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The hypothesis that greater safety from cavitation by air-seeding through inter-vessel pits comes at the cost of less porous pit membranes with greater flow resistance was tested . Sixteen vessel-bearing species were compared: 11 from the Rosaceae, four from other angiosperm families, and one fern. Unexpectedly, there was no relationship between pit resistance (and hence the prevailing membrane porosity) and cavitation pressure. There was, however, an inverse relationship between pit area per vessel and vulnerability to cavitation (r2 = 0.75). This suggests that cavitation is caused by the rare largest membrane pore per vessel, the average size of which increases with total pit area per vessel. If safety from cavitation constrains pit membrane surface area, it also limits vessel surface area and the minimum vessel resistivity. This trade-off was consistent with an approximately three-fold increase in vessel resistivity with cavitation pressure dropping from −0.8 to −6.6 MPa. The trade-off was compensated for by a reduction in the percentage of vessel wall pitted: from 10–16% in vulnerable species to 2–4% in resistant species. Across species, end-wall pitting accounted for 53 ± 3% of the total xylem resistivity. This corresponded to vessels achieving on average 94 ± 2% of their maximum possible conductivity if vessel surface area is constrained.
‘Safe’ xylem transport means sufficient protection against conduit blockage, primarily from cavitation and embolism. ‘Efficient’ transport means low flow resistance for a given investment in vascular tissue. It has long been thought that there is a structural trade-off between these two traits: that efficient xylem comes at the cost of being vulnerable to failure (Zimmermann 1983; Tyree, Davis & Cochard 1994; Martinez-Vilalta et al. 2002). Two observations suggest the trade-off. For one, there is a tendency for plants that experience more cavitation-inducing stress to possess more cavitation-resistant xylem, and vice-versa (Davis et al. 1999; Hacke, Sperry & Pittermann 2000; Pockman & Sperry 2000). Minimum safety margins from cavitation are similarly small across the spectrum (Sperry 1995). If cavitation resistance came at no cost to the plant, safety margins would seemingly be huge and cavitation rare. Second, there is a tendency for the hydraulic resistivity of xylem on a cross-sectional area basis (pressure gradient divided by flux density) to increase with increasing cavitation resistance. This is mirrored by a trend of narrower xylem conduits in safer xylem (Martinez-Vilalta et al. 2002). ‘Trend’ is the appropriate term because the relationship is noisy and may or may not be statistically significant depending on the species sampled (Tyree et al. 1994; Hacke & Sperry 2001).
What could be the cause of a trade-off? The answer is simple for resistance to freezing-induced cavitation. Wider conduits, whether tracheids or vessels, appear to be more vulnerable to freezing-induced cavitation than narrower ones (Ewers 1985; Pittermann & Sperry 2003). This is consistent with wider conduits containing more air in solution per length of conduit, and hence forming larger air bubbles during ice formation. Larger air bubbles nucleate cavitation at less negative pressures during thaw. Narrower conduits are thus safer from freezing-induced cavitation, but also less efficient in transporting water per cross-sectional area.
This paper concerns the more complex question of an efficiency trade-off with safety from water-stress induced cavitation. There is considerable evidence that the cause of cavitation by negative pressure is the failure of the pit ‘valves’ between conduits (Crombie, Hipkins & Milburn 1985; Cochard, Cruiziat & Tyree 1992; Salleo et al. 1996; Sperry et al. 1996). These pits prevent air in already damaged conduits from entering functional ones and ‘air-seeding’ cavitation. Accordingly, any trade-off between safety and efficiency should depend on the functioning of these valves.
Based on a modelling study of the valve action of pits we proposed that an air-tight pit comes at the cost of high flow resistance through the pit (Sperry & Hacke 2004). For a uniformly porous (‘homogenous’) pit membrane typical of angiosperms and many seedless vascular plants (as opposed to the torus-margo organization of many conifer pits), the air-seeding pressure (P) through pre-existing pores can be predicted by the capillary equation: P = −4T/d, where T is the surface tension of water and d is the diameter of the largest membrane pore in the conduit. Greater safety from air-seeding requires a smaller maximum pore diameter per conduit. If the average porosity of the membrane is correlated with the maximum pore diameter, the flow resistance through the membrane should increase with increasing air-seeding protection, because flow resistance through pores in a thin membrane is inversely proportional to the pore diameter raised to the third power (equation 15 in Sperry & Hacke 2004).
Here we test this hypothesis by estimating the flow resistance through pits of species with widely varying resistance to cavitation by water stress. We limited our study to inter-vessel pits with homogenous pit membranes and to pits without vestures, recognizing that vestures, torus-margo membranes, or other modifications may represent qualitatively different solutions to a trade-off. We sampled intensively within a single family (Rosaceae) to minimize qualitative differences that might occur even within the homogenous type of pit membrane between diverse clades. For broader representation we also sampled a smaller number of disparate angiosperm families as well as a vessel-bearing fern. Many of the necessary measurements had already been made on these species from a related study (Sperry, Hacke & Wheeler, 2005), making it natural to include them.
