Tree Physiology 24, 865– 878
© 2004 Heron Publishing—Victoria, Canada
Stomatal control and hydraulic conductance, with special reference to
tall trees
PETER J. FRANKS 1,2
1
2
School of Tropical Biology, James Cook University, P.O. Box 6811, Cairns, Queensland, 4870, Australia ([email protected])
Temporary address: Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138, USA
Received September 16, 2003; accepted February 15, 2004; published online June 1, 2004
Summary A better understanding of the mechanistic basis
of stomatal control is necessary to understand why modes of
stomatal response differ among individual trees, and to improve the theoretical foundation for predictive models and manipulative experiments. Current understanding of the mechanistic basis of stomatal control is reviewed here and discussed
in relation to the plant hydraulic system. Analysis focused on:
(1) the relative role of hydraulic conductance in the vicinity of
the stomatal apparatus versus whole-plant hydraulic conductance; (2) the influence of guard cell inflation characteristics
and the mechanical interaction between guard cells and epidermal cells; and (3) the system requirements for moderate versus
dramatic reductions in stomatal conductance with increasing
evaporation potential. Special consideration was given to the
potential effect of changes in hydraulic properties as trees grow
taller. Stomatal control of leaf gas exchange is coupled to the
entire plant hydraulic system and the basis of this coupling is
the interdependence of guard cell water potential and transpiration rate. This hydraulic feedback loop is always present, but its
dynamic properties may be altered by growth or cavitation-induced changes in hydraulic conductance, and may vary with
genetically related differences in hydraulic conductances. Mechanistic models should include this feedback loop. Plants vary
in their ability to control transpiration rate sufficiently to maintain constant leaf water potential. Limited control may be
achieved through the hydraulic feedback loop alone, but for
tighter control, an additional element linking transpiration rate
to guard cell osmotic pressure may be needed.
ship is confounded by several factors, including: (1) the many
other signals to which stomata respond; (2) the lack of a
clearly defined set-point for the stomatal control system; and
(3) an unresolved mechanism for stomatal sensing of transpiration rate. Much progress has been made in the pursuit of understanding each of these factors (see reviews by Schroeder et
al. 2001, Buckley and Mott 2002). Although empirical models
of stomatal function have been developed (e.g., Jarvis 1976,
Farquhar and Wong 1984, Ball et al. 1987, Tardieu and Davies
1993, Leuning 1995, Dewar 2002), improvements in the mechanistic modeling of stomatal function are necessary to explain observed differences among individual trees and to provide a theoretical basis for predictive models or manipulative
experiments. Because the stomatal apparatus is ultimately a
hydraulically operated mechanical device, mechanistic models of stomatal function require an understanding of both the
hydraulic fluxes and the mechanical forces that govern the
movement of guard cells. This review focuses on these key elements and evaluates the influence of hydraulic conductances
at both the stomatal apparatus and the whole-plant scale. Special attention is given to the effect of potential changes in the
hydraulic system of growing trees, and to the extreme conditions that prevail in tall trees.
Keywords: hydraulic conductivity, stomatal conductance, stomatal models, transpiration control, transpiration rate, tree
water use.
The steady-state flow of fluids through wood is described by
Darcy’s Law (Scheidegger 1974), which, substituting molar
for mass units, equates molar flux (Js; flow rate/sapwood area)
with the product of sapwood-area-specific hydraulic conductivity (Ks; mol m –1 s –1 MPa –1) and the hydraulic pressure gradient (∆P/∆L; MPa m –1):
Theory
Definition of terms for hydraulic conductance and
conductivity
Introduction
Although the link between stomatal and hydraulic conductance has long been realized, its mechanistic basis has been
difficult to unravel. With increasing transpiration rate, the sensitivity of the stomatal response and the extent of the decline in
leaf water potential are both determined by hydraulic conductance (Raschke 1970). The characterization of this relation-
 ∆P 
Js = – K s 

 ∆L
(1)
A complete list of symbols and units is provided in Table 1.
When viewing flow through the tree as a whole, the hydraulic
pressure gradient may be approximated as ∆Ps–l /∆Ls–l, where
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FRANKS
∆Ps–l is the hydraulic pressure difference between soil and leaf
(pressure in leaves minus pressure in soil) and ∆Ls–l is the
length of the flow path between soil and leaf. The hydraulic
pressure difference between soil and leaf that is due to the vertical column of xylem water hanging from the plant’s internal
surfaces, ρgh, must be added to ∆Ps–l, giving:
 ∆P + ρgh 
Js = – K s  s – l

∆L s – l 

(2)
where ρ is the density of water, g is acceleration due to gravity,
and h is the height of the water column (e.g., vertical distance
from soil to sites of evaporation in canopy leaves). In the case
of water flowing up a tree, i.e., against gravity, ρgh acts to oppose Js, although in many gravity-fed hydrological systems it
assists flow.
To maintain compatibility with the leaf-area-specific units
for leaf transpiration rate (E; mol m –2 leaf s –1), liquid-phase
flux of water through the xylem is conveniently expressed as
molar flux of water relative to leaf area fed (JL; mol m –2 leaf
s –1):
 ∆P + ρgh 
J L = – K L  s –1

∆ L s –1 

(3)
where KL is the leaf-area-specific hydraulic conductivity (mol
m–1 s –1 MPa –1), and KL is obtained from Ks as:
 A 
KL = Ks  s 
 AL 
(4)
where As is sapwood cross-sectional area and AL is leaf area
fed. For a given hydraulic path length, e.g., from soil to tree
canopy, the leaf-area-specific whole-plant hydraulic conductance from soil to leaf, kL(s–l), is given by:
k L (s –1) =
KL
∆L s –1
(5)
Table 1: Symbols, description and units for terms used. * = gauge pressure, i.e., with reference to atmospheric pressure.
