A stomatal optimization theory to describe the effects of atmospheric CO2 on leafphotosynthesis and transpiration

Gabriel Katul1,2, Stefano Manzoni1,2, Sari Palmroth1,* and Ram Oren1

1Nicholas School of the Environment, Box 90328, Duke University, Durham, NC 27708, USA and 2Department of Civil andEnvironmental Engineering, Duke University, Durham, NC 27708, USA

* For correspondence. E-mail sari.palmroth@duke.edu

Received: 4 August 2009 Returned for revision: 21 October 2009 Accepted: 10 November 2009 Published electronically: 8 December 2009

† Background and Aims Global climate models predict decreases in leaf stomatal conductance and transpirationdue to increases in atmospheric CO2. The consequences of these reductions are increases in soil moisture avail-ability and continental scale run-off at decadal time-scales. Thus, a theory explaining the differential sensitivityof stomata to changing atmospheric CO2 and other environmental conditions must be identified. Here, theseresponses are investigated using optimality theory applied to stomatal conductance.† Methods An analytical model for stomatal conductance is proposed based on: (a) Fickian mass transfer of CO2

and H2O through stomata; (b) a biochemical photosynthesis model that relates intercellular CO2 to net photosyn-thesis; and (c) a stomatal model based on optimization for maximizing carbon gains when water losses representa cost. Comparisons between the optimization-based model and empirical relationships widely used in climatemodels were made using an extensive gas exchange dataset collected in a maturing pine (Pinus taeda) forestunder ambient and enriched atmospheric CO2.† Key Results and Conclusion In this interpretation, it is proposed that an individual leaf optimally and autono-mously regulates stomatal opening on short-term (approx. 10-min time-scale) rather than on daily or longer time-scales. The derived equations are analytical with explicit expressions for conductance, photosynthesis and inter-cellular CO2, thereby making the approach useful for climate models. Using a gas exchange dataset collected in apine forest, it is shown that (a) the cost of unit water loss l (a measure of marginal water-use efficiency) increaseswith atmospheric CO2; (b) the new formulation correctly predicts the condition under which CO2-enrichedatmosphere will cause increasing assimilation and decreasing stomatal conductance.

Key words: Economics of gas exchange, free air CO2 enrichment, marginal water-use efficiency, photosynthesis,Pinus taeda, stomatal conductance, stomatal optimization.

INTRODUCTION

Two controls of stomatal behaviour are drawing attentionbecause they are relevant to future climate and fresh-waterresources. The first control relates stomatal response to vari-ations in atmospheric CO2 (ca) and other environmentalfactors, which ‘regulate’ stomatal conductance (Ainsworthand Rogers, 2007) and the second deals with the ‘regulatory’role stomata play in limiting water losses in the context ofleaf economics (Scarth, 1927). As summarized by Scarth’sreview, the reductions in the partial pressure of atmosphericCO2 cause stomata to open while increasing ca leadsstomata to close. Although this phenomenon was observedas early as the late 19th century by Sir Francis Darwin(Darwin, 1898), the mechanistic reasons for this firstcontrol on stomata and the variation in the intensity of theresponse are complex and not fully understood (Ainsworthand Rogers, 2007, and references therein). The secondcontrol reflects the link between CO2 absorbed in photosyn-thesis and water vapour loss through the stomata in transpira-tion. Studies on leaf transpiration have a long history –beginning perhaps with the seminal experiments of EdmeMariotte around 1660 (Meidner, 1987). However, still perti-nent is a statement made in Scarth’s review on the interactionbetween leaf transpiration and CO2 uptake (Scarth, 1927):

‘when stomata regulate one process they must regulate theother also but the question remains as to which of theseactions represents the real role of the stomata in theeconomy of the plant’.

Today, the interaction between the processes of photosyn-thesis and transpiration, and stomatal response to CO2 andother environmental factors are recognized to affectglobal-scale phenomena. For example, global climate modelspredict future acceleration of continental scale run-off primar-ily because plant stomata open less as atmospheric CO2 con-centrations increase, thereby reducing transpiration rates(Gedney et al., 2006; Betts et al., 2007). Reduced stomatalconductance is also predicted to reduce the potential uptakeof CO2 by plants, contributing to increased atmospheric CO2

concentration (Cox et al., 2000). When less water is lostthrough transpiration in water-limited ecosystems, the CO2

assimilation period may be extended because of improvedwater balance of plants (Volk et al., 2000). Alternatively,canopy leaf area may increase (Woodward, 1990), leading toa more productive ecosystem (Oren et al., 1987). Eitherresponse would provide a feedback between vegetation andthe climate system. Thus, a theory based on first-principlesexplaining the observed variation in stomatal response toCO2 must be urgently identified.

# The Author 2009. Published by Oxford University Press on behalf of the Annals of Botany Company. All rights reserved.

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Annals of Botany 105: 431–442, 2010

doi:10.1093/aob/mcp292, available online at www.aob.oxfordjournals.org

Two broad modelling approaches have been employed todescribe stomatal conductance as a function of environmentalstimuli. One approach focuses on environmental ‘regulation’of stomatal opening and is based on semi-empirical formulationsrelating stomatal conductance to environmental parameters(Jarvis, 1976) and the rate of photosynthesis (e.g. Ball et al.,1987; Collatz et al., 1991; Leuning, 1995), and is routinelyemployed in ecological, hydrological and climate models(Sellers et al., 1995, 1996; Baldocchi and Meyers, 1998; Laiet al., 2000; Siqueira and Katul, 2002; Juang et al., 2008). Thesecond approach is based instead on the economy of the plantand focuses on the ‘regulatory’ role of stomata. In thesemodels, stomatal opening is optimized to maximize carbongain for a unit water loss; thus water loss is considered a costto the plant (Givnish and Vermeij, 1976; Cowan, 1977, 1982;Cowan and Farquhar, 1977; Hari et al., 1986; Berninger andHari, 1993; Makela et al., 1996). The appeal of this approachis that by virtue of its construction, it addresses the intrinsic inter-actions between assimilation and transpiration, and thussuggests an answer to Scarth’s questions (Scarth, 1927).Despite the appeal and capability of models based on the stoma-tal optimization approach to simulate field conditions, thisapproach has not been employed in operational climate or eco-logical models (Hari et al., 1986, 1999, 2000; Berninger andHari, 1993; Berninger et al., 1996; Makela et al., 1996, 2004,2006; Aalto et al., 2002; Thum et al., 2007).

