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Keywords:

  • guard cell;
  • stomata;
  • vapour;
  • water potential

ABSTRACT

  1. Top of page
  2. ABSTRACT
  3. INTRODUCTION
  4. STOMATAL RESPONSES TO ΔW AND TL
  5. PREVIOUSLY PROPOSED MODELS OF STOMATAL CONDUCTANCE
  6. COMPARING CONDUCTANCE MODELS
  7. HELOX
  8. DISCUSSION
  9. IMPORTANT ABBREVIATIONS
  10. ACKNOWLEDGMENT
  11. REFERENCES
  12. Appendices

A new mechanism for stomatal responses to humidity and temperature is proposed. Unlike previously-proposed mechanisms, which rely on liquid water transport to create water potential gradients within the leaf, the new mechanism assumes that water transport to the guard cells is primarily through the vapour phase. Under steady-state conditions, guard cells are assumed to be in near-equilibrium with the water vapour in the air near the bottom of the stomatal pore. As the water potential of this air varies with changing air humidity and leaf temperature, the resultant changes in guard cell water potential produce stomatal movements. A simple, closed-form, mathematical model based on this idea is derived. The new model is parameterized for a previously published set of data and is shown to fit the data as well as or better than existing models. The model contains mathematical elements that are consistent with previously-proposed mechanistic models based on liquid flow as well as empirical models based on relative humidity. As such, it provides a mechanistic explanation for the realm of validity for each of these approaches.


INTRODUCTION

  1. Top of page
  2. ABSTRACT
  3. INTRODUCTION
  4. STOMATAL RESPONSES TO ΔW AND TL
  5. PREVIOUSLY PROPOSED MODELS OF STOMATAL CONDUCTANCE
  6. COMPARING CONDUCTANCE MODELS
  7. HELOX
  8. DISCUSSION
  9. IMPORTANT ABBREVIATIONS
  10. ACKNOWLEDGMENT
  11. REFERENCES
  12. Appendices

Stomatal responses to air humidity and leaf temperature have been studied for many years and in many species. Despite a wealth of phenomenological data, however, there are no commonly accepted mechanisms for these responses. This situation has arisen in part because the two responses are highly interrelated, and in part because it is unclear exactly what is being sensed as stomata respond to air humidity and leaf temperature. Most mechanisms that have been proposed for the stomatal response to air humidity assume that the response is actually caused by a change in the water potential of the leaf (or some component of the leaf), which in turn is caused by a change in the rate of diffusion of water vapour from the leaf to the air. This diffusion of water vapour, transpiration (E), is driven by the difference in the chemical activity of water vapour between the inside and outside the leaf, which for typical conditions reduces to the difference in the mole fractions of water in air (Δw). For this reason, stomatal conductance (gs ≡ Ew) is plotted against Δw in most studies on the stomatal response to humidity. The internal mole fraction (wL) is assumed to be close to saturation at the temperature of the leaf, TL.

It is important to recognize that Δw (= wL − wa) can be altered either by changing the ambient water vapour mole fraction, wa, or by changing TL to change the internal mole fraction, wL and that the observed response of gs to changes in Δw is much different for the two procedures (Ball, Woodrow & Berry 1987; Grantz 1990; Fredeen & Sage 1999). The interpretation of the latter experiment is complicated by any direct response of stomata to TL that might exist, and for that reason, most studies on the stomatal response to Δw use changes in wa (at constant TL) to effect changes in Δw. Furthermore, most existing models rely heavily on these data for validation of the Δw response and then simply add a temperature response to the model coefficients to account for temperature effects. This is an unfortunate situation because although it is easy to produce variable wa at constant TL in a gas-exchange chamber, this situation probably occurs only rarely in a natural environment. The more likely scenario in a natural environment is that TL varies (because of changes in air temperature or irradiance) at constant wa. It is therefore critical that mechanisms for predicting stomatal responses to the environment capture this response in a robust and mechanistic way.

In this study we present a new mechanistic model for stomatal responses to leaf temperature and humidity. The model is fundamentally different from previous models in that the water supply to the guard cells is assumed to be primarily via the vapour phase rather than via the liquid phase, and the guard cells are assumed to be in near equilibrium with the water vapour in the intercellular spaces and the stomatal pore, respectively. The conductivity for liquid water flow between the xylem and the guard cells is assumed to be low, as is that between the epidermal cells and guard cells. This allows the epidermal and guard cells to equilibrate at different points in the leaf with different water potentials. Our interest in a new mechanism was based on the inability of existing mechanisms to adequately and concisely account for all aspects of the observed responses to humidity and temperature. Our approach was motivated by the fact that stomata in isolated epidermes open and close in response to changes in air humidity (Lange et al. 1971; Shope, Peak & Mott 2008), and that these stomatal movements have been shown to be consistent with water potential equilibrium between the guard cells and the air near the bottom of the stomatal pore (Shope et al. 2008). We show that a model based on this mechanism is consistent with observed responses to wa and TL and that it can explain many of the aspects of these responses that are not easily accounted for by existing mechanisms and models.

STOMATAL RESPONSES TO ΔW AND TL

  1. Top of page
  2. ABSTRACT
  3. INTRODUCTION
  4. STOMATAL RESPONSES TO ΔW AND TL
  5. PREVIOUSLY PROPOSED MODELS OF STOMATAL CONDUCTANCE
  6. COMPARING CONDUCTANCE MODELS
  7. HELOX
  8. DISCUSSION
  9. IMPORTANT ABBREVIATIONS
  10. ACKNOWLEDGMENT
  11. REFERENCES
  12. Appendices

There is substantial variation in stomatal responses to Δw and TL among species and even among plants of a single species grown under different conditions. There are, however, a number of characteristics that are qualitatively conserved and therefore must be captured by any proposed mechanism. These are explained in the further discussion.

Stomata close in helium : oxygen mixture at constant wa

The diffusion coefficient for water vapour in helium : oxygen mixture (helox) is 2.33 times that in normal air (nitrogen : oxygen) (nitox). Replacing air with helox at a constant wa causes stomata to close by approximately the same amount as increasing Δw by a factor of 2.3 (Mott & Parkhurst 1991). The usual interpretation of this result is that the stomatal response to wa is actually a response to the rate of water loss from the leaf.

At constant TL, stomatal close as wa is lowered

This experimental protocol is the most common way of determining the response of stomata to wa or Δw because by keeping leaf temperature constant the response to wa or Δw can be unequivocally separated from the response to TL (Buckley 2005). In most studies, gs declines monotonically as Δw increases. The degree of curvature in the relationship is highly variable, however.

Stomatal closure in response to decreasing wa is sometimes sufficient to result in constant E or even declining E as Δw increases

Simple feedback mechanisms demand that E continues to increase as Δw increases, and this is the case in most studies, particularly at low values of Δw. In some studies, however, stomatal closure in response to declining wa (increasing Δw) at constant TL is sufficient to maintain a constant E (Saliendra, Sperry & Comstock 1995; Tardieu & Simonneau 1998; Fisher et al. 2006), or even cause decreases in E (Farquhar 1978; Monteith 1995), as Δw increases. The former response has been termed isohydric behavior and the latter has been termed feedforward. There has been some controversy as to whether feedforward responses are reversible (Franks, Cowan & Farquhar 1997), and it has been suggested that these responses might be caused by xylem cavitation at high transpiration rates (Buckley 2005).

At constant wa, stomata close only slightly as TL is increased

There are only a few studies in the literature that report these data (Ball et al. 1987; Grantz 1990; Fredeen & Sage 1999; Mott & Peak 2010), and this is a difficult experiment to interpret because two variables (TL and Δw) are varied simultaneously. The increase in Δw should cause a large reduction in gs (see earlier discussion), and the fact that gs decreases only slightly is usually attributed to an independent opening response to TL, which counteracts the closing response to Δw.

At constant, non-zero Δw, stomata open substantially as TL is increased

This experiment is rare in the literature because it is technically difficult to maintain Δw constant (by changing wa) as wL varies with TL. There are several studies that show a large increase in gs in response to TL (Fredeen & Sage 1999; Mott & Peak 2010), however, and this supports the idea that the slight stomatal opening in response to increasing TL is because of a large direct effect of temperature on guard cells that overcomes the closing response to increased Δw (see earlier discussion).

