A new model of stomatal conductance is proposed which combines the essential features of the Ball–Berry–Leuning (BBL) and Tardieu–Davies (TD) models within a simple spatially aggregated picture of guard cell function. The model thus provides a coherent description of stomatal responses to both air and soil environments. The model also presents some novel features not included in either the BBL or TD models: stomatal sensing of intercellular (rather than leaf surface) CO2 concentration; an explanation of all three observed regimes (A, B and C) of the stomatal response to air humidity (Monteith Plant, Cell and Environment 18, 357–364, 1995); incorporation of xylem embolism; and maintenance of hydraulic homeostasis by combined hydraulic and chemical signalling in leaves (in which leaf epidermal hydraulic conductivity plays a key role). Significantly, maintenance of hydraulic homeostasis in the model does not require a direct feedback signal from xylem embolism, the predicted minimum leaf water potential being independent of xylem hydraulic conductivity. It is suggested that stomatal regulation through combined hydraulic and chemical signalling in leaves and/or roots provides a general mechanism enabling plants to maintain their water potentials above a minimum value. Natural selection of the key stomatal parameters would then set the minimum potential to a specific value determined by the most vulnerable plant process under water stress (e.g. cell growth, protein synthesis or xylem cavitation), depending on species and growth conditions.
Empirical models of stomatal conductance play an important scientific role in summarizing commonly observed trends in stomatal behaviour, and thus guiding the formulation of more mechanistic models. One aim of the latter is to explain, from a mechanistic standpoint, not just the common trends but also the observed departures from those trends, with a view to extending the range of validity of empirical models. The objective of this study is to extend the range of validity of two previously published empirical models (Tardieu & Davies 1993; Leuning 1995), by combining them within a simple spatially aggregated model of guard cell function (based on Dewar 1995).
where g is the stomatal conductance for CO2 diffusion, An is the net leaf CO2 assimilation rate, Ds and cs are the vapour pressure deficit (VPD) and CO2 concentration at the leaf surface, respectively, Γ is the CO2 compensation point, g0 is the value of g at the light compensation point, and a1 and D0 are empirical coefficients.
This model encapsulates two empirical trends reported in the literature. First, through the correlation between g and An, Eqn 1 predicts that the ratio (ci − Γ)/(cs − Γ) (where ci is the leaf intercellular CO2 concentration) is largely independent of leaf irradiance and cs (Wong, Cowan & Farquhar 1979) except near the light and CO2 compensation points, and declines linearly as Ds increases (Morison & Gifford 1983; Leuning 1995). Secondly, through the relation E = 1·6 gDs for the transpiration rate, the hyperbolic function of Ds in Eqn 1 is equivalent to a linear decline of g with increasing E (Dewar 1995; Leuning 1995), consistent with the commonly observed behaviour referred to as regime A by Monteith (1995) in his review of stomatal responses to air humidity. Equation 1 can also mimic Monteith's (1995) regime B, in which E decreases at higher VPD (Dewar 1995; Leuning 1995).
The main limitation of the Ball–Berry–Leuning (BBL) model is that it does not describe stomatal closure under soil drying. Also, the factor An/(cs − Γ) is inconsistent with the experiments of Mott (1988) showing that stomata sense ci, not cs. Mott found that the stomata of sunflower (Helianthus annuus L.) and cocklebur (Xanthium strumarium L.) closed when ci was increased at fixed cs; under these conditions Eqn 1 would predict stomatal opening rather than closure. Furthermore, in principle the predicted value of g could become negative below the light compensation point (when An is negative). Finally, the BBL model does not predict Monteith's (1995) regime C, where g is relatively insensitive to VPD in very moist air.
Soil drying experiments on maize (Zea mays L) led Tardieu & Davies (1993) to formulate a model of stomatal response to soil water deficit involving combined hydraulic and chemical signalling through abscisic acid (ABA). According to the Tardieu–Davies (TD) model,
where gmin and gmax are the minimum and maximum stomatal conductances, respectively, [ABA] is the xylem ABA concentration, β (chosen here to be positive) is the basal sensitivity of stomatal conductance to [ABA] at zero foliage water potential (ψf = 0), and δ (negative value) describes the increase in stomatal sensitivity to [ABA] as ψf falls (Tardieu & Davies 1992). ABA was assumed to be produced by the roots at a rate proportional to root water potential, and diluted in the transpiration stream.
As Tardieu & Davies (1993) showed, their model explains two commonly observed and contrasting behaviours of the diurnal minimum foliage water potential (ψf,min). First, when δ < 0 the predicted ψf,min changes little with the soil water reserve or the soil-to-plant hydraulic conductivity. This phenomenon, called hydraulic homeostasis (or isohydric behaviour), has been observed in many plant species including trees (Oren et al. 1999; Magnani, Mencuccini & Grace 2000). Hydraulic homeostasis is somewhat analogous to the homeostatic behaviour of ci/cs in the absence of water stress (incorporated in the BBL model but not in the TD model). Secondly, if the interaction between hydraulic and chemical signals is absent (δ = 0), isohydric behaviour is replaced by a marked decrease in ψf,min in droughted plants (anisohydric behaviour), as observed in sunflower and other species (Tardieu, Lafarge & Simonneau 1996; Tardieu & Simonneau 1998).
It has been proposed that the homeostatic value of ψf,min in isohydric woody species lies close to the threshold potential for catastrophic xylem failure due to runaway cavitation (Tyree & Sperry 1988), reflecting an optimization strategy of maximum leaf gas exchange that would involve some loss of xylem conductivity (Jones & Sutherland 1991). In their modelling study, Jones & Sutherland (1991) also examined a second possibility, that ψf,min coincides with the onset of xylem embolism, reflecting a more conservative (cavitation-avoidance) strategy.
However, in coupling Eqn 2 to a description of soil-to-leaf water flux, Tardieu & Davies (1993) did not incorporate the possible effects of xylem embolism and assumed a fixed root-to-leaf hydraulic conductivity. Therefore, the possible role of hydraulic/chemical signals in regulating xylem embolism has yet to be explored by modellers. The relevance of root chemical signals to stomatal regulation in large woody species has been questioned, because of the long transport time from roots to the stomata (Schulze 1991). This criticism is misplaced in the case of the TD model, where the sensitivity of g to the root signal (ABA) is modulated by short-term changes in foliage water potential (Eqn 2). Nevertheless, it has been suggested that rapid stomatal closure in response to shoot xylem cavitation may instead involve hydraulic and/or chemical signals localized in leaves (Salleo et al. 2000; Hubbard, Ryan & Sperry 2001).
