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Ecology |
Nikko Botanical Garden, Graduated School of Science, University of Tokyo. Nikko Botanical Garden, Graduated School of Science, The University of Tokyo, Nikko, Tochigi, 321-1435
Received for publication December 16, 2003. Accepted for publication August 26, 2004.
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMODEL DESCRIPTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXLITERATURE CITED |
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Key Words: Angiosperms • gymnosperms • hydraulic conductivity • mechanical stability • tree architecture
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMODEL DESCRIPTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXLITERATURE CITED |
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; Rogers and Hinkley, 1979
; White, 1983
; Cornelissen et al., 1996
; Poorter and Nagel, 2000
; Niklas and Enquist, 2002
). Leaf mass is proportional to the leaf area, which governs the extent of carbon gain by photosynthesis and water loss through transpiration; the stem mass contributes to the stem length and diameter, which determine the capacity for the two primary stem functions, water transport and mechanical stability.
The pipe model theory of Shinozaki et al. (1964)
proposes that shoot biomass partitioning patterns are constrained by those primary stem functions. To maintain a high stomatal conductance and a high CO2 assimilation rate, the water transport function of a stem must compensate for the transpiration demand of the leaves (Jarvis, 1975
; Whitehead et al., 1984
; Givnish, 1986
; Cannell and Dewar, 1994
; Magnani et al., 2000
; McDowell et al., 2002
; Mencuccini, 2003
). To preserve spatial positioning and maximize light capture, the mechanical strength of a stem must be sufficient to support the plant's weight and the physical load due to wind, rain, and snow, but not so high as to risk mechanical failure due to stem breakage, buckling, or lodging (McMahon, 1973
; Tateno and Bae, 1990
; Niklas, 1992
; Spatz and Bruechert, 2000
; Niklas and Enquist, 2002
). Many studies have described the impacts of the water transport process and mechanical support on shoot morphology, but only a few studies have addressed the most plausible shoot biomass partitioning criteria attributable to the two functions in actual plants (however, see Farnsworth and van Gardingen, 1995
; Domec and Gartner, 2002
).
Shoot biomass partitioning patterns should depend on stem properties that are related to the primary stem functions. For example, in a shoot that is constrained by water transport criterion, the greater the conductive capacity of the stem, the greater leaf area that can be fed, which should result in the partitioning of more biomass into the leaves than into the stem (Whitehead et al., 1984
; Givnish, 1986
; Magnani et al., 2000
). This result should also be seen in the case of a shoot with a stem that exhibits high mechanical strength, which can support more leaves (Tateno, 1991
; Givnish, 1995
). The stem properties are attributable to the form and anatomical features of the stem. The number of conduits and their diameter determine the conductive capacity of a stem, as the flow rate through a capillary (e.g., xylem conduits) is proportional to the fourth power of the capillary diameter (Hagen-Poiseuille's law; Tyree et al., 1994
). The mechanical strength of a stem increases with increasing stem diameter and rigidity, according to the mechanical properties of the materials (Niklas, 1992
). These stem features vary among plants of different sizes, species, and environments (Carlquist, 1975
; Niklas, 1992
; Gartner, 1995
; Mencuccini et al., 1997
; Niklas, 1997
) and should result in intraspecific and interspecific variation of shoot biomass partitioning patterns.
In this study, we determined the criteria for shoot biomass partitioning patterns, which result from the stem properties of the current shoot. Important in tree architecture and biomass production, current shoots are new plant units that provide more sunlit leaves than do many older shoots. The stems of current shoots have the lowest conductive capacity (Zimmermann, 1978
; Tyree and Sperry, 1988
; Gartner, 1995
; Zotz et al., 1998
) and the weakest mechanical strength (Bertram, 1989
; Mencuccini et al., 1997
; Niklas, 1997
) of any part of a woody plant. Therefore, we have proposed that the growth of current shoots is tightly governed by specific criteria for biomass partitioning.