A trade-off at the pit level does not necessarily translate to a trade-off at the whole conduit or xylem level. If the diameter and length of a vessel is uncoupled from the air-seeding properties of its pits, safe xylem does not necessarily have to be inefficient. High-resistance pits can be placed in wide and long conduits that render any pit bottleneck negligible (Sperry & Hacke 2004). However, our recent work suggests that this does not happen. Across very diverse xylem types we found that approximately 50% of the xylem flow resistivity is attributable to interconduit pits (Sperry et al. 2005). For an unknown reason, vessels are not long enough to compensate for their end wall resistance. A large end wall resistivity is consistent with a linkage between air-seeding pressure and conduit size, an issue we consider in detail in this investigation.
MATERIALS AND METHODS
Eleven species of native Rosaceae of diverse habitat and growth form (Table 1) were collected from a variety of locales in the Wasatch Mountains near Salt Lake City, Utah. Each species was collected from a single limited population to minimize variation. Stems were cut, wrapped in plastic bags, and brought to the laboratory. Five other species from the related study (Table 1) included native and cultivated species of diverse families that were collected locally as described elsewhere (Sperry et al. 2005).
Table 1. Study species, their symbols used in figures, and associated measurements not in figures
Species and figure symbol
rP (MPa s m−1)
The five species from families other than the Rosaceae are listed last. Double capital symbols distinguish these non-Rosaceae species in figures. ‘Contact fraction’ is the portion of the vessel wall area in contact with an adjacent vessel. ‘Pitfield fraction’ is the portion of the interconduit pitfield occupied by pits. The product of these two fractions gives the ‘pit fraction’ (FP), the total pit area per vessel area. ‘Length fraction’ is the fraction of the total vessel length contacting adjacent vessels. The ratio L′/L (Fig. 1) was calculated from the length fraction and average vessel length. Grand means from four to six stems per species ± SE.
Amelanchier alnifolia M. Roemer
0.14 ± 0.018
0.37 ± 0.020
0.41 ± 0.11
0.80 ± 0.09
626 ± 183
A. utahensis Koehne
0.13 ± 0.008
0.33 ± 0.020
0.40 ± 0.05
0.80 ± 0.04
112 ± 35
Cercocarpus ledifolius Torr. & Gray
0.05 ± 0.008
0.44 ± 0.018
0.16 ± 0.06
0.92 ± 0.14
39 ± 14
C. montanus Raf.
0.06 ± 0.002
0.26 ± 0.009
0.18 ± 0.05
0.91 ± 0.10
60 ± 12
Holodiscus dumosus (Hook.) Heller.
0.07 ± 0.006
0.44 ± 0.063
0.19 ± 0.06
0.91 ± 0.11
328 ± 84
Physocarpus malvaceus (Greene) Kuntze
0.08 ± 0.008
0.50 ± 0.036
0.23 ± 0.09
0.89 ± 0.14
337 ± 104
Purshia tridentata (Pursh) DC.
0.08 ± 0.007
0.28 ± 0.012
0.21 ± 0.04
0.89 ± 0.08
84 ± 24
Rosa nutkana C. Presl
0.10 ± 0.013
0.46 ± 0.025
0.25 ± 0.09
0.88 ± 0.13
276 ± 75
Rubus leucodermis Torr. & Gray
0.15 ± 0.011
0.53 ± 0.008
0.37 ± 0.06
0.81 ± 0.05
285 ± 120
R. parviflorus Nutt.
0.19 ± 0.006
0.43 ± 0.018
0.42 ± 0.05
0.79 ± 0.04
78 ± 15
Sorbus scopulina Greene
0.10 ± 0.011
0.39 ± 0.033
0.30 ± 0.13
0.85 ± 0.16
205 ± 73
Vitis vinifera L. (Vitaceae)
0.14 ± 0.015
0.75 ± 0.111
0.33 ± 0.13
0.83 ± 0.13
168 ± 63
Sambucus caerulea Raf. (Adoxaceae)
0.25 ± 0.024
0.56 ± 0.025
0.54 ± 0.08
0.73 ± 0.05
371 ± 92
Acer negundo L. (Aceraceae)
0.15 ± 0.011
0.63 ± 0.018
0.42 ± 0.06
0.79 ± 0.06
66 ± 15
Salix exigua Nutt. (Salicaceae)
0.22 ± 0.006
0.73 ± 0.013
0.44 ± 0.04
0.78 ± 0.05
72 ± 19
Pteridium aquilinum (L) Kuhn (Polypodiaceae)
0.35 ± 0.013
0.22 ± 0.017
1.0 ± 0.0
0.5 ± 0
31 ± 7
The centrifuge method was used for measuring vulnerability to water-stress induced cavitation. Stems 14.2 cm long and 1–5 years of age were flushed with a filtered (0.2 µm) 20 m m KCl at approximately 0.1 MPa to remove native embolism. The 20 m m KCl solution was used to control for any ion effect on pit membrane conductivity (Ieperen, Meeteren & Gelder 2000; Zwieniecki, Melcher & Holbrook 2001b). Flushed stems were spun on a custom-built rotor (Alder et al. 1997) to progressively more negative pressure, with their hydraulic conductivity (flow rate per pressure gradient) being measured between spins to assess the loss in conductivity from cavitation. A Weibull function (Neufeld et al. 1992) was fitted to the conductivity versus negative pressure relationship for each stem, and the pressure inducing 50% loss of conductivity calculated. We refer to the absolute value of this pressure as the ‘P50’ and use it to represent the safety of the xylem from water-stress-induced cavitation. A minimum of six stems per species were used to obtain the average P50 for each species.
