of theoretical A–C_i curves obtained by combining a single A_n versus C_c curve with different g_i values. Details about the curve fits and the parameters (mostly taken from Wullschleger 1993) used to construct the A_n versus C_c curve are given in the figure legend. In reference to the above‐mentioned analysis of the A–C_i curves of Sun & Ehleringer (1986) and Siebke et al. (1990) which yielded underestimates of V_cmax of 54 and 58%, respectively, Fig. 4a shows that, given an initial ‘true’V_cmax value of 56 µmol m⁻² s⁻¹, such an underestimation would be expected given g_i values close to 0.1 mol m⁻² s⁻¹ and that the K_m(CO₂)_i value required to remove that underestimation would be approximately 880 µbar. This is over twice the original K_c(1 + O/K_o) value used by Wullschleger (1993). None of the recommended K_c and K_o combinations mentioned previously give a K_c(1 + O/K_o) value as high as this despite that the K_c and K_o values of von Caemmerer et al. (1994) and Bernacchi et al. (2001) shown in Table 2 were derived from in vivo measurements of K_m(CO₂)_i and that the in vitro K_c value of Farquhar et al. (1980) is likely to be overestimated by approximately 85% due to a calculation error (wrong choice of dissociation constant (pK_a) for carbonic acid versus CO₂– see Yokota & Kitaoka 1985).

Figure 4

Influence of CO₂ transfer conductance (g_i) on (a) the maximal carboxylation rate (V_cmax) and apparent Michaelis–Menten constant for CO₂ evaluated at C_i[K_m(CO₂)_i] and (b) the maximal electron transport rate (J_max) and the J_max/V_cmax ratio. Values of V_cmax and J_max were estimated by fitting Eqns 14 and 15 (standard A–C_i curve fitting method – see text for details) to a series of ideal A–C_i curves generated for various g_i values from a set of initial parameters valid at C_c. For Rubico‐limited photosynthesis (Eqn 8)Γ*, K_c, and K_o were taken from Jordan & Ogren (1984) (at 25 °C and 210 mbar O₂; see Table 2) and V_cmax and R_d were 56 and 0.6 µmol m⁻² s⁻¹, respectively. For RuBP‐limited photosynthesis (Eqns 11 and 12)φ = 0.18 mol e‐ mol quanta⁻¹, θ = 0.9, I = 1500 µmol quanta m⁻² s⁻¹, and J_max = 112 µmol e‐ m⁻² s⁻¹. Values of K_m(CO₂)_i were determined by fitting Eqn 16 to the A–C_i curves, setting V_cmax and the mitochondrial respiration in the light (R_d) to their ‘true’ value. The ‘+’ symbols indicate values of (a) V_cmax and (b) J_max obtained when fitting Eqn 10 to the same set of A–C_i curves starting with K_c(1 + O/K_o) and Γ* values that overestimate their ‘true’ value by 20 and 10%, respectively.

