A model of stomatal conductance was developed to relate plant transpiration rate to photosynthetic active radiation (PAR), vapour pressure deficit and soil water potential. Parameters of the model include sensitivity of osmotic potential of guard cells to photosynthetic active radiation, elastic modulus of guard cell structure, soil-to-leaf conductance and osmotic potential of guard cells at zero PAR. The model was applied to field observations on three functional types that include 11 species in subtropical southern China. Non-linear statistical regression was used to obtain parameters of the model. The result indicated that the model was capable of predicting stomatal conductance of all the 11 species and three functional types under wide ranges of environmental conditions. Major conclusions included that coniferous trees and shrubs were more tolerant for and resistant to soil water stress than broad-leaf trees due to their lower osmotic potential, lignified guard cell walls, and sunken and suspended guard cell structure under subsidiary epidermal cells. Mid-day depression in transpiration and photosynthesis of pines may be explained by decreased stomatal conductance under a large vapour pressure deficit. Stomatal conductance of pine trees was more strongly affected by vapour pressure deficit than that of other species because of their small soil-to-leaf conductance, which is explainable in terms of xylem tracheids in conifer trees. Tracheids transport water by means of small pit-pairs in their side walls, and are much less efficient than the end-perforated vessel members in broad-leaf xylem systems. These conclusions remain hypothetical until direct measurements of these parameters are available.
Simulation of terrestrial ecosystem responses to future climatic change necessitates reliable models of stomata behaviour under variable climatic and soil conditions (Raich et al. 1991, Running & Coughlan 1988). A candidate stomata model for this purpose should be reasonably mechanistically based in order to minimize the possible risks of extrapolation into future environmental regimes. Yet the model has to be mathematically simple enough to be embedded into simulation models so that large-scale simulations are still computationally feasible.
Detailed mechanics analyses of guard cell structure were carried out in engineering or mechanics fields (Cooke et al. 1976; Rand 1987; Upadhyaya, Rand & Cooke 1988). The finite element analysis by Cooke et al. (1976) provided important insight into the mechanism that controls the stomata dynamics. It was shown that an increase in guard cell turgor pressure widens stomata opening largely because of the elliptic shape of the guard cell structure. If the two guard cells join together to form a circular torus, an increase in hydrostatic pressure in guard cells would only decrease, rather than increase, the stomata opening. Furthermore, a decrease in surrounding subsidiary cell turgor tends to increase stomata aperture. The most important parameter that controls stomata responses to variation in guard cell turgor is thus the ratio of long to short axes of the guard cell ellipse. However, the complicated mathematical formulation and numerical procedures of these mechanics models make them almost impossible to embed into ecosystem simulation.
The objective of the present study was to develop a simple model of stomatal conductance to quantify the relationship between plant transpiration and micro-environmental factors such as radiation, vapour pressure, and soil water stress. The model was applied to three data sets for 11 plant species that belong to three functional types in subtropical southern China. The dependence of stomatal conductance on soil water stress, photosynthetic active radiation (PAR), and vapour pressure deficit, and the drought-resisting and -tolerating properties of plants, were discussed and explained in the light of the intrinsic structural and functional characteristics of plant species.
Our basic assumptions for model formulation are the following:
1Water flow from soil to leaf cells is driven by water potential gradients, and leaf water potential (ψleaf) is in equilibrium with guard cell water potential (ψg), or ψleaf=ψg. The total flow rate from soil to unit leaf area, Tr′, is the product of the difference between soil water potential ψs and guard cell water potential ψg, and gz, a conductance coefficient termed soil-to-leaf conductance hereafter, i.e.
Tr′ = (ψs − ψg) gz(1)
2Water loss by means of transpiration is driven by vapour pressure deficit. The transpiration per unit leaf area Tr is the product of relative vapour pressure deficit dvp (defined as absolute vapour pressure deficit Dvp divided by atmospheric pressure Pa) and stomatal conductance gs. That is,
Tr = gsdvp(2)
At equilibrium, or steady state, the flow into and out of unit leaf area should be balanced. Hence Tr′ = Tr gives
(ψs − ψg) gz = gsdvp(3)
3Stomatal conductance was assumed to be directly proportional to guard cell turgor pressure pg, i.e.
gs = pg/β(4)
where β is elastic modulus of guard cell structure (kPa mmol−1 m2 s). This assumption is the same as the one used by Dewar (1995) to explain the model by Leuning (1995). Writing guard cell water potential ψg as sum of osmotic potential πg and turgor pressure pg, or ψg = πg + pg, and using the relationship in Eqn 4, Eqn 3 can be solved for gs to give
where kψ = 1/β is the elastic compliance of guard cell structure, and kβg = 1/(βgz) is a parameter signifying the sensitivity of stomatal conductance to vapour pressure deficit.
