Modelling stomatal conductanceof field-grown sunflower under varying soil water content and leafenvironment: comparison of three models of stomatal response toleaf environment and coupling with an abscisic acid-based modelof stomatal response to soil drying


V. P. Gutschick.Fax: + 1 505 6465665; e-mail:


Stomatal conductance, gs, responds both tothe immediate or local environment of the leaf, such as CO2 partialpressure and irradiance, and to root-sourced signals of water stress,particularly abscisic acid (ABA). Two models for the combined controlof gs were formulated and tested in sunflower(Helianthus annuus). First, several empirical models weretested for the local control, demonstrating that the Ball–Berrymodel [Ball, Woodrow & Berry (in Progress in PhotosynthesisResearch Vol. 4, pp. 5.221–5.224: M. Nijhoff,Dordrecht, The Netherlands) 1987] is consistently amongthe most accurate. A problem of statistical non-independence inthis model is shown to be minor. The model offers regularity ofparameter values among most species and, despite an oversimplicationin representing known humidity-response mechanisms, it incorporates othersignalling loops from CO2 and assimilation. In the firstcombined model, ABA as its concentration in xylem sap, [ABA]xy,down-regulates the slope, m, in the Ball–Berry modelby the factor gfac = exp(– β[ABA]xy).The ABA-induced reduction in gs decreases CO2 assimilation andsurface humidity, thus appearing to induce the local-control mechanismto amplify the ABA-induced stomatal closure. In the second combinedmodel, gs is estimated as the minimum of the local(Ball–Berry) response and the product gfac gs,max,with gs,max as a maximal unstressed conductance.Both models can predict gs from the external environmentalvariables with good accuracy (r2 near 0·8 over20-fold variations in gs). Further analyses showthat gs responds to humidity almost quadraticallyrather than linearly. It also responds to assimilation as a powerlaw with an exponent that is significantly less than 1. These limitations,shared by other models, suggest more research into biochemical signalling.


Stomatal conductance responds to two distinct environments. Thelocal or aerial environment of the leaf is defined by the irradiance,temperature, humidity, CO2 content and boundary-layercondition. The distal environment, particularly that of the roots,commonly generates root-sourced signals of water stress or perhapssignals of hydraulic conductivity (Sperry et al.1998). (It also contributes to determining leaf water potential,to which gs responds in the short term (Zeiger, Farquhar & Cowan 1987; Assmann 1993). The rapid response is not modelledhere.) Accurate models of both local and distal responses have beendeveloped. Local responses are modelled with a variety of empiricalmodels, such as those of Jarvis (1976) and,more recently, of Ball, Woodrow & Berry (1987), Leuning (1995), Jarvis &Davies (1998) and others. Responses to root-sourced signalsof water stress have been described successfully with partly mechanisticmodels (Tardieu & Davies 1993; Tardieu, Zhang & Gowing 1993; Tardieu & Simonneau 1998). To date, thecombined response to both environments has only been described empirically,as by Tenhunen et al. (1990,1994). In the latter model, the effects are multiplicative,

gs     =     gs,BB     ×     gfac     +     b     ≈   gs,BBe     +     b.(1)

Here, the first or ‘Ball–Berry’ (BB)term gs,BB has the form mAhs/Cs,with A as the rate of CO2 assimilation and hs, Cs asthe relative humidity and CO2 mixing ratio at the leaf surface,beneath the leaf boundary layer. We denote the form without theslope parameter m as the Ball–Berry index. The secondterm, gfac, depends on the water potential ofleaf (ψL), root (ψR),or soil (ψS), variously chosen as deemeduseful. When expressed as an exponential function of ψR or ψs, gfac incorporatesdistal responses of gs to root environment. Theintercept, b, is typically small. This combination of responsesis not fully satisfactory. In particular, it has been shown thatthe hormone, abscisic acid (ABA) is a major proximate control ormessenger in stomatal response to water-stress (e.g. Cowan et al.1982; Davies & Zhang 1991; Tardieu et al. 1991). Thehormone IAA may also contribute (Dunleavy &Ladley 1995.) Further, in herbaceous species and tree seedlings,the synthesis of ABA is closely related to root water potential ψR (Simonneau, Barrieu & Tardieu 1998). Theultimate action of ABA may be amplified by ψL insome species (Tardieu & Davies 1993; Tardieu & Simonneau 1998). The relationof root water potential, ψR to that ofsoil depends upon water flux and soil hydraulic conductivity, afunction of both soil texture and water potential. Formulated asa conductivity to root surfaces, it also depends upon root clumping(Tardieu, Bruckler & Lafolie 1992).Both a richness in interpreting the behaviour of field-cultivatedplants and potential robust simplifications derive from these explicitlinkages.

Herein we devised and tested a combined model similar to thatin Eqn 1 but with the factor gfac resolvedas explicitly ABA-dependent. For anisohydric species such as sunflower,this is well approximated with the simple form exp(– β[ABA]),with β as a parameter that is species-specific (at least)and [ABA] as the concentration of ABA in the xylem sapdelivered to the leaf. We analysed two data sets on sunflower (Helianthusannuus L); one data set was used to test the ABA model on sunfloweras an anisohydric species (Tardieu, Lafarge &Simonneau 1996), and a new set was obtained in 1999 withcloser attention to sampling and precise measurement of aerial conditions.In each data set, we measured both [ABA] and theaerial environmental variables needed to compute the Ball–Berrybehaviour. While testing this particular combination model, we alsotested alternative models of aerial responses (Leuning1995). These models reflect more directly (but not alwayswith greater quantitative accuracy) the origin of leaf responsivenessto humidity as a response to epidermal transpiration rate (Dewar 1995; see ‘Discussion’).We also tested an alternative form in which aerial and ABA responsesare not multiplicative; rather, gs is the minimumof BB and ABA responses. The most accurate form is found to be gs = gs,BB gfac([ABA]).This has implications for potential signalling mechanisms withinthe leaf. We discuss these, along with the more complicated signallingin woody plants and the possible sensing of plant hydraulic conductancein the control of stomatal conductance.