Comparisons of the two approaches to modelling stomatalconductance (i.e. semi-empirical and based on optimization)using the same dataset have rarely been conducted and therehas been no study of the effects of elevated ca on their parame-terization. In particular, the effect of elevated ca on the optim-ization parameter representing the trade-off betweenphotosynthetic carbon gain and transpirational water loss hasnot been assessed so far. The aim here is to address this omis-sion and to compare the performance of the two modellingapproaches. To achieve this aim, a simple analytical modelof leaf gas exchange based on short-term (approx. 10 min)optimization of stomatal conductance is proposed, comple-mented by transport equations for CO2 and water vapour,and a non-linear photosynthesis model. Side-by-side modelcomparisons are conducted using an extensive gas exchangedataset collected in a pine forest growing under ambient andenriched CO2 atmosphere. The new short-term optimizationtheory and a unique dataset are used to assess how elevatedca might impact the trade-off parameter, the most pertinentto the economy of gas exchange.

THEORY

Basic equations

Mass transfer of CO2 and water vapour between leaves and thebulk atmosphere can be described by Fickian diffusion throughstomata when the boundary layer resistance is negligible:

fc ¼ gðca � ciÞ

fe ¼ agðei � eaÞ � agDð1Þ

where fc is the CO2 flux, fe is the water vapour flux, g is thestomatal conductance to CO2, ca is ambient and ci intercellular

CO2 concentrations, respectively, a ¼ 1.6 is the relative diffu-sivity of water vapour with respect to carbon dioxide, ei is theintercellular and ea the ambient water vapour concentrationand D is the vapour pressure deficit representing ei – ea

when the leaf is well coupled to the atmosphere, as isusually the case when gas exchange of narrow-leavedspecies is measured in a ventilated cuvette.

When respiration is small with respect to fc, the biochemicaldemand for CO2 is mathematically well described by thephotosynthesis model of Farquhar et al. (1980):

fc ¼a1ðci � cpÞ

a2 þ ci

ð2Þ

where cp is the CO2 compensation point, a1 and a2 are selecteddepending on whether the photosynthetic rate is light orRubisco limited. Under light-saturated conditions, as in allgas exchange measurements used in this study, a1 ¼ Vcmax

(maximum carboxylation capacity) and a2 ¼ Kc(1 þ Coa/Ko),where Kc and Ko are the Michaelis constants for CO2 fixationand oxygen inhibition and Coa is the oxygen concentration inair (see Table 1 for definitions, temperature dependenciesand units). Hence, expressed in terms of conductance, eqns(1) and (2) can be combined to yield

ci

ca

¼1

2

þ�a1� a2 gþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½a1þða2� caÞg�

2þ 4gða1cpþ a2cagÞ

q2gca

ð3Þ

and

fc¼1

2

a1þða2þcaÞg�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½a1þgða2�caÞ�

2þ4gða1cpþa2cagÞ

q !

ð4Þ

when g . 0.

TABLE 1. Biochemical model parameters for eqn (2) andtheir temperature adjustments (Katul et al., 2000)

Parameter Value or temperature adjustment Units

Vcmax,25 59 mmol m22 s21

Kc,25 300 mmol mol21

Ko,25 300 mmol mol21

Co,a 210 mmol mol21

cp 36.9 þ 1.18(T – 25) þ 0.036(T – 25)2 mmol mol21

a1 (i.e. Vcmax) Vcmax;25

exp 0�088 T � 25ð Þ½ �

1þ exp 0�29 T � 41ð Þ½ �

mmol m22 s21

Kc Kc,25 exp[0.074(T – 25)] mmol mol21

Ko Ko,25 exp[0.015(T – 25)] mmol mol21

a2 Kc 1þCoa

Ko

� �mmol mol21

Katul et al. — Stomatal optimization and elevated CO2432

The closure problem

Equations (3) and (4) are not mathematically ‘closed’ andrequire independent formulations of g so that fc and ci canbe computed. Two conductance formulations, the so-calledBall–Berry (g1, see Ball et al., 1987) and the Leuning (g2,see Leuning, 1995) models are commonly used in climate(Sellers et al., 1995, 1996) or detailed biosphere–atmospheremodels (Baldocchi and Meyers, 1998; Lai et al., 2000;Siqueira and Katul, 2002; Juang et al., 2008). The formu-lations, respectively, are:

g1 ¼m1

ca � cp

fc H þ b1 ð5aÞ

and

g2 ¼m2

ca � cp

fc 1þD

Do

� ��1

þ b2 ð5bÞ

where H is the mean air relative humidity, Do is a vapourpressure deficit constant and m1, m2, b1 and b2 are empiricalfitting parameters. Equations (5a) and (5b) therefore providethe necessary mathematical closure (i.e. three equations withthree unknowns: fc, ci and g). Equations (5a) and (5b) are semi-empirical, mainly determining a priori the effects of environ-mental stimuli (e.g. D or H ) on conductance and thus fall intothe first category of stomatal regulation models.

