Journal of Experimental Botany, Vol. 50, No. 332, pp. 385–392, March 1999
Optimization theory of stomatal behaviour
I. A critical evaluation of five methods of calculation
D.S. Thomas1, D. Eamus1,3 and D. Bell2
1 School of Biological and Environmental Sciences, Northern Territory University, Darwin, N.T. Australia, 0909
2 School of Mathematical and Physical Sciences, Northern Territory University, Darwin, N.T. Australia, 0909
Received 11 May 1998; Accepted 14 October 1998
Abstract
There are two principal aims in this first manuscript,
first, to compare five methods for calculating the marginal unit water cost of plant carbon gain (∂E/∂A) of
leaves of two Australian tropical tree species, and
second, to test the hypothesis that ∂E/∂A of tropical
tree leaves is constant when leaf-to-air vapour
pressure difference (D) varies. Few differences in the
absolute values of ∂E/∂A or the form of the response
were found between species. However differences did
exist between methods of calculation. Importantly,
∂E/∂A was rarely constant with changing D. Stomatal
limitations of net photosynthesis caused by a reduction
in internal carbon dioxide concentration as stomatal
conductance declined caused ∂E/∂A to increase when
D was increased. Some methods are applicable to
canopy scale field measurements whilst others could
only be used in laboratory settings where sufficient
control of the environment is possible. The best
method of calculation of ∂E/∂A varied according to the
criteria used to judge best. However, overall, despite
the large numbers of independent data sets required,
Method 1 was judged best for calculating ∂E/∂A for
individual leaves as, although it relies on the largest
number of independently derived relationships, it
has the fewest assumptions. Methods 2 and 3 were
applicable to the field when a number of simplifying
assumptions were made.
Key words: Stomatal optimization theory, marginal unit
water cost.
Introduction
Several theories and different types of models exist
describing the influence of changes in a number of
environmental variables on stomatal behaviour. Von
Caemmerer and Farquhar (1981) detailed a mechanistic
model of photosynthesis incorporating biochemical
approaches, while Jarvis (1976) developed an empirical
model of stomatal behaviour. This and many other later
empirical models of stomatal behaviour ignored correlations between assimilation (A) and stomatal conductance
(G). Ball et al. (1987) relied on this correlation, but more
specifically on the proportionality between ambient and
internal CO concentrations, and others (Leuning, 1990,
2
1995; Lloyd, 1991; Aphalo and Jarvis, 1993) have extensively modified their approach. Cowan (1997) and Cowan
and Farquhar (1977) proposed that stomata behave in a
manner such that the sensitivities of the rates of transpiration (E ) and carbon assimilation (A) to changes in
stomatal conductance (G) [i.e. (∂E/∂G)/(∂A/∂G)] remain
constant over a specified time frame. This will occur if
the marginal unit water cost of plant carbon gain (∂E/∂A)
is a constant, i.e. the ratio of the sensitivities of E and A
to changes in G remain constant (Cowan, 1977; Cowan
and Farquhar, 1977). Under certain circumstances for
∂E/∂A to be a constant, stomata must behave in a
feedforward or direct manner to the changing environment as only then can stomata respond directly to changes
in the environment (Cowan, 1982).
Experimental evidence supporting a constant ∂E/∂A
with changes in the environment are largely restricted to
laboratory studies where responses to leaf-to-air vapour
pressure difference (D) predominate ( Farquhar et al.,
1980; Hall and Schulze, 1980; Meinzer, 1982; Mooney
et al., 1983). Furthermore, in all but a few studies, the
range of D examined was 1–3 kPa. For tropical environments this range is far too small to be representative of
the environment to which leaves are routinely exposed
(Duff et al., 1997).
3 To whom correspondence should be addressed. Fax: +61 8 8946 6847. E-mail: [email protected]
© Oxford University Press 1999
386 Thomas et al.
∂E/∂A of Macadamia integrifolia was relatively stable,
although it did increase at higher D, if the relationships
were examined assuming the relationship between A and
internal CO concentration (C ) was independent of D
2
i
(Lloyd, 1991). Lloyd and Farquhar (1994) redefined the
model to calculate ∂E/∂A from the ratio of C to ambient
i
CO concentration (C ), i.e. C /C . They observed that
2
a
i a
biomes with regular or periodic water had a good linear
relationship between ∂E/∂A and D. However, the slope
of the relationship between ∂E/∂A and D was reduced,
and C /C was lower, in biomes with episodic water
i a
supply or in cultivated plants growing outside their
naturally occurring environment. Episodic water supply
is a defining characteristic of the wet–dry tropics, a
substantial biome incorporating parts of India, Africa,
Central and South America, and Australia. Lloyd et al.
