Do pathways of water movement and leaf anatomical dimensions allow development of gradients in H218O between veins and the sites of evaporation within leaves?

Authors

  • M. M. BARBOUR,

    1. Environmental Biology Group, Research School of Biological Sciences, Institute of Advanced Studies, Australian National University, GPO Box 475, Act 2601, Australia
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      Current address: Landcare Research, PO Box 69, Lincoln, New Zealand.

  • G. D. FARQUHAR

    Corresponding author
    1. Environmental Biology Group, Research School of Biological Sciences, Institute of Advanced Studies, Australian National University, GPO Box 475, Act 2601, Australia
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G. D. Farquhar. Fax: + 61 26125 4919; e-mail: farquhar@rsbs.anu.edu.au

ABSTRACT

The oxygen isotope enrichment of bulk leaf water (ΔL)  is often observed to be poorly predicted by the Craig–Gordon-type models developed for evaporative enrichment from a body of water (Δe). The discrepancy between ΔL and Δe may be explained by gradients in enrichment within the leaf as a result of convection of unenriched water to the sites of evaporation opposing the diffusion of enrichment away from the sites; a Péclet effect. However, this effect is difficult to quantify because the velocities of water movement within the leaf are unknown. This paper attempts to model the complex anatomy of a leaf, and hence such velocities, to assess if the gradients in H218O required for a significant Péclet effect between the vein and the evaporation sites are possible within a leaf. Published dimensions of cells in wheat leaves are used to calculate the cross-sectional areas perpendicular to the flow velocities of water through assumed pathways. By combining the ratio of actual to ‘slab’ velocities with anatomical lengths, equivalent lengths (L) emerge. In this way, it is concluded that if water moves only through the cell walls, or from cell to cell via either aquaporins or plasmodesmata, and evaporates from mesophyll cells, or the substomatal cells, or from the peristomatal region (a total of 15 combinations of assumptions), then the 15 central estimates of the values of L are between 9 and 200 mm. Each of these central estimates is subject to uncertainty, but overall their magnitude is important and estimates of L are comparable with those made from fitting to isotopic data (8 mm for wheat). It is concluded that significant gradients in enrichment between the vein and the evaporation sites are likely.

INTRODUCTION

The oxygen isotope enrichment of leaf water has been of interest to plant physiologists for some time because (1) it influences the isotopic composition of atmospheric CO2 and O2, and (2) it may record both leaf evaporative conditions and plant responses to these conditions. However, problems in both measuring and predicting leaf water Δ18O have meant that the full potential of the technique is yet to be realized. Recent improvements in measurement (Farquhar, Henry & Styles 1997), in terms of time taken for analysis, will improve the through-put of samples and should allow larger experiments with statistical treatment to be attempted.

However, there remain a number of gaps in the theoretical understanding of leaf water Δ18O. Enrichment at the sites of evaporation seems to be broadly predicted using a model developed for surface waters such as lakes (Craig & Gordon 1965), but bulk leaf water enrichment has been observed to depart from this simple function in many experiments. This departure has been variously explained by: (1) pools of water within the leaf (e.g. Yakir, DeNiro & Gat 1990); (2) unenriched water within veins lowering the bulk leaf water enrichment (e.g. Roden & Ehleringer 1999); (3) a string of interconnected pools within the leaf (Gat & Bowser 1991; Helliker & Ehleringer 2000), and (4) the ratio of convection of unenriched water towards the sites of evaporation to back diffusion of enrichment from the sites (e.g. Farquhar & Lloyd 1993). Strong, although indirect, supporting evidence for the latter Péclet effect at the whole leaf level has been presented by Barbour et al. (2000b; 2003). The current paper investigates the Péclet effect at the scale of (4) above, that is in terms of anatomical dimensions and pathways of water movement within leaves from veins to the stomata.

Leaf water Δ18O

Leaf water is enriched in comparison with water taken up from the soil because the heavier H218O evaporates and diffuses more slowly through the stomata during transpiration than does H216O. The enrichment of water at the sites of evaporation within the leaf above source (soil) water (Δe) may be described by the equation (Craig & Gordon 1965; Dongmann et al. 1974; Farquhar et al. 1989; Farquhar & Lloyd 1993):

Δe = e* + ek + (Δv − ek)ea/ei(1)

where e* is the proportional depression of water vapour pressure over H218O compared with the vapour pressure over H216O (and is related to temperature), ek is the kinetic fractionation as water vapour moves through the stomata and leaf boundary layer, Δv is the oxygen isotope composition of water vapour in the atmosphere relative to source water, and ea and ei are ambient and intercellular vapour pressures, respectively. Equation 1 predicts general trends in leaf water Δ18O well, but does not describe observed variability in Δ18O within a leaf (e.g. Yakir, DeNiro & Rundel 1989; Wang & Yakir 1995).

The Péclet effect

The isotope composition of leaf water as a whole (ΔL) has often been shown to be poorly predicted by Eqn 1, often being less enriched (e.g. Allison, Gat & Leaney 1985; Bariac et al. 1989; Walker et al. 1989; Walker & Brunel 1990; Yakir et al. 1990; Flanagan, Bain & Ehleringer 1991a; Flanagan, Comstock & Ehleringer 1991b; Walker & Lance 1991; Flanagan 1993; Flanagan, Marshall & Ehleringer 1993; Flanagan et al. 1994; Wang, Yakir & Avishai 1998) but also, particularly when veins are removed, more enriched (e.g. Bariac et al. 1994; Wang & Yakir 1995; Helliker & Ehleringer 2000). Farquhar & Lloyd (1993) suggested that these discrepancies may be partly explained by the transpirational convection of unenriched water to the evaporating sites opposing the enrichment at the sites of evaporation diffusing back into the leaf; a Péclet effect (Ikeda 1983). Early work reported an increasing discrepancy between ΔL and Δe with increasing transpiration in barley (Walker et al. 1989), and a similar trend was found in bean (Flanagan et al. 1994). Such a relationship can be explained by an increase in the ratio of convection to diffusion as transpiration rates increase.

Barbour et al. (2000b) found strong indirect evidence of a Péclet effect by comparing the Craig–Gordon model of leaf water enrichment with the 18O enrichment of the water with which sucrose exchanged. The latter enrichment was taken to be Δsuc − ewc, where Δsuc is the enrichment of sucrose above source water and ewc is the equilibrium fractionation factor between carbonyl oxygen and water (ewc has been shown to be approximately 27‰ by Sternberg & DeNiro 1983). A strong positive relationship was found between the transpiration rate and the fractional difference between Δsuc − ewc and Δe.

The Péclet effect is characterized by a dimensionless number, , given by the product of velocity of water movement (v, in m s−1) and the distance from the evaporating surface (l, in m), divided by the diffusivity of H218O in water (D, 2.66 × 10−9 m2 s−1):

℘ = vl/D(2)

The average enrichment of leaf water over the distance l is given by (Farquhar & Lloyd 1993)

image(3)

The theoretical relationship between and the fractional difference between ΔL and Δe is shown in Fig. 1. It can be shown that (Farquhar & Lloyd, unpublished) in flow through a series of pipes, the total Péclet effect will be the sum of the Péclet effects for the individual portions.

Figure 1.