At a constant Δw value of zero, stomata open only slightly as TL is increased

There is only one report of this experiment in the literature (Mott & Peak 2010), and in that study the response at Δw ≈ 0 was shown to be considerably smaller than that at a moderate value of Δw. In addition, Fredeen & Sage (1999) showed that the response to temperature was smaller at a vapour pressure difference of 1 kPa than at vapour pressure differences of 2 or 3 kPa. The results are inconsistent with the idea that the slight stomatal opening in response to increasing TL at constant wa is because of a large independent effect of temperature on guard cells that overcomes the closing response to increased Δw (see earlier discussion).

When wa is lowered for one surface of an amphistomatous leaf but maintained constant for the other surface, stomata on the first surface close, but stomata on the other surface are unaffected

Although this result has been shown in only one study, it was demonstrated in two herbaceous species in that study (Mott 2007). Furthermore, epidermal turgor pressure was reduced on the surface for which humidity was lowered, but it was constant for the other surface. This result places some important constraints on possible mechanisms, and these are discussed in the following.

PREVIOUSLY PROPOSED MODELS OF STOMATAL CONDUCTANCE

  1. Top of page
  2. ABSTRACT
  3. INTRODUCTION
  4. STOMATAL RESPONSES TO ΔW AND TL
  5. PREVIOUSLY PROPOSED MODELS OF STOMATAL CONDUCTANCE
  6. COMPARING CONDUCTANCE MODELS
  7. HELOX
  8. DISCUSSION
  9. IMPORTANT ABBREVIATIONS
  10. ACKNOWLEDGMENT
  11. REFERENCES
  12. Appendices

Empirical models

Empirical models are of limited interest in the present analysis because they do not postulate specific mechanisms for the stomatal response to wa or TL. Nevertheless, they have been shown to accurately predict stomatal responses under a wide variety of conditions, and it useful to compare their predictions with those of mechanistic models.

The most widely used empirical model was proposed by Ball et al. (1987) who suggested that responses of gs to light, temperature and humidity can be usefully approximated by

  • image

where A is the CO2 assimilation rate, cs is the CO2 concentration at the leaf surface, h is the relative humidity at the leaf surface, and a and b are constants. In practice, A and cs do not vary substantially with wa or TL, and most of the stomatal response to wa and TL is contained in h. We therefore consider a generalization of the Ball et al. (1987) model of the form

  • image(1)

in which gs depends linearly in h and where GBWB is a temperature-dependent coefficient. Because of its similarity to the empirical model earlier, we call Eqn 1 the ‘BWB’ model. Like the empirical ratio aA/cs in the original formula, the function GBWB is assigned no mechanistic explanation. For most well stirred gas-exchange chambers, ws ≈ wa so that h = wa/wL = 1 − Δw/wL, where Δw = wL − wa.

Leuning (1995) pointed out that humidity responses are often better described by replacing h in the Ball et al. (1987) formula by 1/(1 + ZΔw), where Z is a dimensionless empirical constant. This leads to the conductance model that we will call the ‘BBL’ model:

  • image(2)

where GBBL is a temperature-dependent empirical coefficient.

Mechanistic models

All previously proposed mechanistic models rely on reductions in water potential somewhere in the leaf that are caused by evapouration. These mechanisms fall into two categories. In the first category, evapouration occurs directly from the guard cells into the transpiration stream, and water is supplied to the guard cells via liquid flow. The water potential of the guard cells is therefore directly reduced by water loss from the guard cells, and high values of Δw produce high rates of water loss, resulting in stomatal closure (Farquhar 1978; Maier-Maercker 1983; Dewar 1995; Buckley 2005). In the second category, guard cell osmotic pressure (and therefore turgor pressure) is assumed to respond to a chemical signal that causes stomatal closure and originates from some other evapourating site within the leaf. The strength of this signal is inversely proportional to the water potential (or turgor pressure) of the cells of this evapourating site (Buckley, Mott & Farquhar 2003). High values of Δw produce low water potentials at the evapourating site, creating a strong closing signal for guard cells. Both of these mechanisms create a feedback system in which the turgor pressure of the guard cells is controlled by transpiration rate, and in their simplest form, these models predict that stomatal responses to Δw should be identical when Δw is varied with wa or wL. This means that the observed small stomatal response to Δw when Δw is varied with wL must be attributed to an independent effect of temperature on gs.

Both of these mechanistic models (Dewar 1995; Buckley et al. 2003) can be cast in the simplified form

  • image(3)

where the function gL0 is equal to gs when Δw = 0, and describes the temperature dependence of guard cell osmotic pressure. The term ZL(TLw (both parts of which are dimensionless) arises from the assumption that guard cell turgor pressure decreases with increasing transpiration rate (either because guard cell water potential is reduced directly, or because of a signal from other cells in the leaf). The temperature dependence of ZL(TL) has been suggested to reside in the effective resistance to liquid phase water flow (arising from physical and/or biochemical processes) somewhere along the path from the soil to the evapourating site(s) (Fredeen & Sage 1999; Matzner & Comstock 2001; Sack, Streeter & Holbrook 2004; Sack & Holbrook 2006). We call Eqn 3 the ‘L’ model.

Although the liquid phase models described earlier have been shown to explain much of the data for stomatal responses to humidity in both light and darkness, they have several weaknesses. Firstly, because they depend on feedback between transpiration and stomatal conductance, they are incapable of producing true isohydric behavior or feedforward without invoking other mechanisms. This deficiency has been addressed previously by noting that a strong feedback system can produce results that are experimentally indistinguishable from isohydric behavior, and processes such as cavitation in the xylem can produce irreversible feedforward responses (Buckley 2005) (note that the BBL model, Eqn 2, also cannot produce true, reversible feedforward, but the BWB model, Eqn 1, can. We return to this point in further discussion.).

A second problem for liquid phase models is that it is difficult to account for data in amphistomatous leaves in which wa is changed for one surface but not the other. Stomatal aperture and epidermal pressure respond on the surface for which wa is changed but not on the other surface (Mott 2007). Supposing that the water supply from the soil to the leaf epidermes is depicted as a T-shaped resistor network, as in Fig. 1, where Rs is the resistance from the soil, and ReU and ReL are the resistances from the leaf xylem (X in the Figure) to the upper and lower evapourating sites, then this result suggests that Rs << ReU, ReL, that is, most of the resistance to water flow is between the xylem and the evapourating sites. This idea is contrary to the prevailing consensus that the resistance of the xylem is of the same order of magnitude as the resistance between the xylem and the evapourating site (Sack & Holbrook 2006).

image

Figure 1. Resistance diagram for water flow in an amphistomatous leaf. Water flows from source (S), which is outside the leaf, through the xylem to point X where it leaves the xylem and flows to the evapourating sites for the upper and lower surfaces (eU and eL, respectively).

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A new, vapour phase mechanistic model

Our proposed model deviates from previous mechanistic models in that the guard cells are supplied with water primarily through the vapour phase rather than the liquid phase. Water is assumed to evapourate from sites within the leaf, and the water potential of the guard cells is assumed to be in near-equilibrium with the water potential of the air adjacent to them. The air adjacent to the guard cells has a different concentration of water vapour (because of the water vapour gradient to the outside air) and a different temperature (because of evapourative cooling) from the air adjacent to the evapourating site. As wa and TL vary, the water potential of the air adjacent to the guard cells changes, and the guard cells shrink and swell as they equilibrate with the water vapour in the air.

The pathway of water movement through leaves is controversial, and the exact location of the evapourating sites within the leaf is not specified in the model. They must, however, be in close hydraulic contact with the xylem water, and there must be a substantial amount of heat transfer to them via air (either external or intercellular). For simplicity, we assume that epidermal and mesophyll cells exchange liquid water and are in liquid phase equilibrium with nearby xylem.

