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WHOLE PLANT AND ECOPHYSIOLOGY

The Effect of Leaf-Level Spatial Variability in Photosynthetic Capacity on Biochemical Parameter Estimates Using the Farquhar Model: A Theoretical Analysis^1,[W],[OA]

Department of Plant Biology, University of Illinois, Urbana, Illinois 61801

	ABSTRACT

TOP ABSTRACT RESULTS DISCUSSION MATERIALS AND METHODS LITERATURE CITED

 
Application of the widely used Farquhar model of photosynthesis in interpretation of gas exchange data assumes that photosynthetic properties are homogeneous throughout the leaf. Previous studies showed that heterogeneity in stomatal conductance (g_s) across a leaf could affect the shape of the measured leaf photosynthetic CO₂ uptake rate (A) versus intercellular CO₂ concentration (C_i) response curve and, in turn, estimation of the critical biochemical parameters of this model. These are the maximum rates of carboxylation (V_c,max), whole-chain electron transport (J_max), and triose-P utilization (V_TPU). The effects of spatial variation in V_c,max, J_max, and V_TPU on estimation of leaf averages of these parameters from A-C_i curves measured on a whole leaf have not been investigated. A mathematical model incorporating defined degrees of spatial variability in V_c,max and J_max was constructed. One hundred and ten theoretical leaves were simulated, each with the same average V_c,max and J_max, but different coefficients of variation of the mean (CV_VJ) and varying correlation between V_c,max and J_max (). Additionally, the interaction of variation in V_c,max and J_max with heterogeneity in V_TPU, g_s, and light gradients within the leaf was also investigated. Transition from V_c,max- to J_max-limited photosynthesis in the A-C_i curve was smooth in the most heterogeneous leaves, in contrast to a distinct inflection in the absence of heterogeneity. Spatial variability had little effect on the accuracy of estimation of V_c,max and J_max from A-C_i curves when the two varied in concert ( = 1.0), but resulted in underestimation of both parameters when they varied independently (up to 12.5% in V_c,max and 17.7% in J_max at CV_VJ = 50%; = 0.3). Heterogeneity in V_TPU also significantly affected parameter estimates, but effects of heterogeneity in g_s or light gradients were comparatively small. If V_c,max and J_max derived from such heterogeneous leaves are used in models to project leaf photosynthesis, actual A is overestimated by up to 12% at the transition between V_c,max- and J_max-limited photosynthesis. This could have implications for both crop production and Earth system models, including projections of the effects of atmospheric change.

One of the basic premises of the Farquhar model, as modified by Sharkey (1985), is that steady-state photosynthesis is limited by either (1) the maximum rate of carboxylation governed by Rubisco, termed V_c,max; (2) the rate of ribulose-1,5-bisP (RuBP) regeneration, which is assumed to be limited by the maximum rate of electron transport, known as J_max; or (3) capacity for triose-P utilization, V_TPU. Once these three are known, the leaf photosynthetic rate can be calculated given light flux, CO₂ and O₂ concentrations, and temperature. An implication of this is that the A-C_i response of a leaf will show an abrupt decrease in dA/dC_i, with increasing C_i when A passes from Rubisco- to RuBP-limited photosynthesis and, in turn, to TPU limitation. Represented graphically, this will be evident as inflections in the A-C_i response. Photosynthetic CO₂ uptake is routinely measured by enclosing a whole leaf or a few square centimeters of a leaf in a gas exchange cuvette. The A-C_i response measured this way rarely shows the abrupt transitions predicted by the Farquhar model (e.g. Riddoch et al., 1991; Wullschleger, 1993). Further, metabolic control analysis using leaves with transgenically decreased quantities of specific photosynthetic proteins suggest that, for a given CO₂ concentration, control is shared between Rubisco and proteins limiting the rate of RuBP regeneration (Quick et al., 1991; Price et al., 1998; Raines, 2003). Understanding the basis for these inconsistencies between assumptions of the model and observations may be critical to the application of the model both as an in vivo method of determining biochemical limitations and in projecting photosynthesis at scales from crop canopies to the globe.

