 © 2008 American Society of Plant Biologists
Abstract
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 CO_{2} uptake rate (A) versus intercellular CO_{2} 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}), wholechain electron transport (J_{max}), and trioseP 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 AC_{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} (Ω). 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 AC_{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 AC_{i} curves when the two varied in concert (Ω = 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%; Ω = 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.
The Farquhar model of photosynthesis is a mechanistic, biochemical model that is widely used to describe steadystate CO_{2} assimilation in leaves (Farquhar et al., 2001). Applications of this model range from analysis of transgenic plants to projection of the gross primary production of the terrestrial biosphere under global change (Cramer et al., 1999; Farquhar et al., 2001; Gielen et al., 2005). The model also provides a widely used practical method of quantifying the key biochemical limitations to steadystate C_{3} photosynthesis in vivo from the response of leaf photosynthetic CO_{2} uptake per unit leaf area (A) to intercellular CO_{2} concentration (C_{i}) as measured in gas exchange systems (Wullschleger, 1993; Long and Bernacchi, 2003).
One of the basic premises of the Farquhar model, as modified by Sharkey (1985), is that steadystate photosynthesis is limited by either (1) the maximum rate of carboxylation governed by Rubisco, termed V_{c,max}; (2) the rate of ribulose1,5bisP (RuBP) regeneration, which is assumed to be limited by the maximum rate of electron transport, known as J_{max}; or (3) capacity for trioseP utilization, V_{TPU}. Once these three are known, the leaf photosynthetic rate can be calculated given light flux, CO_{2} and O_{2} concentrations, and temperature. An implication of this is that the AC_{i} response of a leaf will show an abrupt decrease in dA/dC_{i}, with increasing C_{i} when A passes from Rubisco to RuBPlimited photosynthesis and, in turn, to TPU limitation. Represented graphically, this will be evident as inflections in the AC_{i} response. Photosynthetic CO_{2} uptake is routinely measured by enclosing a whole leaf or a few square centimeters of a leaf in a gas exchange cuvette. The AC_{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 CO_{2} 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 leaflevel heterogeneity in stomatal conductance (g_{s}) on the AC_{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 AC_{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 leaflevel 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 nitrogenuse efficiency with respect to CO_{2} uptake (Delucia et al., 1991; Hikosaka and Terashima, 1995, 1996). The effect of leaf crosssectional gradients in electron transport rate on the light response curve has previously been modeled (Terashima and Saeki, 1985); however, its effect on the AC_{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 AC_{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 leaflevel measurements of the AC_{i} response in the presence of withinleaf 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 withinleaf heterogeneity in V_{TPU} is present, the AC_{i} response assumes a shape close to that commonly observed in practice. For such AC_{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 wholeleaf photosynthetic CO_{2} uptake, A is underestimated, the largest error occurring at the transition between V_{c,max} and J_{max}limited photosynthesis.
RESULTS
Effect of Heterogeneity in Leaf Biochemistry on the AC_{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 Ω (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.
A leaf with homogeneous V_{c,max} and J_{max} (CV_{VJ} = 0% and, by definition, perfect correlation; Ω = 1.0) showed an AC_{i} response, with a clear inflection marking the transition between Rubiscolimited and RuBP regenerationlimited photosynthesis (Fig. 2 ). Raising CV_{VJ} to 50% while keeping perfect correlation (Ω = 1.0) resulted in an AC_{i} response nearly identical to the AC_{i} curve of the homogeneous leaf at low C_{i}, but with a lower A at high C_{i}, even though the underlying average V_{c,max} and J_{max} were unchanged from the first curve (Fig. 2). Decreasing the correlation between V_{c,max} and J_{max} while keeping the CV low (CV_{VJ} = 10%; Ω = 0.3) likewise produced little change in the AC_{i} curve from the homogeneous leaf. However, decreasing the correlation while also increasing the CV (CV_{VJ} = 50%; Ω = 0.3) produced an AC_{i} curve that deviated markedly from the curve for the homogeneous leaf and one that underestimated A at all C_{i} values, even though the average V_{c,max} and J_{max} were unchanged.
Determining the Error in V_{c,max} and J_{max} Estimates for Biochemically Heterogeneous Leaves
When the correlation between V_{c,max} and J_{max} was perfect (Ω = 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 Ω = 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 Ω = 1.0. When CV is increased and correlation decreased, J_{max} was progressively underestimated, the error reaching 17.7% at CV = 50%; Ω = 0.1 (Fig. 3B).
Effect of Variation in TPU Limitation
Adding heterogeneity in V_{TPU} produced a large effect on the wholeleaf CO_{2} response curve (Fig. 4 ). A leaf with homogeneous V_{c,max}, J_{max}, and V_{TPU} (CV_{VJ} = 0%; Ω = 1.0, CV_{TPU} = 0%) shows the characteristic AC_{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 AC_{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%; Ω = 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%; Ω = 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.
Effect of Variation in g_{s}
There was a very minor effect of heterogeneity in g_{s} on the wholeleaf CO_{2} 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%; Ω = 0.3; CV_{gs} = 50%) produced a similar result; the AC_{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).
Heterogeneity in Light Environment and Light Acclimation
Heterogeneity in withinleaf 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 threelayer leaf simulations, there was a significant difference in wholeleaf photosynthesis between simulated leaves with and without light acclimation (Fig. 6, B and A , respectively). The AC_{i} response of the threelayer 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 singlelayer 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 singlelayer 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 AC_{i} curves and the bottom layer producing lower A at C_{i}s 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).
Adding variability in V_{c,max} and J_{max} in the three paradermal layers produced a significantly lower A at all C_{i} values (Fig. 6C). The effects of heterogeneity in V_{c,max} and J_{max} on the CO_{2} response curve that were observed in Figure 2 were seen within each layer (i.e. a smoothing of inflection points).