The model assumes that there are several sites along the transpiration pathway at which liquid-phase water is in approximate isothermal equilibrium with vapour phase water. This equilibrium is represented mathematically by:

  • image(4)

where the left side of the equation represents the vapour phase, and the right side represents the liquid phase; R is the ideal gas constant, T is the air temperature (K) at the point of interest, w and wsat are the mole fraction of vapour and the saturated mole fraction of water vapour, respectively, P is the turgor pressure of the cell, and c is the concentration of solutes in the cell. The term cRT equals the osmotic pressure of the cell. Although the different sites along the transpiration pathway are at different temperatures, Eqn 4 is locally valid at each site as long as the liquid and vapour phases are isothermal. As shown in Appendix A, this is because the temperature dependence of wsat exactly compensates for the difference in the change of entropies between the liquid and vapour phases, and Eqn 4 follows immediately from Eqn A2 in Appendix A. Because RT/vL is roughly 140 MPa for temperatures near 300 K (i.e. for temperatures relevant to laboratory experiments), small fractional changes in w or wsat can produce large absolute changes in water potential. For example, near 300 K, a 1% decrease in w produces about a 1.4 MPa decrease in water potential, whereas a 1% increase in wsat also produces about a 1.4 MPa decrease in water potential. In our discussions of the vapour phase model, we assume that TL (the measured temperature of the leaf) is the temperature of the epidermis and guard cells. Again for simplicity, we assume the mesophyll and epidermis are in good thermal contact and the mesophyll is also at TL. Evapourating sites are cooler. Note that at temperature TL, wsat = wL.

Equation 4 is used firstly to determine the mole fraction of water in the air, wi, adjacent to the evapourating sites (together, designated i) based on the liquid-phase of water potential at those sites. In most gas-exchange calculations, wi is assumed to be equal to the saturated vapour pressure (wsat) at the temperature of the leaf, that is, the water potential of the evapourating sites is assumed to be zero. This approximation is valid for gas-exchange calculations because the reduction in mole fraction from saturation is small for the moderately negative water potentials that are likely to exist for the leaf (−1.4 MPa equates to approximately 1% reduction in saturated mole fraction of water vapour). However, in our model the evapourating sites are cooled slightly by evapouration, and the corresponding small reduction in wi has important consequences.

Water vapour is assumed to diffuse from the air adjacent to the evapourating sites to the outside air in response to the mole fraction difference between the two places. The gradient in water vapour along this pathway is determined by the diffusional resistance along the pathway (i.e. through the intercellular spaces, the stomatal pore and the boundary layer). Contributions to the diffusive flux caused by temperature gradients are assumed to be negligible, and it is therefore possible to calculate the mole fraction of water at any point along the pathway by knowing the proportion of the total diffusive resistance to that point. We use this concept to calculate a mole fraction of water in air (wp) at some hypothetical point p, which is adjacent to, and isothermal with, the guard cells. The value of wp is calculated by assuming that p exists at a constant fraction (represented by σ) of the resistance along the water vapour gradient between wi and wa such that

  • image(5)

In practice the proportion of the total resistance from i to p is only a few percent so σ is less than 0.1, and the difference in mole fraction between wi and wp is small but still important for this analysis. Using Eqn 4, the guard cells are assumed to exchange water with the air in the pore until their water potential is equal to that in the pore air.

The structure and function of a typical stomatal unit in the proposed model is schematically represented in Fig. 2. The essential features of the diagram are: (1) liquid water is drawn from source S (with flux E) and ends up as vapour in the external air (a); (2) in between, liquid water is exchanged in equibrium at mesophyll (M) and epidermal (e) sites; (3) point (i) is the principal ‘evapourating site’ (though, of course, evapouration is likely to be distributed over the flow); (4) in steady state, liquid water in the guard cell (g) is in equilibrium with vapour in the stomatal pore (p).

image

Figure 2. Schematic representation of water flow in the vapour phase model. Solid lines indicate liquid water flow; dashed lines indicate water vapour flow. Water evapourates at site i and diffuses with flux E through the stomatal pore (p) into the surrounding air (a). The diffusive flow draws liquid water from source S with flux E. In steady state, liquid water in the epidermis is in equilibrium with liquid water in the mesophyll and in the xylem at site M. Liquid water in the guard cells (g) is in equilibrium with vapour in the pore at site p.

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Stomatal responses

To understand how the model works, it is useful to consider how the model produces stomatal responses to wa and temperature. We first examine the stomatal response to a decrease in wa at constant leaf temperature. In a typical gas-exchange experiment, this would be accomplished by reducing the mole fraction of the air entering the leaf chamber and increasing the temperature of the air in the chamber to maintain leaf temperature constant as the rate of evapourative cooling increases with transpiration. Empirically, we know that this treatment causes a substantial reduction in stomatal aperture, and in our model, this reduction occurs primarily for two reasons. Firstly, reducing wa directly reduces the value of wp as per Eqn 5. Secondly, reducing wa increases Δw, which in turn increases the transpiration and lowers the temperature of the evapourating site relative to the epidermis (which is maintained constant in the gas-exchange chamber by changing the air temperature). As a result, wi also decreases as wa decreases, contributing an additional reduction in wp. At constant epidermal temperature (TL) wL is constant, so wp/wL is reduced, and so is the water potential in the pore (per Eqn 5). Water therefore flows out of the guard cells and aperture is reduced. It is noteworthy that in our model, changes in water potential drawdown through the liquid phase from the source to the evapourating site can contribute to the stomatal response, but for the model parameterization described here, they are assumed to be negligible.

Related events occur at the epidermis, but the magnitude of the changes is much smaller because the resistance to liquid flow between S and M is assumed to be small. Increased flow rate produces a small drawdown in water potential at sites M and e. These changes have the opposite effects on stomatal aperture (i.e. reductions in water potential in the epidermis tend to open the pore rather than close it) and if these changes occur before those in the guard cells, it would result in the observed ‘wrong-way’ response of stomata to humidity. In the steady state, however, the effects on guard cell water potential dominate, and even after accounting for the mechanical advantage of the epidermis, stomata close when wa is decreased.

We next consider an increase in leaf temperature at a constant wa. This experiment is rarer in the literature than the one earlier, and is typically performed by raising air temperature to affect leaf temperature while adjusting the humidity of the air entering the chamber to maintain wa constant as transpiration increases with leaf temperature. Empirically, stomata close as temperature is increased, but only slightly. Our model produces this result because there is very little effect on the ratio wp/wL when leaf temperature is changed at a constant wa. As leaf temperature increases, both wi and wL increase by roughly the same amount. Thus, whereas the minor contribution to wp/wL– that is, σwp/wL– decreases, the major contribution – (1 − σ)wp/wL– is almost constant. The net effect is that because the ratio wp/wL declines only slightly as TL increases, stomata close only slightly in response to this treatment. As shown earlier, related events occur at the epidermis, but the magnitude of the changes is much smaller and the effects on guard cell water potential dominate.

The model

Using the assumptions and theory described earlier, we have derived a closed-form expression that describes stomatal responses to wa and leaf temperature. A full derivation of the model is presented in Appendix B (Derivation of the Vapour Phase Model), where we show that conductance in our model can be expressed as

  • image(6)

where Θ (with units of conductance per K) and ZV (dimensionless) are constants, the values of which depend on various aspects of internal leaf anatomy. Again, as in Eqn 3, g0 is equal to gs when Δw = 0. We call Eqn 6 the ‘V’ model.

Equation 6 is the central result of the work reported here. Rewriting Eqn 6 in terms of humidity,

  • image

shows that the vapour model contains elements of both the BWB and BBL models. It therefore potentially explains why some plants show stomatal responses that are best described by the BWB model, whereas others show responses that are best described by the BBL model, and yet others show intermediate behavior. For plants with neglibible ZVΔw, the vapour model is linear with h and approximates BWB with b = gV0 − ΘTL and GBWB = ΘTL, and admits strong feedforward. For plants with negligible ΘTL, the vapour model approximates BBL with b = 0 and GBBL = gV0 and is incompatible with feedforward (provided, as we assume, that g0 does not vary as wa varies). For cases in which neither ZVΔw nor ΘTL is negligible the model shows intermediate behavior. As both ZV and Θ are related to leaf anatomical properties in our vapour phase model, the model provides many easily testable hypotheses related to these issues. Later we show, as well, that the vapour phase model naturally explains why transpiration responses to changes in humidity over the surfaces of amphistomatous leaves are independent of one another and that the model is compatible with experiments in which normal air (nitox) is replaced by helox.

