effect of leaf-level spatial variability

The Effect of Leaf-Level Spatial Variability inPhotosynthetic Capacity on Biochemical ParameterEstimates Using the Farquhar Model:A Theoretical Analysis1[W][OA]

Charles P. Chen, Xin-Guang Zhu, and Stephen P. Long*

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

Application of the widely used Farquhar model of photosynthesis in interpretation of gas exchange data assumes thatphotosynthetic properties are homogeneous throughout the leaf. Previous studies showed that heterogeneity in stomatalconductance (gs) across a leaf could affect the shape of the measured leaf photosynthetic CO2 uptake rate (A) versusintercellular CO2 concentration (Ci) response curve and, in turn, estimation of the critical biochemical parameters of this model.These are the maximum rates of carboxylation (Vc,max), whole-chain electron transport ( Jmax), and triose-P utilization (VTPU). Theeffects of spatial variation in Vc,max, Jmax, and VTPU on estimation of leaf averages of these parameters from A-Ci curves measuredon a whole leaf have not been investigated. A mathematical model incorporating defined degrees of spatial variability in Vc,maxand Jmax was constructed. One hundred and ten theoretical leaves were simulated, each with the same average Vc,max and Jmax,but different coefficients of variation of the mean (CVVJ) and varying correlation between Vc,max and Jmax (V). Additionally, theinteraction of variation in Vc,max and Jmax with heterogeneity in VTPU, gs, and light gradients within the leaf was also in-vestigated. Transition from Vc,max- to Jmax-limited photosynthesis in the A-Ci curve was smooth in the most heterogeneousleaves, in contrast to a distinct inflection in the absence of heterogeneity. Spatial variability had little effect on the accuracy ofestimation of Vc,max and Jmax from A-Ci curves when the two varied in concert (V 5 1.0), but resulted in underestimation of bothparameters when they varied independently (up to 12.5% in Vc,max and 17.7% in Jmax at CVVJ 5 50%; V 5 0.3). Heterogeneityin VTPU also significantly affected parameter estimates, but effects of heterogeneity in gs or light gradients were comparativelysmall. If Vc,max and Jmax derived from such heterogeneous leaves are used in models to project leaf photosynthesis, actual A isoverestimated by up to 12% at the transition between Vc,max- and Jmax-limited photosynthesis. This could have implications forboth crop production and Earth system models, including projections of the effects of atmospheric change.

The Farquhar model of photosynthesis is a mechanis-tic, biochemical model that is widely used to describesteady-state CO2 assimilation in leaves (Farquharet al., 2001). Applications of this model range fromanalysis of transgenic plants to projection of the grossprimary production of the terrestrial biosphere underglobal change (Cramer et al., 1999; Farquhar et al.,2001; Gielen et al., 2005). The model also provides awidely used practical method of quantifying the keybiochemical limitations to steady-state C3 photosyn-thesis in vivo from the response of leaf photosyntheticCO2 uptake per unit leaf area (A) to intercellular CO2concentration (Ci) as measured in gas exchange sys-tems (Wullschleger, 1993; Long and Bernacchi, 2003).

One of the basic premises of the Farquhar model, asmodified by Sharkey (1985), is that steady-state pho-tosynthesis is limited by either (1) the maximum rateof carboxylation governed by Rubisco, termed Vc,max;(2) the rate of ribulose-1,5-bisP (RuBP) regeneration,which is assumed to be limited by the maximum rateof electron transport, known as Jmax; or (3) capacity fortriose-P utilization, VTPU. Once these three are known,the leaf photosynthetic rate can be calculated givenlight flux, CO2 and O2 concentrations, and tempera-ture. An implication of this is that the A-Ci response ofa leaf will show an abrupt decrease in dA/dCi, withincreasing Ci when A passes from Rubisco- to RuBP-limited photosynthesis and, in turn, to TPU limitation.Represented graphically, this will be evident as in-flections in the A-Ci response. Photosynthetic CO2uptake is routinely measured by enclosing a wholeleaf or a few square centimeters of a leaf in a gas ex-change cuvette. The A-Ci response measured this wayrarely shows the abrupt transitions predicted by theFarquhar model (e.g. Riddoch et al., 1991; Wullschleger,1993). Further, metabolic control analysis using leaveswith transgenically decreased quantities of specificphotosynthetic proteins suggest that, for a given CO2concentration, control is shared between Rubisco andproteins limiting the rate of RuBP regeneration (Quick

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

* Corresponding author; e-mail [email protected] author responsible for distribution of materials integral to the

findings presented in this article in accordance with the policydescribed in the Instructions for Authors (www.plantphysiol.org) is:Stephen P. Long ([email protected]).

