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C4-type photosynthesis is known to vary with growth and measurement temperatures. In an attempt to quantify its variability with measurement temperature, the photosynthetic parameters – the maximum catalytic rate of the enzyme ribulose 1·5-bisphosphate carboxylase/oxygenase (Rubisco) (Vcmax), the maximum catalytic rate of the enzyme phosphoenolpyruvate carboxylase (PEPC) (Vpmax) and the maximum electron transport rate (Jmax) – were examined. Maize plants were grown in climatic-controlled phytotrons, and the curves of net photosynthesis (An) versus intercellular air space CO2 concentrations (Ci), and An versus photosynthetic photon flux density (PPFD) were determined over a temperature range of 15–40 °C. Values of Vcmax, Vpmax and Jmax were computed by inversion of the von Caemmerer & Furbank photosynthesis model. Values of Vpmax and Jmax obtained at 25 °C conform to values found in the literature. Parameters for an Arrhenius equation that best fits the calculated values of Vcmax, Vpmax and Jmax are then proposed. These parameters should be further tested with C4 plants for validation. Other model key parameters such as the mesophyll cell conductance to CO2 (gi), the bundle sheath cells conductance to CO2 (gbs) and Michaelis–Menten constants for CO2 and O2 (Kc, Kp and Ko) also vary with temperature and should be better parameterized.
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The C4 photosynthesis pathway, discovered by Hatch & Slack (1966), is a series of anatomical and biochemical modifications that concentrate CO2 around the carboxylating enzyme ribulose 1·5-bisphosphate carboxylase/oxygenase (Rubisco). This elevated CO2 concentration at the site of Rubisco carboxylation allows C4 plants to have a higher nitrogen and water use efficiency, and therefore a higher production potential (Long 1998). Changes in atmospheric composition associated with global warming (IPCC 2001) have a direct effect on plant growth (Harley et al. 1992) and on the relative distribution of C4 and C3 vegetation by affecting the key physiological parameters that control photosynthesis (Sage & Kubien 2003). The optimal temperature needed for C4 photosynthesis is typically 10 °C greater than that of C3 photosynthesis, which explains why C4 plants are rare in cold, temperate climates and are not found in cold climates (Long 1998). Photosynthesis is highly dependent on temperature under light-saturating conditions, reflecting the temperature dependence of the maximal rate of carboxylation (Berry & Farquhar 1978). The factors affecting CO2 assimilation at the leaf level are qualitatively well known today. However, one of the major challenges is to quantitatively predict the changes in carbon fluxes between the atmosphere and the continental biosphere in response to a changing environment.
Three photosynthetic enzymes control C4-type photosynthesis at saturating light: Rubisco, phosphoenolpyruvate carboxylase (PEPC) and pyruvate, orthoPhosphate DiKinase (PPDK) (Matsuoka et al. 2001). These key photosynthetic enzymes are not only regulated by metabolites, but also by environmental factors. In the C4 photosynthesis models, the rates of phosphoenolpyruvate (PEP) and Rubisco carboxylation (Vp and Vc, respectively) are major determinants in calculating the net rate of CO2 assimilation. The rates of Rubisco and of PEP carboxylation depend among others, upon the Michaelis–Menten constants for O2 and CO2 (Ko and Kc, Kp), the maximal rate of Rubisco carboxylation (Vcmax), the maximal rate of PEP carboxylation (Vpmax), the maximal electron transport rate (Jmax) and the relative specificity of Rubisco (Sc/o). Several authors (Badger, Andrews & Osmond 1974; Farquhar & von Caemmerer 1982; Hudson et al. 1990; Sage 2002; Chinthapalli, Murmu & Raghavendra 2003) pointed out that variation in the optimum temperature for C3 and C4 photosynthesis among species or growth conditions could result from variation in the relative values of Ko, Kc, Kp, Vcmax, Vpmax and Jmax. As a consequence, it seems logical to introduce these parameters as temperature functions in the photosynthesis models, and that these functionsare different for C3 and C4 species. The C4 photosynthesis model seems to be relatively sensitive to Vcmax, Vpmax and Jmax (von Caemmerer & Furbank 1999) and less sensitive to Kc, Ko and Kp. Furthermore, the difference for Rubisco's relative specificity (Sc/o) between C3 and C4 species is less pronounced (Tcherkez, Farquhar & Andrews 2006). Based on this, we chose to concentrate in this study on deriving a technique for estimating values of Vcmax, Jmax and Vpmax for C4 plants.