The theoretical relationship between the Péclet number () and the proportional difference between 18O enrichment of bulk leaf water (ΔL) and water at the evaporating sites (Δe) (redrawn from Farquhar & Lloyd 1993).

The velocity of water movement within the leaf is related to the transpiration rate per unit leaf area, E (mol m−2 s−1). If E is divided by the molar density of water (C = 55.5 × 103 mol m−3), the result is the equivalent velocity of water if it were moving perpendicular to the leaf surface as a slab. The true velocity is faster than the slab velocity, E/C, because the water actually moves in a tortuous path through only a portion of the leaf. We denote the ratio of the true velocity to the slab velocity by a scaling factor k. Thus:

image(4)

When Eqn 4 is substituted into Eqn 2,

image(5)

and we call the product, kl, the effective length, L, so that:

image(6)

Flanagan et al. (1994) estimated an effective length of 8.5 mm in Phaseolus vulgaris, whereas Wang et al. (1998) calculated a range in L from 4 to 166 mm for a large number of species. Barbour & Farquhar (2000) found that an effective length of 8 mm accurately predicted variation in organic matter δ18O from cotton plants, and Barbour et al. (2000b) found that L varied between 9 and 14 mm in castor bean. If the actual distance between the vein and the stoma was 100 µm, a scaling factor of 80 is required to produce an effective length of 8 mm. To produce a significant Péclet effect, that is a fractional difference between ΔL and Δe of more than, say, 0.1, a velocity of about 10 µm s−1 is required. Or, in terms of evaporation rates and effective lengths, at evaporation rates of 5 and 10 mmol m−2 s−1, L must be about 6 and 3 mm, respectively, for a significant Péclet effect to be observed. All of the above estimates used projected leaf areas to describe E, as if transpiration was from one surface only. If the leaves were amphistomatous, the transpiration rates when dealt with separately for the two sides would be roughly halved. To maintain the same (fitted) values of the values of effective lengths appropriate for the separate treatment of the transpiration from the two sides, L2, is double those listed above for L.

Not all recent research supports the importance of a Péclet effect in leaf water enrichment. For example, data and models published by Roden and Ehleringer (Roden & Ehleringer 1999; Roden, Lin & Ehleringer 2000) did not show a relationship between transpiration rate and the difference between measured and Craig–Gordon-modelled stable hydrogen isotope ratios. However, subsequent recalculation of leaf water and cellulose isotope ratios as enrichment above source water showed evidence of a Péclet effect (Barbour et al. 2003). These authors propose that wide variation in source water δ18O in the Roden and Ehleringer experiments masked the relationship between transpiration rate and the deviation of observed isotope ratios from Craig–Gordon modelled values.

There is evidence of gradients of isotopic enrichment on larger spatial scales than those discussed here so far. This includes observations of increasing enrichment towards the outside and tip of leaves (Bariac et al. 1994; Wang & Yakir 1995; Helliker & Ehleringer 2000; Gan et al. 2002). Gradual increases in Δ18O in water expressed from the petioles of excised leaves (Yakir et al. 1989) also point to gradients of enrichment along a leaf. Such observations suggest that the kind of Péclet effect discussed above extends a short distance into the vein allowing some evaporative enrichment from central and basal portions of the leaf to be carried along the leaf to the edge and tip (Farquhar & Gan 2003).

Measurements of vein and mesophyll water enrichment in cotton leaves (Gan et al. 2002) have identified the need for a model describing enrichment along a series of pools of water within the leaf (the string-of-lakes model proposed by Gat & Bowser 1991) to be combined with a Péclet effect. Such a model is described by Farquhar & Gan (2003) and, when tested (Gan et al. 2003) was found to predict progressive enrichment of both leaf and vein water towards the tip of a maize leaf. For improved predictions see Barnes, Farquhar & Gan (2003).

From such observations of progressive enrichment along leaves Gan et al. (2002) suggest that the longitudinal Péclet number (describing the Péclet effect within the veins in the longitudinal direction) must be large in comparison with the radial Péclet number (describing the Péclet effect in laminar mesophyll cells between the veins and the stoma in the radial direction). An important point raised in the treatment by Farquhar & Gan (2003) is that the longitudinal Péclet effect (of order 107) affects the rate of increase in the enrichment of vein water along a leaf but not the difference between bulk leaf water Δ18O and the Craig–Gordon model. The radial Péclet effect (of order one) is most relevant to the diminution of average leaf water enrichment below the Craig value and to effective lengths estimated from bulk leaf water Δ18O measurements. Therefore the radial Péclet number is estimated in this article, although from hereon we drop the ‘radial’ descriptor.

If the Péclet effect may be accepted as having some relevance to leaf water Δ18O, the obvious question raised is whether the dimensions within the leaf will allow the required velocities; namely, do scaled effective lengths of the order of several millimetres make any sense? To address this question, we need to quantify the velocity of water movement within the leaf in relation to slab velocities. To do this we must have some knowledge of, or make justifiable assumptions about, the pathways for water movement in leaves.

Pathways of water movement in leaves

Historically, water movement in leaves was thought to be via the cell walls. Strugger (1943) proposed the Extended Cohesion Theory that described a continuous column of water under tension from the roots through the xylem and along cell walls to the evaporating surfaces, namely apoplastic water. The leaf mesophyll cells were bathed in the water column, but water did not pass through the plasma membrane unless adjustments in pressure were necessary. Water movement across membranes was suggested to be diffusive (Dainty 1963), even though the apparent diffusion coefficient observed experimentally was rather high given the low solubility of water in plasma membrane lipid bilayers (House 1974).

More recently Altus, Canny & Blackman (1985) and Canny (1986, 1988) followed the pathway of transpired water in a wheat leaf using a fluorochrome dye (sulphorhodamine G) that had been shown to be limited to the apoplast. At the veins a bright fluorescence was found away from the xylem elements, along narrow pathways (‘nanopaths’) in the walls of mestome sheath cells (the small cells immediately adjacent to the outermost xylem vessels, Altus et al. 1985). The dye was shown to collect along the cell walls of parenchyma sheath cells (the cells surrounding the vascular tissue and the mestome sheath cells), but to penetrate no further into the leaf. Canny (1986, 1988, 1990) proposed that transpiration-stream water moved into the symplasm (i.e. within the plasma membrane) at the mestome sheath/parenchyma sheath interface.

The plausibility of symplastic water movement came with the discovery of water channels or aquaporins in plant cells, first in the tonoplast (Höfte et al. 1992) and then in the plasma membrane (Daniels, Mirkov & Chrispeels 1994; Kammerloher et al. 1994; Qui, Tai & Wasserman 1995). Kaldenhoff et al. (1998) used antisense technology to create Arabidopsis thaliana plants with reduced expression of a plasma membrane aquaporin, which resulted in a three-fold decrease in protoplast permeability. However, the overall water use (on a fresh weight basis) was no different between antisense and control plants, and unfortunately stomatal conductance and transpiration rate were not measured. More recently, evidence of the potential importance of aquaporins to whole plant water balance has been presented by Frange et al. (2001), who demonstrated high densities of a vacuolar membrane aquaporin in bundle sheath cells of Brassica napus. This finding supports the hypothesis proposed by Canny (1986) that these cells are important in facilitating water fluxes. In order to estimate the velocity of water movement through such a channel, or indeed whether a Péclet effect could even apply at such a scale, it is necessary to know how water moves through aquaporins.