Application of the Farquhar model assumes that the photosynthetic properties of the leaf are spatially homogeneous (von Caemmerer, 2000). Based on this assumption, V_c,max, J_max, and V_TPU are derived by fitting the Farquhar model to measured leaf responses of A to C_i (Long and Bernacchi, 2003; Sharkey et al., 2007). Given complex gradients of leaf development, impact of heterogeneous environments, disease, and pest attack, it seems unlikely that the assumption of homogeneity is often met in nature (Terashima, 1992; von Caemmerer, 2000; Aldea et al., 2006), but does this matter for fitting V_c,max, J_max, and V_TPU and for accurate modeling of photosynthetic carbon assimilation?

The effect of leaf-level heterogeneity in stomatal conductance (g_s) on the A-C_i response curve has previously been studied (Cheeseman, 1991; Buckley et al., 1997). Increased stomatal heterogeneity was found to decrease the initial slope of the A-C_i curve with the result that V_c,max determined from such curves underestimated the true value. However, simulations showed that stomatal heterogeneity could not fully explain the observed leaf-level variability in photosynthetic activity (Cheeseman, 1991; Buckley et al., 1997). Variation in biochemical parameters (i.e. V_c,max, J_max, and V_TPU) has been implicated as a possible explanation; however, the effects of spatial heterogeneity in these parameters have not yet been investigated.

In addition, there is an exponential decline in light from the upper to lower surface when the leaf is illuminated from above. Photosynthetic capacity (i.e. V_c,max and J_max) may decline with this vertical gradient, a phenomenon known as light acclimation (Terashima and Hikosaka, 1995). This acclimation maximizes nitrogen-use efficiency with respect to CO₂ uptake (Delucia et al., 1991; Hikosaka and Terashima, 1995, 1996). The effect of leaf cross-sectional gradients in electron transport rate on the light response curve has previously been modeled (Terashima and Saeki, 1985); however, its effect on the A-C_i response, with regard to fitting the Farquhar et al. (1980) model, has not been studied.

It is difficult to measure variability in these various photosynthetic parameters at high resolution across a leaf. But, it is relatively easy to determine the effects of biochemical variability via a mathematical model incorporating defined degrees of spatial variability in V_c,max, J_max, V_TPU, g_s, and light, while knowing the exact average of these parameters. This study constructs and applies such a model to determine the effect of simulated spatial variance in V_c,max, J_max, V_TPU, g_s, and light on the measured A-C_i response curve, and estimates of V_c,max and J_max derived from that curve. It addresses the question: Is the Farquhar model able to accurately predict leaf photosynthetic performance from leaf-level measurements of the A-C_i response in the presence of within-leaf biochemical heterogeneity?

This study shows that biochemical variability within a leaf has little effect on the accuracy of the Farquhar model in predicting photosynthetic capacity (i.e. the average V_c,max and J_max of a leaf) so long as V_c,max and J_max remain closely coupled and V_TPU is nonlimiting or uniform throughout the leaf. When V_c,max and J_max become uncoupled or when significant within-leaf heterogeneity in V_TPU is present, the A-C_i response assumes a shape close to that commonly observed in practice. For such A-C_i responses, J_max and, to a lesser extent, V_c,max are underestimated. When these parameters are, in turn, used in models to project whole-leaf photosynthetic CO₂ uptake, A is underestimated, the largest error occurring at the transition between V_c,max- and J_max-limited photosynthesis.

	RESULTS

TOP ABSTRACT RESULTS DISCUSSION MATERIALS AND METHODS LITERATURE CITED

 

Effect of Heterogeneity in Leaf Biochemistry on the A-C_i Response Curve

A total of 110 theoretical leaves, each with different levels of univariate variability in V_c,max and J_max (defined by the coefficient of variation [CV_VJ]) as well as (statistical correlation between V_c,max and J_max), were generated. Figure 1 shows sample distributions of V_c,max and J_max for three simulated portions of leaves of varying heterogeneity.

View larger version (77K): [in this window] [in a new window] [as a PowerPoint slide]   Figure 1. Sample distributions of Vc,max and Jmax in three simulated leaves exhibiting different degrees of within-leaf biochemical heterogeneity. Each pixel represents the Vc,max or Jmax of a square section of a theoretical leaf, as, for example, might be enclosed in a gas exchange cuvette (400 sections/leaf). CVVJ is the coefficient of variation of both Vc,max and Jmax, and is the correlation between Vc,max and Jmax (1.0 is perfect correlation); CVVJ is expressed in percent.

