Comparison of an Optimal Stomatal Regulation Model and a Biochemical Model in Explaining CO 2 Exchange in Field Conditions

Gas exchange of Pinus sylvestris L. was studied in subarctic fi eld conditions. Aspects on optimal control of the gas exchange were examined using approach by Hari et al. (Tree Phys. 2: 169–175, 1986). Biochemical model by Farquhar et al. (Planta 149: 78–90, 1980) was utilized to describe the photosynthetic production rate of needles. The model parameters were determined from fi eld measurements. The results from the optimization approach and biochemical model were compared and their performance was found quite similar in terms of R2 calculated using measured exchange rates (0.89 for optimization model and 0.85 for biochemical model). Minor differences were found in relation to responses to intercellular carbon dioxide concentration and temperature.


Introduction
Models of gas exchange rates of leaves in fi eld conditions are required for ecosystem studies.There exist different approaches which can be utilised to predict exchange rates.Photosynthetic production has been studied in detail by Farquhar et al. (1980) who introduced a biochemical model of photosynthesis.This model allows leaf scale fl uxes to be calculated if conductances are known or modelled by other means.Today the model by Farquhar et al. (1980) is widely used for estimations of leaf level gas exchange and further as a submodel in large ecosystem models.Cowan (1977) presented an insight into optimization in gas exchange.Optimal control of the gas exchange of leaves is based on fi nding the mode of stomatal functioning which leads to minimal loss of water and maximal gain of carbon di oxide.Hari et al. (1986) applied the ideas of optimal control to develop a predictive gas exchange model which also results in the photosynthetic production of leaves.
Comparison of the two approaches, optimal stomatal control and photosynthetic production by a biochemical model, adds a new perspective to the relationships between CO 2 uptake and environment.It is useful to study whether differences in responses of the exchange rate to environmental variables could arise due to different approaches, since the leaf level results may affect larger scenarios.The model parameters are often fi tted by utilizing fi eld measurements infl uenced by varying environmental conditions.It is therefore reasonable to use fi eld measurements also in the comparison of the models.
In this work, the biochemical approach was utilized as a description of the photosynthetic reactions with input information like conductance taken directly from transpiration measurements instead of modelling it.Optimization approach was used in a form where a simple formulation of photosynthetic production, transpiration and conductance were closely linked together and CO 2 assimilation rate and conductance were predicted by the model.Relevant parameters of the biochemical model and optimal stomatal regulation model were determined from fi eld measurements.Gas exchange of Pinus sylvestris L. was studied in remote subarctic fi eld conditions in northern Finland using automated measuring system which enabled recording of gas exhange rates of intact shoots at 12 min intervals throughout the growing season.Results of simulated gas exchange rates of Scots pine using both models were compared to the measured rates.

Measurement Set-up
Gas exchange rate of Scots pine (Pinus sylvestris L.) was registered continuously at 12 min intervals during summer months in the Värriö envi-ronmental measuring station (67°46'N, 29°35'E, 390 m a.s.l.).Monitoring was performed using well-established automated cuvette system developed by the personnel on the station and in the University of Helsinki (Hari et al. 1999).The system consisted of trap-like cuvettes, a pneumatic system for controlling cuvettes, gas fl ow controllers and gas analyzers (Hartmann and Braun URAS 4 infrared absorption analyzers).The system was controlled by a computer.
An intact pine shoot (150-250 needles) was installed inside the ventilated cylindrical acrylplastic cuvette (volume 3.5 dm 3 ).The lid of the cuvette was closed for only 70 s once every 12 minutes so that the temperature increase during closing stayed in insignifi cant level and ambient conditions prevailed for most of the time.Cuvette air was mixed with a fan.The change in CO 2 and water vapour concentrations was registered once in every ten seconds while the cuvette was closed and the exchange rates were determined by methods presented in Aalto (1998).During measurements the outgoing sample air and incoming replacement air fl ows were balanced so that no pressure drop occurred inside the cuvette.The sample was transported to the gas analyzers by heated tefl on pipes.There was a constant fl ow of air through the pipes also between measurements in order to avoid condensation of water and exchange of CO 2 with the walls of the tubing.Calibration and pressure corrections were performed for each measurement.Temperature (CuKo thermoelements) and photosynthetically active radiation PAR (LiCor 190 SB sensors) were measured both inside and outside the cuvette.The cuvettes were installed in unshaded positions in the top of the trees and the shoots consisted of one-and two-year-old needles.The reliable functioning of the system has been confi rmed by laboratory and fi eld studies (e.g.Hari et al. 1999).A variety of meteorological and other forest ecological measurements are also performed at the station (Hari et al. 1994, Ahonen et al. 1997).