Water movement through some aquaporins is reversibly inhibited by mercuric chloride (Chrispeels & Agre 1994), while simple diffusion across a lipid bilayer is not. Water transport through aquaporins is characterized by a low Arrhenius activation energy (Ea < 21 kJ mol−1) compared with lipid bilayers lacking channels (Ea > 42 kJ mol−1). Tyerman et al. (1999) suggest that the low activation energy of water moving across a channel is because the water molecules are able to move almost as freely as by diffusion through bulk water. Water molecules are thought to move in single file through a pore in the protein formed by an intracellular and an extracellular loop (Schäffner 1998).

If water moves in a single file, the propagation of gradients in isotopic enrichment from the evaporation sites back to the veins may be blocked, or at least poorly described by the equations we have used. However, it is possible that water moves through aquaporins in a fashion similar to Knudsen diffusion (Reinecke & Sleep 2002), with some molecules moving counter to the net flow, in which case gradients could be propagated.

There is some evidence of control of the permeability of membranes in response to changes in the osmotic gradient via some form of gating (in the aquatic species Elodea densa, Steudle, Zimmerman & Zillikens 1982) or by phosphorylation of the aquaporin protein (e.g. Arabidopsis thaliana, Daniels et al. 1994; spinach, Johansson et al. 1996) or by altering the abundance of aquaporins in the membrane (e.g. Robinson et al. 1996). Control of aquaporin function by any of these three methods could allow fine control of rates of water movement across membranes. If water is assumed to move across membranes, then cell-to-cell water movement must involve the cell wall at the interface between adjacent cells.

An alternative route for cell-to-cell movement of water involves movement within the membrane connections between cells, the plasmodesmata. Ding, Turgeon & Parthasarathy (1992) found gaps (presumed transport channels), in the structure of the plasmodesmata from tobacco leaves. Fisher (1999) estimated the diameter of these channels to be 4 nm. Ding et al. (1992) report between eight and 10 channels per plasmodesma. The possibility of water flow through such tiny channels, stretching between cells over a distance of about 0.4 µm, is debatable. As pointed out by Fricke (2000), the diameter of water molecules (about 0.26 nm) is significant compared with the plasmodesmatal channel diameter, so that conventional treatment of diffusion and convection (Poiseuille's law of laminar flow through a tube, for example) is of questionable applicability (Tyree 1970). Indeed, Fricke concluded that symplastic water flow (via plasmodesmata) between leaf epidermal cells of barley was negligible.

Steudle, Murrmann & Peterson (1993) suggest that water in tissues is best thought of as moving through three parallel pathways; symplastic movement through plasmodesmata, transcellular movement across cell membranes (aquaporins) and apoplastic flow in cell walls that are not suberized. The relative importance of each pathway may differ between tissues. The three possible pathways will be considered individually in terms of their ability, or lack thereof, to explain the observed increasing discrepancy between ΔL and Δe with increasing evaporation rate. With the high level of uncertainty in estimates of velocities of water movement in individual pathways, we have not attempted to estimate velocities in parallel pathways.

Given the three possible pathways of water movement within leaves, the next point to consider is where these paths end; that is, where evaporation occurs.

Sites of evaporation within leaves

Although it is clear that almost all water evaporates from inside the leaf, with only a small amount evaporating directly from the epidermis, the exact sites of evaporation within leaves are not known. A number of theoretical models have provided support for the hypothesis, first suggested by Tanton & Crowdy (1972), that most of the evaporation occurs from the cells nearest the stomata. Using a scaled-up physical model of a substomatal cavity, Meidner (1976) found that wet filter paper lining a hemispherical cavity lost 37% of the total evaporated water from the area close  to the ‘stomatal’ pore. Cowan (1977) created a two-dimensional electrical model of the substomatal cavity with semiconducting graphite as the vapour phase, and silver paint to simulate cell surfaces within the substomatal cavity. By this method he suggested that 77% of water evaporates from the guard cells, subsidiary cells and epidermal cells bordering the stoma. Tyree & Yianoulis (1980) and Yianoulis & Tyree (1984) created a mathematical model of a cavity by dividing a hemisphere or a cylinder into concentric volume amounts. Patterns of evaporation were calculated using Fick's law of diffusion. These models predicted that between 60 and 80% of water evaporates from the cells near the stomatal pore, depending on the way in which heat and mass transfer were considered.

Re-interpreting data presented by Farquhar & Raschke (1978), Boyer (1985) suggested that evaporation occurred close to the vascular system and therefore deep within the leaf. Recently Pesacreta & Hasenstein (1999) reported the existence of cuticle on the inner walls of epidermal cells surrounding the stomata in thistle (Cirsium horridulum). This cuticle seemed to be of similar composition to external cuticle, which led Pesacreta and Hasenstein to suggest that such cuticles may reduce water loss from cells close to the stoma. However, the permeability of the interior cuticle has not been measured. Similar interior cuticles have been reported by Norris & Bukovac (1968), Appleby & Davies (1983) and Wullschleger & Oosterhuis (1989). If extensive interior cuticles such as reported by Pesacreta & Hasenstein (1999) are found to provide similar barriers to evaporation as external cuticles, the bulk of evaporation may occur from mesophyll cells.

From theoretical and experimental evidence there seem to be three possible sites for evaporation within the leaf: (1) from all mesophyll and epidermal cells exposed to the vapour phase (IAS: intercellular air space); (2) from the mesophyll and epidermal cells bordering the substomatal cavity (CAV: substomatal cavities); and (3) most (75%) of evaporation from the peristomatal region and the remainder from the other mesophyll and epidermal cells bordering the substomatal cavity (PER: peristomatal region). These three possibilities are explored in turn to assess differences between them in terms of propagation of gradients of H218O enrichment between evaporation sites and the vein.

MODEL DEVELOPMENT

This paper attempts to resolve the problem that effective lengths of the order of several millimetres are required to generate significant gradients in H218O enrichment, while the actual distance between the vein and the stomata in many leaves is about 0.1 mm. More specifically, we reported a value for L in wheat of 8 mm (Barbour et al. 2000a), making L2 16 mm, whereas the distance from the veins to the stomata, l, is about 90 µm. The scaling factor (k2 = L2/l) therefore needs to be about 180 and the actual water velocity needs to be about 180 times the slab velocity, or water flow must be restricted to about 1/180 of the volume of the leaf. Tortuosity in the path from the vein to the stomata increases l and reduces the requirement for greater velocity (for example a tortuosity factor of 1.5 would mean that the actual velocity has to be 120 times the slab velocity to achieve the 16 mm effective length, L2).

One way of assessing the effective lengths possible inside leaves is to scale the cross-sectional areas through which the water moves in the leaf to the leaf surface area. We know that actual velocity is greater than slab velocity because the leaf is not solid, but has air spaces. If air fills 35% of the leaf internal volume, for example, then the scaling factor between the slab and actual velocities, k, could not be less than 1.5. This would be the scaling factor for water moving through the entire cross-sectional area (i.e. cell wall, cytosol and vacuole) of all cells. A scaling factor of 1.5 gives an effective length of 0.13 mm (assuming an actual length of 90 µm), and no significant H218O gradient from the vein to the evaporating surface. However, as described above, current understanding of water movement in leaves suggests that flow may be restricted to much smaller pathways.