View larger version (13K): [in this window] [in a new window] [as a PowerPoint slide]   Figure 2. The effect of heterogeneity in Vc,max and Jmax on simulated leaf CO2 uptake (A) to intercellular CO2 concentration (Ci) curves. All curves were generated with the same average Vc,max and Jmax (90 µmol CO2 m–2 s–1 and 170 µmol electrons m–2 s–1, respectively, taken from the approximate mean values from 109 C3 species surveyed in Wullschleger (1993). VTPU was assumed to be nonlimiting. CVVJ and are as described and defined in Figure 1.

When the correlation between V_c,max and J_max was perfect ( = 1.0), there was practically no error in the estimation of V_c,max at all levels of CV (0%–50%), and only a 3% error in J_max at the highest CV used, 50% (Fig. 3, A and B ). Similarly, when the correlation between V_c,max and J_max was decreased while keeping the CV low, the error in estimated V_c,max was negligible (<1%; Fig. 3A). However, when the CV of V_c,max and J_max was increased while simultaneously lowering the correlation between the two, V_c,max was underestimated by as much as 12.5%, where CV_VJ = 50% and = 0.1 (Fig. 3A). When the CV was kept below 10%, estimation error of J_max was also near zero (Fig. 3B). In contrast to estimates of V_c,max, estimates of J_max showed an error of up to 2.9% at CV_VJ = 50%, even when = 1.0. When CV is increased and correlation decreased, J_max was progressively underestimated, the error reaching 17.7% at CV = 50%; = 0.1 (Fig. 3B).

View larger version (38K): [in this window] [in a new window] [as a PowerPoint slide]   Figure 3. A, The percent underestimation of Vc,max that would result from fitting the Farquhar model to A-Ci curves generated from leaves with varying degrees of biochemical heterogeneity. Heterogeneity was generated by increasing the CV of both Vc,max and Jmax on the one axis and decreasing the correlation () of Vc,max and Jmax on the other. CV and are as described in Figure 1. B, As in A, except the z axis is the percent underestimation of Jmax. VTPU was nonlimiting in the simulations to simplify estimation of Jmax.

Adding heterogeneity in V_TPU produced a large effect on the whole-leaf CO₂ response curve (Fig. 4 ). A leaf with homogeneous V_c,max, J_max, and V_TPU (CV_VJ = 0%; = 1.0, CV_TPU = 0%) shows the characteristic A-C_i response with two clear inflections in the curve as control of photosynthesis progresses from Rubisco, through RuBP regeneration, to TPU, as predicted by the Farquhar model and the modifications from Sharkey (1985; Fig. 4, black squares). Estimating the parameters V_c,max, J_max, and V_TPU by curve fitting the illustrated A-C_i curve (Eqs. 1–7 in Supplemental Appendix S1) gave values of 90.8 µmol m^–2 s^–1, 170.1 µmol m^–2 s^–1, and 10.0 µmol m^–2 s^–1, respectively, essentially returning the exact values that were used to generate the data (see "Materials and Methods"). Increasing the heterogeneity in V_TPU in the absence of heterogeneity in V_c,max and J_max (CV_VJ = 0%; = 1.0; CV_TPU = 50%) caused A to decline very substantially relative to the curve without heterogeneity in V_TPU even though the plateau considered characteristic of TPU limitation was lost (Fig. 4). Estimating the three biochemical parameters from this curve gave values of 86.5 µmol m^–2 s^–1, 141.1 µmol m^–2 s^–1, and 8.8 µmol m^–2 s^–1, respectively (a 3.9% underestimate of V_c,max, 17% underestimate of J_max, and 12% underestimate of V_TPU). Increasing variation in V_TPU in the presence of heterogeneity in V_c,max and J_max (CV_VJ = 50%; = 0.3; CV_TPU = 20%) produced a greater decrease in A than variation in V_c,max and J_max alone (Fig. 4). Parameter estimates for this combined heterogeneity were 70.0 µmol m^–2 s^–1, 110.9 µmol m^–2 s^–1, and 7.2 µmol m^–2 s^–1, underestimating the true means of V_c,max, J_max, and V_TPU by 22.2%, 34.8%, and 28%, respectively.