The new A–C_i curve fitting method we propose rests on the point emphasized throughout the theory section, that is, for a given V_cmax, the curvature of a CO₂ response function is entirely determined by its effective Michaelis–Menten constant. Since for A–C_i curves the curvature of the non‐rectangular hyperbola is affected by both K_c(1 + O/K_o) and g_i, specifying K_c(1 + O/K_o) when fitting Eqn 10 to the initial A–C_i curve portion (as for Eqn 14 in the standard method) will automatically constrain the value of g_i to match the curvature but will not change the value of the ordinate asymptote (V_cmax − R_d). Figure 4a (‘+’ symbols) shows an example of the relative constancy of the V_cmax parameter, with respect to g_i, estimated from curve fits based on Eqn 10 and initiated from overestimated Γ* (10%) and K_c(1 + O/K_o) (20%) values. In this case, the V_cmax overestimates varied by less than 1% (59.3 to 59.8 µmol m⁻² s⁻¹) across the g_i range explored (0.6 to 0.05 mol m⁻² s⁻¹); the corresponding variation of the V_cmax estimates obtained from Eqn 14, following the standard A–C_i curve fitting method, is 57% (56.4 to 24.5 µmol m⁻² s⁻¹). Graphical examples of this simulation are shown in Fig. 5. Note the 17% underestimation of V_cmax through Eqn 14 in Fig. 5a (g_i = 0.2 mol m⁻² s⁻¹) despite the 20% overestimation of the K_c(1 + O/K_o) parameter and the apparent validity of the curve fit (compare with Figs 5b, 2 & 3). In fact, when the correct Γ* and K_c(1 + O/K_o) values are used to estimate V_cmax from Eqn 14, the latter can be underestimated by as much as 30% (g_i = 0.17 mol m⁻² s⁻¹) from curve fits that still appear visually valid. Similar results are found for g_i of 0.1 mol m⁻² s⁻¹ (29%V_cmax underestimation) when the ‘true’V_cmax value used to generate the A–C_i curves is reduced by half (data not shown). Thus, according to these simulations, there could be considerably more significant V_cmax underestimates than we originally anticipated in the survey of Wullschleger (1993) and other C₃ species parameterizations which used the same Rubisco kinetic constants and curve fitting method (e.g. Harley & Baldocchi 1995; Centritto & Jarvis 1999; Le Roux et al. 1999; Bunce 2000; Warren & Adams 2001; Wilson et al. 2001).

Figure 5

Examples of least‐squares regression fits of Eqns 14 and 15 (standard method – see text for details; continuous lines) versus Eqn 10 (new method; broken lines) performed on ideal A–C_i curves generated for g_i values of (a) 0.2 and (b) 0.05 mol m⁻² s⁻¹ from a set of initial parameters valid at C_c (see Figure 4 legend for details). The number beside each curve indicates the maximal carboxylation rate (V_cmax) or the maximal electron transport rate (J_max) obtained by fitting the appropriate equation to the Rubisco‐ limited (Eqn 14 versus Eqn 10; circles) and RuBP‐limited (Eqn 15 versus Eqn 10; diamonds) portion of the curves, respectively. Open symbols represent data points not used for the curve fits. All curve fits were done with K_c(1 + O/K_o) and Γ* parameters that overestimated their ‘true’ value by 20 and 10%, respectively.

The A–C_i curve fitting method based on Eqn 10 is essentially insensitive to the above‐mentioned biases (Fig. 5). This can easily be verified on any, reasonably well‐defined, measured A–C_i curve simply by increasing the leaf internal resistance (1/g_i) in steps and by re‐fitting the resulting A–C_i curves (re‐computed from the original data using Eqn 9) according to the two methods described above. Another advantage of our approach is that it reduces the sensitivity of the fitted value of V_cmax relative to the value of K_c(1 + O/K_o) chosen. For example, our estimates of V_cmax in the above‐mentioned simulation remain within 10% of the ‘true’V_cmax used initially to construct the A–C_i curves even when the K_c(1 + O/K_o) value chosen to initiate the curve fits is up to ± 30% of the correct value; the corresponding uncertainty of the rectangular hyperbola model, corrected for g_i according to Eqn 16, is twice as high (Fig. 6).

Figure 6

Errors in the estimates of the maximal carboxylation rate (V_cmax) caused by using up to ± 30% of the actual value of K_c(1 + O/K_o) or K_m(CO₂)_i when fitting Eqns 10 or 16, respectively, to the initial portion of an ideal A–C_i curve generated as described in the legend of Fig. 4 with a CO₂ transfer conductance (g_i) value of 0.2 mol m⁻² s⁻¹ (similar results were obtained using other g_i values). The correct value for K_m(CO₂)_i was estimated to 635.3 µbar from the data shown in Fig. 4a.