Optimal Stomatal Regulation Model (OSRM)
The optimal stomatal regulation model was obtained from Hari et al. (1986) (See also Berninger et al. 1996 andMäkelä et al. 1996).According to the formulation, CO 2 exchange (A) and transpiration (E) rates depend on stomatal conductance g: where c and w are CO 2 and H 2 O concentrations (g/m 3 ) respectively, inside stoma (i) and in ambient air (a).α is an empirical parameter to be fi tted from measurements (see Table 1) and I is irradiance.As a solution of the optimization, conductance can be expressed followingly: where λ is the 'cost of water' also to be fi tted from the measurements.Minimum value of g is restricted to 0 and maximum to g 0 describing the state of fully open stomata.

Biochemical Model for Photosynthetic
CO 2 Assimilation (BPM)

Model Structure
The model used was based on the original idea by Farquhar et al. (1980) as modifi ed by Harley and Baldocchi (1995) and Lloyd et al. (1995).
The functional form of the model is identical to Aalto and Juurola (2001), modifi cations are due to values of the major parameters concerning carbon assimilation (J max , q, Θ and V c(max) ) and respiration (R d ) and their temperature dependences.
In short, the net rate of CO 2 exchange is (e.g.Lloyd et al. 1995 fi rst expression being the RuBP regeneration limited exchange rate and second the net Rubiscolimited rate (c i in µmol/mol).For more detailed information and numerical values used for Michaelis-Menten constants k c , k o and Γ * , CO 2 compensation point in the absence of mitochondrial respiration, we refer to Aalto and Juurola (2001).
The dependence of J on irradiation can be written as (e.g Lloyd et al. 1995) where J max is the maximum of the electron transport rate, q is the effectivity factor for the use of light and Θ is the convexity factor of the curve.
The following equation was used for J max (Lloyd et al. 1995, Farquhar et al. 1980): where E j is activation energy, H j curvature parameter and S j temperature response parameter for RuBP-regeneration limited assimilation rate, and B is a constant The Arrhenius type of relationship (e.g.Harley and Baldocchi 1995) was utilized for temperature dependences of the maximum carboxylation rate V c(max) and the rate of non-photorespiratory respiration R d : where f T is the value of a given parameter at temperature T and E f are activation energies.R is the gas constant and T is temperature in K.

Methods of Evaluation
Measurements of shoot gas exchange during August 1997 were utilized in order to evaluate the parameters of the biochemical model.CO 2 and H 2 O fl uxes, conductances and intercellular CO 2 concentrations were obtained from cuvette measurements as in Aalto (1998).Measurements before noon were selected to attain high conductance conditions.The irradiance response of the CO 2 fl ux was modelled using only the expression for RuBP regeneration limited exchange rate (Eq.4).Additional selection criterion was therefore applied to the research material in order to keep the stomatal concentration of CO 2 (c i ) at high level, i.e ratio of stomatal to ambient concentration was required to exceed 0.5.Also, measurements at PAR values exceeding 600 µmol m -2 s -1 were abandoned.
The initial slope of the irradiance curve, q (Eq.5), was solved fi rst keeping Θ constant and focusing on good fi t near the irradiance compensation point.The curvature factor Θ was solved using the estimation for q and freely varying J max .The value of q was then re-estimated using the new solution for Θ.Values obtained for q and Θ (Table 1) were then used throughout the work assuming invariability over the temperature range examined (6-20°C).Results were similar to Aalto (1998) using measurements from the same pine stand.
J max (Eq.5) was solved using fi xed q and Θ.This was done in order to minimize the number of free variables and thus enable studies of the temperature dependence of the parameter.The temperature-induced variation was examined by solving J max for each four-degree intervals.J max was found to increase relatively slowly with the temperature.Eq. 6 was fi tted to the results for J max (Table 1).J max was assumed to be constant above 20°C because there were only few measurement points at high temperatures.
V c(max) was fi tted to before-noon measurements at irradiance levels exceeding 800 µmol m -2 s -1 using expression for the net Rubisco-limited rate (Eq.4).The temperature dependence of V c(max) (at 8-24°C) was solved from Eq. 8 (see Table 1) for four-degree intervals as in J max .V c(max) was constantly increasing with temperature showing relatively similar results to the ones obtained by Aalto (1998).
Respiration was solved from measurements during night.Respiration increased with rising temperature (Table 1).Non-photorespiratory respiration in dark was assumed to be suppressed in light by factor 1.45 as suggested by Wang (1996) for Scots pine.
In the case of OSRM (optimal stomatal regulation model) conductance was determined for every measurement from Eq. 3 and the exchange rate of CO 2 from Eq. 1. Measurements during fi ve fi rst days of August were utilized for estimation of α and λ (Table 1) by minimizing the residual sum of squares.Exchange rates during the rest of the month were predicted by the model using the obtained values.