(1995) redefined the model developed by Lloyd (1991)
and applied it to a productivity model of an Amazon
rainforest system. They found that a constant value of
∂E/∂A could not explain the variation in gas exchange
associated with changing canopy-to-air vapour pressure
difference (analogous to D).
∂E/∂A is typically calculated from the ratio of the
partial derivatives ∂E/∂G and ∂A/∂G. The literature
provides several methods to calculate or estimate these
partial derivatives (Cowan and Farquhar, 1977; Farquhar
et al., 1980) or to calculate ∂E/∂A more directly from A,
G and C (Lloyd et al., 1995). There are two principal
i
aims in this first paper; first to compare five methods for
calculating ∂E/∂A of leaves of two Australian rainforest
tree species, and second, to test the hypothesis that ∂E/∂A
of tropical tree leaves is constant when D varies. Two
rainforest species were chosen in this study because of
the importance of patches of rainforest to the diversity
and conservation value of Australian savannas and
because the use of two species increases the confidence
with which statements can be made pertaining to the
merits and demerits of different methods of calculating
∂E/∂A.
Materials and methods
Methods of calculating ∂E/∂A
Three methods of calculating ∂E/∂A are taken directly from the
literature. Method 1 uses the equations outlined in Cowan and
Farquhar (1977); Method 2 uses the approach outlined by
Farquhar et al. (1980) where a regression of A on G is used to
calculate ∂A/∂G directly, and the calculation of ∂E/∂G is
simplified; and Method 3 uses a further linearization of the gas
exchange relationships as outlined in Lloyd et al. (1995). A
summary of the equations used in the various methods are
given in the Appendix. Values for the CO compensation point
2
used in this study had previously been calculated for each
species using Equation 11 (Appendix) when calculating ∂E/∂A
for Method 3.
Farquhar et al. (1980) acknowledge that the calculation of
∂E/∂A is sensitive to the form of equations used to linearize the
partial derivatives. Therefore, two models were developed using
modified forms of some equations in an attempt to improve
certain relationships between A and C (Method 1M ) or A and
i
G (Method 2M ).
Method 1 requires the calculation of the change in A with
changing leaf temperature (T ), i.e. dA/dT . The temperature
l
1
response given in Reed et al. (1976) was used to describe the
response of A to T : ( Equation 7, Appendix).
l
In Method 1M the equation describing the A/C response to
i
a rectangular hyperbola ( Equation 11, Appendix) was modified,
but otherwise used the same equations as outlined by Cowan
and Farquhar (1977).
In Method 2M the regression of A/G was modified to allow
the relationship not to pass through the origin (Equation 18,
Appendix), but otherwise used the equations outlined in
Farquhar et al. (1980).
Data were only used from a particular relationship (for
example, the effect of increasing D at 33 °C in the wet season)
to calculate the ∂A/∂G and ∂E/∂G partial derivatives and hence
∂E/∂A for that particular relationship except when other
responses were required, e.g. the ∂A/∂T partial derivative was
l
required when using Method 1 (Cowan and Farquhar, 1977).
All constants describing an equation were fitted using the
Newton-Gauss algorithm within Statistica v.5.0.
Plant material
Maranthes corymbosa Blume, and Myristica insipida R.Br. were
studied. Both are monsoon rainforest evergreen species (Brock,
1993). All plants were grown in shadehouses (30% light
transmission) in 6 l pots of sand5peat5vermiculite5perlite
(2525151, by vol.) irrigated daily in Darwin, N.T., Australia.
Plants were regularly fertilized with liquid (Aquasol ) and slow
release fertilizer (Osmocote), with regular pest and disease
control being undertaken when appropriate. Measurements
were conducted in October (end of dry season) on newly
expanded leaves after the canopy of each tree had been pruned
in the previous May (end of wet season).
