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Keywords:

  • humidity;
  • hydraulics;
  • model;
  • stomata;
  • water relations

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods and materials
  5. Results and discussion
  6. Conclusion
  7. Acknowledgments
  8. References
  9. Appendix

The feasibility of two hypothetical mechanisms for the stomatal response to humidity was evaluated by identifying theoretical constraints on these mechanisms and by analysing timecourses of stomatal aperture following a step change in humidity. The two hypothetical mechanisms, which allow guard cell turgor pressure to overcome the epidermal mechanical advantage, are: (1) active regulation of guard cell osmotic pressure, requiring no hydraulic disequilibrium between guard and epidermal cells, and (2) a substantial hydraulic resistance between guard and epidermal cells, resulting in hydraulic disequilibrium between them. Numerical simulations of the system are made possible by recently published empirical relationships between guard cell pressure and volume and between stomatal aperture, guard cell turgor pressure, and epidermal cell turgor pressure; these data allow the hypothetical control variables to be inferred from stomatal aperture and evaporative demand, given physical assumptions that characterize either hypothesis. We show that hypothesis (1) predicts that steady-state πg is monotonically related to transpiration rate, whereas hypothesis (2) suggests that the relationship between transpiration rate and the steady-state guard to epidermal cell hydraulic resistance may be either positive or negative, and that this resistance must change substantially during the transient phase of the stomatal response to humidity.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods and materials
  5. Results and discussion
  6. Conclusion
  7. Acknowledgments
  8. References
  9. Appendix

The mechanism by which stomata open under high atmospheric humidity and close under low atmospheric humidity remains unknown. However, the field of viable hypotheses is limited by a few key empirical facts. First, steady-state stomatal conductance responds to transpiration rate, rather than to any measure of humidity or the humidity gradient per se (Mott & Parkhurst 1991). Second, stomatal aperture is controlled by a balance between guard cell turgor, which opens the pore, and epidermal turgor, which closes it (Cowan 1994). Third, epidermal turgor is more effective in controlling stomatal aperture (it has a ‘mechanical advantage’), so an equal decrease in guard and epidermal turgors will open the pore (Sharpe, Wu & Spence 1987; Franks, Cowan & Farquhar 1998). Fourth, mesophyll and epidermal turgor pressures decline with increasing transpiration rate (Shackel 1987; Nonami, Schulze & Ziegler 1990). Any valid hypothesis must therefore take an increase in the rate of evaporative water loss from cells in the leaf and turn it into a substantially greater decrease in guard cell turgor than in epidermal cell turgor. Such hypotheses can be divided into two general categories. The first category, hereafter called the ‘osmotic regulation model’, suggests that the increase in water loss rate triggers a metabolically induced decline in guard cell osmotic pressure (via solute efflux) and therefore in guard cell turgor pressure (e.g. Meidner 1986; Grantz 1990; Buckley & Mott 2001). The second, hereafter called the ‘drawdown model’, suggests that these responses are the result of a water potential gradient, or drawdown, between epidermal and guard cells (e.g. Raschke & Kuhl 1970; Lange et al. 1971; Dewar 1995). It is important to note that in both models, guard cells lose turgor passively, but in the osmotic regulation model, passive water loss is not enough to overcome the epidermal mechanical advantage and cause stomatal closure.

Early studies favoured the drawdown hypothesis (Raschke 1970; Lange 1971), and considerable effort was made to explain stomatal responses to humidity in terms of evaporation from the guard cells and epidermis (e.g. Appleby & Davies 1983; Maier-Maercker 1983; Sheriff 1984). However, experiments with leaves and epidermes bathed in solutions of different osmotic pressure suggest that a metabolic signal from the mesophyll is involved (Grantz & Schwartz 1988), which implies an active regulation of guard cell osmotic pressure. Furthermore, the similarities in kinetics between responses to light and humidity also suggest that active regulation of guard cell osmotic concentration is responsible for humidity responses (Grantz 1990).

Although there is no unequivocal evidence for either of these models, recently published data make it possible to formalize and quantify the mechanistic constraints on each model based on the physics of stomatal movements. The goals of the present study were to provide a mathematical context that permits evaluation of the theoretical plausibility of each hypothesis, and to generate, from this mathematical context, testable criteria for each hypothesis. A numerical solution of the mathematical system required to achieve these goals was made possible by recent data characterizing the hydromechanics of stomata in Vicia faba L. Franks, Cowan & Farquhar (1995, 1998) characterized the effects of guard cell turgor pressure (Pg) and epidermal cell turgor pressure (Pe) on stomatal aperture (a) in V. faba and Tradescantia virginiana L., by manipulating Pg with a pressure probe and by altering Pe via the water potential of the bathing medium. These relationships, together with one of the hypothetical mechanisms for the stomatal response to humidity (discussed above) and a set of empirical constraints (discussed in the Appendix, under Mathematical development) close the system mathematically. This makes it possible to infer relationships among any arbitrary subset of the variables that control or influence stomatal dynamics. In the present study, this mathematical framework, along with measurements of stomatal aperture in response to changes in humidity, was used to infer either the dynamics of guard cell osmotic pressure or the properties of the epidermal-to-guard cell water potential gradient that must exist for either hypothesis to be consistent with observed stomatal responses.