Therefore, a possible extension of the TD model would be to incorporate xylem embolism and the potential regulatory role of leaf hydraulic and chemical signals, as an alternative or in addition to root signals. Indeed, Tardieu & Davies (1993) emphasized that their model contained many oversimplifications, but suggested that combined hydraulic/chemical feedback regulation could provide a base for further models.
In a previous paper (Dewar 1995) the BBL model was interpreted mechanistically in terms of a simple spatially aggregated picture of guard cell function. The main objective of the present paper is to propose a more complete description of stomatal behaviour than either the BBL or TD models alone, based on an extension of the guard cell model (Dewar 1995) to incorporate a role for chemical signalling. The specific aims are:
1to combine the essential features of the BBL and TD models (Eqns 1 & 2) within a common mechanistic framework;
2to remove the shortcomings of the BBL model with respect to the factor An/(cs − Γ);
3to extend the TD model to include leaf as well as root chemical signals;
4to couple the new model to a simple water balance scheme involving xylem embolism;
5to compare the generic behaviour of the new model with that of the BBL and TD models; and
6to explore the roles played by root and/or leaf hydraulic and chemical signals in regulating leaf water potential and xylem embolism.
The structure of the paper is as follows. The next section presents the assumptions of the guard cell model, leading to a new expression for the steady-state stomatal conductance (g) in terms of net leaf CO2 assimilation (An), leaf intercellular CO2 concentration (ci), leaf xylem ABA concentration ([ABA]) and leaf epidermal water potential (ψe) (aims 1–2). Then the assumptions regarding the ABA and water balances, including xylem embolism, are presented (aims 3–4). The behaviour of the model is then examined with regard to the stomatal response to VPD (regimes A–C), and regulation of the ci/cs ratio and leaf epidermal water potential (ψe) (aim 5). Finally, the implications of the model for the stomatal regulation of xylem embolism are discussed in the light of these results (aim 6).
Guard cell model assumptions (aims 1–2)
Figure 1 depicts the overall structure of the model. All conductances and fluxes are expressed per unit leaf area. Symbol definitions and default parameter values are given in Tables 1 and 2, respectively.
Table 1. Symbol definitions and units
Definition (relevant equations)
Compartment labels: e = bulk leaf epidermis, g = guard cells, r = roots, s = soil. m2 refers to leaf area.
ABA sequestration rate (8)
mol H2O m−2 s−1
Parameter combination (7b)
Leaf xylem ABA concentration (8)
mol ABA m−3
Leaf net CO2 assimilation rate (10b, c)
mol CO2 m−2 s−1
Soil characteristic (9c)
Leaf intercellular, surface CO2 partial pressures
mol CO2 (mol air)−1
Ion diffusion rate, minimum value (5a, d)
Water vapour partial pressure deficit at leaf surface
Parameter combination (7c)
Leaf transpiration rate (6c)
mol H2O m−2 s−1
Stomatal conductance (7a)
mol CO2 m−2 s−1
Carboxylation conductance (10c)
mol air m−2 s−1
Value of g at light compensation point (1)
mol CO2 m−2 s−1
Xylem water flux (9a)
mol H2O m−2 s−1
Leaf e → g hydraulic conductivity, maximum (4b)
mol H2O m−2 s−1 MPa−1
Xylem hydraulic conductivity, maximum (9b)
mol H2O m−2 s−1 MPa−1
Turgor pressure of compartment i = e, g (3b, c)
Leaf photosynthetically active irradiance (10c)
mol quanta m−2 s−1
Leaf dark respiration rate (10c)
mol CO2 m−2 s−1
Xylem hydraulic resistance, 1/Lre
MPa (mol H2O)−1 m2 s
Soil-root hydraulic resistance, minimum value (9c)
MPa mol−1 H2O m2 s
Partial molal volume of water
m3 mol−1 H2O
Photochemical efficiency (10c)
mol CO2 mol−1 quanta
ABA sensitivity parameter (2)
m3 mol−1 ABA
ABA sensitivity parameter (2)
Ion pumping coefficient (5c)
MPa mol−1 CO2 m2 s
CO2 compensation point (1)
Stomatal mechanical coefficient (3a)
mol H2O m−2 s−1 MPa−1
Osmotic pressure gradient πg − πe (5b)
Water potential gradient ψe − ψg (4a)
ABA synthesis constants for compartments i = e, r
mol ABA MPa−1 m−2 s−1
Rate of active ion pumping into guard cells (5a, c)
Rate of ion diffusion out of guard cells (5a)
Osmotic pressure of compartment i = e, g
Parameter combination (A3.8)
Water potential of compartment i = e, g, r, s
Value of ψs at field capacity (9c)
Threshold potentials for decline in K (4b)
Threshold potentials for decline in Lre (9b)
Table 2. Parameter values. See Table 1 for definitions
Values when ABA synthesis switched on (zero otherwise).
b For variable K (K=Kmax by default). The value of Kmax is not given explicitly, but is implicit in the value of D0 (see Eqn 7c).
It has been shown both experimentally and theoretically that stomatal aperture is a linear combination of the turgor pressures of the guard cells and their adjoining subsidiary cells, with subsidiary cell turgor having a greater mechanical influence than guard cell turgor (Edwards, Meidner & Sheriff 1976; Wu, Sharpe & Spence 1985; Sharpe, Wu & Spence 1987). The present model, however, considers the relationship of leaf stomatal conductance to the mean turgor pressures of the guard cells and the bulk leaf epidermis (Dewar 1995), which are assumed to have the same mechanical influence on g (but opposite in sign):
where g is the leaf conductance for CO2 diffusion, Pg and Pe are the mean turgor pressures of the guard cells and bulk leaf epidermis, respectively, and χ is a mechanical coefficient. In contrast to the local description of stomatal mechanics involving the mechanical advantage of subsidiary cells, Eqn 3a represents a spatially averaged description. These two descriptions should be considered not as logically exclusive alternatives but rather as potentially equivalent descriptions on different spatial scales.
As shown in Appendix 1, the two descriptions may be reconciled in principle if it is supposed that the guard/subsidiary cell complex is coupled mechanically to the rest of the epidermis (cf. Meidner & Edwards 1975). Equation 3a is qualitatively consistent with the expectation that the mechanical influence of epidermal cells declines with distance from the guard cell (Stålfelt 1966). Quantitatively however, Eqn 3a represents a working hypothesis whose mechanistic justification would require a scaling-up analysis of cell pressure–volume relations across the epidermis. The main reason for adopting Eqn 3a in this study, as previously (Dewar 1995), is that it is mathematically consistent with the empirical BBL and TD models (as shown below), a ‘top-down’ philosophy that is reflected in the opening sentence of the Introduction. In contrast, Appendix 2 shows that spatially explicit models describing the known mechanical advantage of subsidiary cells, as currently formulated – specifically, the model of Haefner, Buckley & Mott (1997) which incorporates hydraulic interactions across the leaf – are not yet fully consistent with the BBL and TD models. A spatially explicit ‘bottom-up’ explanation of the BBL and TD models may require a comprehensive description of mechanical, osmotic and hydraulic interactions across the leaf (see Appendix 2 for further details). This lies outside the scope of the present study.