We developed two different quantitative models, a water transport model and a mechanical support model. The water transport model predicts shoot biomass partitioning values into stem tissue to ensure water transport sufficient for the vigorous transpiration that maximizes the productivity of the shoot on the basis of the biochemical model of CO2 assimilation (Farquhar et al., 1980
; von Caemmerer and Farquhar, 1981
). The mechanical support model predicts the shoot biomass partitioning values into the stem to ensure the mechanical stability of the shoot on the basis of Euler's buckling formula (Tateno, 1993
). Both models predict a relationship between the biomass partitioning and the stem properties related to a stem function. To determine the carbon economy of a current shoot, we used the mass-specific hydraulic conductivity and the stem mass density as surrogates for the degree of water transport and the strength of mechanical support, respectively.
Next, we determined the theoretical criteria for biomass partitioning in shoots with given stem properties. When the biomass partitioning values based on each of the two models was in agreement, the partitioning was designated as optimum biomass partitioning. Optimum biomass partitioning provides both static mechanical support and adequate water transport without functional redundancy. In contrast, when the biomass partitioning values predicted by the models are different, one of the two criteria, mechanical support or water transport, must govern current shoot biomass partitioning. Because both criteria are essential for survival and carbon gain, biomass must be partitioned to allow both functions, even if this results in excess capacity for one of the functions. For example, if the biomass partitioning in the current shoot is governed by the water transport criterion, sufficient biomass must be allocated to the stem to provide the required water transport capacity, even if this results in a stem with greater mechanical support than is necessary. Alternatively, if biomass partitioning is governed by the mechanical support criterion, biomass sufficient for the required mechanical strength must be partitioned to the stem despite the excess water transport capacity that may result. Following the above logic, theoretical criteria were determined for varying stem properties.
Finally, we compared the stem properties predicted for optimum partitioning with the stem properties observed in a wide range of actual plants. We present the most plausible criteria for biomass partitioning in the current shoot for these plants.
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MODEL DESCRIPTION |
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TOPABSTRACTINTRODUCTIONMODEL DESCRIPTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXLITERATURE CITED |
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The shoot dry mass is expressed as
 | (1) |
The object of this model is to predict the optimal p, which maximizes shoot productivity (SP) (Givnish, 1986
; Mencuccini, 2003
). SP is factored into the leaf area per shoot (LA) and the CO2 assimilation rate per leaf area (A) such that
 | (3) |
 | (4) |
A depends on the stomatal conductance to water vapor (gs) at a constant leaf nitrogen content, because a high gs leads to a high intercellular partial pressure of CO2. The relationship between A and gs is presented based on previous models of the biochemistry of CO2 assimilation (Farquhar et al., 1980
; von Caemmerer and Farquhar, 1981
, and see Appendix for details). By definition, gs is expressed as
 | (6) |
):
 | (7) |
The mechanical support model
In the mechanical support model, the same shoot form is assumed as in the water transport model. To estimate the shoot biomass partitioning necessary for adequate mechanical support, we used the model reported by Tateno (1993)
. Tateno and Bae (1990)
focused on the safety factor against lodging (SF), which is expressed as
expressed SF as 
obtained 
), and is written as 
is defined as the ratio of Ms to the fresh stem volume. As E is proportional to (Niklas, 1992
), a can be expressed as a function of only (i.e., a 
). Although a should be used as the surrogate of mechanical strength per biomass investment into the stem, we use in the present study because can be easily measured and directly reflects the features of the xylem. Using Eqs. 14 and 15, we determine the value of p necessary to obtain a given SF, and analyze the effect of on p. The units and abbreviations of all parameters are shown in Table 1.
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MATERIALS AND METHODS |
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TOPABSTRACTINTRODUCTIONMODEL DESCRIPTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXLITERATURE CITED |
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A-gs relationship
The A-gs relationship was obtained in September 2000 for the following five species, all of which have broad leaves amenable to measurement of the CO2 assimilation rate: Helianthus annuus L., an angiosperm herb; Prunus sargentii Rehder, a deciduous angiosperm tree; Quercus myrsinaefolia Blume, an evergreen angiosperm tree; Ginkgo biloba L., a deciduous gymnosperm tree; and Podocarpus nagi (Thunb.) Zollinger et Moritzi, an evergreen gymnosperm tree. A and gs were measured using a CIRAS 1 system (PP Systems, Hitching, UK) on sunny days at intervals of 1 hr from 0800 to 1500, with shoots set in water and exposed to sunlight. The shoots experienced various VPDs, and so the gs varied to some extent. To measure A at very small gs levels, the shoots were removed from the water. The measurement conditions were as follows: leaf temperature, 30°C; CO2 concentration, 355 ppm; relative humidity, 70%; and photon flux density, 2000 µmol photons · m–2 · s–1. The predicted A is dependent on the maximum CO2-supply-limited rate of RuBP carboxylation, Vmax (see Appendix). To fit the predicted A to the measured A, the values of Vmax for the theoretical A-gs relationship were determined by the least square regression. The immediate values of Vmax obtained were used as parameter values in this study.