Gas exchange
Leaf gas exchange of three or four replicate plants was measured
using leaves which had expanded in the dry season using a
laboratory-based photosynthesis system under a variety of
environmental conditions at C of 360 mmol mol−1. Initial
a
environmental conditions in any measurement period were set
while slowly increasing PPFD at the leaf surface from 0 to
1000 mmol m−2 s−1 over 90 min. The effect of increasing D
from 1.4 to 4.3 kPa was studied when T was maintained at
l
33 °C, and from 2–6 kPa when T was maintained at 38 °C. The
l
response to D was studied from the lowest value to the highest
value. The effect of T from 25–38 °C was studied while D was
l
maintained at 2 kPa. The response of A to C was studied by
i
altering C from 800 to 150 mmol mol−1. In all studies three or
a
four plants were moved to the laboratory the night before
measurements were taken. Leaf gas exchange was measured on
10–20 cm2 of the leaf near the central portion of mature leaves
in an open gas exchange system (Eamus et al., 1995) with three
or four replicate leaf chambers. Calculations of gas exchange
parameters were measured according to Long and Hallgren
(1985). During the experimental period plants were kept well
watered by irrigating every 2 h and by maintaining a small
reservoir of water in saucers positioned under the pots. All
measurements were completed during the period 10.00 h to
16.00 h as previous studies determined leaf gas exchange
remained stable during this period.
Stomatal optimization theory 387
Results
∂E/∂A varied between methods used to calculate ∂E/∂A,
but not between species, and values ranged from approximately 200 to 1500 mol mol−1. At higher leaf temperature
(38 °C ) ∂E/∂A was higher than at the lower leaf temperature (33 °C ). Data at 33 °C tended to be more tightly
grouped than those at 38 °C ( Figs 1, 2).
Values of ∂E/∂A calculated by Method 1 are similar to
those calculated by other methods, although the ∂A/∂G
and ∂E/∂G partial derivatives were lower than those
calculated by Methods 2 and 2M (data not shown). The
∂E/∂G and ∂A/∂G partial derivatives using Methods 2
and 2M show similar values and responses to D to those
published in the literature (Hall and Schulze, 1980;
Lloyd, 1991).
Leaf-to-air vapour pressure difference
∂E/∂A calculated by all methods, except Method 3,
generally (70% of cases) increased with increasing D
regardless of species (Figs 1, 2). It was found that ∂E/∂A
below a D of about 3 kPa could be incorrectly interpreted
as being constant, but when the full range of D was
Fig. 2. The marginal unit water cost of plant carbon gain (∂E/∂A)
(mol mol−1) calculated by various methods as a function of leaf-to-air
vapour pressure difference (kPa) when the leaf temperature was
maintained at 38 °C. Scale of ∂E/∂A axis differs between calculation
methods.
Fig. 1. The marginal unit water cost of plant carbon gain (∂E/∂A)
(mol mol−1) calculated by various methods as a function of leaf-to-air
vapour pressure difference (kPa) when the leaf temperature was
maintained at 33 °C. Scale of ∂E/∂A axis differs between calculation
methods.
considered, ∂E/∂A showed a significant (P≤0.05)
increase. ∂E/∂A calculated by Method 3 could be considered constant over the entire D range ( Figs 1, 2).
Although the typical ∂E/∂A range was from 500 to
1500 mol mol−1 ( Figs 1, 2) in some cases, particularly
when ∂E/∂A was calculated by Methods 2 and 2M, values
nearer 3000 mol mol−1 were observed (Fig. 2).
C generally showed a negative linear correlation with
i
∂E/∂A in Methods 1, 1M, 2, and 2M (mean correlation
coefficient from −0.75 to −0.40 depending on the method
of calculation of ∂E/∂A (data not shown). This indicates
the stomatal limitations of C may limit A and this may
i
contribute to the less efficient marginal unit water cost of
plant carbon gain ratio at higher D. ∂E/∂A calculated by
Method 3 showed nil or positive relationships with C ,
i
but this may be biased as the equation used to calculate
∂E/∂A in Method 3 was derived in part from equations
relating C to A and G (Lloyd et al., 1995).
i
The ∂E/∂G partial derivatives calculated by all methods
increased strongly with D (data not shown). This was as
expected because of the increased driving force of water
loss with increasing D. ∂A/∂G showed little variation with
D when calculated directly from the regression equation
388 Thomas et al.
(Methods 2 and 2M ) (data not shown). Thus A and G
changed in unison with changing D. However, ∂A/∂G
showed a positive relationship with D when it was calculated less directly (Methods 1, 1M ) (data not shown).