COMPARING CONDUCTANCE MODELS

  1. Top of page
  2. ABSTRACT
  3. INTRODUCTION
  4. STOMATAL RESPONSES TO ΔW AND TL
  5. PREVIOUSLY PROPOSED MODELS OF STOMATAL CONDUCTANCE
  6. COMPARING CONDUCTANCE MODELS
  7. HELOX
  8. DISCUSSION
  9. IMPORTANT ABBREVIATIONS
  10. ACKNOWLEDGMENT
  11. REFERENCES
  12. Appendices

To assess the validity of our model, we compare it with the BWB, BBL and L models using experimental data reported in a previous study for Tradescantia pallida (Mott & Peak 2010). These data were taken in darkness, with fixed [CO2] and [O2], so the influence of temperature on photosynthesis and ci was negligible. It is reasonable, then, to assume that stomatal responses to changes in Δw and TL in these experiments are independent of responses to A or CO2. It should be noted, however, that for the data of Mott & Peak (2010) responses in darkness were shown to be similar to those in light, which suggests that the mechanisms operating in the dark are similar to those in the light and that the influence of photosynthesis and ci on the responses in the light were small in that study.

Strictly speaking the empirical model of Ball, Woodrow and Berry cannot be applied to stomatal responses in darkness because A/cs will be negative under this condition. For the purposes of this analysis therefore we assume that Eqn 1 substitutes for the BWB empirical formula when stomata are open in darkness. We use these data to compare the models of Eqns 1–3, 6. We find that the BBL model, Eqn 2, provides a much poorer fit than the other models for temperature variations conducted at finite Δw. For this reason we compare only the BWB model, the liquid phase model (L), and the vapour phase model (V).

Mott & Peak (2010), report four separate experiments: (a) temperature dependence of stomatal aperture at high humidity, h ≈ 1, Δw ≈ 0; (b) Δw dependence of gs at constant temperature, 25 °C, achieved by varying wa; (c) temperature dependence of gs at a moderate humidity, Δw = 0.017; and (d) temperature dependence of gs at constant wa = 0.020. As described in Appendix C, we normalized these data and then parameterized the BWB, L and V models with the results of experiments (a)–(c). We then used each parameterized model to ‘predict’ results of experiment (d).

Figure 3 shows the response of gs to TL at near-saturating humidity. There is little if any transpiration at high humidity, so stomatal apertures (rather than stomatal conductance) were measured directly in this experiment, and gs was assumed to be linearly proportional to aperture. These data are important because they define the response of the guard cells to temperature independent of any effects of transpiration, and can therefore be used to define the temperature function of each model [GBWB(TL), gV0(TL) and gL0(TL)]. We approximate the relatively flat stomatal response at high humidity shown in Fig. 3 by the linear function shown by the solid line in that figure. Figures 4 and 5 show that all three models can be fit to the data when Δw is varied at constant TL and when TL is varied at constant Δw. However, fitting the L model requires a large temperature dependence for ZL. As discussed earlier, ZL is thought to arise from the resistance to liquid water flow to the evapourating site, and the temperature dependence of ZL is therefore thought to be caused by changes in viscosity of water with temperature, which change the effective resistance. However, the temperature dependence of ZL required for the liquid model to fit the constant Δw data is much larger than the temperature dependence of water viscosity (Fig. 6).

image

Figure 3. Stomatal apertures as a function of TL when Δw = 0. Gray squares show the data of Mott & Peak (2010) normalized to TL = 25 °C and Δw = 0.017 as described in the text. Error bars show one standard error on either side of the mean. All models are constrained to the linear fit shown.

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image

Figure 4. gs as a function of Δw when TL = 25 °C. Gray squares show the data of Mott & Peak (2010) normalized to TL = 25 °C and Δw = 0.017 as described in the text. Error bars show one standard error on either side of the mean. Open diamonds, filled triangles and filled circles show the fits of the vapour phase, BWB and L models, respectively.

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image

Figure 5. gs as a function of TL when Δw = 0.017. Gray squares show the data of Mott & Peak (2010) normalized to TL = 25 °C and Δw = 0.017 as described in the text. Error bars show one standard error on either side of the mean. Open diamonds and filled triangles show the predictions of the vapour phase and BWB models, respectively.

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image

Figure 6. Temperature dependence of the parameter ZL(T) in the L model and temperature dependence of water viscosity. Both values are normalized to 1 at 20 °C.

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Finally, we use models as parameterized in Figs 3–5 to ‘predict’ the conductance values from the fourth experiment (d), of Mott & Peak (2010), in which Δw is varied at constant wa. In the original data for experiment (d) of Mott & Peak (2010), TL ranged from 20 to 30 °C and Δw ranged from 0.007 to 0.028. To provide an even more stringent test of the models, we followed the protocol of Mott & Peak (2010), to extend the temperature range to 35 °C and the Δw range to 0.045. The results are shown in Fig. 7. In the parameterization range, the BWB, L and V models fit the data indistinguishably, as would be expected. Outside the parameterization range, on the other hand, the models exhibit differences. At the highest temperatures the L model significantly overestimates the experimental data and the BWB model slightly underestimates it. The vapour phase model fits the conductance data better than either the BWB or the L model at high values of Δw and TL.

image

Figure 7. gs as a function of Δw at constant wa of 0.020. Δw was changed by varying TL, and TL values are given for each data point. Gray squares show the data of Mott & Peak (2010) normalized to TL = 25 °C and Δw = 0.017 as described in the text. Error bars show one standard error on either side of the mean. Open diamonds, filled triangles and filled circles show the predictions of the vapour phase, BWB and L models, respectively. The data in the region labeled ‘parameterize’ were used to parameterize the three models.

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HELOX

  1. Top of page
  2. ABSTRACT
  3. INTRODUCTION
  4. STOMATAL RESPONSES TO ΔW AND TL
  5. PREVIOUSLY PROPOSED MODELS OF STOMATAL CONDUCTANCE
  6. COMPARING CONDUCTANCE MODELS
  7. HELOX
  8. DISCUSSION
  9. IMPORTANT ABBREVIATIONS
  10. ACKNOWLEDGMENT
  11. REFERENCES
  12. Appendices

Any model of stomatal response to humidity also has to explain the nature of the observed aperture closure when nitox is replaced with helox at constant leaf temperature and constant wa. Not only do apertures close when helox is introduced, they close by approximately the same amount as when Δw is increased to 2.3Δw in nitox. In other words, the two experiments appear to reduce the turgor pressure of the guard cells by about the same amount. This is easily explained in a liquid phase model because these models depend on a transpiration-induced drawdown in water potential, either directly to the guard cells or to another part of the leaf, to effect changes in stomatal aperture. Thus, switching to helox should close stomata by exactly the same amount as increasing Δw by a factor of 2.3 (by decreasing wa) in nitox. Therefore, in a liquid phase model there is perfect symmetry between helox and ‘2.3Δw’.

This symmetry does not automatically exist in the vapour phase model, however. In that model the guard cell water potential is determined by the water potential of the vapour in the pore, and that in turn is determined by the water vapour mole fraction in the pore, wp = σwa + (1 − σ)wi. Because of this, increasing Δw by decreasing wa leads to a change in the guard cell water potentials through two effects. Firstly, lowering wa has a direct effect on wp, and second, lowering wa causes wi to decrease because the associated increase in transpiration rate lowers the temperature of the evapourating site. On the other hand, introduction of helox changes wp only by the first of the two mechanisms,that is by changing wi; it does not change wa. Since E increases (initially) by the same amount in both experiments, it is tempting to conclude that wi should be reduced by the same amount (initially) in both experiments and that therefore the reduction in wp is less in helox than in ‘2.3Δw.’ In fact, invoking this assumption and using the parameter values of the vapour phase model we have extracted for Tradescantia pallida, we estimate that guard cell water potential should decrease by only a factor of 1.3 when helox is introduced, not 2.3. This difference would seem to unambiguously exclude the vapour phase model. But, as we argue in further discussion, the introduction of helox has two effects: it increases E and it also decreases the rate of heat transport to the evapourating site. The combined effect is to lower the temperature of the evapourating site by more than transpiration alone and consequently to reduce wi by more than would be expected.