An optimality model

The second approach to stomatal conductance modellingemploys an optimality principle originally proposed byGivnish and Vermeij (1976) and retained in the work ofCowan and Farquhar (1977), Hari et al. (1986), Berningerand Hari (1993) and more recently of Konrad et al. (2008).In this approach, an objective function (i.e. the net carbongain for the leaf) is defined as

f ðgÞ ¼ fc � lfe ¼1

2

a1þ ða2þ caÞg

�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½a1þ gða2� caÞ�

2þ 4 gða1cpþ a2cagÞ

q !� lðagDÞ

ð6Þ

By maximizing the gross carbon gain fc while minimizing thewater loss lfe (in units of carbon) for a species-specific costparameter l, an expression for conductance can be derived.Throughout, the notation of Hari et al. (1986) is used; thesymbol l in Cowan (1977) indicates the inverse of the cost par-ameter employed here. Such a maximization condition can beexpressed as df(g)/dg ¼ 0 [f(g) is guaranteed to have amaximum given the expected concavity in eqn (6) off(g)with respect to g]. The optimization problem framed ineqn (6) does not a priori assume a functional response of gto D or ca. Instead, it assumes (a priori) that excessive waterlosses do induce stomatal closure, which is consistent withthe experimental findings of Mott and Parkhurst (1991) who,

using a ‘helox’ gas medium, demonstrated that stomatarespond to transpiration rates rather than measures of airhumidity. The original formulae of Givnish and Vermeij(1976), Cowan (1977), Cowan and Farquhar (1977), Hariet al. (1986) and Berninger and Hari (1993) define eqn (6),or a variant of it, for an integration interval bounded bytimes t0 and t1. Since in this optimization problem there isno dynamic component (i.e. stomatal conductance isassumed to adapt instantaneously to changing environmentalconditions), it can be shown that this arbitrary integrationinterval does not alter the mathematical expression foroptimal conductance, provided that the integration limits areheld constant. In other words, the maximization of the cumu-lative C gain over an arbitrary time interval is equivalent tomaximizing eqn (6) at each instant in time (Pontryaginmaximum principle applied to this disjoint optimizationproblem; see Denn, 1969).

Assuming that stomatal behaviour is optimal (and thus l is aconstant with respect to g) and differentiating, eqn (6) yields

df ðgÞ

dg¼

1

2

a2þcaþa1ð�a2þca�2cpÞ�gða2þcaÞ

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½a1þða2�caÞg�

2þ4gða1cpþa2cagÞ

q �2lagD

0B@

1CAð7Þ

When setting df(g)/dg ¼ 0 and solving for g, the following isobtained:

g¼�a1ða2�caþ2cpÞ

ða2þcaÞ2

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiaDla2

1ðca�cpÞða2þcpÞða2þca�2aDlÞ2ða2þca�aDlÞ

qaDlða2þcaÞ

2ða2þca�aDlÞ

ð8Þ

Similar to the two formulations of the first approach (eqns 5a and5b), this formulation is explicit in relating conductance to ca andD. However, unlike these formulations, both relating conduc-tance to the photosynthetic rate leaving two (g1 formulation)or three (g2 formulation) unknown parameters, eqn (8) relatesconductance to the three parameters of the photosynthesismodel leaving only one unknown physiological parameter (l).Moreover, the model formulation in eqn (8) suggests that lcannot be entirely ‘free’ and must be bounded between zeroand (ca – cp)/(aD) to ensure real and positive conductance.This theoretical bound is in contrast to the less constrainedchoices of m1 and m2 in the conductance models of eqn (5).

It is worth emphasizing that for eqn (8) to be applicable, ldoes not need to be exactly constant for a given leaf and undercertain external conditions. Suboptimal conditions are stilladmissible in eqn (8) provided jdl/lj � jdfe/fej. Recallingthat fe ¼ agD (eqn 1), this necessary condition can beexpressed as l/g �jdl/dgj. Future experimental studies mightemploy this constraint to test how close to optimality a singleleaf is.

Katul et al. — Stomatal optimization and elevated CO2 433

A linearized optimality model

Linearizing the biochemical demand function results in amuch simpler and informative model for optimal g.Linearizing the model requires the assumptions that cp� ci

and that the variability of ci affects only marginally thedenominator of eqn (2), leading to the approximation a2 þ

ci ¼ a2 þ (ci/ca)ca ¼ a2 þ sca. As a result,

fc ¼a1ci

a2 þ sca

ð9Þ

where s is treated as a constant set equal to the long-term meanof ci/ca. We stress that s is treated as a model constant only inthe denominator of eqn (9), while in eqn (1) ci/ca is allowed tovary. Note also that the geometric interpretation of the groupof parameters a1/(a2 þ sca) in eqn (9) is simply the slope ofthe fc(ci) curve. Combining this linearized photosynthesismodel with eqn (1) results in

ci

ca

¼ðaþ scaÞ

a1=gþ a2 þ ca sð10Þ

and

fc ¼g a1ca

a1 þ gða2 þ scaÞð11Þ

The objective function in eqn (6) now simplifies to

f ðgÞ ¼ fc � lfe ¼g a1ca

a1 þ gða2 þ scaÞ� lðagDÞ ð12Þ

and upon differentiation with respect to g yields

df ðgÞ

dg¼

caa21

a1 þ gða2 þ scaÞ½ �2� alD ð13Þ

By setting df(g)/dg ¼ 0 and solving for g results in

g ¼a1

a2 þ sca

� ��1þ

ca

alD

� �1=2� �

ð14Þ

This expression is identical to that in Hari et al. (1986), thoughderived without the use of arbitrary time integration. Replacingeqn (14) into eqns (10) and (11) provides closed formexpressions for ci and fc given by

ci

ca

¼ 1�al

ca

� �1=2

D1=2 ð15Þ

and

fc ¼a1

a2 þ sca

ðca �ffiffiffiffiffiffiffiffiffiffiffiffiffialDca

pÞ ð16Þ

Hereafter, the solution in eqn (8) is referred to as the ‘non-linear model’ and the result in eqn (14) as the ‘linear model’(denoted by LI). The linear model was recently shown byKatul et al. (2009) to be consistent with (a) stomatal responses

to changes in vapour pressure deficit and (b) the non-linearitiesin ci/ca variations with D.