Methods and materials

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods and materials
  5. Results and discussion
  6. Conclusion
  7. Acknowledgments
  8. References
  9. Appendix

Synopsis of the mathematical technique

Using the standard equations of plant cell water relations and gas exchange, a mathematical system was developed to relate stomatal aperture to guard cell osmotic pressure and epidermal-guard cell hydraulic resistance. The complete development of the necessary mathematical system was made possible by recent data, obtained from experiments using a cell pressure probe and a confocal microscope, that provide empirical relationships between (i) stomatal aperture and the turgor pressures of epidermal and guard cells (Franks et al. 1998) and (ii) guard cell turgor pressure and guard cell volume (Franks et al. 2001). The relationships used in the present study for these fundamental relationships of stomatal hydromechanics are shown in Figs 1 and 2, respectively (equations are given the Appendix). Their main features are: (i) stomatal aperture increases with guard cell turgor in a saturating fashion when epidermal turgor is zero; (ii) aperture increases with guard cell turgor in a sigmoidal fashion when epidermal turgor is high; and (iii) guard cell volume increases in a weakly saturating fashion with guard cell turgor.

imageimage

Figure 1. Relationships between stomatal aperture (a, µm) and the turgor pressures of guard and epidermal cells (Pg and Pe, respectively, MPa). Panel (a) shows aperture as a function of Pg and Pe. Superimposed on this surface are two trajectories that would result if water potential were increased from −0·6 to 0 MPa in the absence of decoupling between Pg and Pe, for two values of guard cell osmotic pressure (πg, 2·0 and 4·0 MPa); both of these trajectories show a continuous decline in aperture with increasing water potential, highlighting the need for a mechanism by which Pg and Pe can be decoupled to produce the observed increase in stomatal aperture with water potential. Panel (b) shows the residual mechanical advantage of the epidermis (m − 1, where m is unitless), both as a shaded contour plot (marked with boundaries between regions where m − 1 is negative and positive, and where it is less or greater than 10) and as a surface plot.

image

Figure 2. Relationship between guard cell turgor pressure (Pg, MPa) and volume (Vg, µm3), redrawn from Franks et al. (2001).

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This mathematical system, which is developed fully in the Appendix, can be distilled to two equations (A9 and A12b). Two numerical parameters appearing in these equations required determination by experiment. These parameters are: (1) rse · γ, the product of rse (the hydraulic resistance from the water source, assumed to be at constant water potential, to the epidermal cells) and γ (the scaling factor between stomatal aperture and conductance); and (2) πe, the epidermal osmotic pressure (also assumed to be constant). Because the terms rse and γ always appear in Eqns A8 and A12b as the product, rse · γ, it was necessary only to determine a value for this product, rather than for each of these parameters.

To obtain approximate values for rse · γ and πe, we determined the purely hydraulic effects of D (the mole fraction gradient for water vapour from the substomatal cavity to the atmosphere) on epidermal turgor pressure (Pe) and stomatal aperture (a). This was done by equilibrating leaves at a low value of D and then measuring Pe and a as D was increased. These measurements were made rapidly – during the quasi-steady state that is achieved as stomata open in response to an increase in D, before the closing response is initiated.

The procedures for measuring stomatal apertures and epidermal turgor pressures in intact leaves were similar to those described in Mott, Denne & Powell (1997) and Mott & Franks (2001). Briefly, a fully expanded leaf of Vicia faba L. was secured to a microscope stage and the ambient humidity was controlled by flowing gas of known composition over the leaf through a 0·5 cm internal diameter latex tube. The leaf was secured to the stage with the adaxial surface facing up. The abaxial surface of the leaf was sealed with clear plastic to prevent gas exchange through that surface. Light was provided by a 500 W xenon bulb and delivered to the adaxial surface by two 0·5 cm fibre-optic bundles. To view stomata on the adaxial surface, light above 700 nm was applied to the abaxial surface through the microscope’s condenser. The transmitted light was imaged using a CCD camera (NEC model TI-324 A; NEC Technologies, Woodale, IL, USA) mounted on the microscope. Epidermal turgor was measured with a cell pressure probe mounted to the microscope stage.

It was impossible to monitor the oil–water meniscus of the pressure probe and stomatal apertures simultaneously. Therefore, the effects of D on Pe and a were determined in separate experiments on separate leaves. In all experiments, the leaf was brought to steady state at a temperature of approximately 23 °C, a D of approximately 9 mmol H2O mol−1 air, and a photon flux density of approximately 800 mol photons m−2 s−1. To determine Pe, the pressure probe was inserted in an epidermal cell, and a stable pressure was recorded for about 5 min. D was then increased in steps, and the pressure was recorded continuously over time. For each step in D, the pressure was allowed to stabilize at the new value before D was increased again. The process was continued until D reached 25 mmol mol−1 or until the seal around the pressure probe was lost. Five experiments using different leaves were performed to obtain at total of 25 data points, and most experiments were complete within about 15 min. To determine the hydraulic effect of D on a, an image containing one or two stomata was captured at D= 9 mmol mol−1 and the D was then increased to 14·5, 18·5, or 21·5 mmol mol−1. Six experiments using different leaves were performed to obtain a total of 12 data points. Images were captured at 1 min intervals for about 20 min, and apertures were measured from these images using image analysis software. A stable aperture was achieved after about 10 min, and this value was used in the analysis below.