For C3 plants, Vcmax can be obtained from the initial slope of the response of the assimilation rate (An) to substomatal carbon dioxide concentration (Ci), whereas Jmax is calculated from the plateau of the response of the assimilation rate (An) to photosynthetic photon flux density (PPFD). Concerning C4 plants, and because of the complex biochemical mechanisms, calculation of Vcmax, Vpmax, and Jmax cannot be done graphically. This is why, so far, C3-derived functions have been used in C4 photosynthesis modelling.
In this paper, we present a method for calculating Vcmax, Vpmax, and Jmax by combining an inversion of the C4 photosynthesis model (von Caemmerer & Furbank 1999) and experimental curves of An versus Ci, and An versus PPFD at different measurement temperatures on maize leaves. We then suggest parameters for a fitted Arrhenius function describing the variation of the photosynthesis C4 parameters (Vcmax, Vpmax and Jmax) with temperature. The effects of choosing C3-derived temperature functions for other photosynthesis parameters (Kc, Ko and Kp) are also addressed in a sensitivity study of this inversion method.
MATERIALS AND METHODS
Seeds of maize (Zea mays var. Chambord) were germinated in mini-greenhouses and then replanted in 25 L pots containing a commercial potting mixture of white peat, black peat and sand (Floradur A; Floragard, Oldenburg, Germany). Germinations were spread out over several weeks in a way to get similarly aged plants for every series of measurement. Plants were grown in phytotrons with day/night average temperatures of 23.5 ± 1.2/19.8 ± 0.6 °C, day length of 12 h (from 0700 to 1900 h), average PPFD of 440 ± 36 µmol m−2 s−1, average relative humidity of 52 ± 6% during the day and 87 ± 6% during the night and average CO2 concentration of 433 ± 19 µmol mol−1.
A thermocouple (copper/constantan), humidity (HMP45AC, Vaisala, Helsinki, Finland) and quantum (Li-190SZ; Li-Cor, Lincoln, NE, USA) sensors, and a CO2 analyser (model 6262; Li-Cor) were installed in the phytotron to permanently control growth conditions. All these variables were recorded by a Campbell data logger (CR10X, Campbell Scientific, Leicester, UK). Sensors were sampled every 5 s, and averaged and stored on 15 min intervals.
Soil water content was maintained at field capacity by a regular surface irrigation. Fertilizers based on N-P-K (15-10-30) and oligo-elements were added systematically twice a week after the fourth week of growth.
Gas exchange measurement
Photosynthesis, transpiration and leaf conductance were measured with a portable gas exchange system (model 6400; Li-Cor). Gas measurements were performed on the eighth fully developed leaf of each plant. Each plant was used for six measurements over 2 d (three curves per day). The Li-Cor 6400 was calibrated every day before the measurements and matched twice a day (between the curves). The plants were placed in a separate phytotron 1 h before the start of the measurements at a temperature of 20 or 35 °C according to the block measurement temperature needed because the Li-Cor can only regulate the temperature within ±6 °C of the ambient temperature. Six temperatures ranging from 15 to 40 °C were studied. A high relative humidity was maintained in the phytotron by injections of water vapour at ambient temperature of 35 °C. This humidity caused a slight difference between the measured leaf temperatures and the prescribed temperatures. Two kinds of curves were assayed: net photosynthesis (An) versus intercellular airspace CO2 concentrations (Ci), and An versus PPFD.
An versus Ci curves were measured using a light intensity of 1800 µmol m−2 s−1 quanta and a variable CO2 concentration (400, 350, 300, 240, 180, 120, 80, 40 and 0 µmol mol−1) in the sample chamber. Three gas-exchange measurements were made 15 min after having reached the desired CO2 concentrations at 30 s intervals.
An versus PPFD curves were measured using a constant CO2 partial pressure of 350 µmol mol−1 in the sample chamber and a variable light intensity (500, 150, 100, 50, 0, 500, 1000, 1500 and 2000 µmol m−2 s−1 quanta). Three gas measurements were made at 30 s intervals 20 min after prescribing the desired light intensity.