EXPERIMENTS

The experiments, including details of the free-air CO2 enrich-ment (FACE) facility at Duke Forest, the specifics of the por-table gas exchange system and the data analysis are presentedelsewhere (Ellsworth, 1999, 2000; Hendrey et al., 1999; Katulet al., 2000; Maier et al., 2008). Gas exchange measurementswere made at the Duke University FACE site, Orange County,North Carolina (358980N, 79880W, elevation ¼ 163 m) at aneven-aged loblolly pine (Pinus taeda) forest planted at 2.0 �2.4 m spacing in 1983 following clear cutting and burning.The site index is between 20 m and 21 m at age 25 years.The FACE system consists of eight 30-m-diameter plots,four of which are fumigated with CO2 to maintain atmosphericCO2 concentration at ambient þ 200 mmol mol21 while theother four serve as controls and receive ambient air only.Two datasets, described below, were employed in this study.

Dataset 1 (1996–1999)

Steady-state gas exchange of leaves was measured in 1996–1999 in plots 1–6, at two growth CO2 concentrations (ambientand ambient þ 200 mmol mol21 CO2 concentration) and nearambient temperature using a portable infrared gas analysersystem for CO2 and water vapour (CIRAS-1, PP-Systems,Amesbury, MA, USA). The system was operated in openflow mode with a 5.5-cm-long leaf chamber and an integratedCO2 supply system. The light-saturated photosynthetic rate andstomatal conductance were measured on upper canopy foliage,at 11–12 m above ground. For further details see Ellsworth(1999, 2000).

Dataset 2 (2005)

Light-saturated photosynthetic rates and leaf conductancewere measured in 2005 in plots 1–8 on detached foliageusing a LI-6400 (Li-Cor Inc., Lincoln, NE, USA). Gasexchange measurements at growth CO2 concentrations werecompleted within 30 min of removing the fascicles. Withinthis time period, the light-saturated leaf gas exchange ratesremained stable with little change in stomatal conductance(see figure 1 in Maier et al., 2008). Chamber temperaturewas maintained at 20 8C (spring) and 25 8C (summer andautumn). Details are given in Maier et al. (2008).

The combined dataset, resulting in 193 gas exchangemeasurements, 73 of which were collected under elevated ca,is used in the following analyses. All measured gas exchangerates are reported per unit leaf area, where the leaf area isexpressed as half of the total needle surface area (i.e. one-sidedleaf area).

MODEL PARAMETER ESTIMATION

Before comparing the four stomatal conductance model for-mulations and discussing the effects of elevated ca on theirparameters, measurement errors and how these errors mightimpact parameter estimation are briefly discussed.

Katul et al. — Stomatal optimization and elevated CO2434

Measurement errors

Measurement errors during leaf gas exchange experimentsmay be due to (a) uncertainties in the measured CO2 andH2O concentrations in the gas exchange chamber, (b) unac-counted leaks from the chamber and (c) uncertainties in thecalculated needle leaf area when results are scaled to a unitleaf area. Gas concentration measurement errors are in theorder of 0.001 % for CO2, 0.1 % for H2O and 0.1 % for temp-erature. Diffusive leaks are not expected to be an issue instudies where gas exchange measurements are performed atexternal CO2 and water vapour concentrations. Hence, thesetwo sources of error should not significantly contribute tothe variability in the observations.

Leaf area estimates may be an important source of uncer-tainty. These estimates are calculated from needle width,which is measured to the nearest 0.1 mm and therefore canintroduce an error on the order of 10 %. However, the erroris the same for photosynthesis, leaf transpiration and stomatalconductance estimates, and thus propagates to some but not allparameters, as described later.

Ball–Berry and Leuning models

The effective model parameters for g1 and g2 (eqns 5a and5b) and their frequency distributions were computed from thetwo gas exchange datasets. The values for m1, m2, b1 and b2were computed using an ordinary least squares (OLS)approach (Fig. 1, left panels, and Table 2). Hereafter, these

best-fit values from OLS are referred to as effective parametervalues. Because the interest here is in the responses toenvironmental drivers, the focus is on the sensitivity par-ameters m1 and m2. Variations in m1 and m2 among leavesand trees were also computed by inverting eqns (5a) and(5b) for given values of the intercepts b1 and b2. To accountfor uncertainty in the intercept estimates (due to both naturalvariability and measurement errors induced by leaf areameasurements), a Monte-Carlo approach was employed byselecting b1 and b2 from normal distributions having meanvalues identical to the effective parameter values in Table 2and standard deviations commensurate with the standarderror of estimate determined from OLS (see Table 2). Hence,for each measurement point in Fig. 1, 100 b1 and b2 realiz-ations were generated and for each of these m1 and m2 werecalculated. These estimated m1 and m2 parameters are sum-marized as frequency distributions, grouped by ambient andelevated ca (Fig. 1, right panels). Increasing the number ofrealizations for the intercepts did not change the resulting fre-quency distributions. It is noted that estimates of m1 and m2 arenot affected by leaf area measurement errors affecting conduc-tance and photosynthesis identically and thus cancelling out inthe ratio.

Optimal conductance models

The optimal conductance models in eqns (8) and (14)require the parameters a1, a2, cp and l. To determine these

0 0·01 0·02 0·03 0·040

0·05

0·10

0·15

0·20

(fc/ca) RH

A

0

0·05

0·10

0·15

0·20C

0 2 4 6 8

0 0·01 0·02 0·03 0·04 0 2 4 6 8

0

0·1

0·2

0·3

0·4

0·5

m1

(fc/ca) (1+D/Do)−1 m2

g 2g 1

Fre

quen

cy d

istr

ibut

ion

B AmbientElevated

0

0·1

0·2

0·3

0·4

0·5

Fre

quen

cy d

istr

ibut

ion

D

FI G. 1. Left panels: estimation of the effective parameters via regression analysis for the stomatal conductance models in eqns (5a) and (5b), the Ball–Berrymodel g1 ¼ m1 fcRH/ca þ b1 (A) and the Leuning model g2 ¼ m2 fc[(1 þ D/D0)ca]

21þ b2 (C) using gas exchange data collected under ambient (light circles) and

elevated (heavy circles) ca. The parameters m1, m2, b1 and b2, determined from linear regression analysis (lines), are shown in Table 1. Right panels: the dis-tributions of m1 and m2 (B and D, respectively), computed with fixed b1, b2 and Do, are shown for ambient and elevated ca.