The parameters rse · γ and πe were estimated from these data in a three-step procedure. First, the data for D versus a were used to create a regression function a(D); second, this function was applied to the data for D versus Pe to infer a from the values of D measured in that experiment; third, the product of the measured values of D and inferred values of a were regressed against values of Pe measured in the same experiment. The slope of the resulting regression (Fig. 3) then provided an estimate of rse · γ, and the intercept provided an estimate of πe (assuming that the source water potential was equal to zero).

image

Figure 3. Data relating epidermal turgor pressure (Pe, MPa) with rate of water loss (expressed as the product of stomatal aperture [a, µm] and evaporative demand [D, mmol H2O mol  air−1]), collected for the purpose of estimating the source-epidermis hydraulic resistance factor, rse · γ. See text for discussion.

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Results and discussion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods and materials
  5. Results and discussion
  6. Conclusion
  7. Acknowledgments
  8. References
  9. Appendix

Empirical relationships of stomatal hydromechanics

Figure 1a shows the empirical relationship (Eqn A1) between stomatal aperture and the turgor pressures of epidermal and guard cells, which is based on the measurements reported by Franks et al. (1998) for Vicia faba L. The most evident feature of the relationship between aperture and the turgor pressures of epidermal and guard cells is its shift from a sigmoidal relationship between a and Pg at high Pe to a simpler, saturating curve at low Pe; note this is essentially a three-dimensional expansion of Fig. 7 in Franks et al. (1998), with different parameter values for V. faba. Although it is possible for the physical state of a stomatal pore to occupy any position on this surface, this would require guard-cell and epidermal-cell turgor pressures to be independent. In a normally functioning leaf, these two pressures are hydraulically coupled through the effect of transpiration on epidermal water potential (Eqn A5). To demonstrate this, two dotted lines superimposed on the surface in Fig. 1a show the trajectories of stomatal aperture that would result as water potential increased from −0·6 MPa to 0·0 MPa with no decoupling of epidermal and guard cell turgors (i.e. constant guard cell osmotic pressure and equal epidermal and guard cell water potentials), for two different values of guard cell osmotic pressure (2 and 4 MPa).

Figure 1a shows that aperture declines with increasing water potential if guard cell and epidermal cell water potentials are perfectly coupled. This is a result of the epidermal mechanical advantage, m (Eqn A14). The value of m determines the amount by which the pore will open in response to an equal drop in turgor pressure of both guard and epidermal cells, so the amount by which m exceeds unity – the residual mechanical advantage (m – 1, plotted in Fig. 1b) – determines how much decoupling must occur between guard and epidermal cell water potentials (either by guard cell osmoregulation or a water potential gradient) to make aperture decline with decreasing water potential, and thus with increasing transpiration rate. Note that if the residual mechanical advantage is negative, passive equilibration of guard- and epidermal-cell water potentials should produce the correct steady-state humidity response, but Fig. 1b shows that, empirically, m − 1 is negative only at very low values of Pg.

The relationship between guard-cell volume and turgor pressure used in the present study was obtained by confocal microscopy and pressure probe measurements (Franks et al. 2001), and is described by a second-order polynomial function (Eqn A4; Fig. 2). Guard cell volume increased in a saturating fashion with turgor pressure for the three cells measured in that study.

Mathematical criteria for the two hypothetical mechanisms

Stomatal aperture and conductance are observed to decline as the leaf-to-air evaporative gradient increases (see reviews by Grantz 1990; Monteith 1995; Buckley & Mott 2001; direct observations by Kappen & Haeger 1991; Kappen, Schultz & Vanselow 1994; Mott et al. 1997). In the absence of ‘apparent feedforward’ (wherein transpiration rate declines at very low humidity; Franks et al. 1997; Farquhar 1978) stomatal aperture and conductance also decline with increasing transpiration rate. Since this ‘apparent feedforward’ response occurs only at very high values of D and may be hysteretic (Franks, Cowan & Farquhar 1997), it is excluded from the discussion below. As noted above, in most plants for which data are available, stomatal aperture is more sensitive to epidermal turgor than to guard cell turgor (i.e. the ‘mechanical advantage’ is greater than unity, as is true for V. faba under most conditions; Franks et al. 1998). For aperture to decline with increasing transpiration rate in such a leaf, an increase in transpiration rate must lead to a greater reduction in turgor in guard cells than in epidermal cells. In other words, guard cell turgor must be actively decoupled from epidermal turgor to overcome the mechanical advantage. This decoupling may be achieved either by decoupling the water potentials of guard and epidermal cells, or by decoupling guard cell turgor pressure from water potential. The first possibility demands that the hydraulic resistance between epidermal and guard cells is substantial. The second possibility demands that guard cell osmotic pressure is actively regulated in response to changes in humidity. Mathematical criteria for each of these hypotheses are presented below; detailed derivations are presented in the Appendix. The general criterion for aperture to decrease with increasing transpiration rate is given by Eqn 1 (Eqn A20 in the Appendix):

  • image(1)

In Eqn 1, πg is guard cell osmotic pressure, aD is the transpiration rate, reg* is the hydraulic resistance between epidermal and guard cells (multiplied by two factors assumed constant: the ratio, K, of evaporation rates from guard and epidermal cells, and the scaling factor, γ, between aperture and conductance), rse is the resistance from a hydraulic ‘source’ to the epidermal cells, and m is the epidermal mechanical advantage. If hydraulic supply is assumed to be insensitive to transpiration rate (that is, if increasing transpiration does not affect either the source water potential or the resistance of the hydraulic supply pathway to the epidermis; this is not strictly correct, but for low and moderate evaporative demands, rse is fairly insensitive to leaf water potential, and thus to E; see Sperry 2000): the criterion can be simplified to Eqn 2 (A20b):

  • image(2)

The two alternative hypotheses (guard cell osmoregulation or water potential drawdown between epidermal and guard cells) are represented by degenerate versions (Eqns 3 and 4, respectively; A20c and A20d) of Eqn 2:

  • image(3)
  • image(4)

Equation 3 says that, for the osmoregulation hypothesis to be correct, steady-state changes in guard cell osmotic pressure (πg) must outpace changes in epidermal turgor (Pe) by a factor equal to the residual mechanical advantage (m – 1).