The plateau of some curves was clearly differentiated from the rest. Upon examining the stomatal conductance to H2O calculated by the Li-Cor apparatus for these curves at 25 and 30 °C and at an intercellular air space CO2 concentration of 100 µmol mol−1, we noticed that they show a low conductance. This low conductance indicates stresses encountered by the plant during the measurements, and thus introduce a non-desired parameter in the experiment probably because of a ‘pot’ effect. A threshold stomatal conductance of 0.25 mol m−2 s−1 (Rochette et al. 1991) was established. All plants having had a conductance below this threshold at 25 or 30 °C for the series of measurements the concerned day were eliminated.
A statistical analysis was performed using the analysis of variance (anova) and the Duncan multiple comparisons test (SPSS v10.0; SPSS, Inc., Chicago, IL, USA). For each measurement temperature, pair-wise comparisons were made with all measurement temperatures. Statistical significance was assumed to occur for P < 0.05 after adjustment for multiple comparisons.
The model can be summarized by a couple of equations describing the carbon balance in the mesophyll and in the bundle sheath cells (Fig. 1) based on the simplifying assumption that oxygen concentration in the bundle sheath cells is equal to oxygen concentration in mesophyll cells. This assumption is particularly true for maize, an NADP–ME species, because bundle sheath cells produce little or no O2 (Hatch 1987). However, for other applications, O2 concentration in the bundle sheath cells could be modelled according to Berry & Farquhar (1978).
In the mesophyll cells, the net carbon assimilation (An) is the carbon fixed to PEP, defined by the rate of PEP carboxylation (Vp), minus the leakage (L) from the bundle sheath cells and minus the carbon released by mitochondrial respiration of the mesophyll cells (Rm) (see Appendix for a full list of abbreviations and symbols):
In the bundle sheath cells, the net carbon assimilation is the carbon fixed by Rubisco defined by the rate of Rubisco carboxylation (Vc) minus the carbon released by the photorespiratory cycle (half the rate of Rubisco oxygenation, Vo) and minus mitochondrial respiration (Rd):
where gbs is the bundle sheath cell conductance to CO2, Cs and Cm the CO2 partial pressure in the bundle sheath cells and in the mesophyll cells, respectively, and γ* half of the reciprocal of Rubisco specificity (Sc/o). The original version of the model is forced with values of light intensity, temperature and CO2 partial pressure in the mesophyll cells (Cm), and it calculates the resulting net photosynthesis (An). Vcmax, Vpmax and Jmax are given parameters in the model.
In this model, photosynthesis can either be enzyme limited or electron transport limited. In each case, the rate of PEP (Vp) and Rubisco (Vc) carboxylation are described as a function of the maximal rates of Rubisco (Vcmax) and of PEP (Vpmax) carboxylation for the former and of the maximal electron transport rate (Jmax) for the latter.
The mathematical solution of the model is given by taking the minimum of the enzyme limited (Ac) or electron transport limited (Aj) photosynthesis. Table 1 describes and gives the values at 25 °C of the photosynthetic parameters used in the model.
Rubisco, ribulose 1·5-bisphosphate carboxylase/oxygenase; PEP, phosphoenolpyruvate; PSII, photosystem II.
650 µbar or variable with temperature
Michaelis–Menten constant of Rubisco for CO2
450 mbar or variable with temperature
Michaelis–Menten constant of Rubisco for O2
80 µbar or variable with temperature
Michaelis–Menten constant of PEPcase for CO2
Photosynthetically active irradiance absorbed by PSII
80 µmol m−2 s−1
PEP regeneration rate
Leaf mitochondrial respiration
Mesophyll mitochondrial respiration
3 mmol m−2 s−1
Bundle sheath conductance to CO2
2 mol m−2 s−1
Mesophyll conductance to CO2
Partitioning factor of the electron transport rate
Empirical curvature factor
CO2 compensation point
Half the reciprocal of Rubisco specificity
For the inversion model concerning experimental gas measurements, we tried to have conditions of either enzyme-limited or electron transport-limited photosynthesis. By doing so for the calculation of Vcmax and Vpmax, we used the enzyme-limited expressions of Vc and Vp, whereas for the calculation of Jmax we used the electron transport limited expressions of Vc and Vp. The inversion method consists of forcing the model with values of lighting (PPFD), measurement temperature, intercellular air space CO2 partial pressure (Ci) and photosynthesis (An) and calculating Vcmax, Jmax and Vpmax.
