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Analysis of circular bordered pit function I. Angiosperm vessels with homogenous pit membranes¹

Biology Department, University of Utah, 257 South 1400 East, Salt Lake City, Utah 84112 USA

Received for publication July 1, 2003. Accepted for publication October 21, 2003.

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS MODEL DESCRIPTION RESULTS DISCUSSION LITERATURE CITED

 
A model predicted pit and vessel conductivity, the air-seed pressure for cavitation, and the implosion pressure causing vessel collapse. Predictions were based on measurements from 27 angiosperm species with circular bordered pits and air-seed pressures of 0.2–11.3 MPa. Vessel implosion pressure exceeded air-seed pressure by a safety factor of 1.8 achieved by the increase in vessel wall thickness per vessel diameter with air-seed pressure. Intervessel pitting reduced the implosion pressure by 20 to 40%. Pit hydraulic conductivity decreased by 30-fold from low (<1 MPa) to high (>10 MPa) air-seed pressure primarily because of decreasing pit membrane conductivity. Vessel conductivity (per length and wall area) increased with vessel length as higher lumen conductivity overcame low pit conductivity. At the "saturating vessel length," vessel conductivity maximized at the Hagen-Poiseuille value for the lumen per wall area. Saturated vessel conductivity declined by sixfold with increasing air-seed pressure because of increased wall thickness associated with increased implosion resistance. The saturated vessel length is likely the optimal length because: (a) shorter vessels have lower conductivities, (b) longer vessels do not increase conductivity when functional yet decrease it more when cavitated, (c) observed pit structure most closely optimized vessel conductivity at the saturated length, and (d) saturated lengths were similar to measured lengths.

Key Words: functional wood anatomy • hydraulic architecture • plant biomechanics • plant water transport • xylem cavitation • xylem hydraulic conductivity

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS MODEL DESCRIPTION RESULTS DISCUSSION LITERATURE CITED

 
Xylem conduit walls must perform three important functions: (1) permit water flow between adjacent conduits, (2) prevent air entry from embolized (gas-filled) conduits to adjacent water-filled functional ones, and (3) prevent implosion under the significant negative pressure of the transpiration stream (Zimmermann, 1983 ; Carlquist, 1988 ). These functions are fulfilled by a thick lignified wall for strength that is punctuated with thin areas (pits) to allow water flow (Fig. 1). The pits lack thick secondary wall layers, and the thin compound middle lamella of adjacent primary cell walls is modified to form a relatively porous pit membrane. The surrounding secondary wall arches over the pit membrane, forming a pit border with an aperture opening to the inner pit chamber (Fig. 1C, D). The first function is in direct conflict with the other two because the wall features that prevent air entry (strong pit membranes with narrow pores) and implosion (thick, lignified wall layers with few pits) also inhibit water flow.

View larger version (56K): [in this window] [in a new window]   Fig. 1. Relevant features of angiosperm vessel wall structure. (A) Transverse view of two adjacent conduits and pitted wall. Negative pressure in water-filled conduits induces circumferential hoop stresses in the conduit wall. Larger bending stresses are induced when the adjacent conduit is air-filled and the common wall bends towards the water-filled side. b = width of the pitted common wall. (B) Face view of pitted wall. Da = aperture diameter; Dm = membrane diameter; s = spacing between pits; Le = ligament efficiency (ratio of dimensions l/l'). (C) Transverse view of pitted wall showing pit aperture, border, and membrane. ry = radius of curvature of membrane deflected distance y from flat position; rya = radius of curvature of inner chamber wall; tw = double wall thickness; ta = thickness of single aperture. (D) Median transverse section through a single pit: y = membrane deflection from flat position; ya = membrane deflection for radius of curvature = rya; yl = distance from flat membrane to inner edge of aperture; = angle of membrane deflection from flat position.

Intuitively, a pit that is more resistant to air-seeding should also be less conductive to water. If so, this could be one reason for the observed correlation between a species' air-seed pressure and the severity of water stress it must endure (Davis et al., 1999 ; Hacke et al., 2000 ; Pockman and Sperry, 2000 ). The xylem should be no more resistant to air-seeding than it has to be if in doing so it unnecessarily sacrifices hydraulic conductivity. The variation in pit shape and membrane structure may be adaptive in optimally balancing the conflict between conductivity and safety from air-seeding.