a- relationship
a and were measured in September 2001. To obtain a broad range of a and , 10 species were used. The angiosperm herbs examined were Achyranthes fauriei Lev. et Van., Bidens frondosa L., Chenopodium album L., Polygonum cuspidatum Sieb. et Zucc., and Solidago altissima L. The deciduous angiosperm trees used were Fagus japonica Maxim., Hamamelis japonica var. obtusata (Matsumura) Sugimoto, Hypericum chinense L. var. salicifolia (Sieb. et Zucc.), Quercus serrata Murray, and Rhododendron obtusum (Lindley) Planchon var. kaempferi (Planchon) Wilson. Stem segments of 20–40 cm long located near the base of the current shoot were collected, in order to use the almost constant diameter and elasticity through a stem axis. The critical load against lodging was measured according to Tateno and Bae (1990)
. Unlike Tateno and Bae (1990)
, the point load was weighted at the free end of the stem to obtain the mechanical property of stem. This is because the stem segments used in the measurement had been weighted mainly by the distal leaves and stems in the field condictions. a was calculated from Eqs. 11 and 12. The of the stem segments used in the critical load measurement was calculated as the ratio of Ms to the stem fresh volume. The stem fresh volume was determined by measuring the length and the diameters at both ends of a segment. Four or more shoots per species were used for measurements. The a- relationship, in which a is proportional to the–1 power of (see Eq. 15), was determined by least squares regression using the averaged values of a and for each species.
Measurements of ksm and
Measurements of ksm and were carried out from July to September 2001 using eight species of angiosperm herbs, 11 species of deciduous angiosperm trees, seven species of evergreen angiosperm trees, and 14 species of evergreen and deciduous gymnosperm trees. Details are summarized in Table 2. Most of these species are native to warm and cool temperate zones. ksm was measured according to Sperry et al. (1988)
. The water used for the measurements had been distilled, adjusted to a pH of about 2 with HCl, and filtered through a 0.22 µm-filter. Segments of 4, 10, and 20 cm long located near the bases of current shoots were used for gymnosperm trees, angiosperm trees, and herbs, respectively. After refilling via a positive pressure of 175 kPa for at least 3 min, values for kh were obtained at a pressure of 8 kPa. Values for were determined as described in the previous section. Segments were then dried for 2 d at 80°C and their Ms values were determined. At least four shoots per species were examined to determine ksm and .
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To determine M at ls = 0.4 m and the allometric relationships between M and ls, M vs. ls was regressed as the power function M = 
·lsß, using least squares regression after the raw data were subjected to base 10 log transformation (LaBarbera, 1989
). The following species were used: Helianthus annuus (M = 28.6 ls2.12, r2 = 0.97, P < 0.01), Prunus sargentii (M = 8.25 ls1.25, r2 = 0.84, P < 0.01), Quercus myrsinaefolia (M = 11.0 ls1.05, r2 = 0.56, P < 0.01), and Abies firma (M = 25.3 ls1.29, r2 = 0.90, P < 0.01).
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RESULTS |
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TOPABSTRACTINTRODUCTIONMODEL DESCRIPTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXLITERATURE CITED |
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and favorable environmental conditions (summarized in Tables 3 and 4) because the water transport model predicts shoot biomass partitioning that is optimal for productivity rather than for persistence. ksm for a current shoot is assumed to be constant and peculiar to species.
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). The measured A-gs relationships fit well to those predicted for the given values of Vmax for five woody and herbaceous species (r2 = 0.63–0.97 and P < 0.01).
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; Mencuccini, 2003
). We address the effect of this simplification in the Discussion section. To explore the effect of shoot size on the ksm-p* relationship, we altered the shoot size using the allometric equation derived from Prunus sargentii: M = 8.25 ls1.25 (Fig. 3c). The value of p* increases with ls, which reflects the inverse relationship between Js and ls (see Eqs. 7–9). At the same ls value, a high ksm value has a small p* value, indicating the generality of the above ksm-p* relationship.