4Increases in photosynthetic active radiation Ip (PAR) induced increases in potassium cation concentration in guard cells, so that osmotic potential of guard cells decreases and that of the subsidiary cells increases. The decrease in the osmotic potential of guard cells was assumed to be proportional to Ip, or
πg = π0 − αIp(6)
where π0 is the osmotic potential at dark, an intrinsic parameter related to the drought tolerance of plants; and α (kPa µmol−1 m2 s) is a constant coefficient describing the sensitivity of osmotic potential to PAR. As a result of the decreased guard cell osmotic potential, additional water leaves subsidiary epidermal cells and enters guard cells in order to maintain the equilibrium between guard cells and subsidiary epidermal cells. Consequently, turgor pressure inside guard cells increases, and subsidiary cell turgor decreases, both helping increase stomata aperture and conductance (Cooke et al. 1976).
or gs as an explicit function of soil water potential
where kαβ = α/β is a parameter defined as changes in stomatal conductance induced by unit change in PAR, hence signifying the sensitivity of stomatal conductance to photosynthetic activities; g0m = –kψπ0 is the maximum possible stomatal conductance at dark with zero soil water potential (saturated soil water content); g0= kψ(ψs − π0) = g0m + kψψs, is the stomatal conductance at dark with 100% relative humidity. Equations 7 and 8 show that stomatal conductance increases linearly with PAR and soil water potential, but decreases with vapour pressure deficit.
The expression of kαβ indicates that kαβ is larger for more sensitive guard cell osmotic potential to PAR, as a higher osmotic sensitivity means a greater decrease in osmotic potential, and thus a greater increase in guard cell turgor, and consequently a greater increase in stomata aperture and conductance, for a given increase in PAR. Parameter kαβ is smaller for stiffer guard cell structure because stiffer guard cells deform less, and consequently the stomata aperture and conductance increases less, for a given increase in turgor. Parameter kβg is larger for a softer guard cell structure, as a softer guard cell structure is easier to deform, and thus stomata aperture and conductance decreases more with a given decrease in guard cell turgor. Furthermore, kβg is smaller for larger soil-to-leaf conductance, because a larger conductance means more efficient water supply from soil to guard cells, and thus less decrease in guard cell turgor and stomatal conductance, for a given increase in vapour pressure deficit.
We can then write transpiration rate by unit leaf area Tr as
or Tr as an explicit function of soil water potential
Equations 7–10 indicate that stomatal conductance and transpiration increase linearly with soil water potential and PAR, that stomatal conductance decreases monotonically with vapour pressure deficit, and that transpiration is a hyperbolic function of relative vapour pressure deficit. Depending on the values of the parameters, our model should be able to describe the observed daily and seasonal courses of stomata behaviour for most plant species of trees and shrubs.
Our model has mathematically the same dependence of stomatal conductance on vapour pressure deficit as the model by Leuning (1995) with mechanistic explanation by Dewar (1995), and also a similar relationship between stomatal conductance and photosynthetic activities to the models by Ball et al. (1987). However, the parameters in the present model have more precise physical definition and significance. The present model also includes explicitly the effects of soil water stress on stomatal conductance and transpiration, so that the model should be more suitable for applications with wide ranges of soil water regimes. The model may be less accurate for extreme soil water stress and for some plants that only respond to small PAR ( Nobel 1983) because of the assumptions of linear responses of stomata conductance to turgor pressure, and guard cell osmotic potential to PAR.
Application and parameterization
The present model of stomatal conductance and transpiration was applied to the ecophysiological studies in Heshan Ecological Station in southern China. The site is geographically located at (24°41′ N, 112°54′ E), and the area is warm and wet from May to September, but dry from October to April Annual precipitation is about 1700 mm, and annual mean temperature is 21·7 °C, with 28·7 and 13·1 °C as the monthly means in the warmest July and coldest January, respectively. Monsoon evergreen broad-leaf forests have been recognized as regionally climax vegetation (Hou 1988). However, the area has been deforested because of economical pressure and mismanagement during the last few decades. Heshan Ecological Station was established in 1984 to lead the effort to reforest the area. Trees were planted since the early 1980s and studied physiologically under various climatic and environmental conditions. The application of our model used data sets originated from three field studies conducted from 1992 to 1995. The studies involved 11 species that belong to three functional types: (1) evergreen coniferous trees (Pinus massoniana, Pinus elliottii and Pinus caribaea) (2) evergreen broad-leaf trees (Schima wallichii, Schima superba, Cinnamomum burmani, Cinnamomum camphora and Castonopsis hickellii), and (3) evergreen shrubs (Evodia lepta, Clerodendron fortunatumm and Dianella ensifolia) (Cai et al. 1995; Zeng et al. 1995; Zeng, Zhao & Peng 1999a; Zeng et al. 1999b). These studies included hourly or bi-hourly measurements of transpiration rates or stomatal conductance within a few consecutive days with typical representative weather and soil conditions during both wet and dry seasons. Air temperature, PAR and relative humidity or vapour pressure, were also simultaneously recorded.