[W] The online version of this article contains Web-only data.[OA] Open Access articles can be viewed online without a sub-

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

Plant Physiology, October 2008, Vol. 148, pp. 1139–1147, www.plantphysiol.org � 2008 American Society of Plant Biologists 1139

et al., 1991; Price et al., 1998; Raines, 2003). Under-standing the basis for these inconsistencies betweenassumptions of the model and observations may becritical to the application of the model both as an invivo method of determining biochemical limitationsand in projecting photosynthesis at scales from cropcanopies to the globe.

Application of the Farquhar model assumes that thephotosynthetic properties of the leaf are spatiallyhomogeneous (von Caemmerer, 2000). Based on thisassumption, Vc,max, Jmax, and VTPU are derived by fit-ting the Farquhar model to measured leaf responses ofA to Ci (Long and Bernacchi, 2003; Sharkey et al., 2007).Given complex gradients of leaf development, impactof heterogeneous environments, disease, and pestattack, it seems unlikely that the assumption of ho-mogeneity is often met in nature (Terashima, 1992; vonCaemmerer, 2000; Aldea et al., 2006), but does thismatter for fitting Vc,max, Jmax, and VTPU and for accuratemodeling of photosynthetic carbon assimilation?

The effect of leaf-level heterogeneity in stomatalconductance (gs) on the A-Ci response curve has pre-viously been studied (Cheeseman, 1991; Buckley et al.,1997). Increased stomatal heterogeneity was found todecrease the initial slope of the A-Ci curve with theresult that Vc,max determined from such curves under-estimated the true value. However, simulationsshowed that stomatal heterogeneity could not fullyexplain the observed leaf-level variability in photosyn-thetic activity (Cheeseman, 1991; Buckley et al., 1997).Variation in biochemical parameters (i.e. Vc,max, Jmax,and VTPU) has been implicated as a possible explana-tion; however, the effects of spatial heterogeneity inthese parameters have not yet been investigated.

In addition, there is an exponential decline in lightfrom the upper to lower surface when the leaf is il-luminated from above. Photosynthetic capacity (i.e.Vc,max and Jmax) may decline with this vertical gradient,a phenomenon known as light acclimation (Terashimaand Hikosaka, 1995). This acclimation maximizesnitrogen-use efficiency with respect to CO2 uptake(Delucia et al., 1991; Hikosaka and Terashima, 1995,1996). The effect of leaf cross-sectional gradients inelectron transport rate on the light response curve haspreviously been modeled (Terashima and Saeki, 1985);however, its effect on the A-Ci response, with regard tofitting the Farquhar et al. (1980) model, has not beenstudied.

It is difficult to measure variability in these variousphotosynthetic parameters at high resolution across aleaf. But, it is relatively easy to determine the effects ofbiochemical variability via a mathematical model in-corporating defined degrees of spatial variability inVc,max, Jmax, VTPU, gs, and light, while knowing the exactaverage of these parameters. This study constructs andapplies such a model to determine the effect of sim-ulated spatial variance in Vc,max, Jmax, VTPU, gs, and lighton the measured A-Ci response curve, and estimates ofVc,max and Jmax derived from that curve. It addressesthe question: Is the Farquhar model able to accurately

predict leaf photosynthetic performance from leaf-level measurements of the A-Ci response in the pres-ence of within-leaf biochemical heterogeneity?

This study shows that biochemical variability withina leaf has little effect on the accuracy of the Farquharmodel in predicting photosynthetic capacity (i.e. theaverage Vc,max and Jmax of a leaf) so long as Vc,max andJmax remain closely coupled and VTPU is nonlimiting oruniform throughout the leaf. When Vc,max and Jmaxbecome uncoupled or when significant within-leafheterogeneity in VTPU is present, the A-Ci responseassumes a shape close to that commonly observed inpractice. For such A-Ci responses, Jmax and, to a lesserextent, Vc,max are underestimated. When these param-eters are, in turn, used in models to project whole-leafphotosynthetic CO2 uptake, A is underestimated, thelargest error occurring at the transition between Vc,max-and Jmax-limited photosynthesis.