View larger version (15K): [in this window] [in a new window] [as a PowerPoint slide]   Figure 4. The effect of heterogeneity in VTPU on CO2 response curves of theoretical leaves with and without heterogeneity in Vc,max and Jmax. The y axis has been rescaled to reveal the differences at high Cis. CVVJ is the coefficient of variation of both Vc,max and Jmax, is the correlation between Vc,max and Jmax (1.0 is perfect correlation), and CVTPU is the coefficient of variation of a normal distribution of VTPU values across the leaf; CVVJ and CVTPU are expressed in percent.

There was a very minor effect of heterogeneity in g_s on the whole-leaf CO₂ response curve (Fig. 5 ). The addition of heterogeneity in g_s (CV_gs = 50%) caused a small, but visible, decrease in A at all C_i above 150 µmol mol^–1 (Fig. 5). Parameter estimates of V_c,max, J_max, and V_TPU from this curve gave values of 90.5 µmol m^–2 s^–1, 167.0 µmol m^–2 s^–1, and 9.9 µmol m^–2 s^–1, respectively, resulting in a nearly perfect estimation of V_c,max, but very slight underestimation of J_max and V_TPU by 1.8% and 1.0%, respectively. Increasing variation in g_s in the presence of heterogeneity in V_c,max and J_max (CV_VJ = 50%; = 0.3; CV_gs = 50%) produced a similar result; the A-C_i response curve showed a minimal decrease in A at most C_i values when compared to the curve generated with variation in V_c,max and J_max alone (Fig. 5).

View larger version (15K): [in this window] [in a new window] [as a PowerPoint slide]   Figure 5. The effect of heterogeneity in stomatal conductance (gs) on simulated leaf CO2 uptake (A) to intercellular CO2 concentration (Ci) curves of theoretical leaves with and without heterogeneity in Vc,max and Jmax. CVVJ and are as described in Figure 4, and CVgs is the coefficient of variation of a normal distribution of gs values across the leaf; CVVJ and CVgs are expressed in percent. The simulated leaves were assumed to be perfectly heterobaric; mean leaf Ci at each point was calculated as the average Ci across all sections of the leaf at each ambient CO2 concentration (Ca).

Heterogeneity in within-leaf light environment, in the form of decreasing light from the upper to lower surface of the leaf, was simulated by dividing the theoretical leaf into three layers of equal thickness. Each layer was ascribed a photon flux according to the exponential decline observed in actual leaves (Vogelmann and Evans, 2002). In the three-layer leaf simulations, there was a significant difference in whole-leaf photosynthesis between simulated leaves with and without light acclimation (Fig. 6, B and A , respectively). The A-C_i response of the three-layer leaf without light acclimation (i.e. uniform V_c,max, J_max, V_TPU, and R_d across all layers; see Supplemental List of Abbreviations S1 for complete list of abbreviations and definitions) showed lower A than the single-layer leaf at C_i > 250 µmol mol^–1. However, when parameters were assumed to diminish with depth into the leaf, in concert with the light gradient, the response was virtually identical to that of the single-layer leaf (Fig. 6, A and B). In the nonacclimated leaf, the contribution of each of the three layers to overall photosynthesis was relatively equal, with the top two layers producing nearly identical A-C_i curves and the bottom layer producing lower A at C_is above 200 µmol mol^–1 (Fig. 6A). In contrast, in the acclimated leaf, each layer contributed significantly different amounts to the overall leaf photosynthesis, approximately proportional to the amount of light absorbed by each layer (50%, 30%, and 10%, respectively; Fig. 6B).

View larger version (8K): [in this window] [in a new window] [as a PowerPoint slide]   Figure 6. Comparison of the simulated leaf CO2 uptake (A) to intercellular CO2 concentration (Ci) curves for leaves exhibiting a simulated decline in light through the leaf, with and without simultaneous decline in its biochemical parameters (Vc,max, Jmax, VTPU, and Rd) and with and without paradermal variation in Vc,max and Jmax. The CO2 response curve for a homogeneous leaf consisting of a single leaf layer with uniform light absorption and biochemical parameters is shown in black squares () in each plot. This is compared to a leaf consisting of three transdermal layers of equal thickness, in which 50%, 30%, and 10% of the incident photon flux (2,000 µmol m–2 s–1) is absorbed, respectively. A-Ci response curves are given for layers 1 (), 2 (), and 3 (), and for the sum of the three layers (stars). A, Leaf with no light acclimation (i.e. all three leaf layers have the same Vc,max, Jmax, VTPU, and Rd regardless of light environment). B, Leaf with light acclimation (i.e. the Vc,max, Jmax, VTPU, and Rd of each leaf layer is proportional to the fraction of light absorbed by each layer). C, Leaf with light acclimation, as in B, but also with paradermal heterogeneity in Vc,max and Jmax (CVVJ = 50%; = 0.3, as in Fig. 2).