Implications for Modeling Photosynthesis from Heterogeneous Leaves
Figure 7 compares the AC_{i} response curve for a simulated leaf exhibiting heterogeneity in V_{c,max} and J_{max} (CV_{VJ} = 50%; Ω = 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 Rubiscolimited to RuBPlimited 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).
DISCUSSION
Spatial heterogeneity in V_{c,max} and J_{max} within leaves was found to have an effect on the ability of the Farquhar model to accurately characterize and predict photosynthesis at the leaf level. The most apparent effect is that, while an AC_{i} curve derived from a leaf with homogeneous biochemical properties (i.e. constant V_{c,max} and J_{max}) across the leaf shows a distinct inflection point, curves derived from heterogeneous leaves show a smoother transition from Rubiscolimited to RuBPlimited photosynthesis (Fig. 2). This smooth transition resembles many reported AC_{i} curves measured both in the field and in controlled conditions, suggesting that photosynthesis in real leaves is more often heterogeneous than not (e.g. Wullschleger, 1993). Why does heterogeneity lead to a smoother AC_{i} response curve? Theoretically, when the V_{c,max} to J_{max} ratio varies between patches in a leaf, the C_{i} at which the transition occurs must also vary (Farquhar et al., 1980). Thus, the observed overall AC_{i} response of the leaf is the average of a range of photosynthetic CO_{2} response curves, each with inflection points at different C_{i} values. This could explain an apparent inconsistency between metabolic control predicted by the Farquhar model and that determined by control analysis of transgenic plants. Implicit in the Farquhar model is that, at low C_{i}, metabolic control will reside entirely with Rubisco (i.e. a control coefficient of 1), whereas above the inflection point of the curve, control will reside entirely with regeneration of RuBP. However, transgenic alteration of the amount of individual photosynthetic proteins suggests that control is shared between Rubisco and proteins involved in RuBP regeneration (for review, see Raines, 2003). Incomplete coupling of spatial variation in the amount of Rubisco and proteins controlling RuBP regeneration could explain the apparent contradiction between the theory of Farquhar et al. (1980) and observed metabolic control.
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 AC_{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 AC_{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 AC_{i} curve to estimate V_{c,max} and J_{max} should avoid the transition area between Rubisco and RuBPlimited 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 AC_{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 CO_{2} 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 AC_{i} response curve significantly (Fig. 5). A CV of 50% in g_{s} produced an AC_{i} response that was virtually identical to the homogeneous case at lower C_{i}s and caused a marginally lower A at higher C_{i}s. 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 worstcase 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 AC_{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 AC_{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 crosssection of the leaf were acclimated to their respective light environments in the threelayer simulations, the nitrogenuse efficiency of the leaf was maximized (Fig. 6B). However, when all chloroplasts were assumed to be equal through the crosssection of the leaf (i.e. nonacclimated), the AC_{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 CO_{2} 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 leaflevel measurements of the AC_{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 CO_{2} 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 Rubiscolimited and RuBP regenerationlimited sections of the curve (Fig. 7). This could have substantial consequences because most photosynthesis in nature occurs near the inflection point of the AC_{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 withinleaf 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 leaflevel 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 longterm 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 withinleaf variability of TPU; V_{TPU} has always been considered as a wholeleaf 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 leaflevel 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 AC_{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. Ω < 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 withinleaf heterogeneity of V_{c,max}, J_{max}, and V_{TPU} in actual leaves.
MATERIALS AND METHODS
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; σ_{x} and σ_{y} correspond to the sd for V_{c,max} and J_{max}, respectively; and Ω 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_{i}s 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 CO_{2} m^{−2} s^{−1} and J_{max} of 170 μmol electrons m^{−2} s^{−1} for a typical C_{3} 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 CO_{2} 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 (Ω; i.e. how wellconserved 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 Ω of 1.0 represented perfect correlation (i.e. a constant V_{c,max} to J_{max} ratio across all areas of a leaf).
AC_{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 AC_{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 wholeleaf response (Eqs. 3 and 5 in Supplemental Appendix S1). V_{c,max} was estimated using C_{i}s 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 wholeleaf CO_{2} 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 CO_{2} between the intercellular spaces of different sections of the leaf). Consequently, C_{i} for each section at each ambient CO_{2} concentration (C_{a}) was calculated based on the intersection of the demand function (the AC_{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 wholeleaf CO_{2} response curves.
Modeling CrossSectional Light Acclimation and Photosynthetic Heterogeneity
To investigate the effect of transdermal variation in light environment and photosynthetic capacity on the AC_{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 onethird the value found in the singlelayer 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 AC_{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 wholeleaf CO_{2} 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 Ω of 0.3. The resulting AC_{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 AC_{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 AC_{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.
Footnotes

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).

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

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

↵[W] The online version of this article contains Webonly data.
 Received June 4, 2008.
 Accepted August 13, 2008.
 Published August 20, 2008.