The turgor pressures of the guard cells and leaf epidermis are calculated from the corresponding water potentials (ψg and ψe) and osmotic pressures (πg and πe):
The epidermal osmotic pressure (πe) is assumed to be constant.
Previous studies have suggested that evaporation of water into substomatal cavities occurs mainly in the vicinity of the guard cells (e.g. Meidner 1975; Meidner & Edwards 1975; Tyree & Yianoulis 1980). This so-called peristomatal transpiration implies that the water potential difference between the bulk leaf epidermis and the guard cells is proportional to the transpiration rate (E):
where K is the hydraulic conductivity between the bulk leaf epidermis and the guard cells (Dewar 1995). In the default version of the model, K is assumed fixed at its maximum value (Kmax). However, a later section will examine the hypothesis that K declines as ψe falls, describing the effect of local leaf dehydration on membrane hydraulic conductivity. A simple ramp function will be assumed:
in which ψtK is the threshold potential at which K starts to decline, and ψxK is the value of ψe at which K falls to zero.
Guard cell solute balance
It is assumed that ions are actively pumped from epidermal cells to the guard cells at a rate denoted by πin. In addition, guard cell solutes are assumed to diffuse back to epidermal cells at a rate πout that is proportional to the difference in osmotic pressures, Δπ = πg − πe. The rate of change of the guard cell osmotic pressure is therefore
where d is a diffusion rate coefficient. In the steady state,
It is assumed that the active ion loading rate πin is proportional to the local (guard cell) free energy pool, which is itself assumed to be proportional to the driving force for the CO2 reduction cycle in mesophyll chloroplasts (in the form of ATP and NADPH). The underlying assumption here is that ATP dynamics in the guard cells and mesophyll cells involve two analogous sets of reactions running in parallel (Zeiger, Farquhar & Cowan 1987; Kearns & Assmann 1993). This assumption also forms the basis of the stomatal model of Farquhar & Wong (1984). A convenient measure of the mesophyll free energy pool is the ratio (An + Rd)/ci, where An is the net leaf CO2 assimilation rate and Rd is the dark respiration rate, An + Rd being the rate of gross photosynthesis (e.g. Thornley & Johnson 1990; p. 225, eqn 9·7e). Thus,
where µ is an empirical loading rate constant. Although πin has been interpreted here as an influx of osmotica from outside the guard cell, this does not exclude the possibility that a component of πin may involve guard cell sucrose metabolism (Talbott & Zeiger 1998). It should be stressed, however, that Eqn 5c is largely empirical and that the detailed nature of the underlying mechanisms, and their relationship to guard cell ATP, lies outside the immediate scope of this study.
It is assumed that ABA regulates stomatal conductance by stimulating the rate of ion diffusion out of the guard cells (d), and that this effect is enhanced as the epidermal water potential (ψe) falls. In anticipation of Eqn 2, an exponential response is assumed:
where dmin is the minimum diffusion rate, [ABA] is the leaf xylem concentration of ABA, β (positive value) is the basal sensitivity of ion diffusion to [ABA] at zero leaf water potential (ψe = 0), and δ (negative value) describes the increase in the sensitivity of ion diffusion to [ABA] as ψe falls. It has been suggested that the apparent interaction between foliage water potential and [ABA] reflects the effect of xylem sap pH on the partitioning of ABA between guard cells and mesophyll cells (Wilkinson & Davies 1997).
The above assumptions may be combined to derive an expression for the steady-state stomatal conductance. For turgor pressures in the normal operating regime Pg > Pe > 0, the turgor difference ΔP = Pg − Pe which determines g (Eqn 3a) can be decomposed into hydraulic (‘hydropassive’) and osmotic (‘hydroactive’) components (Eqns 3b & 3c):
Equation 6b is consistent with the empirical hydraulic feedback relationship g/gm = 1 − E/Em (Monteith 1995; regime A), and provides a mechanistic interpretation of the empirical parameters gm and Em as gm = χπin/d and Em = Kπin/d.
For example, the form of Eqn 6b suggests a possible mechanistic interpretation of Monteith's (1995) re-analysis of data from experiments by Turner, Schulze & Gollan (1985) on stomatal responses in Helianthus annuus. In Monteith's re-analysis, the fitted value of gm increased with extractable soil water content (EWC) over the range 2–8% but was approximately independent of EWC above 8%; Eqn 6b suggests that this behaviour could reflect ABA-induced changes in the ion diffusion rate d over the range 2–8% EWC. In contrast, Monteith (1995) found that the fitted value of Em increased over the entire experimental range of EWC (2–94%); this would then be consistent with Eqn 6b if the leaf hydraulic conductivity K increased with EWC over the entire range 2–94%. Variation in K with soil water content might be modelled through the intermediary of leaf water potential, as in Eqn 4b, possibly representing an effect of leaf water status on cell membrane hydraulic conductivity. The implications of a variable K will be explored in a later section.
A more detailed expression for g follows by combining Eqn 6b with the expression for the transpiration rate,
and by substituting Eqns 5c and 5d for πin and d. After rearrangement, we find
where the coefficients a1 and D0 are given in terms of the guard cell parameters by
Equation 7a−7c is the central result of the paper; the steady-state stomatal conductance is the product of two factors encapsulating, respectively, the essential features of the BBL and TD models (cf. aim 1).
It will also be noted (cf. aim 2) that in replacing the factor An/(cs − Γ) of the BBL model by (An + Rd)/ci, Eqn 7a is consistent with stomatal sensing of ci (and not cs), and correctly predicts stomatal closure when ci is increased at fixed cs (Mott 1988). The factor (An + Rd)/ci also predicts that g remains positive at and below the light and CO2 compensation points and, moreover, leads to a reduction in the number of model parameters (Rd in place of g0 and Γ). Leuning (1995) found that the values of a1 and D0 obtained from empirical fits of the BBL model were negatively correlated, a possible explanation for which is suggested by Eqns 7b and 7c in terms of variation in the mechanical coefficient χ.