Symbol
Description
Units
AL
As
a
c
Leaf area
Sapwood cross-sectional area
Stomatal pore width
Constant that is proportional to diffusive conductance to the direct component
of the D-response stimulus
Leaf-to-air vapor pressure difference
Transpiration rate
Representative of bulk stomatal elastic modulus
Stomatal conductance to water vapor
Acceleration due to gravity
Open loop gain of the stomatal hydraulic negative feedback system
Tree height
Molar flow rate of water per unit leaf area
Molar flow rate of water per unit sapwood area
Leaf-area-specific hydraulic conductivity
Sapwood-area-specific hydraulic conductivity
Leaf-area-specific hydraulic conductance from soil to leaf
Leaf-area-specific hydraulic conductance from soil to epidermis
Leaf-area-specific hydraulic conductance from soil to guard cell
Leaf-area-specific hydraulic conductance from epidermis to guard cell
Hydraulic path length from soil to leaf
Ratio of sensitivities of a to Pe and a to Pg
Difference in hydraulic pressure over path length (∆L)
Difference in hydraulic pressure between soil and leaf
Epidermal cell turgor pressure
Guard cell turgor pressure
Atmospheric pressure
Epidermal cell osmotic pressure
Guard cell osmotic pressure
Maximum guard cell osmotic pressure
Leaf water potential
Soil water potential
Guard cell water potential
Epidermal cell water potential
Density of water
m2
m2
µm
MPa
D
E
ε*
gsw
g
H
h
JL
Js
KL
Ks
k L(s–l)
k L(s–e)
k L(s–g)
k L(e–g)
∆Ls–l
m
∆P
∆Ps–l
Pe
Pg
Pa
Πe
Πg
Πg(max)
Ψl
Ψs
Ψg
Ψe
ρ
TREE PHYSIOLOGY VOLUME 24, 2004
kPa
mol m –2 s –1
MPa
mol m –2 s –1
m s –2
–
m
mol m –2 s –1
mol m –2 s –1
mol m –1 s –1 MPa –1
mol m –1 s –1 MPa –1
mol m –2 s –1 MPa –1
mol m –2 s –1 MPa –1
mol m –2 s –1 MPa –1
mol m –2 s –1 MPa –1
m
–
MPa *
MPa *
MPa *
MPa *
kPa
MPa
MPa
MPa
MPa
MPa
MPa
MPa
kg m –3
STOMATAL CONTROL AND HYDRAULIC CONDUCTANCE
Both Ks and KL are independent of the length of the hydraulic
pathway, whereas kL(s–l) decreases with increasing ∆Ls–l (tree
height) if all else remains constant. Below (see Discussion),
the effect of increasing hydraulic path length on the stomatal
control mechanism is considered for the case of a growing tree
where, although all else may not remain constant, it has been
shown that kL(s–l) sometimes declines with height.
If JL is known, and ∆Ps–l is measured as the difference between soil and leaf water potential, Ψl – Ψs, then kL(s–l) may be
calculated as:
k L (s –1) =
JL
Ψs – Ψ1 – ρgh
(6)
Equations 1–5 apply only when total rate of water uptake by
the plant (mol s –1) is equal to total rate of water lost (steady
state), otherwise the capacitive effect of adjacent tissues must
be considered. When the steady-state condition is satisfied,
then:
J L (s –1) = E
(7)
Equation 6 can be adapted to represent the leaf-area-specific
hydraulic conductance between any two points along the plant
hydraulic pathway. The analysis here uses leaf-area-specific
whole-plant hydraulic conductance from soil to guard cell
(kL(s–g)), and leaf-area-specific whole-plant hydraulic conductance from soil to epidermal cells (kL(s–e)):
k L (s – g) =
E
Ψs – Ψg – ρgh
(8)
k L (s – e) =
E
Ψs – Ψe – ρgh
(9)
and
where Ψg and Ψe are guard cell and epidermal cell water potential, respectively.
Stomatal hydraulics
Stomata are coupled hydraulically to each other and to the rest
of the leaf (Raschke 1970, Mott et al. 1997, Buckley and Mott
2000, Mott and Franks 2001). Hydraulic coupling of stomata
may play a role in their sensitivity to perturbations in evaporative demand (see below). A consequence of this linkage is that
stomatal regulation of leaf gas exchange is directly influenced
by the water status and hydraulic structure of the whole plant.
Any change in transpiration rate will be translated almost immediately into a change in the water status of every leaf cell,
including stomatal guard cells. Similarly, a change in the hydraulic properties of the plant, such as a reduction in kL(s–e),
will alter the water status of guard cells, and therefore stomatal
conductance, if a compensating mechanism is not built into the
stomatal control system. It can be seen from Equation 6, for
example, that for any given JL, Ψl is increasingly more nega-
867
tive the smaller the hydraulic conductance kL(s–l), so higher
stomatal conductances to water vapor must be supported by
higher hydraulic conductances, if Ψl is to remain above some
threshold. This positive correlation between hydraulic and
stomatal conductances has been observed in several studies
(Meinzer and Grantz 1990, Meinzer et al. 1995, Saliendra et
al. 1995, Comstock 2000). Furthermore, when xylem hydraulic conductance is reduced by transverse cuts to the xylem or
induced air embolisms, stomatal conductance decreases also
(Sperry et al. 1993, Hubbard et al. 2001, Cochard et al. 2002),
indicating that hydraulic conductance to transpirational flux is
an integral and dynamic component of the stomatal control
mechanism.
Total leaf transpiration rate (E), which is driven by evaporative demand (or more precisely, the difference in water vapor
pressure between leaf interior and exterior (D)), includes a
component that bypasses the stomatal pores and diffuses to the
atmosphere directly across the epidermal cuticle. It is possible
that part of this cuticular transpiration occurs directly across
the externally exposed cuticle of the guard cells (Maier-Maercker 1979), although this component has proven difficult to
quantify. If the cuticular transpiration from stomatal guard
cells is great enough to significantly reduce guard cell water
potential, then a substantial reduction in stomatal aperture
could occur in response to increasing evaporative demand.