Non-linear regression analysis was used to obtain parameters g0, kαβ and kαg in Eqns 7 and 9 using Tr as the dependent variable in the transpiration model, Eqn 9, and measured PAR and vapour pressure deficit as independent variables. In order to test the differences within and between functional types, the same non-linear analysis was also applied to each of the three functional types and every two functional types by regarding all species within each functional type or functional type pair as one single ‘virtual’ species. Due to the lack of synchronized measurement on soil water content and soil water potential, we made an assumption that the magnitude of seasonal variation of soil water was much larger than interannual variation so that long-term mean seasonal soil water contents averaged over different slope positions (upper, middle and lower slopes) can be used. Based on this assumption, soil clay content and typical volumetric soil water contents in wet and dry seasons were obtained from Li, Fang & Lu (1995) and Yu & Peng (1996), and soil water potentials were calculated from soil water contents in the light of the relationship between soil clay content and water retention characteristics (Campbell & Norman 1998). We obtained ψsw = −88 kPa for the wet season and ψsd = −724 kPa for the dry season. Note that subscripts ‘w’ and ‘d’ denote wet and dry seasons, respectively, hereafter. Parameters g0m and kψ can then be calculated from non-linearly estimated g0w (g0 for wet season) and g0d (g0 for dry season) using kψ = (g0w − g0d)/(ψsw − ψsd) and g0m = g0w – ψsw(g0w − g0d)/(ψsw − ψsd) for each functional type. Since kψ = 1/β, parameters α and gz can also be evaluated.
Results and discussion
Tables 1 and 2 give results and all parameters of non-linear statistical estimation from field transpiration measurements of 11 plant species and three functional types, and Fig. 1 plots predicted transpirations by the present stomatal conductance and transpiration model against those observed in the field for the 11 species. The model described the transpiration rates and hence stomata behaviour well with R2 > 70% for all species and functional types. Table 2 indicates that the differences in regression analyses within each functional type are not significant, as the three F statistics are all less than the corresponding critical values at 0·05 confidence level. In other words, species within one functional type can functionally be regarded as one single species. The tests crossing functional types shows that all functional types are significantly different from each other at the 0·05 confidence level, as the F statistics of crossing functional types are larger than their corresponding table values. The observed and predicted transpiration rates (a, b, c) and calculated hourly transpiration rates in a common typical day (d, e, f) for the three functional types were plotted inFig. 2. The difference in magnitudes of transpiration rates between upper three panels (a, b, c) and lower three panels (d, e, f) is because site-specific weather data were used for the non-linear regression fitting (a, b, c), whereas transpiration rates in the bottom three panels were calculated based on a common set of hourly observations on PAR and vapour pressure deficit for a typical day in the wet season in Heshan Meteorological Station. Broken lines were calculated by setting kβg = 0, in order to see the effect of vapour pressure deficit on stomatal conductance. Figure 2 shows that vapour pressure deficit did not affect stomatal conductance of broad-leaf trees and shrubs, but significantly decreased the stomatal conductance of pines so that transpiration were suppressed by approximately five-fold.