Katul et al. — Stomatal optimization and elevated CO2 435

parameters, the gas exchange data were analysed as follows.Parameters a2 and cp were determined using the standardtemperature formulations summarized in Table 1 (i.e. assum-ing no variability across leaves). This assumption seems justi-fied given the consistency of these two parameters in a widerange of species. Parameter a1 was determined by re-arrangingeqn (2) as

a1 ¼fcða2ðTÞ þ ciÞ

ci � cpðTÞð17Þ

and then using each of the 193 measured fc, ci and leaf temp-erature (T ) values. It is emphasized that a1 was not determinedfrom typical fc(ci) curves. Hence, the uncertainty in a1 orig-inates from errors incurred when measuring fc and fe (fromwhich g, and subsequently ci, are estimated) and/or fromnatural variability among leaves and individual trees.Because ci is computed from the ratio of photosynthesis toconductance, A/g, it is unaffected by uncertainties in leafarea; however, some of the variability in a1 might be tracedto errors in fc caused by uncertainties in leaf area estimates.

Values of a1 computed from eqn (17) were then compared inFig. 2 to the standard temperature formulation for themaximum carboxylation capacity (Table 1) using a value ofVcmax,25 ¼ 59 mmol m22 s21 obtained from independent data(Katul et al., 2000). The agreement in leaf temperatureresponses for a1 is reasonable considering that thegas-exchange was measured under ambient and enriched ca

using two different systems, and over a wide range of con-ditions, as described earlier. Such an agreement suggeststhat, regardless of some uncertainty introduced by measure-ment errors, the estimation method (eqn 17) is sufficientlyreliable for the purposes of this work.

When it is assumed that stomata are optimally controlled,the parameter l can be estimated using the definition of mar-ginal water-use efficiency found by differentiation of eqn (6)(see also Hari et al., 1986),

l ¼dfc=dg

dfe=dgð18Þ

The definition in eqn (18) can be used to compute l from a1,a2, cp (previously described) and the measured D and g,

l ¼a1 ca � a2 � 2cp

þ ca þ a2ð Þ y� g ca þ a2ð Þ½ �

2aDyð19Þ

where

y ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2

1 þ 2a1 a2 � ca þ 2cp

gþ a2 þ cað Þ

2 g2

qð19aÞ

was defined.

10 15 20 25 30 35 400

50

100

150

T (ºC)

Vcm

ax(T

) (μ

mol

m−

2 s−

1 )

ElevatedAmbient

FI G. 2. Estimated maximum carboxylation capacity (or a1 forRubisco-limited photosynthesis) from 193 gas exchange measurements as afunction of leaf temperature (eqn 17) for needles exposed to ambient and elev-ated atmospheric CO2 (sampled from 1996 to 1999). The standard temperaturesensitivity function (from Campbell and Norman, 1998) with Vcmax,25 ¼ 59 is

also shown for reference (continuous line). For formulation, see Table 1.

TABLE 2. Effects of elevated ca on the statistics of the stomatal model parameters, estimated using ordinary least-square regression(OLS) or by inverting the stomatal conductance equations (see Figs 1 and 3)

Parameter Units Estimation method

Mean (s.d.)

Ambient CO2 Elevated CO2

m1 – OLS 3.42 (0.30) 3.56 (0.44)Inversion of eqn (5a) 3.38 (1.22) 3.58 (1.39)

b1 mol m22 s21 OLS 0.04 (0.01) 0.03 (0.01)m2 – OLS 3.52 (0.27) 4.71 (0.49)

Inversion of eqn (5b) 3.51 (0.86) 4.70 (1.06)b2 mol m22 s21 OLS 0.016 (0.006) –0.006 (0.010)l mmol mol21 kPa21 OLS 21.12 (1.81) 28.12 (3.00)

Eqn (19) 20.06 (8.27) 27.64 (14.75)lLI mmol mol21 kPa21 OLS 23.69 (1.84) 34.02 (3.13)

Eqn (20) 22.64 (9.73) 36.88 (17.79)

For each parameter, mean and standard deviation (in parenthesis) are given.Do in eqn (5b) is assumed equal to 3 kPa.

Katul et al. — Stomatal optimization and elevated CO2436

In the linear model, l can be directly determined frommeasured ca, D, fc and g (or ci),

lLI ¼1

aDca

A

gC

� �2

¼ca

aD1�

ci

ca

� �2

ð20Þ

Note that measurement errors due to the leaf area estimate (themost relevant in the present dataset) do not affect lLI and l,which depend on the ratios A/g or a1/g.

The effective (or best-fit) values of l and lLI along withstandard error of estimates, determined from the OLS usingeqn (18) (Fig. 3A and 3C) are reported in Table 2. For eachgas exchange measurement, the uncertainty in the estimatedl for the linear model originates from measurement errors infc, g and D. Likewise, for the non-linear model, additionaluncertainty originates from the same variables used tocompute a1. However, much of the variability in l stemsfrom needle-to-needle variations in measured fc/g, ratherthan measurement errors, as discussed above.