The theoretical simulations described above suggest that it is possible for leaves of similar biochemical photosynthetic capacity (V_cmax) and intercellular operating point (C_i) to achieve very different A_n through significant variation in g_i. Yet, the commonly evoked positive relationship between A_n and g_i(Fig. 7a; von Caemmerer & Evans 1991; Lloyd et al. 1992; Epron et al. 1995; Evans & Loreto 2000; Warren et al. 2003) may lead one to conclude that V_cmax correlates conservatively with g_i among C₃ plant species. If so, one would expect the g_i limitation to photosynthesis to be similar among species, which would imply that the K_m(CO₂)_i value required to compensate for that limitation is also conserved. However, the apparent convexity of the A_n versus g_i relationship (Fig. 7a) suggests a general tendency towards ‘over‐investment’ in photosynthetic capacity as g_i falls below 0.2 mol m⁻² s⁻¹; this is clearly indicated by the corresponding increasing drop in CO₂ concentration between C_i and C_c (Fig. 7b). Hence, the g_i limitation to photosynthesis can only be similar among species if the operating C_i is inversely proportional to A_n and g_i (in order to equalize C_c). On the contrary, the operating C_i is usually found to be remarkably conserved among species despite large differences in A_n (e.g. Yoshie 1986) or else positively related to A_n (e.g. Franks & Farquhar 1999). Thus, the correlation between A_n and g_i among species does not necessarily reflect the relationship between V_cmax and g_i for in theory A_n is fundamentally dependent on g_i but V_cmax is not (Eqn 10), at least not necessarily to the same degree. This is shown explicitly in Fig. 7c for those data in Fig. 7a for which g_s was given alongside A_n and g_i (filled circles), thereby allowing the estimation of V_cmax on the basis of C_c via Eqns 8 and 9 (see figure legend for details). Here, the tobacco Rubisco kinetic constants of von Caemmerer et al. (1994) determined in vivo on the basis of C_c (see next section) were used to calculate V_cmax so the corresponding K_m(CO₂)_i values estimated from Eqn 16 could be compared with the value found by von Caemmerer et al. (1994) (730 µbar at 25 °C and 200 mbar O₂; see Table 2). The results clearly show that K_m(CO₂)_i is not a conserved property among C₃ plant species (Fig. 7d) – no single value can therefore be recommended for properly estimating V_cmax in all species. For example, the K_m(CO₂)_i value of von Caemmerer et al. (1994) produced V_cmax estimates (Eqn 14) that varied from −24 to +10% (−8 ± 10%, mean ± SD, n = 24) of the ‘true’ V_cmax  (Eqn  8),  as  opposed  to  from −57  to −8% (−29 ± 14%, mean ± SD, n = 29), for leaves with a g_i above and below 0.2 mol m⁻² s⁻¹, respectively.

Figure 7

Relationship between the CO₂ transfer conductance (g_i) and (a) the net CO₂ assimilation rate (A_n); (b) the CO₂ concentration drawdown between the intercellular spaces subtending the stomata and the chloroplasts’ carboxylation sites (C_i − C_c; from A_n/g_i); (c) the maximal carboxylation rate (V_cmax); and (d) the apparent Michaelis–Menten constant for CO₂ evaluated at C_i[K_m(CO₂)_i] among the C₃ plant species listed in Table 1 (n = 262; see References given therein for the data source). Filled circles in (a) and (b) represent the data subset (n = 53) for which the stomatal conductance (g_s) was given alongside g_i and A_n. From these, C_c was estimated at a reference leaf surface CO₂ concentration of 340 µbar [i.e. C_c = (340 − 1.56A_n/g_s) − A_n/g_i] and V_cmax was then solved using Eqn 8 (R_d = 0.8 µmol m⁻² s⁻¹ at 25 °C; otherwise adjusted for leaf temperature using the temperature response function of Bernacchi et al. 2001) parameterized with the Rubisco kinetic constants of von Caemmerer et al. (1994) evaluated at C_c (K_c = 259 µbar, K_o = 179 mbar, and Γ* = 38.6 µbar at 25 °C; otherwise adjusted for leaf temperature using the temperature response functions of Bernacchi et al. 2002); K_m(CO₂)_i was subsequently estimated at 200 mbar O₂ from V_cmax using Eqn 16 (with C_i* determined from Eqn 9). The final V_cmax and K_m(CO₂)_i values shown in (c) and (d) were adjusted to 25 °C using the temperature response functions of Bernacchi et al. (2001); other estimates derived from different leaf surface CO₂ concentrations gave different absolute values but similar trends (not shown). The dotted line in Fig. 7d represents the K_m(CO₂)_i value found by von Caemmerer et al. (1994) for transgenic tobacco plants (see Table 2).