Calculation of Vcmax and Vpmax
From the experimental gas measurements, under saturating light conditions, we obtained carbon dioxide assimilation rate (An) as a function of substomatal CO2 concentration (Ci). Because in the von Caemmerer & Furbank (1999) photosynthesis model, calculations are based on CO2 concentrations in the mesophyll cells (Cm), we can add an equation to the two equations (Eqns 3 & 4) described earlier that allows us to calculate the corresponding Cm for every Ci measured, assuming a value for the mesophyll cell conductance (gi) is given.
Considering that the measurements were made in conditions where photosynthesis is enzyme limited, we can use the equations given as follows for Vc and Vp.
where Kc, Ko and Kp are the Michaelis–Menten constants for Rubisco carboxylation, oxygenation and PEP carboxylation, respectively, and O is the oxygen concentration in the mesophyll and bundle sheath cells.
Considering the three Eqns 5, 8 and 9: gi, gbs, Rm, Kp, Kc, Ko, O, Rd and γ* are constant parameters at a given temperature (see Table 1 for values at 25 °C); An and Ci are measured values; and Cm, Cs, Vcmax and Vpmax are unknowns.
In this system, we have only three equations and four unknowns; there is therefore no unique solution. To overcome this problem, we take advantage of the fact that Vcmax and Vpmax are constant for different Ci and thus different Cs and Cm. Taking two sets of experimental points of the An versus Ci curve (An1; Ci1) and (An2; Ci2) and replacing in Eqns 5, 8 and 9, we end up with six equations and six unknowns (Vcmax, Vpmax, Cs1, Cs2, Cm1 and Cm2).
Our experiments were based on nine different external CO2 concentrations (400, 350, 300, 240, 180, 120, 80, 40 and 0 µmol mol−1). Because a pair of experimental points is needed for every calculation of Vcmax and Vpmax, we end up with four independent values for a given temperature. We therefore chose to replace the experimental points by an equation that best fits those points. A fourth-degree polynomial was chosen as a best fit according to the least square methods using Stanford graphics. Figure 2a illustrates an example of experimental points measured at 25 °C together with the fitted polynomial. This polynomial allows us to have values of An for any value of Ci at chosen intervals. Two couples of (An, Ci) are then used in two sets of Eqns 5, 8 and 9 to calculate Vcmax, Vpmax, Cs1, Cs2, Cm1 and Cm2. With this system of resolution for every experimental An versus Ci curve at a given temperature, we can get a Vcmax versus Ci, Vpmax versus Ci and two confounded Cs versus Ci curves as illustrated by an example in Fig. 2b–d.
Upon analyzing the Vcmax and Vpmax values, we notice that those are not constant especially for small values of Ci. This can be explained by instability in the model calculation because calculations are done numerically and not analytically. Furthermore, small values of Ci might hold some errors because of limitations in measurement techniques. As a result, Vcmax and Vpmax were taken in the constant part of the Vcmax versus Ci, and Vpmax versus Ci curves, respectively; that is for values of Ci limited between 40 and 80 µmol mol−1. Although the parameters were rather constant at high Ci values, we chose to limit ourselves by an upper boundary because we were not sure to be in saturating light conditions.
Calculation of Jmax
For the calculation of Jmax, we consider the measurements made with a constant Ci and a variable PPFD which correspond to electron transport-limited photosynthesis and saturating CO2 partial pressures. We therefore use the electron transport-limited expressions of Vc and Vp.
where x is a partitioning factor of electron transport and Jt is the electron transport rate and is modelled according to Farquhar & Wong (1984) as an empirical non-rectangular hyperbolic function:
where I2 is the total absorbed irradiance and is a function of the incident irradiance I, and θ is an empirical curvature factor.
Similar to Vcmax and Vpmax calculations, Cm can be obtained from Eqn 5.
Considering these two equations (Eqns 13 & 14) together with Eqns 12 and 5: θ, O, x and γ* are constant parameters at a given temperature (see Table 1 for values); An, Ci and I (PPFD) are measured values (Ci is a fixed value for all measurements); and Jmax, Cm and Cs are unknowns.
In this case, we have a set of three equations and three unknowns. We can therefore, for a given temperature and using one value of An and PPFD, obtain a unique value of Jmax, Cm and Cs. It is simple in this case to calculate values of Jmax, Cm and Cs for every measurement point of the An versus PPFD experimental curves.