Finally, note that in the new model the stomatal response to air humidity (Ds) occurs through two mechanisms inherited, respectively, from the BBL and TD models: (i) hydraulic feedback via peristomatal transpiration, giving the factor 1/(1 + Ds/D0) in Eqn 7a, and (ii) modulation of the stomatal sensitivity to [ABA] via changes in ψe (for δ < 0). In experiments by Assmann, Snyder & Lee (2000), stomatal conductance was as sensitive to Ds in ABA-deficient and ABA-insensitive mutants of Arabidopsis as in wild-type plants, appearing to rule out an obligate role for ABA in the stomatal response to air humidity, and implicating the involvement of a non-ABA-related mechanism. By combining the BBL and TD models, the new model includes both non-ABA- and ABA-related feedback mechanisms – respectively, peristomatal transpiration and sensitizing of stomata to ABA – with the former (latter) dominating when the parameter D0 is small (large). Referring to Eqn 7c, the relative importance of these two mechanisms would be affected by the leaf epidermal-to-guard cell hydraulic conductivity (K).
Aba and water balance assumptions (aims 3–4)
Tardieu & Davies (1993) considered root-synthesized ABA only. The present model allows for the possibility that ABA synthesis occurs in the roots and/or the leaves (Hartung & Davies 1991). Extrapolating the approach of Tardieu & Davies (1993), it is assumed that the ABA synthesis rates in both roots and leaves increase linearly as their respective water potentials decrease. Although a linear relationship is the simplest assumption here, and a non-linear relationship may be more realistic (Pierce & Raschke 1981), the precise form of this relationship does not affect the main conclusions of the present study.
In the steady state, the leaf xylem ABA concentration is the balance between total ABA synthesis, dilution by the water flux (Jw), and an effective ABA sequestration rate (a):
where λr and λe are root and leaf ABA synthesis coefficients (positive values), and Vw is the partial molal volume of water. This formulation allows for the possibilities that ABA is synthesized by the leaves only (λr = 0, λe > 0, the default case), by roots only (λr > 0, λe = 0), or by both (λr > 0, λe > 0) (cf. aim 3). Equation 8 gives the steady-state value of [ABA]; this may be a reasonable approximation to the dynamic value for the case of leaf-synthesized ABA (λr = 0, λe > 0), but will be less accurate in the case of root-synthesized ABA (λr > 0) on account of lags due to the transport time from roots to the stomata (days for some trees). Such dynamic effects lie outside the scope of the present study.
The soil-to-leaf water flux (Jw) is described by an Ohm's law analogue:
where ψs is the soil water potential, and Rsr and Rre are the soil-to-root and root-to-leaf epidermis hydraulic resistances, respectively. The value of Rre is assumed to increase as xylem vessels become embolized (cf. aim 4). Following Jones & Sutherland (1991), a simple ramp function is used to describe the relationship between the xylem hydraulic conductivity Lre = 1/Rre and leaf epidermal water potential (cf. Eqn 4b):
in which Lmax is the maximum root-to-leaf hydraulic conductivity, ψtL is the threshold potential at which Lre starts to decline and ψxL is the value of ψe at which Lre falls to zero. Sperry et al. (1998) have discussed a more realistic description of xylem embolism, which could also be implemented in the present model. Equation 9b is adopted here for simplicity. As discussed below, the conclusion of the present study concerning the stomatal regulation of xylem embolism (aim 6) does not depend on the particular function used in Eqn 9b.
where Rsr,min and ψsmax are, respectively, the minimum soil hydraulic conductivity and maximum soil water potential at field capacity, and b is a dimensionless constant usually lying in the range 2 < b < 18 and depending on soil type (b= 2 for sand, b= 18 for clay). The root water potential is then given by
The steady-state solution of the coupled stomatal-xylem system involves finding five unknowns: g, An, ci, [ABA] and ψe. The central relationship between these variables is given by the stomatal conductance model (Eqn 7a). A second relationship is given by the expression for [ABA] (Eqn 8), in which Jw and ψr are related to ψe via Eqns 9a, 9b and 9d. A third relationship is given by the steady-state balance between the rates of xylem water transport and leaf transpiration (Jw = E) which, from Eqns 9a and 6c, gives:
Two further relationships are required. These are given by the CO2 diffusion equation (the ‘CO2 supply function’):
and the biochemical relationship between gross photosynthesis (An + Rd) and the intercellular CO2 concentration (ci) (the ‘CO2 demand function’). Among the numerous photosynthesis models available, I chose the simplest for present purposes (Thornley & Johnson 1990; Eqn 9·7i):
where α is the photochemical efficiency, Q is leaf photosynthetically active irradiance, and gx is the carboxylation conductance. Equation 10c ignores photorespiration. The conclusions of the present study do not depend significantly on the particular form of the CO2 demand function, and alternative forms could be used (e.g. Farquhar, von Caemmerer & Berry 1980; von Caemmerer & Farquhar 1981).
The steady-state solution may then be obtained by numerically solving Eqns 7a, 8 and 10a−10c, although some analytical simplification is possible under the approximation that Rd is negligible compared to An (see Eqn 11 below).
Water supply and demand curves
The steady-state solution may be represented graphically (Fig. 2a) as the intersection of two curves describing the dependencies of, respectively, the xylem water flux (Jw, the ‘water supply curve’) and the leaf transpiration rate (E, the ‘water demand curve’) on epidermal water potential (ψe). The point of intersection of the water supply and demand curves represents the steady state where Jw = E (Eqn 10a). The water supply and demand curves can be constructed by expressing each side of Eqn 10a in terms of ψe alone (i.e. after eliminating the other unknowns g, An, ci and [ABA] using the remaining four relationships).
As Fig. 2a shows, the water supply curve (Jw versus ψe) falls to zero at ψe = ψs (zero soil-to-leaf water potential gradient) and ψe = ψxL (complete loss of hydraulic conductivity through xylem embolism). The intermediate point at which Jw has a maximum represents the water potential below which catastrophic xylem failure through runaway cavitation would occur in the absence of stomatal regulation of ψe (Tyree & Sperry 1988; Jones & Sutherland 1991).
The shape of the water demand curve (E versus ψe) reflects the role of ψe in regulating the sensitivity of the guard cell ion diffusion rate (d) to ABA, as well as the production and dilution of ABA itself. As ψe decreases, increased sensitivity of d to ABA leads to a decline in E (Fig. 2a). When ψe approaches ψxL this decline in E is enhanced by an increase in [ABA] as Jw(and thus ABA dilution) becomes small due to xylem embolism. The hydraulic feedback via peristomatal transpiration does not affect the overall shape of the water demand curve, but instead raises the entire curve as VPD increases, because this feedback is regulated through the epidermal-guard cell water potential difference (Δψ) rather than the absolute value of ψe.
Model behaviour (aim 5)
In this section, I examine the behaviour of the new model with regard to the stomatal response to VPD (regimes A–C), and regulation of the ci/cs ratio and foliage water potential (ψe), and compare this with the BBL and TD models (aim 5). The results are presented for the default case where ABA is synthesized in the leaves only (λe > 0, λr = 0), and the leaf epidermal hydraulic conductivity (K) is fixed at its maximum value (Kmax). The value of Kmax is not given explicitly, but is implicit in the value of D0 (Eqn 7c). Other parameter values are given in Table 2. A subsequent section will examine the effects on leaf water potential of root versus leaf ABA synthesis and of variable K with a view to gaining theoretical insights into the stomatal regulation of xylem embolism (cf. aim 6).