Implications for Modeling Photosynthesis from Heterogeneous Leaves

Figure 7 compares the A-C_i response curve for a simulated leaf exhibiting heterogeneity in V_c,max and J_max (CV_VJ = 50%; = 0.3) to the predicted curve generated by the Farquhar model based on the estimates of V_c,max and J_max fitted to the heterogeneous leaf. There was a significant difference between the initial A and modeled A that resulted from using parameters derived from the Farquhar model to describe the heterogeneous leaf (Fig. 7A). The fitted curve agreed well with the original curve at low C_i, but then progressively overestimated A as C_i increased, reaching a maximum overestimation of 12.5% at C_i = 350 µmol mol^–1, in the transition region from Rubisco-limited to RuBP-limited photosynthesis (Fig. 7B). With further increase in C_i, the overestimation diminished progressively and then underestimated A at C_i > 800 µmol mol^–1 (Fig. 7A).

View larger version (16K): [in this window] [in a new window] [as a PowerPoint slide]   Figure 7. An example of the result of modeling photosynthesis using Vc,max and Jmax calculated from a heterogeneous leaf under the assumption that it is homogeneous. A, The leaf CO2 uptake (A) to intercellular CO2 concentration (Ci) curve for a leaf of known Vc,max and Jmax and of defined heterogeneity (CVVJ = 50%; = 0.3; ) and the modeled curve derived from the Vc,max and Jmax estimated from the first curve under the assumption of homogeneity (). B (inset), Overestimation of the actual A (%). VTPU is assumed to be not limiting.

	DISCUSSION

TOP ABSTRACT RESULTS DISCUSSION MATERIALS AND METHODS LITERATURE CITED

Heterogeneity in biochemical properties of the leaf led to underestimation of V_c,max and J_max, but only when the two were uncoupled (Fig. 3). This implies that, providing the two parameters vary in concert within actual leaves, estimates of V_c,max and J_max made from A-C_i curves will not be in error due to this heterogeneity. However, spatial variability in the two parameters (CV_VJ) interacts with decreased coupling to amplify the error (Fig. 3). Estimation of V_c,max was affected less by a given amount of heterogeneity than J_max. This is because V_c,max is estimated from the initial slope of the A-C_i curve. As C_i approaches 0, dA/dC_i will be unaltered. Increasing heterogeneity will simply cause the initial slope of dA/dC_i to decline from that projected by Rubisco kinetics at a progressively lower C_i. As long as V_c,max is estimated from the true initial slope, it will not be underestimated. However, heterogeneity causes A to be lower at all higher values of C_i. As a result, J_max will be underestimated (Fig. 2). Lower A at high C_i occurs because, by simulating variation in V_c,max independent of J_max, some of the patches contributing to the simulated average A will be limited by low amounts of Rubisco, even at high C_i. Consequently, one practical application of this finding is that the data points chosen from the A-C_i curve to estimate V_c,max and J_max should avoid the transition area between Rubisco- and RuBP-limited photosynthesis to minimize estimation errors due to heterogeneity. However, in leaves exhibiting TPU limitation, finding points that are exclusively RuBP limited could be difficult, if not impossible. This task is made even more difficult when even a little variation in V_TPU is introduced (Fig. 4). The A-C_i response curve appears to be very sensitive to variation in TPU, showing significant decreases in A at even low CV_TPU (Fig. 4). Of greater concern, perhaps, is the loss of a clear plateau in the CO₂ response curve in the presence of TPU heterogeneity because this makes it appear as if the leaf is not limited by TPU at all. Under these conditions, J_max would be underestimated and V_TPU might be assumed to not be limiting at any of the measurement values of C_i.

