Leuning model

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abridged from Leuning (1995). A critical appraisal of a combined stomatal-photosynthesis model for C3 plants. Plant, Cell and Environment, Vol. 18, No. 4. (1995), pp. 339-355.


Gas-exchange measurements on Eucalyptus grandis-leaves and data extracted from the literature were used to test a semi-empirical model of stomatal conductance for CO2,

where A is the assimilation rate;

Ds is the humidity deficit at the leaf surface;
Cs is the CO2 concentration at the leaf surface;
g0 is the conductance as A → 0 (when leaf irradiance → 0);
D0 and a1 are empirical coefficients.

This model is a modified version of

first proposed by Ball, Woodrow & Berry (1987), in which hs is relative humidity. Inclusion of the CO2 compensation point Γ improved the behaviour of the model at low values of cs. A hyperbolic function of Ds for humidity response correctly accounted for the observed hyperbolic and linear variation of gsc and ci / cs as a function of Ds. Here ciis the intercellular CO2 concentration.

In contrast, use of relative humidity as the humidity variable led to predictions of a linear decrease in gsc and a hyperbolic variation in ci / cs as a function of Ds. This did not agree with the data from E. grandis -leaves. The revised model also successfully described the response of stomata to vanations in A, D and c for published responses of the leaves of several other species. Coupling of the revised stomatal model with a biochemical model for photosynthesis of C3 plants synthesizes many of the observed responses of leaves to light, humidity deficit, leaf temperature and CO2 concentration. The best results are obtained for well-watered plants.


Stomata have evolved physiological control mechanisms to satisfy the conflicting demands of allowing a net carbon gain by leaves while restricting water loss to acceptable levels, under a range of environmental conditions.

Stomata respond to environmental stimuli in a complex way and the physiological mechanisms of stomatal behaviour are not yet fully understood. In the absence of a mechanistic description, semi-empirical models are often used to describe stomatal response to environmental and physiological factors. Such models are required to predict the response of photosynthesis and plant water use under present and future CO2 concentrations and temperatures. Relatively simple phenomenological models of stomatal behaviour are also required to describe plant water use in response to diurnal and seasonal variations in environmental conditions.

Here we critically examine a semi-empirical model for stomatal behaviour first proposed by Ball et al. (1987) and developed further by Leuning (1990). Stomatal conductance is assumed in the model to be a function of relative humidity at the leaf surface, assimilation rate, and surface CO2 concentration, but does not include the effects of plant water stress on conductance. The model is modified by replacing surface relative humidity with a hyperbolic function of humidity deficit, as this provides a superior description of the reponses of both conductance and intercellular CO2 concentration to varying humidity.

stomatal model

physical approach

Many modellers have followed Jarvis (1976) in assuming that environmental factors act independently in determining stomatal conductance, gsc. Stomatal response is then described by


where fi are non-linear functions of

Q: the absorbed short-wave irradiance;
Tl: the leaf temperature;
Ds: the water vapour deficit;
Cl: the CO2 concentration, and
Θ: the soil moisture content.

The conductance to water vapour gsw = 1·56 gsc where gsc is the conductance to CO2. The numerical factor 1.56 accounts for the ratio of diffusivities for water vapour and CO2 in air. Equation 1 describes a five-dimensional response surface. The possibility of other dimensions is indicated. The response surface will be complex because the responses of gsc to individual variables are usually non-linear. Stylized responses of stomata to individual factors were presented by Jarvis & Morison (1981). These functions have been determined by many workers using leaf gas-exchange measurements under controlled conditions or by using boundary-line analyses of measurements under natural conditions. The results indicate that gsc

  • increases rapidly as irradiance increases from zero, and that
  • saturates at relatively low values of Q.

Conductance shows a peaked response to temperature, although the optimum is usually fairly broad. Increasing atmospheric humidity deficits and CO2 concentrations both cause gsc to decrease. Stomatal conductance is relatively insensitive to decreasing soil moisture availability until a threshold is reached, whereupon gsc decreases approximately linearly.

introducing leaf physiology

An alternative approach to modelling stomatal behaviour only in terms of physical variables (as in Eqn 1) is to incorporate leaf physiology by combining models of stomatal behaviour and photosynthesis. This recognizes that stomatal control has evolved to maximize carbon gain while minimizing water loss. The apparent close link between stomatal conductance and assimilation led Farquhar & Wong (1984) to propose an empirical model relating stomatal conductance to the ATP content and photosynthetic capacity of leaves. Ball et al. (1987) presented an empirical relationship which incorporates the often-observed correlation between A and gsc and includes the effects of humidity and ambient CO2 concentrations on conductance, namely



g0: the conductance as A → 0 (when leaf irradiance → 0);
a1: a coefficient, and
hs: the relative humidity at the leaf surface;
cs: the CO2 concentration at the leaf surface;