Farquhar (1978), building on earlier hypotheses of stomatal
control that invoke cuticular transpiration (Maercker 1965,
Raschke 1970, Lange et al. 1971), presented this idea more
generally, showing mathematically that stomatal response to
cuticular transpiration could provide a mechanism for regulation of leaf transpiration rate that is independent of bulk leaf
water status, thereby allowing transpiration rate to decrease
despite increasing evaporative demand, as is occasionally observed. Given the likelihood of at least some cuticular transpiration from stomatal guard cells and epidermis, however
small, the question is not so much whether this mechanism exists, but what contribution it makes to the overall stomatal response to evaporative demand. So far it has been difficult to
demonstrate a dominant and consistent role for cuticular transpiration, whether from guard cells or epidermis in general.
For example, Meinzer (1982) showed that partially removing
the cuticle of Douglas-fir (Pseudotsuga menziesii (Mirb.)
Franco) needles with a hexane wash increased stomatal sensitivity to D, proving the potential for cuticle-mediated stomatal
response to changes in D. However, Frensch and Schulze
(1988), using simultaneous pressure probe and gas exchange
measurements on Tradescantia virginiana L., a plant with a
well-documented but typically mild stomatal response to D,
observed no significant change in epidermal turgor or transpiration rate in response to a step change in D when stomata
were closed, suggesting that in T. virginiana the stomatal response to D could not be mediated via the cuticle. It therefore
seems that cuticular transpiration may in some cases play a
role, but is not always an essential component of the mechanism regulating stomatal conductance in response to D. For
these reasons, most of the analysis here focuses on the case
where guard cell or epidermal cuticular transpiration contrib-
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868
FRANKS
utes only minimally to stomatal control. Later, this is contrasted with the response characteristics of a stomatal control system that is sensitive to cuticular transpiration.
All mechanistic models of stomatal regulation of transpiration rate require assumptions to be made about the pattern of
hydraulic fluxes in the vicinity of stomatal guard cells. Theoretical analyses of Meidner (1976), Cowan (1977), Tyree and
Yianoulis (1980) and Pickard (1982) all show that for a uniformly wet sub-stomatal chamber, most of the transpirational
flux will emanate from wet surfaces closest to the stomatal
pore. However, it is not clear how far the exterior waxy cuticle
of the leaf surface extends into the throat of the stomatal pore
(or even further, into the substomatal cavity), and so it is difficult to generalize about the extent of wet cell surfaces in the vicinity of the stomatal pore. There are two conceptual models
that divide a broad spectrum of possibilities regarding the
pathways taken by the transpiration stream once it leaves the
xylem and flows toward the stomatal pores: the first, shown in
Figure 1A(i), is based on the assumption that the cuticular covering of stomatal guard cells is extensive, and that wet surfaces
of the leaf interior extend only as far as the adjacent subsidiary
or epidermal cells. This means that these cells support the bulk
of the transpirational water flux, and that because there is negligible evaporative loss from guard cells, Ψg = Ψe in the steady
state. The second model, shown in Figure 1A(ii), assumes that
cuticular covering of guard cells is not extensive, and that the
surfaces of the guard cells that face the leaf interior, being devoid of a substantially impermeable cuticular covering, are
wet and support the bulk of the transpirational water flux. In
this case, Ψg and Ψe are not equal in the steady state, and may
be expressed by rearrangement of Equations 8 and 9, as:
 E 
 – ρgh
Ψg = Ψs – 
 k L (s – g) 
(10)
and
 E 
Ψe = Ψs – 
 – ρgh
 k L (s – e) 
(11)
In this second model, the hydraulic conductance between subsidiary cell and guard cell, kL(e–g) has a strong influence on the
water potential of the guard cell for any given steady state E. If
kL(e–g) is relatively low, then Ψg will be much lower than Ψe. If
kL(e–g) is relatively high, then kL(s–e) ≈ kL(s–g) and Ψg ≈ Ψe. kL(s–g)
is the sum of kL(s–e) and kL(e–g):
k L (s – g)
 1
1 

=
+
 k L (s – e) k L (e – g) 
Figure 1. Possible patterns of transpiration-driven water flux (A) and
interacting mechanical forces (B) in the vicinity of guard cells. Two
alternatives are shown for bulk transpirational flux (thick black arrow): (i) guard cells are mostly covered by cuticle and evaporation is
mainly from adjacent epidermal or subsidiary cells; and (ii) guard
cells are not entirely covered in cuticle and their walls support most of
the transpirational flux. See Table 1 for symbol descriptions and units.
turgor of surrounding epidermal cells, Pe: an increase in Pe can
push guard cells closer together to reduce pore size, and decreasing Pe can act in the reverse (Figure 1B). At any point in
time, Pg and Pe are defined by:
Pe = Ψe + Π e
–1
(12)
and
Pg = Ψg + Π g
Stomatal mechanics
The opening and closing of stomatal pores results primarily
from increasing or decreasing guard cell turgor, Pg. In addition, stomatal pore size a may be influenced by changes in the
(13)
(14)
where Πg and Πe are guard cell and epidermal cell osmotic
pressures (MPa), respectively. In the short term, Π e is kept relatively constant, and Pe changes passively with Ψe. In contrast,
TREE PHYSIOLOGY VOLUME 24, 2004
STOMATAL CONTROL AND HYDRAULIC CONDUCTANCE
Πg is actively regulated by guard cells in direct response to
both internal (within-leaf ) and external (environmental) stimuli. Guard cell turgor pressure, and hence stomatal conductance, may change relatively quickly in response to stimuli
that alter Πg or Ψg (Grantz and Zeiger 1986).
The characteristics of guard cell inflation have been measured and described quantitatively for several species (Meidner and Edwards 1975, Franks et al. 1998, 2001, Franks and
Farquhar 2001). Figure 2 is typical of the form of the relationship between a, Pg and Pe, and shows how Pe can reduce
stomatal pore size over the entire range of Pg. In this example,
the reduction in a due to Pe ranges from 100% at low Pg to
about 50% at high Pg, but across species, the effect may not always be this great. Increasing a as a result of decreasing Pe
means that elaborate countermeasures need to be built into the
stomatal control mechanism to limit transpirational water loss
under conditions that tend to decrease Pe, such as increasing
evaporative demand or decreasing soil water potential.