RESULTS AND DISCUSSION
An versus Ci curves at the different measurement temperatures and for a PPFD of 1800 µmol m−2 s−1 are represented in Fig. 3. The repetitions show relatively little variability within a measurement temperature. As expected, experimental photosynthesis curves show that not only the maximal rate of photosynthesis (asymptote) of the curves changes with temperature, but also the carboxylation efficiency (slope at the origin). We also noticed that photosynthesis shows strong temperature dependence under high light with an optimum at a temperature between 30 and 40 °C as illustrated in Table 2. Noting that for the maximal rate of photosynthesis, the plateaus for measurement temperatures of 30, 35 and 40 °C are not significantly different from one another; this leads us to say that our measured maximum capacity of photosynthesis on maize is between 30 and 40 °C.
Table 2. Average carboxylation efficiencies and maximal rates of photosynthesis for different average leaf temperatures
Block T (°C)
Average leaf T (°C)
Carboxylation efficiency (mol m−2 s−1)
Maximal rates of photosynthesis (µmol m−2 s−1)
Gas exchange parameters were measured at a photosynthetic photon flux density (PPFD) of 1800 µmol quanta m−2 s−1 and ambient CO2 concentrations varying between 0 and 400 µmol mol−1. Values are means ± standard deviations for five to six measurements per assigned block temperature. Means in each column denoted by different superscripts are significantly different (P < 0.05).
An versus PPFD curves at the different measurement temperatures are represented in Fig. 4. Upon analyzing the light response curves, we observed that the mean slope of the initial linear portion of the curve called the ‘quantum efficiency for CO2 uptake’ does not vary with measurement temperatures. However, the asymptote which is the maximal or ‘light-saturated’ rate of photosynthesis varies with temperature.
Measured quantum efficiencies are in the range stated in the literature, which is between 0.052 and 0.069 mol CO2 mol−1 absorbed photons (Pearcy & Ehleringer 1984), and the maximal rates of photosynthesis confirm that C4 photosynthesis is optimal between temperatures of 30 and 35 °C as noticed for An versus Ci curves. The absence of a temperature dependence of the quantum yield can be explained by the fact that the maximum quantum yield of photosystem II is insensitive to leaf temperatures up to 45 °C (Ehleringer & Bjorkman 1977; Crafts-Brandner & Salvucci 2002). However, under conditions of high radiation, photosynthesis rates are controlled by the temperature dependence of Vc (Leegood & Edwards 1995), which explains the variation of the maximal rates of photosynthesis observed.
Model inversion results
The two kinds of curves obtained, An versus Ci and An versus PPFD, allowed us to calculate by model inversion the parameters Vcmax, Vpmax and Jmax. Because C4 plants normally operate at saturating or near saturating concentrations of CO2, Rubisco functioning is thought to be more sensitive to temperature than in C3 plants (Leegood & Edwards 1995) because it is the limiting factor. This temperature dependence of Rubisco is reflected by the influence of temperature on the maximum velocity of carboxylation (Berry & Farquhar 1978). Figure 5a represents the different values of Vcmax calculated at different temperatures compared to functions of Vcmax for C3 plants from the literature (Farquhar et al. 1980; Harley et al. 1992; Bunce 2000; Bernacchi et al. 2001). Our inversion results correspond to a modified Arrhenius type equation that has deactivation energy because we measured an optimum for Vcmax. Calculated values of Vcmax increased with leaf temperature from 15 to 30 °C, and then tended to remain constant at 35 °C. This optimum for Vcmax above 30 °C conforms to Bunce (2000) for measurements he made on eight C3 species and with Kubien et al. (2003) for in vivo measurements made on Flaveria bidentis. However, the values we calculated are much lower than those calculated by Harley et al. (1992) which could be explained by the fact that Vcmax depends on the quantity of Rubisco (Stitt & Schulze 1994) and that C4-type plants have less Rubisco than C3 plants (Long 1998; Kubien et al. 2003).
Figure 5b represents values of Vpmax modelled at different leaf temperatures compared to functions in literature. These values show a constant increase until 40 °C. Chinthapalli et al. (2002) found that the temperature optimum of PEPC is around 40–45 °C and that its activity sharply drops below the optimal temperature which is in accordance with our results. We noticed that the curve proposed by Chen et al. (1994) fits rather well the calculated values of Vpmax. However, we could not obtain measurements for temperatures above 40 °C and therefore were not able to detect the optimum for PEPC.