RESULTS

Effect of Heterogeneity in Leaf Biochemistryon the A-Ci Response Curve

A total of 110 theoretical leaves, each with differentlevels of univariate variability in Vc,max and Jmax (de-fined by the coefficient of variation [CVVJ]) as well as V(statistical correlation between Vc,max and Jmax), weregenerated. Figure 1 shows sample distributions ofVc,max and Jmax for three simulated portions of leaves ofvarying heterogeneity.

A leaf with homogeneous Vc,max and Jmax (CVVJ 5 0%and, by definition, perfect correlation; V 5 1.0) showed anA-Ci response, with a clear inflection marking thetransition between Rubisco-limited and RuBP regen-eration-limited photosynthesis (Fig. 2). Raising CVVJto 50% while keeping perfect correlation (V 5 1.0) re-sulted in an A-Ci response nearly identical to the A-Cicurve of the homogeneous leaf at low Ci, but with alower A at high Ci, even though the underlying aver-age Vc,max and Jmax were unchanged from the firstcurve (Fig. 2). Decreasing the correlation betweenVc,max and Jmax while keeping the CV low (CVVJ 510%; V 5 0.3) likewise produced little change in theA-Ci curve from the homogeneous leaf. However,decreasing the correlation while also increasing theCV (CVVJ 5 50%; V 5 0.3) produced an A-Ci curve thatdeviated markedly from the curve for the homogeneousleaf and one that underestimated A at all Ci values, eventhough the average Vc,max and Jmax were unchanged.

Determining the Error in Vc,max and Jmax Estimates for

Biochemically Heterogeneous Leaves

When the correlation between Vc,max and Jmax was per-fect (V 5 1.0), there was practically no error in theestimation of Vc,max at all levels of CV (0%–50%), andonly a 3% error in Jmax at the highest CV used, 50% (Fig.

Chen et al.

1140 Plant Physiol. Vol. 148, 2008

3, A and B). Similarly, when the correlation betweenVc,max and Jmax was decreased while keeping the CVlow, the error in estimated Vc,max was negligible (,1%;Fig. 3A). However, when the CV of Vc,max and Jmax wasincreased while simultaneously lowering the correla-tion between the two, Vc,max was underestimated by asmuch as 12.5%, where CVVJ 5 50% and V 5 0.1 (Fig.3A). When the CV was kept below 10%, estimationerror of Jmax was also near zero (Fig. 3B). In contrast toestimates of Vc,max, estimates of Jmax showed an error ofup to 2.9% at CVVJ 5 50%, even when V 5 1.0. WhenCV is increased and correlation decreased, Jmax wasprogressively underestimated, the error reaching 17.7%at CV 5 50%; V 5 0.1 (Fig. 3B).

Effect of Variation in TPU Limitation

Adding heterogeneity in VTPU produced a largeeffect on the whole-leaf CO2 response curve (Fig. 4).A leaf with homogeneous Vc,max, Jmax, and VTPU (CVVJ 50%; V 5 1.0, CVTPU 5 0%) shows the characteristicA-Ci response with two clear inflections in the curve ascontrol of photosynthesis progresses from Rubisco,through RuBP regeneration, to TPU, as predicted bythe Farquhar model and the modifications fromSharkey (1985; Fig. 4, black squares). Estimating theparameters Vc,max, Jmax, and VTPU by curve fitting theillustrated A-Ci curve (Eqs. 1–7 in Supplemental Ap-pendix S1) gave values of 90.8 mmol m22 s21, 170.1mmol m22 s21, and 10.0 mmol m22 s21, respectively,essentially returning the exact values that were used togenerate the data (see ‘‘Materials and Methods’’).Increasing the heterogeneity in VTPU in the absenceof heterogeneity in Vc,max and Jmax (CVVJ 5 0%; V 5 1.0;CVTPU 5 50%) caused A to decline very substantiallyrelative to the curve without heterogeneity in VTPUeven though the plateau considered characteristic ofTPU limitation was lost (Fig. 4). Estimating the three

biochemical parameters from this curve gave values of86.5 mmol m22 s21, 141.1 mmol m22 s21, and 8.8 mmolm22 s21, respectively (a 3.9% underestimate of Vc,max,17% underestimate of Jmax, and 12% underestimate ofVTPU). Increasing variation in VTPU in the presence ofheterogeneity in Vc,max and Jmax (CVVJ 5 50%; V 5 0.3;CVTPU 5 20%) produced a greater decrease in Athan variation in Vc,max and Jmax alone (Fig. 4). Parameterestimates for this combined heterogeneity were 70.0mmol m22 s21, 110.9 mmol m22 s21, and 7.2 mmol m22

s21, underestimating the true means of Vc,max, Jmax, andVTPU by 22.2%, 34.8%, and 28%, respectively.