These methods (Methods 1 and 1M ) use the ∂E/∂G
partial derivative in the ∂A/∂G equation, thus there would
be some biasing in these correlations. ∂A/∂C is also used
i
in the estimation of ∂A/∂G. ∂A/∂C would be larger at
i
higher D thereby contributing to the increase in ∂A/∂G
as D increases.
Discussion
This paper addresses two aims. First, to answer the
question which of the methods of calculating ∂E/∂A
proved to be the most satisfactory? Second, to address
the question how did ∂E/∂A vary with increasing D over
the range normally encountered in the wet–dry tropics?
Comparing methods of ∂E/∂A
Several issues arise in considering the first question. First,
what constitutes a good method? Three criteria were
identified as being the most important. First, a minimum
number of assumptions should be present in the calculations, and those assumptions that are present need to be
acceptable. Second, the methodology that is required to
determine the various parameters need to be available for
not too much cost and effort. Finally, the question of
leaf scale versus canopy-scale needs to be addressed.
Methods 1 and 2, based upon Cowan and Farquhar
(1977) and Farquhar et al. (1980) assume that temperature is optimal for assimilation so that the slope of A
versus G as D varies represents ∂A/∂G at the optimal
temperature. When temperature is far from optimal, the
methods are applicable but it is possible that ∂A/∂G will
yield very different values from those at optimum temperature because the sensitivities of A and G to temperature
may not be the same ( Eamus et al., 1983). Consequently,
values of ∂A/∂G so obtained are likely to have little
applicability at temperatures close to optimum. Farquhar
et al. (1980) further state that the simplified approach for
deriving ∂E/∂A (whereby a regression of A on G is used
to calculate ∂A/∂G; Method 2) is appropriate only at
temperatures optimal for CO assimilation. However,
2
maintaining leaf temperature at the optimum for assimilation in a laboratory gas exchange system is not a problem.
Methods 1 (and 1M ) are the most theoretically satisfactory methods, but require large numbers of separate data
sets, including determinations of the relationships between
A and C , A and leaf temperature, and the dependence of
i
A and G on D. As such this method requires a long time
in the laboratory to collect all those data. It is not readily
applicable to canopy-scale measurements as temperature,
D and C cannot be controlled for canopies.
i
Method 2 is more applicable to canopy-scale considerations because of the simplifications made. Thus A/C and
i
A/T relationships are not required. However, the simplil
fied approach of Method 2 requires that leaf temperature
be close to optimum for A. At optimal leaf temperature,
∂A/∂T can be ignored if the form of the relationship
between ∂E/∂A and D is required rather than the absolute
value of ∂E/∂A. At temperatures close to optimal, and in
temperature-controlled cuvettes, ∂A/∂T is constant but
not zero. Again this will influence the absolute value of
∂E/∂A but not the form of the relationship between
∂E/∂A and D. This condition may not be met in many
field situations, although choice of sampling time can, to
some extent, overcome this. Method 2 does not require
the assumption that transpiration does not affect leaf
temperature as temperature corrections for variables such
as boundary layer conductance are used. This is particularly important in the tropics where radiation loads may
be largest when soil water content is lowest (e.g. dry
season savannas) as in the dry season, the soils are dry
and radiation load is high. This can have a large influence
on leaf temperature as the effects of evapotranspirational
cooling on modifying leaf temperature are reduced.
Correction of variables used in Method 2 for changes in
leaf temperature are therefore required. These corrections
are available for some, but not all, variables.
Method 3, based upon Lloyd et al. (1995) ignores the
effect of transpiration on leaf temperature. This to us
represents a serious impediment to the use of this method,
since transpiration can clearly modify leaf temperature
(Prior et al., 1997). However, it is recognized that field
measurements of canopy-scale processes require simplifying assumptions. Method 3 also assumes that boundary
layer resistance is sufficiently small for it to be ignored,
so that the assumption that the partial derivative dE/dG
is equal to D, can be satisfied. In a closed canopy, this
condition is unlikely to be met, with large numbers of
leaves likely to have a significant boundary layer resistance, particularly when wind speed is low and/or leaf size
is large (Nobel, 1991). A further assumption of Method
3 is that carboxylation efficiency remains constant as C
i
varies when G varies. This is probably reasonable for
small fluctuations in C but will not be valid when C
i
i
increases to values above the ‘shoulder’ on a standard
A/C curve. Such events occur when, for example, temperi
ature or water availability become limiting to assimilation
( Eamus and Cole, 1997; Eamus et al., 1995).