Materialsand methods

Plantsand growth conditions

Fieldexperiments (1994 and 1999)

Sunflower plants (Helianthus annuus, hybrid Albena) were sownon 6 May 1994 and 17 May 1999 in a field with a deep clay–sandy–loamsoil near Montpellier, France. Part of the field was irrigated with18 mm of water every third day, whereas the rest was not.Plants located at the boundary between irrigated and non-irrigatedzones received intermediate irrigation depending on their distancefrom the sprinklers. Measurements were performed in July while the flowerswere opening on plants of the three zones in 1994 and in the intermediatezone in 1999.

Glasshouseexperiments (1994)

Plants of the same hybrid (Albena) as in the field experimentswere sown in a glasshouse in Montpellier on 5 May and 15 June 1994,in 6·3 dm3 pots. Soil was maintainedat retention capacity by daily irrigation (Hoagland N/10nutrient solution) until experiments. Irrigation was then withheldand plants were sampled on the following days at various levelsof soil dehydration. Measurements were performed in late June (forthe first sowing date) and late July (second sowing date), whenflower buds had appeared.

ABAfeeding in the field and in the glasshouse (1994)

ABA solutions at concentrations ranging from 0·005 to 1 mol(+)-ABA m−3 were fed to well-wateredplants from both the glasshouse and the field as previously described (Tardieu et al. 1996). Briefly,synthetic (±)-ABA (Lancaster synthesis) was dissolved indegassed artificial sap, poured into a funnel sealed around thebasis of the stem, and allowed to enter the xylem stream via a 1 mmperforation drilled in the stem below the surface of the artificialsap.

Measurementsof aerial environment and of plant variables

In the 1994 field experiment, air temperature, relative humidityand mean windspeed were measured every hour (CE 184, CE 190 andCE 155 sensors; Cimel, Paris, France) at a meteorological stationlocated 500 m away from the field. In all other experiments,windspeed (A100R; Campbell Scientific Ltd, Shepshed, UK), air temperatureand relative humidity (HMP35A; Vaisala Oy, Helsinki, Finland) weremeasured at 2 m above the soil surface every 20 s, averagedand stored in a data logger every 600 s.

Gas exchange of one of the four youngest fully developed leaveswas measured at different times of the day using a ventilated closedcuvette coupled to a gas-exchange analyser (LI-6200; LI-COR, Lincoln,NE, USA; volume, 1 dm3). In the field, measuredleaves were chosen either at the top of the canopy, or deeper inthe canopy where they were shaded by upper leaves. In 1999, a shadingshelter (3 m × 3 m)was installed at approximately1·5 m above the topof the canopy, allowing for the upper leaves to be measured witha 30% reduction of photosynthetic photon flux density (PPFD).One-third of the measured plants were located below this shelter.Immediately after stomatal conductance to water vapour, gs,was measured, the leaf was excised and enclosed in a plastic bagfor measurement of leaf water potential using a pressure chamber.Then, approximately 150 mm3 of sap were extractedfrom the same leaf by increasing the pressure to about 0·5 MPa abovethe balancing pressure. The sap was stored at −80 °C pendingABA analysis. ABA concentration was analysed in crude samples ofxylem sap by radio-immunoassay (Quarrie et al.1988) as previously described (Barrieu& Simonneau 2000). The specificity for ABA of the monoclonal antibody(MAC 252, provided by Dr S.A. Quarrie, Cambridge Laboratory, JohnInnes Centre, Cambridge, UK) was verified in xylem sap by comparinga radioimmunoassay (RIA) of crude sap samples with RIA of sap fractions recoveredfrom thin layer chromatography (Tardieu et al. 1996).

Calculationof components of the Ball–Berry index

We calculated A, hs and Cs fortwo different conditions: those occurring in the cuvette duringgas-exchange measurements (superscripted as ‘gx’)and those projected to occur before the emplacement of the cuvette(superscripted ‘0’). The gas-exchange system reported Agx directly.We had to compute hs, which is defined as theratio of two partial pressures of water vapour, in air at the leafsurface and in the leaf interior, es/ei.The value of ei can be well approximated as thatfor pure water at the leaf temperature, TL; thesmall correction factor for leaf water potential, exp(ψ V/RT)(Nobel 1991) can be ignored. The valueof es was obtained by equating two expressionsfor transpiration rate,

gs(ei  −  es)  =  gb(es  −  ea)(2)

with ea being the partial pressure in freeair outside the leaf boundary layer. The conductance of this layeris denoted as gb. One can readily obtain


The gas-exchange system reported eagx and TLgx,and it was routinely used to estimate gbgx undercuvette conditions according to the manufacturer's directions(Li-Cor, Inc. 1990). The value of Cs wasapproximated closely as:

Cs     =     Ca     −     A     ×    1·37/gb(4)

with Ca as the mixing ratio of CO2 infree air.

If stomatal conductance changes slowly, the Ball–Berry indexshould be computed using initial conditions just before    leaf    emplacement    into    the    cuvette.    Although    gs is   assumed   unchanged,   leaf   temperature,   external   CO2 and   ea  and   gb  all   change.   One   must   recalculate   A0,   hs0 and Cs0.The contrast of initial and gx conditions is not marked; thus, thecalculations are omitted here; they are detailed    in    a    ­document    available    on    our    Web    page,

Dataquality control

Some ranges of data were eliminated from analyses. Specifically,we excluded data points for which the CO2 mixing ratiowithin the cuvette of the gas-exchange system was below 250 or above380 µmol mol−1.At both extremes, the leaf environment has been changed markedlyfrom free-air conditions. It is not certain how rapidly the stomatalconductance re-equilibrates under all conditions, so we do not feelconfident in using these points. Their inclusion does not changeany of our conclusions, while adding a small additional variance.In addition, some individual data points were excluded. First, transportphysics demands that Eqn 4 be satisfied.We rejected data for which the value of Ci reportedby the gas-exchange system differed by more than 10 µmol mol−1 fromthe value calculated with Eqn 4. Such deviationscan occur when a bolus of air of very different composition is suddenlyintroduced into the cuvette, or if mixing is poor, etc.