The values of Jmax at different temperatures compared to functions stated in literature are represented in Fig. 5c. Our calculated Jmax increased continuously until a leaf temperature of 30–35 °C and then tended to decrease, suggesting that a modified Arrhenius-type equation would best fit this curve. Calculated values are contradictory with Bernacchi, Pimentel & Long (2003) who measured for experiments done on tobacco a continuous increase of Jmax until a leaf temperature of 40 °C. The estimates of Jmax he calculated at 40 °C (300 µmol m−2 s−1) are therefore higher than the ones we calculated (225 µmol m−2 s−1). However, Jmax calculated by Harley et al. (1992), Medlyn et al. (2002a) and June, Evans & Farquhar (2004) for C3 species are conforming to the optimum we found. No experimental values of Jmax for C4 species were found in literature. Knowing that because of low levels of photorespiration in C4 plants, the activity of electron transport is closely linked to CO2 fixation; we should therefore expect differences in Jmax values compared to C3 plants.
Suggestions of temperature-dependent curves for Vcmax, Vpmax and Jmax
Based on the discussion before and following Farquhar et al. (1980) and Leuning (1997), the temperature dependence of Vcmax, Vpmax and Jmax can be described using a modified Arrhenius equation, which introduces an optimum temperature:
where k25 is the value of Jmax, Vcmax or Vpmax at temperatures 25 °C; the activation energy (Ea) gives the rate of exponential increase of the function below the optimum (and is analogous to parameter Ea in the Arrhenius function); the deactivation energy (Hd) describes the rate of decrease of the function above the optimum and ΔS is known as an entropy factor but is not readily interpreted. The values of Vcmax, Vpmax and Jmax at 25 °C to which the curves are normalized are 32, 125 and 191 µmol m−2 s−1 respectively.
Leuning (2002) presents a review of existing data sets for these parameters for C3 species. Having a temperature function of Vcmax, Vpmax and Jmax with parameters (Ea, Hd and ΔS) adapted to C4 plants will contribute in better integrating temperature responses in photosynthesis models. The temperature responses of Vcmax, Vpmax and Jmax were fitted to Eqn 15 by the non-linear regression method (SPSS 10.0.1). Figure 6a–c represents the obtained Vcmax, Vpmax and Jmax functions, respectively.
The fitted function of Jmax (Fig. 6c) might be questioned at high temperatures because of the fact that there is a relatively strong dispersion of these values. This leads us to think that the fitted Jmax at high temperatures might be underestimated.
Effect of parameter choice on model inversion results
Calculating the temperature dependence of Vcmax, Jmax and Vpmax from experimental curves requires assumptions about the temperature dependence of other kinetic parameters of PEPC, Rubisco and the carbon cycle, such as the Michaelis–Menten constants for CO2 and O2, the bundle sheath (gbs) and mesophyll conductance to CO2 (gi) and the mitochondrial respiration rates (Rd).
Effect of mesophyll cell conductance to CO2 (gi)
The diffusion path for CO2 in C4 species is from the intercellular air space to the cytoplasm where it is fixed by PEPC. Therefore, gi is likely to be proportional to mesophyll surface area (von Caemmerer & Furbank 1999). At present, there is no reliable estimate of gi. Pfeffer & Peisker (1998) used a derivative equation of the von Caemmerer & Furbank (1999) model along with measurements of PEPC activity to estimate values of gi. However, their estimate of a value of gi = 0.87 mol m−2 s−1 is based on the assumption that gi does not change with growth conditions including temperature. We tested the inversion model for two values of gi (1 and 3 mol m−2 s−1), the resulting Vcmax, Vpmax and Jmax were compared to the Vcmax, Vpmax and Jmax calculated for a gi of 2 mol m−2 s−1. We noticed that for Jmax, the variation resulting from different values of gi is less than 1% of the reference Jmax value (at gi = 2 mol m−2 s−1). Sensitivity analysis of results of Vcmax and Vpmax are represented in Fig. 7. This sensitivity of the inversion model is expected because the original model (von Caemmerer & Furbank 1999) is sensitive to these parameters. Moreover, according to the sensitivity analysis done by von Caemmerer & Furbank (1999), when placed in substomatal CO2 concentrations (Ci) between 40 and 80 µmol mol−1, the model is more sensitive to Vpmax than to Vcmax. This could explain the higher sensitivity of Vpmax than Vcmax to gi in the inversion model.
Effect of bundle sheath cell conductance to CO2 (gbs)
At present, little is known also about the variation of bundle sheath conductance (gbs) with growth conditions. Bundle sheath conductance is the product of the diffusion conductance across the bundle sheath cell wall interface and the bundle sheath surface area (von Caemmerer & Furbank 1999). We based our parameter choice on estimated values of gbs in literature.