Table 1. Result of non-linear regression of transpiration data for 11 species using the present model of stomatal conductance. Units of the parameters are: g0 (mmol m−2 s−1), kαβ(mmol µmol−1), kβg (dimensionless), SS, sum of residual squares (mmol2 m−4 s−2); R2 (%)
Table 2. Results of non-linear regression of transpiration data for each of the three functional types and for every two functional types. Units of the parameters are the same as Table 1 . The F-statistics to test the significance of differences among species within each functional type and the differences between two functional types, and the corresponding critical value (Fc) values, are provided
Functional type/ functional type pair
F6068 = 1.07
F100116 = 1.25
F4856 = 1.28
F1844 = 107.7
F1244 = 42.46
F1724 = 4.489
The 11 species can be classified into two classes in the light of the dependence of their stomatal conductance and transpiration on vapour pressure deficit. Pines had kβg > 100, whereas kβg values for broad-leaf trees and shrubs were less than 25. A larger kβg means a stronger dependence of stomatal conductance and transpiration on vapour pressure deficit, and an exceedingly large vapour pressure deficit may causes excessive water loss in guard cells and decreases in xylem potential and stomata aperture. Field studies have reported that pine trees showed mid-day depression in transpiration and photosynthesis activities in summer (Yu & Peng 1996; Zhao, Yu & Zeng 1996). Our model attributed this depression to the combination of a possible large vapour pressure deficit and large kβg parameter values. As kβg = 1/(βgz), a large kβg can be the result of either a softer guard cell structure (smaller guard cell elastic modulus), or a smaller soil-to-leaf conductance, or both. In contrast, broad-leaf trees and shrubs did not show the dependence of stomata conductance on vapour pressure deficit (Cai et al. 1995; Zeng et al. 1999a). The possible explanations are because of their larger soil-to-leaf conductance compared with pine trees, or stiffer guard cell structure, or both (see discussion later).
Table 2 shows that broad-leaf trees had smaller kαβ values than other functional types. Note that kαβ = α/β, the parameter is directly proportional to the sensitivity of osmotic potential and inversely proportional to the stiffness of guard cell structure. Thus broad-leaf trees can either have stiffer guard cell walls, or lower osmotic sensitivity to PAR, than other functional types.
Plant capability to resist and to tolerate drought can be visualized by comparing parameters in Table 3, in which the parameters g0m, π0, α, β, and 1/gz for the three functional types, estimated in the light of the average soil water contents/potentials for dry and wet seasons, are given. Although g0m is the possible maximum stomatal conductance at dark with saturated soil water (zero photosynthesis and transpiration), kψ describes the slope of decline of stomatal conductance with soil water stress (negative soil water potential), and partially represents the capability of a functional type to resist incremental drought (or increases in water stress). Pine trees were the most, and broad-leaf trees the least, resistant to increases in soil water stress, leaving shrubs in between.
Table 3. Estimated parameters for three functional types using the present stomatal conductance model
g0m(mmol m−2 s−1)
kψ = 1/β (mmol m−2 s−1 kPa−1)
α(kPa µmol−1 m2 s)
1/gz(µmol−1 m2 s kPa)
Dark osmotic potential π0 was calculated in light of π0 = –g0m/kψ. Parameter π0 is equivalent to the soil water potential at which stomata closure occurs. The parameter should thus reflect the capability of plants to tolerate drought. The actual wilting water potential in daytime may be much lower than this value, since stomatal conductance is strongly affected by photosynthetic activity, the relationship between stomatal conductance and soil water potential is likely to be exponential, especially for the long and narrow-shaped stomata structure of monocotyledon gramineal grasses (Kemp et al. 1997). Nevertheless, we can still use this parameter as an index to compare the drought tolerance of plants (Table 3). Again pine trees showed the highest, but broad-leaf trees the lowest, tolerance to drought, with shrubs in the middle. The great drought tolerance of pine trees and shrubs made them well-known pioneer plants for subtropical ecosystems in southern China (Yu & Peng 1996).
The estimated α and gz in Table 3 indicate that the insensitivity of broad-leaf trees to vapour pressure deficit was largely because of their greater soil-to-leaf conductance (the infinite gz for these species was because of the limited capability of statistical regression to obtain an accurate small value of kβg from field data), since they had the smallest guard cell elastic modulus compared with other functional types. On the other hand, the sensitivity of pines to vapour pressure deficit could be a result of their small soil-to-leaf conductance, as they had the largest guard cell elastic modulus. The lower sensitivity of broad-leaf stomatal conductance to photosynthetic radiation was because of their low sensitivity of osmotic potential to PAR (α), since broad-leaf trees had the lowest guard cell elastic modulus. The lowest sensitivity of guard cell osmotic potential to PAR explains the heiophyte features of these trees (Yu & Peng 1996).
Pertinent observational and anatomical evidence
Parameters α, β, and gz reflect the sensitivity of osmotic potential to variations in PAR, the sensitivity of stomata aperture to changes in guard cell turgor pressure, and the sensitivity of guard cell water potential to variations in soil water stresses. Although we cannot provide direct experimental measurements on these parameters to validate our hypothetic conclusions, the following observational and anatomic evidences are provided here to substantiate the discussion in the previous section.