Structureof the data sets

In all cases, measurements of gs, the stomatalconductance for water vapour were coupled with analysis of the ABA concentrationin xylem sap of the same leaf. From primary measurements of gasexchange, we have the variables needed to estimate with good precisionthe components A, hs and Cs ofthe Ball–Berry model, both as values occurring during gas-exchangemeasurements (superscripted as ‘gx’) and as valuesoccurring in free air just before measurements began (superscriptedas ‘0’). More precise and time-intensive measurementsof pre-measurement conditions were not practical. The same calculationmethods (described in ‘Materials and Methods’)can also be used to calculate vapour-pressure deficits from leafto air, D = ei − ea,that are used in the model of Dewar (1995).In the earlier data set from 1994, four treatments were included: field;greenhouse; greenhouse with transparent chambers to reduce air circulationand raise humidity around the leaf; and injection of plants withABA. The four treatments showed no significant differences in behaviourand are considered as a group with 193 valid members (all variables measured;CO2 within acceptable limits during gas exchange). Thesecond data set from 1999 contained 145 valid members, obtainedon four separate days of experimentation. In contrast to the 1994data set, all data were obtained under field conditions. Furthermore,the treatments cover a smaller range of stress (smaller range of [ABA])but a wider range of irradiance and in more detail.

Initialtests: forms of local-response models

We have noted the form of the Ball–Berry model, following Eqn 1 Its possible limitations are analysedin the ‘Discussion’. Two other forms were alsotested. The first, proposed by Leuning (1995),differs principally in posing the humidity response in terms ofthe vapour-pressure deficit from leaf to air, D:


The parameter D* is adjusted for the best least-squaresfit. The factor Csx may be taken variously as Cs itselfor as Cs − Γ.We found negligible differences among the two forms for Csx inour data sets and used Cs only in results reported herein.This model has been analysed by Dewar (1995). Lhomme et al. (1998) havealso shown that it effectively incorporates the responsiveness of gs tothe transpiration rate that was found by Mott& Parkhurst (1991) and Monteith (1995).The actual responsiveness is to epidermal transpiration rate (Bunce 1996), which commonly parallels whole-leafrate. The empirical fit is observed to vary with leaf temperature(Mott & Parkhurst 1991). Matzner& Comstock (2001) provide an interpretation in termsof leaf hydraulic conductance. The net response to transpiration andtemperature does not scale exactly as relative humidity (Matzner & Comstock 2001) as in the Ball–Berrymodel, but is often close.

An older empirical form (Jarvis 1976; Stewart 1988; Noilhan &Planton 1989) also uses the vapour-pressure deficit but inthe numerator:


Without incorporating a factor for stress response, we wouldnot expect a strong fit to the data, although ABA-induced stomatalclosure appears to induce the local response (lower gs reduces A and hs)and thus mimics an aerial response. The local environment and thus A, hs orD and Cs changes upon inserting a leaf into thegas-exchange cuvette. It is possible (see ‘Materials andMethods’) to estimate the original environment from thegas-exchange environment. If stomata respond rapidly, one may expectthat the use of gas-exchange environment (‘gx’)is more appropriate than the original environment (‘0’)and will yield a better statistical fit to the data. We tested theuse of both environments in composing the indices in the Ball–Berry, Leuningand Jarvis models. The differences were minor, with the exceptionof the 1999 data analysis with the Leuning or Jarvis models (Eqns 5a & 5b); r2 improvesfrom 0·551 to 0·653 and from 0·637 to0·691, respectively, upon using initial leaf conditions.For the other four combinations of years and models, use of initialconditions reduces r2 by an amount ranging from −0·022to −0·078.

Table 1 showsthat the Ball–Berry model with gx conditions is appreciablysuperior to the Leuning and Jarvis models, for both data sets. Thefits of both the Leuning and Jarvis models to the 1999 data werenotably improved by using estimated initial environmental conditions.At best, the Jarvis model with such estimates was slightly superiorto the Ball–Berry model using conditions during gas exchange.We have similarly found moderate statistical superiority of the Ball–Berrymodel over the other two models in diverse woody species in a mixeddeciduous-coniferous forest (Gutschick et al. inprep.) and in two riparian tree species in New Mexico, USA (Catalan-Valencia et al. inprep.) It was not possible to make a firm selection among the local-response modelsbased on the comparisons here. However, the Ball–Berrymodel was always in the acceptable group; thus, we used this modelin the combined or ‘BB + ABA’ model,in the next section. In the case of the Leuning and Jarvis models,the goodness of fit was rather insensitive to the value of D*.Commonly, a variation of 50%, relatively, changed r2 byabout 0·020. The slope, m, in the Ball–Berry model,differed significantly between the 1994 and 1999 data sets; seethe ‘Discussion’ below. For all models, the fits forthe 1994 data set were always markedly worse than for the 1999 dataset. This is expected, because several aerial variables were estimatedby less accurate methods in the 1994 data set. In particular, allvariables are given as their averages over three observations ratherthan as the most reliable single observation, as used in the 1999data set. This is typically the second observation, after changesin CO2 in the cuvette have stabilized but before assimilationhas decreased CO2 as radically.

Table 1.  Comparisonof goodness-of-fit of three models of stomatal conductance to watervapour, gs, as responding to the aerial environment
YearModelSlope (m)Intercept (b)(mol m−2 s−1)D * (Pa)r2
  1. Only data with low ABA concentrationsin the xylem sap (< 100 nm)were used (127 data points for 1994, 136 for 1999). Note that thescaling parameter D*doesnot occur in the Ball–Berry model. Values of the coefficientof determination, r2, in boldface are superioramong the three alternative models and exceed 0·5; valuesin italic are acceptable (> 0·5) or, if less thanbut close to 0·5, are superior among the three models.