Perhaps the most striking difference between pit types is the contrast between the typical "homogenous" pit membrane of wide phylogenetic distribution that is uniformly thin and microporous vs. the torus-margo structure of many gymnosperm tracheids. The two membrane types represent different solutions to the same problem; but it seems likely that they may have quite different hydraulic conductivities for the same air-seed pressure (Lancashire and Ennos, 2002 ).

In addition to being influenced by pit structure, the hydraulic conductivity of a conduit is also determined by the width and length of the conduit lumen. According to the Hagen-Poiseuille equation, hydraulic conductivity of the lumen should increase with the fourth power of the conduit width. This gain in conductivity can only be realized, however, if the lumen is sufficiently long so that the pit conductivity is not limiting. The work of Gibson and colleagues (Calkin et al., 1986 ; Schulte et al., 1987 ; Schulte and Gibson, 1988 ) has shown that tracheid length must increase with tracheid diameter if there is to be a net gain in total tracheid conductivity, and this is presumably why tracheid diameter and length are correlated. The same principle applies for vessels where length and diameter are also positively related (Ewers and Fisher, 1989 ).

Adjustments in conduit diameter and length may compensate for the changes in pit conductivity associated with different air-seed pressures. Thus, a species capable of avoiding cavitation by air-seeding and necessarily having low-conductivity pits does not necessarily have to have a low overall conductivity if the conduit width and length increase to overcome the added pit resistance. The interaction between pit and lumen conductivity may underlie the variable relationship between cavitation resistance and xylem conductivity, with some researchers showing a trade-off of varying significance and others showing no relationship at all (Sperry and Saliendra, 1994 ; Tyree et al., 1994 ; Davis et al., 1998 ; Pockman and Sperry, 2000 ; Hacke and Sperry, 2001 ; Martinez-Vilalta et al., 2002 ).

In this series of two papers, we present a model that predicts the three functions of pitted conduits—hydraulic conductivity, air-entry pressure (= "air-seed pressure"), and mechanical strength—from conduit structure. For simplicity, we confine ourselves to species with circular bordered pits. The model extends Petty's mechanical analysis of pit membranes (Petty, 1972 ), our previous study of conduit wall strength against implosion (Hacke et al., 2001a ), and several previous studies of tracheid conductivity (Calkin et al., 1986 ; Schulte and Gibson, 1988 ; Lancashire and Ennos, 2002 ) to account for all three conduit functions with a relatively simple set of calculations.

In this first paper, we describe the model in detail and apply it to the uniformly thin and homogenous pit membrane structure typical of intervessel pits of angiosperms. We compare the model to data from root and stem xylem of 27 angiosperm species with measured air-seed pressures ranging from 0.2 to 11.3 MPa, hence, reflecting the broad range of negative pressure exhibited by vascular plants (Table 1). We evaluated the trade-offs between the three functions and considered the extent to which vessel structure has optimized conductivity per investment in conduit structure and per air-seed pressure.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS MODEL DESCRIPTION RESULTS DISCUSSION LITERATURE CITED

 
The data set and air-seed pressure (P_a)
The data set was largely compiled from previous studies, including a comparative analysis of cavitation resistance and wood density (Hacke et al., 2001a ). The most important criterion was to have a wide range of cavitation resistance for comparison with conduit structure. The cavitation resistance of each xylem sample in the data set is represented by the "air-seed pressure (P_a)" (Table 1), which is the pressure difference (a positive value) required to force air into the xylem conduits and cause cavitation. It is equal and opposite to the negative sap pressure at cavitation. It was obtained from vulnerability curves measured with the centrifugal force method (Pockman et al., 1995 ) on n 6 xylem samples per species and organ (usually stems or roots). Segments 14 cm long were flushed at 100 kPa with deionized, filtered (0.2 µm) water to remove any xylem embolism. The maximum hydraulic conductivity of each segment was subsequently measured with a conductivity apparatus (Sperry et al., 1988 ). Then, segments were spun in a centrifuge to a known xylem pressure (Pockman et al., 1995 ). After spinning, the loss of hydraulic conductivity was measured and plotted vs. the corresponding xylem pressure to obtain vulnerability curves. Shoots were 5–13 mm in diameter. Root diameter ranged from 3 to 13 mm. The air-seed pressure of a xylem sample was represented by the pressure required to cause a 50% loss of hydraulic conductivity by cavitation (the absolute value of P₅₀ sensu Hacke et al., 2001a ). By including species from numerous diverse habitats, we were able to obtain a wide range of P_a from a minimum of 0.2 MPa in Alnus incana roots to a maximum of 11.3 MPa in Larrea tridentata stems (Table 1).

