PGEN: an integrated model of leaf photosynthesis, transpiration, and conductance

ECOLOGltnl IllODELllllG ELSl?VIER

Ecological Modelling 77 (1995) 233-255

PGEN: an integrated model of leaf photosynthesis, and conductance

transpiration,

A.D. Friend Institute of Terrestrial Ecology, Edinburgh Research Station, Bush Estate, Penicuik, Midlothian EH26 OQB, UK

Received 6 April 1993; accepted 27 October 1993

Abstract A detailed model of leaf-scale photosynthesis, respiration, transpiration, stomata1 conductance, and energy balance is described. The model, PGEN ~2.0 ‘, is designed for use in larger-scale ecosystem, climate and hydrological models concerned with fluxes of CO,, water, and heat. Given a set of environmental and biological (mostly leaf) parameters, PGEN calculates instantaneous rates of net photosynthesis and transpiration, and associated conductances to CO, and water. The model is intended to predict species-specific behaviour with minimal need for empirical parameterisation. The biochemical model of photosynthesis is derived from the models of Farquhar and co-workers. This biochemical model is embedded in a model of the leafs energy balance, which is based on the work of Monteith and Jones. Stomata1 conductance is calculated using an optimisation concept. In this concept there is an assumed tradeoff between CO, entering and water leaving the leaf, resultirig in a single stomata1 conductance for each set of environmental conditions, that maximises a function including the costs and benefits. Predicted responses of stomata1 conductance, net photosynthesis, transpiration rate, and the ratio of CO, concentration in the leaf to that outside the leaf boundary layer, to key environmental factors, closely match observed responses. A sensitivity analysis of PGEN ~2.0 shows that predicted net photosynthesis is most sensitive to the degree of co-limitation between carboxylation- and ribulose-1,5-bisphosphate regeneration-limited photosynthesis, the Rubisco carboxylation kinetic parameters, the atmospheric concentration of CO,, and leaf nitrogen content. Predicted stomata1 conductance is most sensitive to relative humidity, the critical leaf water potential for plant dry matter production, the hydraulic resistance between the root and the leaf, and the atmospheric concentration of CO,. Keywords:

Conductance;

PGEN; Photosysthesis;

Transpiration

1. Introduction This paper presents a complete description of PGEN ~2.0, a model of leaf photosynthesis, tran-

’ The code is available from the author upon request. 0304-3800/95/$09.50

0 1995 Elsevier

SSDI 0304-3800(93)E0082-E

Science

spiration and stomata1 conductance. PGEN predicts fluxes and conductances of CO,, water, and heat between a single leaf and the atmosphere outside the leafs boundary layer. The accurate prediction of CO,, water, and heat fluxes is an essential component of ecosystem and climate models. PGEN (~2.0) is designed to be used as a

B.V. All rights reserved

234

A.D. Friend /Ecological

subroutine of larger-scale ecosystem models, but can also be used to assess single-leaf scale observations. The modelled leaf is assumed to be exposed to the sky and at some angle. The model is driven by commonly measured biological and physical parameters, and has as its principal aims the prediction of stomata1 conductance and photosynthesis. An earlier version of PGEN (~1.0) was described by Friend (1991). The present paper is a full description of PGEN ~2.0, which includes many new developments. The modelling of the partitioning of leaf nitrogen between Rubisco and chlorophyll has not been developed further since PGEN ~1.0, and so is not included in this paper. Results of a sensitivity analysis are presented, along with some general results indicating the relationships between environmental parameters and predictions. The assumption in PGEN ~1.0 that leaf water potential has a direct effect on photosynthesis has been removed from PGEN ~2.0. This is due to increasing evidence that photosynthesis is not affected by leaf water status until severely low relative water contents are reached (e.g. Kaiser, 1987). In addition, observations of a purported direct effect based on the interpretation of “A / Ci” curves may in fact be artifacts due to patchy stomata1 closure in short-term drought experiments (Sharkey and Seeman, 1989). There is, however, clear evidence for an effect of drought on photosynthetic capacity over longer periods of soil drying. But it would appear that the reduction in capacity in these instances may be due to down regulation of the biochemistry of photosynthesis, as a longer-term result of stomata1 closure (Martin and Ruiz-Torres, 1992). Vassey et al. (1991) studied the effect of water stress on sucrose-phosphate synthase (SPS) activity in Phaseolus vulgaris. They found that, under mild water stress, changes in the biochemical capacity for photosynthesis could be attributed to changes in SPS activity, and that these, in turn, were the result of reductions in internal CO, concentrations caused by stomata1 closure. Reduced SPS activity is known to lead to feedback inhibition of the biochemistry of photosynthesis. Vassey et al. (1991) concluded that “, . . there is rzo effecr of

Modelling 77 (1995) 233-255

water stress on photosynthesis except by way of stomata1 closure until relatively severe water stress occurs”. Graan and Boyer (1990) and Quick et al.

(1992) cite many of the references pertinent to this debate. The decision to omit a direct effect from PGEN ~2.0 is based on the clear direction away from a direct effect in the development of this subject. Nevertheless, due to the success of the former approach in predicting stomata1 behaviour, the mathematical structure of the optimisation scheme has been maintained as close as possible to that used in PGEN ~1.0 (see below). A comparison is included in this paper between predictions made with and without a direct effect. PGEN ~1.0 has been shown to predict realistically the measured responses of leaf photosynthesis and stomata1 conductance to environmental variables (Friend, 1991). Its use in a canopy-scale application has also yielded realistic results (Friend et al., 1993). Dewar and Friend (unpublished) derived an analytical solution to a simplified version of the stomata1 model in PGEN ~1.0. They showed that this solution is consistent with the observed conservative behaviour of intercellular concentration, and highlighted some interesting insights that the model provides into the way in which environmental and physiological factors may interact to determine stomata1 responses. A simplified version of the full PGEN ~1.0 model was used by Friend and Cox (1994) to investigate the importance of vegetation-climate feedbacks in modelling the effects of atmospheric CO, change on climate and productivity. In PGEN ~2.0, SI units are used throughout. It is hoped that this will help in the transferability and clarity of the equations, predictions, and parameters presented here. Fluxes are expressed in mol me2 s-l, resistances in s m-l, conductances in m s-l, concentrations in mol rne3, and leaf contents in kg m-* or mol m-‘. The area referred to is vertically projected area for the leaf, and ground area for shortwave radiation input. Conversion of leaf properties from kg to moles occurs in converting from leaf Rubisco to leaf catalytic site content (Eq. 24) and in leaf chlorophyll content to irradiance-saturated potential electron transport rate (Eq. 29). There has been

A. D. Friend /Ecological

some debate concerning the relative merits of using mol m-* s-l as opposed to m s-l for conductance units in plant ecophysiology (e.g. Pearcy et al., 1989). The former is preferred by some because of its lower temperature sensitivity (although it is still affected by temperature and pressure). However, in PGEN ~2.0, m s-l are used throughout, so that temperature and pressure dependencies are made explicit. Thus PGEN can be used to fully assess the implications of temperature changes on all components of the leaf-environment system. Pressure dependencies are included so that questions relating to the effect of altitude on leaf processes can be addressed (e.g. Friend and Woodward, 1990; Smith and Donahue, 1991). In addition, the effect of temperature and pressure on the solubilities of CO, and 0, in the chloroplast is explicitly considered. This has the consequence that the measured effects of temperature on the K, values of Rubisco for CO, and 0, are used directly. The treatment of the biochemistry of photosynthesis in PGEN follows from the frequently used models of Farquhar et al. (1980) and Farquhar and von Caemmerer (1982). Stomata1 conductance is calculated using an optimisation scheme. There are many different models of stomata1 conductance; the majority are empirical or use some type of goal-seeking approach. Examples of the former include the models of Jarvis (1976) and Ball et al. (1987). These models are derived from measured responses to environmental variables. However, it is not clear whether they remain valid when used to simulate responses to conditions different from those from which they are parameterised. Also, the details of the model may be circumspect. For instance, the Jarvis model assumes that the environmental variables influence stomata1 conductance independently, which is unlikely to be true. The model of Ball et al. is driven by leaf surface concentration of CO, and leaf surface relative humidity; however, it is more likely that stomata respond to internal leaf CO, concentration and leaf water status or transpiration rate (Kearns and Assmann, 1993). This has significant consequences for the effects of ambient CO, concentration, leaf-to-air vapour density deficit, and leaf bound-