Leuning18·190·478  500·375
Jarvis18·010·49  800·377

Testsof the multiplicative model, BB × ABA

Our combined model proposed that ABA modifies the Ball–Berryslope downward from its initial value, mi:

gs   =   gfacmiAhs / Cs   +   b =   e β [ABA]miAhs / Cs   +   b(6)

We assumed that the Ball–Berry model is good for describingthe local response, as discussed above. The form of ABA responseis appropriate to anisohydric species, of which sunflower is a representative(Tardieu & Simonneau 1998). The simplesttest of the combined model is a linear regression of gs againstthe new index, gfacAhs/Cs,also denoted as gfacIBB. Weneed to find the optimal value of β that maximizes r2 forthe linear regression.

Table 2 showsthe results, using cuvette conditions for the local environmentin composing the Ball–Berry index. The combined model wasquite successful, explaining nearly 80% of the variancein gs, in both the 1994 and 1999 data sets. The numericalvalues of the Ball–Berry parameters are well prescribedby non-linear least-squares fitting, as shown by the small standarderror. The 95% confidence interval (Wald) in slope m is8% (10%) above and below the mean for 1994 (1999)data. The error bounds for intercept b are larger on a relativescale but change gs itself by only about 10% orless under average leaf conditions. The ABA response parameter, β,has a larger relative error, which would affect calculation of gs morestrongly under strong water stress. Inclusion of more high-stressdata points in determining β would probably help,as might a search for a more robust functional form for the ABAresponse; see the Discussion. Figure 1 (opencircles) shows that there is no significant systematic bias in thefit according to range of gs in the 1994 or 1999data; the few outliers do tend to be over-predictions at high gs.There are some subtleties in comparing this performance with thatof the Ball–Berry model alone or the ABA model alone. Mostsignificantly, the Ball–Berry model should not be appliedto the entirety of either data set. Those data points obtained withhigh [ABA] include a partial mimicking of theBall–Berry response by ABA, which reduces gs,in turn reducing both A and the Ball–Berry index. Thus,we offer the limited comparison from Table 1 earlier.There, the Ball–Berry model alone explains only about 47and 67% of the variances in 1994 and 1999, respectively.The ABA model alone explains 58% of the variance in 1994(when ABA dominated in the control of gs).and8% in 1999. It is notable that virtually the same valueof the parameter β applies to both data sets. Bothof these values require that we adjust β to be significantlyhigher than in the combined model; we shall return to this pointlater. The improvement in r2 afforded by the combinedmodel was on the order of 12 to 20%. This is significantfor the practical application of predicting gs.It is, admittedly, limited, for several reasons. Foremost, ­perhaps,   there   is   limited   room   for   improvement,   given theinherent random errors in measuring gs. The relative errorsare largest at low conductance (low signal-to-noise in added watervapour, in the gas-exchange system) or very high conductance (minorerrors in gb require large, ­artifactualchanges in gs to preserve the measured total conductance).

Table 2.  Improvedfit of gs with combined models of ABA action andBall–Berry behaviour
YearSlope (m0)Intercept (b)(mol m−2) β (nm−1)r2
  1. ABA action reduces (by the factorexp(–β[ABA]) the slope, m0, inthe Ball–Berry model, Eqn 6. The Ball–Berry portionis expressed in variables reflecting the environment during gas exchange,as in Table 1. Allparameters are determined by non-linear least-squares fitting withequal weighting. Values in parentheses following the parameter valuesare adjusted standard errors; 95% confidence bounds areclosely twice as large. All of the fits are highly significant statistically(P << 0·000001).

199428·08 (1·11)0·157 (0·054)0·0030 (0·0010)0·770
199915·46 (0·77)0·200 (0·073)0·0030 (0·0010)0·795
Figure 1.

Stomatalconductance (gs) predicted from two models ofcombined response to local environment and to root-sourced ABA signal:the multiplicative model (○) and the model of the minimumof local and ABA responses (+). Data for 1994 (a) and for1999 (b) includeall valid data with CO2 mole fraction in the range (250,380) µmol mol−1 (seetext).

Consistent with the combined model gaining explanatory powerby incorporating both the local and ABA models, we find that typicallyall the components of the combined model are statistically significant.We evaluate this on data transformed to the form

lngs = –β[ABA] + c1ln(A) + c2ln(hs) − c3ln(Cs)(7)

(One may also adjust this to ln(gs − b).)With this linearization, we may apply the simplest and most robusttests of significance of each factor ([ABA], ln(A),ln(hs) and ln(Cs)). We may alsotest that each factor in the Ball–Berry index is raisedto the appropriate power: A and hs as firstpowers and Cs as the −1 power. This wouldbe indicated by the coefficients c1, c2 and c3 beingclose to unity.

Table 3 presentsresults obtained: (1) with all the data points in each year; thisgenerates some bias toward high gs that ‘anchors’ theregression there; (2) consequently, re-sampling data randomly tohave approximately equal numbers of points in four ranges of ln(gs).Here, we chose the lower values, ln(gs) < −1(gs < 0·37 mol m−2 s−1).For these data points, the slope of ln(A) against ln(gs)appears to be nearly constant (Fig. 2);this is useful in analysing the control coefficient of gs upon A forother analyses, presented later. This subsampling yielded 59 datapoints for the 1999 data set and 48 for the 1994 data set. We alsodiscarded all data points in the 1994 data set with very large valuesof [ABA], which also ‘anchor’ theregressions in a region where: (1) the ABA effect is far beyondsaturation; (2) ln(gs) is less reliably measured;and (3) the effect of a minimal or ‘floor’ valueof gs should be included but is difficult to model.