Another important point is that water moves radially out from the vein. This means that the cross-sectional area perpendicular to the flow is expected to vary considerably within the leaf. For example, near the xylem vessels, all the water must travel through the few cells immediately adjacent, whereas further away from the vein the number of cells increases dramatically. Such complex geometry is difficult to model accurately, so we stress that our attempt is very approximate.

To calculate the cross-sectional area perpendicular to the flow of each component of the pathway from the vein to the evaporating site, we divide the cells within the leaf into classes according to their distance from the vascular tissue. A simple way to do this is to consider just a portion of a leaf. Cells within a transverse section of this portion are classified, then the number of cells in each class is estimated by scaling the number in each class in the transverse section to the entire leaf portion of interest using individual cell lengths and the length of the portion, as described below. The cross-sectional areas of assumed pathways can be estimated from the dimensions of each cell class. An overall scaling factor (k) is given by the length-weighted ratio of each component cross-sectional area to leaf surface area, as described in detail below.

We chose a wheat leaf to do the calculations presented in this paper for three reasons. First, wheat leaf anatomy is rather simple, and the leaf may be divided easily into regular portions separated by veins (Altus et al. 1985). Second, good transverse and longitudinal sections of wheat leaves, with dimensions of different cells, are available (Parker & Ford 1982). Third, an estimate for the projected area-based effective length (L) of about 8 mm is available for wheat (Barbour et al. 2000a).

Wheat leaf dimensions

Altus et al. (1985) showed that a wheat leaf may be divided into portions 2.5 mm long and 0.25 mm wide, bordered on the two long sides by lateral or intermediate veins and on the short sides by transverse veins, giving a two-sided leaf area (LA) of 1.25 mm2[we assume evaporation occurs equally from both sides of the leaf, and calculate a two-sided scaling factor (k2), then divide k2 by two to get a projected area-based scaling factor (k)]. For the purposes of these calculations, the portion of leaf is taken to be the same size as that described by Altus et al. and having an intermediate vein running along the centre of the portion, as shown in Fig. 2. Figure 2 is an idealized representation of a wheat leaf transverse section, in which four stomata coincide on the same section. Such an arrangement is not realistic but was chosen so that distances between the vein and the stomata were minimized and the scaled effective lengths calculated are underestimates.

Figure 2.

An idealized diagram of the section of the wheat leaf to be considered. Cell classes shown in the transverse section are those used in IAS Model: X, xylem; MS, mestome sheath cells; PS, parenchyma sheath cells; M1, mesophyll cells adjacent to the PS cells; M2, evaporating mesophyll and epidermal cells. The orientation of the cut in the transverse and longitudinal section and the revealed surface used to calculate the cross-sectional area of cell walls is indicated. Not to scale.

Classification of cells

As described above, we wish to assess gradients in H218O enrichment within leaves under three assumed patterns of evaporation, and with three possible pathways of water movement. In the IAS model we assume that all mesophyll and epidermal cells exposed to intercellular spaces are evaporating (as suggested by Boyer 1985). We classify these cells M2 in this model. From Fig. 3 we count 53 cells in this class in the transverse section. All mesophyll cells that are not exposed to intercellular space are classified M1 in the IAS model. From Fig. 3 there are 13 cells in the M1 class in the transverse section. Furthermore, from the transverse section in Fig. 3, there are nine parenchyma sheath (PS) cells, and nine mestome sheath (MS) cells, of which three are assumed to conduct water from the adjacent xylem vessels in 0.2 µm thick nanopaths (Canny 1988).

Figure 3.

An idealized diagram of a transverse section of a wheat leaf to show the cell classes used in the CAV and PER models. MS, mestome sheath cells; PS, parenchyma sheath cells; M1, mesophyll cells adjacent to parenchyma sheath cells; M2 to M5, mesophyll and epidermal cells at different distances from the vein; SS, mesophyll and epidermal cells bordering the substomatal cavity (SS1, 8, 13 and 19 are the subsidiary cells adjacent to the guard cells, and important in the PER model).

In the CAV and PER models we assume that only the mesophyll and epidermal cells bordering the substomatal cavity are evaporating (labelled SS in Fig. 3). These cells are then classified according to the number of cells water must pass through to reach an evaporating cell. Mestome sheath and parenchyma sheath cells have the same classification in all three models. As shown in Fig. 3, there are five evaporating cells for which water only passes through one non-evaporating mesophyll cell (SS4, 5, 6, 10 and 11). There are six evaporating cells for which water must pass through two non-evaporating cells (SS3, 7, 9, 16, 17 and 21), six for which water passes through three non-evaporating cells (SS2, 8, 12, 18, 20 and 22), three for which water passes through four non-evaporating cells (SS1, 15 and 19), and one each for which water must pass through five and six cells (SS14, and 13, respectively). In the PER model, most of the evaporation occurs from the subsidiary cells (SS1, 8, 13 and 19 in Fig. 3), which are assumed to be fed water by the evaporating (SS) cells adjacent to them.

Having divided the cells in the transverse section into classes, the total number of cells in each class may be calculated by dividing the length of the leaf portion by the average length of the cells in each class, and multiplying by the number of cells present in the transverse section. From the longitudinal sections presented by Parker & Ford (1982) the average dimensions of the various types of cells may be estimated. Mestome sheath cells are long and thin, with an average length of 130 µm, and an average diameter of 7 µm. Parenchyma sheath cells are much larger, but half the length, with an average length of 68 µm, and an average diameter of about 25 µm. Mesophyll cells in wheat leaves are lobed, but tend to be about 25 µm in diameter, and about 67 µm in length. The total number of cells of a class within the portion of leaf (total ♯) is given by:

image(7)

The total numbers of cells in each class for all three models are presented in Table 1.

Table 1.  Total number of cells (Total ♯) of each cell class in the leaf portion for the different models of assumed evaporation sites, their total surface area (Total SA), and the surface area of the cell class interfacing with the previous cell class (Interface SA) in a wheat leaf
Cell
class
ModelTotal ♯Total SA
(µm2)
Interface SA
(µm2)
  1. The IAS model assumes that evaporation occurs from all mesophyll and epidermal cells exposed to the vapour phase, the CAV model assumes that evaporation is evenly spread between mesophyll and epidermal cells bordering the substomatal cavity, and the PER model assumes that 75% of evaporation occurs from cells close to the stomatal pore, the remainder evenly shared between other mesophyll and epidermal cells that border the substomatal cavity. In all models MS, mestome sheath cells; PS, parenchyma sheath cells; M1, mesophyll cells adjacent to parenchyma sheath cells. In the IAS model M2 are the mesophyll and epidermal cells exposed to the intercellular spaces, and so assumed to be evaporating. In the CAV and PER models, M2 to M6 are the mesophyll and epidermal cells at different distances from the vein and SS are the mesophyll and epidermal cells bordering the substomatal cavity, and so assumed to be evaporating.