Equation 4 is more difficult to interpret. The central element in this equation is reg*, which we call the ‘epidermal-guard cell hydraulic drawdown factor.’ This quantity, which is the product of reg (the resistance between epidermal and guard cells) with K (the ratio of evaporation rates from guard and epidermal cells) and γ (the proportionality between aperture and conductance), must be large enough to draw down guard cell water potential relative to epidermal water potential, in order to overcome the epidermal mechanical advantage. Note that this could be accomplished either by large K (a large proportion of evaporation occurring from guard cells) or by large reg (a large resistance in water supply to guard cells). The formal mathematical requirement is not simply reg* > (m − 1), however. There are two other factors. First, reg is normalized to rse, which is the supply resistance for epidermal cells; this shows that it is the balance of water supplies to guard and epidermal cells that is critical, and this makes sense intuitively because the goal is to decouple the water potentials of these two cells. Second, either K or reg may, in principle, change with transpiration rate (either passively or actively), which provides an additional way for drawdown to overcome the mechanical advantage. Even if the balance of water supplies to guard and epidermal cells is small to begin with (i.e. even if K · reg/rse is small), a large change in that balance would also decouple epidermal and guard-cell water potentials by making them depend differently on E. Therefore, the sensitivity of reg* to changes in transpiration rate [strictly, to normalized changes, hence the natural logarithm: dreg*/d(lnaD)] also contributes to the effort of overcoming the mechanical advantage.

What is required for either decoupling factor to produce the steady-state response?

We have taken two approaches to evaluating the feasibility of these hypotheses. First, the mathematical system described in the Appendix was constrained by implementing each hypothesis separately. This allowed us to generate the three-dimensional relationships (Figs 4 and 5) between aperture, evaporative gradient, and each decoupling factor (πg or reg*) implied by each hypothesis. These relationships, in turn, permit simple visual evaluation of the feasibility of each hypothesis for producing the observed steady-state response of stomata to humidity, as discussed below. Second, the mathematical system was constrained with measured timecourses of stomatal aperture following a step change in humidity. This allows inference of the timecourses (Fig. 6) of the hypothetical decoupling factor implied by each hypothesis, which in turn reveal the ability of each decoupling factor to produce the observed transient dynamics and kinetics of the stomatal response to humidity.

image

Figure 4. Model output showing the effect of guard cell osmotic pressure (πg, MPa) and evaporative demand (D, mmol H2O mol−1 air) on stomatal aperture (a, µm). Arrows superimposed on the surface show two hypothetical trajectories, one of which is consistent with observation (the white arrow, which has a declining with increasing D) and one of which is inconsistent with observation (the black arrow, which has a increasing with increasing D).

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image

Figure 5. Model output showing the effect of the epidermal/guard cell drawdown factor (reg*, MPa µm−1) and evaporative demand (D, mmol H2O  mol−1 air) on stomatal aperture (a, µm), for each of three values of guard cell osmotic pressure (πg, MPa). Arrows superimposed on the surface for πg = 3·0 MPa represent trajectories that are consistent (white) and inconsistent (black) with the observation that aperture declines with increasing evaporative demand.

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image

Figure 6. Timecourses, following a step increase in evaporative demand, observed for stomatal aperture (panels a and d) and inferred, using the model, from these aperture data for guard cell osmotic pressure (πg, panels b and e) and the epidermal/guard cell drawdown factor (reg*, panels c and f).

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Figure 4 shows how stomatal aperture is affected by guard cell osmotic pressure (πg) and evaporative gradient (D) in the limiting case of hydraulic equilibrium between epidermal and guard cells, and Fig. 5 shows how aperture is controlled by the epidermis-guard cell drawdown factor (reg*) and D in the limiting case where guard cell osmotic pressure is constant. In Fig. 4, aperture always increases with both πg and D except at extremely high values of πg and D, where aperture decreases with D. This small planar section at the far upper corner of the surface corresponds to zero epidermal turgor; that is, these values of πg and D produce a transpiration rate that draws down ψe to be equal to –πe, below which further water loss causes plasmolysis. The requirement that aperture must decline as D increases has a very simple topological interpretation on each of these surfaces: any empirically plausible response of the decoupling factor to D must produce a trajectory of steady states that move progressively downhill. Examples of implausible and plausible trajectories are overlaid on the surfaces in Figs 4 and 5.