Adding stomatal heterogeneity to the simulations did not alter the A-C_i response curve significantly (Fig. 5). A CV of 50% in g_s produced an A-C_i response that was virtually identical to the homogeneous case at lower C_is and caused a marginally lower A at higher C_is. This minimal effect was the same regardless of variability in V_c,max and J_max, indicating that there was no interaction between heterogeneity in V_c,max and J_max, with g_s. However, stomatal heterogeneity did visibly lower the C_i at each C_a compared to the uniform leaf. This should be considered a worst-case scenario because the leaf was assumed to be entirely heterobaric (i.e. the substomatal chambers were assumed to be not connected). In reality, heterogeneity in g_s would be partially offset by lateral diffusion between areas of high and low C_i. The finding here should not be interpreted as a contradiction of the simulations of Cheeseman (1991) and Buckley et al. (1997), which ascribed a higher importance to heterogeneity in g_s. These studies found significant effects of stomatal heterogeneity on the A-C_i curve, but only when either very low g_s values made up a large proportion of the distribution or the distributions were heavily skewed or bimodal (Cheeseman, 1991; Buckley et al., 1997). Here, use of a maximum CV of 50% showed the A-C_i response to be far less sensitive to heterogeneity in g_s than in V_c,max and J_max (Fig. 5).

The effects of varying the light environment within the leaf were in agreement with previous analyses (Fig. 6; Terashima and Hikosaka, 1995). When the chloroplasts within the cross-section of the leaf were acclimated to their respective light environments in the three-layer simulations, the nitrogen-use efficiency of the leaf was maximized (Fig. 6B). However, when all chloroplasts were assumed to be equal through the cross-section of the leaf (i.e. nonacclimated), the A-C_i response curve was lower than its potential maximum (Fig. 6A). This was due to the fact that the lower layers were not light saturated and, as a result, the actual mean V_c,max and J_max of the leaf was slightly underestimated by curve fitting the Farquhar model (Fig. 6A). This error could be significant for leaves that are naturally near vertical in orientation and lit from both surfaces. Such leaves are unlikely to show acclimation of photosynthetic capacity from the adaxial to abaxial surface, but, if artificially lit only from above, as in a conventional gas exchange chamber (Smith et al., 1998; Johnson et al., 2005), then the error shown in Figure 6A would apply. However, this effect was small compared to the effects of lateral heterogeneity in V_c,max and J_max on the overall CO₂ response curve (Fig. 6C).

Landscape and regional models of terrestrial carbon assimilation are commonly scaled from the Farquhar model of leaf photosynthesis (for review, see Cramer et al., 1999). This, in turn, is parameterized from leaf-level measurements of the A-C_i curve from which V_c,max and J_max are derived. Under conditions where the measured leaves exhibit biochemical heterogeneity, this could result in an overestimation of modeled CO₂ uptake (Fig. 7). The modeled response curve, generated here from the estimated values of V_c,max and J_max, showed the greatest deviation from the actual curve in the transition area between the Rubisco-limited and RuBP regeneration-limited sections of the curve (Fig. 7). This could have substantial consequences because most photosynthesis in nature occurs near the inflection point of the A-C_i curve (Drake et al., 1997; Bernacchi et al., 2005).

How prevalent is heterogeneity of V_c,max and J_max, and how well coupled are they in nature? The V_c,max to J_max ratio is generally well conserved within species, even under a variety of conditions such as variable nutrient availability and water stress (Wullschleger, 1993; Wohlfahrt et al., 1999; Medlyn et al., 2002; Nogues and Alegre, 2002; Bruck and Guo, 2006). To date, there have been no studies of within-leaf variability of both V_c,max and J_max, so it is difficult to assess the degree to which heterogeneity in these parameters actually affects leaf-level photosynthetic studies that employ the Farquhar model. However, it is conceivable that certain environmental stresses could induce conditions where heterogeneity could be a significant factor. One example is the effect of tropospheric ozone stress. In species such as wheat (Triticum aestivum) and oak (Quercus robur), under both short- and long-term ozone exposure, V_c,max is decreased to a greater extent than J_max (Farage et al., 1991; Farage, 1996), possibly because decreased Rubisco activity is often the first symptom of ozone damage (Pell et al., 1997; Long and Naidu, 2002). Effects of ozone are known to be heterogeneous across the leaf surface, (Nie et al., 1993), suggesting that decrease in V_c,max relative to decrease in J_max would not be uniform across the leaf. Senescence within a leaf is often patchy and, because Rubisco protein is catabolized before the membrane proteins of the electron transport apparatus (Weng et al., 2005), uncoupled variation in V_c,max and J_max could result during the latter part of a leaf's lifespan.