Modelling 77 (1995) 233-255

235

ary layer conductance on predicted stomata1 conductance in the Ball et al. model. Examples of goal-seeking models of stomata1 conductance include those of Cowan (19771, Givnish (19861, and Friend (1991). The Cowan model uses the principle that, over a given period of time, stomata behave so as to minimise the amount of water lost for a given amount of carbon gained. However, it is difficult to use in practice because it is not possible to know a priori what this value of carbon gained will be. Also, there is evidence that stomata do not, in fact, behave in this way, especially in response to varying irradiance (Dewar and Friend, unpublished). There are also some semi-mechanistic models of stomata1 conductance (e.g. Farquhar and Wong, 1984; Johnson et al., 1991; Tardieu and Davies, 1993). These are potentially useful approaches, but we do not yet have sufficient knowledge of the mechanisms whereby stomata respond to the environment to enable these models to be elevated to the status of complete mechanistic descriptions. Consequently they must be used with caution, and should be treated as tools with which mechanistic hypotheses can be evaluated. We have much to learn about the functioning of stomata; a clear and concise account of the current state of knowledge is provided by Kearns and Assmann (1993). PGEN ~2.0 follows from the work of Givnish (1986) and Friend (1991). Givnish (1986) described a model of optimal stomata1 conductance and allocation of energy between leaves and roots, assuming that the cost of transpiration to the plant was an effect of leaf water potential on the CO, photosynthetic compensation point and the apparent mesophyll resistance. Friend (1991) used the same approach to calculate stomata1 conductance, but included a more mechanistic biochemical treatment of photosynthesis and assumed that net photosynthesis is affected by leaf water potential (this model was PGEN ~1.0). He scaled the predictions up to make quantitative predictions of the responses of photosynthesis and stomata1 conductance to the environment. The purpose of this paper is to give explicitly all the equations used in the model PGEN ~2.0,

236

A. D. Friend /Ecological

Modeliing 77 (1995) 233-255

to demonstrate some general predictions, and to indicate the parameters to which the model is most sensitive. It is not the intention here to provide a rigorous validation of PGEN.

2. The model PGEN combines biochemical and physical processes at the single leaf scale. The scaling from the biochemical to the whole leaf level is based largely on the work of Farquhar and co-workers (e.g. Farquhar et al., 1980; Farquhar and von Caemmerer, 1982); the more physical and micrometeorological aspects are derived from the work of Monteith (1973) and Jones (1983). For each set of conditions, three fluxes of CO, are calculated. Two of these, the Rubisco carboxylation-limited rate (Acar& and the ribulose-15bisphosphate (RuBP) regeneration-limited rate (A nuni,), are at the chloroplast level (scaled to the whole leaf). The third rate, the stomata1 resistance-limited rate (A,=; N.B. resistances are used to mathematically describe the model, resistance is the inverse of conductance), is directly at the whole leaf level. For a given value of stomata1 resistance, PGEN calculates the two leaf CO, concentrations (Ci,carb and Ci,nunP) that balance Arc with Acar,,and ARUBP,respectively (i.e. Acar,, = Arcat Ci = Ci,carb, and ARuBP= Arcat Ci = rates so obCi,RuBP 1. The two chloroplast-level tained then determine the final predicted rate of net photosynthesis at that value of stomata1 resistance, assuming an element of co-limitation. Stomata1 behaviour is calculated using an optimisation approach. Photosynthesis is clearly a “benefit” to the plant, and observations of stomatal closure associated with higher leaf-to-air vapour pressure deficits and/or lower amounts of soil water indicate that transpiration represents a “cost”. In PGEN ~2.0, stomata1 behaviour is assumed to balance these opposing effects. Instead of assuming a direct effect of leaf water potential on photosynthesis (as in PGEN ~1.0; see Friend, 1990, in PGEN ~2.0 it is assumed that leaf water potential limits plant growth. In order to maintain the mathematical structure of the optimisation scheme used to cal-

culate stomata1 conductance in PGEN ~1.0, the function maximised in PGEN ~2.0 is still a factor related to intercellular CO, concentration multiplied by a factor related to the plant’s water status. However, photosynthesis is identified with only the first of these factors, rather than their product. Leaf water potential (q,ea,) is used to represent plant water status, but another index such as root water potential or whole plant relative water content could be used instead. The function being maximised is then identified with dry matter production rather than photosynthesis. Below a critical leaf water potential (yrC) it is assumed that there is no dry matter production CR’-,;kg s-l>. As ‘I’,eaf increases above qc, dry matter production increases up to some maximum at q,eaf = 0. This functional dependence between W, and ‘Pilaf is denoted by fV',,& and varies between 0 and 1. Since increasing transpiration reduces qleaf, fq,, represents the cost associated with transpiration. These assumptions imply that:

Wd=fqleti.kX

(1)

where A = net photosynthesis (molccoz, m-’ s-l; Eq. 18); and k = a constant of proportionality (kg m2 mol-‘). k is not simply leaf area since it must also include respiration and litterfall components. Its value is not important in the present context since it is assumed constant. Consequently, at optimum stomata1 resistance to CO, flux (I,,,; s m-l):

(2) The test of the stomata1 resistance component of PGEN ~2.0 is whether a unique function f*,,,, when used in Eq. 2, is sufficient to predict stomatal behaviour with the degree of accuracy required. I will show here that realistic predictions are made with Vc set to - 1650 J kg-‘, and f*,,, increasing linearly to 1 at *i,,r = 0. In PGEN ~1.0 a direct effect of leaf water potential on photosynthesis was included, making predicted A in PGEN ~1.0 equivalent to A *fw,, in PGEN ~2.0. An optimisation concept for predicting stomatal behaviour is potentially very useful since it

A.D. Friend/Ecological

237

Modelling 77 (1995) 233-255

enables complex behaviour to be predicted without the need to parameterise detailed mechanisms. Potential weaknesses in its application include the possibility that plants do not operate optimally (likely, but selection is very powerful and so deviation from optimality may be small; see Friend, 1991, regarding arguments on optimality), its use in conditions to which plants have not had a chance to become adapted (a similar criticism could be lodged at empirical models when applied outside the range of conditions under which they were parameterised), the possibility that Yc is not fixed for all conditions in a given species (implying that fvleaf is not a linear function or, more seriously, that it is not a unique, species-specific, function), and the difficulty in defining physiologically exactly what is being optimised (not necessarily important if the model is used purely as a predictive tool). Nevertheless, the stability of qc can be examined by comparing predictions with observations across as wide a range of species and under as many conditions as possible. PGEN finds the optimal stomata1 resistance numerically. All model inputs, intermediate parameters, and outputs are defined in Tables 1, 2 and the Appendix.

and 0, respectively (leaf air space equivalents; mol m-3); Oi = concentration of O2 in leaf air spaces (assumed equal to concentration in air outside leaf boundary layer, O,i,; mol mV3); and R, = mitochondrial (“dark”) respiration (mol~coz, mm2 s-‘; Eq. 26).

2.1. Main CO, balance relations

f Plea‘= 1 - ( *,:,,t/*c

Carboxylation-limited photosynthesis

The carboxylation-limited rate of net photosynthesis is given by (Farquhar et al., 1980): A carb

Kmax(Ci,carb = ci

,carb

-

+K,(l+o,/K,)

r*

>

_

Ribulose-bisphosphate synthesis

regeneration-limited

photo-

The ribulose-bisphosphate (RuBP) regeneration-limited rate of net photosynthesis is given by (Farquhar and von Caemmerer, 1982): A

J(Ci,RuBP-

r*)

RuBP= 4.5Ci,R”BP + lO.W*

-R,,

(4)

rate where ARuBP = RuBP regeneration-limited of net photosynthesis (mol~coz, mW2 SK’; Eq. 18); J = potential electron transport rate (mol rnA2 S-‘; Eq. 28); and Ci,nuBP= concentration of CO, in leaf air spaces for RuBP regeneration-limited net photosynthesis (mol rnp3; Eq. 7). The concentrations in the leaf air spaces are adjacent to the chloroplasts. Effect of leaf water potential

PGEN assumes a linear response of W, to down to a critical water potential, so that zLI:‘is given by: 17

(5)

where Vl,eaf= leaf water potential (J kg-‘; Eq. 20); and Vc = critical leaf water potential below which dry matter production is zero (J kg-‘). If the leaf water potential is below ‘PC,fqleaf is set to zero.