In our vapour phase model, the evapourating site within the leaf is slightly (a few tenths of a degree C) cooler than the epidermis. This difference in temperature is possible we assume because there is poor liquid phase contact between the two places. Heat transfer between the epidermis and the evapourating site is therefore largely through the vapour phase. The thermal conductivity of helium is about six times higher than nitrogen, so it seems reasonable to suggest that in helox the temperature difference between the epidermis and evapourating sites should be smaller in helox than in nitox and that the vapour phase model should predict almost no change in guard cell water potential upon switching helox. But, the thermal conductivity of a gas only measures the rate of transfer of energy from molecule-to-molecule in the gas, not the more important process of transfer across the gas–liquid interface. Essential to the latter is the mismatch between the effective masses of the molecules in the two different media. In the surface, the molecules are densely packed, so that when a projectile gas molecule collides with a target molecule in the surface it effectively interacts with many molecules simultaneously. In other words, a target molecule in the liquid is effectively much heavier than if it were in the gas phase (Velic & Levis 1997). This mismatch causes the energy transport to be much less efficient than if the molecules were closer in mass (as in gas-to-gas or liquid-to-liquid transfer). When nitox is replaced by helox the mass mismatch between gas and liquid molecules increases and heat transfer actually decreases (a simple molecular collision argument explaining this is given in Appendix E). The result is that the temperature difference between the epidermis and the evapourating site is actually greater in helox than in nitox for a given transpiration rate, producing a larger decrease in conductance.

To see whether experimental data support this theoretical argument we attached a thermocouple to an aluminum plate in an enclosed chamber. One surface of the attached thermocouple was exposed to nitox or helox (depending on the experiment); the other was in contact with the plate. A second thermocouple was embedded in the plate to measure plate temperature, and a third was placed at a distance from the plate surface beyond the thermal boundary layer to measure the temperature of the gas. In this experimental setup, the temperature of the first thermocouple was always intermediate between the temperatures of the plate and that of the gas, and the precise temperature was determined by the heat transfer rates between the plate and the thermocouple and between the gas and thermocouple. With plate temperature fixed at 25 °C, the thermocouple temperature was found to be closer to the temperature of the plate in helox than in nitox in all cases (data not shown), and the data show that heat transfer from gas to the thermocouple was more than twice as effective in nitox as in helox. Such a reduction in interface heat transport when nitox is switched to helox must occur in the vapour model as well and results in a commensurate increase in the temperature difference between the epidermis and the evapourating site. We therefore find that for Tradescantia pallida, the vapour phase model predicts aperture reductions when nitox is switched to helox and when Δw is switched from 0.010 to 0.023 that are within about 10% of one another. In other words, the helox response in Tradescantia pallida can be explained in terms of vapour phase physics only.

DISCUSSION

  1. Top of page
  2. ABSTRACT
  3. INTRODUCTION
  4. STOMATAL RESPONSES TO ΔW AND TL
  5. PREVIOUSLY PROPOSED MODELS OF STOMATAL CONDUCTANCE
  6. COMPARING CONDUCTANCE MODELS
  7. HELOX
  8. DISCUSSION
  9. IMPORTANT ABBREVIATIONS
  10. ACKNOWLEDGMENT
  11. REFERENCES
  12. Appendices

We present a new mechanistic model for stomatal responses to humidity and temperature that is based on the transport of water via the vapour phase within the leaf. In most previous mechanistic models, changes in water potential within the leaf (either to the guard cells or to the evapourating site) were assumed to be caused by liquid phase resistance within the leaf. In our model, however, changes in water potential of the guard cells are caused by two factors. The first is a small temperature difference between the evapourating sites and the epidermis that is caused by evapouration of water. To produce the necessary changes in water potential, temperature differences need only be a few tenths of a degree. The second is vapour phase resistance between the evapourating sites and the guard cells. As water diffuses from the evapourating sites to the atmosphere, there is a gradient in water vapour concentration along the pathway, which is determined by resistance to diffusion. We assume that guard cell water potential equilibrates with the water vapour at some point along this pathway. For the model parameterization reported here, the diffusion resistance from the evapourating sites to the guard cell is approximately 2% of that between the evapourating sites and the atmosphere (i.e. through the stomata).

We note that a very different view of stomatal and epidermal hydration via the vapour phase has recently been suggested as an explanation for apparent stomatal responses to radiation (Pieruschka, Huber & Berry 2010). In that study, the water status of the epidermis was assumed to be controlled by the difference between radiation-induced evapouration from the mesophyll and the amount of water vapour exiting the stomata. The mesophyll was assumed to be warmer than the epidermis and guard cells, and the study does not address how stomata adjust to differences in the water status of the epidermis.

We demonstrate that our vapour phase model can be parameterized to fit the data for stomatal responses to TL and wa published by Mott & Peak (2010). In addition, the model does a slightly better job of predicting data outside the range for which it was parameterized than does the Ball–Berry model or liquid-phase mechanistic models. This alone should make the model worth further investigation, but there are a number of other aspects of the model that make it superior to existing models.

First and foremost, the model provides a mechanistic explanation for why both the empirical Ball–Berry model and the mechanistic liquid-phase models do a reasonable job of predicting stomatal responses to wa and TL. It also explains why neither is perfect for many plants. The vapour phase model contains elements of both the Ball-Berry and the hydraulic (i.e. liquid phase) models, and it can therefore produce behavior similar to either of them or it can produce behavior that is intermediate between the two, depending on the parameter values. Examining Eqn 6, we can see that when the parameter Θ is small, the model approximates the hydraulic model and predicts a gradual, curved decrease of gs as Δw increases at constant temperature. Neither isohydric behavior nor feedforward is observed. When ZV is small, the model approximates the Ball-Berry model and predicts a linear decrease of gs with increasing Δw. As a result, E increases with Δw at low values of Δw but it becomes approximately constant (isohydric behavior) or even declines (feedforward) at higher values of Δw (see Appendix B). This provides a mechanistic explanation for isohydric and feedforward responses without relying on secondary effects such as ABA production or xylem cavitation. Although it is possible that in some cases, isohydric behavior or feedforward might be caused by these secondary (and irreversible in the short term) effects, our model provides a simple, reversible mechanism for these behaviors. When neither Θ nor Zv is small, the model produces intermediate behavior. It seems reasonable to speculate that the relative values of Θ and Zv could vary among plants, and this could lead to the diversity in observed responses of gs to Δw and TL in the literature.

Secondly, the model produces the experimentally observed responses of gs to TL (at constant Δw and at constant wa) without relying on large intrinsic responses of gs to temperature, which are difficult to explain mechanistically. The data of Mott & Peak (2010) show that if Δw is held constant as TL is varied, gs is steeply dependent on TL at nonzero values of Δw, but when Δw ≈ 0, the effect of TL on gs is much smaller. This suggests that the temperature response of gs is in some way dependent on Δw and places constraints on possible mechanisms. In liquid phase models, the temperature dependence of gs arises in two places. The first is an inherent response of gs to TL. This is assumed to be a metabolic response of the guard cells or of some other process in the leaf that influences gs, and it seems reasonable to conclude that this response would be present even when Δw = 0. Based on the data of Mott & Peak (2010), however, this response is small. This means that most of the temperature dependence in liquid phase models and in our model arises only when Δw is non-zero. In liquid phase models this temperature dependence must therefore be contained in the function, ZL, which is related to the resistance for water flow to the evapourating sites. The temperature dependence in ZL has been attributed to changes in hydraulic resistance caused by changes in the viscosity of water with temperature. As has been noted previously (Fredeen & Sage 1999) and as we show in Fig. 6, however, the temperature dependence of ZL is larger than can be explained by the decrease in viscosity of water with increasing temperature, and hydraulic models must therefore rely on an as-yet unidentified temperature effect such as an effect on aquaporins (Sack & Holbrook 2006) to produce the observed responses. In our model most of the temperature effect is contained in the parameter wL, which appears in both the Θ and the Zv terms. The direct dependence of gs on temperature is small, as shown in Fig. 3.

Thirdly, the model also explains why stomata on the two surfaces of amphistomatous leaves respond to humidity independently of one another (see Appendix D). Stomatal aperture is controlled by differences in temperature and water vapour concentration between the evapourating site and the guard cell rather than by differences in water potential caused by liquid flow. If we assume that the evapourating sites are separate for the two surfaces and that small temperature differences can exist between them, then stomatal responses will be independent for the two surfaces.