We tested the inversion model for two different values of gbs (1.5 and 6 mmol m−2 s−1) and compared the resulting Vpmax, Vcmax and Jmax to values of Vpmax, Vcmax and Jmax calculated for a reference value of gbs of 3 mmol m−2 s−1. Results indicate that the inversion model is sensitive to gbs especially at high temperatures. We noticed that for Jmax, the variation resulting from different values of gbs is less than 1% of the Jmax value for a gbs of 3 mmol m−2 s−1. Sensitivity analysis results of Vcmax and Vpmax are represented in Fig. 8. As opposed to what was observed for gi, where Vcmax and Vpmax vary in opposite directions, Vcmax and Vpmax vary in the same sense when gbs varies. This is explained by the fact that gbs, unlike gi, not only affects Vp and Vc, via its effect on CO2 concentrations in the mesophyll and in the bundle sheath cells, but also via its effects on the leakiness value (L).
Effect of Michaelis–Menten constants for Rubisco (Kc, Ko) and for PEP (Kp)
The Michaelis–Menten constant of Rubisco for CO2 (Kc), for O2 (Ko) and of PEPC for CO2 (Kp) are known to vary with temperature (Medlyn, Loustau & Delzon 2002b). The in vivo and in vitro temperature dependence of these coefficients was measured by several authors. Bernacchi et al. (2001) measured these parameters in vivo, while Badger & Collatz (1977) and Jordan & Ogren (1984) measured these parameters in vitro. After testing the sensitivity of the inversion model to these variable parameters, we noticed that the values of Jmax, Vcmax and Vpmax are less sensitive to the chosen equations of Kc, Ko and Kp. Because our inversion model functions with in vivo measurements, it is logical to choose to parameterize with measurements done in vivo. However, the measurements of Bernacchi et al. (2001) are done in the case of tobacco, a C3 plant, and therefore are not applicable to PEPC. Moreover, our calculated Vcmax, Vpmax and Jmax results do not really agree with values he found. As a consequence, we chose to parameterize the inversion model with values used by Chen et al. (1994) which he uses in his C4 model and which are derived from measurements made by Jordan & Ogren (1983).
Effect of mitochondrial respiration (Rd)
At the leaf level, Collatz et al. (1991) model dark respiration (Rd) as a function of Vcmax; a typical value being Rd equals 0.015 times Vcmax. Their assumption implies that Rd is a function of leaf nitrogen. Farquhar et al. (1980) state that Rd is a strong, non-linear function of temperature. They described the temperature dependency of Rd as an Arrhenius function similar to the one they attribute to the Michaelis–Menten constants. On the other hand, Chen et al. (1994) used a Q10 function to describe the temperature dependency of Rd. To test the sensitivity of our inversion model to mitochondrial respiration, we tried the function of Chen et al. (1994) and the Vcmax function of Farquhar et al. (1980). Resulting Vcmax, Vpmax and Jmax were not significantly different than those obtained with values of Rd = 0 (von Caemmerer & Furbank 1999). We therefore used values of zero for the rest of our results. This is particularly true because we have high values of An and therefore the values of Vcmax, Vpmax and Jmax are not very sensitive to Rd.
Comparison of Vpmax values between two different methods
Another method for calculating Vpmax for C4 plants can be deduced from the initial slope of the An versus Ci curve (Pfeffer & Peisker 1998; von Caemmerer & Furbank 1999). The approximated equation used to calculate Vpmax from the slope is illustrated as follows, noting that gi and Kp (PEP kinetic constant for CO2) should be given.
Figure 9 shows the comparison between Vpmax obtained from Eqn 16 and Vpmax obtained by inversion. We noticed that the predicted Vpmax calculated with Eqn 16 agrees with values of Vpmax calculated from the model inversion. The linear regression between slope calculated and model calculated Vpmax resulted in an R2 value of 0.93. This validation of Vpmax values not only validates our inversion method, but also the value of gbs and Rd taken because Eqn 11 is independent of both those parameters. Moreover, because our inversion method calculates a couple, Vcmax and Vpmax, each time, and that this couple is tightly correlated, this suggests that values of Vcmax are correct.