R

d’

(3)

where Acarb = carboxylation velocity-limited rate of net photosynthesis (mol(co2, m-’ SK’; Eq. 18); Vc,max= maximum rate of carboxylation (mol(,,,, m-2 s-l; = concentration of CO, JQ. 22); Ci,carb in leaf air spaces for carboxylation-limited net photosynthesis (mol rnp3; Eq. 7); I* = photosynthesis compensation concentration in leaf air spaces in absence of mitochondrial respiration (mol rnp3; Eq. 27); K, and K, = Michaelis-Menten constants of Rubisco for CO,

Stomata1 resistance-limited photosynthesis

The stomata1 resistance-limited rate of net photosynthesis at the whole leaf level (Arc; mol (co,) m -2 s- ‘> is given by (after von Caemmerer and Farquhar, 1981): Arc =

‘air

-

r,

ci

-

‘air

+

2

ERT

‘i

x

P’

(6)

where Cair = concentration of CO, in air outside leaf boundary layer (mol mW3); Ci = concentration of CO, in leaf air spaces (i.e. Ci,carbor Ci,RuBPin

238

A. D. Friend /Ecological

Eqs. 3 and 4) (mol me3); rc = total resistance to CO, flux from outside leaf boundary layer to outside mesophyll liquid phase, a component of which is stomata1 resistance (s m-‘; Eq. 30); . . Eq. 19); E = transpiration (moluizo, m -2 s-1. R = gas constant (8.3144 J K-i mdl-I); T= average of leaf and air temperatures (K); and P = atmospheric pressure (Pa). RF/P converts CO, concentration (mol rne3) to mole fraction (molccoz, mol;,‘,,).

Modelling

77 (1995)

233-255

Then the parameters

in Eq. 7 are given by:

a=g,+y

(13)

and 6carb~~c~~v~Cair~~~~~v~Cair~ +

K,nm

-

%

(14)

and b RuBP =gc’(z-c,i,)+Y’(z+c,i,) +x.(J-4.5R,)

(16)

Solving for internal leaf CO, concentration

and

Stomata1 resistance affects three primary physiological variables: leaf air space CO, concentration, leaf water potential, and leaf temperature. In order to find the two steady-state chloroplastlevel rates of photosynthesis at a given value of rc, it is necessary to solve the balance conditions A carb =Acond and ARum, =Acond (using Eqs. 3, 4, and 6) for internal leaf CO, concentrations Ci,carb and ‘i RuBP’ The solutions can be found analytically: ’

d carb= v ’ Cair ’ ( Y - g,> - vc,,,

-6, ‘i,x

=

+ \/6,2- 4ad, 2a

’

(7)

where x is either carb or RuBP. Equations to calculate the parameters for Eq. 7 are given below. If the total conductance to CO, flux (g,; m s-‘) is given by: g, = l/r c

(8)

(15)

and d RuBP =z’c,i,*(Y

x=25

I

( 10)

+ lOS.I’,

.Rd).

(17)

These values of Ci are then used in Eq. 6 to give the two steady-state rates (Acarb and ARUB,) for a given value of total resistance, rc. Calculation of co-limited photosynthetic rate at a given stomata1 resistance

In some models of photosynthesis the minimum of the carboxylation- and RuBP-limited rates is taken as the actual rate. However, this does not allow for any degree of co-limitation, and results in discontinuous response curves due to a sudden change in the limiting factor. Here the following is used (after Collatz et al., 1990): A=

and

-g,)

-x*(J.I’,

and

(9)

. r* + R, ’ 1,

6-m 28

’

(18)

where A = final predicted rate of net photosynthesis at a given stomata1 resistance (molooz, mV2 s-l); 6 =Acarb +An”ar; c =Acarb XAnuap; and 8 = empirical co-limitation coefficient (0 G 8 G 1; dimensionless).

and Y=y

E

.-

2.2. Main H,O balance relations

RF P

(11)

and 2 = (10.5/4.5)f*.

(12)

Transpiration

The rate of transpiration is predicted from the water vapour concentration difference between the leaf air spaces and the air outside the leaf

A. D. Friend /Ecological

boundary deficit): EC_

layer (i.e. the leaf-to-air

vapour

density

wi - w, (19)

’

rw

where Wi = concentration of water vapour in leaf air spaces (mol me3); W, = concentration of water vapour in air outside the leaf boundary layer (mol mP3); and r,,, = total resistance to water flux across the leaf boundary layer and leaf surface (s m -l; Eq. 32). Wi is calculated as the saturated air water vapour concentration at leaf temperature (ysCr,,; Eq. 35). W, is given by the product of the relative humidity and WgT,). Wi - W, is the leaf-to-air vapour density defilit.

Leaf water potential Leaf water potential Honert, 1948): *

leaf =

*root

-

is given by (after van den

O.OlSr,,,E,

(20)

(J kg- ‘; Eq. where Yrroot= root water potential 21); and r,,, = resistance to water flux between root and leaf (kg-’ m4 SC’). The constant 0.018 converts transpiration from molar to mass units. rr, is species specific and increases with the height of the leaf above the root (Friend, 1993) though this dependence is not included here. Root water potential is given by: 1I’rOOt= qsoir - O.O18r,,,E,

(21)

where *soi, = soil water potential (J kg-‘; Eq. resistance to water flux between = 36); and ra,r bulk soil and root interior (kg-’ m4 SC’; Eq. 33). The equation for rs.r is given below in the section on resistances. 2.3. Auxiliary relations

Rub&o parameters Vc,max(Eq. 3) is a function

of the carboxylation turnover number and the number of catalytic sites (after Farquhar et al., 1980):

Vc,max= kCE, 9 where

k, = Rubisco

(221 carboxylation

turnover

num-

Mode&g

77 (1995) 233-25.5

239

ber (molC,oZ, mol&) s-r); and E, = leaf Rubisco catalytic site content (mol m-‘; Eq. 24). Vo,max, the maximum rate of oxygenation by Rubisco (molCo2, mW2 SC’; needed to calculate r* in Eqs. 3 and 4; see Eq. 27 below) is calculated in similar fashion to Vc,max:

v,,,,

= k,E, 9

(23)

where k, = Rubisco oxygenation turnover number (molCoz, molC&sj s- ‘1. The leaf Rubisco catalytic site content, E,, depends on leaf Rubisco content. Full activation of Rubisco is assumed. Since the molar mass of Rubisco is 550 kg mol- ‘, and there are 8 active sites per molecule (after Farquhar et al., 1980):

E, = 8a,,,/550,

(24)

where uRub = leaf Rubisco content (kg mm2). Leaf Rubisco content is calculated from leaf nitrogen, assummg 0.16 kg,,, kglkl,bisco) (after Farquhar et al., 1980): uRub

=

6.25f,,,,&

(25)

where fN,Rub = fraction of leaf nitrogen bound in Rubisco (dimensionless); and N = leaf nitrogen content (kg m-“). N needs to be specified as an external parameter. Friend (1991) calculated fN,Rubassuming optimal partitioning of N between Rubisco and chlorophyll, but here fN,Rub is an input to PGEN v2.0.