Fourthly, the model potentially explains data for rehydration kinetics of leaves that suggest the existence of two hydraulic compartments within leaves (Tyree et al. 1981; Zwieniecki, Brodribb & Holbrook 2007). These studies show two distinct water uptake phases with distinctly different time constants as leaves rehydrate from low water potentials, and this suggests that there are at least two hydraulic compartments within leaves. Although it is clear that these two compartments have very different effective resistances for rehydration, it is unclear whether the two compartments are connected via a serial or parallel pathway (Tyree et al. 1981). The size of the compartment associated with the faster of the two rehydration phases suggests that it consists only of the xylem and the bundle sheath extensions in some species, but includes the epidermis and spongy mesophyll in others (Zwieniecki et al. 2007). We speculate that the faster phase might be associated with portions of the leaf that are connected via liquid pathways and the slower phase with those that are supplied primarily through the vapour phase. This might also provide an explanation for the apparent dependence of leaf hydraulic conductivity on illumination (Sack et al. 2002; Nardini, Salleo & Andri 2005). If illumination changes the temperature profile inside the leaf, it could easily change the apparent resistance to water flow if some of the flow is through the vapour phase.

One of the most important arguments against a mechanism that is based on water potential equilibrium rather than water loss rate is the effect of helox on stomatal aperture (Mott & Parkhurst 1991). As helox closes stomatal aperture by approximately the same amount as a 2.3× increase in Δw, the simplest explanation for the helox effect is that stomata respond to the water loss rate. However, we present a primarily physical alternative explanation for this effect that is based on the effectiveness of heat transfer between the gas and liquid phases in the leaf. This is a fundamentally different interpretation of the helox phenomenon, and we expect to expand and test this idea in a subsequent study.

Because the model is mechanistic in nature, it can be used to generate testable hypotheses relating to leaf anatomy and stomatal responses. For example, the parameter Θ contains two plant-dependent components. The first is the scaling factor, χ, between stomatal conductance and the turgor pressures of the guard and epidermal cells (see Appendix B), which will depend primarily on stomatal density and size. The second, σ, is a ratio of two resistances for water vapour diffusion: that between the evapourating site and the guard cells and that between the evapourating site and the atmosphere. The value of σ should therefore depend on the position of the guard cells relative to the evapourating site, and all other things being equal, sunken stomata would be expected to have smaller values of σ, and exerted stomata should have larger values. Larger values of σ will cause larger responses of gs to Δw at constant TL. Similarly, the parameter Zv is related to the temperature differences between evapourating site and the guard cells. In Tradescantia pallida, these appear to be determined by heat transport across air-liquid interfaces, but in other plants with higher cell densities, liquid–liquid transport might be more important. Our model predicts that in those cases, stomatal responses to Δw and helox will be reduced.

In summary, we propose a new mechanism for stomatal responses to humidity and temperature that is based on a near-equilibrium in water potential between the guard cells and the air at the base of the stomatal pore. We construct and test a model based on this mechanism and show that it contains mathematical elements of previously proposed mechanistic models based on liquid flow as well as empirical models based on relative humidity. Because of this, the model provides a mechanistic explanation for the success and failure of the two approaches under different conditions and in different species, and it explains the variation among plants in observed responses. There are undoubtedly other factors such as xylem embolism and root signals that can affect stomata, and these might be responsible for some stomatal responses such as those to supply side perturbations. We suggest that our proposed mechanism could be responsible for most of the short term, reversible effects of humidity and temperature on stomata. Despite confronting the model presented here with a number of empirical challenges, we have not been able to convincingly falsify it nor reject the feasibility of a vapour phase hydration mechanism. We hope that our discussion will stimulate further research in this area.

IMPORTANT ABBREVIATIONS

  1. Top of page
  2. ABSTRACT
  3. INTRODUCTION
  4. STOMATAL RESPONSES TO ΔW AND TL
  5. PREVIOUSLY PROPOSED MODELS OF STOMATAL CONDUCTANCE
  6. COMPARING CONDUCTANCE MODELS
  7. HELOX
  8. DISCUSSION
  9. IMPORTANT ABBREVIATIONS
  10. ACKNOWLEDGMENT
  11. REFERENCES
  12. Appendices
ETranspiration ratemol m−2 s−1
TLLeaf (epidermis) temperature°C or K
wLSaturated mole fraction of water vapour in air at TLmol water/mol air
waMole fraction of water vapour in air in ambient airmol water/mol air
ΔwMole fraction difference between leaf and airmol water/mol air
gsStomatal conductancemol m−2 s−1
wsatSaturating mol fraction of water in airmol water/mol air
wiMol fraction of water at the evapourating sitemol water/mol air
pPoint at which guard cells equilibrate with the air in the pore 
wpMol fraction of water in air at point pmol water/mol air
BWBBall, Woodrow, Berry (model) 
BBLBall, Berry, Leuning (model) 
LLiquid (model) 
VVapour (model) 
σFractional conductance of point p along the diffusion pathway for water vapour 
hRelative humidity 
g0Stomatal conductance at Δw = 0mol m−2 s−1
ΘConstant that depends on leaf internal anatomymol m−2 s−1 K−1
ZVDimensionless constant that depends on leaf internal anatomy 
GTemperature-dependent factormol m−2 s−1

REFERENCES

  1. Top of page
  2. ABSTRACT
  3. INTRODUCTION
  4. STOMATAL RESPONSES TO ΔW AND TL
  5. PREVIOUSLY PROPOSED MODELS OF STOMATAL CONDUCTANCE
  6. COMPARING CONDUCTANCE MODELS
  7. HELOX
  8. DISCUSSION
  9. IMPORTANT ABBREVIATIONS
  10. ACKNOWLEDGMENT
  11. REFERENCES
  12. Appendices

Appendices

  1. Top of page
  2. ABSTRACT
  3. INTRODUCTION
  4. STOMATAL RESPONSES TO ΔW AND TL
  5. PREVIOUSLY PROPOSED MODELS OF STOMATAL CONDUCTANCE
  6. COMPARING CONDUCTANCE MODELS
  7. HELOX
  8. DISCUSSION
  9. IMPORTANT ABBREVIATIONS
  10. ACKNOWLEDGMENT
  11. REFERENCES
  12. Appendices

APPENDIX A – DERIVATION OF EQN 4

The condition for equilibrium between vapour phase water outside a cell and liquid phase water inside is that the chemical potentials for water in the two phases are equal. Assuming the two phases are at the same temperature this condition can be expressed as

  • image((A1))

In this expression, the Ps are total pressures in the two phases, xWV is the mole fraction of water in the gas phase (the same as w in the text), and xWL is the mole fraction of water in the cellular liquid solution. For conditions relevant to leaves a good approximation is to treat both vapour and liquid as ideal binary mixtures. In that case, µi(T,Pi,xi) = µi0(T,Pi) + RTln(xi), where µi0 is the chemical potential for pure water in phase i (Berry, Rice & Ross 1980).

The vapour phase chemical potential for a pure ideal gas is (Berry et al. 1980)

  • image

Here nQ and ZI are the ‘quantum concentration’ (in mol m−3) and the ‘internal partition function’ (dimensionless), respectively, and ε0 is the energy of the molecular ground state. Applying this to the case of water vapour in a ‘carrier’ gas, and adding and subtracting RT ln(psat), yields

  • image

where p is the water vapour partial pressure, p = xWVPV, and psat is the saturated water vapour pressure. The second term on the right is the chemical potential for pure water vapour at the saturated pressure at temperature T.

For pure liquid water, µ0L(T,PL) depends linearly on pressure because the molar volume of liquid water, vL, is approximately constant. Consequently,

  • image

For liquid water in an ideal solution

  • image

Though water is the minority component in the vapour phase in leaves, it is the majority component in the liquid phase. Replacing xWL by 1 − xsolute, and assuming xsolute << 1, leads to

  • image

Finally, because psat is defined as the pressure at which pure liquid water is in equilibrium with its vapour

  • image

As a consequence, Eqn A1 reduces exactly to

  • image((A2))

where c = xsolute/vL. Eqn A2 is Eqn 4 because p/psat = w/wsat and PL >> psat.