Figure 4 shows that πg must decline monotonically with D to produce the correct steady-state response of aperture to humidity; in particular, πg must decline with D more than the planar contours that transect the surface in Fig. 4, as shown by the ‘plausible’ trajectory that is drawn on the surface. Another interesting property of Fig. 4 is that at high evaporative demands, stomatal aperture is more sensitive to guard cell osmotic pressure (excluding the zero–Pe region). As D rises, smaller increments in πg are required to cause a given change in a. This means that stomata can respond more rapidly and more efficiently to changes in other environmental variables under conditions of high evaporative demand (as reported by Mott, Shope & Buckley 1999 for stomatal responses to light) – consistent with the idea that stomata are adapted to optimize the carbon–water tradeoff when water is limiting (Cowan & Farquhar 1977). Another property of the system defined by the osmoregulation hypothesis is the implicit maximum steady-state aperture that occurs at the high-πg–low-D corner of the surface (using the hydromechanical parameters reported by Franks et al. (1998, 2001), this value is about 8·1 µm). Because steady-state aperture declines monotonically with increasing D, but increases monotonically with increasing πg, steady-state aperture will never occupy the region of the surface that rises higher than this corner (i.e. all points with a > 8·1 µm in this example), although this region can be occupied during transients.

Figure 5 contains three different surfaces, each for a different constant value of πg (2·5, 3·0, and 3·5 MPa), showing how aperture would vary with reg* and D if the drawdown hypothesis were correct. All three surfaces have the same basic shape, in that aperture increases with D when reg*= 0 (as one would expect), but decreases with D for sufficiently large reg*. However, the surfaces differ in two ways. First, the predicted aperture increases with the imposed value of πg for any given set of values for reg* and D. Second, the critical value of reg* (above which the drawdown hypothesis produces the correct response, that is, aperture declines with increasing D for all values of D) increases with πg, and is 0·22, 0·30, and 0·50 (MPa mol air mmol H2O−1 µm−1) when πg is 2·5, 3·0 and 3·5 MPa, respectively.

Can either decoupling factor produce the observed transients and response kinetics?

Figures 6(a) and (d) show timecourses of stomatal aperture measured before and after a step change in D from 9 to 18·5 mmol mol−1. These timecourses are consistent with previously published data and show the typical rapid transient opening followed by a slower closing response. The inferred timecourses of πg (Fig. 6b, e) and reg* (Fig. 6c, f) are presented below the aperture measurements. Figures 6(b) and (e) show that, according to the osmoregulation hypothesis (i.e. if reg* is zero, meaning the ψg = ψe), πg must decrease continuously and monotonically over time following a step increase in D to produce the observed dynamics of stomatal aperture. Because it is already well established that guard cell osmotic pressure can be actively and continuously regulated across a broad range, there is little reason to question the plausibility of the inferred πg timecourses in Fig. 6(b, e). Furthermore, a continuous and monotonic decline of πg during the approach to a new steady-state is easily explained by the kinetics of metabolically controlled solute uptake by guard cells, if the ‘target’ steady-state value of πg is hypothesized to be controlled by direct feedback to either epidermal or mesophyll water potential (Grantz 1990; Haefner, Buckley & Mott 1997).

Figures 6(c) and (f) reveal an important and novel implication of this analysis: the ‘pure’ drawdown hypothesis, which requires that πg play no role in the response to humidity and therefore must remain constant, requires that reg* change substantially during the response if this hypothesis is to explain the observed transient dynamics of aperture. The nature and magnitude of the variation in reg* depend on the assumed value of πg. (Since πg was unknown in the plants for which timecourse data are presented, we imposed three values of πg to infer numerical values of reg*.) Although it has never been proposed to our knowledge, regulation of reg* is not inherently implausible. Regulation of reg* could be effected by changes in reg, K or γ. The analysis of Tyree & Yianoulis (1980) suggests that K is relatively constant, and γ is a function of stomatal topology and density. However, at least one mechanism for regulating cell-to-cell hydraulic resistance is known to exist – aquaporins, which are potentially regulatable water and ion channels in cell membranes (Tyerman et al. 1999; Johansson et al. 2000). Aquaporins are known to exist in guard cells, but their function in these cells has not been ascertained.

Conclusion

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods and materials
  5. Results and discussion
  6. Conclusion
  7. Acknowledgments
  8. References
  9. Appendix

This paper provides a theoretical analysis of two hypothetical mechanisms by which guard cell turgor pressure may be decoupled from epidermal turgor to overcome the epidermal mechanical advantage and produce a decline in stomatal aperture with decreasing humidity. These decoupling mechanisms are: (a) regulation of guard cell osmotic pressure, and (b) a water potential drawdown from epidermal to guard cells. The analysis has yielded novel insights concerning these two hypotheses. First, the drawdown hypothesis demands that one or more of the factors controlling the gradient in water potential between epidermal and guard cells must vary substantially as stomata respond to humidity. Such changes could, in principle, be effected by aquaporins in the guard cell membrane (Tyerman et al. 1999; Johansson et al. 2000). Second, the osmoregulation hypothesis predicts a continuous and monotonic change in guard cell osmotic pressure as stomata respond to humidity. Third, the drawdown hypothesis predicts no consistent relationship between the steady-state values of humidity and the putative drawdown-controlling factors, whereas the osmoregulation hypothesis predicts a monotonic steady-state relationship between guard-cell osmotic pressure and humidity. Both predictions of the osmoregulation hypothesis are consistent with a feedback response of guard-cell osmotic pressure to epidermal water potential or turgor, which is in turn consistent with observed short-term responses of stomatal conductance to xylem cavitation and soil water potential (Buckley & Mott 2001). The predictions of the drawdown hypothesis, however, cannot be explained by a simple hydraulic feedback loop. Mathematical models of stomatal functioning that attempt to predict the stomatal response to humidity should incorporate the mathematical and empirical constraints on this response revealed by recent experiments (Franks et al. 1998, 2001) and by our analysis.