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–1 (r2 = 0.77, P < 0.01) is used as the relationship between a and . Here, we analyzed the effect of on pmin using Eq. 14 and the above relationship between a and .
Values of pmin vs. are shown in Fig. 4a. At a constant value, a high SF value requires a high p value. It is notable that pmin increases with , indicating that species with high values require greater p values to obtain a given SF than do those with low values. This is because species with high values have thinner stems that show less resistance to lodging than do those with low values, at the same values of Ms and ls (Givnish, 1995
). This tendency is maintained when the shoot size is altered using the allometric relationship shown in Fig. 3b (Fig. 4b).
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that produces optimal biomass partitioning, and the partitioning criteria of actual plants values that result in p* = pmin, which represents optimum partitioning because there is not excess capacity for either water transport or mechanical support. The solid line shown in Fig. 5a represents the combinations of ksm and for which p* corresponds to pmin to obtain an SF of 1. That is, a shoot with a high ksm value is predicted to have a low p*. To obtain an SF of 1, this small value for p* requires a low value, which represents a high mechanical support capacity per biomass investment in the stem. The converse is also true. Thus, combinations of high ksm/low and low ksm/ high are predicted to be the values required for optimum partitioning.
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that result in values for p* that are less than values for pmin (Fig. 5a). These stem properties can be represented by the solid line in Fig. 5b and have an SPmax at SF values lower than the minimum value. According to the water transport model, shoots continue to assimilate even if they are unable to stand on their own. However, in practice, mechanical failure results in a marked decrease in productivity because lodged plants are shaded by their neighbors. Therefore, the mechanical support criterion governs shoot biomass partitioning, and p must follow pmin.
In contrast, the region below the solid line represents the combinations of ksm and that result in values for p* that are greater than values for pmin. These stem properties can be represented by the broken line in Fig. 5b and have an SPmax at SF values higher than the minimum value. The shoot biomass partitioning for SPmax requires a value of p that is sufficient to achieve the minimum SF. In this case, the water transport criterion governs shoot biomass partitioning, and p should follow p*. Figure 5a indicates that water transport criterion is the most important under conditions of very low ksm or very low values, whereas mechanical support criterion is the most important under conditions of comparatively high ksm and high values.
To determine the most important factor in the partitioning of biomass in the current shoot, we compared the theoretical stem properties for optimal partitioning with those from actual plants (listed in Table 3 and Fig. 6a). The plants were classified into four functional groups based on their xylem properties and leaf phenologies: (1) angiosperm herbs, (2) deciduous angiosperm trees, (3) evergreen angiosperm trees, and (4) gymnosperm trees. To most closely represent the actual situation, the calculations were carried out with ls = 0.4 m and the values of Vmax, SLA, c, and M derived from Helianthus annuus, an angiosperm herb (Fig. 6b); Prunus sargentii, a deciduous angiosperm tree (Fig. 6c); Quercus myrsinaefolia, an evergreen angiosperm tree (Fig. 6d); and Abies firma, an evergreen conifer (Fig. 6e). The values for the parameters are summarized in Table 4. Although the present model assumes SF to be 1, these plants generally had SF values greater than 1. For example, mulberry trees (Morus bombycis) planted in low individual density conditions maintain SF values from 3 to 4 during vegetative growth (Tateno and Bae, 1990
), and many studies have reported plants with a certain safety factor against mechanical failure such as buckling and stem breaking (McMahon, 1973
; Mattheck et al., 1993
; Niklas, 1994
; Spatz and Bruechert, 2000
). These observations suggest that a value for SF of 1 is insufficient for plant survival under natural conditions in which plants are exposed to mechanical stresses, such as strong winds, and that SF values of around 4 are the minimum required for survival in the field. Therefore, we examined the effects of SF values of 1 and 4 on the specific partitioning criteria.