APPENDIX B – DERIVATION OF THE VAPOUR PHASE MODEL

As with other mechanistic models of stomatal conductance, the proposed new model is based on the relation

  • image((B1))

where Pg and Pe are guard and epidermal cell turgor pressures (in MPa), respectively, m (unitless, typically about 2) represents the mechanical advantage of the epidermis, and χ is an empirical coefficient (in mol/MPa·m2·s) related to stomatal size and density that scales the turgor-based expression to a value of stomatal conductance. In the proposed model, guard cells exchange water primarily with vapour in the stomatal pore and intercellular spaces, and under steady-state conditions, the liquid-phase water potentials within the cells are equal to the respective vapour phase water potentials outside them. Because of this, Eqn B1 is more usefully written as the equivalent expression

  • image((B2))

where the ψs are liquid phase water potentials and Δπ = πg − e is a weighted difference of osmotic pressures.

The structure and function of a typical stomatal unit in the proposed model is schematically represented in Fig. 2. In this model, the evapourative cooling rate (per unit area) of site i is LWE, where LW is the latent heat of vapourization of water (4.37 × 104 J mol−1 at 300 K). The heat supplied to balance this energy loss comes from the surroundings of the respective site, that is, from adjacent mesophyll tissue or from the air. Both such contributions are expressible as κΔT/L, where κ is an effective thermal conductivity across the water film interface and L is an effective heat conduction length. We assume that, in either case, ΔT equals the difference between TL (the assumed temperature of the epidermis and mesophyll and the air in the pore) and the temperature of the evapourating site. This leads to

  • image((B3))

where λ (with units K-m2·s mol−1) is of the form λ = LW/(κair/Lair + κmes/Lmes). We assume that λiE<<TL. We now turn to evaluating ψe and ψg, in Eqn B2.

Epidermal cell water potential

In steady state, in this model, the water potential of the epidermis is given by

  • image((B4))

In this expression, RS is the resistance to liquid water flow from site S to the branch point M in Fig. 2.

Guard cell water potential

In steady state, the guard cell water potential satisfies inline image. We have to derive an expression for the mole fraction wp. There is a diffusive resistance between the evapourating site, i, and the point of water exchange in the pore, p. In steady state, water vapour diffusion into the pore equals diffusion out. Thus, we have

  • image((B5))

where gi and gp are conductances into and out of the pore, respectively, and wi is the water vapour mole fraction in the intercellular space just below the pore. Solving for wp leads to

  • image((B6))

where σ = gp/(gi + gp). To complete specification of ψp (and hence ψg) requires solving for wi in terms of TL, E and ψS. We assume that ψi = ψm = ψS − RSE and also inline image, where wis is the saturated mole fraction at temperature Ti. Eliminating ψi and solving for wi yields, wi = wisexp[vL(ψS − RSE)/RTi]. A useful closed form approximation for saturated mole fraction of water at temperature T is

  • image((B7))

where T is measured in K, w0 is 2.6607 × 105 at sea level (and 3.0939 × 105 at the elevation of Logan, Utah), TW = 4270.7919 K, and T0 = 30.3800 K. Inserting Ti = TL − λiE into (B7) (and assuming λiE is small) leads to wis = wL[1 − TWλiE/(TL − T0)2]. Noting that vL(ψS − RSE)/RTi<<1 permits writing wi = wL[1 − TWλiE/(TL − T0)2 + vL(ψS − RSE)/RTL]. Finally, substituting wi into ψg produces

  • image((B8))

where

  • image((B9))

In Eqn B8, h = wa/wL is the ambient relative humidity calculated at leaf temperature TL. Rg is partly caused by liquid resistance up to i (RS) and partly caused by ‘interfacial resistance’ across i (Rg). The resistor network shown in Fig. 8 summarizes the steady state scenario outlined here.

image

Figure 8. Resistance diagram for the vapour phase model. S is the source water potential and RS is the liquid phase resistance through the xylem. i is the evapourating site for water vapour transfer to the guard cells. The quantity Rafter is the total effective resistance to vapour phase flow from i to the pore p, including diffusive resistance into the pore plus ‘interfacial resistance’ associated with converting liquid to vapour at i. Rp is the vapour phase resistance out of the stomatal pore. p and g represent the water potential of the air in the stomatal pore, M and e represent the water potential in the mesophyll and epidermal cells, and a represents the water potential of the air.

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Closed form expression for stomatal conductance

Combining Eqns B2, B4 and B7 leads to

  • image((B10))

Equation B10 is only an implicit expression for gs because E depends on gs as well. Using Eqns B5 and B6 shows that E = gp(1 − σ)(wi − wa). Given that λiE<<TL, wi ≃ wL, so that E = gsΔw = gs(wL − wa), with gs = gp(1 − σ). We assume that diffusive flows both into and out of the stomatal pore require the pore to be open and that therefore gi = cia and gp = cpa, where the cs are proportional to water vapour diffusivity, DW, and depend on geometric aspects of the stomatal pore, and a is stomatal aperture. As a result, σ = cp/(ci + cp) depends only on geometry and is constant for a given plant. In addition, we assume that in the dark, at least, the epidermal and guard cell osmotic pressures do not vary much with changing stomatal aperture. Assembling these pieces allows us to finally write

  • image((B11))

Equation B11 is the fundamental result of the new model. To cast Eqn B11 into Eqn 6 define gV0 ≡ χπ + (1 − m)ψS] (the temperature dependence of gV0 is the temperature dependence of Δπ), Θ ≡ χσR/vL and ZV ≡ χ(Rg − mRe). (The parameter values for Tradescantia pallida, in the dark, suggest that about one-fourth of the water potential drop from i to p is caused by the evapourative resistance and about three-fourth caused by the diffusive resistance associated with the term σRTL(1 − h)/vL.) Note that because m is typically about 2 (DeMichele & Sharpe 1973; Franks, Cowan & Farquhar 1998), the term (1 − m)ψS in the numerator of Eqn B11 represents a wrong-way response to changes in source water potential, ψS. The right-way response to changes in ψS has been postulated to result from ABA production somewhere along the transpiration stream (Buckley 2005), which, in turn, causes guard cell osmotic pressure (and therefore Δπ) to decrease.

Feedforward

Equation 6 predicts E = gsΔw = (g0 − ΘTLΔw/wLw/(1 + ZVΔw). At fixed temperature, this expression has a maximum at

  • image((B12))

or, more experimentally useful, at a wa value of

  • image((B13))

For the parameters derived in the following for Tradescantia pallida, in the dark, the leaf temperature for which wa(max) has its largest value is 20 °C, but even at that temperature wa(max) is only about 0.0041. This value is lower than any for the data in Mott & Peak (2010) and is consistent with the fact that feedforward was not observed in that study. In plants with higher values of γ than Tradescantia pallida, feedforward should be more easily observed.

APPENDIX C – NORMALIZATION AND FITTING

Because there was considerable variability among experiments in gs and stomatal apertures (despite all attempts to prepare plants identically for each day's experiments), meaningful comparisons between experiments require some kind of normalization procedure. The one adopted here is predicated on the assumption that, all things being equal, a plant should have a unique value of conductance at a given reference leaf surface temperature, TR, and a given reference ΔwR. For the data of Mott & Peak (2010), it is convenient to take TR = 25 °C and ΔwR = 0.017 (or, equivalently, hR = 0.53). We then normalize all measured conductances so that

  • image((C1))

which, by construction, equals 1 in the reference state.

Each of the models to be tested depends on temperature and humidity differently. It is convenient to write

  • image((C2))

where x stands for one of the models, Ax is a model-dependent constant coefficient, and Fx is a corresponding model-dependent function of temperature and humidity. All models are required to agree with the temperature dependence of normalized conductance for Δw = 0 (h = 1). This leads to identifying GBWB + b with g0 in (1) and writing

  • image((C3))

The quantities Ax and Fx determined for Eqns 3, 6 and C3 (with their respective normalization conditions) are displayed in Table C1. The table introduces several unitless quantities. They are f(TL) = g0(TL)/g0(TR), inline image, K = g0(TR)/b − 1 and γ = ΘTR/g0(TR).

Table C1.  Values for the parameters Ax and Fx for the BWB, L and V models
xAxFx
  1. See text for discussion.