Acknowledgments

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods and materials
  5. Results and discussion
  6. Conclusion
  7. Acknowledgments
  8. References
  9. Appendix

Many thanks to Rand Hooper for his invaluable technical work and to Peter Franks for useful discussions.

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods and materials
  5. Results and discussion
  6. Conclusion
  7. Acknowledgments
  8. References
  9. Appendix
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Received 12 July 2001;received in revised form 11 October 2001accepted for publication 11 October 2001

Appendix

  1. Top of page
  2. Abstract
  3. Introduction
  4. Methods and materials
  5. Results and discussion
  6. Conclusion
  7. Acknowledgments
  8. References
  9. Appendix
Empirical models for aperture and volume versus turgor pressures

Observations of stomata of Tradescantia virginiana by Franks et al. (1998) allow stomatal aperture (a, µm) to be estimated from empirical functions of epidermal and guard cell turgor pressure (Pe and Pg, respectively, MPa):

inline image (A1)

The functions f1(Pg)and f2(Pg), which relate stomatal aperture to guard cell turgor at zero and maximum epidermal turgor, respectively (maximum epidermal turgor is considered equal to epidermal osmotic pressure, πe, MPa) are given by Eqns A2 and A3:

inline image (A2)

inline image (A3)

The parameters ζ1 and ζ2 (both MPa) control the curvature of a versus Pg; am is the maximum aperture (µm); ξ is the unitless proportion by which maximum stomatal aperture is reduced at full epidermal turgor (πe, MPa) relative to zero epidermal turgor; P̂g pis the value of guard cell turgor (MPa) at which the curvature of a versus Pg becomes negative at high epidermal turgor (when Pe = πe); and the second term in parentheses in Eqn A3 ensures the pore is closed when both guard cell and epidermal turgor are zero.

The data of Franks et al. (2001) allow guard cell volume (Vg, µm3) to be inferred from Pg, using an empirical model of the form:

inline image (A4)

where the terms c1, c2 and c3 are constants for a given guard cell pair. Values for all of the parameters in Eqns A1–A4 are given in Table 1, and our parameter estimation procedures are discussed below under the heading Parameter estimation. We assume the relationship between Pg and Vg is not affected by Pe; no data exist to test this hypothesis, which is applied here for simplicity. Note this is equivalent to specifying a value of unity for Cowan’s (1977) variable σ.

Table 1.  Parameter values
Parameter nameSymbolValue
  1. Sources in bracketed superscripts to the right of numerical values: aestimated from the data of Franks et al. (1998) and bFranks et al. 2001); see Parameter estimation in the Appendix for details. cMeasurements by the authors; see Materials and Methods for details.

Maximum stomatal apertuream18·11 µma
a versus Pg curvature parameter, low Peζ11·598 MPaa
a versus Pg curvature parameter, high Peζ20·7694 MPaa
Relative reduction in am by high Peξ0·509 (unitless)a
Inflection point in a versus Pg at high Peinline imageg2·015 MPaa
Polynomial constants for Vg versus Pgc188·469 µm3 MPa−2 b
Polynomial constants for Vg versus Pgc2726·846 µm3 MPa−1b
Polynomial constants for Vg versus Pgc34813·476 µm3b
Source water potentialψs0 MPaa
[source-to-epidermis resistance]·[aperture to conductance scaling factor]rse·γ0·001309 mol air mmol−1 H2µm−1c
Epidermal osmotic pressureπe0·5252 MPac
Gas constantR8·31441 MPa µm3 pmol−1 K−1
Leaf temperatureT298 K
Mathematical development

We assumed that the water potential of the epidermal cells (ψe, MPa) is in equilibrium with the water potential of other parallel evaporating sites (Nonami et al. 1990), and that this common water potential is determined by a balance between xylem supply (liquid-phase flow from a source at potential ψs[MPa] through a resistance rse[MPa (mmol H2O mleaf-2 s-1)-1]) and transpiration (vapour-phase flow at a rate E[mmol H2O mleaf-2 s-1]) from inner surfaces of cells in the leaf. E is defined as the product of conductance (g, mol air mleaf-2 s-1) and the evaporative gradient (D, mmol H2O mol−1air; formally, the mole fraction gradient of water vapour from leaf to atmosphere). Stomatal conductance (gs, mol air mleaf-2 s-1) is assumed to be proportional to aperture by a constant factor γ (mol air mleaf-2 s−1 µm−1), and the boundary layer resistance is considered zero for simplicity (g = gs). Collectively, these assumptions (Eqn A5) imply that ψe is a decreasing linear function of the rate of water loss (γaD), with slope given by -rse (Eqn A6):

inline image (A5)

inline image (A6)

Equation A6 can also be expressed in terms of epidermal turgor pressure (Pe) and epidermal osmotic pressure (πe, MPa, assumed constant in the simulations presented here), for the purposes of predicting water flow across isothermal phase boundaries:

inline image (A7)

This equation emerges from standard water relations, given the assumptions listed above. An independent expression for Pe can be obtained by solving the empirical/mechanical equation (Eqn A1) of Franks et al. (1998) that relates aperture to Pe and Pg:

inline image (A8)