Stomatal response to air humidity: regimes A, B and C
Figure 2a illustrates the water demand curves for a range of air humidity deficits (Ds), with other parameter values given in Table 2. As Ds increases, the height of the point of intersection with the water supply curve at first increases, then reaches a peak before decreasing due to the progressive loss of hydraulic conductivity from xylem embolism.
Figure 2b shows the corresponding plot of g against E. Three regimes of behaviour can be identified, similar to the experimentally observed regimes A, B and C reviewed by Monteith (1995). In regime A (moderate values of Ds), g declines linearly with increasing transpiration rate (E). This behaviour reflects the two hydraulic feedback mechanisms through peristomatal transpiration and sensitizing of stomata to ABA. In regime B (high Ds), E decreases as Ds is increased. Farquhar (1978) suggested that such behaviour was a reflection of non-stomatal (cuticular) transpiration. Mott & Parkhurst (1991) suggested that patchy stomatal closure was responsible. The present model provides an additional/alternative explanation in which the decline in E at high Ds is due to xylem embolism (corresponding to steady states lying to the left of the peak in the water demand curve, Fig. 2a). The possibility that regime B might reflect a decline in hydraulic conductance with increasing Ds, caused by xylem cavitation, has been noted previously (Oren et al. 1999; Buckley & Mott 2002). The observation by Franks, Cowan & Farquhar (1997) that the short-term stomatal response to VPD in regime B was irreversible (i.e. involved hysteresis) lends indirect support to this explanation (Buckley & Mott 2002).
Monteith (1995) also noted a third regime C at low Ds, less frequently observed perhaps because of experimental difficulties in avoiding condensation. In this regime, stomatal conductance is relatively insensitive to humidity or transpiration rate (e.g. Thomas & Eamus 1999). This regime may be identified with the local maximum in the g–E plot at high humidity (Fig. 2b), and reflects the effect of ABA dilution by the water flux Jw (Eqn 8). ABA dilution is most sensitive to Jw at low Jw(i.e. low Ds) where it dominates the hydraulic feedback effect, resulting in an increase in g as E increases at low Ds. The local maximum in g occurs when the dilution and hydraulic feedback effects cancel, resulting in no response of g to humidity. With further increase in E, the hydraulic feedback effect dominates, stomatal closure occurs, and the system passes into regime A. Regime C becomes more pronounced as the ABA sequestration rate (a) tends to zero (data not shown), or when ABA synthesis occurs in the roots (Fig. 2b).
The new model therefore explains not only regimes A (hydraulic feedback) and B (xylem embolism), but also regime C (ABA dilution), and thus extends the range of validity of the BBL model.
Behaviour of ci/cs
Let us consider the behaviour of ci/cs under the approximation that Rd is negligible compared to An, which is reasonable except in the vicinity of the light or CO2 compensation points (cf. Fig. 1 in Leuning 1995). This approximation allows ci/cs to be calculated analytically. By substituting Eqn 10b into Eqn 7a and re-arranging, we find:
This result predicts that ci/cs declines with increasing Ds (Fig. 3) but is independent of Q and cs (and of the choice of photosynthesis model, Eqn 10c). As Q and cs approach the light and CO2 compensation points, respectively, Eqn 11 is replaced by ci/cs = 1.
Thus, in the absence of water stress, the new model predicts the same generic behaviour of ci/cs as the BBL model. Quantitatively, the decline of ci/cs with increasing Ds in the new model at low water stress (Fig. 3) would be practically indistinguishable from the negative linear relationship between ci/cs and Ds predicted by the BBL model. Thus, use of the factor (An + Rd)/ci overcomes the aforementioned difficulties associated with the factor An/(cs − Γ) in the BBL model, while retaining the same qualitative and quantitative behaviour of ci/cs in the absence of water stress.
Let us now consider the behaviour of ci/cs in the presence of water stress. Experimentally, the observed ci/cs ratio sometimes declines under soil drying (Korol et al. 1999), although not always (McMurtrie et al. 1992). Equation 11 predicts the former response (via the ABA component of the new model). This behaviour is equivalent to a decrease in the apparent slope of the unstressed model (a1). The new model does not account for the possibility of patchy stomatal closure, corresponding to a constant a1, a constant ci/cs ratio, and a parallel reduction in g and An. Leuning (1995) showed that variation of a1 in the BBL model under water stress would produce a family of curves for ci/cs versus Ds that are roughly parallel, whereas patchy stomatal closure would cause ci/cs to converge at low Ds and diverge at high Ds, thus suggesting a method of experimentally distinguishing these two modes of stomatal closure under water stress.
As shown in Fig. 3, the new model predicts a diverging family of curves of ci/cs versus Ds under different soil water potentials (ψs). This picture is similar to what one would expect from patchy closure in the BBL model, even though patchy closure is not included in the new model. This is because the decrease in the apparent slope of the unstressed model (described by the ABA-dependent factor in Eqn 7a, as inherited from the TD model) is not the same at all values of Ds, being smallest at high ψe (low Ds) and greatest at low ψe (high Ds) through sensitizing of the stomatal response to ABA. The implication of the new model therefore is that it may not be straightforward to infer the existence of patchy closure under water stress from plots of ci/cs versus Ds in the manner suggested by Leuning (1995), because of the ABA × ψe interaction.
Regulation of leaf water potential
Figure 4 shows the predicted response of leaf epidermal water potential (ψe) to air and soil humidity deficits. As Ds increases, ψe at first decreases but then reaches a minimum value (ψe,min) that is insensitive to further increases in Ds (Fig. 4a) and changes little with soil water potential (ψs) (Fig. 4b). Therefore the new model, like the TD model, leads to homeostatic regulation of leaf water potential.
What are the key model features that account for this hydraulic homeostasis? To answer this question it is instructive to consider how stomata would have to behave in order to maintain the foliage water potential (ψe) strictly constant under variations in Ds and ψs (‘perfect homeostasis’). From the steady-state balance between water supply and demand,
we see that if stomatal conductance were inversely proportional to Ds and directly proportional to the supply rate Jw,
(where f represents any explicit dependence of g on ψe) then we would have
For this hypothetical plant therefore f and hence ψe would be constant under variations in air and soil humidity deficit, and the constant value of ψe would depend only on the functional form of f. The new model (Eqn 7a) is not exactly of the form required for perfect homeostasis (Eqn 12b). Nevertheless, the humidity function 1/(1 + Ds/D0) in Eqn 7a becomes more nearly proportional to 1/Ds as Ds increases. Furthermore, for a given value of ψe, there is a strong positive relationship between g and Jw through the effect of ABA dilution (Eqn 8). We conclude that the two key model features accounting for the maintenance of hydraulic homeostasis under variations in air and soil humidities are: (i) hydraulic feedback through peristomatal transpiration, and (ii) xylem ABA dilution by the transpiration stream. These features ensure that an approximate homeostasis is maintained, particularly under the moderate to severe air and soil humidity deficits for which hydraulic homeostasis is likely to be of greatest advantage to the plant.