BWBinline image1 + h[(1 + K)f(TL) − 1]
L1 + ZL(TRwRinline image
Vinline imageinline image

Figure 3 shows the high humidity data [experiment (a)] used to determine f(TL) required by each model. There is little if any transpiration at high humidity, so stomatal apertures were measured directly for these data. Assuming that, in general, conductance is proportional to aperture, we approximate the relatively flat stomatal response at high humidity shown in Fig. 3 by the linear function

  • image((C4))

Temperature is measured in kelvins in (C4) and TR = 298 K.

The single parameter K of the BWB model and the parameter γ of the V model are found by fitting the models to the constant (25 °C) temperature data [experiment (b), above] and the constant Δw = 0.017 data [experiment (c), above]. The resulting values are K = 1.79 and γ = 0.52. We find that adequate fits to the data (including experimental uncertainties) for the V model can be achieved for a range of ZV values. A reasonable value that allows the V model to also agree with observed changes in aperture when nitox is replaced by helox (see further discussion) is ZV = 5 ± 1. The parameter ZL(TR) of the L model is found by fitting the L model to the constant temperature data; its best-fit value is 31.7. (All parameter values reported here, except ZV, have an uncertainty of about ±10%.) The essentially identical fits of the three models to the constant temperature data are shown in Fig. 4.

The essentially-identical fits of the BWB and V models to the constant Δw data are shown in Fig. 5. We place the L model on equal footing with the other two by solving the L model for ZL(TL), ZL(TL) = {[1 + Z(TRwR]f(TL)/ĝ(TLwR) − 1}/ΔwR, and using values of ĝ from the BWB and V fits. This leads to the approximate functional relation, ZL(TL) = −2.33TL + 728 (where TL is in kelvins), required to complete the L model. As stated previously, ZL in the liquid model is proportional to the effective resistance to liquid water flow. A comparison of the temperature dependence of ZL, required for the liquid model to fit the constant Δw data, with the temperature dependence of water viscosity is shown in Fig. 6. Clearly, the required ZL of the L model cannot be attributed to water viscosity only. This further supports our previously stated misgiving about the physical origin of resistance in a liquid phase model and suggests that the putative validity of such a model must hinge on the existence of some as yet unidentified biochemical processes. In summary, parameter values can be found that cause the BWB, L and V models to indistinguishably fit the experimental results (a)–(c) of Mott & Peak (2010).

Finally, we force the parameterized models to ‘predict’ the conductance values from the fourth experiment (d), of Mott & Peak (2010), namely, one in which Δw is varied at constant wa. The temperature range for experiments (a)–(c) is 20 to 30 °C and the Δw range is 0.010 to 0.025. In the original data for experiment (d) of Mott & Peak (2010), the Δw range was extended from 0.007 to 0.028. To provide an even more stringent test of the models, we followed the protocol of Mott & Peak (2010), to extend the temperature range to 35 °C and the Δw range to 0.045. The results are shown in Fig. 7. In the parameterization range, the BWB, L and V models fit the data indistinguishably, as would be expected. Outside the parameterization range, on the other hand, the models exhibit differences, especially at higher temperatures and Δws. At the highest temperatures and Δws the BWB model significantly underestimates conductance and the L model significantly overestimates it. The vapour phase model fits the conductance data much better in the extended region.

APPENDIX D: ORDERS OF MAGNITUDE

The model derived here is very specific about the origins of its parameters. Many of the ingredients needed to calculate them are available from other measurements that complement the four experiments used to test the competing conductance models. In other words, the quantitative aspects of the model are severely constrained. Here we examine these constraints and, by estimating order of magnitude values, draw from them a few implications for Tradescantia pallida, in the dark (in the discussion ‘a few’ means a value roughly in the range of 3 to 7).

  • 1
    At typical transpiration rates, guard cell water potential (relative to ψS) is about −(a few) MPa, whereas epidermal cell water potential is about −(a few) × 10−1 MPa. The latter is −RSE, whereas the former is of the form −(Rafter + RS)E, where Rafter is the total vapour resistance (evapourative plus diffusive) after the site i (see Fig. 8). It must be that Rafter + RS ∼ 10RS. This explains why, in this model, the two surfaces of amphistomatous leaves are hydraulically independent.
  • 2
    (a) Measured values of E are about 10−3 mol/m2·s. This implies that the quantity Rafter earlier must be about (a few) × 103 MPa-m2·s mol−1. We have previously estimated that Rg is about one-fourth of Rafter, so Rg must be about 103 MPa-m2·s mol−1. (b) From Eqn B9, Rg = λi(RTL/vL)[TW/(TL − T0)2]. The quantity (RTL/vL)[TW/(TL − T0)2] is about (a few) MPa/K for leaf temperatures near 300 K. As a result λi must be about (a few) × 102 K-m2·s mol−1. (c) λi is of the form λi = LW/(κ/L), where κ is a thermal conductivity and L a temperature relaxation length. If the evapourating pool is in good thermal contact with the mesophyll then we might expect κ ∼ (a few) × 10−1 W/K·m and L ∼ (a few) × 10−5 m, or λ ∼ (a few) K-m2·s mol−1. If, on the other hand, the evapourating pool receives most of its heat from the air then κ ∼ (a few) × 10−2 W/K·m, L ∼ (a few) × 10−4 m and λ ∼ (a few) × 102 K-m2·s mol−1. Thus, in this model, the large resistance from the evapourating site i to the stomatal pore requires that the evapourating pool at i exchanges heat primarily with the air in the intercellular spaces.
  • 3
    With E ∼ 10−3 mol/m2·s, TL − Ti ∼ (a few) × 10−1 K: in this model, the temperature of the evapourating site is only a fraction of a degree lower than the leaf surface.
  • 4
    The term σRTL(1 − h)/vL in Eqn B8 cannot be more than (a few) MPa. Because RTL/vL ∼ 102 MPa, it must be that σ ∼ (a few) × 10−2. Because, σ = gp/(gi + gp), it must be that, in this model, the conductance out of the pore is only a few percent of the conductance into the pore.
  • 5
    Finally, it is clear why the model parameters γ and ZV have the values found in our experimental data fits. To see this, note that conductances from measurements of E and Δw are ∼(a few) × 10−2 mol/m2·s. Because χ = gs/(Pg − mPe), it must be that χ ∼ 10−2 mol/MPa-m2·s. The model parameter γ = χσRTR/g0(TR)vL must be about 1 – the fit value is 0.52 – and the model parameter ZV = χ(Rg − mRe) must be about (a few) – the fit value is 5.

APPENDIX E – GAS TO LIQUID HEAT TRANSPORT

To estimate the resistance for heat flow at a gas–liquid interface consider a generic collision between a ‘projectile’ (P) molecule and a ‘target’ (T) molecule, observed in the latter's initial rest frame. Assuming the collision is head-on and elastic, the kinetic energy transferred to the target molecule is KT = 4K0mPmT/(mP + mT)2, where K0 is the projectile's initial kinetic energy and the ms are the respective molecular masses. The total rate of transfer of energy to an average target molecule from a ‘beam’ of projectiles of concentration nP is nPσPTvPKT, where σPT is the cross section for a P–T collision and 〈vP〉 is the mean thermal speed of a projectile. The ratio of transfer rates for two projectile beams of different molecular mass but the same temperature and density is

  • image

Now, suppose that the projectiles are in the gas phase and the targets are in the liquid phase. In that case, mT is an effective molecular mass, because of its roughly 10 times closer proximity with other such masses. Depending on details of the liquid structure mT can easily be a factor of 10 or more times the mass of an isolated molecule of the same type (Velic & Levis 1997). On the other hand if the projectiles are in the gas phase mP is the true molecular mass. In other words, for nitox and helox molecules impinging on a liquid water pool

  • image

In the previous equation, the ms are the average molecular masses in nitox and helox, and the σs are the corresponding cross-sections for gas molecules colliding with a water molecule. Assuming that the rate of heat flow from the (higher temperature) gas to the (lower temperature) liquid is  = (Tgas − Tliquid)/Rtotal, where Rtotal is the sum of gas phase, liquid phase and interface resistances, r12 is the ratio Rtotal helox/Rtotal nitox. Because mnitox = 3mhelox and σnitox ≈ 4σhelox/3 (el Nadi 1951), we find Rtotal helox/Rtotal nitox ≈ 2.3 (approximately what we measured in our plate experiments).