Carbon isotope theory
where Ra and Rl are the 13C/12C ratios of CO2 in the air and carbon in the leaf, respectively. Carbon isotope fractionation occurs in C3 plants via two processes. 13CO2 diffuses more slowly across the stomatal pathway, resulting in a fractionation (a) of 4·4‰. The second fractionation occurs during carboxylation by Rubisco (b), where 13C is effectively discriminated against by about 27‰. The fractionations are weighted by the CO2 partial pressure imposed at the step involved. Δ13Cl is modelled ( Farquhar et al. 1982 ; Farquhar & Richards 1984) by:
where pi and pa are the CO2 partial pressures of intercellular and ambient air, respectively. Equation 2 predicts a positive linear relationship between pi/pa and Δ13Cl in C3 plants. Δ13Cl values typically lie between 17 and 23‰.
To predict Δ13Cl from Eqn 2, pi/pa must be known. In the present experiment, pi/pa was estimated using a rectangular hyperbolic fit of pi/pa to gs (mol m−2 s−1) in cotton plants, where variation in conductance was caused by changing vapour pressure deficit (J. Read and J. Yong, unpublished results), and yielded the curve:
Oxygen isotope theory
Evaporation enriches water at the sites of evaporation in the leaf because the heavier H218O vapour has a lower vapour pressure than H216O and because H218O diffuses more slowly than H216O. The heavier molecule diffuses 1·028 times more slowly than the lighter molecule in air and through stomata, but 1·019 times more slowly through the laminar boundary layer, from Pohlhausen analysis ( Farquhar & Lloyd 1993). The kinetic fractionation factor (ɛk, in‰) becomes:
where rs and rb are stomatal and boundary layer resistances to water flux, respectively.
The proportional depression of vapour pressure by the heavier water molecule (ɛ*) is related to temperature (T, in K) by ( Bottinga & Craig 1969):
so ɛ* is 9·2‰ at 25°C and 9·0‰ at 20°C.
Following notation developed for carbon isotope discrimination, oxygen isotope enrichment at the sites of evaporation (Δ18Oe) is presented in relation to the isotope ratio of source water:
where Re and RS are the 18O/16O ratios at the sites of evaporation and for source water, respectively. In steady-state conditions Δ18Oe is related to kinetic and vapour pressure fractionation factors and the leaf evaporative environment ( Craig & Gordon 1965; Dongmann et al. 1974 ; Farquhar & Lloyd 1993) by:
where Δ18Ov is the isotopic composition of water vapour in the air, relative to source water, and ea and ei are ambient and intercellular vapour pressures, respectively.
A large discrepancy between Δ18Oe and measured leaf water is commonly found, and this discrepancy tends to increase with transpiration rate ( Flanagan et al. 1991 ; Walker & Lance 1991; Flanagan et al. 1994 ). Farquhar & Lloyd (1993) suggested that the discrepancy is due to gradients of isotopes within the leaf. These gradients are a result of diffusion of enriched water away from the sites of evaporation, opposed by convection of source water to those sites. In the steady state, the isotope ratio of water (Δ18Od) at some distance l from the evaporative site can be expressed as:
where v is the velocity of water movement (m s−1) and D is the diffusivity of H218O in water (2·66 × 10−9 m2 s−1). The velocity of water movement is related to the evaporation rate (E) by a scaling factor (k; between 102 and 103), which converts the velocity of water moving through a leaf as a slab to that of movement through a porous medium, i.e.
where C is the concentration of water (55·5 × 103 mol m−3). The dimension-less term EL/(CD) is the Péclet number, ℘, so that the average value of Δ18Od (Δ18OL) over the scaled effective length L (L = k× actual length) becomes ( Farquhar & Lloyd 1993):
Evaporative enrichment of water in the leaf is passed on to organic material due to exchange of carbonyl oxygen with water, resulting in a 27‰ enrichment of the organic oxygen compared to water at equilibrium. The most important exchange occurs in triose phosphates, as two of the three oxygen are in carbonyl groups, and the half time to equilibration is known to be rapid ( Sternberg et al. 1986 ; Sternberg 1989; Farquhar et al. 1998 ). We expect sucrose to be in equilibrium with cytoplasmic water, which we take here to be Δ18OL ( Barbour et al., in press b ).