The existence of two independent expressions for Pe allows its elimination, removing one degree of freedom. We equate the right-hand sides of Eqns A7 and A8 and solve for aperture:

inline image (A9)

Equation A9 shows that the theoretical hydraulic relationship between a, Pe and D (Eqn A7) reduces the empirical mechanical relationship between a, Pe and Pg (Eqn A1) to a semi-empirical hydromechanical function allowing a to be inferred from only Pg and D (Eqn A9); (Pg and D are considered the only dependent variables because all other terms in Eqn A9 are assumed constant). The hydraulics and mechanics of guard cells also represent an independent set of theoretical and empirical constraints on D, Pg and a, and link these variables to ψe, guard cell water potential (ψg, MPa), the hydraulic resistance between guard and epidermal cells (reg, MPa (mmol H2O m−2 s−1)−1), guard cell volume (Vg), and guard cell osmotic content (ng, pmol). One fundamental theoretical relationship among these variables is the equality of ψg with Pg − πg. A gradient/resistance model of water flow provides another constraint: at steady-state; any flow from epidermal to guard cells (F, mmol H2O  mleaf-2 s−1) is balanced by evaporation from guard cells, and is determined by reg and the difference between ψg and ψe. Defining the rate of evaporation from guard cells (Eg, mmol H2O  mleaf-2 s−1) as a fraction (K, unitless) of the total rate of transpiration, E, the steady-state flow through the guard cell evaporating site is given by Eqn A10:

inline image (A10)

The magnitude of K is a key factor distinguishing the two hypotheses under discussion in this study: the ng-regulation hypothesis does not require any drawdown between ψe and ψg, so K can be assumed to equal zero, but the reg-regulation hypothesis does require a drawdown, and therefore positive F and K. However, for the purposes of this study, neither K nor the aperture/conductance scaling factor, γ, need to be considered in isolation from reg, because these three terms only appear as a product (K·reg·γ) in the critical equations. In solving Eqn A10 for guard cell water potential or turgor pressure, the product of K, reg and γ can thus be replaced by a single variable, reg*:

inline image (A11)

Applying Eqn A9–A11 and rearranging yields:

inline image (A12)

It is more illuminating in the present context to express πg in terms of guard cell osmotic content, ng, because, whereas πg is influenced by a suite of hydraulic factors, ng is directly controlled by metabolic processes. Recognizing that Vg can be inferred from Pg using the data of Franks et al. (2001) (Eqn A4), and that πg is a simple function of Vg and ng in ideal dilute solutions, πg can be expressed as a function of ng and Pg (Eqn A13):

inline image (A13)

Equation A13 allows πg to be replaced by ng in Eqn A12, yielding Eqn. A12b:

inline image (A12b)

We now have seven variables (a, ψs, Pg, D, rse, reg*, and ng) and two constraints among these variables (Eqns A9 and A12b), leaving five degrees of freedom. To perform the semi-empirical simulations presented in this study, we closed the system by using adding four empirical constraints (measurements of a, D, and ψs, and estimates of rse · γ, in intact plants; details are provided in the Materials and Methods section in the main text) and one hypothetical constraint, the latter representing either the ng- or reg-regulation mechanism for the humidity response. To represent the guard cell osmotic regulation hypothesis, reg* was set equal to zero, and to represent the drawdown hypothesis, πg was held constant at each of three values (2·5, 3·0 and 3·5 MPa). (These choices of hypothetical constraints are explained in the next section of the Appendix, where the mathematical criteria for each hypothesis are developed.) The remaining variable (ng or reg*, respectively) was then determined by iterative solution of Eqns A9 and A12b, as described below, and plotted against time and other system variables to show the inferred behaviour of the hypothetical control variable during the stomatal response to humidity.

Care is needed in implementing Eqns A9 and A12b in a numerical simulation, because, although Pe is not explicitly constrained to be non-negative in the preceding derivation, Pe remains implicit in Eqns A9 and A12b. In conditions where plasmolysis occurs, Pe will implicitly become negative unless it is explicitly calculated by the computer code and constrained to be non-negative; note that in this case, epidermal osmotic pressure must also implicitly be allowed to increase with further decreases in epidermal water potential. However, when Pe is zero, Eqn A9 may be replaced by Eqn A2, so the code never needs to change πe explicitly.

Derivation of the mathematical criteria for the mechanism of the feedback humidity response

Stomatal aperture declines as the evaporative demand increases. Under most conditions, this also causes transpiration rate (E) to increase with D, consistent with evidence that stomata respond directly to the rate of water loss from leaves (Mott & Parkhurst 1991). This is often described as a feedback response of g to E, or of E to D, which contrasts with occasional reports of an apparent ‘feedforward’ response in which E declines with increasing D at high values of D (Farquhar 1978; Monteith 1995; Franks et al. 1997). In the following analysis, we consider only the ‘feedback’ domain of the response, because the mathematical criteria for producing a decline of aperture with increasing E are much simpler.