It was important to sample broadly across phylogenetic lineages, growth form, and organ to include these potential sources of variation in structure that might be independent of variation in cavitation resistance. The 27 species are in 12 families of widespread phylogenetic affinity, including herbs, woody shrubs, and trees (Table 1). Stem and root measurements were available, as well as leaf data from one species (Oryza sativa). Roots tend to be more vulnerable to cavitation than stems (Sperry and Saliendra, 1994 ; Mencuccini and Comstock, 1997 ; Kolb and Sperry, 1999a ; Hacke et al., 2000 ). This was also seen in the present data set where root P_a averaged 1.9 MPa less than stem P_a for the four species with data for both organs.

Anatomical measurements
Pit and conduit dimensions (Fig. 1) were measured on the same or similar xylem samples used to determine the P_a. Measured parameters included the hydraulic mean conduit diameter: D_c = d_c⁵/d_c⁴, where d_c = individual conduit diameter (Kolb and Sperry, 1999b ). The D_c represents the size of a conduit cavitating at P_a under the following conditions. (1) A normal distribution of hydraulic conductivity vs. d_c for a xylem sample. This means that 50% of the hydraulic conductivity will occur in conduits greater than D_c and 50% will occur in conduits narrower than D_c. (2) Wider conduits have lower air-seed pressures than narrower ones within a xylem sample. Thus, the pressure causing a 50% loss of hydraulic conductivity (the P_a) will be the cavitation pressure of a vessel of diameter D_c. Both conditions are consistent with observations (Salleo and Lo Gullo, 1989 ; Lo Gullo and Salleo, 1993 ; Hargrave et al., 1994 ).

Conduit dimensions were measured in transverse freehand sections stained with phloroglucinol-HCl. To determine the D_c, lumen area was measured for all vessels in radial sectors of recent growth rings from each xylem sample used to determine the P_a. The d_c was calculated as the diameter of a circle with the same lumen area, and D_c was calculated for each xylem sample before the mean D_c was obtained for all samples. The thickness of intervessel walls (t_w; Fig. 1C) was measured only for conduit pairs that averaged within ±3 µm of D_c. The width of the common wall (b, Fig. 1A) was assumed equal to D_c. This proved to be a better approximation for vessels than using the side of a square of equal area to the conduit lumen as used previously (Hacke et al., 2001a ).

Pit dimensions were measured in longitudinal freehand sections stained with phloroglucinol-HCl. All species had circular bordered pits. Photographs were taken with a digital camera attached to a light microscope under oil immersion at 1000x magnification. The diameter of the pit membrane (D_m) and pit aperture (D_a; Fig. 1B) was measured with standard image analysis software on n 10 pits per plant organ and species. For irregular pit aperture shapes, the aperture diameter was calculated as the diameter of a circle with an area equal to the aperture area. Definitions of symbols are provided in Table 2.

	MODEL DESCRIPTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS MODEL DESCRIPTION RESULTS DISCUSSION LITERATURE CITED

Conduit implosion pressure (P_i)
The negative water pressure in functional conduits pulls inward on the conduit wall, creating mechanical stress. If this stress exceeds the wall strength, the wall will buckle inwards, or implode. The largest wall stress occurs when a water-filled conduit abuts an air-filled one, causing the common wall to bend towards the water-filled side (Fig. 1A). These bending stresses are greater than the circumferential hoop stresses girdling the entire wall (Hacke et al., 2001a ) and are the focus of the implosion analysis. The implosion pressure (P_i) was defined as the pressure difference between water and air on either side of the wall (a positive number) that was necessary to cause the bending stress to exceed the wall strength. While the implosion pressure can exceed the air-seed pressure, the reverse seems unlikely in lignified conduits because as the wall implodes it should trigger air-seeding and cavitation—thus eliminating the pressure difference. If so, conduit walls should exhibit a P_i/P_a ratio (the implosion safety factor) of one or more.