Modification of Method 1, i.e. Method 1M, describes
the A/C relationship mathematically as a rectangular
i
hyperbola whilst Method 1 (Cowan and Farquhar, 1977)
uses the physiological relationship of A=(C −C )G/1.6.
a
i
The mathematical modification gives more flexibility in
the curve-fitting process, ignores any autocorrelation
between parameters in the A=(C −C )G/1.6 equation.
a
i
Stomatal optimization theory 389
The equation A=(C −C )G/1.6, although reasonably
a
i
sound and reliable is imprecise as it does ignore the effect
of transpiration upon assimilation that occurs due to
pressure changes in the leaf as large volumes of water
diffuse from the leaf (von Caemmerer and Farquhar,
1981). Modification of Method 2 (i.e. Method 2M ),
removed the constraint of forcing the A/G relationship
through the origin, again increasing the flexibility of
fitting curves to less than perfect data.
To summarize, it is considered that Method 1 is the
most theoretically satisfactory, rigorous and precise
method of finding ∂E/∂A. This is because it has the fewest
assumptions and relies on the largest collection of
independently derived relationships (A/T ; A/C ; A/D;
l
i
G/D). However, it is not applicable to canopy-scale studies
and requires the longest time in the laboratory. The
minor modification present in Method 1M improves the
applicability of the method slightly. Methods 2 and 2M
are simpler than Method 1 but requires leaf temperature
to be optimum for A. In the field this can be a problem.
Method 3 is clearly field applicable, but has the major
limitations of assuming no impact of transpiration upon
leaf temperature and boundary layer resistance is sufficiently small to be ignored.
The response of ∂E/∂A to increasing D
The ranges of ∂E/∂A calculated were similar to those
previously published (Farquhar et al., 1980; Hall and
Schulze, 1980; Meinzer, 1982; Mooney et al., 1983; Lloyd
and Farquhar, 1994; Lloyd et al., 1995). Givnish (1986)
summarized the earlier data to show that ∂E/∂A varied
from a mean of 1000 mol mol−1 for herbaceous species
to 500 in shrubs and 360 mol mol−1 in coniferous trees.
These values ranged from approximately 200 to
2000 mol mol−1, with occasional values up to
4000 mol mol−1.
A horizontal line for ∂E/∂A as D increases supports
the hypothesis that stomata are behaving optimally with
respect to water and carbon dioxide fluxes. In 70% of all
cases ∂E/∂A had a significant positive slope ( Figs 1, 2),
with correlation coefficients ranging from 0.57 to 0.99.
For Methods 1 and 1M, a positive slope was observed in
100% of cases. A significant positive slope for ∂E/∂A
against D means that as D increases, the marginal unit
cost (i.e. water transpired) is increasing for each unit of
carbon fixed. The larger the slope, the less optimally are
stomata behaving. The slopes are not large and thus in
this respect stomata are functioning almost completely
optimally. Is there any evidence from other studies of
such behaviour? Monteith (1995), in a review of published
literature, identified three regions of stomatal response to
increasing D. At low values of D, as D increases, transpiration increases and stomatal responses are small. At
moderate values of D, stomatal regulation increases and
as D increases, transpiration is held approximately constant. Finally, at large values of D, stomatal regulation
is unable to prevent E increasing as D increases. Thus,
optimization of stomatal function breaks down, or the
gain of the feedback loop linking E and G is insufficient
to limit E. It is apparent that when only small ranges
of D are used experimentally (1.5–3.0 kPa), apparent
constancy of ∂E/∂A is seen, but over the full range of D
to which tropical trees are exposed, ∂E/∂A is not constant
and unit marginal costs increase with increasing D.
On average, the relationship between ∂E/∂A and D was
not constant, and to some extent was dependent on the
method of calculation. At optimal leal temperature
( Fig. 1; T =33 °C ) ∂E/∂A had a positive slope in all
1
methods in both species. The method of calculation did
not influence the relationship between ∂E/∂A and D. This
supports the view that the positive slope, so consistently
observed, is a physiologically based (as opposed to
methodological artefact) response. It is speculated that
the positive response arises because of the difference in
sensitivity of the response of assimilation and conductance
to transpiration rate. At supra-optimal temperatures
( Fig. 2, T =33 °C ) the method of calculation appears to
1
be critical in determining the response of ∂E/∂A to D.