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values of actual plants are plotted in Figs. 6. We distinguished the most plausible criterion as follows: If the plots (±1 SD) for the majority of species in a given functional group fell within the region of pmin > p*, the group was considered to be constrained by the mechanical support criterion. If the plots (±1 SD) for the majority of species in a given functional group were within the region of p* > pmin, the group was considered to be constrained by the water transport criterion. If an equal number of plots (±1 SD) from a functional group belonged to each region, the group was designated as obeying the optimum criterion. It became evident that the partitioning of biomass in the current shoot in almost all of the species examined is governed by the mechanical support criterion rather than by the optimum values. This tendency is typical of angiosperm species. However, note the dashed and dotted lines in Fig. 6, which represent the stem properties at SF = 1 and at SF = 4 with a change of ±40 % in , respectively. These lines indicate that at low and SF values, water transport is the most important criterion for gymnosperms. A comparison between the p values measured in the current shoots native to the field and those predicted from the two models revealed that the measured p values were near the pmin values for angiosperm species (Fig. 7), but were closer to the p* values for gymnosperm trees (Fig. 7d). The measured p values were determined under field conditions. This suggests that under field conditions, the biomass partitioning in the current shoots of angiosperm species is governed by the mechanical support criterion and that of gymnosperm trees is governed by the water transport criterion.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMODEL DESCRIPTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXLITERATURE CITED |
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). However, some gymnosperms did not follow this pattern (Fig. 6e); comparisons of predicted and actual p values for gymnosperms indicated that the water transport function is the major criterion for these plants (Fig. 7d). This difference between angiosperms and gymnosperms would be due to the difference in conductive capacity. Extremely low ksm of gymnosperms causes high sensitivity of biomass partitioning to the environmental conditions in the water transport model. For example, assuming SF = 4, the present models predict that the criterion for gymnosperms shifts from mechanical support to water transport at = 0.9 (Fig. 6e), or VPD = 0.025 (data not shown), which is a condition usually experienced by plants in growing season of a temperate zone. In contrast, angiosperms remain constrained by the mechanical support criterion even under severe conditions of = 0.35, or VPD = 0.06 (data not shown), which seldom occur in temperate regions. These differences are probably attributable to differences in conductive capacities of angiosperms and gymnosperms (Fig. 6a and Table 3). The evolution of vessels may have provided angiosperms with high conductive efficiency, reducing the need for a greater biomass investment in the water transport function relative to the mechanical support function.
Using the present models, we can examine the criteria under extreme environmental conditions. Plants native to alpine or coastal regions are constantly exposed to strong winds and thus should have a current shoot SF value greater than 4 to prevent mechanical failure from these physical stresses. Tateno (1991)
showed that mechanically stimulated Morus bombycis shoots have SF values of about 8 throughout their growing period. In this case, even shoots of gymnosperm trees should be governed predominantly by the mechanical support criterion. In contrast, herbs in dense stands often have SF values of near 1 (H. Nagashima, and M. Tateno, unpublished data), because they have highly elongated stems to facilitate competition for light. Figure 6b indicates that herb stem properties almost fall above the broken line representing p* = pmin for SF = 1 and that, even in this condition, the criterion for herbs is the mechanical support function. Climbing plants would have SF values of far less than 1 because they cannot support even their own weight. Although the stems of climbing plants may possess very high conductive capacities, the water transport criterion would govern biomass partitioning in the current shoots of these plants. The present models predict that angiosperms with comparatively high ksm values would continue to be constrained by the mechanical support criterion even under conditions of high evaporative demand (high VPD), as discussed above, and low soil water availability (low , Fig. 6). Even if a reversible embolism develops during the daytime in these angiosperms, reducing the water transport capacity of the stem (Canny, 1997
; McCully, 1999
; Zwieniecki et al., 2000
), the mechanical support function should continue to govern the biomass partitioning of the current shoot. The water transport criterion may constrain the angiosperm xerophytes occurring in areas with xeric conditions, as observed in gymnosperms, because angiosperm xerophytes such as desert shrubs that are adapted to xeric environments possess narrow vessels (Carlquist, 1975
; Baas, 1987
), their conductive capacities may be lower than those of temperate angiosperms.