Figure 1. Sample distributions of Vc,max

and Jmax in three simulated leavesexhibiting different degrees of within-leaf biochemical heterogeneity. Eachpixel represents the Vc,max or Jmax of asquare section of a theoretical leaf, as,for example, might be enclosed in agas exchange cuvette (400 sections/leaf). CVVJ is the coefficient of variationof both Vc,max and Jmax, and V is thecorrelation between Vc,max and Jmax

(1.0 is perfect correlation); CVVJ is ex-pressed in percent.

Figure 2. The effect of heterogeneity in Vc,max and Jmax on simulated leafCO2 uptake (A) to intercellular CO2 concentration (Ci) curves. Allcurves were generated with the same average Vc,max and Jmax (90 mmolCO2 m22 s21 and 170 mmol electrons m22 s21, respectively, taken fromthe approximate mean values from 109 C3 species surveyed inWullschleger (1993). VTPU was assumed to be nonlimiting. CVVJ andV are as described and defined in Figure 1.

The Farquhar Model and Leaf Biochemical Heterogeneity

Plant Physiol. Vol. 148, 2008 1141

Effect of Variation in gs

There was a very minor effect of heterogeneity in gson the whole-leaf CO2 response curve (Fig. 5). The addi-tion of heterogeneity in gs (CVgs 5 50%) caused asmall, but visible, decrease in A at all Ci above 150 mmolmol21 (Fig. 5). Parameter estimates of Vc,max, Jmax, andVTPU from this curve gave values of 90.5 mmol m22 s21,167.0 mmol m22 s21, and 9.9 mmol m22 s21, respectively,resulting in a nearly perfect estimation of Vc,max, butvery slight underestimation of Jmax and VTPU by 1.8%and 1.0%, respectively. Increasing variation in gs in thepresence of heterogeneity in Vc,max and Jmax (CVVJ 550%; V 5 0.3; CVgs 5 50%) produced a similar result;the A-Ci response curve showed a minimal decrease in

A at most Ci values when compared to the curvegenerated with variation in Vc,max and Jmax alone (Fig. 5).

Heterogeneity in Light Environment andLight Acclimation

Heterogeneity in within-leaf light environment, inthe form of decreasing light from the upper to lowersurface of the leaf, was simulated by dividing thetheoretical leaf into three layers of equal thickness.Each layer was ascribed a photon flux according to theexponential decline observed in actual leaves (Vogelmannand Evans, 2002). In the three-layer leaf simula-tions, there was a significant difference in whole-leafphotosynthesis between simulated leaves with andwithout light acclimation (Fig. 6, B and A, respec-tively). The A-Ci response of the three-layer leaf with-out light acclimation (i.e. uniform Vc,max, Jmax, VTPU,and Rd across all layers; see Supplemental List ofAbbreviations S1 for complete list of abbreviations anddefinitions) showed lower A than the single-layer leafat Ci . 250 mmol mol21. However, when parameterswere assumed to diminish with depth into the leaf, inconcert with the light gradient, the response wasvirtually identical to that of the single-layer leaf (Fig.6, A and B). In the nonacclimated leaf, the contributionof each of the three layers to overall photosynthesiswas relatively equal, with the top two layers produc-ing nearly identical A-Ci curves and the bottom layerproducing lower A at Cis above 200 mmol mol21 (Fig.6A). In contrast, in the acclimated leaf, each layercontributed significantly different amounts to theoverall leaf photosynthesis, approximately propor-

Figure 3. A, The percent underestimation of Vc,max that would resultfrom fitting the Farquhar model to A-Ci curves generated from leaveswith varying degrees of biochemical heterogeneity. Heterogeneity wasgenerated by increasing the CV of both Vc,max and Jmax on the one axisand decreasing the correlation (V) of Vc,max and Jmax on the other. CVand V are as described in Figure 1. B, As in A, except the z axis is thepercent underestimation of Jmax. VTPU was nonlimiting in the simula-tions to simplify estimation of Jmax.

Figure 4. The effect of heterogeneity in VTPU on CO2 response curves oftheoretical leaves with and without heterogeneity in Vc,max and Jmax. They axis has been rescaled to reveal the differences at high Cis. CVVJ is thecoefficient of variation of both Vc,max and Jmax, V is the correlationbetween Vc,max and Jmax (1.0 is perfect correlation), and CVTPU is thecoefficient of variation of a normal distribution of VTPU values across theleaf; CVVJ and CVTPU are expressed in percent.