The wide variation in overall structure and physical arrangement of guard cells and adjacent epidermal cells in different
species suggests that the magnitude of the mechanical interaction between the two is likely to vary considerably across species. This is now confirmed by a growing number of direct
measurements of a at different Pg and Pe with the cell pressure
probe (Franks 2003). What has also emerged from these pressure probe studies is that the mechanical interaction varies
considerably over the range of Pg and Pe for a given stomatal
complex. Therefore, the mechanical relationship, m, between
guard cells and epidermis is specific to a given combination of
Pg and Pe, and for any such combination is best defined by the
ratio of the sensitivities of a to Pe at constant Pg, and a to Pg at
constant Pe (Cooke et al. 1976):
Figure 2. Typical relationships between stomatal pore width (a) and
guard cell hydrostatic pressure (Pg ) at different epidermal turgor pressures (Pe; 0 and 0.9 MPa), as measured directly with a pressure probe.
To show the dependence of the mechanical interaction m on Pe and Pg,
four example values are given for the points marked with a dot (m calculated with Equation 15). The greater the magnitude of m, the harder
it is for guard cells to open against the opposing force of the epidermal
cells. Adapted from Franks et al. (1998).
m=–
 ∂a 


 ∂Pe  Pg
869
(15)
 ∂a 


 ∂Pg  Pe
When m > 1 (the usual case), there will be a net reduction in
stomatal aperture for the same increase in Pg and Pe. Under
these conditions, the epidermal cells are said to have a “mechanical advantage” over the guard cells (m is sometimes referred to as the mechanical advantage). The larger the value of
m, the more difficult it is for guard cells to open against the opposing force of the adjacent epidermal cells. If there is no mechanical interaction between guard cells and epidermis, then
m = 0.
If the dependencies of a on Pg with zero epidermal turgor
and maximum epidermal turgor, Pe(max), are defined by functions f1(Pg ) and f2(Pg ), respectively, and the reduction in a due
to Pe is assumed to be proportional to Pe, then a may be expressed as (Franks et al. 1998):
a = f1 Pg –
Pe ( f1 Pg – f2 Pg )
Pe(max)
(16)
If stomatal pore size is unaffected by Pe, then Equation 16 reduces to:
a = f1Pg
(17)
The functions f1(Pg ) and f2(Pg ) are as yet unknown for any forest tree species, but it could be assumed that the aperture versus guard cell turgor relationship for the stomata of trees is
qualitatively similar to that for the stomata in herbaceous
plants. These qualitative features, as illustrated in Figure 2,
are: (1) in the absence of any mechanical interaction with epidermal cells, the relationship between a and Pg approximates
an exponential decay function; (2) in the presence of mechanical interaction with epidermal cells, the relationship between a
and Pg is sigmoidal; and (3) maximum stomatal aperture for
any given epidermal turgor pressure is achieved with guard
cell turgor pressures of about 4.0 MPa. For simplicity, the simulations presented here use the parameters for f1(Pg ) and f2(Pg )
given in Franks et al. (1998), and for illustrative purposes, an
arbitrary multiplier of 0.04 is used in all simulations to scale
from stomatal aperture to stomatal conductance, gsw (gsw =
0.04 × a). The magnitude of the scaling factor, 0.04, which accounts for differences in stomatal size and density, was chosen
because it yields stomatal conductances in the range typical of
forest trees.
Interaction between leaf gas exchange and leaf hydraulics
When the leaf boundary layer conductance is high (leaves well
coupled to the atmosphere), the stomatal conductance to water
vapor on a leaf area basis, gsw, is given by:
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870
FRANKS
gsw =
E
D/Pa
(18)
where D is the leaf-to-air vapor pressure difference (kPa) and
Pa is atmospheric pressure (kPa). When Equations 7 and 18
hold, and if transpiration is sustained mainly from the moist
surfaces of guard cells just inside the stomatal pore (Figure 1A
(ii); see Stomatal hydraulics), then:
gsw =
k L (s – g)( Ψs – Ψg – ρgh )
E
=
D/Pa
D/Pa
(19)
Although Equation 19 accurately describes the relationship
between gsw, E, D, kL(s–g), Ψs, Ψg and h in the steady state, it
cannot be used as a predictive tool for studying the response of
gsw to D or kL(s–g), because the physical structure of the system
makes gsw, E and Ψg interdependent. For example, an instantaneous change in D will change E, which will change Ψg, which
will change Pg, which will change gsw, which will change E.
This circular flow of information forms part of the elaborate
feedback system that controls stomatal movement, and is illustrated in Figure 3. A more formal mathematical description of
this system is summarized in the block diagram of Figure 4,
which is adapted from Franks and Farquhar (1999) and built
on the general principles of feedback in the stomatal mechanism as outlined by Cowan (1977). This model captures the
basic biophysical linkages between stomatal guard cells and
the attached plant hydraulic system. It is coupled to the other
stoma stimuli (light flux, CO2 concentration, temperature)
through Πg, adopting the premise that these stimuli set a maximum Πg (Πg(max)), and hence a maximum gsw, to which the
(usually negative) response to D is added. Use of Πg(max) is
analogous to the common practice in empirical models of
adopting gsw(max) as a reference point (Jarvis 1976, Lohammar
et al. 1980), except Πg is a more basic physiological driver.
Use of the above model by Franks and Farquhar (1999) was
limited to a demonstration of the constraints imposed on stomatal sensitivity to D in a system where stomatal control of
transpiration rate was achieved exclusively by negative feedback of transpiration rate. Here the model is used to examine
the effect of two key aspects of plant hydraulic architecture on
stomatal control of transpiration rate: (1) the relative role of
hydraulic conductances in the vicinity of the guard cells; and
(2) the potential consequences of decreasing whole-plant hydraulic conductance with increasing tree height. Lastly, to
demonstrate a mechanism by which the limitations of an exclusively negative hydraulic feedback system can be overcome, a feedforward element is added to the feedback model,
in the form of Πg responding directly to cuticular transpiration
rate. For the feedback model, which is employed in the simulations presented here, Equations 10, 11, 13, 14, 16 and 18 are
solved simultaneously using an iterative procedure to obtain
Ψe, Ψg, Pe, Pg, a, gsw and E, using assigned values for D, Πe,
Πg, kL(s–g), kL(s–e), kL(e–g), h and Ψs.