According to Eqn 10 parameterized with the Rubisco kinetic constants of Jordan & Ogren (1984), the g_i value required to match the curvature of the real A–C_i curves shown in Fig. 3 is 0.1 and 0.15 mol m⁻² s⁻¹ for Sun & Ehleringer (1986) and Siebke et al. (1990), respectively. For the A–C_i curve of the Douglas‐fir shoot shown in Fig. 2, the matching g_i value is 0.12 mol m⁻² s⁻¹ (corresponding to a V_cmax estimate of 44 mol m⁻² s⁻¹, that is 1.6 times the value found with Eqn 14)– a close match to the g_i value found for this shoot based on the measured difference between C_i* and Γ* (0.15 mol m⁻² s⁻¹; see Warren et al. (2003) for details about the technique). These estimates compare very well with the expected g_i value obtained from theory using the same Rubisco kinetic constants and assuming steady‐state Rubisco activity (Fig. 4a). In Douglas‐fir, we found that the curvature of the A–C_i curves sometimes increased significantly as C_i approached Γ and that we needed to remove these data from the curve fits to obtain reasonable g_i estimates (data not shown). This likely indicated a significant loss of Rubisco activity since it is known that the activation state of the enzyme can decrease when C_i falls below 100 µbar (e.g. von Caemmerer & Edmondson 1986; Sage, Sharkey & Seemann 1990). Such changes in the activation state of Rubisco may also explain the usual apparent linearity of the initial A–C_i curve portion (e.g. von Caemmerer & Farquhar 1981; Brooks & Farquhar 1985). However, we emphasize that one consequence of the presence of significant g_i limitation in a leaf is that it effectively ‘linearizes’ the initial portion of its A–C_i curve (for example, compare Fig. 5a & b– note also the reduction of the overall slope of that initial A–C_i curve portion). In any case, when seeking estimates of g_i with this method, we recommend using a higher degree of measurement resolution for the Rubisco‐limited A–C_i curve portion above a C_i of 100 µbar. In theory, if the K_c(1 + O/K_o) value chosen to initiate the least‐squares fit of Eqn 10 remains within ± 30% of its true value, the sensitivity of the g_i estimate to K_c(1 + O/K_o) is equivalent to that of the chlorophyll fluorescence technique when the Γ* value chosen to estimate g_i with this method is within ± 10% of its true value (data not shown).

The effect of g_i on values of J_max estimated using the standard A–C_i curve fitting method (Eqn 15) is marginal over much of the range covered by the simulation (Fig. 4b). This is because: (1) J_max is estimated from the saturating portion of the A–C_i curve; and (2) the entire domain of the A_j  function  is  almost  entirely  specified  since  its  root  (Γ*, –R_d) is set a priori. There is therefore little need, other than improving the least‐squares fit to better the delineation of the transition zone between the Rubisco‐limited and the RuBP‐limited portions of the A–C_i curve (see Fig. 5b), to account for varying g_i through Eqn 10 when estimating J_max. However, failure to correct V_cmax for g_i will increase the J_max/V_cmax ratio significantly when g_i is low relative to the leaf photosynthetic capacity (Fig. 4b).