This equation predicts that stomatal conductance increases with assimilation rate and with relative humidity at the leaf surface when cs is held constant. Conductance will decrease with rising ambient CO2 concentration, provided that A increases more slowly than cs. Equation 2 satisfies the requirements of Eqn 1 in that the dependence of gsc on Q, c and possibly Tl is incorporated in the dependence of A on these variables. The apparent simplicity of Eqn 2 has led to its adoption by modellers working at the scale of individual leaves, canopies, and landscape, and in some global climate models.

Equation 2 has no apparent mechanistic basis because the elegant experiments by Mott (1988) showed conclusively that stomata respond to substomatal CO2 concentration, ci, rather than to the surfave concentration cs. Despite this conclusion, formal justification for the term A/cs in Eqn 2 can be obtained by examining the simplified equation for supply of CO2 by diffusion through the stomata,


and rewriting it in the form


This simple equation ignores interactions between molecules of water leaving and entering the stomata. Use of cs rather than CO2 concentration outside the leaf boundary layer, ca , eliminates complications of the equations arising from transfer through the boundary layer. We see that gsc will be correlated to the ratio A/cs, provided that the ratio ci/cs remains relatively constant when the leaf is subject to varying environmental conditions. While Eqn 3b defines gsc, it does not mean that conductance is linked mechanistically to A/cs.

a general humidity term

The BWB model has caused further concern because it is widely accepted that stomata respond to humidity deficit rather than to surface relative humidity. Stomata respond to atmospheric humidity through evaporation from the leaf, rather than to humidity deficit itself. Because there is a close link between transpiration, Et and humidity deficit (Et = gsw.Ds) we shall continue to use Ds, rather than Et as the humidity variable in subsequent developments. This avoids the need to include transpiration (and hence the full leaf energy balance) explicitly in the equations for gse and A.

Succesful application of Eqn 2 suggests that this equation may still be useful if hs is replaced by a more general humidity function f(D), i.e



BWB model

Jarvis (1976) describes f(D) at constant leaf temperature, but the slope 1/D0 depends on temperature.

Lohammer et al. (1980) developed from analysis of measurements of leaves in the field, where variation in Ds is largely caused by changes in leaf temperature rather than by changes in the absolute humidity of the air


e*: the saturation water vapour pressure at leaf temperature.
D0: an empirically determined coefficient

In its present form, Eqn 4 is not capable of describing stomatal behaviour at low CO2 concentrations, since conductance increases to maximal values as cs approaches the CO2 compensation point, Γ while A → 0. Equation 4 predicts that gsc → 0 under these circumstances. Leuning (1990) accounted for these observations by replacing cs with cs - Γ, i.e.


This modification ensures that gsc remains large as A → 0, while cs → Γ. Note that gsw is not defined by this equation when cs = Γ.

The implication of Eqn 8 for the expected value of ci can be seen by combining it with Eqn 3 to give


This equation was called the supply-constraint function by Leuning (1990). A plot of Eqn 9 shows that leaves that conform to the BWB model maintain ci almost constant at the asymptotic value, cs - (cs - Γ) / (a1.f(D)) over a wide range of A, when Q varies while Ds and cs are held constant (Fig.).

Note that g0 cannot be negative physically, and g0 ≠ O, otherwise ci/cs would be constant for all values of Q. The rate of convergence of ci to cs at low A (low Q) depends on the value of g0: when g0 is large, ci deviates from the asymptotic value at higher values of A than when g0 is small.


Figure 1. Hypothetical A-ci curves showing the photosynthetic demand function (Eqn 10) at Q=1500 and 150 μmol quanta at Tl = 25 °C. Examples of the supply-constraint function (Eqn 9) are also shown for cs= 350 and 700 μmol/mol, and Ds= 1 and 3 kPa, with g0 = 0.01 mol m-2s-1 and Ds0=0.35 kPa. The 'operating point' of the leaf is given by the intersection of the demand and supply-constraint curves. A is more sensitive to changes in Ds, at lower values of ci and when Q is high. The model predicts that ci remains almost constant as A decreases with diminishing Q for any given values of Ds and cs.