The implosion pressure was estimated from conduit dimensions using standard mechanical engineering equations (Young, 1989 ). These equations assume that the solid cell wall material has the same mechanical properties in all directions (isotropic) and that the wall structure conforms to relatively simple geometries for which there are analytical approximations for maximum stresses and strains. Neither condition is completely true but it is appropriate to start with a simple approach.

We build on an earlier analysis of implosion pressure (P_i) that ignored pits and estimated wall stress, assuming the wall is a flat solid plate of finite width b and effectively infinite length (Young, 1989 ; Hacke et al., 2001a ):

(1)

where W is wall strength, ß is a coefficient that depends on the width-to-length ratio of the wall (=0.25 for a ratio of 0.5 or less), and t_w is the thickness of the double wall. Here we add the weakening effect of pits based on the analysis by O'Donnell and Langer (1962) on the bending stresses of perforated plates. Regularly spaced perforations increase the stress in a solid plate in inverse proportion to their "ligament efficiency" (L_e). The ligament efficiency is the minimum distance between the edges of the holes divided by the distance between the hole centers (Fig. 1B). In terms of the circular bordered pit,

(2)

where s is the minimum horizontal distance between pit edges (Fig. 1B). P_i becomes

(3)

Equation 3 neglects the presence of a pit chamber. The bulk of the chamber volume is located near the neutral plane of bending (the pit membrane), and it should not weaken the wall much compared to a solid wall. Nevertheless, to account for its effect, we calculated the ratio of the second moment of area (I) for a median section through a circular bordered pit with (I_h) and without (I_s) a chamber present. Equation 3 assumes that P_i WI_s, so multiplying the right-hand side by the (I_h/I_s) ratio corrects for the presence of a chamber. The irregular geometry of the pit section required a numerical calculation of I. The I_h/I_s ratio was calculated for the median section where it would be smallest and represents a liberal estimate of the weakening effect of a chamber. The complete calculation of P_i was:

(4)

We used Eq. 4 to calculate the P_i from measured values of L_e, the "thickness-to-span ratio" (t_w/b)², and estimates of the minimum "moment ratio" (I_h/I_s).

Air-seed pressure (P_a)
Two modes of air-seeding were considered (Fig. 2). "Capillary-seeding" occurs by the failure of the air–water meniscus in a preexisting pit membrane pore (Fig. 2A). It occurs at a pressure difference, P_c, sufficient to overcome the capillary force of the meniscus (Pickard, 1981 ; Zimmermann, 1983 ).

View larger version (27K): [in this window] [in a new window]   Fig. 2. Modes of air-seeding in angiosperm vessels. Air-seeding occurs when the pressure difference (Pa) between air (left) and water (right) across the pit membrane is high enough to force air through to nucleate cavitation. (A) Air-seeding by capillary failure ("capillary-seeding") occurs when the air-water meniscus fails at preexisting pores in the pit membrane. The Pa thus equals the pressure required to displace such a meniscus (Pc). (B) Air-seeding by membrane rupture ("rupture-seeding") occurs when new holes open up in the membrane owing to its structural failure. The Pa equals the pressure causing the membrane to fail (Pr)

View larger version (20K): [in this window] [in a new window]   Fig. 3. Pressure difference across pit membrane vs. membrane deflection (y) illustrating how the model computed air-seed pressure. Heavy Py curve shows pressure required to displace membrane of ns = 100 spokes by distance y from its flat position. Solid portion of curve is before membrane aspiration, dashed portion of curve is after aspiration. Dash-dotted lines show the capillary-seed pressure (Pc), rupture-seeding pressure (Pr), and air-seed pressure (Pa) for the ns = 100 pit. This pit air-seeded by capillary failure (Pa = 1.4 MPa) because Py reached Pc before Pr. The light Py curve is for a membrane of ns = 60 spokes; the fewer the spokes the lower the air-seed pressure (Pa = 0.8 MPa). A given air-seed pressure was achieved in the model by iterating the spoke number until the calculated Pa reached the desired value