Mtochondrial respiration R, (Eqs. 3 and 4) is assumed function

of leaf nitrogen

(after

to be a linear Ryan, 1991):

R, = R+N,

(26)

Table 1 Constants required to calculate temperature-dependent parameters using Eq. 54. The parameters for k,, k,, and IV~.~,,, were derived to fit the data of Jordan and Ogren (1984); the parameters for RdT are fitted to give RdT= 4.1 x 10M4 mol~,oz, kg&\ s-’ at 25°C with a Q,,, value of 2.1. Units for aT are as for parameter Parameter

kc k, K c,chl R dT

E, (J mol-‘)

ar (mol (co,) mol&) SC’) (mo& mol&) s-t) (mol m-s) (mol (co,) kgi’ s-‘J

2.897x

1Or4

4.397x 10’ 3.076 x 10s 1.658 x 106

82000 44000 60 100 54836

240

A. D. Friend /Ecological Modelling77 (1995) 233-255

where R,+ = rate of mitochondrial respiration (mol(co*, kg;,: s-l). R+ is calculated as a function of temperature using Eq. 54 and the constants in Table 1. Photorespiratory compensation point

The CO, flux compensation concentration in the leaf air spaces in the absence of mitochondrial respiration (F,; Eqs. 3 and 4) is a function of the rates of oxygenation and carboxylation (van Caemmerer and Farquhar, 1981): Kc”i r* = o’5K,max

vc,*ax

K,

*

(27)

Electron transport

The potential electron transport rate (J; Eq. 4) is the rate which would occur if the acceptor molecule NADP+ were at its maximum concentration (Farquhar and von Caemmerer, 1982). J is dependent on absorbed irradiance, leaf temperature, and leaf chlorophyll content. The irradiance dependence is given by (after von Caemmerer and Farquhar, 1981): J=

Jmax IAPAR IAPAR

+ 2*1 Jm

’

potential elecwhere J,,,,, = irradiance-saturated tron transport rate (mol m-* s-r; Eq. 29); and I APAR = absorbed photosynthetically active irradiante (molcquanta)rnp2 s-l; Eq. 38). The irradiance-saturated potential electron transport rate is dependent on the leaf chlorophyll content: J max=

jmaxfN,chlN

0.056

’

potential elecwhere j,, = irradiance-saturated tron transport rate per mole of chlorophyll (mol molt& s - ‘); and fr.+,,, = fraction of leaf nitrogen bound in chlorophyll (dimensionless). The denominator in Eq. 29 is the product of the molar mass of nitrogen (0.014 kg mol-‘1 and the number of moles of nitrogen per mole of chlorophyll (i.e. 4). fN,ch, and N are inputs to PGEN. j,, is strongly temperature dependent (Eq. 58).

Resistances to CO, and water jluxes The total resistance to CO, flux between the

air outside the leaf boundary layer and the air in the leaf adjacent to the mesophyll cells (r,; Eq. 6) is calculated assuming that the boundary layer resistance, stomata1 resistance (including cuticular resistance), and leaf internal resistance act in series: rc = rc,a + r,,, +

(30)

rc,iy

where rca = resistance to CO, flux across leaf boundary layer (s m-‘; Eq. 41); rcS = resistance to CO, flux across leaf surface (s ‘m-l; Eq. 2); and rc,i = resistance to CO, flux from inside leaf surface to outside mesophyll liquid phase (s m-‘; Eq. 31). is varied until its optimum is determined (sa&ying Eq. 2) to a given precision (see below). rci is approximated from leaf thickness:

(31) where q = a constant (83 X lo3 s me2); w = leaf thickness (m); T, = leaf temperature (K; Eq. 42); T, = standard temperature (293.15 K); and P, = standard atmospheric pressure (101325 Pa). The coefficient in Eq. 31 is calculated assuming that a leaf with a thickness of 300 x low6 m (average value given by Parkhurst et al., 1988) has an internal resistance of 25 s m-l (Nobel, 19911, though there is evidence that internal resistances may be substantially more than this in at least some species (D.F. Parkhurst, pers. commun., 1990). r,,, (Eq. 19) is calculated from the CO, flux resistances: rW= 0.607r,,,

+ 0.704r,

s.

(32)

The coefficients in Eq. 32 are adapted from Monteith (19731, assuming still air across the leaf surface and laminar flow through the leaf boundary layer. rs,r (Eq. 21) is calculated using the semi-empirical expression of Johnson et al. (1991): (33)

‘(III) uo~Suaw~p a!ls!JalaE -.Ileqa3EaI=P pue f(,_SLU)paadsPU!M= tl ‘XC3qM

*(sE-w 2Q) '2 30 an~EA2h~lEm~es= mw‘sx alayM

(LS)

:(E861 ‘Sa’JOffEL61 ‘ql’WOI4 Dye) paads PUlM aqJ pue 3”q aql 30 uO!SUaUup ays~~alae~eqa ayl ~013 pa~elnaleasy (zc pue 0~ .sba fe"l)xng zo3 01 aauEls!sa~ _~arle~ hepunoq 3Eal‘snolEwols!ydrut? sj3eala~ leyl%punssV az4v~y1si~4 ~a&~ k.mpunoqJva~

'(ES67‘sauof)2'0~IllW~ s! q'sd‘saaE3lns palElaiIaA 10d *(ssap~oysuaw!p) a3uE -!pey a~e~llqloy103 puno.Qacq JO luay33aoa UO!laaua.l = “'d pUE f((p""o;8_)Ul M) QS UlOl3 a3UB -!peg a~wtl_~oqs = p‘elf(aal%ap)Qs ~1013aaule!p -??.I.I! ]Uap!aU!pUE Jl?al UaaMlaq a@Ik?= /.L alaqM (09)

‘(S%+

‘CEO1x8EP)/Sz=~vdz

‘

WIU‘S e =yJ

= “x

E+ ‘!_qzi-1 “e

:LquaA@ s!4~Ayanpuoa aynerpllypas ~(ssa~uo!suaunp)~alauwEd a!3pads (‘a3”M~~) paw -110sleaurdrua = ‘!OSqpue '((I!"s)~ . . * E I!os=wwsfj -riles pas uaqM lualuoaIaxM aA!lEIaJ f('l!+ (ra'em$u) lualuoa Salem aA!llelaJ I!OS= "0 f(,_%yr>““& 30 an~eAu.uuu~tu ="7"rE. alaqM

:dqUaAg s! Ieyualod IaleM 1'0s'(166~)'1~$a uosu~of dq uaA!fIsuo!ssa.uixa aql %ursn palEInaIe3an2 &A!1 -anpuoaaywpQ 110spue pylualodIaieM QOS ,@cly?puo3 zgnvlp4y pas puv pyualod AaluMpas salqv.uvn ~uauruo.yu~

~)"‘"~(1L)S03 ="I

:dpuanbasuoa‘EalE punol% l!un lad passaldxa ale saauE!peJq ow asaqL *puno.@aEq ayl ~0.13aauE!pw! pa$aaaaJ ayl pue kjs aql 111013 aaue!pE~~! alrcwoys agl30 uxnsayl s!11Tale 3EallrunJad s!"1‘65'bg UI 'HVd 01 aAEM -I.roys pit101 1.11013 1laAuoa01 g-~30 .zolae3 E sapya -II!pue ‘suoloqd30 saI0u.I 01 slle~ u10.13 sl!un uo!lE!pw sllaAuoa6E:'bg u! luap!33aoaaqL XOP 'ba +~al)UIM) 3ealuo luap!au!aauerpelq aAEAwoys ="I alarIM (6'C)

:(zy, a1qm ‘~861 ‘sauof Iage) se aAeAwoqs luappu! ~013 paA!Japs! Xvdz '(luauoduroa laIo!ArzlIn ayl &upnpur 1ou)Lysayl U0rJEypE.l aAEMJ.IOyS 6q UaA;r.Ip S! NTJf-jd

‘p-2

-(g)a.~nlE.~adu~al =J. pue :&III 10~)~ w lnOdEA IaleM 30 uogequaauoa uogtzuws =(J'sh aw.jM

(SC>

‘

S8'S'C-L (SKLZ

dxax

-L)69Z'LT

:(~961‘IcE.LIn~ Iage) UOgEnba pza!qdtua 2kI!MOIlO3 aIj$Aq UaAi% S! (Es pUE ‘69 ‘61 *sbg 103 PaSIl) UO!lE.IlUaaUOa lnOdEA .lalEMUO!lEInlES pue amle~adwal uaaM)aq d!~suo!lEIa.I ayJ. uoytwuawo~ lnodvn lait%u uoyvwvg

‘(w)qldap

8u!loo~ =‘p

alarIM

UIOl3

aaq.wis3eaIuo '(6s'ba f,_sz-~ @Juenb)~~tu) mapy XV’d 1e301= avdz pue f(ssaluo!suauup) XVd 103luag!33aoa uopdlosqE3EaI= Xvd~ a_IayM ( XVdz

(8s)

XVd x) = WdVz

XVd 1~101u1ol3pawInaII?as! (82 'ba fXvaz :aVd) aXIe!pe~I!a~yE d~pza!laqluLsoloyd paq.IosqV asuv~pvLl~