Table 3.  Multiplelinear regression of ln(gs) against [ABA] and thenatural logarithms of A, hs and Cs
YearData setCoefficients ofr2 value
  1. The nature of the data sets forthe two different years is described in the text, as also is the mannerin which data sets were re-sampled for even coverage in ranges ofln(gs). As a subscript to each partial regressioncoefficient is its corresponding Students’t-value. Coefficientsthat are statistically significant at the P = 0·05level are indicated in italics; those that are significant at the P = 0·005level are in boldface. **P < 0·005.

1994All data0·00037(−3·03)0·628(8·80)2·643(15·92) −0·404(−0·56)0·848 * *
Re-sampled −0·00108(−3·72)0·886(7·78)2·429(7·63) −0·275(−3·41)0·887 * *
1999All data0·00091(−2·35)0·582(8·53)2·235(15·64)0·102(0·46)0·891 * *
Re-sampled −0·00003(−0·13)0·669(5·34)3·093(8·48) −0·654(−0·96)0·837 * *
Figure 2.

Nearlylinear relation between logarithms of A[CO2 assimilationrate (µmol m−2 s−1)] and gs[ stomatalconductance (mol m−2 s−1)],for 1999 data set, over entire range of both variables.

Table 3 shows,first, that β in the ABA response is statisticallysignificant (∣t∣ > 1·8,yielding P < 0·05) for allcases except the full 1999 data set, which is dominated by low-[ABA] datapoints, giving an unreliable fit. Second, the factors A and hs arealso highly significant statistically. The coefficients of ln(A)are near 1, but those of ln(hs) are always greaterthan 1, in the range 2–3. This discrepancy is revisitedin the Discussion. The coefficient of ln(Cs) isnot statistically significant; very small variance in Cs issampled in either data set, because measurement conditions were nearambient CO2 levels.

Consistent with the ABA effect being amplified by the local response,increasing levels of ABA appears to decrease the Ball–Berryslope progressively. We may partition the data into smaller rangesof [ABA], or, similarly, into ranges of gfac = exp(– β[ABA])for gfac in the range 0·5–1·0,the effective slope is 26·54 in the 1994 data. For progressivehalvings of the gfac range (0·25–0·5;0·125–0·25; 0·0625–0·125),the slope decreases to 18·84, then 9·67 and finally7·74. One might similarly use multiple linear re­-gressionon ln(gs) = a ln(IBB) − β[ABA] + b toexpress this effect (see later), although the functional effectis less readily comprehended (and the data are ln-transformed, changingthe statistical fit). The 1999 data do not have ­adequatesamples in high-[ABA] ranges for a comparable analysis.

It appears that the best fit to data in the combined model requiresa value of ABA-responsiveness, βdirect = 0·003 nm−1, thatis only about one-half as large as in the model using only the ABAresponse (βtotal = 0·006 nm−1).We interpret this as the local response causing an amplificationof the ABA effect. That is, the ABA-induced decreases in leaf assimilation A andin leaf-surface relative humidity hs induce theappropriate Ball–Berry-like response in gs.A simplified model of A and of hs supportsthis interpretation. Let us resolve a direct effect of ABA upon gs,as a factor G (a ‘direct-only’gfac)multiplying the BB slope. The total effect of ABA, including thefeedback multiplication via the local response, is proposed to bea power of the direct effect: gs ≈ gs0 Gn,as a change from the original con­ductance gs0.Similarly, assimilation is proposed to change as a power of gs itself: A ≈ A0(gs/gs0)C. Wealso take surface relative humidity as a power function, hs ≈ hs0(gs/gs0)Q. Then,in the Ball–Berry model of local response, we may write

gs   ≈  (m0G)(A0 Gn*C)(hs0 Gn*Q)  =  (m0 A0 hs0)G(1  +  n*[C  +  Q])  =  gs0G(1  +  n*[C  +  Q])((8a))


Gn     =     GGn * C   Gn * Q

This yields simply that

n     =    1  +  n(C  +  Qn  =  1/(1  −  [C  +  Q])((8b))

The amplification factor, n, is expected to vary withthe range of gs or, equivalently, [ABA].The factor is larger if C and Q are larger and thisoccurs at small values of gs. We may see thisby approximating carboxylation kinetics with a model having tworesistances, 1/gs and an effective biochemicalor mesophyll resistance, 1/g. Taking the ambient partialpressure of CO2 as Ca, one obtains A = gm gs Ca/(gs + gm).The exponent C is simply the logarithmic derivative,   ln(A)/∂  ln(gs),which equals 1/(1 + gs/g).It is large when gs is small.

By using the whole data set, we average these effects and obtainmean values of C, Q and n. We must determine if these valuesare consistent with the apparent value n ≈ 3. Forthis, we obtain the exponent C as the slope of ln(A) againstln(gs), and analogously Q as the slopeof ln(hs) against ln(gs). Equation 8b enables us to predict the amplificationfactor as npred. The apparent or observed amplificationfactor, nobs, is the ratio of the βtotal to βdirect.We obtain βtotal via single regressionof ln(gs) against [ABA], and βdirect viamultiple regression that includes ln(A), ln(hs)and ln(Cs) in addition to [ABA].For both data sets, each regression was done in two ways: (1) withall valid data points (A > 0, so that ln(A)is defined); and (2) re-sampled to put approximately equal emphasison different regions of ln(gs), as described earlier.The predicted and observed multipliers are comparable, as shownin Table 4. Thevalues of C are consistent with a coupled model of assimilation(Farquhar, von Caemmerer & Berry 1980)and CO2 transport, for a mean gs ofapproximately one-fifth the maximal value of gs (modeldetails not shown; they are readily derived). The values of Q arespecific to the boundary conditions for radiative and convectiveheat transfer and cannot be estimated with a single model.