Figure 3. Summary of the relationship between main components that
determine stomatal conductance. See Table 1 for description of symbols. Solid lines represent direct links, dotted lines less-direct links.
Discussion
The mode of stomatal response to increasing evaporative
demand
As air becomes drier, the leaf-to-air vapor pressure difference
(D) increases, and so does the potential rate of transpiration.
Stomata cannot protect the plant from declining water potentials associated with drying soil, or from the static negative water potentials imposed by the water columns hanging vertically
in the xylem, but by regulating their aperture, and therefore,
the transpiration rate, they can minimize the additional drop in
tissue water potentials associated with the hydraulic drag on
the transpiration stream. It is this role that is at the core of stomatal action, and it is best studied through examination of stomatal response to changes in D.
Almost without exception, studies of the stomatal response
to an increase in D report a decline in stomatal conductance
(Darwin 1898, Lange et al. 1971, Schulze et al. 1972, Hall et
al. 1976, Whitehead et al. 1981, Monteith 1995, Oren et al.
1999, Buckley and Mott 2002). The mode of this response is
often described as resembling an exponential decay or negative log function of D (e.g., Lohammar et al. 1980, Granier and
Bréda 1996, Oren et al. 1999), and this often yields functions
accounting for a significant proportion of the variance in large
data sets. However, the true nature of the relationship between
gsw and D, revealed under more carefully controlled conditions, may deviate somewhat from this simplification, and the
underlying mechanism has remained a puzzle for some time.
One of the first studies to illustrate both the generality of the
overall response (declining gsw with D) and the variability of
the form of this response across woody species was that of
Waring and Franklin (1979). Their results, reproduced in Figure 5, highlight two characteristics that vary across different
species. The first relates to a region of the curve at low values
TREE PHYSIOLOGY VOLUME 24, 2004
STOMATAL CONTROL AND HYDRAULIC CONDUCTANCE
871
Figure 4. Steady-state block diagram
of the main elements of the feedback
loop involving stomata, epidermis and
the plant hydraulic system. The model,
adapted from Franks and Farquhar
(1999), is divided into hydraulic, mechanical and osmotic components, and
simulates the change in transpiration
rate in response to a change in leaf-toair vapor pressure difference (D). The
actual change in transpiration rate
(∆E ), is the sum of the initial change
at constant stomatal conductance (gsw),
(∆E )g, and the change due to the feedback effect of ∆E on stomata, (∆ E)D.
of D where gsw is relatively insensitive to D, or sometimes actually increases (Farquhar 1978, Monteith 1995). This characteristic may be virtually absent in some species, while in others, it may occur for values of D approaching or even exceeding 1.0 kPa. It could be argued that if the primary function of
stomata is to protect against dangerously low tissue water potentials, then this region may represent the conditions where
water potential is above the danger threshold (threshold for onset of cavitation-induced air embolisms in the xylem) and
therefore no stomatal closure is required. However, the greater
the range of D for which stomatal closure is unnecessary, the
greater the under-utilization of the conductive potential of the
hydraulic system, because higher gsw and potentially higher
CO2 assimilation rates could have been sustained.
The second, and more important, characteristic of the gsw
versus D curve that varies across species is the rate of decline
of gsw with D, or ∂gsw /∂D. Due to the nonlinear dependence of
gsw on Pg, and the interaction between guard and epidermal
cells (see Theory section), this sensitivity changes with D. In
most observations, ∂gsw /∂D is negative. The steeper the slope
of the tangent at any point on the gsw versus D curve, the more
negative ∂gsw /∂D, and the smaller the increase in E for any increment in D. Under certain conditions, ∂gsw /∂D is sufficient
to result in no change in E with an increase in D (i.e., constant
E) and, in some cases is so steep that E actually decreases with
an increase in D. Therefore, the sensitivity of ∂gsw /∂D will determine the additional water potential draw down that a plant
experiences in association with an increase in evaporative demand. With increasing E, leaf water potential will decrease;
with decreasing E, leaf water potential will increase; when E
and JL are both constant, leaf water potential will remain constant. The magnitude of the change in leaf water potential for
any change in E will depend on the hydraulic conductance.
The lower the value of kL(s–l), the greater the change in leaf water potential. However, if kL(s–l) is very high, the change in bulk
leaf water potential with changing E may be too small to detect, and may therefore appear to be zero. This effect is evident
in the results of Nonami et al. (1990), where the water poten-
tial of epidermal, subsidiary and mesophyll cells of Tradescantia virginiana declined significantly with increasing E,
even though xylem water potential appeared to remain constant.
Stomatal sensitivity to D falls into three general categories:
(1) highly sensitive, to the point of maintaining constant E
with increasing D; (2) moderately sensitive, allowing some increase in E with increasing D; and (3) oversensitive, whereby
E decreases with an increase in D. Examples of these categories are shown in Figure 6 for leaves of three different rainforest canopy tree species. Although it would seem ideal that
stomatal control kept E, and hence Ψl, constant and close to the
Figure 5. One of the first field surveys of the response of stomatal conductance (gsw) to increasing leaf-to-air vapor pressure difference (D)
in forest trees and shrubs of the Pacific Northwest, USA, adapted
from Waring and Franklin (1979): (1) Pseudotsuga menziesii (Mirb.)
Franco; (2) Tsuga heterophylla (Raf.) Sarg.; (3) Cornus nuttallii
Aud.; (4) Acer macrophyllum Pursh; (5) Castanopsis chrysophylla
(Dougl.) A. DC.; (6) Acer circinatum Pursh; (7) Rhododendron macrophyllum G. Don); (8) Gaultheria shallon Pursh. Plants typically
show a decline in stomatal conductance with increasing D, even when
well supplied with water. However, the relative magnitude and pattern
of this decline varies greatly across species.
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FRANKS
threshold of cavitation in the xylem so as to gain the most from
a given investment in xylem (Tyree and Sperry 1988, Sperry et
al. 1993), this degree of control does not appear to be the general case. Why some species appear capable of very tight control of E (e.g., A. peralatum, Figures 6A and 6B) whereas others do not (e.g., S. sayeri, Figures 6C and 6D), has yet to be
fully explained.