To produce the observed steady-state and transient ‘wrong-way’ responses of stomatal aperture to humidity, we must express these responses mathematically, in the context of the epidermal mechanical advantage. Formally, changes in epidermal turgor pressure must be overcompensated by changes in guard cell turgor to cause a decline in aperture with transpiration rate. The magnitude of this overcom-pensation is given by the ratio of the sensitivities of aperture to epidermal and guard cell turgors:

inline image (A14)

inline image (A15)

Note that the mechanical advantage, formally defined by Eqn A15, is highly sensitive to both Pg and Pe (Fig. 1b). From Eqns A14 and A15, the total dependence of aperture on transpiration rate, E, can be expressed in differential form as:

inline image (A16)

Noting that E=γaD, with γ assumed constant in the present study, we can rewrite Eqn A16 as:

inline image (A16b)

This transformation will clarify later steps in the derivation, by preserving the linkage of K, reg and γ in the variable reg*, and of rse and γ as a product (which was estimated as a single parameter; see Materials and Methods). Therefore, since ∂a/∂Pg is always positive (Sharpe et al. 1987; Franks et al. 1998) in Fig. 1a, the parenthetical term in Eqn. A16b must be negative if aperture is to decline with increasing transpiration rate:

inline image (A17)

The derivatives in Eqn. A17 are obtained by differentiating Eqns A11 and A7:

inline image (A18)

inline image (A19)

By combining Eqns A17, A18 and A19, we identify the following general criterion:

inline image (A20)

Equation A20 states that, in order for guard cell turgor to overcome the residual mechanical advantage (m − 1), at least one of the following two quantities must always be sufficiently large: (i) a metabolically controlled decline in guard cell osmotic pressure (–dπg/d[aD]), or (ii) an epidermal-guard cell drawdown factor (reg*) and its sensitivity to relative changes in transpiration rate (dreg*/d[lnaD]). These requirements are weighted inversely by the hydraulic supply resistance to the epidermis and its sensitivity to normalized transpiration (the denominator on the left side of Eqn A20).

Equation A20 can be greatly simplified under conditions where transpiration does not substantially alter hydraulic supply, either by causing source water potential (ψs) to decrease or by drawing down leaf water potential enough to cause cavitation in the hydraulic supply pathway to the epidermis, decreasing rse. If ψs and rse are constant, then from Eqn A5,

inline image (A21)

  • The parenthetical quantity in Eqn A21 can be multiplied by the first term in Eqn A20 to give

inline image (A22)

Invariance of rse also eliminates the second term in the denominator of Eqn A20, which, together with Eqn A22, simplifies Eqn A20 to

inline image (A20b)

The hypothesis that the humidity response is due solely to regulation of guard cell osmotic pressure, πg (with no need for a water potential gradient between epidermal and guard cell) is best represented by the assumption that reg* is zero, which can be interpreted in two ways: no evaporation occurs from guard cells (K = 0) or the resistance between epidermal and guard cells is negligible (reg = 0). This hypothesis reduces the criterion to Eqn. A20c, which states that changes in guard cell osmotic pressure must outpace changes in epidermal turgor by a factor equal to the residual mechanical advantage (m − 1):

inline image (A20c)

The alternative hypothesis, that the humidity response is due solely to non-zero reg (with no assistance from active regulation of guard cell osmotic pressure), is best represented by the assumption that πg is constant (dπg = 0). In this case, the criterion reduces to Eqn. A20d,

inline image (A20d)

which states that the sum of the epidermal-guard cell hydraulic drawdown factor (reg*) and its sensitivity to relative changes in transpiration rate, both expressed relative to the source-to-epidermis hydraulic resistance factor (rse · γ), must exceed the residual mechanical advantage.

Iterative solution procedure

For the model output shown in Figs 4 and 5, the system was solved by specifying values for D and either ng (Fig. 4) or reg*(Fig. 5), and adjusting Pg incrementally until the ratio of values for ψe specified by two independent functions (Eqns A9 and A12b) was within 10−4 of unity. If an increment in Pg overshot the target value, the increment was reversed and the step size cut in half. To infer πg from empirical measurements of aperture (Fig. 6), Pe was determined from Eqn A8 using the measured values for a and D, and Pg was incremented as described above until the estimate of aperture given by Eqn A12b agreed with the empirically measured aperture with a rational precision of 10−4. Hypothetical dynamics of reg* were inferred from aperture data (Fig. 6) by the same method used to infer ng, except that Eqn A12 was used instead of Eqn. A12b, and a constant value of πg (2·5, 3·0 or 3·5 MPa) was imposed.

Parameter estimation for Eqs A1–A4

Franks et al. (1998) did not present data relating a and Pg at low Pe for V. faba, so we estimated this relationship from other data given in their paper. First, we fit Eqn A3 to the a versus Pg data given for high Pe (Fig. 1 from Franks et al. 1998) using the ‘solver’ feature in Microsoft Excel to identify values for P̂g, ζ2, and [(1 − ξ)am]. Second, we assumed that the ratio [(1 − ξ) in Eqn A3] of maximum apertures (am) at high and low Pe (0·491 for T. virginiana in Franks et al. 1998), and the ratio of the curvature parameters (dx in Franks et al. 1998; ζ2 and ζ1 in Eqns. A2 and A3) at high and low Pe (0·30 for T. virginiana in Franks et al. 1998) were the same for V. faba as for T. virginiana.

Guard cell pressure versus volume data were presented by Franks et al. (2001) for three guard cells of V. faba. We chose the data set with the highest resolution in Pg, doubled all volumes so that the model described in this paper represents a pair of guard cells, and fit a second-order polynomial (r2 = 0·992) to those data. These parameters are given in Table 1.