We have identified the link between g and Jw as the key feature underlying hydraulic homeostasis with respect to variation in the soil water reserve. Tardieu & Davies (1993) suggested that the key feature is the ABA × ψe interaction, because suppression of this interaction (δ = 0) leads to anisohydric behaviour. In fact there is no real conflict between these two views. As discussed in the following section, homeostasis is still possible when δ = 0, although at an unrealistically low value of ψe,min (below −100 MPa). Thus, the g − Jw correlation ensures the possibility of homeostasis, while the ABA × ψe interaction (δ < 0) ensures that the actual value of ψe,min lies within physiologically reasonable bounds. Varying δ is analogous to changing the functional form of f in Eqn 12c.
How does the degree of hydraulic homeostasis predicted by the new model compare with that predicted by the TD model? As already mentioned, the air humidity response in the TD model (Eqn 2) occurs entirely through the dependence of g on ψe and [ABA], whereas in the new model an additional hydraulic feedback response occurs through the factor 1/(1 + Ds/D0) reflecting the role of peristomatal transpiration. With reference to Eqn 12bet seq., it follows that the value of ψe is more stable with respect to changes in Ds in the new model than in the TD model (as may be verified numerically – data not shown). By including the effect of peristomatal transpiration (inherited from the BBL model), the new model maintains hydraulic homeostasis under variations in both air and soil humidity.
Common basis for homeostatic behaviours
As noted in the Introduction, hydraulic homeostasis is analogous to the homeostatic behaviour of ci/cs in the absence of water stress. We have seen how the former arises from a positive correlation between g and Jw (via ABA dilution) whereas the latter arises from a positive correlation between g and An (via ATP). These two behaviours therefore have a common basis; in either case, approximate homeostasis is maintained through a positive correlation between the diffusive conductance (for H2O or CO2) and the corresponding (H2O or CO2) flux.
REGULATION OF LEAF WATER POTENTIAL BY LEAF/ROOT CHEMICAL and HYDRAULIC SIGNALS (AIM 6)
In the previous section it was argued that maintenance of hydraulic homeostasis under variations in air and soil humidity depends on two key features: (i) hydraulic feedback through peristomatal transpiration, and (ii) xylem ABA dilution by the transpiration stream. In this final section I examine more quantitatively how root and/or leaf chemical and hydraulic signals affect the tightness of homeostatic control under soil drying, as well as the actual value of the minimum leaf water potential that is attained (i.e. its dependence on parameters such as δ).
Figure 4b shows how ψe depends on soil water potential at fixed VPD, for the default case (ABA synthesized in leaves only, fixed leaf epidermal hydraulic conductivity K), as well as the effects of switching ABA synthesis to the roots and/or allowing K to vary as a function of ψe (Eqn 4b). Two main conclusions may be drawn from Fig. 4b: (i) the model is able to maintain a certain degree of hydraulic homeostasis under soil drying, irrespective of whether ABA synthesis occurs in the roots or the leaves, and irrespective of whether K is constant or varies with ψe, but (ii) the tightest regulation of ψe occurs when K varies with ψe (which represents an extra hydraulic feedback), particularly when ABA synthesis occurs in the leaves.
What are the key parameters affecting the minimum leaf water potential (ψe,min) that is attained? The value of ψe,min corresponds to the lowest turning point in the ψe versus ψs plot (Fig. 4b). As shown in Appendix 3, an implicit equation may be derived for ψe,min (Eqn A3·7) in the limit of saturating light (Q → ∞) and under the approximation that the ABA sequestration rate (a, Eqn 8) is small relative to the dilution flux (Jw). Even without solving this equation (which must be done numerically), the analytical form of the equation itself shows us which parameters affect the value of ψe,min, and which parameters do not.
For example, in the default case where ABA is synthesized in leaves only (λr = 0, λe > 0), and in the limit of high VPD (Ds → ∞, corresponding to the lowest possible value of ψe,min), the equation is particularly simple. From Eqns A3·9 and A3·11 (Appendix 3), ψe,min is given in this case by the value of ψe satisfying
where Ω is given by the parameter combination
Equation 13a implies that, in the default case, ψe,min depends on just two factors: δ (ABA-sensitivity parameter) and Ω. Equation 13b shows that Ω depends on (i) the ABA-related parameters dmin, β and λe (ii) the ATP-related parameters µ and gx, and (iii) the leaf epidermal hydraulic conductivity (K). Realistic values of ψe,min of the order of 1–3 MPa are obtained with the parameter values of Table 2 (in which δ = −2 MPa). In contrast, when δ = 0 the value of ψe,min lies below −100 MPa. Therefore, although hydraulic homeostasis is still possible when δ = 0, the value of ψe,min is unrealistically low unless δ < 0.
An interesting conclusion that emerges from Eqn 13 is that ψe,min is independent of the soil-to-root hydraulic conductivity (as found by Tardieu & Davies 1993) as well as the xylem hydraulic conductivity (Lre) as illustrated in Fig. 4b. The implication of this conclusion for the stomatal regulation of xylem embolism is discussed below.
Synthesis and extension of BBL and TD models
The new model combines the main features of the BBL and TD models within a simple spatially aggregated picture of guard cell function. Mathematically, the new model (Eqn 7a) is simply the product of the BBL and TD models, and so their respective properties combine quasi-independently. This is why the homeostatic behaviours of the ci/cs ratio and of leaf water potential are preserved. At the same time, the new model presents four novel features not included in either the BBL or TD models.
First, and most obviously, the new model combines stomatal responses to above-ground factors (from BBL) with responses to soil drying (from TD). Secondly, the model captures all three regimes of the VPD response (Monteith 1995): regime A (hydraulic feedback, from BBL and TD), regime C (ABA dilution, from TD), and regime B (through the inclusion of xylem embolism).
Thirdly, the integration of chemical and hydraulic signals in the TD model has been extended to include (i) leaf ABA synthesis, and (ii) extra hydraulic signals through peristomatal transpiration (from BBL) and leaf epidermal hydraulic conductivity (K). As a result, tighter homeostatic control of leaf water potential is achieved in the new model under variations in VPD (through the extra hydraulic signal from peristomatal transpiration) as well as under soil drying (through the extra coupling of hydraulic and chemical signals via K and leaf ABA).