The requirements for a decrease in gsw with increasing D
If there were no mechanical interaction between guard and
epidermal cells, then according to the system outlined in Figure 4, Pg would always decline with increasing D, and therefore so would gsw (Figure 7, line 1). The mechanical interaction between epidermal and guard cells presents a problem; in
many instances, the mechanical relationship m is such that an
equal reduction in Pg and Pe will lead to a net increase in
stomatal aperture because of the strong influence of Pe (see
Figure 2). An equal reduction in steady state Pg and Pe with increasing D would occur if guard cells supported no transpiration (Figure 1A(i)), or if the guard cells did support transpiration and the hydraulic conductances kL(s–e) and kL(s–g) were similar. This effect is simulated in Figure 7, line 2, where gsw increases with D, until the draw down in Ψe is sufficient to reduce Pe to zero, after which further increases in D result in
decreasing gsw. This effect can be induced artificially when Pe
(not D) is manipulated directly in vitro with epidermal strips
bathed in different osmotic solutions (Glinka 1971). The mechanical interaction between guard and epidermal cells has not
been quantified for many species, but, for example, it is known
to be substantial for Tradescantia virginiana (Figure 2), which
always appears to exhibit a reduction in gsw with increasing D
(Shackel and Brinckmann 1985, Nonami et al. 1990, Franks
and Farquhar 2001). In a purely hydraulic feedback model
where, as a first approximation, Πg is maintained constant, it is
possible to overcome the tendency for stomata to open with
declining epidermal water potential if: (1) guard cells sustain a
substantial portion of the evaporative flux, as per Figure 1A
(ii); and (2) kL(s–g) is much less than kL(s–e). Line 3 in Figure 7
simulates the effect of these conditions, using a value for kL(s–e)
that is ten times higher than that for kL(s–g). It is therefore possible to explain moderate reductions in gsw with D in terms of
this latter set of conditions. However, more dramatic reductions in gsw with D may be achieved with this feedback system
if Πg does not remain constant, but is sensitive, even slightly, to
some component of the hydraulic feedback loop, such as Pe.
Grantz and Zeiger (1986) have proposed that because of the
similarity in many cases between the kinetics of stomatal responses to light and D, it is highly probable that the response to
D involves a metabolic component. Line 4 in Figure 7 shows
how, in a stomatal control system incorporating hydraulic negative feedback, where guard cells sustain a large portion of the
Figure 6. Response of stomatal conductance
(gsw; A, C, E) and transpiration rate (E; B, D,
F) in three tropical rainforest canopy trees to a
step change in leaf-to-air vapor pressure difference (D) from 1.0 to 2.0 kPa. Arrows indicate
the time at which the step change was applied.
The results illustrate three categories of response, as indicated by the position of the new
steady state transpiration rate (E) relative to E
at 1.0 kPa: (A and B), the reduction in gsw is
such that E is held almost constant; (C and D),
the reduction in gsw is insufficient to maintain
E constant and E increases with the increase in
D (most typical type of response); (E and F),
the reduction in stomatal conductance to water
vapor (gsw) is so pronounced that E declines to
a value below what it was before the step
change in D (less common response). Measurements were carried out in situ with an openflow leaf gas exchange analyzer (LI-6400, LiCor, Lincoln, NE). Before the step change,
leaves were brought to steady state at 1.0 kPa
vapor pressure deficit, ambient CO2 concentration 350 µmol mol –1, 30 °C and 1000 µmol
m –2 s –1 PAR. Soil was wet from recent sustained heavy rains. Data were collected from
sun leaves at the top of the canopy on 35–40 m
tall trees. (A and B) = Argyrodendron peralatum (Bailey) Edlin; (C and D) = Syzigium sayeri (F. muell.) B. Hyland; and (E and F) =
Dysoxylum pettigrewianum Bailey. Data were
collected at the Australian Rainforest Canopy
Crane facility, Cape Tribulation, Australia.
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STOMATAL CONTROL AND HYDRAULIC CONDUCTANCE
H=–
Figure 7. Simulation of four variations in the properties of the stomatal control mechanism that result in decreasing stomatal conductance (gsw) with increasing leaf-to-air vapor pressure difference (D):
(1) no mechanical interaction between guard cells and epidermal cells
(m = 0), constant guard cell osmotic pressure (Πg; 3.5 MPa), and no
difference between the leaf-area-specific hydraulic conductance from
soil to leaf (kL(s–g)), and from soil to epidermis (kL(s–e)); (2) with mechanical interaction between guard cells and epidermal cells, constant
Πg (3.5 MPa), and no difference between kL(s–g) and kL(s–e); (3) as for
2, but with kL(s–e) ten times that of kL(s–g); and (4) as for 3, but with Πg
decreased from a maximum of 3.5 MPa, in direct proportion to Pe.
The conditions in 4, which combine low epidermal-to-guard cell hydraulic conductance with decreasing Πg, produce a response that most
closely resembles that typified in Figure 5 for forest trees.
transpirational flux, the combined effect of low kL(s–g), and Πg
decreasing from its set point at D = 0 (Πg(max)) in proportion to
Pe, results in a more dramatic reduction in gsw with increasing
D. Although kL(s–g) has yet to be measured, there is much indirect evidence to suggest that this last set of conditions is met in
many cases in plants that exhibit a substantial reduction in gsw
with increasing D. Figure 8 is one typical example: despite a
smooth decline in gsw following a step increase in D, the presence of a mechanical influence of Pe on gsw is seen when D is
returned to its original low value and the sudden restoration in
Pe causes a momentary reduction in gsw (marked as *). The
subsequent rise in gsw is only slight, suggesting not only that
Πg must have declined during the period of higher D, but that
after D was returned to its former value, Πg could not do so in
the short to medium term.