(5)

At equilibrium, this acting force is balanced by a re-acting tensile force in the membrane. Petty assumed that the most important load-bearing elements in the membrane are the radial "spokes" of microfibrils running across the membrane center (Fig. 4). The total resisting force is then the sum of the tensile force vector in each radial spoke that is directed normal to the membrane:

(6)

where n_s is the number of radial spokes, T is the tensile force in each spoke, and is the angle of membrane deflection from the flat position (Fig. 1D). The tensile force is in turn equal to:

(7)

where E is the modulus of elasticity of the spokes, e is their strain, and A_f is their cross-sectional area (we used A_f = 707 nm² from Petty, 1972 ). As reported in the results section, we performed a sensitivity analysis to arrive at provisional settings for E and the microfibril strength, F.

View larger version (35K): [in this window] [in a new window]   Fig. 4. Representation of pit membrane. (A) A single sheet of parallel microfibrils, each spaced distance sf apart. One strand from each sheet formed a pair of radial spokes (numbered heavy line) assumed to bear the load on the stressed membrane. (B) A membrane composed of six sheets and ns = 12 radial spokes (numbered heavy lines). Dp = maximum pore diameter; Dpe = equivalent pore diameter giving same membrane conductivity if all pores were of equal size; Dm = membrane diameter; sf = spacing between strands of a single sheet

(8)

where the subscript "y" denotes a value at displacement y, and r is the radius of curvature of the membrane (Fig. 1C). This equation makes the simplifying approximation that the membrane curvature is spherical, allowing D_m/2r_y to be substituted for sin . Continuing with the assumption of circular membrane curvature, r_y equates to

(9)

and e_y is

(10)

where _y = Acos[(r_y – y)/r_y]. By incrementing y and solving for P_y, a pressure vs. displacement relationship can be found (Fig. 3, solid P_y line). When the membrane reached the pit chamber wall, aspiration occurs (Fig. 3, arrow). The y at aspiration (y_a, Fig. 1D) was calculated from a membrane strain at aspiration e_a = 0.03, as estimated from the measurement of several published micrographs of circular bordered pits of different sizes (Panshin and de Zeeuw, 1970 ; Siau, 1971 ; Bauch et al., 1972 ; Core et al., 1979 ; Siau et al., 1984 ). This calculation assumed a constant radius of curvature of the inner chamber wall of r_ya (Fig. 1C), which was solved numerically from the equality (derived from Eqs. 9 and 10):

(11)

For deflection beyond aspiration (Fig. 3, heavy dashed line), Eq. 8 was modified to account for the deflection of the membrane through the pit aperture:

(12)

where P_asp is the pressure causing aspiration, D_a is the aperture diameter, r is the radius of curvature of the membrane deflecting through the aperture, and e is the additional strain caused by deflection through the aperture. This strain was calculated assuming that the stretch across the aperture after aspiration was distributed throughout the entire length of the microfibril spokes.

Equations 8 and 12 were used to generate a P_y vs. y relationship for the pit membrane (Fig. 3). This curve allowed us to predict the air-seed pressure based on which limiting pressure came first during membrane deflection: the rupture-seed pressure (P_r) or the capillary-seed pressure (P_c). The P_r was equal to the P_y where spoke stress (T/A_f) equaled the spoke strength (F). In the example in Fig. 3, P_r is equal to 1.8 MPa (Fig. 3, dash-dotted P_r line). The P_c was given by the capillary equation:

(13)

where is the surface tension of water, a is the contact angle between meniscus and wall, and D_p^' is the stretched diameter of the membrane pores. The stretching of the pores during membrane deflection caused a decrease in P_c with increasing deflection y (Fig. 3, dash-dotted P_c curve). For the pit membrane in Fig. 3, the P_a was 1.42 MPa (dash-dotted P_a line), and air-seeding occurred by capillary seeding because the deflection pressure reached P_c before P_r.