Fourthly, the factor (An + Rd)/ci avoids the shortcomings of the BBL factor An/(cs − Γ). Use of the factor (An + Rd)/ci, being more directly related to mesophyll ATP than An/(cs − Γ), is also more consistent with the ATP-based stomatal model of Farquhar & Wong (1984), which was developed largely to explain the observed behaviour of ci/cs. Thus the new model actually represents a synthesis of three previous models (Farquhar & Wong 1984; Tardieu & Davies 1993; Leuning 1995). As an aside, we note that although the BBL factor An/(cs − Γ) mimics regime B at high VPD (due to a reduction in ci), this behaviour is lost when using (An + Rd)/ci, so inclusion of xylem embolism (or some other mechanism such as cuticular transpiration) is then necessary to explain this regime.
Regulation of xylem embolism
It has been suggested that the observed minimum value of ψe in woody plants is related to the threshold potential for the onset of xylem cavitation (Tyree & Sperry 1988; Jones & Sutherland 1991; Sperry & Pockman 1993). Subsequently, various authors have suggested a role for chemical and hydraulic signals in the stomatal control of xylem embolism (e.g. Cochard, Bréda & Granier 1996; Salleo et al. 2000). Salleo et al. (2000) speculated that minor drops in water flow in stem and/or leaf veins caused by cavitation might induce major changes in water potential at the level of guard cells, and/or trigger localized chemical signals such as release of apoplastic ABA or pH changes, leading to stomatal closure.
The present model demonstrates that stomatal regulation of xylem embolism could indeed operate successfully through a combination of changes in leaf epidermal hydraulic conductivity (K, which affects the epidermal-guard cell water potential difference, Δψ) and leaf [ABA]. Significantly, it is just this combination of leaf hydraulic and chemical signals which achieves the tightest control of ψe in the model. To this extent, the model supports the suggestion of Salleo et al. (2000).
However, as was shown in Appendix 3 and as illustrated in Fig. 4b, in the model the minimum leaf water potential (ψe,min) does not depend explicitly on the xylem hydraulic conductivity (Lre). As shown in Appendix 3, the key parameters determining ψe,min include the threshold potentials (ψtK and ψxK) defining the vulnerability curve of the leaf epidermal hydraulic conductivity (K, Eqn 4b), and the ABA-related parameters δ, β and λe. Although the model does not exclude the possibility that some of these parameters (e.g. λe) are linked dynamically to xylem cavitation events, the model suggests that there need be no dynamic link between xylem cavitation and stomatal closure in order to regulate the extent of xylem embolism (cf. Salleo et al. 2000). Rather, for a given species the key parameters determining the minimum leaf water potential (e.g. ψtK, ψxK, δ, β and λe) may simply be selected so as to maintain ψe,min at a value appropriate to the successful functioning of that species, a value which might imply either some decline in Lre(optimization strategy) or avoidance of embolism altogether (conservative strategy) (Jones & Sutherland 1991).
Indeed, the value of ψe,min in the model need not be related to xylem embolism at all, and might correspond to a threshold level of water status that is critical to some other process sensitive to water stress, such as growth. The suggestion here is that combined hydraulic and chemical signalling in leaves and/or roots provides a general stomatal mechanism by which plants may control their water status independently of the nature of the physiological process which is primarily affected by water stress. The minimum leaf water potential might then be set through natural selection of the key stomatal control parameters (e.g. ψtK, ψxK, δ, β and λe), at a value determined by the most vulnerable process under water stress, which might be cell growth, protein synthesis, respiration, xylem cavitation, photosynthate translocation or some other process sensitive to water stress, depending on species and growth conditions (Hsiao 1973).
Key model features requiring further critical evaluation
The BBL × TD model (Eqn 7a) and its derivation represent, respectively, a practical proposal for improving the predictive ability of the BBL and TD models, and an underlying theoretical justification of that synthesis in terms of guard cell function. The new model can be criticised on both levels, and this is useful: a stomatal model based on a theoretical description of guard cell function, no matter how crude that description might be, has the advantage that when the model fails at the empirical level, as all models eventually do, the cause of failure can be identified and corrected in a more systematic way than is possible with a purely empirical model (Thornley 1997).
On the practical level, the key feature to be tested experimentally is the multiplicative nature of the BBL × TD model. Contingent on this feature being verified, Eqn 7a could be incorporated within larger-scale models of plant functioning, as is currently done with the BBL model. Although the full model is sufficiently simple to solve numerically, for some practical applications various approximation schemes might also be tested. For example one might set ψe to a constant (‘perfect homeostasis’) so that, through the dependence of [ABA] on Jw in Eqn 8, the ABA-dependent factor in Eqn 7a simplifies to a function of soil and (maximum) xylem hydraulic conductivities (respectively, 1/Rsr and Lmax) and of soil water potential (ψs). Furthermore, dark respiration might reasonably be ignored in many applications, thus allowing Eqn 11 to be used for the ci/cs ratio.
On the theoretical level, a key feature requiring further evaluation is the spatially aggregated description of the mechanical interaction between guard cells and the bulk leaf epidermis (Eqn 3a). This description is currently neither proven nor disproven. As discussed in Appendices 1 and 2, reconciliation of Eqn 3a with the known mechanical advantage of neighbouring subsidiary cells is possible in principle if the guard cell/subsidiary cell complex is mechanically coupled to the rest of the epidermis, for which there is some evidence (Meidner & Edwards 1975). The outcome of this collective mechanical behaviour – as expressed in terms of the relationship between g, Pg and Pe– would depend on the transmission of pressure across the epidermis, and thus on the common areas and elastic properties of the walls dividing neighbouring epidermal cells. Further theoretical justification of the BBL × TD model, incorporating the known mechanical advantage of subsidiary cells, may require a scaling-up analysis of mechanical and osmotic relationships across the epidermis, possibly through extension of the spatially explicit model of hydraulic interactions proposed by Haefner et al. (1997).
A synthesis and extension of the BBL and TD models has been proposed, which describes combined stomatal responses to air and soil environments. The new model explains a number of common trends reported in the literature, and shows how stomatal regulation of xylem embolism and other processes sensitive to water stress may be achieved through combined hydraulic and chemical signalling in leaves and/or roots. Although over-simplified in many respects, the underlying guard cell picture from which this synthesis has been derived is physiologically meaningful, testable and useful as a guide for further model development. The new model is sufficiently simple to be implemented within plant growth models.
I thank Maurizio Mencuccini, Ray Leuning and Jérôme Ogée for useful comments on this work, and an anonymous reviewer for valuable criticism leading to significant improvement of the manuscript. This work was supported by the European project CARBO-AGE (EVK2 CT 1999 00045).