(PS> :yldap%U!IOOJaqi 119sser.u LIP ]Emlanlls 100~ ayl du!p!~!p1(qpauyqo s!lll!suap 1008 '(I _S zm gOI X l-j-~) luWSu0~ E = J‘y/ pue f((Punom z_~ i3~)ssm lcrpIwnlanqs 100~ = “9~ :(LE'bg fs E_~ 83) Q!Ayanpuoa aynwp -,Q 110s= 'x t(pc 'bg f@$u ('a1'ew hp'3y) Q!s -uap 1001=Jd f(,r.u 0.~)luelsuw E =sfsy alaqM

242

A. D. Friend /Ecological

The coefficient in Eq. 41 is adapted from that used in eq. 3.30 of Jones (1983), with a correction to agree with observations (X 1.5; Jones, 1983, p. 541, and conversion to CO, units from heat assuming laminar flow in the leaf boundary layer (Monteith, 1973). For large hypostomatous leaves, r c.a must be doubled. Leaf temperature

The calculation of leaf temperature from eq. 9.6 of Jones (1983): T, = T, +

is adapted

(42)

where T, = temperature of air outside leaf boundary layer (K); y = psychrometer “constant” (mol me3 K-l); @‘ni= isothermal net radiation of leaf (W m$n; Eq. 45); pa = density of dry air (kg rnp3; Eq. 48); cp = specific heat of air (1012 J kg- ’ K-l); SW = water vapour concentration deficit of air outside leaf boundary layer (mol rne3; Eq. 49); rHR = parallel resistance to heat loss from leaf by convection and radiation (s m-‘; Eq. 50); and s = slope of saturation vapour concentration curve (mol me3 K-l; Eq. 53). The psychrometer “constant” is given by (after Jones, 1983): PC,

-.’ = 0.622A

1 RT, ’

(43)

where A = latent heat of vapourisation of H,O (J kg-‘; Eq. 44). The effect of air temperature on A is calculated from a linear interpolation of the values given by Jones (1983): A = 3.1512 x lo6 - 2.38 x 103T,.

The difference between T, and T, depends on the amount of cloud. Under clear skies, T, is about 20 K below T,. This difference is reduced to about 2 K in cloudy conditions (Jones, 1983). The following empirical relationship to short-wave irradiance is used: T, = T, - 0.825 exp(3.54 X 10-3Z,,d).

(46)

The leaf absorption coefficients for shortwave and photosynthetically active radiation (PAR) are related empirically by (Pearcy et al., 1989): (Y,= 0.73a,,,

rwY *ni

------6W P&p

Modelling 77 (I 995) 233-255

- 0.119.

(47)

The density of dry air is calculated from the data in appendix 3 of Jones (1983): pa = 2.42 - 4.12 x 10-3T,.

(48)

SW is calculated from:

6W= K(T.Jl -fw),

(49)

where WNTa,= saturation concentration at air temperature (mol rnp3; Eq. 35); relative humidity of air (dimensionless). The total thermal resistance to heat the leaf by convection and radiation given by (Monteith, 1973): 1 -=

1 -+-,

1

'FIR

'aH

'R

of water and f, = loss from (rHR) is

(50)

where ran = leaf boundary layer resistance to convective heat flux (s m-‘; Eq. 52); and rR = radiative “resistance” to heat loss (s m-‘; Eq. 51). rR is given by (Monteith, 1973): rR = (~ac,)/(4G7.

(51)

Isothermal net radiation is based on eq. 2.12 of Jones (1983):

r aH is calculated from rc,+ (assuming laminar air flow) using the coefficient given by Jones (1983; after Monteith, 1973):

Qni = a,Z, + UT; + UT; - 2aT:,

r aH = 0.76r,,,.

(44)

(45)

where (Y,= fraction of shortwave absorbed by leaf (dimensionless); u = Stefan-Boltzmann constant (5.6703 X 10e8 W me2 Kp4); T, = apparent radiative temperature of atmosphere (K); and Tb = background temperature (assumed equal to air temperature; K>.

(52)

s is defined as: S=

WG”d - wk, T, - T,

(53)

’

where K(T,) = saturation concentration at leaf temperature

(mol rne3; Eq. 35).

of water

A. D. Friend /Ecological

Modelling 77 (1995) 233-255

243

2.5. Leaf temperature responses of parameters

Table 2 Constants required to calculate solubilities using Eq. 56. m and n are derived from the data in table 10.2 of Lawlor (1987)

Jordan and Ogren (1984) found that the Michaelis-Menten constant of Rubisco for oxygen at the chloroplast level (K,,,,; mol me31 did not change with temperature, hence it is assumed to remain constant at the average value measured (i.e. 0.535 mol m-3). Conversely, the MichaelisMenten constant of Rubisco for CO, at the chloroplast (i.e. dissolved) level (K,,,,; mol mP3) has been observed to increase by an order of magnitude between 5°C and 35 “C (Jordan and Ogren, 1984). Conversion between chloroplast concentration values and air space equivalents is given below. The parameters k, (Eq. 22), k, (Eq. 231, K+, (Eq. 551, and RdT (Eq. 26) increase with leaf temperature. The following equation (Thornley and Johnson, 1990) was fitted (values given in Table 1) to measured temperature responses (Jordan and Ogren, 1984; Rd, was assumed to have a Q,, of 2.1):

Parameter

m (mol me3)

n (J mol-‘1

Sco2 S02

1.881 x lo-” 2.690x 10F3

24100 15200

x=a,exp

Ea RTl

( 1 --

,

where x = value of parameter (k,, k,, &,,,, or at temperature T,; aT = parameter-specific constant (units as for parameter); and E, = activation energy (J mol- ‘I. The effect of solubility on the (air space equivalent) values of K, and K, (Eqs. 3 and 27) must be allowed for. This is done using their chloroplast-level values:

Rd,)

Ki = (KchlDi)/S,

(55)

where Ki = parameter value in air space equivalents (K, or K,; mol me3); K,,, = parameter value in chloroplast (Kc,chl or K, & mol rne3); Di = concentration of air in leaf air spaces (mol -3; Eq. 57); and S = solubility (mol rne3; Eq. :I. K, is calculated from Ko,ch, using the solubility of oxygen, and K, is calculated from Kc,chl using the solubility of COz. The observed temperature dependence of the solubilities (Lawlor,

1987) was fitted to the following equation which pressure dependence is also included): S=

(in

m

exp( -n/RT,)

(56)

*

where m and IZ= empirically derived constants. The constants used to calculate the solubilities of CO, and 0, are given in Table 2. Di is assumed to be dependent on temperature and pressure according to the ideal gas law:

(57)

imx

is also temperature dependent, and there is good evidence that its response has a maximum (Farquhar et al., 1980). The function used here is based on the expected response to temperature of a reaction that involves an enzyme that becomes deactivated at high temperatures (Thornley and Johnson, 1990), unlike the temperature dependence of the other parameters given by Eq. 54:

Lax =

aT

exp( - Ea/RTl)

1 + exp(( AST, - J&)/RT,)

’

(58)

where aT = a constant (see below); AS = entropy (J K-’ mol-‘); and E, = deactivation energy (J mol-‘1. With aT set to 3.486 x 1013 mol++ctronsj mol~&orophylljs -l, E, set to 79500 J mol-‘, E, set to 199000 J mol-‘, and AS set to 650 J K-’ mol-‘, it was found that the response closely matched that measured by Nolan and Smillie (19761, and so these values were used for all simulations. Eq. 58 should be viewed as an empirical relationship. The parameter values can be interpreted mechanistically in terms of enzyme activation and deactivation; however, such an interpretation should not be taken literally in view

244

of the complexity volved.

A. D. Friend /Ecological

of the actual processes

in-

2.6. Optimisation Optimal rC,S(satisfying Eq. 2) is found numerically to within a precision of 0.0002 s m-‘, although this precision will not be required for most applications of the model. To start the maximisation, rCs is initialised at 2030 s m-‘, and -A is calculated. rC,S is then increased by a f \Y,eaf step of 10 s m-i, and fqlcaf *A re-calculated. This continues if each successive value of f*,,, *A is higher than the last. When the value decreases, the sign of the step is reversed, and the step value is halved. When the step size is below 0.0002 s m-l the search is stopped. If the number of iterations exceeds 200, it is assumed that there is no solution and the search is terminated. rCS is not bounded in the simulations presented here, although a maximum value of between 2000 and 10000 s m-i is reasonable (Jones, 1983, p. 123: cuticular resistance).