Comparison of Models
where Y i,est is the modelled and Y i measured CO 2 fl ux, was calculated for different cases.When the whole month was examined R 2 was 0.85 for BPM (biochemical model for photosynthetic CO 2 assimilation) and 0.89 for OSRM.Sensitiv- T is temperature in K. *: R d,298 must be divided by 1.45 when irradiance level exceeds 50 µmol m -2 s -1 .J max was assumed to equal 112 µmol m -2 s -1 in temperatures higher than 293K.
1.15 µmol m -2 s -1 q 0.14 Θ 0.7 ity analysis revealed only minor changes; when the most important single parameter in the BPM, J max , was increased by 10% the resulting value for R 2 was 0.83.J max was also modifi ed to experience steeper increase with increasing temperature according to earlier results for Scots pine growing in southern Finland (Wang 1996).This resulted in R 2 of 0.83.In comparison to the J max fi tted for the northern Scots pine, residuals suggested slightly more overestimations of exchange rates in high temperatures and underestimations in low temperatures.
Residual plot as a function of irradiance (Fig. 1) shows that largest errors according to OSRM occur during very high irradiation levels, while the majority of large errors in BPM is in slightly lower irradiances (it is supposed that these errors are overestimations by the models since the fl uxes are positive during daytime when irradiance levels are high).Residuals are rather well centered around zero when the 95% majority is examined.Values during night at low irradiance show moderate overestimation of respiration rate for OSRM and underestimation for BPM (signs opposed).
Temperature plot (Fig. 2) shows largest errors for BPM in higher temperatures than for OSRM.Otherwise the plot is quite evenly scattered.A weak trend can perhaps be observed from overestimations in low temperatures to underestimations in high temperatures for OSRM and vice versa for BPM.This suggests that the function for J max (Eq.6) in BPM might have given a better fi t with even weaker response to increasing temperature.
Plot as a function of c i (Fig. 3) shows largest errors in the typical mid-range of c i for daytime CO 2 exchange.On the lowest range of c i there are some large errors for BPM and during the near-nightime high c i OSRM produces more scattering which seems to be somewhat clumped.c i for BPM is deduced directly from measurements using transpiration rates and ambient CO 2 concentration and c i for OSRM is solved using the optimization scheme.
Studies of diurnal dependence (Fig. 4) show similar decreasing trend to both models.Exchange rates during mornings are slightly underestimated and afternoons overestimated.

Discussion
The fi rst cold period after summer occurred in the middle of August (Fig. 5).In general, both models underestimated the exchange rates before the low temperature period and overestimated during late August (Fig. 6) thus suggesting coldinduced changes in the photosynthetic responses of the needles.The minimum diurnal temperatures were 2-4°C during the cold period, but daytime maximum temperatures reached only 6-10°C with simultaneous high irradiation levels.
The largest overestimations by BPM occurred right after the cold period when the daytime temperatures rapidly elevated to over 20°C.During the cold period OSRM overestimated fl uxes, while BPM showed slight underestimations during that time.The results are supported by for example Korpilahti (1988) and Hollinger et al. (1999), who reported a drawdown on carbon uptake of a boreal coniferous forest after fi rst autumn frost.Leverenz and Öquist (1987) found a decrease in quantum yields of photosynthesis for Scots pine when temperature decreased below 5°C.The effect was most profound in the middle of the summer and decreased towards autumn.Some underestimations by BPM occur during extremely low c i (in the beginning of the month) which indicates specifi c effects due to the model structure not taking into account possible partial activation of the carboxylase-oxygenase enzyme, as presented by Farquhar and von Caemmerer (1982).In the diurnal pattern these low values are located around midday, while during afternoons overestimations of fl uxes are more usual than underestimations.
Generally results are good and rather similar for both approaches OSRM showing slightly higher value for R 2 with current parametrizations.Both models can be used to predict CO 2 exchange of pine needles in northern, relatively cold but humid, environment with suffi cient accuracy.However, it has to be noticed that the photosynthetic capacity may change according to canopy height and there exists different approaches to scale from leaf photosynthesis to canopy photosynthesis (de Pury and Farquhar 1997).This study does not take into account those changes and their possible effects to the model comparison result.

Fig. 1 .
Fig. 1.Residuals (estimated -measured CO 2 exchange rate) as a function of irradiance.95% of the results lay between dashed lines.

Fig. 3 .
Fig. 3. Residuals (estimated -measured CO 2 exchange rate) as a function of intercellular CO 2 concentration.95% of the results lay between dashed lines.

Fig. 2 .
Fig. 2. Residuals (estimated -measured CO 2 exchange rate) as a function of temperature.95% of the results lay between dashed lines.

Fig. 6 .
Fig. 6.CO 2 exchange according to optimal stomatal regulation model, biochemical model and measurements before and after cold period in Värriö.