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tional to the amount of light absorbed by each layer(50%, 30%, and 10%, respectively; Fig. 6B).

Adding variability in Vc,max and Jmax in the threeparadermal layers produced a significantly lower A atall Ci values (Fig. 6C). The effects of heterogeneity inVc,max and Jmax on the CO2 response curve that wereobserved in Figure 2 were seen within each layer (i.e. asmoothing of inflection points).

Implications for Modeling Photosynthesis from

Heterogeneous Leaves

Figure 7 compares the A-Ci response curve for asimulated leaf exhibiting heterogeneity in Vc,max andJmax (CVVJ 5 50%; V 5 0.3) to the predicted curvegenerated by the Farquhar model based on the esti-mates of Vc,max and Jmax fitted to the heterogeneousleaf. There was a significant difference between theinitial A and modeled A that resulted from usingparameters derived from the Farquhar model to de-scribe the heterogeneous leaf (Fig. 7A). The fittedcurve agreed well with the original curve at low Ci,but then progressively overestimated A as Ci increased,reaching a maximum overestimation of 12.5% at Ci 5350 mmol mol21, in the transition region from Rubisco-limited to RuBP-limited photosynthesis (Fig. 7B).With further increase in Ci, the overestimation dimin-ished progressively and then underestimated A at Ci .800 mmol mol21 (Fig. 7A).

DISCUSSION

Spatial heterogeneity in Vc,max and Jmax within leaveswas found to have an effect on the ability of theFarquhar model to accurately characterize and predictphotosynthesis at the leaf level. The most apparenteffect is that, while an A-Ci curve derived from a leafwith homogeneous biochemical properties (i.e. con-stant Vc,max and Jmax) across the leaf shows a distinctinflection point, curves derived from heterogeneousleaves show a smoother transition from Rubisco-limited to RuBP-limited photosynthesis (Fig. 2). Thissmooth transition resembles many reported A-Cicurves measured both in the field and in controlledconditions, suggesting that photosynthesis in realleaves is more often heterogeneous than not (e.g.

Figure 5. The effect of heterogeneity in stomatal conductance (gs) onsimulated leaf CO2 uptake (A) to intercellular CO2 concentration (Ci)curves of theoretical leaves with and without heterogeneity in Vc,max

and Jmax. CVVJ and V are as described in Figure 4, and CVgs is thecoefficient of variation of a normal distribution of gs values across theleaf; CVVJ and CVgs are expressed in percent. The simulated leaves wereassumed to be perfectly heterobaric; mean leaf Ci at each point wascalculated as the average Ci across all sections of the leaf at eachambient CO2 concentration (Ca).

Figure 6. Comparison of the simulated leaf CO2 uptake (A) to intercellular CO2 concentration (Ci) curves for leaves exhibiting asimulated 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 leafconsisting of a single leaf layer with uniform light absorption and biochemical parameters is shown in black squares (n) in eachplot. This is compared to a leaf consisting of three transdermal layers of equal thickness, in which 50%, 30%, and 10% of theincident photon flux (2,000 mmol m22 s21) is absorbed, respectively. A-Ci response curves are given for layers 1 (s), 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 isproportional to the fraction of light absorbed by each layer). C, Leaf with light acclimation, as in B, but also with paradermalheterogeneity in Vc,max and Jmax (CVVJ 5 50%; V 5 0.3, as in Fig. 2).



Wullschleger, 1993). Why does heterogeneity lead to asmoother A-Ci response curve? Theoretically, when theVc,max to Jmax ratio varies between patches in a leaf, theCi at which the transition occurs must also vary(Farquhar et al., 1980). Thus, the observed overallA-Ci response of the leaf is the average of a range ofphotosynthetic CO2 response curves, each with inflec-tion points at different Ci values. This could explain anapparent inconsistency between metabolic controlpredicted by the Farquhar model and that determinedby control analysis of transgenic plants. Implicit in theFarquhar model is that, at low Ci, metabolic controlwill reside entirely with Rubisco (i.e. a control coeffi-cient of 1), whereas above the inflection point of thecurve, control will reside entirely with regeneration ofRuBP. However, transgenic alteration of the amount ofindividual photosynthetic proteins suggests that con-trol is shared between Rubisco and proteins involvedin RuBP regeneration (for review, see Raines, 2003).Incomplete coupling of spatial variation in the amountof Rubisco and proteins controlling RuBP regenerationcould explain the apparent contradiction between thetheory of Farquhar et al. (1980) and observed meta-bolic control.