MSall58 0.02 × 106
PSall331 2.09 × 1060.02 × 106
M1all485 3.78 × 1060.83 × 106
M2IAS197815.43 × 1063.39 × 106
M2CAV, PER597 4.66 × 1061.02 × 106
M3CAV, PER410 3.20 × 1060.70 × 106
M4CAV, PER149 1.16 × 1060.26 × 106
M5CAV, PER149 1.16 × 1060.26 × 106
M6CAV, PER379 0.29 × 1060.06 × 106
SSCAV, PER821 6.40 × 1061.41 × 106

Total cross-sectional area of the cell-to-cell interface

To calculate the cross-sectional area perpendicular to the flow as water moves from one cell to another through the cell wall, the surface area of the cell wall interface between two adjacent cells is estimated as a proportion of the total cell surface area.

Parker & Ford (1982) calculate that the average surface area of a single mesophyll cell is 7803 µm2. The surface areas of other cells types may be calculated by assuming that they are cylinders with flat ends. This gives a surface area of a mestome sheath cell of 2936 µm2, and that of a parenchyma sheath cell 6323 µm2, using the average length and diameters presented above. The total surface area of all cells in each class in the leaf portion is then given by the surface area of one cell multiplied by the total number of cells in the class within the leaf portion. The total surface area for each cell class in each model is presented in Table 1.

The proportion of cell wall that interfaces with the next cell was estimated from transverse sections presented by Parker & Ford (1982). The proportion of cell wall surface area of M1 cells that interfaces with M2 cells was estimated to be 0.22, as was the proportion of cell wall surface area of parenchyma sheath cells that interface with M1. The proportion of parenchyma sheath cell wall area that interfaces with the mestome sheath cells that conduct water (three of the nine) was estimated to be 0.07.

The total surface area of the interface between cells is calculated by multiplying the total surface area of each cell class by the proportion of the cell wall that interfaces with the cell wall of the previous cell type. This allows calculation of water velocities through the cell wall at the cell-to-cell interfaces. Surface areas are presented in Table 1.

Cross-sectional areas perpendicular to the flow

As we wish to assess scaled effective lengths for water movement through three possible pathways, the cross-sectional area perpendicular to the flow for each of these pathways must be calculated. This means calculating the cross-sectional area of the cell wall through which the water moves, in the first instance, for each of the cell classes for each of the models. The cross-sectional area of cell walls may be visualized by cutting a core through the leaf around the vascular tissue midway along the cell class of interest. An example of this section is given in Fig. 2, where the parenchyma sheath cells are cut to expose the longitudinal section parallel to the axis of the vein. In a similar way, the cross-sectional area of the cytosol may be visualized. Walls of parenchyma sheath and mesophyll cells are assumed to be 0.2 µm thick (Evans 1983), and the cytosol 2 µm thick. Cell walls exposed to intercellular air spaces have been shown to be about 0.1 µm thick, but those in contact with neighbouring cells are usually somewhat thicker (Evans 1983). Mesophyll sheath cell walls are usually thicker than 0.2 µm, but water movement is confined to 0.2 µm thick nanopaths within the wall (Canny 1988). Variation in cell wall thickness will be explored more thoroughly later. The calculations are described in more detail in Appendix A. The cross-sectional areas of the cell wall and the cytosol for each cell class are presented in Table 2.

Table 2.  Cross-sectional areas of pathways of water movement perpendicular to the flow for: cell walls (XSACW), cytosol (XSAC), vacuole (XSAV), aquaporins (XSAA) and plasmodesmata (XSAP) for various classes of cells in three models of evaporation sites within the leaf
Cell
class
ModelXSACW
(µm2)
XSAC
(µm2)
XSAV
(µm2)
XSAA
(plasma membrane)
XSA
(tonoplast)
XSAP
(µm2)
rand.
(µm2)
max.
(µm2)
rand.
(µm2)
max.
(µm2)
  1. The assumptions for each model are described in the text, and the cell classes are outlined in Figs 2 and 3.

MSall1.58 × 103
PSall4.05 × 103 60.22 × 1030.50 × 106167418335 83711.8
M1all5.88 × 103 87.31 × 1030.72 × 106333757666151423.5
M2IAS
M2CAV, PER7.23 × 103107.46 × 1030.88 × 106410932820186329.0
M3CAV, PER4.97 × 103 73.88 × 1030.61 × 106281640564128119.9
M4CAV, PER1.81 × 103 26.86 × 1030.22 × 106102232205 466 7.2
M5CAV, PER1.81 × 103 26.86 × 1030.22 × 106102232205 466 7.2
M6CAV, PER0.45 × 103  6.71 × 1030.06 × 106 26 58 51 116 1.8
SSCAV, PER

Given the relatively high abundance of aquaporins commonly observed in the vacuolar membrane (e.g. Frange et al. 2001), it seems likely that water within the symplast of the cell also moves through the vacuole. With this in mind, an alternative version of the symplastic water pathways is included. In the alternative models, water moves through the cytosol to the vacuole at the cell-to-cell interface, through the aquaporins in the vacuolar membrane, then through the vacuole as a whole (at very low velocities) and in reverse order at the cell-to-cell interface on the exit side of the cell. These versions are called: Aquaporin/cytosol for cell-to-cell movement via aquaporins, then water movement within the cytosol only; Aquaporin/vacuole for cell-to-cell movement via aquaporins, then water movement through the cytosol at both entry and exit sides of the cell and movement through the vacuole between. Similarly, plasmodesmata/cytosol and plasmodesmata/vacuole refer to cell-to-cell movement via plasmodesmata, then movement only through the cytosol or through the vacuole, respectively.

As water moves from cell to cell in the symplastic pathways of water movement, it moves through either the aquaporins in the membrane or the plasmodesmata at the cell interface, so that both may be considered to be ‘pipes’ leading either across the membrane or across the cell wall. The cross-sectional area of these ‘pipes’ can be calculated from their diameters, the abundance of the ‘pipes’ per unit cell surface area, and the surface area of the cell-to-cell interface. Assumptions made about ‘pipe’ dimensions, and a description of the calculations may also be found in Appendix A. The cross-sectional areas of aquaporin and plasmodesmatal connections for each cell class are presented in Table 2.

Lengths of pathways

Cross-sectional areas of the various assumed pathways must be weighted by the length over which a particular area occurs. The length of the pathway through the cell walls or cytosols is given by πØ/2, where Ø is the diameter of the cell, so the length of the pathway through the nanopaths in the mestome sheath cell wall is 11.0 µm, and the length of the pathway through parenchyma sheath or mesophyll cell walls is 39.3 µm. The length of the pathway for water movement across the cell-to-cell interface (unless otherwise stated), either through the cell wall or via plasmodesmata, is 0.4 µm, which is twice the thickness of the cell wall. The length of the pathway through the vacuole is assumed to be the diameter of the cell, minus the thickness of the cell walls and the cytosol (i.e. the vacuolar pathway length of the parenchyma sheath cells is 20.6 µm). Membranes are often about 5 nm thick (Alberts et al. 1989), and this is assumed to be the length of the pathway for aquaporin water movement. For movement through the symplasm we assume a tortuosity factor of 1.4, and multiply the above path lengths by 1.4. This is equivalent to assuming that on average the water molecules move at 45° to the direction of flow because of other impeding molecules (Cowan 1986).