Methods 2 and 3 consistently gave a constant ∂E/∂A as
D increased, but Method 1 gave an increasing ∂E/∂A as
D increased. Method 2 should not be applied when leaf
temperature is not optimal ( Farquhar et al., 1980).
Method 3 assumes a constant carboxylation efficiency as
C varies. However, when leaf temperature is suprai
optimal this is unlikely to be so. Futhermore, Lloyd
(1991) has shown that the relationship between A and C
i
(which determines carboxylation efficiency) is highly
dependent upon the method of varying C . ∂A/∂C is
i
i
larger when C was varied (thereby changing A and C )
a
i
than when D was varied. This dependence of the relationship between A and C on D directly affects ∂A/∂C and
i
i
∂A/∂T (Lloyd, 1991). Consequently, Method 3 has a
1
critical dependency on the method employed to vary A
and C and hence G. Thus again, this study views Method
i
1 as the method of choice.
In conclusion, it was found that several methods are
available to calculate dE/dA. Method 1 is the most robust
and gives consistent results, whilst Methods 2 and 2M
gave results consistent with Method 1 and are simpler to
apply than Method 1. When the full range of D
(1.5–6 kPa) was applied, ∂E/∂A was not found to be
constant and unit marginal cost of carbon fixation
increased with increasing D.
In a subsequent paper ( Thomas et al., 1998) a variation
of the method outlined by Cowan and Farquhar (1977)
will be used (i.e. Method 1M ) to examine if stomata of
six tropical tree species behave in an optimal manner
when D, leaf temperature, PPFD, and soil drought are
390 Thomas et al.
altered. It was decided to use Method 1 owing to its more
rigorous nature in the calculation method, and because
this was a laboratory-based study.
Acknowledgements
This research was supported by an Australian Research Council
grant (A19532684). The authors would like to thank the
reviewers and editors of the Journal of Experimental Botany for
their helpful critique of the manuscript.
Appendix
Method 1 used the full equations as outlined by Cowan and
Farquhar (1977)
∂E/∂G=−r2×(∂E/∂r )=r2×E/(r +r +r *
l
l
l
l b b
×(L/C )×(∂w/∂T ))
p
l
[Equation 15 in Cowan and Farquhar, 1977]
where r * equals
b
r *=1.12×r /(1+9×r ×s×T3/C )
b
b
b
p
[Equation 16 in Cowan and Farquhar, 1977]
and r =leaf stomatal resistance and equals
l
r =(Dw/E)−r
l
b
[Equation 3 in Cowan and Farquhar, 1977]
(1)
(2)
(3)
(8)
(9)
Therefore ∂A/∂C =−G/1.6
(10)
i
The modified version of Method 1 used the following mathematical function to describe the A−C response.
i
A=A [(C −C )/(K +C −C )]
(11)
max i
ci
i
where predicted maximum A (A ), CO compensation point
max
2
(C ), and a constant describing the curvature of the relationship
(K ). The partial derivative of this equation was
ci
∂A/∂C =(A ×K )/(C +K −C )2
(12)
i
max
ci
i
ci
In all other respects this method was identical to Method 1
described above (from Cowan and Farquhar, 1977).
∂E/∂G=Dw/(1+G/b)
(13)
[Equation A6 from Farquhar et al., 1980]
The mole fraction of water vapour as a function of temperature
was
(4)
where b=1/(e×rH)
b
[From Farquhar et al., 1980]
(14)
and 1/rH=1/(1.12×r )+8×s×T3/C
b
b
p
[From Farquhar et al., 1980]
(15)
e equals the rate of increase of latent heat content of saturated
air with increase of sensible heat content. e equals 4.2 at 33 °C
and 5.2 at 38 °C and is assumed constant with small changes in
temperature (Cowan and Troughton, 1971). The second order
polymonial calculated by fitting an equation to the data given
by Cowan and Troughton (1971) can be used to calculate the
response of e to T (°C ).
e=0.743+0.01241×T+0.002836×T2(r2=0.99)
therefore
dw/dT =0.06717−0.0006×T +0.0002184×T 2
l
l
l
∂A/∂G=−r2×(∂A/∂r )={1.6×r2×A+r *×(L/C )
l
l
l
b
p
×[∂A/∂T )/(∂A/∂C )]×(∂E/∂G)}/{1.35
l
i
×r +1.6×r +1/(∂A/∂C )}
b
l
i
[Equation 17 in Cowan and Farquhar, 1977]
The partial derivative of ∂A/∂T was:
l
∂A/∂T ={A /[(T −T )×(T −T )b]}
l
max
o
m
x
o
×[(T −T )b−(T −T )b×(T −T )b−1]
x
l
l
m
x
l
A and C were related by the physiological relationship
i
A=(C −C )×G/1.6
a
i
[Equation 7 in Cowan and Farquhar, 1977]
Method 2 uses a partial linearization of equations as described
by Farquhar et al. (1980).