Heterogeneity in biochemical properties of the leafled to underestimation of Vc,max and Jmax, but onlywhen the two were uncoupled (Fig. 3). This impliesthat, providing the two parameters vary in concertwithin actual leaves, estimates of Vc,max and Jmax madefrom A-Ci curves will not be in error due to thisheterogeneity. However, spatial variability in the twoparameters (CVVJ) interacts with decreased coupling to

amplify the error (Fig. 3). Estimation of Vc,max wasaffected less by a given amount of heterogeneity thanJmax. This is because Vc,max is estimated from the initialslope of the A-Ci curve. As Ci approaches 0, dA/dCiwill be unaltered. Increasing heterogeneity will simplycause the initial slope of dA/dCi to decline from thatprojected by Rubisco kinetics at a progressively lowerCi. As long as Vc,max is estimated from the true initialslope, it will not be underestimated. However, hetero-geneity causes A to be lower at all higher values of Ci.As a result, Jmax will be underestimated (Fig. 2). LowerA at high Ci occurs because, by simulating variation inVc,max independent of Jmax, some of the patches con-tributing to the simulated average A will be limited bylow amounts of Rubisco, even at high Ci. Consequently,one practical application of this finding is that the datapoints chosen from the A-Ci curve to estimate Vc,maxand Jmax should avoid the transition area betweenRubisco- and RuBP-limited photosynthesis to mini-mize estimation errors due to heterogeneity. However,in leaves exhibiting TPU limitation, finding points thatare exclusively RuBP limited could be difficult, if notimpossible. This task is made even more difficult wheneven a little variation in VTPU is introduced (Fig. 4). TheA-Ci response curve appears to be very sensitive tovariation in TPU, showing significant decreases in A ateven low CVTPU (Fig. 4). Of greater concern, perhaps, isthe loss of a clear plateau in the CO2 response curve inthe presence of TPU heterogeneity because this makesit appear as if the leaf is not limited by TPU at all.Under these conditions, Jmax would be underestimatedand VTPU might be assumed to not be limiting at any ofthe measurement values of Ci.

Adding stomatal heterogeneity to the simulationsdid not alter the A-Ci response curve significantly (Fig.5). A CV of 50% in gs produced an A-Ci response thatwas virtually identical to the homogeneous case atlower Cis and caused a marginally lower A at higherCis. This minimal effect was the same regardless ofvariability in Vc,max and Jmax, indicating that there wasno interaction between heterogeneity in Vc,max andJmax, with gs. However, stomatal heterogeneity didvisibly lower the Ci at each Ca compared to the uniformleaf. This should be considered a worst-case scenariobecause the leaf was assumed to be entirely hetero-baric (i.e. the substomatal chambers were assumed tobe not connected). In reality, heterogeneity in gs wouldbe partially offset by lateral diffusion between areas ofhigh and low Ci. The finding here should not beinterpreted as a contradiction of the simulations ofCheeseman (1991) and Buckley et al. (1997), which as-cribed a higher importance to heterogeneity in gs.These studies found significant effects of stomatalheterogeneity on the A-Ci curve, but only when eithervery low gs values made up a large proportion of thedistribution or the distributions were heavily skewedor bimodal (Cheeseman, 1991; Buckley et al., 1997).Here, use of a maximum CV of 50% showed the A-Ciresponse to be far less sensitive to heterogeneity in gsthan in Vc,max and Jmax (Fig. 5).

Figure 7. An example of the result of modeling photosynthesis usingVc,max and Jmax calculated from a heterogeneous leaf under theassumption that it is homogeneous. A, The leaf CO2 uptake (A) tointercellular CO2 concentration (Ci) curve for a leaf of known Vc,max andJmax and of defined heterogeneity (CVVJ 5 50%; V 5 0.3; n) and themodeled curve derived from the Vc,max and Jmax estimated from the firstcurve under the assumption of homogeneity (s). B (inset), Overesti-mation of the actual A (%). VTPU is assumed to be not limiting.

Chen et al.