Calculating the scaling factor

Farquhar and Lloyd (unpublished) have shown that in flow through a series of pipes, the total Péclet effect will be the sum of the Péclet effects for the individual sections. This allows the two-sided scaling factor (k2) for water moving from the vein to the evaporating sites through cross-sectional areas calculated above to be estimated (assuming a single pathway in series) by adding the ratio of two-sided leaf surface area to cross-sectional area perpendicular to the flow for each component of the pathway, and weighting each component ratio by the length over which it occurs. In a formal sense this is given by:

image(8)

where LA2 is the two-sided leaf surface area, XSA1 and l1 are the cross-sectional area and the length of the first component, respectively, XSA2 and l2 are the cross-sectional area and the length of the second component, respectively, and so on, from the vein to the stomata on both sides of the leaf. The projected area-based scaling factor (k) is simply k2/2, assuming that leaf anatomy is symmetrical, and evaporation occurs equally from both sides of the leaf, to a first approximation. Equation 8 does not hold for water movement in parallel through two or more pathways. A model including parallel water movement has not been attempted.

As an example, assuming an apoplastic water pathway for the simplest model (the IAS model), the overall value of k2 is given by the ratio of the leaf surface area to the cross-sectional areas through the mestome sheath cell wall, multiplied by the length of the pathway, plus the equivalent ratio through the parenchyma sheath cell walls and the non-evaporating mesophyll cell walls, both multiplied by the length of that component of the total pathway. Formally this is:

image

and k= 163.

Projected area-based scaling factors are then multiplied by the actual length between the vein and the stoma (k × l, where l = 89.5 µm, in this case) to get the corresponding scaled effective length (L). Projected area-based scaling factors (k) and effective lengths (L) for each model and for each assumed pathway are presented in Table 3, and may be compared with the published estimate for L of 8 mm.

Table 3.  One-sided scaling factors (k), weighted lengths of the pathways (lw) and one-sided scaled effective lengths (L) for wheat leaves with assumed water pathways and evaporating sites
PathwayIAS modelCAV modelPER model
klw
(µm)
L
(mm)
klw
(µm)
L
(mm)
klw
(µm)
L
(mm)
  1. Pathway refers to the assumed pathway of water from cell to cell

Apoplastic 32689 29.2 323 73 23.5 394101 39.8
Aquaporin/cytosol 11590 10.4  83155 12.9  73200 14.6
Aquaporin/vacuole 12870  9.0  88118 10.4  69151 10.5
Plasmodesmata/cytosol168290152.4 819155127.11006200200.9
Plasmodesmata/vacuole215470150.91029118121.41262151190.4

In both the CAV and PER models, the water passes through the nanopaths in the cell walls of the mestome sheath cells before moving into the symplast at the parenchyma sheath plasma membrane via the aquaporins. The water then moves through the cytosol of the parenchyma sheath cells, and into the adjacent mesophyll cells either via aquaporins or plasmodesmata. These patterns of water movement are consistent with patterns of dye deposition found by Canny (1986).

In the CAV and PER models a scaling factor for each type of evaporating substomatal cell (e.g. those fed by pathways that go through four mesophyll or epidermal cells are one ‘type’) is first calculated, then all scaling factors are weighted by the proportion of evaporation modelled to occur from that type. The weighting procedure is described in more detail in Appendix B.

RESULTS

The calculated projected area-based effective lengths (L) for each model and assumed pathway are given in Table 3. As Table 3 shows, when an apoplastic pathway is assumed the estimate of L is between 23 and 40 mm, depending on where in the leaf evaporation is assumed to occur. If the water is assumed to move symplastically via the membrane aquaporins past the parenchyma sheath cells, estimates of L range between 9 and 15 mm, depending on whether or not the water is assumed to move through the vacuole and on where evaporation occurs. The third possible pathway for water movement, that is from cell to cell via plasmodesmata past the parenchyma sheath cells, produces values of L between 121 and 201 mm.

The estimates of L vary between the different models of evaporation sites. The IAS model (where all cells exposed to intercellular spaces are evaporating) produces the lowest values of L for a given water pathway. The PER model (where 75% of the water is assumed to evaporate from the peristomatal region) produces the highest estimates of L for a given pathway, while values of L produced by the CAV (where evaporation is assumed to be evenly divided among all cells bordering the substomatal cavity) are intermediate.

DISCUSSION

This paper set out to assess whether scaled effective lengths within wheat leaves would allow the development of significant gradients in enrichment of H218O between the evaporating sites and the veins. Effective lengths need to be in the order of millimetres for a significant radial Péclet effect to occur. We have shown that although the actual distance between the vein and the stoma is only about 100 µm, projected area-based scaled effective lengths (L) over this distance are between 9 and 200 mm, depending on the assumptions made regarding pathways of water movement and sites of evaporation within leaves. This suggests that the anatomy of the wheat leaf is such that significant radial Péclet effects could occur at normal transpiration rates.

As well as a radial Péclet effect, Farquhar & Gan (2003) also include a longitudinal Péclet effect in their model of spatial variation in Δ18O of leaf water. The longitudinal Péclet number determines the spatial pattern of enrichment along the leaf via gradients in enrichment within the veins. In contrast, the radial Péclet number (with associated effective length) determines the average leaf water enrichment across the whole blade. Therefore, values of effective lengths calculated from bulk leaf water measurements are more closely related to radial Péclet numbers than longitudinal Péclet numbers, so the comparison between fitted L from isotopic data (Barbour et al. 2000a) and L calculated in this paper is valid. The Farquhar & Gan (2003) model is, by their own admission, a simplification of even the very simple venation of monocot leaves. Complex patterns of leaf water enrichment observed in dicotyledoneous leaves have been related to their reticulate venation (Gan et al. 2002). However, it seems likely that the radial Péclet effect will be even more important than longitudinal effects in dicots, when compared to monocots.

Pathways of water movement

As described in the introduction, the pathway for water movement within leaves is unclear. Conventional theory, proposed by Strugger (1943) suggests that the transpiration stream flows only within the apoplast. If this were the case, and cellular water reaches isotopic equilibrium over time with the water flowing around it, then the scaling factors and effective lengths calculated here are within a factor of three to five of the value for L of 8 mm estimated for wheat from whole leaf δ18O values (Barbour et al. 2000a). Given the uncertainties in our calculations this could be taken as reasonable consistency. The diffusive processes involved in generating Péclet gradients will occur within cell walls, and it seems entirely possible for significant Péclet effects to be so generated in leaves.

However, Canny (1986) suggested that the transpiration stream moves only within narrow pathways in the cell walls of the two or three mestome sheath cells immediately adjacent to the xylem vessels, and that when it reaches the parenchyma sheath cells it moves across the plasma membrane at the interface and into the symplast. This suggestion is supported by recent work by Frange et al. (2001), who found high densities of aquaporins in the bundle sheath cells of Brassica napus.

There are a number of possible paths for further movement once the stream reaches the plasma membrane at the interface between the parenchyma sheath cells and the mesophyll cells. The water may move from the parenchyma sheath cells to the mesophyll cells across the plasma membranes via aquaporins and the intervening cell wall at the interface. Once inside the cytosol of the parenchyma sheath cells, the water may then move to the parenchyma sheath–mesophyll interface within the cytosol only, or may move through the vacuole via tonoplast aquaporins. Alternatively, the water may move across the cell-to-cell interface via plasmodesmatal connections, then either stay in the cytosol or move through the vacuole (again via tonoplast aquaporins).