The aim of this study was to define temperature dependent functions of Vcmax, Vpmax and Jmax relevant to C4 species to improve modelling temperature effects on photosynthesis. In general, the obtained temperature-dependent functions give satisfactory results when compared to experimental measurements. However, the patterns of Vcmax, Vpmax and Jmax represented, while suitable for modelling the temperature response of photosynthesis, have an ambiguous biochemical interpretation at present.
Concerning Vpmax, the values calculated seem to be adequate based on the comparison with different calculation methods and with literature. Moreover, the effect of Vpmax on the initial slope of the An versus Ci curve facilitates the interpretation of gas measurement curves and the deduction of parameter values.
With regard to Vcmax, values obtained are a little doubtful. The inversion method can be responsible for this. On one hand, because of the polynomial fitted to the experimental curves which could be biased by light limitation; on the other hand, by choosing the values of Vcmax in the intermediate part of the An versus Ci curve where Vcmax and Vpmax influence the response of photosynthesis. Validation of Vpmax values allows in a certain way to validate Vcmax values because for each curve, a couple (Vcmax, Vpmax) was chosen. Further investigations on the part of the curve used in the inversion model and on light limitation of these curves would ensure a more rigorous calculation of Vcmax. These uncertainties can be caused by being limited by electron transport, therefore having a biased polynomial for the model inversion or to being limited by the rate of PEP regeneration and therefore by Vpr and not Vcmax. However, these uncertainties are minimized by choosing to invert the model on the median part of our An versus Ci curve and consequently avoiding the plateau effect.
As for Jmax, the inversion method is simpler and more accurate because no polynomial fitting is required and we obtained one value of Jmax for every gas measurement. However, the plateau of An versus PPFD curves is rarely reached, and we might be in enzyme-limiting conditions.
Relying on this sensitivity to parameter choice, the proposed Arrhenius function parameters must be used carefully. One should keep in mind that comparing values of Vcmax, Vpmax and Jmax between different studies can be misleading. One reason is that parameter values obtained from data can differ according to the method used to derive them; another is the choice of the model parameters such as gs, gi, and the Michaelis–Menten constants.
The database of temperature response of Vcmax, Vpmax and Jmax has the potential to expand in the near future, given recent improvements in temperature control in commercially available gas-exchange systems, and given that graphical and mathematical interpretation of these measurements expands likewise.
Another key requirement for future research highlighted by this study is the need for more information on the temperature dependence of the Michaelis–Menten constants for Rubisco and PEP, and on the values of gi and gbs which influence the model inversion results. This is why in the absence of good knowledge of the temperature dependence of these parameters, it is important, particularly when modelling, to ensure that the parameter sets are consistent.
This study was part of the ‘BioPollAtm’ project. It was supported by the French Ministry of Ecology and Sustainable Development and by the Agence de l'Environnement et de la Maîtrise de l'Energie (ADEME) (French Environment and Energy Management Agency).
We thank Professor Graham Farquhar and two anonymous referees for suggestions that greatly improved the manuscript.
List of symbols and abbreviations
Empirical curvature factor
Half of the reciprocal of Rubisco specificity
An, Ac, Aj
Net, enzyme-limited and electron transport rate of CO2 assimilation, respectively
Ci, Cm, Cs
Intercellular air space, mesophyll and bundle sheath cells CO2 partial pressures, respectively
Stomatal conductance to CO2
Mesophyll cells conductance to CO2
Bundle sheath cells conductance to CO2
Photosynthetically active irradiance absorbed by PSII
Electron transport rate and maximum electron transport rate, respectively
Rubisco kinetic constant for CO2 and O2, respectively
PEP kinetic constant for CO2
Leak rate of CO2 out of bundle sheath cells
Nicotinamide adenine dinucleotide–malic enzyme
O2 partial pressure in the bundle sheath and mesophyll cells
Parameter at 25 °C
Photosynthetic active radiation
Pyruvate, orthoPhosphate DiKinase
Photosynthetic photon flux density
Universal gas constant
Mitochondrial respiration of mesophyll cells, and bundle sheath cells in the light, respectively
Dark mitochondrial respiration
Ribulose 1·5-bisphosphate carboxylase/oxygenase
Ribulose 1,5 bisphosphate
Rubisco carboxylation rates and maximal carboxylation rates, respectively
Rubisco oxygenation rate
PEP carboxylation rates and maximal carboxylation rates, respectively
PEP regeneration rate
Partitioning factor of electron transport rate between the C4 and the C3 cycle