In the OLS estimates of l, the abscissa was aD and theordinate was computed analytically by differentiating withrespect to g in eqn (4) (in the non-linear model) and eqn(9) (in the linear model), using the gas exchange data(including D) as input in the evaluation. Figures 3B and3D also show the frequency distributions of l and lLI ofindividual leaves based on eqns (19) and (20). Figure 4 com-pares l with lLI of all 193 data points. The correlation coef-ficient (R) between these two l estimates is large (R ¼ 0.96),

yet the linear model overestimated l by about 20 %. Thiscomparison suggests that lLI can provide an approximationof l obtained from the more realistic, non-linear biochemicalmodel.

0 2 4 60

50

100

150

200

250A

∂fe/∂g

∂fc/

∂g(∂

f c/∂

g)LI

0

50

100

150

200

250C

0 50 1000

0·1

0·2

0·3

0·4

0·5B

Fre

quen

cy d

istr

ibut

ion

λ

λ0 2 4 6

∂fe/∂g0 50 100

LI

AmbientElevated

0

0·1

0·2

0·3

0·4

0·5D

Fre

quen

cy d

istr

ibut

ion

FI G. 3. Left panels: linear regression-based estimates of the parameter l from eqn (18) for the non-linear (A) and the linear optimization models (C), using gasexchange measurements collected under ambient (light circles) and elevated (heavy circles) ca (see Table 2). Right panels: frequency distribution of l (B) and lLI

(D) for needles exposed to ambient and elevated ca. Note the increase in the mean (not mode) and standard deviations in l and lLI with increasing ca.

0 20 40 60 80 1000

20

40

60

80

100

λ

ElevatedAmbient

λLI

FI G. 4. Comparison between l (non-linear model, denoted by NL; eqn 19)and lLI (linearized model, LI; eqn 20) for needles exposed to ambient andelevated ca. To estimate l, a1 was estimated from data points in Fig. 2. Theregression model l ¼ 0.8lLI þ 1.5 (continuous line) describes the data witha coefficient of determination R2 ¼ 0.92. The dot–dashed line is the

one-to-one line.

Katul et al. — Stomatal optimization and elevated CO2 437

RESULTS

Effects of elevated CO2 on the parameters of Ball–Berryand Leuning models

When using the effective values of the OLS, both Ball–Berryand Leuning models described the gas exchange data well,although they were collected from plots with a wide range ofsoil nitrogen over years with a wide range of environmentalconditions (McCarthy et al., 2007) and at different timesduring the growing season (Ellsworth, 2000), all of whichaffect leaf nitrogen concentration and physiological state.The coefficient of variation (CV ¼ standard deviation/mean),was fairly large for m1 and m2 among individual needles(approx. 35 % and approx. 10 %, respectively; see frequencydistributions in Fig. 1). The mean of m1 in the Ball–Berrymodel did not differ between ambient and elevated ca

(Table 2), as was also found by Medlyn et al. (2001). In con-trast, in the Leuning model, the effective m2 was significantlydifferent when comparing ambient and enriched CO2 con-ditions. In both models, the standard deviations of m1 andm2 was higher at elevated ca. A Kolmogorov–Smirnoff testhas also been conducted on differences between the frequencydistributions of m1 and m2 shown in Fig. 1, and again, therewas a significant CO2-induced effect on the parameters inthe Leuning model at the 95 % confidence level but not inthe Ball–Berry model.

Effects of elevated CO2 on l

Having determined l, the effect of elevated ca on l and lLI

was assessed. Elevated CO2 increased the effective values forl in both the linear and the non-linear model (Table 2).Frequency distributions of l estimates based on eqns (19)and (20) at the leaf scale are compared in Fig. 3B and 3D (ana-logous to the histogram comparisons conducted for m1 and m2

in Fig. 1). Elevated ca increased the mean l and lLI, but due tosimultaneous increase in the standard deviations, their CVremained unaffected (see Table 2). The CV values of l andlLI are comparable to those of m1 and m2.

Model inter-comparison

Given the large variation in a1 among leaves (e.g. Fig. 2),the two optimization models and the two semi-empiricalmodels are compared in an ‘ensemble-averaged’ sense toassess how well each reproduced expected effects of elevatedatmospheric ca on photosynthesis, g and ci/ca.

For this comparison, eqns (1) and (2) were combined witheqns (5a) and (5b) (Ball–Berry and Leuning stomatalmodels) and numerically solved to provide estimates for ci, gand fc. Equations (5a) and (5b) were parameterized with thecoefficients in Table 1. The parameter a1 in eqn (2) was para-meterized using the continuous line in Fig. 2 (i.e. the standardmodel with Vcmax,25 ¼ 59 mmol m22 s21) assuming that thisformulation represents the ensemble fc(ci) behaviour in allcases and is independent of elevated ca (meaning that thereis no down-regulation). The constant Vcmax,25 from the1996–1998 dataset was used because later studies of down-regulation at the site were inconclusive (Rogers andEllsworth, 2002; Crous et al., 2004, 2008; Springer et al.,

2005; Maier et al., 2008). If the magnitude of the down-regulation is precisely known for each foliage element, it canbe incorporated into parameter a1. However, this informationis not available, and attempting to incorporate down-regulationwould introduce additional and uneven uncertainty to theresults from the four models.

Two calculations were made, each based on the non-linearand linear optimization models. The first assumed that ldoes not vary with changing ca and was set equal to themean value in Table 2 for ambient ca conditions. The otherassumed that l � 0.089ca/Dref, where Dref ¼ 1 kPa. The coef-ficient 0.089 was intended to reproduce the estimated variationof l with elevated ca, commensurate with the change in meanvalues reported in Table 2 for lLI, and is assumed to be anupper-envelope of plausible l variations.

The inputs to all models were leaf temperature, vapour pressuredeficit or relative humidity and ca ¼ 360 mmol m22 s21 forambient and ca ¼ 560 mmol m22 s21 for elevated conditions.The ensemble model calculations and comparison to data arereported in Figs 5 and 6. When setting l to a constant independentof ca, increasing ca led to an increasing conductance and ci/ca

(Fig. 5). The increase in g contrasts with evidence suggestingunchanging or slightly decreasing conductance with increasingca. The mathematical structure of the optimization modelssuggest that when ca! þ1, the linear model predicts that gscales as ca

21/2, while the non-linear predicts g scales as ca21.