E, transpiration; G, stomatal conductance; A, net assimilation;
C , internal CO concentration; C , ambient CO concentration;
i
2
a
2
r , boundary layer resistance. (=1/3000 m2 s mmol−1 for the
b
equipment used in this study); L, molar heat of vaporization of
water (=10.45 kcal mol−1 which is equivalent to 43.72 kJ mol−1
at 25 °C ); C , specific heat of air at constant pressure (=
p
1.012×103 J kg K−1 at 100 kPa pressure); w, mol fraction of
water vapour (mol mol−1); T, leaf temperature ( K ); T , leaf
l
temperature (°C ); s, Stefan-Boltzmann constant (=
5.6696×10−8 W m−2 K−4)
w=0.525579+0.06717×T −0.0003×T 2
l
l
+0.0000728×T 3
l
ature, T , optimum leaf temperature (°C ), T , maximum leaf
o
x
temperature (°C ), and T , minimum leaf temperature (°C ).
m
(5)
(6)
(16)
The mathematical function used by Farquhar et al. (1980) to
describe the relationship between A and G was not stated.
However, the relationship was graphically displayed and it was
stated that all lines were extrapolated to the origin. Therefore,
it can be assumed that the function was of the form of a
hyperbolic equation such as
A=K ×G/(K +K ×G) (17)
1
2
3
The partial derivative of this function would be
(17)
The relationship between A and T was described by the
l
following relationship given by Reed et al. (1976).
∂A/∂G=K /(K +K ×G)−K ×K ×G/(K +K ×G)2
1 2
3
1
3
2
3
×[(T −T )×(T –T )b]/[(T –T )×(T –T )b (7)
max
l
m
x l
o m
x o
where T , measured leaf temperature (°C ), b, [(T –T )/(T –T )],
l
x o
o m
and A , maximum net photosynthesis at optimum leaf tempermax
where K , K and K are constants fitted by non-linear least
1 2
3
squares regression analysis using the Newton-Guass algorithm
within Statistica v. 5.0.
A=A
(18)
Stomatal optimization theory 391
The modified version of Method 2 uses a different mathematical
function which has a better physiological basis and also greater
mathematical flexibility to describe the response of A to G. The
function used was
A=(K ×G)/(K +G)+K
(19)
4
5
6
where predicted maximum A (K ), a constant describing the
4
curvature of the relationship (K ) and A when G=0 (K ).
5
6
The partial derivative of this equation was
∂A/∂G=(K ×K )/(G+K )2
(20)
4
5
5
In all other respects this method was identical to Method 2
described above (from Farquhar et al., 1980).
Method 3 uses the following formula presented by Lloyd and
Farquhar (1994). ‘By linearizing the curvilinear relationship
between carboxylation efficiency and chloroplastic mole fraction
of CO and ignoring both leaf boundary layer and leaf internal
2
conductance to CO diffusion Lloyd (1991) showed that a
2
constant l was associated with stomatal response of the form’
(Lloyd et al., 1995)
C /C =1−√ [1.6×D (C −C )×P/lC 2]
(21)
st a
c a
a
[Equation 12 in Lloyd et al., 1995]
where C is the CO concentration in the substomatal cavities,
st
2
i.e. analogous to C . D , canopy to air water vapour pressure
i c
mole fraction difference. It has been assumed that this is
analogous to Dw. P, pressure; l is a lagrange multiplyer representing the marginal water cost of plant carbon gain, i.e. ∂E/∂A.
Taking C =C −1.6×A/G the above equation can be redefined
st
s
as
G=A√[1.6×l×P/(C −C )×D ]
(22)
a
c
[Equation 13 in Lloyd et al., 1995]
Rearranging this equation to make l the subject and substituting
Dw for D yields
c
l=∂E/∂A=G2×Dw×(C −C )/1.6×A2×P
(23)
a
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