The conductivity was measured by fitting the stem to tubing filled with the filtered KCl solution. A gravity-induced pressure difference across the stem drove flow into a reservoir on an electronic balance. Stems generally absorbed a small amount of water in the absence of applied pressure. This negative flow was subtracted from the flow under pressure to obtain the net pressure-driven flow rate. The net flow rate was divided by the pressure gradient to obtain the conductivity.
Methodology of determining area-specific pit resistance (rP)
We estimated inter-vessel pit resistance using an approach similar to Schulte & Gibson's (1988) study on tracheids (see also Lancashire & Ennos 2002). The pit resistance was determined from flow measurements through stem segments and the dimensions of the vessels and pits in the segments. The method assumes the xylem of a stem can be represented by longitudinal files of overlapping vessels of average size (Fig. 1). Lumen and end-wall are arranged in series with a unit length of L′ (Fig. 1), which is shorter than the vessel length (L) by one end wall. The average vessel resistivity (RC, pressure gradient divided by volume flow rate) is the sum of the lumen (RL) and end wall (RW) resistivity for the average-sized vessel. The end-wall resistivity (like any resistivity) is a length-independent measure that equals the end wall resistance (rW, pressure divided by volume flow rate) divided by the length L′ between successive end walls. Measuring L′ thus yields the end wall resistance: rW = RWL′. The end wall resistance itself is a function of the extent of the end wall – more specifically the number and area of pits it contains. From measurements of the area of pits in one end wall, the end wall resistance can be converted to a pit resistance on an area basis (rP = rW × pit area). The cavitation pressure of the average vessel corresponding to the resistance measurements was assumed to be near the P50, allowing safety from cavitation to be compared with flow resistances. Tables 1 and 2 summarize the major variables used in the analysis.
Table 2. List of major variables with definition and units employed
A ‘resistivity’ (R) is length-independent (pressure gradient per volume flow rate) and may be further standardized by cross-sectional area of flow as in the RCa term. A ‘resistance’ (r) is length-dependent (pressure difference per volume flow rate) but may be standardized by cross-sectional area of flow as in the rP term.
absolute value of xylem pressure causing 50% loss of hydraulic conductivity
MPa s mm−4
MPa s mm−4
end wall resistivity
MPa s mm−4
vessel resistivity on lumen area basis
MPa s m−2
resistance of one vessel end wall
MPa s mm−3
resistance of inter-vessel pits on pit area basis
MPa s m−1
Anatomical measurements were made using Image Pro Plus (Media Cybernetics Inc., Silver Springs, MD, USA) and ImageJ software (free-ware available from http://rsb.info.nih.gov/ij/). An overview of calculations is provided in the Appendix. The parameters RC, RL, RW, and FW(below) were already available for the non-Rosaceae species using an analogous methodology (Sperry et al. 2005).
Vessel lengths (L)
Stems were flushed to remove native embolism and injected with silicone (Rhodorsil RTV-141; Rhodia USA, Cranbury, NJ, USA; imported by Walco Materials, Escondido, CA, USA) coloured with red pigment (Silastic LSPRD11; Dow Corning, Kendallville, IN, USA) at 0.5 MPa for 1 h before curing in an oven for approximately 1 h at 70 °C. Earlier work indicated that this silicone does not penetrate pit membranes (André 2002) and fills the vessels completely, making them easier to recognize than the traditional paint pigment injection (Sperry et al. 2005). The cured stems were cut at several distances from the injection surface, and the average number of silcone-filled vessels per xylem area was counted. Most vessels were short (Zimmermann & Jeje 1981), so we used an exponential scale to concentrate the vessel counts near the injection surface
where Li is the ith distance counted starting at i = 1 at Lmin, which was the shortest non-zero distance (0.005 m) and ending at i = N for Lmax, which was the longest length counted. The Lmin was constant at 0.005 m, whereas Lmax was varied according the vessel length of the species.
Rather than using the double-difference algorithm (Zimmermann 1983) for calculating vessel lengths we extended the approach originated by Cohen, Bennick & Tyree (2003). An exponential decay function was fitted to the decrease in density of silicone-filled vessels (NL) with distance L from the injection surface:
where N0 is the total vessel density and k the best fit extinction coefficient. The fraction of conduits of length L (PL) is given by L/N0 times the second derivative of Eqn 2
Integrating the product of PL and L over the length distribution (∫PLL dL from L = 0 to infinity) gives the mean vessel length. As the length distributions were short-skewed, the mean of the log-transformed length distribution was used to represent the average-sized vessel (Fig. 1, L). Each species was represented by the grand mean of three to seven stems. Vessel lengths were already available for the five non-Rosaceae species using similar methods (Sperry et al. 2005). (Note: Eqn 3 incorrect in Sperry et al. 2005 as Eqn 2.)