Hence, both models eventually predict a decline in g with increas-ing ca, as long as ca is high enough (3-fold current concen-trations), even if l is held constant. This indicates that tocorrectly predict changes in conductance within a realisticrange of ca, the dependence of l on atmospheric CO2 cannot beneglected.

When l was increased with elevated ca, the non-linearmodel calculations agreed well with the gas exchange datafor the 1996–1998 period (Fig. 5), correctly predicting aslight decrease in g and ci/ca and markedly increased assimila-tion rate under elevated CO2 in the Duke Forest dataset. Also,the linearized optimization approach provides reasonable esti-mates of photosynthesis, g and ci/ca when l varies with ca,similar to the Ball–Berry and Leuning model predictions(Fig. 6). Independent datasets from other experiments onloblolly pine under elevated ca (ranging from chamber toFACE) were also included for reference in Fig. 6 and somedo show g hardly changing or even exhibiting an increasewith ca. Changes in stomatal conductance, however, were notstatistically significant in these experiments and thus cannotbe reconciled with the 25 % increase predicted by the optimiz-ation model when l was assumed independent of ca (Fig. 5).Finally, it is noted that the increase in l with ca is consistentwith an independent analysis based on the sensitivity of g toD for ambient and enriched ca by Katul et al. (2009) usingpublished experiments reported by Heath (1998), Bunce(1998) and Medlyn et al. (2001).

DISCUSSION

The stomatal conductance optimization hypothesis assumesthat the regulatory role of stomata is to simultaneously maxi-mize the carbon gain rate while minimizing the rate of waterlosses. Accepting this as the stomatal role, it was possible to

Katul et al. — Stomatal optimization and elevated CO2438

predict how certain environmental stimuli control stomatalconductance thereby unifying the two controls discussed inScarth’s review (Scarth, 1927), i.e. ‘regulation’ by environ-mental conditions (atmospheric CO2 concentration andvapour pressure deficit) and ‘regulatory’ in the economics ofleaf gas exchange. The mathematical equivalence betweenthe maximization of daily and short-term assimilation leadsus to interpret the optimization problem differently from theearlier theory. We argue that optimization operates on time-scales commensurate with opening and closure of stomatalaperture (approx. 10 min, below which the dynamic, delayedresponses of stomata to external stimuli should be accountedfor) and each leaf optimally and autonomously regulates sto-matal conductance. Perhaps two well-investigated examplesin support of this argument are the responses of leaves to tran-sient sunlight due to canopy shading (Pearcy, 1990; Naumburget al., 2001; Naumburg and Ellsworth, 2002) and to variable

cloud cover in dry ecosystems (Knapp and Smith, 1989;Knapp et al., 1989; Knapp, 1993).

In the first case, light penetration through the canopy illumi-nates a small area of a leaf for a brief period resulting in rapidopening of the stomata in the illuminated region while those inshaded regions remain relatively closed. This action wouldallow the entire plant to use light in photosynthesis as itbecomes available without unnecessarily losing water(Hetherington and Woodward, 2003). Hence, while this ‘adap-tive strategy’ may have evolved to satisfy the carbon and watereconomies of the entire plant, it is achieved by the rapidresponse of stomata of individual leaves (or areas of a leaf)to light availability. Leaf scale optimization thus leads to anoptimal strategy for managing the carbon–water economy ofthe whole plant.

In the second example, plants subjected to drying soil andhighly variable light during the growing season face a

0·5

1·0

1·5

g/g a

mb

1·0

1·5

2·0

2·5NL modelLeuningBall−BerryDuke−FACE (1996−1998)Liu and Teskey (1995)Thomas et al. (1994)Teskey (1995)Murthy et al. (1997)Tissue et al. (1996)Tissue et al. (1997)

1·0 1·5 2·00·5

1·0

1·5

An/

An,

amb

Ca/Ca,amb

(Ci/C

a)/(

Ci/C

a)am

b

FI G. 6. The effects of elevated ca on conductance g (top), photosynthesis An

(middle) and ci/ca (bottom) as modelled by the non-linear approach with vari-able l, the Ball–Berry model and the Leuning model. For reference, theensemble-averaged gas exchange data (i.e. the ratio of time- ortreatment-average elevated to average ambient values) from a number of

experiments conducted on loblolly pine are also shown (symbols).

0·8

0·9

1·0

1·1

1·2

1·3

g/g a

mb

1·0

1·2

1·4

1·6

1·8

2·0

An/

An,

amb

1·0 1·2 1·4 1·60·90

0·95

1·00

1·05

1·10

1·15

1·20

Ca/Ca,amb

(Ci/C

a)/(

Ci/C

a)am

b

NL, variable λLI, variable λLI

NL, constant λLI, constant λLI

Duke−FACE (1996−1998)

FI G. 5. The effects of elevated ca on conductance g (top), photosynthesis An

(middle) and ci/ca (bottom): a comparison between the non-linear (NL) andlinearized (LI) optimality results for a constant (set at ambient) and variablel forced by ensemble-averaged leaf temperature and relative humidity. Forreference, the ensemble-averaged gas exchange observations (in terms ofmean elevated to mean ambient) for 1996–1998 are also shown (symbols).

Katul et al. — Stomatal optimization and elevated CO2 439

similar (although more complex) optimization problem. Their‘objective’ is to conserve water when uptake of CO2 is lightlimited (Knapp et al., 1989), rather than maximizing assimila-tion during the sun fleck periods as described above. Strikingdifferences among leaf responses occur, depending on theplant growth form (Knapp and Smith, 1989) and photosyn-thetic pathway (Knapp, 1993). Shallow-rooted grasses, forexample, are more prone to water stress than trees, resultingin greater ability to adjust stomatal conductance in responseto rapid changes in meteorological and light conditions(Knapp and Smith, 1989). This example again shows that evol-ution has lead to short-term response strategies that are consist-ent with the hypothesis of short-term optimization.Mathematically, this hypothesis translates to maximization ofthe objective function f in eqn (6), without needing temporalintegration (although it is noted again that the two formu-lations are mathematically equivalent due to Pontryaginmaximum principle).