ESTIMATION OF RUBISCO KINETIC CONSTANTS

There are still very few published in vivo estimates for the kinetic constants of Rubisco that have been properly evaluated at C_c. This is because the domain of the A_c function is normally considered too small to allow an accurate extrapolation of the effective Michaelis–Menten constant for CO₂: K_c(1 + O/K_o). Von Caemmerer and colleagues (von Caemmerer et al. 1994; Bernacchi et al. 2002) circumvented this problem by using transgenic tobacco plants which had a reduced Rubisco content and were limited by Rubisco activity, as opposed to RuBP regeneration capacity, even at high CO₂ concentrations. The method they used for estimating K_c(1 + O/K_o) for leaves exposed to a range of O₂ concentrations is based on the same principle described above. They fitted Eqn 6 to the CO₂ response curves (evaluated at C_c) having determined g_i, C_i*, and R_d a priori. However, as we show earlier, Eqn 7 can be used to solve Γ, K_c(1 + O/K_o), and V_cmax − R_d simultaneously from A_c versus C_c curves without having to rely on prior knowledge of Γ* and R_d. This can also be accomplished by replacing g_i in Eqn 10 for the measured value. Figure 8 shows an example of this using the data of von Caemmerer et al. (1994) reproduced from their Fig. 4a. The O₂ response obtained for the three fitted parameters: Γ, K_c(1 + O/K_o), and V_cmax − R_d is shown in Fig. 9 (compare with Fig. 5 of von Caemmerer et al. 1994). As predicted from theory, the V_cmax − R_d values estimated are essentially the same across the O₂ concentration range (Fig. 9c).

Figure 8

Least‐squares regression fits of Eqn 10 (as derived from Eqns 7 and 9) to the A–C_i curves of Fig. 4a from von Caemmerer et al. (1994). Measurements were done on a transgenic tobacco leaf with a reduced Rubisco content at 25 °C (I = 1500 µmol quanta m⁻² s⁻¹) under five different O₂ concentrations. The CO₂ transfer conductance (g_i) for such leaves was estimated to be 0.27 ± 0.2 mol m⁻² s⁻¹ (Evans et al. 1994).

Figure 9

O₂‐response of (a) the CO₂ compensation point (Γ); (b) the effective Michaelis–Menten constant for CO₂[K_c(1 + O/K_o)]; and (c) the effective maximal carboxylation rate in the presence of mitochondrial respiration in the light (V_cmax − R_d) estimated from the least‐squares regression fits of Eqn 10 to the A–C_i curves shown in Fig. 8.

According to Michaelis–Menten theory, the linear dependence of Γ on O (Fig. 9a) originates from the O₂‐dependence of Γ* (Eqn 5) and K_c(1 + O/K_o) (Fig. 9b) and is given as

(17)

The y‐intercept of Eqn 17 represents the residual CO₂ compensation point in the absence of RuBP oxygenation and photorespiration. It can be used to derive R_d from the estimated values of V_cmax − R_d and K_c (y‐intercept of Fig. 9b). Following this, the value of V_cmax is readily obtained and can be used to solve S_c/o from the slope of Eqn 17 (K_o is given by the slope of Fig. 9b); Γ* can then be obtained from Eqn 5. The final sets of values derived from the least‐squares fits of Eqn 10 to the A–C_i curves shown in Fig. 8 are given in Table 3 for g_i values of 0.27 ± 0.2 mol m⁻² s⁻¹ determined for these transgenic tobacco plants using the isotopic method (Evans et al. 1994). The final K_c(1 + O/K_o) value we obtained for this particular leaf (550 to 566 µbar at 200 mbar O₂) compares very well with the average value reported by von Caemmerer et al. (1994) (549 µbar at 200 mbar O₂).

Table 3. Complete set of parameters obtained from estimating Γ, K_c(1 + O/K_o), and V_cmax − R_d from least‐squares fits of Eqn 10 to A–C_i curves measured at different O₂ concentrations. Data from von Caemmerer et al. (1994) (see Figs 8 & 9)

	g i = 0.25 (mol m^−2 s^−1 bar^−1)	g i = 0.27 (mol m^−2 s^−1 bar^−1)	g i = 0.29 (mol m^−2 s^−1 bar^−1)	Units
Γ	54.6	54.9	55.1	µbar
V cmax − Rd	34.07	34.19	34.28	µmol m^−2 s^−1
K c(1 + O/Ko)^a	549.7	558.5	566.0	µbar
K c	319.3	327.0	333.7	µbar
K o	277.1	282.6	287.3	mbar
R d	0.63	0.66	0.69	µmol m^−2 s^−1
V cmax	34.70	34.85	34.97	µmol m^−2 s^−1
Sc/o	2275	2291	2304	bar bar^−1
Γ*^a	44.0	43.7	43.4	µbar

• ^a Evaluated at O = 200 mbar.