Conduit (RC), lumen (RL), and end wall (RW) resistivities
The RC was measured on stems that were longer than the vast majority of the vessels. Measured stems were 14 cm and longer than 96% of the vessels for all species except Holodiscus dumosus. Open vessels in stem segments will cause RC to be under-estimated, but by less than 7% (even in H. dumosus) assuming the linear relationship between R and open vessel percentage observed in the related study (Sperry et al. 2005). Stems were flushed to remove native embolism and their resistivity measured with the same apparatus that was used for vulnerability curve measurement. The stem was then perfused with basic fuchsin (0.1% w/w) or safranin (0.1% w/w) to mark the functional xylem. The resistivity was multiplied by the stained xylem area to obtain the xylem area-specific resistivity. This was multiplied by the vessel density of the same stem to obtain the RC of the average vessel. To assess the conducting efficiency of vessels across species, RC was also expressed on the basis of the cross-sectional area of a lumen of average diameter as calculated in the following paragraph (Rca; Table 2).
The same stem was sectioned and the lumen resistivity (RL) estimated from vessel diameter measurements. The area of each vessel lumen in a radial sector of functional xylem was measured, and the equivalent circle diameter calculated before summing the Hagen–Poiseuille conductivities to obtain the total conductivity per area of the radial sector. Several sectors per stem (2–4) were measured to obtain an average, and the value expressed as the average xylem area-specific lumen resistivity. This was multiplied by the vessel density to obtain the RL of the average vessel. Previous work indicated the Hagen–Poiseuille value accurately estimates the actual lumen resistivity (Zwieniecki, Melcher & Holbrook 2001a; Sperry et al. 2005). The diameter of the average vessel was calculated from the average RL according to the Hagen–Poiseuille equation.
The average end wall resistivity on a vessel basis (RW) was calculated for each stem as the total resistivity minus the lumen resistivity (RW = RC − RL). The fraction of the total resistivity in the end wall (FW) was FW = RW/RC (Table 2). Each value (RC, RL, RW, FW) was determined for five to six stems per species, with the species being represented by the grand mean of each parameter.
End wall resistance (rW)
According to the assumed vessel geometry (Fig. 1), rW = RWL′. To obtain L′ we estimated the L′/L ratio and multiplied this by the average length L. To get L′/L we measured the ‘length fraction’ (Table 1), which was the length of a vessel in contact with other vessels relative to the total vessel length. One way to estimate this length fraction would be to serially section a vessel from end-to-end at regular length intervals, and note the fraction of the total number of sections that the vessel is not in contact with another one. This would approximate the fraction of the vessel length that was not in contact with adjacent vessels. One minus this quantity is the length fraction. Instead of making serial sections, we assumed that the end walls were randomly distributed longitudinally. Accordingly, a single cross-section across all vessels will contain the equivalent of the serial sections arrayed in one plane. The length fraction was estimated from one cross-section as the ratio of solitary vessels to total vessel groupings, where the latter is the sum of solitary vessels and clusters of contiguous vessels (each cluster counting as 1).
The length fraction was estimated from counts of all vessel groupings in three radial sectors of xylem per stem. Each stem was represented by the mean of the three subsample fractions. The species was represented by the grand mean from the same six stems used in the other measurements. From the length fraction, the ratio L′/L was calculated as: L′/L = 1 − (length fraction)/2 (Table 1). The length fraction is divided by 2 because we assumed that half of the total overlap per vessel was in one end wall. This assumes no lateral contact between adjacent files of vessels, and so our L′ and rW estimates represent maximum values for the vessel.
Area-specific pit resistance (rP)
This was calculated as the product of the average end wall resistance and half of the average pit membrane area per vessel (AP): rP = rWAP/2 (Table 2). Half of the pit area was used because vessel efflux will be through just one end wall, the other half handling the influx through the other end wall (Fig. 1). The rP represented the maximum for the vessel because its calculation assumed all pitting was in the end walls with none on lateral walls.
Determining the pit membrane area per vessel (AP) began with measuring the ‘contact fraction’ which was the portion of the vessel wall area that is in pitted contact with other vessels (Table 1). Similar to the length fraction, the contact fraction could be determined from serial sections of a vessel at regular intervals, measuring in each section the perimeter of the vessel in contact with another as well as the total vessel perimeter. The contact fraction would be the sum of the contact perimeter across all sections over the sum of the total perimeter. Assuming that end-walls are randomly distributed, a single section across all vessels contains the equivalent of the serial sections for all vessels in one plane. The contact fraction was estimated from the ratio of contact perimeter to total perimeter, both being first summed across all vessels in section. The fraction represented an average for all vessel sizes in the section.