Table 4.  Apparentamplification of ABA action on gs by itsaction to reduce assimilation A and surface relative humidity hs,thus reducing the Ball–Berry factor
  C, in fitA ∝ (gs)C.Q, in fiths ∝ (gs)Q.Amplification factor
Predicted,n = 1/(1 − C − Q)Observed βtotal/βdirect
  1. Same data points are used as for Table 3. The power-lawdependences of A and hs are analysed asdescribed in the text. Predicted amplification factor is from Eqns 8a and 8b in the text. Apparentcoefficients of ABA response, βtotal and βdirect arederived by fitting stomatal conductance gs toABA response only or as part of ABA + Ball–Berryresponse, respectively. Dashes indicate that an analysis is notvalid, because βdirect isnot statistically significant for this data set.

1994All data0·4120·1631·72·6 = 0·00390/0·00150
Re-sampled0·4760·1322·62·8 = 0·00320/0·00115
1999All data0·5660·1413·37·7 = 0·00177/0·00023
Re-sampled0·6840·1225·2 –   =  0·00125/ –

Alternativemodel: gs as the minimum of ABA and Ball–Berryresponses

An alternative coupling may be proposed, as

gs     =    min[mAhs/Cs  +  b, gs,maxe − β[ABA]  +  gs,min](9)

The alternative model is in the spirit of a transition between twolimiting factors or two controlling factors. It contrasts with themultiplicative model, which suggests a sequential biochemical actionof ABA and some internal factor corresponding to the Ball–Berry ‘signal’.Transitional models are common, as in the widely applicable modelof photosynthetic carboxylation (Farquhar et al.1980 ff.). Commonly, they incorporate a smoothing of thetransition with a convexity parameter (Collatz et al.1990.). At present, we have little direct informationon the biochemistry of stomatal control (see ‘Discussion’),and thus, no inherent preference for the multiplicative model overthis model of the minimum, as we will call it. We tested the goodnessof fit of this model to the same two data sets, with varied choicesof m and of gs,max. We let b = gs,min bethe intercept of a regression incorporating the remaining factors.The fit was good, but not as good as that for the multiplicativemodel detailed above. For the 1994 data set, the optimal values m = 28, gs,max = 1·6yielded r2 = 0·739,compared with 0·770 for the multiplicative model. For the1999 data set, the optimal values are m = 16, gs,max = 2·0,giving r2 = 0·791.These regression coefficients are nearly as good as those for the multiplicativemodel. Thus, the choice between the two models is not clear on thebasis of statistics. Neither is a functional basis readily apparent. Figure 1 indicatesthat the two models show similar patterns of goodness of fit accordingto range of gs. A modest distinction is that the modelof the minimum (symbol ‘+’) shows somesystematic convexity at low gs, in the fit to1994 data.


Practical,empirical prediction of stomatal conductance

There has been significant discussion in the literature on theability of various empirical models to predict stomatal conductanceunder broad ranges of environmental conditions and for diverse plantspecies. The most accurate and most appropriate form for the localresponse has been debated (Aphalo & Jarvis1993). The Ball–Berry form in particular has beencharacterized as a correlation rather than as a mechanistic equation.This is certainly true, while also true of all existing models.Some mechanistic aspects of stomatal control have been clarified,particularly that: (1) the stomata respond to internal CO2 partialpressure and not to that at the surface (Mott1988); (2) they respond to transpiration and more specificallyto epidermal transpiration, as noted in the Introduction; a separateresponse is to leaf temperature (Matzner &Comstock 2001); (3) modification of PEP carboxylase enzymeby a kinase is involved (Du, Aghoram &Outlaw 1997); (4) there appear to be separate control loopsresponding to humidity and to assimilation rate (Santrucek& Sage 1996); (5) a number of elements in the molecularsignalling cascade for ABA responses are known, whereas the ABAreceptor itself is elusive; the same is true for direct sensingof osmotic stress independent of ABA (Luan 2002).However, no mechanistic synthesis is anywhere near at hand (Assmann 1999). Consequently, we need useempirical models. Accuracy of a chosen model in representing a specificdata set is a strong criterion, although not a sole criterion. TheBall–Berry, Leuning and Jarvis models variously are superiorfor specific data sets.

Ability to estimate behaviour for partially characterized speciesand conditions is another significant criterion, for example, inclimate modelling, in which large regions are poorly characterizedphysiologically or not at all (e.g. Sellers et al.1996).We propose that the Ball–Berry model is useful,in that it requires only two parameters and these are commonly stable.The slope, m, is often near 10 among diverse C3 speciesin various biomes (Ball et al.1987; Leuning 1990; Collatz et al.1991; de Pury 1995; Schultz& Lebon 1995). We note here that there is a significantdifference in unstressed slope, m0, between ourtwo data sets (1994, 1999) that is not readily explained other thanby unmeasured or unanalysed differences in growth conditions. However,both slopes are so large that total (stomatal plus boundary-layer)conductance is dominated at low stress by the boundary layer andis accurately predicted with either value of slope.

It is worth noting that the Ball–Berry index is computed usingthe measured value of gs itself (see Eqn 3 for hs).Thus, the Ball–Berry equation may be regarded in one senseas an implicit equation for gs, computationally,even if hs approximates an independent mechanisticdriving variable. In any event, the use of simple regression of gs on IBB may beregarded as including an artifactual correlation. One may removethis artefact by recasting the Ball–Berry equation as aquadratic in gs and solving for gs,then doing non-linear regression of gs on theremaining environmental ­variables. We may write


One may then multiply both sides by [1 + gs/gb] andgather terms into a quadratic equation, which has the formal solution


with A* = 1/gb2, B* = 1 − b/gb − mA(Csgb), C* = b + mAha/Cs, ha = ea/ei

One may then regress gs against the right-hand-sideof Eqn 11. For the 136 data points of 1999at low [ABA] (< 100 nm;negligible ABA control; best test of the Ball–Berry response),one obtains a fit with r2 = 0·618.This may be compared with a fit to the unmodified Ball–Berryequation, which yields r2 = 0·676.