Dgsw
ε * k L (s – g)
873
(20)
where ε* is a measure of bulk guard cell elasticity (smaller ε*
means greater elasticity). The higher the magnitude of H, the
higher (in magnitude) the stomatal sensitivity to D. Based on
this principle, a plant with lower kL(s–g) would show greater
sensitivity to D, all else being equal. Equation 20 also predicts
that plants with higher stomatal conductances will show greater stomatal sensitivity to D, provided their hydraulic properties are not too dissimilar. In accordance with this, Oren et al.
(1999) found, from a survey of published data for a range of
trees and shrubs, that plants with higher gsw at low D did tend
to have greater stomatal sensitivity to D.
The simple hydraulic negative feedback mechanism described above can, at best, exhibit sufficient stomatal sensitivity to D to hold E constant, although in reality it will fall far
short of this because of the likely instabilities associated with
the necessarily high loop gain required for nearly constant E.
Furthermore, reliance on low hydraulic conductance for greater stomatal sensitivity could place additional stress on the hydraulic system in the form of potentially damaging low xylem
water potentials. Alternatively, a reduction in kL(s–g) as a result
of cavitation-induced air embolisms in the xylem may temporarily serve to reduce stomatal conductance and curb the increase in transpiration rate (Figure 9A), but the associated fall
in leaf water potential (Figure 9B) will increase the probability
of further cavitation events, perhaps leading to the runaway
embolism pattern described by Tyree and Sperry (1988). However, it was the emergence of data of the form in Figure 6E and
6F, showing that the reduction in gsw in response to increasing
Combined feedback and feedforward
Through the inherent coupling of E and Pg the hydraulic negative feedback loop will always be active in transpiring leaves.
Though it may not always be the only mechanism acting to
down-regulate E in the face of increasing evaporative demand
(see below), it can produce the more moderate and fully reversible patterns of stomatal response to D. Within limits, it is
the hydraulic conductance kL(s–g) that will account for much of
the sensitivity of gsw to D in this feedback loop. Franks and
Farquhar (1999) derived an expression for this sensitivity in
terms of the open loop gain H:
Figure 8. Response of Normambya normambyi stomatal conductance
to water vapor (gsw) to a step change in leaf-to-air vapor pressure difference (D) from 1.0 to 2.0 kPa, and back to 1.0 kPa. Application of
the change is indicated by arrows. Immediately following the return of
D to 1.0 kPa, gsw showed the common transient “wrong-way” response (marked by *), in this case, a slight decrease before increasing,
indicating a mechanical interaction between guard cells and epidermis. Note also that gsw did not return to its original value, suggesting
Πg changed, and does not recover in the short term. Conditions as for
Figure 6.
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FRANKS
D could be large enough to result in a decrease in E (Schulze et
al. 1972; see also references in Farquhar 1978) that meant hydraulic negative feedback could no longer be regarded universally as the mechanism by which gsw responded to D. Several
alternative hypotheses have been proposed, but the simplest is
that stomata are sensitive to a portion of the transpiration flux
that is independent of the stomatal pores. This hypothesis was
first proposed by Cowan (1977), who referred to it as feedforward control because, instead of gsw responding to E, it responds in this case directly to the environmental factors that
promote changes in E. For this reason, Cowan has also referred
to this mechanism as the direct response of gsw to D (Cowan
1994).
Since it was first proposed as a mechanism to explain the
more dramatic reductions in gsw with increasing D, the elements of feedforward control of E, if present, have been difficult to identify. Furthermore, the phenomenon of decreasing E
with increasing D is not easy to induce or reproduce under
controlled conditions, free from the complications of other
changing conditions such as light, temperature or soil moisture, so identifying the feedforward mechanism is all the more
difficult. This unpredictability, and the often sluggish and irreversible nature of the response, is uncharacteristic of the dynamic behavior of most true control systems, and it is perhaps
worth questioning whether in some cases of feedforward-like
behavior the plant is actually exhibiting symptoms more characteristic of a breakdown in control. These irregularities have
been noted elsewhere (Franks et al. 1997, Mencuccini et al.
2000), prompting caution in the interpretation of feedforwardlike behavior.
If a feedforward mechanism is indeed present in some
plants, then data with which to characterize it in detail are limited. However, the mechanism may prove to be relatively simple. If it can be assumed that: (1) the basis of the stimulus for
all modes of response to D is the rate of water loss from the leaf
(Mott and Parkhurst 1991); (2) feedforward control comprises
the sensing of this stimulus directly, i.e., independent of the
component of E that passes through the stomatal pores and
therefore independent of bulk leaf water status; and (3) the response includes change in Πg, then these components may be
linked by the following expression for Πg:
Π g = Π g(max) –
cD
Pa
(21)
where c is a constant directly proportional to a diffusive conductance that is supplying the “direct” portion of the stimulus
to D (e.g., guard cell or epidermal cuticular conductance). The
component cD/Pa in Equation 21, which is the result of translation of the direct stimulus (an evaporative flux) into an osmotic flux, is a feedforward element that can work together
with the ever present hydraulic negative feedback system. This
dual feedback–feedforward system may be more representative of the true nature of the mechanism of the gsw response
to D. The effect of incorporating this element into the feedback
model is shown in Figure 10, using different values of c. These
simulations closely resemble the modes of response observed
Figure 9. Predicted effect of a single cavitation event that reduces leafarea-specific hydraulic conductance from soil to guard cell (kL(s-g)) by
30% when epidermal cell water potential (Ψe) reaches –1.0 MPa. Due
to the effect of this on the hydraulic feedback loop, stomatal conductance to water vapor is reduced and transpiration rate (E ) is lower (A;
dotted line) than it would have been had kL(s–g) remained unchanged
(A; solid line). Although the cavitation event appears to curb the increase in E with leaf-to-air pressure difference, the lower Ψe (B; dotted line) and correspondingly lower leaf and xylem water potentials
increase the probability of further cavitation, making the system unstable.
in many species, but the validation of this simple mechanism
depends on experimental proof of a linkage between a portion
of the cuticular transpiration and Πg.