This fraction was estimated from perimeter measurements of all vessels in three radial sectors of xylem per stem. Each stem was represented by the mean of the three subsample fractions. The species was represented by the grand mean from the same six stems used in the other measurements.
Not all of the contact area between vessels is occupied by pits. In radial longitudinal sections of a subsample of the six stem segments we measured the total pit area per unit contact area. This ‘pitfield fraction’ (Table 1) was multiplied by the contact area fraction to obtain the average pit area per vessel area fraction (FP). The average vessel area (AV) was estimated from the average diameter and length for the species assuming cylindrical geometry. The total pit area per vessel was: AP = FPAV (Table 2).
Standard methods (Bernard & Epp 1995) were used to propagate standard errors from measured values (parameters RC, RL, RW, L, length fraction, pitfield fraction, contact fraction) to calculated parameters (AP, AV, FP, L′/L, rP, rW, FW, average conduit diameter).
Pit aperture and membrane properties
In six species (Amelanchier alnifolia, A. utahensis, Physocarpus malvaceus, Rubus parvifolius, R. leucodermis, Sorbus scopulina) we further subdivided the total pit resistance into aperture and membrane components based on our previous pit modelling work (Sperry & Hacke 2004). Aperture resistance on a pit area basis was estimated from equation 20 in Sperry & Hacke (2004) using measurements of aperture diameter and thickness. Apertures were elliptical so we calculated their hydraulic diameter (the equivalent circle diameter giving the same Hagen–Poiseuille conductivity) from the long and short axis according to Lewis (1992). Membrane resistance on an area basis was calculated as the total pit resistance (rP) minus the aperture resistance. From equation 17 in Sperry & Hacke (2004) we calculated the ‘equivalent pore diameter’ of the membrane from our measurement of the area-specific membrane resistance. The area-specific membrane resistance times the pore density gives the average resistance per membrane pore, the diameter of which is the equivalent pore diameter. Pore density was based on the membrane being composed of multiple sheets of parallel microfibrils superimposed as detailed in Sperry & Hacke (2004).
The xylem resistivity on a conduit area basis (RCa) increased significantly with increasing cavitation resistance over a P50 range from 0.76 to 6.66 MPa (Fig. 2) consistent with a safety versus efficiency trade-off at the whole conduit level. This trend was significant within the 11 Rosaceae species (r2 = 0.36) and reinforced by the five additional species (r2 = 0.51). The trend was also significant for vessel resistivity expressed on a total xylem area basis (not shown).
The increase in resistivity with P50 was nearly equally divided between lumen (RL) and end wall (RW) components (Fig. 3). The average wall resistivity fraction (FW; see Table 2 for summary of symbols) was 53 ± 3%, indicating that lumen and wall were approximately co-limiting. There was no significant trend in FW with P50, nor any difference between the average FW for Rosaceae versus other species. This result means that any safety versus efficiency trade-off involved similar increases in both the end wall and lumen resistance.
The increase in RL with P50 resulted from a decline in average conduit diameter with P50(Fig. 4a; symbols). What caused the increase in RW with P50? The RW is the wall resistance (rW) divided by the length L′ (Fig. 1), so its increase could result from shorter length and/or higher rW with P50. There was no significant relationship between either L′/L (Table 1) or total vessel length (L) and P50 (Fig. 4b; symbols). There was, however, a significant increase in rW with P50(Fig. 5).
The increase in rW could result from higher area-specific pit resistance (rP) and/or lower pit area per vessel (AP). We had hypothesized that rP should increase with P50 because of the less porous pit membranes required to suppress cavitation by air-seeding. Although there was considerable variation in rP, contrary to our expectations, a regression against P50 showed no relationship (see Table 1 for rP values). The rP was also just as variable in the Rosaceae as across the diversity of other families (Table 1). Instead, the increase in rW was a result of dramatically less pit area per vessel with increased P50(Fig. 6; solid regression). Pit surface area per vessel (AP) declined in inverse proportion to increasing P50 both within the 11 Rosaceae (r2 = 0.62) and for all species combined (r2 = 0.75).
Total vessel surface area (AV) also declined with increasing P50, but not as steeply as pit area (Fig. 6; dashed regression). This was because the pit fraction (FP; pit area per vessel area) decreased significantly with increasing cavitation resistance (Fig. 7). All pit fractions were very low, the highest being 15% declining to less than 2% for the most cavitation-resistant species. The low pit fractions were mostly a result of limited overlap between vessels (i.e. low contact fractions in Table 1) rather than sparse pitting at areas of overlap (i.e. relatively high pitfield fractions in Table 1).