The statistical artefact may thus be termed modest. The predictionof gs from environmental conditions would still employthe full Ball–Berry equation, with modestly reduced confidence.It may be noted that alternative models of the local stomatal responseare also implicit in a formal sense. For example, in predicting gs,the vapour-pressure deficit in Eqns 5a and5b is computed using leaf temperature, which is computedin turn using estimated gs in the energy-balanceequation. This only means that feedbacks are operating; the approximaterepresentation of feedbacks differs among the Ball–Berry,Leuning and Jarvis models. The combined model, using the local andstress (ABA) responses multiplicatively, has high accuracy, with valuesof r2 near 0·8, even for data sets thatwere not designed for accurate resolution of both local environment andABA concentration. The gain in explanatory power, as r2,over a model with only the ABA response is significant. The gainover a model with only the Ball–Berry or similar localresponse is modest, but the local models are not useful predictorsin stressed conditions; foremost, the values of A, hs and Cs arestrongly shifted by stress and cannot be computed from the localresponse alone. In explanatory power (as r2),the multiplicative model is superior but not definitively superiorto the alternative model of the minimum. Consideration of both models’ relativesuccess may help to guide mechanistic studies at the biochemicallevel, in seeking to resolve parallel versus serial action whenlocal signals combine with signals from ABA (and IAA, pH and K+).Of course, our model is not intended to resolve rapid, hydraulicresponses to short-term changes in water status (Zeiger et al.1987; Assmann 1993).

The practical use of our combined model, in either the multiplicativeor minimum form, merits some discussion. First, the algorithm iscompletely defined for computing gs in     any     combination     of     local     environment     (irradiance, airtemperature, etc.) and soil environment [soil water ­content   and   texture,   thus,   its   water   potential   and   hydr­-auliccon­ductivity; rooting density and clumping factor (Tardieu et al. 1992)].The xylem ABA concentration is ­computed in this process,adapting the methods of Tardieu &     Davies     (1993).     The     complete     mathematical     model, asa narrative description and as a Fortran program, is available  from  author  Gutschick  and  on  the  Web  site full model includes numerous processes and their parameters,but is robust in the sense that parameters are readily measuredor can be estimated for broad classes of plants and soils. Second,one must determine the time scale on which the local environmentis defined. Does the leaf respond to the irradiance and other variableson a short and essentially instantaneous time scale, or to a (weighted)average over a longer time scale? All stomatal models face sucha challenge. For sunflower here, we find a rapid response, suchthat conditions during the 1·5 min of gas exchangeare better predictors than initial conditions before the measurementsbegan and perturbed the environment. In woody species, the timelags are commonly substantial. Ideally, a fully dynamic model is merited.Dynamic models of response to the local environmental alone havebeen made, for limited plant species and functional types (Gross, Kirschbaum & Pearcy 1991; Kirschbaum et al. 1998).They are most relevant for examining adaptations to contrastingenvironments, such as forest understories. For predicting totaltranspiration or assimilation of plant canopies, it is more practicalto use effective (dynamic average) parameters in our existing model.

Othertheoretical and conceptual implications

Our analyses imply an amplification of ABA action by the stomatalresponse to the local environment. Specifically, ABA induces partialclosure, which decreases both assimilation and surface humidity.These decreases further reduce gs, acting throughunknown signals. From one point of view, this appears counterintuitive.A decrease in internal CO2 partial pressure, Ci,typically induces stomatal opening (Mott 1988).This would counteract and not amplify ABA-induced stomatal closure.Amplification would require that stomatal controls responding to A (or Ci)and to ABA act in series rather than in parallel.

Santrucek & Sage (1996) have analysedstomatal control in terms of two feedback loops, one respondingto Ci directly and another responding to A,which is clearly affected by Ci. In Chenopodiumalbum, the gain of the first loop, gs, wasfound to be near −0·15 at ambient CO2 partial pressures.The gain of the A-responsive loop, GA,was of larger absolute magnitude, near −0·25.If one posits that the Ball–Berry model is an accuratedescription of gs, the same mathematical calculationsindicate a small response to Ci directly (Gs ≈ −0·05)and a larger response to A (GA ≈ −0·2). Suchbehaviour is consistent with our proposed amplification, if theABA control loop is in series with the A-responsive loop. The dropin Ci is less important than the drop in A,such that stomatal closure can be enhanced. It is also worth notingthat other empirical models such as that of Leuning(1995) have similar patterns of feedback-loop gains.   They   differ   from   the   Ball–Berry   model   primarily inthe gain of a parallel feedback loop that responds to transpiration.

Jarvis, Mansfield & Davies (1999)offer another model of combined stomatal responses to assimilationand humidity. The humidity response is consistent with that proposedby Leuning (1995). The formulation of A issomewhat simplified, allowing an intriguing symmetry in A and E responses tooccur. The model does not incorporate the water-stress response,but it does consider acclimation to growth at elevated CO2.It might be profitably combined with an ABA model.

In an action that is equivalent to changing the feedback-loopstrengths, we may loosen some constraints on our model, by representing gs as mA hsQ/Cs + b,where m is again a function of [ABA] but Q neednot be unity. Our earlier analysis of ln(gs) impliedthat Q equals or exceeds 2. This result implies that eitherthe Ball–Berry model should be amended to allow hs toappear to a power significantly different from 1, or that ABA maymediate a higher-order (quadratic) response to humidity (compare Bunce 1996). Our data support the formeridea. We consider only those data with low [ABA],less than 100 nm. Even for these,a multiple linear regression of ln(gs) againstln(A), ln(hs) and ln(Cs)yields a coefficient of ln(hs) = 2·32 ± 0·14 forthe 1999 data set and 2·38 (± 0·20)for the 1994 data set. In the original Eqn 6,before logarithmic transformation, a regression of gs againstthe modified index A hs2/Cs does makea small improvement over use of the normal Ball–Berry indexwith Q = 1. For the low-ABA dataof 1999, r2 improves from 0·676 to 0·735.For the low-ABA data of 1994, the improvement is from 0·473to 0·516. In summary, the evidence for enhanced sensitivityto humidity in our system is modest, especially given the inherentlyempirical nature of the Ball–Berry formulation. It is known,however, that sensitivity to humidity (as hs orVPD) does vary with growth conditions, including CO2 level(Bunce 1998; Heath 1998).