Of greater uncertainty is the variability in other photosynthetic parameters, such as V_TPU or R_d (mitochondrial respiration). There is currently no literature on the within-leaf variability of TPU; V_TPU has always been considered as a whole-leaf parameter. Effects of variation in R_d were not considered in this study outside the light acclimation simulations, but recent studies have suggested that R_d could vary within leaves in a manner coupled to the photosynthetic capacity (Tcherkez et al., 2008).

In conclusion, this study found that leaf-level photosynthetic heterogeneity within the mesophyll could lead to underestimation of V_c,max, J_max, or V_TPU calculated by fitting the Farquhar model to the A-C_i response of photosynthesis. Substantial error, though, would only result if the V_c,max to J_max ratio or V_TPU itself was heterogeneous within the leaf (e.g. < 0.8 or CV_TPU > 10%). Given that it would be expected that variation in V_c,max and J_max would normally be well coupled, we conclude that error caused by heterogeneity in the estimation of these parameters and error resulting in turn from their use in crop production and Earth system models will be small. Proof of this conclusion would require quantification of within-leaf heterogeneity of V_c,max, J_max, and V_TPU in actual leaves.

	MATERIALS AND METHODS

TOP ABSTRACT RESULTS DISCUSSION MATERIALS AND METHODS LITERATURE CITED

 

Construction of the Model

The equations in the Farquhar model of photosynthesis as implemented by Long and Bernacchi (2003) were coded into a graphical modeling environment (Stella 7.0.3; iSee Systems). The equations and constants used are given in Supplemental Appendix S1. The model was designed so that each run simulates a theoretical leaf with a specified range of stochastic variability in V_c,max and J_max. Each theoretical leaf consisted of 400 squares of equal area and for each section a particular V_c,max and J_max was assigned, based on a bivariate normal frequency distribution of the two parameters. V_c,max and J_max were varied around these means based on a probability density function for bivariate normal distributions (according to Ripley, 1987):

(1)

where x and y correspond to V_c,max and J_max, respectively; _x and _y correspond to the SD for V_c,max and J_max, respectively; and corresponds to the statistical correlation between V_c,max and J_max. In all simulations, temperature was 25°C, photosynthetic photon flux density was 2,000 µmol quanta m^–2 s^–1, and oxygen concentration was 210 mmol mol^–1. To isolate the effect of biochemical heterogeneity in the mesophyll from stomatal effects, C_is were treated as uniform within each leaf, except where stated otherwise.

Systematic Variation of V_c,max and J_max

To determine the effects of different degrees of heterogeneity on estimates of V_c,max and J_max based on the Farquhar model, fixed average values of V_c,max and J_max were set for all leaves. Wullschleger (1993) surveyed the measured V_c,max and J_max of a wide range of species and found an average V_c,max of approximately 90 µmol CO₂ m^–2 s^–1 and J_max of 170 µmol electrons m^–2 s^–1 for a typical C₃ leaf. These values were used here. The coefficient of variation (i.e. the ratio of SD to mean) was varied from 0% to 50% at intervals of 10% for both V_c,max and J_max simultaneously; SDs therefore correspondingly ranged from 0 to 45 µmol CO₂ m^–2 s^–1 for V_c,max and 0 to 85 µmol electrons m^–2 s^–1 for J_max. The correlation coefficient between the two parameters (; i.e. how well-conserved the V_c,max to J_max ratio was between sections within a given leaf) was varied from 1.0 to 0.1 at intervals of 0.1 (Fig. 1). A of 1.0 represented perfect correlation (i.e. a constant V_c,max to J_max ratio across all areas of a leaf).

A-C_i responses were generated with 30 values of C_i ranging from 50 to 2,000 ppm for each section. Results for individual sections were combined and means were calculated to obtain the overall A-C_i response for the leaf. V_c,max and J_max estimates for each simulated leaf were then obtained by fitting the equations of Farquhar et al. (1980) to this whole-leaf response (Eqs. 3 and 5 in Supplemental Appendix S1). V_c,max was estimated using C_is of 60, 80, 100, and 150 ppm, and J_max was estimated using points at 700, 800, 1,000, and 1,200 ppm for all leaves. These estimated values of V_c,max and J_max were compared against the original parameters used to generate the theoretical leaf, to determine the percent error in each parameter estimate.