The area-specific pit resistances were much higher than estimated from our pit model (Sperry & Hacke 2004). The model predicted a range from 0.2 to 20 MPa s m−1 (Fig. 9 in Sperry & Hacke 2004) whereas measured values averaged 167 MPa s m−1 excluding an extreme value for Amelanchier alnifolia (Table 1). Although the rP estimates were the maximum in the absence of any lateral pitting and the error in its determination was relatively high because of the number of calculations required (average standard error was 29% of the mean, Table 1), it is unlikely that these factors accounted for the order-of-magnitude discrepancy. Aperture resistance accounted for 1% or less of the total pit resistance in the six species subsampled, indicating the model was under-estimating the membrane resistance. Membranes were much less porous than the model predicted. Equivalent pore diameters based on the measured membrane resistance ranged from 3.5 to 7.6 nm (mean = 5.3 nm). In contrast, the pore diameter expected to air-seed at the P50 values of these species ranged from 44 to 383 nm from the capillary equation (surface tension = 0.073 Pa m).
A proposed basis for a safety versus efficiency trade-off
The results did not support the expected trade-off between area-specific pit resistance and P50. Although our measurement of rP was an indirect and maximum estimate with inevitably high error (Table 1), the pit model predicted a 10-fold increase over the 6 MPa range in P50 in our study species (Sperry & Hacke 2004) whereas we observed no P50-related variation. Apparently the majority of membrane pores were of similar size in all species and this size was not correlated with the largest pore causing air-seeding of cavitation. This was the same conclusion drawn by Choat and others from experimental estimates of membrane porosity (Choat et al. 2003, 2004). Either the air-seeding is caused by yielding or rupture of the pit membrane, or the largest pore that causes the air-seeding is relatively rare and not related to the prevailing porosity that determines membrane resistance.
In place of our rejected hypothesis the results suggest an alternative. The decline in total pit area per vessel with increasing P50 (Fig. 6) suggests that it is the total area of pits in a vessel rather than individual pit structure that is most important in setting a vessel's cavitation resistance and transport efficiency. If the average size of the rare largest pore per vessel increases with total pit area per vessel, the capillary equation predicts an inverse relationship between total pit area and the average air-seeding pressure. Although the probability of mechanical failure might also increase with cumulative pit area, it is not clear why this should be an inverse relationship as was observed (Fig. 6). The most parsimonious explanation, and one offered previously by Choat et al. (2003, 2004) and Hargrave et al. (1994), is that safety from cavitation is determined by the rare largest membrane pore per vessel. We add the assumption that the average size of the largest pore per vessel increases with increasing total pit membrane area per vessel.
There is a basis for this assumption if there is a theoretical pore size distribution that is essentially common to all species with a homogenous type of pit membrane. Vessels with small total pit area represent this distribution with a small sample size. The average pore distribution for a small sample size will have a similar mean pore diameter and rP as the theoretical distribution, but much smaller maximum pore size on average, and hence much greater average cavitation pressure (and P50) according to the capillary equation. Vessels with increasingly larger total pit area sample more of the theoretical distribution, and while the mean pore size and rP stays the same, the average maximum pore size will increase leading to lower average cavitation pressure (and P50). Adding to this inherent vulnerability of vessels with large pit area is the fact that these vessels should also have a greater chance of contacting air-filled vessels.
According to this pore size scenario, species with the highest P50 values and hence the very smallest pit areas will have an average maximum pore size closer to the theoretical mean. The highest P50 in an angiosperm that we know of is 11.3 MPa in the desert shrub Larrea tridentata (Pockman & Sperry 2000). This corresponds to an average maximum pore diameter per vessel of 26 nm, which is approaching the 3–7 nm range predicted for the equivalent pore diameters and the range measured experimentally by others. Conversely, the lowest P50 in our data set was 0.76 MPa in Vitis vinifera, which also had the largest pit area per vessel (Fig. 5). This pressure corresponds to an average maximum pore size per vessel of 383 nm, which should be approaching the maximum of the theoretical pore distribution.
A causal relationship between pit area and cavitation pressure provides a relatively simple explanation for many of the results, beginning with the observation that wall and lumen resistivities are roughly co-limiting (Fig. 3). If a given level of safety from cavitation dictates a limited total pit area per vessel, and if the pit area fraction (FP) is held constant, the total wall area of the vessel is also limited. If the total wall area of the vessel is limited, the product of length and diameter must be constant. Such a vessel can be long and narrow, or short and wide, but its surface area must be preserved. Solving for the wall resistivity fraction that minimizes the conduit area-specific resistivity (RCa) under this surface area constraint (Appendix) yields an optimal fraction of FW = 0.40 (Fig. 8; conductivities = 1/RCa shown for illustrative purposes). The average wall fraction was higher at 0.53, but the breadth of the optimum is such that the species still achieved an average of 94 ± 2% of their maximum conductivity per conduit area (Fig. 8; symbols on curve).
The proposed trade-off also explains the rather low pit fractions (FP) and their decrease with P50 (Fig. 7). A low FP means a larger vessel for the same limiting total pit area. A larger vessel means a lower RCa (with the minimum still at FW = 0.4). The decline in FP with increasing P50 prevents total vessel size from decreasing as much as total pit area (Fig. 6) and keeps the conduit resistivity from increasing with P50 as much as it would if FP were constant.