In the analysis above, the assumption is made that the properties of diffusion of H218O in free water also hold through aquaporins and through the transport channels of plasmodesmata. Both these assumptions are made rather tentatively. We have very little understanding of how water moves through aquaporins, and whether there can be any diffusion of enrichment past the membrane containing aquaporins. However, if some water molecules move counter to the net flux, and there is a two-way Knudsen-like diffusion (Reinecke & Sleep 2002), then gradients in H218O enrichment could be propagated beyond the plasma membrane. Free water may not be present within plasmodesmata, but rather a viscous mix of molecules may fill the channels from cell to cell. If H218O must diffuse through a region with a viscosity higher than water, then D (the diffusivity of H218O) may be lowered, increasing ℘. The diameter of the transport channels in plasmodesmata is very small, so that most water molecules within the channels may interact with lipids and proteins forming the sides of the channels. As such, mass flow may not occur (Fricke 2000).

If Canny (1986) is correct in his interpretation that water moves symplastically past the parenchyma sheath cells, then effective lengths of between 9 and 15 mm are estimated for cell-to-cell movement via aquaporins in the cellular membranes. The range in values arises from different assumptions being made as to the sites of evaporation within the leaf, and whether water moves through the vacuole of the cell or stays within the cytosol. The estimates of L for the aquaporin pathway of water movement are a little lower than for apoplastic movement because the assumed cross-sectional area perpendicular to the flow of the cytosol and/or vacuole of each cell is rather larger than the cell wall cross-sectional area. Although the aquaporins create a small cross-sectional area for water flow at each membrane, the effect is over a very short distance (just 0.005 µm), so the aquaporins have little direct effect on the overall scaling factor, while having a large indirect effect. Two arrangements of aquaporins within the membrane were assessed; either aquaporins were randomly arranged throughout the plasma membrane and tonoplast, or all aquaporins were at either one of the interfaces between cells in the transpiration stream. The different arrangements of aquaporins did not produce significantly different scaling factors, again because the length over which their effect is seen is small.

The second possible symplastic pathway of water movement is via plasmodesmata. Using estimates of plasmodesmatal frequency and assuming mass flow processes hold within the narrow ‘transport channels’ in plasmodesmata, one-sided effective lengths of between 121 and 201 mm are estimated (again, the exact value depending on the assumptions made about sites of evaporation, and if the assumed water pathway includes the vacuole). These values are larger than values of L estimated from other pathways. This is because the cross-sectional area perpendicular to the flow is small for these connections and the length over which the effect occurs is relatively long (0.4 µm). Velocities within these channels would have to be very high if this were the dominant pathway. We consider it unlikely that significant mass flow of water occurs from cell to cell via plasmodesmata. This conclusion is supported by recent evidence from Fricke (2000), that plasmodesmatal water flow does not contribute significantly to water exchange between leaf epidermal cells from barley.

Sites of evaporation

Another area of great uncertainty regarding water movement within leaves is the exact site of evaporation. Several models (e.g. Cowan 1977; Yianoulis & Tyree 1984) suggest that most water evaporates from the surfaces of cells very close to the stomatal pore. On the other hand, Boyer (1985) suggests that evaporation may occur from mesophyll cells deep within the leaf. In order to explore these two opposing views, as well as cover some of the mid-ground between them, we created three models of evaporation. In the IAS model all cells exposed to intercellular spaces evaporate, and so are at Δe (as an extreme case of Boyer's hypothesis). In the CAV model only those cells bordering the substomatal cavity evaporate, but evaporation is equally shared by all these cells (as an intermediate), and in the PER model 75% of all water evaporates from the stomatal subsidiary cells in the epidermis, and the remaining 25% from the other cells bordering the substomatal cavity.

These different models produce significantly different effective lengths for all models of water movement. As might be expected, there was a positive relationship between the number of cells around which the water travelled, on average, and the estimated effective length, so that the PER model had the largest L. However, it should be noted that the values of L were in the order of several millimetres for all models of evaporation, meaning that no information as to the adequacy of one model over the other is available.

Sensitivity analysis

Throughout the text we have attempted to highlight both the complexity of the system modelled, and the assumptions made. We see the most important assumptions made in the calculations as: (a) plasmodesmatal frequency (b) the thickness of cell walls (c) thickness of the cytosol. Accordingly, sensitivity analysis was carried out by altering the assumed values of these parameters and comparing calculated values of L1.

A 10% increase in plasmodesmatal frequency from the assumed frequency gives an 8.5% decrease in L1 for the plasmodesmatal pathway of water movement across all three models of sites of evaporation, showing that the plasmodesmatal model is very sensitive to the frequency of cell-to-cell connections. A 10% increase in the parenchyma sheath cell wall thickness from the assumed thickness decreased L by up to 4% for the model of apoplastic water movement, and increased L by up to 4% for the model of plasmodesmatal movement. The models were also sensitive to the thickness of the mesophyll cell walls. A 10% increase in mesophyll cell wall thickness resulted in a 6% decrease in L for the model describing apoplastic water movement, and an 8% increase in L for the model describing movement through plasmodesmata (for the PER model of evaporation, and slightly smaller changes for the other two models). The thickness of the cytosol had a very small effect on the value of L calculated using the plasmodesmatal water pathway, and no effect on L calculated by either the apoplastic model or the aquaporin model.

This analysis shows that the modelled values of L are fairly sensitive to the plasmodesmatal frequency (in the case of the plasmodesmatal water pathway model), and to the mesophyll cell wall thickness (in the cases of both plasmodesmatal and apoplastic water movement). More tightly constrained values for these parameters would allow greater confidence in the exact values of L calculated, but would be unlikely to alter the overall conclusions from the study.

CONCLUSIONS

Based on the calculations presented in this paper, we suggest that a significant radial Péclet effect may be expected between intermediate veins and the stomata in wheat leaves. That is, the anatomy of the leaf and the assumed pathways of water movement give rise to products of velocities and lengths between the vein and the evaporative sites that are comparable with the diffusivity of H218O in water. The projected area-based scaled effective lengths (L) are in the order of millimetres, whether the water pathway is assumed to be via the apoplast or the symplast. This conclusion was shown to be valid if evaporation was modelled to occur from all cells exposed to the vapour phase, or if only the cells bordering the substomatal cavity evaporated, or if the bulk of the evaporation occurred from the subsidiary cells in the stomatal complex.

ACKNOWLEDGMENTS

We gratefully acknowledge critical review of earlier drafts of the manuscript by Dr J. R. Evans, Dr M. E. McCully, Dr M. J. Canny, Dr S. K. Gan and Mr L. A. Cernusak.

APPENDIX A

Cross-sectional areas perpendicular to the flow

Cross-sectional areas within cell walls

The cross-sectional area of the cell wall of a single cell (XSA) perpendicular to the flow of water is given by the length of the cell multiplied by the diameter of the cell minus the area within the cell wall. The area within the cell wall is given by the cell length minus two times the thickness of the cell wall multiplied by the cell diameter minus two times the thickness of the cell wall. Cell walls are assumed to be 0.2 µm thick (wheat mesophyll cell walls adjacent to intercellular spaces were found to be 0.10 µm thick by Evans 1983; while cell walls adjacent to other cells were up to 0.30 µm thick), and 66% of the cell wall volume is assumed to be water (Gaff & Carr 1961). For example, Fig. 2 shows that a parenchyma cell is 68 µm in length, and 25 µm in diameter, so the cross-sectional area of the cell wall perpendicular to the flow is given by:

XSA  cell  wall = {(68 × 25) − [(68 − 0.4) × (25 − 0.4)]} × 0.66

       = 24 µm2

The total cross-sectional area perpendicular to the flow for each cell class is then the XSA of the cell wall for a single cell, multiplied by the number of cells in the class in the leaf portion (331 for the example above). These cross-sectional areas are presented in Table 2.