In principle, if g_i is known, it should be possible to estimate the overall value of K_c(1 + O/K_o) in wild‐type plants from a least‐squares fit of Eqn 7 to the Rubisco‐limited portion of A_n versus C_c curves (or A–C_i curves using the equivalent Eqn 10 derivation) performed at ambient O₂ concentration. Although the restricted measurable domain of the A_c function remains a concern in wild‐type plants (especially if significant Rubisco deactivation is observed at C_i < 100 µbar), our measurements on Douglas‐fir, as well as those of Makino et al. (1994) performed on rice, are encouraging in that they indicate that the transition C_i between A_c and A_j can be as high as 400 µbar in leaves acclimated to a growth temperature that is lower than the A–C_i curve leaf temperature (e.g. 5–10 °C difference). Needless to say, this technique demands a rigorous calibration of the gas exchange equipment used to generate the A–C_i curves and a high degree of resolution over the domain of interest (e.g. 25 µbar steps).

CONCLUSION

We have shown that conventional A–C_i curve fitting methods can significantly underestimate V_cmax for plants photosynthetically limited by g_i and have provided a new approach to estimate V_cmax that accounts for g_i and which is less sensitive with respect to the absolute accuracy of the K_c(1 + O/K_o) value chosen to parameterize the model of Farquhar et al. (1980). Alternative parameterizations of the conventional method based on a single K_m(CO₂)_i value used for all C₃ plants are not acceptable since the relationship between V_cmax and g_i is not conserved among species.

Our method is based on the non‐rectangular hyperbola version of the model of Farquhar et al. (1980) and gives reasonable estimates of g_i providing the K_c(1 + O/K_o) value chosen to parameterize the model is close to its true value. More C₃ species in vivo estimates of K_c(1 + O/K_o) properly evaluated at C_c are needed in this respect – our proposed A–C_i curve fitting method suggests new ways for obtaining these estimates.

The success of the photosynthesis model of Farquhar et al. (1980) cannot be understated. In the words of the authors, it ‘has had an impact and seen application that far exceeded our expectations’ (Farquhar, von Caemmerer & Berry 2001). The wide use of the model is not only due to its solid theoretical basis, but also to its simplicity, a necessary prerequisite to make any model useful (von Caemmerer 2000). We suggest, however, that many present model parameterizations are incorrect for leaves with low g_i. Of course, such parameterizations remain useful to predict photosynthesis under the conditions used since they are empirically fitted to measured responses. They may, however, mislead us when it comes to evaluating the variability of V_cmax among C₃ plant species or interpreting the fundamental physiological processes underlying the measured photosynthetic responses of plants to various environmental conditions or through time. This, in effect, goes against ‘la raison d’être’ of the theoretical mechanistic basis of the photosynthesis model of Farquhar et al. (1980). Considering the importance given to the V_cmax parameter in global climate change modelling, we believe it deserves better definition. By accounting for g_i through a non‐rectangular hyperbola version of the model of Farquhar et al. (1980), our V_cmax estimation method shows great potential for providing a more realistic view of the species variation and environmental regulation of photosynthetic capacity in C₃ plants. In doing so, the original simplicity of the original rectangular hyperbola version of the model of Farquhar et al. (1980) need necessarily not be sacrificed since, following proper estimation of V_cmax with the non‐rectangular hyperbola model parameterized with a set of Rubisco kinetic constants valid at C_c, it can be adjusted to account for g_i through Eqn 16.

ACKNOWLEDGMENTS

This research was supported by grants from the Natural Sciences and Engineering Research Council of Canada (to N.J.L.).