3. Results and discussion 3.1. Predicted photosynthetic rate

PGEN ~2.0 was tested using the default values listed in Table 1 and the Appendix. The effect on .A (Eq. 2) of forcing stomata1 resistance to f \y,,af vary, with Za,dset to 580 W m-*, is shown in Fig. 1. Clearly there is an optimum stomata1 resistance and associated internal CO, concentration which satisfies Eq. 2. This peaked response is due to the positive effect of leaf internal CO, concentration to the left of the maximum, and the greater relative negative effect of leaf water potential to the right. At the peak, A is 12.38 X 10e6 mol me2 s-l, and rc is 361.4 s m-i. Evans (1983) measured values of A up to 34 X lop6 mol m-* s-l in leaves of Triticum aestivum containing similar amounts of Rubisco and chlorophyll to those simulated here. The possible causes of this discrepancy deserve consideration. The Rubisco activities used for these PGEN simulations are derived from the measure-

Modelling 77 (1995) 233-255

ments of Jordan and Ogren (1984) on Spinacea Evans (1983, 1986) measured Rubisco activity in the leaves of Triticum spp., in which he also made measurements of photosynthesis, and found that the activity was significantly higher than that measured by Jordan and Ogren (1984). Evans (1986) measured values of k, at 23°C of 3.4, rising to 4.8 at 30°C (3.4 x and 2.2 X higher respectively than values used here). These higher estimates of Rubisco carboxylation turnover numbers are sufficient to account for the observed rates of photosynthesis in T. aestivum. oleracea.

3.2. General simulations

Using the default conditions, model behaviour was examined by varying the environmental inputs. The effect of changing Za,d on predicted stomata1 conductance, net photosynthesis, transpiration rate, and the Ci/Cair ratio is shown in Fig. 2. For the presentation of these results, irradiance is expressed as incident PAR on the leaf surface, rather than total shortwave. Optimal stomata1 conductance exhibits a peaked response, with a maximum of 3.36 x 1O-3 m s-l at 166.5 W m-* PAR (Fig. 2a). The reason for the fall in stomata1 conductance as irradiance increases above this level is that leaf temperature increases, and this increases the leaf-to-air vapour density deficit (Eq. 19). Thus the stomata close to reduce water loss, which nevertheless still increases with irradiance (Fig. lc), causing a fall in *,eaf from -270 J kg-’ at 50 W me2 PAR to -510 J kg-’ at 500 W m-* PAR. Net photosynthesis increases with irradiance up to saturated values of about 12.3 X 10e6 mol m-2 s-1 above 200 W rnA2 PAR. Above an irradiance of about 100 W m-*, the ratio between internal leaf CO, concentration and ambient CO, concentration remains almost constant at about 0.7 as irradiance in varied (Fig. 2d). These predictions are consistent with the observations of Wong et al. (1978), Farquhar and Wong (19841, and in other studies (see Farquhar and Wong, 1984). Leaf temperature was altered by varying air temperature, and the results are presented in Fig. 3. Maximum stomata1 conductance occurs at a leaf temperature of 25.6”C (Fig. 3a) and net pho-

A. D. Friend /Ecological

i

Modelling 77 (I 995) 233-255

245

u)

: E

0

1500

1000

500 Stomotol

resistance

2000

(s m-l)

Fig. 1. Effect of varying stomata1 resistance on f,r,,,, .A, the function maximised in Eq. 2, under the standard conditions given in the Appendix. Units for stomata1 resistance refer to CO, flux.

~2_~ 5 i;

E

x 0.0

__100

200

300

400

500

600

600

a::L____________ (d)

j

0.6 -

\

=

;

O.O10 0 100 lrradiance (PAR; Wme’)

0.4

-

0.2

-

0.0

0

100

200 lrrodiance

300 (PAR;

400

500

600

W me')

Fig. 2. Predicted responses of stomata1 conductance to COs (a), net photosynthesis (b), transpiration cc), and the Ci/Cair ratio (d) to incident PAR on the leaf surface (i.e. per unit leaf area) under the standard conditions listed in the Appendix. Incident PAR varied by varying shortwave irradiance from sky. Solid lines using PGEN ~2.0, broken lines using PGEN ~1.0.

246

A. D. Friend /Ecological

Modelling 77 (1995) 233-255

tosynthesis has an optimum leaf temperature of 29.O”C (Fig. 3b). Lloyd et al. (1991) observed stomata1 conductance to have a peaked response to leaf temperature as predicted here. Photosynthesis remains positive up to almost 50°C and is still non-zero at -7°C. These predicted responses are similar to those observed across a wide range of species (e.g. Pisek et al., 1973). The overall response of photosynthesis to temperature in PGEN is determined by many processes. The most important of these are the temperature responses of the Rubisco kinetic parameters, electron transport, and mitochondrial respiration. The role of these parameters in determining the temperature response of photosynthesis, and how it is affected by other environmental factors (such as atmospheric CO, concentration), will be addressed in a future paper. Due to the direct effect of leaf temperature on transpiration, there is not a direct correspondence between stomata1

conductance and transpiration rate (Fig. 3~). The predicted Ci/Cair ratio varies with leaf temperature (Fig. 3d); it has a minimum of 0.66 at 33.7”C, and is higher when net photosynthesis approaches 0 at very high and very low leaf temperatures. Similar responses were observed by Lloyd (1991). The prediction of a minimum Ci/Cair ratio at a higher temperature than the temperature optimum for net photosynthesis has also been found experimentally (see Farquhar and Wong, 1984). At leaf temperatures greater than 45°C optimal stomata1 conductance is predicted to increase. This is due to the influence of temperature on dark respiration (&I. Opening the stomata increases transpiration and so reduces leaf temperature at a given air temperature. At high leaf temperatures, A is dominated by R,, thus the reduction in q,eaf caused by the increased rate of transpiration is less important than the

;z

0 -10

0

10

20

30

40

50

1.2 w 1 .o [

0.8

3 \

0.6

Li

~~: -10 Leaf temperature

(‘C)

0

10

20

Leof temperature

30

40

50

(“C)

Fig. 3. Predicted responses of stomata1 conductance to CO, (a), net photosynthesis (b), transpiration (cl, and the Ci/Cair ratio (d) to leaf temperature under the standard conditions listed in the Appendix. Variation in leaf temperature achieved by altering air temperature. Solid lines using PGEN ~2.0, broken lines using PGEN ~1.0.

A.D. Friend/Ecological

Modelling 77 (1995) 233-255

effect of leaf temperature on R,. In some species, Lange (cited and discussed by Levitt, 1972) observed very similar responses of transpiration to temperature to those predicted here. Also, Chiariello et al. (1987) found that stomata1 conductance increased dramatically in Piper auritum at leaf temperatures above 40°C. Leaf-to-air vapour density deficit (6) was altered by changing ambient relative humidity. The predictions are given in Fig. 4. There is clearly a large effect of S on stomata1 conductance (Fig. 4a). There is a smaller effect on net photosynthesis (Fig. 4b), resulting in a decline in the Ci/Cair ratio as 6 increases (Fig. 4d). Transpiration increases in a curvilinear manner with 6 (Fig. 4~). These responses are similar to those observed (e.g. Turner et al., 1984, 1985; Lloyd, 1991; Dai et al., 1992). Increasing atmospheric CO, concentration results in a reduction in predicted stomata1 conduc-

tance (Fig. 5a). Reduced stomata1 conductance with increasing atmospheric CO, is observed experimental (e.g. Meidner and Mansfield, 1968). The predicted reduction is due to changes in the balance of the trade-off between leaf water potential and internal leaf CO, concentration, as atmospheric CO, changes, and follows from the fact that net photosynthesis is a saturating function of internal leaf CO, concentration (Eqs. 3 and 4; Fig. 5b). Predicted transpiration follows stomata1 conductance (Fig. 5~). The Ci/Cair ratio remains almost constant at 0.65 as atmospheric CO, changes, except at low CO, (Fig. 5d). The increase in this ratio at low atmospheric CO, concentrations is evident in published data (Farquhar and Wong, 1984). Figs. 2 to 5 also show predictions made by PGEN in which q,eaf has a direct effect on net photosynthesis (broken lines). All conditions and model equations were the same as for PGEN

$001 0.0

0.2

0.4

0.6

0.6

‘ioi , , , , 0.0

0.2

0.4 Vapour

0.6 density

0.6 deficit

1.0

1.2

1.4

1.6

,

,

247

z

0.0

1

0.2

0.4

0.6

L

0.6

1.0

1.2

0.6

1.0

1.2

11

1.4

1.6

1.4

1.6

0.2

(

t

0.01 1.0 (mol

1.2 mm’)

1.4

1.6

0.0

I

0.2

I

I

0.4

0.6

Vopour

density

t

deficit

(mol

I

mm’)

Fig. 4. Predicted responses of stomata1 conductance to CO, (a), net photosynthesis (b), transpiration cc), and the C,/C,,, ratio (d) to vapour density deficit under the standard conditions listed in the Appendix. Variation in vapour density deficit achieved by altering relative humidity. Solid lines using PGEN ~2.0, broken lines using PGEN ~1.0.