Parameter β is the overall elastic modulus of guard cell structure, and is determined by the geometric shape of the guard cell structure, the elastic properties of guard cell walls, and the connection with subsidiary cells. Gao, Pitt & Bartsch (1989) measured the bulk modulus of apple and potato parenchyma cells (defined as the amount of turgor pressure per unit volumetric strain) to be 5·3 and 4·2 MPa, respectively. Dong & Zhang (2001) found that the bulk modulus of leaf cells of three desert shrubs in northern China was approximately 20 MPa, a much larger value than those for parenchyma of apple and potato. The bulk modulus reflects the mechanical properties of the cell walls. However, the finite element analysis by Cooke et al. (1976) indicated that a typical stomata opening width increases from 7 to 15 µm when turgor pressure inside guard cells increases from 0 to 700 kPa. If we assume that stomatal conductance is directly proportional to stomata opening, and that stomata openings of 7 and 15 µm approximately correspond to typical stomatal conductance values of 250 and 500 mmol m−2 s−1, respectively, a rough estimate of β can be obtained as 700/(500–250) = 2·8 kPa mmol−1 m2 s, which corresponds to guard cell compliance kψ = 0·357 mmol m−2 s−1 kPa−1. This value is of the same order as those we obtained with our stomatal conductance model. The light sensitivity coefficient α describes the sensitivity of the osmotic potential of guard cells to PAR. Nobel (1983) stated that molar concentration of K+ can increase by more than 0·3 mol L−1 under full sunlight (typical value of 1500 µmol m−2 s−1), which is equivalent to more than 0·3 × 2·437 = 0·731 MPa decrease in osmotic potential. This gives a rough estimate of α > 731/1500 = 0·487 kPa µmol−1 m2 s, again, in the same order of the smallest value in Table 3.
Anatomy of seed plants ( Esau 1977) shows that there were three types of guard cell structure, elliptic-shaped guard cells for dicotyledons, long, narrow-shaped guard cell structure for monocotyledon plants (gramineal grasses), and elliptic-shaped, sunken and suspended guard cell structure with partially lignified cell walls for gymnosperms (conifers). The sunken and suspended guard cell structure of conifer trees make the cells less sensitive to the variation in turgor pressure of subsidiary cells than broad-leaf trees, and lignified cell walls implies that the guard cells deform less with a given increase in turgor pressure. Both of these features imply that the guard cell structure of pines is stiffer than that of broad-leaf trees. Consequently, the only explanation of the large kβg is the smaller soil-to-leaf conductance of pines than broad-leaf trees. If we view soil-to-leaf conductance as xylem conductance and soil-root conductance in series, soil-to-leaf conductance can be small if one of these two (or both) conductance values is small. The anatomy of plant xylem systems ( Esau 1977) also revealed that there are two kinds of tracheid elements in xylem systems of seed plants, imperforated tracheids and end-perforated vessel members. Vessel members conduct water by means of their large perforated holes in the two ends and sometimes side walls, and thus are much more efficient than tracheids which only have small pits in their side walls to provide passages for water. Conifer trees have only tracheids in their xylem systems, thus in general should have much smaller xylem conductance (a major part of soil-to-leaf conductance) than broad-leaf trees which have both vessel members and tracheids in their xylems.
Summary and conclusions
In the present study a new model for plant stomatal conductance and transpiration as a function of soil water stress, vapour pressure deficit and PAR was derived. Model parameters signify the elastic property of guard cell structure (g0, β), the sensitivity of solute concentration and osmotic potential of guard cells to photosynthetic activities (α), and the efficiency of water transportation from soil to leaf (gz). The model was applied to 11 species of shrubs, pines and broad-leaf trees in Heshan Ecological Station, and was shown to be capable of describing the stomata behaviour and transpiration in dry and wet conditions. The following was concluded from the application: (1) stomatal conductance of pines and shrubs decreased the least with soil water stress largely because of their greater elastic modulus or smaller compliance of the guard cell structure, thus these functional types were the most resistant to drought; (2) pine trees had the lowest and shrubs have the second lowest osmotic potential at dark and thus were more tolerant for drought conditions than broad-leaf trees; (3) stomatal conductance of broad-leaf trees respond least to variation in photosynthetic radiation because of their lowest sensitivity of guard cell osmotic potential to photosynthetic radiation; (4) stomatal conductance of pines had the strongest dependence on vapour pressure deficit and decreased more with increased vapour pressure deficit due to their low soil-to-leaf conductance, which may be explained by the inefficient tracheids in their xylem systems; (5) the results were supported by observational and anatomic evidence existing in the literature. However, the conclusions remain hypothetical until direct measurements on these parameters become available.
This research was jointly supported by the Chinese Ministry of Science and Technology under grant number G2000018605, and Chinese National Science Foundation under grant numbers 39899370 and 39725006.
Received 2 February 2002;received inrevised form 8 May 2002;accepted for publication 9 May 2002