Most importantly, a pit area versus cavitation linkage explains the tendency for conduit resistivity to increase with P50 (Fig. 2). Assuming the inverse relationship between total pit area and P50 (Fig. 6), incorporating the observed relationship between pit fraction (FP) with P50 (Fig. 7), and using average values for area-specific pit resistance (rP = 167 MPa s m−1), wall fraction (Fw = 0.53), and L′/L (0.82), the total conduit resistivity increases with P50 much as observed (Fig. 9; ‘observed FP trend’ line versus data regression, see Appendix for calculations). The remaining discrepancies are owing to variation in the actual values of FP, FW, L′/L, and rP around their average. The compensating effect of decreased pit fraction (FP) in more resistant species is demonstrated by holding this value constant at its maximum (0.15) and observing how much more steeply the conduit resistivity increases with P50 (Fig. 9; ‘constant FP’ line). Theoretically, a reduction in FP with increasing P50 can completely eliminate the safety versus efficiency trade-off, but this requires FP to drop drastically to below 0.05%. The disadvantage of extremely low pit fractions would be much higher lateral flow resistance (Orians et al. 2004), a problem that probably limits how much FP can drop and still benefit the plant.
The trade-off is also consistent with vessel diameter being more sensitive to P50 than length (Fig. 4). The solid curves in Fig. 4 are the diameter and length of the vessels having resistivities plotted in Fig. 9 assuming the observed trend in pit fraction and setting other parameters to their averages (Fig. 9; ‘observed FP trend’ line, Appendix). Diameter drops by almost three-fold over the P50 range whereas length drops by just over 1.5-fold.
Finally, the variable relationship between diameter, length and overall resistivity with P50 (Figs 2 & 4) is accommodated by the trade-off hypothesis. The variation is primarily a result of variation in FP, FW, L′/L, and rP– none of which are mechanistically linked to P50, which is set only by total pit area. The greater scatter in the insignificant length versus P50 data (Fig. 4b) versus the significant diameter versus P50 relationship (Fig. 4a) is because length is more sensitive to variation in FP than diameter (Appendix). As a result of these interactions, there was no prevailing allometry between length and diameter across the 16 species as opposed to our earlier more limited survey that suggested that length scaled with the square of the diameter (Sperry et al. 2005). In summary, a single P50 can be associated with a number of different vessel sizes and resistivities depending on variation in FP, FW, L′/L, and rP. Nevertheless, the need to reduce pit area per vessel to achieve greater safety from cavitation is proposed to drive the general trend toward less efficient xylem transport seen over broad ranges in P50.
The safety versus efficiency trade-off that we propose is quite different from what we expected. We had thought that plants acquired greater safety from cavitation by making denser pit membranes that were less porous on average and so with higher flow resistance. Instead, the results show that average pit membrane porosity and flow resistance does not vary systematically with cavitation resistance in the study species. These plants may be achieving greater safety by reducing the pit area per conduit and thus by chance reducing the average size of the largest pore per conduit, which sets the air-seed pressure. Although the plant cannot control the cavitation pressure of a single conduit in this way, it can control the average response of the typically large population of conduits on which it depends. The trade-off with decreased transport efficiency results from the higher resistance of having less total pit area, and from the fact that a limited total pit area ultimately tends to limit the total size of the conduit.
The relatively small size of the majority of pores appears to be controlled by the hemicelluloses and pectins and other matrix components that constitute the ‘filler’ between the strength-providing cellulose microfibrils of the membrane (Dickson 2000; Choat et al. 2003). Because of this filler, membrane strength is not necessarily coupled to porosity as assumed in our model of homogenous pit membranes. The large variation in area-specific pit resistance that we measured (Table 1) and that was nevertheless not related to cavitation resistance may result from variation in the state of the matrix filler between species. Variation in xylem conductivity with ionic strength of the sap has been attributed to change in hydration of the matrix and hence pore size (Zimmermann 1983; Zwieniecki et al. 2001b). Based on limited observations with one species (Acer negundo), sap ionic strength does not alter the P50 (V. Stiller, J. Sperry, unpubl. data). Perhaps the treatment affects the majority of small pores more than the largest pore, leaving cavitation pressure unaltered.
The tendency for lumen and end wall resistivity to be co-limiting in plants (Fig. 3) was difficult to account for under our original hypothesis because it put no limit on vessel size with P50. Large, safe vessels with large total pit area would be possible if the pit membrane porosity was as tightly controlled as we had originally assumed. Such vessels could be sufficiently long to reduce wall resistivity to a small percentage (Sperry & Hacke 2004; Sperry et al. 2005). The new trade-off hypothesis does limit vessel size to within the bounds of the minimum permissible pit area per vessel area fraction (FP). A limit on vessel size results in the minimum resistivity being a compromise between reducing lumen resistivity with wide but short vessels versus reducing wall resistivity with long but narrow ones (Fig. 8).