Sucrose is then transported from the source leaves to sink tissue in the phloem. To form cellulose from sucrose, cleavage to hexose phosphates occurs, allowing 20% oxygen to re-exchange with water. A proportion, y, of hexose phosphates also goes through a futile cycle to triose phosphates ( Hill et al. 1995 ), allowing further exchange of 60% of the oxygen. When the hexose phosphates are reformed, there is a 50% chance that the non-exchangeable oxygen in the previous hexose phosphate will be in the exchangeable position in the new hexose phosphate. The proportion of oxygen in cellulose that has exchanged with water in the developing cell (pex) may be expressed (corrected from that presented in Farquhar et al. 1998 ):
and (1 –y) is the proportion of hexose phosphates that is immediately used to form cellulose without recycling through triose phosphates again. Hence, when all hexose phosphates are recycled through triose phosphates (i.e. if there is no cellulose synthesis), both y and pex are equal to 1. The derivation of Eqn 11 is presented in the Appendix.
The proportion of exchangeable oxygen in cellulose has been found to be 0·47 in carrot by Sternberg et al. (1986) and 0·38 in Lemna gibba ( Yakir & DeNiro 1990). Hill et al. (1995) present measurements of randomization of 14C-labelled hexose phosphates during cellulose synthesis in oak stem tissue. Recalculation of these figures gives a range in pex of between 0·49 and 0·57 (if one quarter of the label is found at the opposite end of the product, i.e. 25% randomization, pex is 0·5). pex may vary with rate of cellulose synthesis and sucrose import; if fluxes are slow, then pex may be high.
Water in the developing cell, with which oxygen exchange occurs, may be a mixture of both xylem water (un-enriched by exchange) and phloem water (initially, on leaving the source leaf, at Δ18OL). Bret-Harte & Silk (1994) suggest that the phloem could potentially supply 80% of the water required for cell expansion in root meristems of corn seedlings. This suggests that the proportion of xylem water in developing cells (px) could, in some conditions, be as low as 0·2. Phloem and xylem water may exchange during sucrose transport to the sink leaf, and water within the sink leaf may also be enriched by transpiration (assuming some cellulose is laid down after stomata become functional), and these processes are also included in the px term.
Assuming carbonyl oxygen have an average equilibrium fractionation factor with water (ɛwc) of 27‰, the oxygen isotope composition of cellulose (Δ18Oc) may be expressed ( Farquhar et al., in press ) as:
Whole-leaf tissue is known to be less enriched than its cellulose by between 4·2 and 9·2‰ ( Farquhar et al., in press ), so Eqn 12 may be re-written to describe the oxygen isotope enrichment of whole tissue compared to source water (Δ18Ol) as:
where ɛcp is the difference in enrichment between whole plant tissue and its cellulose (ɛcp = Δ18Ol–Δ18Oc, so that ɛcp is negative).
Using the models described above, both Δ13Cl and Δ18Ol may be predicted under the conditions applied in this experiment. Figure 1(b) shows predicted relationships between Δ13Cl and Δ18Ol at 43% and 76% relative humidity (RH) for a leaf with a boundary layer resistance (rb) of 0·2 m2 mol−1 s−1, when air temperature is 30°C and photosynthetically active radiation (PAR) is 1200 μmol m−2 s−1 for conductances ranging from 0·4 to 0·9 mol m−2 s−1. The effective scaled length for the Péclet effect is 8 mm, pex·px is set at 0·38 (anticipating the fitted value), and ɛcp is assumed to be − 8‰. In this model, photosynthetic capacity is held constant, but pi/pa is allowed to vary according to Eqn 3. The difference between leaf and air temperatures (ΔT) is given by:
Figure 1. The relationships between (a) modelled pi/pa and ea/ei, and (b) modelled Δ13Cl and Δ18Ol in two humidity environments and when Ta = 30 °C, PAR = 1200 μmol m−2 s−1, Δ18Ov = − 5·5‰, rb = 0·2 m2 mol−1 s−1, ɛwc = 27‰, pex·px = 0·38, L = 8 mm and ɛcp = − 7·5.
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where r*bH is the combined resistance to sensible and radiative heat transfer in parallel, Q0 is the isothermal net radiation, L is the molar heat of vaporization, D is the water vapour concentration deficit, Cp is the molar specific heat constant of air at constant pressure, and ɛ is the change of latent heat content of saturated air with a change in sensible heat content (for details see Barbour et al., in press a ). The relationship between Δ13Cl and Δ18Ol is predicted to be negative and slightly curvilinear (see Fig. 1b). The curvilinear nature of the relationship is due to the non-linear relationship between pi/pa and ea/ei (see Fig. 1a). The slope of the relationship in Fig. 1(b) differs between the two environments, with a slope of − 0·68‰ change in Δ18Ol with a 1‰ change in Δ13Cl at low humidity, and a slope of − 0·54‰ at higher humidity.