For this short-term, leaf-level optimization, the interpret-ation of l also becomes highly localized and thus spatiallyvariable within the canopy and across the individual plantsin a tree stand. Nevertheless, l for a given ca is no more vari-able within a stand than the parameters of models currentlyemployed in climate systems (e.g. m1 and m2, as shown inFig. 1). Clearly, the carbon–water economy of plants is alsoaffected by processes occurring at longer time-scales and atthe whole-plant level. For example, at the weekly time-scale,stomatal conductance and photosynthetic activity are reducedunder water shortage through complex plant-scale hormonalsignalling (Chapin, 1991). It is speculated that such responsesmay help the plant prevent severe stress events (imbalancedmetabolite concentrations or high growth costs) and thus maxi-mize long-term fitness. At even longer time-scales, carbon ispartitioned to optimize water or nutrient absorption by rootsversus carbon uptake by leaves (Givnish et al., 1984;Palmroth et al., 2006; Franklin, 2007). Overall, the short-term,leaf-level optimization can be interpreted as the end memberof a hierarchy of responses to environmental conditions thatspan temporal scales ranging from minutes to years andspatial scales from a patch on a leaf to trees and stands.

The proposed optimization models require one physiologi-cal parameter – the marginal water-use efficiency, l. Basedon the relationship between the linearized and non-linearmodels (Fig. 4), l for loblolly pine needles can be estimatedfrom lLI, which in turn is inferred from measured assimilationrate, stomatal conductance and vapour pressure difference (eqn20). Alternatively, lLI can be estimated from measured ci/ca,which is becoming increasingly available from a stableisotope network [BASIN (Biosphere–Atmosphere StableIsotope Network); http://basinisotopes.org/] bypassing theneed to estimate l from direct gas exchange measurements(see Lloyd and Farquhar, 1994; Palmroth et al., 1999). Dataon additional species and conditions would be necessary toassess the generality of the relationship between l and lLI.

The analysis showed that, unlike the sensitivity parameter ofthe Ball–Berry model (m1), both the sensitivity parameter ofthe Leuning model (m2) and l change with ca (Figs 1 and 3and Table 2). This might limit the practical use of stomataloptimization theories. However, the present results suggestthat l tends to vary predictably. The data and model results

in Fig. 6 suggest that ci/ca is largely invariant with elevatedca, which (a) implies that the linearization in eqn (9) hassome empirical support and (b) explicitly links l and ca

based on eqn (20). For example, noting that ci/ca in C3

plants is bounded between 0.65 and 0.80 (e.g. Jones, 1992),results in lLI � 0.03ca – 0.08ca at a reference D ¼ 1 kPa.Hence, based on the analytical results and the limited sensi-tivity of the effects of ci/ca to increasing ca, it is not surprisingthat l increases with increasing ca. For C4 species, ci/ca isusually bounded between 0.4 and 0.6 (e.g. Jones, 1992), result-ing in lLI � 0.1ca – 0.22ca at a reference D ¼ 1 kPa,suggesting a larger enhancement in the marginal water-useefficiency l with increasing ca of C4 than C3 plants. This isconsistent with a large number of studies of relationships ofphotosynthesis versus conductance (Hetherington andWoodward, 2003). The interpretation of this increase is alsophysiologically meaningful – the marginal water-use effi-ciency is known to increase with increasing ca for both C3

and C4 species, but at different rates. In short, the predictabilityof l with ca removes the potential limitation of the optimalityapproach presented here.

In addition to the explicit link dictating an increase in l withca, l should also increase with drought severity (Cowan andFarquhar, 1977; Cowan, 1982; Berninger and Hari, 1993;Makela et al., 1996). Drought-induced changes in l may beinterpreted as plant responses to altered availability of eithercarbon or water. When water availability decreases at constantca, the marginal cost of water losses (interpreted as risk ofwater stress damage; Berninger and Hari, 1993) becomesmore important. However, in the present study the linkbetween l and leaf water status was not explored.

In conclusion, the set of equations derived from the optim-ization condition stated here has the advantage of havingclosed and explicit form for conductance, photosynthesis andintercellular CO2 vis-a-vis the analogous set of equations forthe Ball–Berry or Leuning models that require numerical sol-ution. In our approach, the numerical solution of the coupledeqns (1), (2) and (5) is not necessary, and the numericalinstabilities that often arise at low assimilation rates (Zhanet al., 2003; Vico and Porporato, 2008) can be avoided.Thus, this approach is suitable for climate models (Sellerset al., 1995), or large eddy simulation models employed to rep-resent the details of complex turbulent flow regimes in bio-sphere–atmosphere mass and energy exchange studies(Albertson et al., 2001). The proposed formulation willbenefit from a theoretical explanation of the changes in ldue to CO2 concentration and soil moisture availability.These theoretical issues are currently being investigated andwill be the subject of a future contribution.

ACKNOWLEDGEMENTS

This study was supported by the United States Department ofEnergy (DOE) through the Office of Biological andEnvironmental Research (BER) Terrestrial Carbon Processes(TCP) program (FACE and NICCR grants:DE-FG02-95ER62083, DE-FC02-06ER64156), by theNational Science Foundation (NSF-EAR 0628342, NSF-EAR0635787), by Bi-National Agricultural ResearchDevelopment (BARD) fund (IS-3861-06) and by the US

Katul et al. — Stomatal optimization and elevated CO2440

Department of Agriculture (USDA grant 58-6206-7-029). Wealso thank three anonymous reviewers and Giulia Vico forhelpful comments.

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