Cross-sectional areas within the cytosol

The cross-sectional area perpendicular to the flow within the cytosol may be calculated in a similar manner to that for the cell wall. The cytosol is assumed to be 2 µm thick, and water is able to move through the entire volume (i.e. no need to multiply by 0.66 as above), although by a tortuous route. For example, the cross-sectional area perpendicular to the flow through the cytosol of a single parenchyma sheath cell is given by:

XSA cytosol = (68 × 25) − [(68 − 4.4) × (25 − 4.4)]

          = 390 µm2

The total cross-sectional area of the cytosol perpendicular to the flow for each cell class is then calculated by multiplying the XSA of the cytosol of a single cell by the total number of cells in the class for the portion of the leaf. These values are also presented in Table 2.

Cross-sectional areas of aquaporins

Aquaporins are thought to comprise about 20% of plasma membrane proteins in leaves (Johansson et al. 1996). However, the exact arrangement of these aquaporins within the membrane is not clear. We decided to test the two extreme arrangements of aquaporins, namely that they are randomly spaced throughout the membrane (‘random’), or that aquaporins only occur at the cell-to-cell interfaces through which water will be moving (‘maximum’). This required two methods of calculation.

In the ‘random’ arrangement the cross-sectional area of aquaporins will be given by the proportion of the surface area of the aquaporin that forms the transport channel, multiplied by the proportion of membrane proteins that are aquaporins, multiplied by the ratio of proteins to lipids in the membrane, multiplied by the ratio of surface areas of proteins and lipids, and all multiplied by the total surface area of cell-to-cell interface for the cell class of interest.

In the ‘maximum’ arrangement the cross-sectional area of aquaporins will be given by the proportion of the surface area of the aquaporin that forms the transport channel, multiplied by the proportion of membrane proteins that are aquaporins, multiplied by the ratio of proteins to lipids in the membrane, multiplied by the ratio of surface areas of proteins and lipids, and all multiplied by the total surface area of the cell class of interest, divided by two (two cell-to-cell interfaces per cell).

The transport channel of an aquaporin is estimated to cover 0.01 of the surface area of the aquaporin protein, up to 0.2 of all membrane proteins have been shown to be aquaporins (Johansson et al. 1996), a ratio of one protein to 50 lipids is common in membranes (Alberts et al. 1989), and an individual protein is assumed to form about 10 times the membrane surface area of each lipid molecule.

The cross-sectional area perpendicular to the flow of aquaporins (XSAAq) in non-evaporating mesophyll cells at the interface with the parenchyma sheath cells is then given by multiplying the above by the total surface area of the cell-to-cell interface:

XSAAq = 0.01 × 0.2 × 0.02 × 10 × 0.83 × 106

   = 333 µm2

for the ‘random’ arrangement of aquaporins, and:

XSAAq = 0.01 × 0.2 × 0.02 × 10 × 3.78 × 106 × 0.5

   = 757 µm2

for the ‘maximum’ arrangement of aquaporins (where the total surface area of parenchyma sheath cells was taken from Table 1). Similar calculations were made for the aquaporins at each interface for all cell classes, and are presented in Table 2. Differences between scaled effective lengths calculated using the ‘random’ and ‘maximum’ arrangements were not significantly different (because the length over which the aquaporin pathway occurs is small), so only those for the maximum arrangement are presented in Table 3.

Cross-sectional area of plasmodesmatal connections

The third possible pathway for water movement within the leaf is through the plasmodesmatal connections between cells. Ding et al. (1992) present evidence of ‘presumed transport channels’ in plasmodesmata of tobacco leaves. These channels are about 4 nm in diameter (Fisher 1999) and there are eight to 10 channels per plasmodesma. The cross-sectional area for transport in one plasmodesma (XSAplas) may be calculated by:

XSAplas = 9 × π(Ø/2)2

where Ø is diameter of the channel, so that XSAplas is 44.1 × 10−6 µm2.

Fisher 2000) suggests that plasmodesmatal frequencies may lie between 0.1 and 10 plasmodesmata per µm2, depending on cell type. Plasmodesmatal frequencies in wheat leaves have been estimated by Kuo, O’Brien & Canny (1974), but unfortunately only for the interface between vascular tissue and mestome sheath cells, making values inappropriate for this study. In the absence of appropriate values for wheat, plasmodesmatal frequencies from interfaces between parenchyma sheath and mesophyll cells in barley leaves are used (Evert, Rustin & Botha 1996). Assuming the sections used were about 0.7 µm thick, then there are about 0.25 plasmodesmata µm−2 at the interfaces. We assume that this frequency also holds for all mesophyll to mesophyll and mesophyll-to-epidermal interfaces (unless otherwise stated). This allows calculation of the total cross-sectional area of transport channels at each interface. For example, the cross-sectional area of plasmodesmatal channels at the PS : M1 interface in the IAS model (XSAP PS : M1) is given by the XSA of channels in a single plasmodesma, multiplied by the number of plasmodesmata per unit surface area of the interface, multiplied by the total surface area of interface between the parenchyma sheath cells and the non-evaporating mesophyll cells (from Table 1):

XSAP PS : M1 = 44.1 × 10−6 × 0.25 × 0.42 × 106

    = 4.6 µm2

Table 2 presents cross-sectional areas perpendicular to the flow for transport channels in plasmodesmata for all cell classes in the three models.

Appendix B

Weighting the scaling factors

As described in the ‘Calculating the scaling factor’ section above, a weighting must be given to the scaling factor calculated for each individual cell class in the CAV and PER models according to the proportion of the water that passes through each class, to allow calculation of an overall scaling factor. For example, in the CAV model 0.227 of the water passes through the M1 pathway because there are five out of 22 SS cells fed by M1 cells. Similarly, 0.273 of the flow passes through each of the M2 and M3 pathways, 0.136 through the M4 pathway, and 0.045 through each of the M5 and M6 pathways.

In the PER model 75% of water is assumed to evaporate from the subsidiary cells (SS1, 8, 13 and 19), for which water must pass through either four mesophyll or epidermal cells (SS1 and 19), three mesophyll or epidermal cells (SS8), or six mesophyll or epidermal cells (SS 13). The remaining 25% of the water is assumed to be equally lost by the other 18 evaporating cells. Therefore, of this 25%, 0.069 of water passes through the M1 pathway (25% divided by 18, multiplied by five of the 22 evaporating cells fed by M1 cells), 0.083 through the M2 pathway, 0.256 through the M3 pathway (0.187 of the water through SS8, plus five other non-subsidiary cells lose 0.069 of the water). Two of the four subsidiary cells are fed by water from the M4 pathway, and one non-subsidiary evaporating cell, meaning that 0.389 of the water moves through these cells, and 0.014 and 0.187 move through the M5 and M6 pathways, respectively.

Ancillary