The hydraulic trade-off we propose here is reinforced by a mechanical trade-off. The mechanical problem of achieving greater safety from cavitation is that it requires a thicker wall for a given diameter to withstand compressive forces and implosion. A greater wall thickness per diameter can translate to higher wood density, particularly where conduits make up a significant portion of the wood volume (Hacke et al. 2001). Higher density represents greater cost of wood growth per volume. A thicker wall per diameter also increases the hydraulic resistivity on an area basis even if there is no change in lumen diameter. A limitation on maximum wall thickness can potentially limit maximum diameter independently of purely hydraulic considerations.
Trade-offs result in a strong selective advantage to mechanisms that at least partially circumvent or resolve the conflict. We have already drawn attention to the advantage of minimizing the pit area per vessel area fraction (FP), particularly as the P50 is increased in safer xylem (Figs 7 and 9). Variation in FP provides an explanation for why there is no 1 : 1 relationship between vessel size and vulnerability – big vessels can be safe from cavitation, but only by reducing their FP to keep their pit area small. There are other candidate mechanisms. The pit area that is presumably influencing the cavitation pressure is the pit area responsible for the capillary sealing of the pit membrane. Any mechanism that reduces the actual sealing area of the membrane relative to the total conducting area will mitigate the trade-off. Pit aspiration is a simple mechanism – if capillary forces are sufficient to displace the membrane against the chamber wall, only the part of the membrane spanning the aperture is responsible for bearing the full brunt of the air–water pressure difference in a sealed pit. Vestured pits could work in the same way – water could be flowing through the entire membrane, but in the sealing position much of the membrane would be deflected against the vestures, reducing the area involved in sealing off air.
Tighter control over membrane pore size and distribution would have a tremendous advantage, as we had initially hypothesized, because total pit area could become less tightly coupled with vulnerability to cavitation. Although we did not see evidence for this in our limited sample, not all angiosperm pit membranes need be the same. Although the membrane porosity studies cited above (Choat et al. 2003, 2004) show no relationship with air-seeding pressure, similar methods applied to different species have shown correspondence between membrane pore size and cavitation (Jarbeau, Ewers & Davis 1995). Certainly some angiosperm pits are quite different in having a torus that may function like a conifer pit (Wheeler 1983; Jansen et al. 2004). In conifers, the dense torus seals the pit, allowing the pores of the surrounding margo to be quite large and conductive without sacrificing safety. Air-seeding occurs by slippage of the torus, and in theory is a function of individual pit membrane strength and structure (Hacke, Sperry & Pittermann 2004) rather than total pit area. Given the potential diversity in pit function it is unlikely that there will be a single structural basis for cavitation pressure across vascular plants.
Funding was provided from National Science Foundation grants IBN-0112213 and IBN-0416297. The authors thank two anonymous reviewers whose thoughtful comments helped to improve the manuscript.
Calculation of end wall and pit resistances
Assuming the vessel geometry diagrammed in Fig. 1 with lumen and wall components in series, vessel RC = RL + RW. In terms of vessel diameter and length (D, L)
RC= 128 η/(πD4) +rW/(LFL)((A1))
where η is viscosity, rW is the resistance of one end-wall, and FL is the ratio L′/L (Fig. 1). By measuring RC, D, L, and FL for the average-sized vessel as described in Materials and Methods we solved Eqn A1 for rW. The pit resistance on a pit membrane area basis (rP) was calculated as
where AP is the total pit membrane area per vessel. This assumes all pit area is located on vessel end walls as opposed to side walls, and that one end wall contains half of the total pit area. The AP was calculated from
where FP is the fraction of the vessel surface area occupied by pit membranes measured from wood sections as described.
Vessel resistivity (RC) as a function of AP
The proposed safety versus efficiency trade-off assumes that P50 is determined by AP (Fig. 6). The ramifications for RC can be explored by expressing Eqn A1 in terms of AP rather than D and L. Efficiency is best expressed as RC on a cross-sectional area basis (RCa), so each resistivity in Eqn A1 was multiplied by the lumen cross-sectional area. To eliminate D in this version of Eqn A1 we solved Eqn A3 for D[D = AP/(FPπL)] and substituted. This gives
To eliminate L we make use of the wall fraction (FW) in terms of RL and RW
This equation is graphed with respect to FW in Fig. 8, showing that the minimum RCa(maximum conductivity per area) occurs at FW = 0.40 when other parameters are constant. Figure 9 shows RCa calculated from Eqn A8 using mean values of rP, FL, FW and assuming AP varies with P50 as in Fig. 6. The FP was assumed either to vary with P50 as in Fig. 7 (Fig. 9, observed FP trend), or to remain constant (Fig. 9, constant FP).
A final equation gives D as a function of AP by substituting Eqn A7 for L into Eqn A3 and solving for D
The curves in Fig. 4 show that for the observed decline in AP and FP with P50, D decreases more than L (calculated from Eqns. A7 and A9 using mean values for the other parameters). Length is more sensitive to variation in FP than D (compare Eqns. A7 and A9), consistent with the greater scatter in the L versus P50 relationship in Fig. 4b.