A.D. Friend /Ecological

248

Modelling 77 (199.5) 233-255

~2.0, except that

tion (Fig. SC), which increases S because the air in the planetary boundary layer becomes less humid. This causes a positive feedback due to the closing response of stomata to increased S (Friend and Cox, 1994).

A was multiplied by f,r,,eaf to give predicted A, and this predicted value of net photosynthesis was maximised. This is the scheme used in PGEN ~1.0. There is no significant effect of this change on predicted stomata1 conductance or transpiration rate as any of the environmental factors are varied, though, as would be expected, net photosynthesis is reduced and Ci/Cair is increased. The effects of CO, concentration and 6 on stomata1 conductance are particularly important in relation to the potential impacts of climate and atmospheric CO, concentration change on vegetation, and the impacts of vegetation on the hydrological cycle. The closing response of stomata predicted by PGEN to increased atmospheric CO, concentrations (Fig. 5a) has been shown to have significant impacts on climate change and CO,fertilization effect predictions (Friend, and Cox, 1994). This closing results in reduced transpira-

3.3. Sensitivity analysis A sensitivity analysis was performed on PGEN ~2.0 to assess those parameters to which predicted net photosynthesis and stomata1 conductance are most sensitive. The following dimensionless index of sensitivity (p) was calculated for each parameter (after Friend et al., 1993):

(59) where R, = predicted net photosynthesis or stomatal conductance when parameter set to P,; R, =

‘E 25 e y

20

0

”

ti

01

p

0.00

-”

0.02

0.04

0 0.00

&___________

4

‘yl

r

:

I \\

E

I

i:/

“;2\

0.04

0.02

\ =

s .e :: s ; 0

0.4

‘z.

I 0.2 t 0.0 0.00

I

0.02

0.00 Atmospheric

CO, (mol

0.04 m-‘)

0.04

0.02 Atmospheric

CO, (mol

m-‘)

Fig. 5. Predicted responses of stomata1 conductance to CO, (a), net photosynthesis (b), transpiration (c), and the Ci/Cair ratio (d) to atmospheric CO, concentration under the standard conditions listed in the Appendix. Solid lines using PGEN ~2.0, broken lines using PGEN ~1.0.

A.D. Friend /Ecological

default predicted rate of net photosynthesis or stomata1 conductance; P, = 1.1 x default parameter value (various dimensions); and PO = default parameter value (various dimensions). The conditions for the sensitivity analysis were as in the Appendix and Table 1, except that Z,,d was reduced to 197.65 W m-* so as to make A carb and ARuBP equal. This results in a default value for A of 11.0 X low6mol me2 s-‘-and rcs of 301.8 s m-‘. Each parameter was then increased in turn by lo%, with other parameters fixed (except 8, which was decreased by 10%). A p value of 1.0 means a directly proportional change in net photosynthesis or stomata1 conductance, a value of - 1.0 a directly proportional decrease. Those parameters giving the 14 highest sensitivity indices are given in Fig. 6. Five parameters have absolute values of p for net photosynthesis greater than 0.5 (Fig. 6a). Pre-

Net

Modelling 77 (1995) 233-255

249

dieted net photosynthesis is most sensitive to 8, the degree of co-limitation between carboxylation- and RuBP-regeneration limited photosynthesis. This parameter is set empirically, and its considerable importance indicates that a critical examination should be made both of the concept of co-limitation, and the natural variation in this parameter. The value of 8 used (0.95) is taken from Collatz et al. (1990); it was derived from a fit to data from Xunthium stnrmarium. This parameter is necessary when using the Farquhar et al. (1980) and Farquhar and von Caemmerer (1982) models of photosynthesis. Without co-limitation, there is an abrupt change in the predicted photosynthesis-irradiance response curve at the point where the two limitations are equal. Clearly this is unrealistic, and the quadratic in Eq. 18 was proposed by Collatz et al. (1990) to produce a smoother response function. The need for an

photosynthesis

(b) Stomata1

conductance

Fig. 6. Sensitivity of predictions of net photosynthesis (a) and stomata1 conductance (h) to variation in the 14 most important model input parameters. Sensitivity calculated using Eq. 59. A sensitivity index of 1 indicates a directly proportional effect. Standard conditions used as given in the Appendix (except la,d = 197.65 W me2). K,, O,,, and k, have negative effects on net photosynthesrs; rr,, and Cair have negative effects on stomata1 conductance, all other parameters shown here have positive effects. T, converted to “C before sensitivity analysis; 0 tested by comparing predictions made using its default value with those made when it is set to 0.9 x its default value.

Resistance to water flux between root and leaf Leaf thickness Root structural dry mass Fraction of PAR absorbed by leaf Entropy parameter to calculate effect of temperature on j,,, Angle between leaf and incident irradiance from sky Co-limitation factor Critical leaf water potential below which there is no dry matter

rr,l w WGJ

Environmental inputs Empirical soil-specific parameter bmil Concentration of CO, in air outside leaf boundary layer Relative humidity of air outside leaf boundary layer ? Shortwave irradiance from sky IW a.d Maximum soil hydraulic conductivity K Concentration of 0, in air outside leaf boundary layer Atmospheric pressure Temperature of air outside leaf boundary layer Wind speed Relative soil water content Maximum soil relative water content Reflection coefficient of background for shortwave irradiance Soil water potential when soil saturated

AS

(YPAR

Rd,

N

ko

kc

Michaelis-Menten constant of Rubisco for oxygenation Rubisco carboxylation turnover number (temperature dependent) Rubisco oxygenation turnover number (temperature dependent) Leaf nitrogen content Rate of mitochondrial respiration (temperature dependent)

K o,chl

dr 4

d

on parameters

except where

indicated)

production

s-l for i,,,) (3.486~ lo’3 mol~electronsr mol,&rophyll) Characteristic dimension of leaf Rooting depth Activation energy parameter to calculate effect of temperature on parameters (79500 for j,,,) Deactivation energy parameter to calculate effect of temperature on j,,, Proportion of leaf nitrogen bound in Rubisco Proportion of leaf nitrogen bound in chlorophyll PAR-saturated potential electron transport rate (temperature dependent) Michaelis-Menten constant of Rubisco for carboxylation (temperature dependent)

effect of temperature

values (all areas are leaf areas,

used to calculate

and default

Definition

Biological inputs Constant aT

Symbol

Appendix Symbols, definitions,

-

5.0 0.0145 0.7 580 a 1.0x 10-s 8.471 101325 298.15 5 0.5 0.5 0.2 - 10.0

mOl&sj

m4&

m&aterr m&, m&rerr m&j dimensionless J kg-’

W m& kg m-3 s mol mm3 Pa K ms-’

dimensionless mol mm3 dimensionless

mol (coz) kg& kg-r m4 s-r m kg mm2 dimensionless J K-r mol-’ degree dimensionless J kg-’

mol me3 mol me3 mol (~0~) mol (~0~) kgm-2

s-l

S-l

Sm *

J mol-’ J mol-’ dimensionless dimensionless mol (electrons) mol$rorophyrr) s -1

see Table 1 199000 0.22 0.02 0.4257 (at 300.3 K) 10.84~10-~ (at 300.3 K) 0.535 1.584 (at 300.3 K) 0.9780 (at 300.3 K) 0.002 480.9x lO-‘j (at 300.3 KJ 10.00 x 106 0.3x10-3 1.0 0.85 650 10 0.95 - 1650

