## Introduction

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).

### Ball–Berry–Leuning model

The empirical model of Ball, Woodrow & Berry (1987), as modified by Leuning (1990, 1995), states that

where *g* is the stomatal conductance for CO_{2} diffusion, *A*_{n} is the net leaf CO_{2} assimilation rate, *D*_{s} and *c*_{s} are the vapour pressure deficit (VPD) and CO_{2} concentration at the leaf surface, respectively, Γ is the CO_{2} compensation point, *g*_{0} is the value of *g* at the light compensation point, and *a*_{1} and *D*_{0} are empirical coefficients.

This model encapsulates two empirical trends reported in the literature. First, through the correlation between *g* and *A*_{n}, Eqn 1 predicts that the ratio (*c*_{i} − Γ)/(*c*_{s} − Γ) (where *c*_{i} is the leaf intercellular CO_{2} concentration) is largely independent of leaf irradiance and *c*_{s} (Wong, Cowan & Farquhar 1979) except near the light and CO_{2} compensation points, and declines linearly as *D*_{s} increases (Morison & Gifford 1983; Leuning 1995). Secondly, through the relation *E* = 1·6 *gD*_{s} for the transpiration rate, the hyperbolic function of *D*_{s} 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 *A*_{n}/(*c*_{s} − Γ) is inconsistent with the experiments of Mott (1988) showing that stomata sense *c*_{i}, not *c*_{s}. Mott found that the stomata of sunflower (*Helianthus annuus* L.) and cocklebur (*Xanthium strumarium* L.) closed when *c*_{i} was increased at fixed *c*_{s}; 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 *A*_{n} 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.

### Tardieu–Davies model

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 *g*_{min} and *g*_{max} 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 (*ψ*_{f} = 0), and *δ* (negative value) describes the increase in stomatal sensitivity to [ABA] as *ψ*_{f} 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 (*ψ*_{f,min}). First, when *δ* < 0 the predicted *ψ*_{f,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 *c*_{i}/*c*_{s} 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 *ψ*_{f,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 *ψ*_{f,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 *ψ*_{f,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.

### Objectives

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
*A*_{n}/(*c*_{s}− 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 CO_{2} assimilation (*A*_{n}), leaf intercellular CO_{2} concentration (*c*_{i}), leaf xylem ABA concentration ([ABA]) and leaf epidermal water potential (*ψ*_{e}) (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 *c*_{i}/*c*_{s} ratio and leaf epidermal water potential (*ψ*_{e}) (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).