The effects of varying the light environment withinthe leaf were in agreement with previous analyses(Fig. 6; Terashima and Hikosaka, 1995). When thechloroplasts within the cross-section of the leaf wereacclimated to their respective light environments inthe three-layer simulations, the nitrogen-use efficiencyof the leaf was maximized (Fig. 6B). However, when allchloroplasts were assumed to be equal through thecross-section of the leaf (i.e. nonacclimated), the A-Ciresponse curve was lower than its potential maximum(Fig. 6A). This was due to the fact that the lower layerswere not light saturated and, as a result, the actualmean Vc,max and Jmax of the leaf was slightly under-estimated by curve fitting the Farquhar model (Fig.6A). This error could be significant for leaves that arenaturally near vertical in orientation and lit from bothsurfaces. Such leaves are unlikely to show acclimationof photosynthetic capacity from the adaxial to abaxialsurface, but, if artificially lit only from above, as in aconventional gas exchange chamber (Smith et al., 1998;Johnson et al., 2005), then the error shown in Figure 6Awould apply. However, this effect was small comparedto the effects of lateral heterogeneity in Vc,max and Jmaxon the overall CO2 response curve (Fig. 6C).

Landscape and regional models of terrestrial carbonassimilation are commonly scaled from the Farquharmodel of leaf photosynthesis (for review, see Crameret al., 1999). This, in turn, is parameterized from leaf-level measurements of the A-Ci curve from whichVc,max and Jmax are derived. Under conditions wherethe measured leaves exhibit biochemical heterogene-ity, this could result in an overestimation of modeledCO2 uptake (Fig. 7). The modeled response curve,generated here from the estimated values of Vc,max andJmax, showed the greatest deviation from the actualcurve in the transition area between the Rubisco-limited and RuBP regeneration-limited sections ofthe curve (Fig. 7). This could have substantial conse-quences because most photosynthesis in nature occursnear the inflection point of the A-Ci curve (Drake et al.,1997; Bernacchi et al., 2005).

How prevalent is heterogeneity of Vc,max and Jmax,and how well coupled are they in nature? The Vc,max toJmax ratio is generally well conserved within species,even under a variety of conditions such as variablenutrient availability and water stress (Wullschleger,1993; Wohlfahrt et al., 1999; Medlyn et al., 2002;Nogues and Alegre, 2002; Bruck and Guo, 2006). Todate, there have been no studies of within-leaf vari-ability of both Vc,max and Jmax, so it is difficult to assessthe degree to which heterogeneity in these parametersactually affects leaf-level photosynthetic studies thatemploy the Farquhar model. However, it is conceiv-able that certain environmental stresses could induceconditions where heterogeneity could be a significantfactor. One example is the effect of tropospheric ozonestress. In species such as wheat (Triticum aestivum) andoak (Quercus robur), under both short- and long-termozone exposure, Vc,max is decreased to a greater extentthan Jmax (Farage et al., 1991; Farage, 1996), possibly

because decreased Rubisco activity is often the firstsymptom of ozone damage (Pell et al., 1997; Long andNaidu, 2002). Effects of ozone are known to be hetero-geneous across the leaf surface, (Nie et al., 1993),suggesting that decrease in Vc,max relative to decreasein Jmax would not be uniform across the leaf. Senes-cence within a leaf is often patchy and, becauseRubisco protein is catabolized before the membraneproteins of the electron transport apparatus (Wenget al., 2005), uncoupled variation in Vc,max and Jmaxcould result during the latter part of a leaf’s lifespan.

Of greater uncertainty is the variability in otherphotosynthetic parameters, such as VTPU or Rd (mito-chondrial respiration). There is currently no literatureon the within-leaf variability of TPU; VTPU has alwaysbeen considered as a whole-leaf parameter. Effects ofvariation in Rd were not considered in this studyoutside the light acclimation simulations, but recentstudies have suggested that Rd could vary withinleaves in a manner coupled to the photosyntheticcapacity (Tcherkez et al., 2008).