We should also examine if the exponent C for the responseof gs to assimilation, A, is appropriatelyunity. Indicative of the challenges in representing this coupling, thecoefficient c1 for ln(A) in the regressionanalysis of ln(gs) changes in magnitude as theother driving variables (hs, Cs, [ABA])are introduced in a regression analysis and also as data with higherranges of [ABA] are included. Table 5 shows that c1 isonly near unity when ABA effects are essentially absent ([ABA] < 10 nm).As data with greater ABA content are included, the value of c1 decreases.This indicates that the multiplicative model is partially confusingsome control loops, but we have little guidance for modifying theformulation, in the absence of further biochemical studies. Table 5 shows thatthe value of c1 also varies with the completenessof the set of driving variables. This sensitivity is a normal resultof significant correlations among the driving variables. For example,a high assimilation rate is linked mechanistically to high gs, andthus to strong humidification of the boundary layer (high hs)and also to depression of Cs. A path analysiscan include the direct linkages (in which the ‘direct’ coefficient c1,d mightbe near unity with any data) and the correlations. An adequatepath analysis is unlikely to be constructed, given that: (1) mechanisticlinks and their directionality should all be known, whereas we lackinformation on some intermediate biochemistry; and (2) the mathematical ­relationsamong variables such as ln(A) and ln(Cs)are non-linear and often of the form of transcendental (non-­algebraic)equations. The results of Table 5 maythen be construed as further argument to pursue deeper biochemicalstudies of stomatal control. It is of interest that the coefficient c2 forln(hs) is very stable, near 2, independent ofthe data range and the inclusion of other variables. This adds tothe evidence that the response to hs is strongerthan in the simple Ball–Berry model. Equivalently, in modelsof gs responding to VPD, the response is strongerthan linear – for example, in the data of Leuning(1995).

Table 5.  Variationof the effective algebraic power of A in the Ball–Berryequation as other driving variables are introduced or data rangesare changed
Data rangeCoefficients ofr2n
ln(A)ln(hs)ln(Cs) [ABA]
  1. Stepwise multiple linear regressionwas performed on 1999 data, in which low [ABA] valuespredominate, allowing the Ball–Berry response to dominate.Note that [ABA] has been put as the last drivingvariable, differing from the order in Table 3. *P < 0·05; **P < 0·005.

[ABA] < 1001·33 ± 0·09**   0·614136
 2·96 ± 0·12**  0·820136
   −3·51 ± 0·36** 0·431136
0·56 ± 0·07**2·25 ± 0·13**  0·882136
0·56 ± 0·07**2·27 ± 0·15**    0·39 ± 0·23 0·882136
0·60 ± 0·07**2·22 ± 0·15**    0·12 ± 0·23 −0·00194 ± 0·000710·888136
[ABA] < 300·75 ± 0·09**1·94 ± 0·17** −0·06 ± 0·32 −0·00730 ± 0·00230*0·915  91
[ABA] < 100·92 ± 0·17**2·06 ± 0·37**    1·46 ± 0·95 −0·00680 ± 0·013400·873  23
All ABA0·58 ± 0·07**2·03 ± 0·14**    0·10 ± 0·22 −0·00091 ± 0·000390·894145


Biochemical studies are progressing and might be aided by testingthe hypothesis that ABA and aerial signals act sequentially. TheABA response appears to have additional dimensions that merit investigation.First, the pH of xylem sap appears to modulate ABA action in theleaf, probably by changing the degree of ionization of ABA and thusits ability to move freely in the apoplast (Hartung,Wilkinson & Davies 1998; Wilkinson et al.1998; Wilkinson 1999; Netting2000). Quantitative prediction of xylem-sap pH and its concurrenteffect on the ABA response is not yet possible, although progressis being made (Auge et al. 2000).Second, K+ nutrition may affect ABA action,given that K+ is intimately involved in ABAaction on guard cells (see, e.g. Assmann 1999; Netting 2000). Ecotypes that differ in K+ dynamicsmay differ in ABA responses. Third, vapour-­pressure deficitsmay modulate ABA action, or, as one may rephrase it, may act inpart through their effect on ABA concentration or delivery rate.It is certainly conceivable that changes in epidermal transpirationrates, linked to VPD, may alter the delivery of ABA to the guardcells (Wilkinson & Davies 2002).

The action of ABA in some trees is similar to its action in herbaceousplants (Niinemets et al. 1999),in that gs responds to xylem [ABA].In other trees, the sensitivity of gs varies withleaf water potential, in a manner that could be consistent withthe isohydric model of Tardieu & Simonneau(1998), but the analysis is not complete (Fuchs& Livingston 1996; Correia et al.1997). Redistribution of ABA stored within leaves may controlgs. This independent route of action would require amajor extension of the latter model.


V.G. gratefully acknowledges the support of a travel grant asa supplement to the Jornada Long-Term Ecological Research grant(DEB-111971) from the National Science Foundation. He also acknowledgesthe National Institutes for Global Environmental Change, a programof the US Department of Energy, for cumulative support of research duringwhich many of the ideas were developed. V.G. thanks Igr. FrancoisTardieu for hosting his visit, for extensive discussions on ABAaction, and for suggesting the model of the minimum; Phillipe Naudinfor setting up computer services, and the other staff and studentsof the ­Laboratorire d’Ecophysiologie des Plantessur Stress ­Environnemontaux for discussions and assistancein field work. The authors thank Tanguy Lafarge for use of the 1994 dataset, and two anonymous reviewers for many substantive comments.

Received 8 March 2002;received inrevisedform 7 June 2002;accepted for publication 11 June 2002