The potential effects of increasing tree height and the role of
stomatal mechanics
From Equation 5, it can be seen that a potential effect of increasing hydraulic path length is to decrease hydraulic conductance. Although this may be offset by an increase in KL
with height (Pothier et al. 1989), several studies have reported
a decline in whole-plant hydraulic conductance with height
(Mencuccini and Grace 1996, Ryan et al. 2000, McDowell et
al. 2002b). These studies were concerned primarily with hydraulic conductances representative of kL(s–l), but it could be assumed that because this is included as part of the series constituting kL(s–e) and kL(s–g), then these too may have decreased in a
proportional manner. In addition to this, there is the effect of
the downward pull of the vertical head of water in the xylem
(ρgh), which appears at the leaf as an additional drop in water
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Figure 10. Simulation of the addition of a feedforward element to the
hydraulic feedback model. Instead of guard cell osmotic pressure remaining constant, it is sensitive to leaf-to-air vapor pressure difference (D). Depending on the magnitude of the sensitivity (corresponding to c in Equation 21), transpiration rate (E) may decrease with increasing D. Stomatal conductance to water vapor = gsw. Other model
parameters are as for Figure 7, line 3.
potential of about 0.01 MPa per meter of vertical height (Scholander et al. 1965, Zimmermann 1983).
Correlation between stomatal and hydraulic conductance to
the transpirational flux has now been widely observed (see
Stomatal hydraulics), but less is understood about how stomatal conductance is affected by a decline in hydraulic conductance as trees grow taller. It has been proposed that the stomatal control mechanism is affected directly by these heightrelated (and by association, age-related) changes to plant hydraulic conductances, and that this could contribute to declining productivity as trees get taller (Ryan and Yoder 1997). The
basis of this argument is the possibility that a reduction in hydraulic conductance will lower the water potential in the canopy. However, others have proposed that in some cases adaptive mechanisms could compensate for these height-induced
changes (reviewed by Becker et al. 2000). Several possible
compensating mechanisms are evident in the combined expression for kL(s–l) (from Equations 4 and 5):
kL(s –1)
KL
As K s
=
=
∆ L s −1 AL ∆ L s −1
(22)
For example, in addition to an increase in KL, an increase in Ks
or an increase in the ratio As /AL may counteract the effects of
875
increasing ∆Ls–l on kL(s–l). Schafer et al. (2000) concluded that
increased As /AL partially compensated for the negative effect
of increased hydraulic path length on gsw in Fagus sylvatica L.
McDowell et al. (2002a) reached a similar conclusion in their
study of Pseudotsuga menziesii and a meta-analysis of data
from 13 whole-tree studies. However, if these compensating
mechanisms do not completely offset the effects of increasing
hydraulic path length, then the decline in kL(s–l) with increasing
tree height is likely to affect gsw.
From the perspective of the stomatal control mechanisms
described above, it is easy to demonstrate how height-related
change in kL(s–e) and kL(s–g) may translate into changes in gsw.
Figure 11 shows one example simulation with fixed KL over
40 m. The basis of these changes in gsw is a predicted increase
in the sensitivity of gsw to D as kL(s–g) decreases (refer to Equation 20). Whether this mechanism is expressed fully, or is compensated by other whole-tree processes, may depend upon environmental and genetic variables. There is clearly much to
learn about the mechanism of height-related effects on leaf gas
exchange control in trees. In the pursuit of this, special attention should be given to the hydro-mechanical properties of the
stomatal apparatus in trees. It can be seen in the simulations of
Figure 11C that the mechanical interaction between stomata
and epidermal cells potentially has a dramatic effect on the
way that height-related alterations to hydraulic conductance
will manifest themselves in the form of changes in gsw. With no
mechanical interaction, the model predicts a steady decline in
gsw with height, the slope of which is steeper the smaller KL
(Figure 11C, dotted line). With strong mechanical interaction,
gsw may actually increase over the initial change in height, until Pe reduces to zero, after which it will decline with height
(Figure 11C, solid line). Furthermore, detecting the effect of
height-induced changes in gsw may depend on the height range
and frequency at which gsw is sampled. Too few sample
heights, or too small a height range, could fail to detect the true
pattern of change in gsw with height, assuming fixed mechanical relationships between guard and epidermal cells. These relationships could themselves change with height. Lastly, any
change that tends to reduce the water potential in the canopy,
including a height-related reduction in hydraulic conductance
or increase in the hydrostatic component ρgh, may be counteracted by osmotic adjustment, e.g., an increase in Πe and
Πg(max). Osmotic adjustment in response to reduced water potentials caused by drought and increased salinity is well documented (Morgan 1984), but little is known at present about the
role of this mechanism in counteracting height-related effects
on canopy water potential and productivity as trees grow taller.
Conclusions
The focus of this review has been the linkage between stomatal
control and the plant hydraulic system. This relationship underlies the nature of leaf gas exchange regulation, and changes
in hydraulic conductances with increasing tree height or adverse environmental conditions may significantly alter gas exchange characteristics. The approach here has been to assume
a maximum guard cell osmotic pressure (constrained ulti-
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FRANKS
ational flux, with kL(s–g) being much lower than kL(s–e) in order
to overcome the counteractive influence of the interaction between guard cells and epidermal cell. This requirement is diminished the more mechanically isolated the guard cells are
from the epidermis. A linkage between this feedback loop and
guard cell osmotic pressure, possibly via the turgor of epidermal or mesophyll cells, could allow for greater sensitivity of
gsw to D. The effect on this system if hydraulic conductances
change with tree height will depend significantly on both the
alteration to the gain of the hydraulic feedback loop and the
mechanical interaction between guard cells and epidermal
cells. Further insights into these mechanisms will be gained
from more quantitative studies of hydraulic conductances at
the scale of the stomata–epidermal complex and the whole
plant, as well as how these and the hydro-mechanical properties of stomata vary with increasing tree height.
Acknowledgments
I thank Rick Meinzer and Guillermo Goldstein for inviting this review. This work was supported by grants from the Australian Research Council RIEF scheme, and a Harvard University Charles Bullard Fellowship. I also gratefully acknowledge the support of N.M.
Holbrook (Harvard University), D. Foster (Harvard Forest) and R.E.
Cook (Arnold Arboretum), and many valuable discussions with G.D.
Farquhar.
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Figure 11. Predicted pattern of change in gsw with increasing tree
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