In conclusion, this study found that leaf-level photo-synthetic heterogeneity within the mesophyll couldlead to underestimation of Vc,max, Jmax, or VTPU calcu-lated by fitting the Farquhar model to the A-Ci responseof photosynthesis. Substantial error, though, wouldonly result if the Vc,max to Jmax ratio or VTPU itself washeterogeneous within the leaf (e.g. V , 0.8 or CVTPU .10%). Given that it would be expected that variation inVc,max and Jmax would normally be well coupled, weconclude that error caused by heterogeneity in theestimation of these parameters and error resulting inturn from their use in crop production and Earthsystem models will be small. Proof of this conclusionwould require quantification of within-leaf heterogene-ity of Vc,max, Jmax, and VTPU 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

Vc,max and Jmax. Each theoretical leaf consisted of 400 squares of equal area and

for each section a particular Vc,max and Jmax was assigned, based on a bivariate

normal frequency distribution of the two parameters. Vc,max and Jmax were

varied around these means based on a probability density function for

bivariate normal distributions (according to Ripley, 1987):

f ðx; yÞ5 1

2psxsy

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 V

2p exp 2

1

2ð1 2 V2Þ

x2

s2

x

1y2

s2

y

22Vxy

sxsy

! !ð1Þ

where x and y correspond to Vc,max and Jmax, respectively; sx and sy corre-

spond to the SD for Vc,max and Jmax, respectively; and V corresponds to the

statistical correlation between Vc,max and Jmax. In all simulations, temperature

was 25�C, photosynthetic photon flux density was 2,000 mmol quanta m22 s21,

and oxygen concentration was 210 mmol mol21. To isolate the effect of

biochemical heterogeneity in the mesophyll from stomatal effects, Cis were

treated as uniform within each leaf, except where stated otherwise.



Systematic Variation of Vc,max and Jmax

To determine the effects of different degrees of heterogeneity on estimates

of Vc,max and Jmax based on the Farquhar model, fixed average values of Vc,max

and Jmax were set for all leaves. Wullschleger (1993) surveyed the measured

Vc,max and Jmax of a wide range of species and found an average Vc,max of

approximately 90 mmol CO2 m22 s21 and Jmax of 170 mmol electrons m22 s21 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 Vc,max and Jmax simultaneously; SDs therefore correspondingly ranged

from 0 to 45 mmol CO2 m22 s21 for Vc,max and 0 to 85 mmol electrons m22 s21

for Jmax. The correlation coefficient between the two parameters (V; i.e. how

well-conserved the Vc,max to Jmax ratio was between sections within a given

leaf) was varied from 1.0 to 0.1 at intervals of 0.1 (Fig. 1). A V of 1.0 rep-

resented perfect correlation (i.e. a constant Vc,max to Jmax ratio across all areas

of a leaf).

A-Ci responses were generated with 30 values of Ci 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-Ci response for the leaf. Vc,max

and Jmax 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). Vc,max was estimated using Cis of 60, 80, 100, and

150 ppm, and Jmax was estimated using points at 700, 800, 1,000, and 1,200 ppm

for all leaves. These estimated values of Vc,max and Jmax were compared against

the original parameters used to generate the theoretical leaf, to determine the

percent error in each parameter estimate.

Heterogeneity in VTPU and gs

Heterogeneity in VTPU, the limitation on carbon assimilation rate imposed

by capacity for TPU, was added to the model for selected simulations by

varying the CVof VTPU (Eq. 1) across all sections while keeping the mean value

constant (VTPU 5 10 mmol m22 s21). VTPU was varied independently of Vc,max

and Jmax. In these selected simulations, VTPU was estimated for each leaf from

the whole-leaf CO2 response curve as defined in the equations in Supple-

mental Appendix S1.

Likewise, gs was varied for selected simulations by controlling the CV of

the population of gs associated with the sections of the leaf. Mean gs was 0.1

mmol m22 s21. When gs 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, Ci for each section at each

ambient CO2 concentration (Ca) was calculated based on the intersection of the

demand function (the A-Ci response curve) and the supply function (1/gs).

Overall Ci for each leaf was calculated as the mean of the Ci 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 andPhotosynthetic Heterogeneity

To investigate the effect of transdermal variation in light environment and

photosynthetic capacity on the A-Ci 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 mmol m22 s21

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

Vc,max, Jmax, VTPU, and Rd (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-Ci relationship was calculated for

each layer separately and fit to the equations of Farquhar et al. (1980) to

estimate Vc,max and Jmax 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 Ca to generate a whole-leaf CO2 response curve. Simulations

were run with and without lateral variation in Vc,max and Jmax 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 Ci was compared to the actual A from a leaf with a Vc,max and Jmax CV of

50% and a V of 0.3. The resulting A-Ci curve was then used to estimate Vc,max

and Jmax as if it had been generated from gas exchange methods, following the

procedures of Long and Bernacchi (2003). A further A-Ci 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 Ci represents the error that

results from assuming homogeneity of the A-Ci 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.

ACKNOWLEDGMENT

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.

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effect of leaf-level spatial variability

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