Received 12 January 2002;received inrevised form 17 April 2002;accepted for publication 22 April 2002
Appendix 1– reconciliation of eqn 3a with the mechanical advantage of subsidiary cells
Theoretical and experimental analyses of stomatal mechanics have shown that subsidiary cell turgor (Ps) has a greater (and opposing) influence on stomatal aperture compared with guard cell turgor (Pg):
In contrast to this local description of stomatal mechanics, the spatially averaged model assumed in the present study (Eqn 3a) has the form
where Pe is the turgor pressure of the bulk epidermis. The subscript ‘ge’ indicates that the value of the spatially aggregated mechanical coefficient χge is in general different from the local mechanical coefficient χgs in Eqn A.1·1.
This Appendix shows that it is possible to reconcile Eqns A1·1 and A1·2 in principle, provided that the guard cells, subsidiary cells and bulk epidermis are mechanically coupled, and that the nature of this coupling takes a particular form. Pressure probe studies have demonstrated that increased turgor in cells other than neighbouring subsidiary cells can cause partial stomatal closure (Meidner & Edwards 1975). If the nature of this mechanical coupling is such that the spatially averaged turgor pressures are related by
for some constant r, then Eqns A1·1 and A1·2 are equivalent, their respective mechanical coefficients being related through
as may be verified by direct substitution. Two further conditions are required for consistency. First, Eqn A1·3 implies that if Ps is to remain positive throughout the range 0 < Pe < Pg then we must have r < 1. Secondly, for water to flow from the bulk epidermis towards the subsidiary cells requires that ψs < ψe, implying that
where we have used Eqn A1·3. In the subrange 0 < Pe < Pg(1 − r)/(m − r) where the turgor difference Ps − Pe is positive, we therefore require that the osmotic gradient Δπse is sufficiently large and positive. In the derivation of the BBL × TD model from Eqn A1·2, πe was assumed to be a fixed parameter; this does not exclude the possibility that πs is larger than πe and thus that Eqn A1·5 can be satisfied.
In summary, consistency between the two mechanical models of stomatal conductance is possible in principle, provided that the mechanical coupling between the average turgor pressures of the guard cells, subsidiary cells and bulk epidermis takes the form of Eqn A1·3 for some r < 1, and that there is a sufficiently positive osmotic pressure gradient from the bulk epidermis to the subsidiary cells. Establishing the validity or otherwise of these conditions would require further theoretical and/or experimental information on the spatial distribution of cell pressure-volume relations and osmotic gradients across the epidermis.
Appendix 2– prospects for a ‘bottom-up’ explanation of the bbl and td models
The main motivation for adopting the spatially aggregated description of stomatal mechanics (Eqn 3a) for this study has been that it is mathematically consistent with the BBL and TD models. Complementary to this ‘top-down’ approach to stomatal mechanics, one would like to explain the BBL and TD models from a ‘bottom-up’ approach based directly on a local description of the known mechanical advantage of subsidiary cells. Haefner et al. (1997) proposed a spatially explicit model of hydraulic interactions across the leaf, consisting of a rectangular grid of stomatal units (guard cells and surrounding epidermal tissue) grouped into areoles surrounded by veins. The stomatal conductance of each grid was modelled using Eqn A1·1:
where Pg and Ps represent the local turgor pressures of, respectively, the guard cells and surrounding epidermal tissue within the grid. Water flows between grid points in response to the local water potential gradient. Haefner et al. assumed that local pumping of solutes into guard cells was proportional to Ps. This Appendix discusses the prospects for such a model to explain the empirical BBL and TD models.
Assuming that the guard cells are in local hydraulic equilibrium with the surrounding epidermal tissue within each stomatal unit (ψg = ψs), it is straightforward to show from Eqn A2·1 that
where µ0 is a pumping coefficient (cf. main text Eqn 5a). Haefner et al. assumed that the value of πs was a fixed constant across the leaf. Combining Eqns A2·2 and A2·3 in the steady state then gives
Within a simple one-dimensional approximation to water flow (as used in the present model based on Eqn A1·2), we have
where Pv is the vein turgor pressure (which was assumed fixed), Kvs. is the hydraulic conductivity between the veins and the guard cells, and we have used Eqn 6c. Combining Eqns A2·4 and A2·5 then gives, after rearranging,
Therefore, in this simple one-dimensional approximation, Haefner et al.'s model predicts a stomatal response to air humidity deficit (Ds) that has the same mathematical form as that of the BBL model (Eqn 1). However g is a saturating function of µ0/d. If the responses to leaf irradiance, CO2 and soil humidity were to enter through the pumping coefficient µ0 and diffusion coefficient d (e.g. via ATP and ABA in analogy with the present model), then we would not obtain the simple multiplicative form of the BBL × TD model (cf. Eqn 7a), and would lose the associated homeostatic behaviour of the ci/cs ratio and leaf water potential. For example, if µ0 were proportional to (An + Rd)/ci(cf. Eqn 5c) then g would not be proportional to (An + Rd)/ci, resulting in a more variable ci/cs ratio.
This simplified analysis does not exclude the possibility that the BBL and TD models can be explained using the full two-dimensional version of Haefner et al.'s model, with an alternative prescription for the pumping parameter µ0, and/or with a spatially varying (rather than constant) value of πs. Appendix 1 suggests that mechanical interactions across the leaf epidermis may also need to be represented.
Appendix 3– the dependence of the minimum foliage water potential (ψe,min) on the model parameters
Here we calculate the value of ψe at the lowest turning point in the ψe − ψs plot (Fig. 4b), in the limit of saturating light (Q → ∞). In this limit, the stomatal conductance model (Eqn 7a) simplifies because (An + Rd)/ci → gx (see Eqn 10c), and the steady-state water balance relationship (Eqn 12a) then gives
At the minimum of the ψe − ψs plot, a small change in ψs leads to an equal change in both sides of Eqn A3·1, and no change in ψe. If ∂/∂ψs denotes a partial rate of change with respect to ψs at constant ψe, then applying ∂/∂ψs to Eqn A3·1 at the minimum gives
and ψe,min is then the value of ψe that satisfies this condition. From Eqn 8 (in which we make the approximation a << Jw, and substitute ψr = ψe + JwRre) we have
where in the second step we have re-used Eqn A3·3. Substituting Eqn A3·4 into Eqn A3·2 and noting some cancellations, we find
In this case therefore the minimum leaf water potential depends on just two parameters: δ and Ω. Finally, the absolute lowest value of ψe,min occurs in the limit of high VPD (Ds → ∞), when the expression for Ω reduces further to
where we have used Eqns 7b and 7c of the main text to substitute for a1 and D0.