Units

Units as for parameter m m

value

see Table 1 0.1 0.5

Default

Specific heat of air Standard atmospheric presssure Gas constant Standard temperature Stefan-Boltzmann constant

Calculated parameters A Net photosynthesis A carb Carboxylation-limited rate of net photosynthesis A cnnd Stomata1 resistance-limited rate of net photosynthesis A R”BP RuBP regeneration-limited rate of net photosynthesis Concentration of CO, in leaf air spaces exterior to mesophyll cell liquid phase Ci Concentration of CO, in leaf air spaces for and Acarb Ci,carb C t,RuBP Concentration of CO, in leaf air spaces for and ARUn, Concentration of air in leaf internal air spaces Di E Rate of transpiration Leaf Rubisco catalytic site content Leaf water potential factor Z.,, Conductance to CO, transfer from air outside the leaf boundary layer to g, mesophyll cell surface I PAR Total photosynthetically active radiation (PAR) incident of leaf I APAR Absorbed PAR Shortwave irradiance incident on leaf 1, J Potential electron transport rate J max PAR-saturated potential electron transport rate Michaelis-Menten constant of Rubisco for carboxylation (in air space) K, Michaelis-Menten constant of Rubisco for oxygenation (in air space) K0 Soil hydraulic conductivity K, Concentration of 0, in leaf air spaces exterior to mesophyll Oi Mitochondrial respiration R, Leaf boundary layer resistance to convective heat transfer raH Resistance to CO, transfer from air outside the leaf boundary layer to rc mesophyll cell surface Resistance to CO, transfer across leaf boundary layer rr,a Resistance to CO, transfer from inside leaf surface to mesophyll cell surface ‘c,i Resistance to CO, transfer across leaf surface rc,s Parallel resistance to heat loss from leaf by convection and radiation ‘HR Radiative ‘resistance’ to heat loss rr r s,r Resistance to water flux between bulk soil and root interior Resistance to water flux from air outside leaf boundary layer to inside the leaf rw s Solubility of pure gas (either CO, or 0,) in pure water S Slope of saturation vapour concentration versus temperature T Average of leaf and air temperature Background temperature Tb

TO cr

cP PO R

Physical constants

75.69x 299.2 298.1

361.4 18.08 23.87 319.5 12.86 200.6 1.010x 235.9

10m3

106

2.767 x 1O-3 1.565x 10-s 1.330x 10-s 685.4 205.4 x 1O-6 304.0x 10-6 15.04 x 10-s 18.33 x lo-’ 1.0x 10-s 8.471 0.9619x 10m6 13.74

12.38x 10m6 13.60x 10mh 12.38x lo-’ 18.65 x 1O-6 9.777x 10-s 9.340x 10-s 7.531 x 1o-3 40.58 2.3213x 10m3 4.0x 10-s 0.7151

1012 101325 8.3144 293.15 5.6703 x lo-’

m

-2

s-1

mol~electrons~

m

-2

sm-’ sm-’ sm-’ sm-’ s m-l sm-’ kg-’ m4 s-t sm-’ mol mm3 mol me3 K-’ K K

mol me3 mol mm3 kg me3 s mol mm3 mol (CO2) m sm-’

s-1

-2

molcquantar m2 ss’ mol (quantar m2 s-l W me2 mol (electrons) m -2

ms-’

mol mm2 dimensionless

mol(H,O)

-1 s-1 mo+coz, m-2 mol (CO>) mm2 s:: mol V=2) m s mot (CO*) m -2 s-1 mol mm3 mol m-s mol mm3 mol me3

J kg-’ K-’ Pa J K-’ mol-’ K W me2 Km4

sm1 smI

parameters

Leaf temperature Apparent radiative temperature of atmosphere Maximum rate of carboxylation by Rubisco Maximum rate of oxygenation by Rubisco Concentration of H,O in air outside leaf boundary layer Dry matter production divided by k in Eq. 1 Concentration of Ha0 in leaf air spaces Saturation concentration of H,O in air at air temperature Saturation concentration of H,O in air at leaf temperature Fraction of shortwave irradiance absorbed by leaf Photosynthesis compensation concentration of CO, in leaf air spaces in the absence of mitochondrial respiration Psychrometer ‘constant’ Water vapour concentration deficit of air outside leaf boundary layer Latent heat of vapourisation of H,O Density of dry air Root density Leaf Rubisco content Isothermal net radiation of leaf Leaf water potential Root water potential Soil water potential

Definition

* 197.65 for Figure 6.

*sail

Q leaf q root

@ni

uRub

Pr

Pa

A

iW

wNr,, w%r~, ffs F*

wi

wd

w,

Vc,max Vo,max

T,

T

Calculated

Symbol

Appendix (continued) Symbols, definitions, and default values (all areas arc leaf areas, except where indicated)

2.145x 10m3 27.24x 1O-3 0.3833 2.422x 1O-6 1.192 2.0 2.75x 1O-3 306.3 - 470.0 - 52.20 - 10.00

300.3 291.7 63.38x 10V6 39.12x 1O-6 0.8944 8.850x 1O-6 1.442 1.278 1.442 0.5015

Default value

m

s

kgo, matter)m&j kg m-* Wm-* J kg-’ J kg-’ J kg-’

mol mm3 mol me3 K-t mol mm3 J kg-’ kg me3

mol mm3 mol m-* ss’ mol me3 mol mm3 mol mm3 dimensionless

molco2)

K K mol (COZ)miz s;

Units

l

A.D. Friend/Ecological

arbitrary constant is unfortunate, but is to be expected when it is realised that the biochemical model, when scaled to the leaf level, is an idealisation of the leafs state. In reality there are gradients in light, CO,, and biochemical capacities throughout the leaf in all directions. Thus different parts of the same leaf may be limited by different processes. Eq. 18 can therefore be viewed as an expression of this within-leaf variation. k,, the turnover number of Rubisco for CO,, is the next most important parameter for A. This parameter has a linear effect on I&,, (Eq. 22). Ambient CO, concentration is next, followed by K, and leaf nitrogen content. Ambient 0, concentration, K,, fN,Rub, f,,,, and k, also have significant effects on predicted net photosynthesis. As would be expected, increasing the parameter K,, Oair or k, has a negative effect on A. The results of the sensitivity analysis for stomatal conductance are given in Fig. 6b. Overall, stomata1 conductance is less sensitive to parameter variation than is net photosynthesis. The clear exception is for f.+,. Increasing relative humidity has very large effect on conductance, with the stomata showing a strong opening response. This was demonstrated in Fig. 4a. The next most important parameters are the critical leaf water potential for dry matter production, and the hydraulic resistance between the root and the leaf. The former determines the slope of the cost component in the tradeoff (Eq. 5) and the latter determines the relative effect of transpiration on leaf water potential (Eq. 20). Also important in determining stomata1 conductance is ambient CO, concentration. This was shown in Fig. 5a. Interestingly, atmospheric pressure is the next most important parameter. P is used in many parts of the model, but further analysis showed that it is particularly important for predicted stomata1 conductance because of its influence on the saturated air concentration of water vapour (Eq. 39, and thus the driving force for transpiration. Increasing the parameter rr,, or Cair has a negative effect on stomata1 conductance. The significance of the results of this sensitivity analysis must be judged in relation to our confidence in the parameters concerned. From

Modelling 77 (1995) 233-255

253

the earlier discussion, it is clear that our knowledge of the Rubisco parameters is incomplete, yet the four V,, and K, values of Rubisco are all in the top ten most critical parameters; the same is also true for the co-limitation factor (e), *c and rr,r. The temperature response parameters were not included in this analysis, but are likely to be of importance also.

4. Conclusions PGEN ~2.0 makes qualitatively correct predictions of the responses of physiological variables (net photosynthesis, stomata1 conductance, transpiration rate, and the Ci/Cair ratio) to variation in environmental parameters. Here the sensitivity of photosynthesis to the degree of co-limitation between carboxylation- and RuBP-regeneration limited photosynthesis and the Rubisco kinetic parameters, and the sensitivity of stomata1 conductance to the critical leaf water potential for dry matter production and plant hydraulic resistance, are highlighted. It is important that more estimates of these parameters are made for different species.

Acknowledgements

I am grateful to Bob Besford, Roddy Dewar, and John Monteith for helpful discussions, and to Melvin Cannell, Roddy Dewar, and John Thornley for comments on the manuscript.

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