Heterogeneity in V_TPU and g_s

Heterogeneity in V_TPU, the limitation on carbon assimilation rate imposed by capacity for TPU, was added to the model for selected simulations by varying the CV of V_TPU (Eq. 1) across all sections while keeping the mean value constant (V_TPU = 10 µmol m^–2 s^–1). V_TPU was varied independently of V_c,max and J_max. In these selected simulations, V_TPU was estimated for each leaf from the whole-leaf CO₂ response curve as defined in the equations in Supplemental Appendix S1.

Likewise, g_s was varied for selected simulations by controlling the CV of the population of g_s associated with the sections of the leaf. Mean g_s was 0.1 mmol m^–2 s^–1. When g_s was varied, the leaf was assumed to be perfectly heterobaric (i.e. there was no diffusion of CO₂ between the intercellular spaces of different sections of the leaf). Consequently, C_i for each section at each ambient CO₂ concentration (C_a) was calculated based on the intersection of the demand function (the A-C_i response curve) and the supply function (1/g_s). Overall C_i for each leaf was calculated as the mean of the C_i of each section, in the same manner as the calculation of A in the construction of the whole-leaf CO₂ response curves.

Modeling Cross-Sectional Light Acclimation and Photosynthetic Heterogeneity

To investigate the effect of transdermal variation in light environment and photosynthetic capacity on the A-C_i response, the leaf model was replicated in triplicate to simulate three leaf layers of equal thickness, which might approximate to upper palisade, lower palisade, and spongy mesophyll. The light absorbed by each layer was 50%, 30%, and 10% of the incident photon flux, respectively, and was approximated from the data of Vogelmann and Evans (2002). For simplicity, the incident photon flux was 2,000 µmol m^–2 s^–1 for all simulations and was assumed to be all blue (approximately 470 nm).

To simulate a leaf exhibiting no acclimation to local light environment, the V_c,max, J_max, V_TPU, and R_d (mitochondrial respiration) were apportioned into each layer equally (i.e. each parameter in each layer was one-third the value found in the single-layer leaf). Light acclimation was then simulated by apportioning the above parameters into the three layers in relative proportion to the light absorbed by each layer. The A-C_i relationship was calculated for each layer separately and fit to the equations of Farquhar et al. (1980) to estimate V_c,max and J_max for each layer, then summed to give the overall photosynthetic capacity of each leaf. The A for all three layers was also summed at each C_a to generate a whole-leaf CO₂ response curve. Simulations were run with and without lateral variation in V_c,max and J_max to check for interaction between transdermal and paradermal heterogeneity.

Modeling Photosynthesis from a Heterogeneous Dataset

To determine the effect of heterogeneity on the ability of the Farquhar model to accurately predict photosynthetic performance, the modeled A for a given C_i was compared to the actual A from a leaf with a V_c,max and J_max CV of 50% and a of 0.3. The resulting A-C_i curve was then used to estimate V_c,max and J_max as if it had been generated from gas exchange methods, following the procedures of Long and Bernacchi (2003). A further A-C_i curve was then generated with these estimates from the Farquhar model, assuming no heterogeneity. The difference between A generated by this modeled curve and actual A for the simulated leaf at any given C_i represents the error that results from assuming homogeneity of the A-C_i response across a leaf that in reality is heterogeneous.

Supplemental Data

The following materials are available in the online version of this article.

Supplemental Appendix S1. The equations of the Farquhar model as implemented here.

Supplemental List of Abbreviations S1. Definitions, units, and, where appropriate, values of all abbreviated terms in the text.

	ACKNOWLEDGMENTS

 
We thank Dr. Fernando Miguez for his advice on the generation of the bivariate distributions.

Received June 4, 2008; accepted August 13, 2008; published August 20, 2008.

	FOOTNOTES

 
¹ This work was supported by a National Science Foundation Graduate Research Fellowship.

The author responsible for distribution of materials integral to the findings presented in this article in accordance with the policy described in the Instructions for Authors (www.plantphysiol.org) is: Stephen P. Long (stevel{at}life.uiuc.edu).

^[W] The online version of this article contains Web-only data.

^[OA] Open Access articles can be viewed online without a subscription.

www.plantphysiol.org/cgi/doi/10.1104/pp.108.124024

^* Corresponding author; e-mail stevel{at}life.uiuc.edu.

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