The ecological implications of diverse stem properties
The present models indicate that the ideal stem should have a low value and a high ksm value. However, this combination of stem properties is only observed in herbs; trees have high and low ksm values (see Table 3 and Fig. 6a). This can be explained by the difference in life span between herbs and trees. In terms of the water transport function, a high is correlated with a high degree of resistance to air embolism (Hacke et al., 2001
), a condition in which conduits are blocked by air bubbles that are pulled into the conduits by excess negative pressure in the xylem. Tree leaves generally have longer life spans than herb leaves, and a higher value would give trees greater protection of the conduit integrity. Moreover, small vessels and tracheids, which lead to low ksm values, are resistant to freeze-thaw xylem embolism, a condition in which conduits are blocked by air bubbles that are emitted as freezing sap thaws. A high resistance to freeze-thaw embolism is important in evergreen species native to cool and cold temperate zones (Sperry et al., 1994
; Feild and Brodribb, 2001
). The mechanical support model in the present study assumes static and instantaneous loads. In the field, however, a stem must tolerate large local loads and small but frequent long-term loads, both of which induce fatigue in the xylem and decrease stem stiffness (Gardiner, 1995
). Under these conditions, the tracheid elements must be narrow and have thick walls, which results in high values of E and . The production of excessively large and numerous vessels in order to maintain a high value, leading to a high ksm value, is unlikely. Furthermore, a high value is directly related to a long lifetime because it confers strong decay resistance (Loehile, 1988
; Shain, 1995
). Therefore, even though high and low ksm values cost more in stem mass, these properties should be favorable for woody species.
Remaining problems and future directions
Although the water transport model focuses on shoot productivity, the relative hydraulic safety of water transport and the prevention of xylem embolism are also important considerations in biomass partitioning. The hydraulic safety factor is defined as the ratio of the minimal water potential necessary to prevent catastrophic conduit dysfunction to the leaf (xylem) water potential observed in the field (Sperry, 1995
; Domec and Gartner, 2002
). Many studies have reported low hydraulic safety factors under normal growth conditions (Tyree and Sperry, 1988
; Sperry, 1995
; Salleo et al., 2000
; Brodribb et al., 2003
) Domec and Gartner (2002)
showed that the hydraulic safety factor is far lower than the mechanical safety factor, suggesting the importance of the water transport function in shoot morphogenesis. However, severe xylem embolisms are not usually observed in plants because the leaf water status is tightly regulated by the stomata (Salleo et al., 2000
; Brodribb et al., 2003
). Thus, hydraulic failure can be prevented through stomatal regulation rather than through the partitioning of biomass into the leaf and stem. In contrast, mechanical stability must be maintained through biomass partitioning because no other physiological processes can rapidly regulate mechanical stability. Therefore, it is reasonable that biomass partitioning in the current shoot is determined by the mechanical support criterion, as shown with the models presented in this report.
The hydraulic limitation hypothesis predicts that the whole-plant hydraulic conductance, the CO2 assimilation rate per leaf area, and the ratio of foliage to sapwood area decrease with increasing in tree height (Ryan and Yoder, 1997
; Magnani et al., 2000
; Ryan et al., 2000
; McDowell et al., 2002
; Mencuccini, 2003
). The water transport model, however, assumes a constant KRL of 1 for all species regardless of stem height, because KRL is predicted to have little effect on p* at the current shoot level, even if KRL varies between 0.01 and 10 (Fig. 3b). The extension of our model to mature trees would be useful in understanding the form developed and the biomass production of trees, because it can be used to evaluate the effects of KRL.
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APPENDIX |
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TOPABSTRACTINTRODUCTIONMODEL DESCRIPTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXLITERATURE CITED |
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. When the carboxylation reaction is limited by the supply of CO2, then the assimilation rate (Ac) is expressed by 
* is the CO2 compensation point, Km is the Michaelis-Menten constant for the carboxylation reaction, and Rd is day respiration rate. Rd is derived from Rd = 0.0089 Vmax (Watanabe et al., 1994
).When the carboxylation reaction is limited by the regeneration of RuBP, then the assimilation rate (Aj) is expressed by 
) 
). Ci is given by (von Caemmerer and Farquhar, 1981
) 
 | (A5) |
 | (A6) |
The units and abbreviations are listed in Table 1 and parameter values are listed in Table 2.
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FOOTNOTES |
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101 2 Corresponding author. Current address: Graduate School of Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka, 560-0043 Japan (taneda{at}bio.sci.osaka-u.ac.jp
)
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LITERATURE CITED |
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TOPABSTRACTINTRODUCTIONMODEL DESCRIPTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXLITERATURE CITED |
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