Membrane structure
To be able to predict D_p^' and also the membrane hydraulic conductivity, we needed to link the number of load-bearing microfibril spokes to membrane porosity. This required making several assumptions about membrane structure (Fig. 4): (a) the membrane was made up of several sheets of microfibrils superimposed on one another (Fig. 4B); (b) each sheet consisted of parallel microfibrils spaced a constant distance s_f apart (Fig. 4A); (c) one fibril of each sheet ran across the center of the membrane, forming a pair of microfibril spokes (oriented 180° from each other), so that n_s = twice the number of sheets composing the membrane (Fig. 4A); (d) the angle between adjacent radial microfibrils was 180° divided by the number of sheets in the membrane, or 360°/n_s (Fig. 4B).

These assumptions at least qualitatively reflect cell wall development, given that microfibrils are laid down in multiple layers, and these layers can be composed of roughly parallel microfibrils, and successive layers can cross each other at various angles (Carpita and Gibeaut, 1993 ). Beyond this similarity to membrane development, the main purpose of these assumptions was to provide a quantitative link between membrane strength, which depended on the number of radial spokes (n_s), and the membrane porosity.

Based on measurements from eight membranes of n_s = 4 to 24, the diameter of the largest circular pore (D_p) inscribed between microfibrils (Fig. 4B) was approximately:

(14)

The stretched pore diameter (D_p') at each membrane deflection y was calculated assuming that membrane stretch was accounted for by expansion of pores without any shrinkage of microfibril thickness. The approximation in Eq. 14 was of less concern than its main purpose of establishing a consistent quantitative link between membrane strength and porosity.

To set s_f, the spacing between microfibrils of a membrane sheet (Fig. 4), we used Petty's measurements from conifer membranes where the porous structure of the margo allows both D_p and n_s to be measured. Petty (1972) reported that a pore size of approximately 0.1 µm corresponded with an n_s of approximately 100. From Eq. 14, this gives an s_f of approximately 1.5 µm. This s_f setting was used as the default for angiosperm and conifer membranes alike to simplify the comparison between pit types. Variation in membrane porosity was thus achieved by varying the number of microfibril sheets rather than the spacing within a sheet. As noted in the results section, we assessed the effect of varying the s_f setting from the default over a range from 0.5 to 3.0 µm.

To match a given membrane structure with a specific air-seed pressure, we increased the number of spokes (n_s) by increments of two (two spokes per microfibril sheet; Fig. 4A) and calculated the air-seed pressure for each spoke setting until the desired air-seed pressure was reached. For example, in Fig. 3 a spoke number of n_s = 60 gave an air-seed pressure of 0.88 MPa (light P_y curve, n_s = 60). If the target air-seed pressure was 1.4 MPa, n_s for this same pit would have to be incremented to 100 (heavy P_y curve, n_s = 100).

Pit and conduit hydraulic conductivity
Calculating the hydraulic conductivity of a single pit began with the equation for hydraulic resistance (R, reciprocal of conductance) of a circular pore of diameter D_p in an infinitely thin plate (Vogel, 1994 ):

(15)

where is viscosity. If the membrane is assumed to have circular pores of equal diameter and to have negligible thickness, the total membrane conductance could be estimated as the sum of the individual pore conductances. However, closely spaced pores interact such that their individual conductances are greater than predicted from Eq. 15. Tio and Sadhal (1994) have modeled this effect and found that the pore resistance is decreased by a fraction equal to the following function of the proportion of the plate area occupied by pores (h):

(17)

where n_po is the number of pores in the membrane.

The pores in the membrane were not of uniform size nor were they circular (Fig. 4B). To simplify the use of Eq. 17, we estimated the equivalent pore diameter (D_pe; Fig. 4B) that gave the same membrane conductivity for the same number of pores as in the actual membrane. We assumed that pores were the largest circle fitting within the membrane openings and that membrane conductivity was proportional to the sum of the pore diameters to the third power (Eq. 17). Based on measurements from the same set of eight membranes used to estimate the maximum pore diameter D_p (Eq. 14), the equivalent pore diameter was approximately 63% of D_p for n_s > 6 (always the case):

(18)

The pore number (n_po) was estimated from the total membrane area divided by the area of a single circular pore including the surrounding microfibril strand (t_f = 30 nm; Petty, 1972 ):

(19)

The D_pe and n_po from Eqs. 18–19 were used in Eq. 17 to estimate the membrane resistance.

The hydraulic resistance of the pit aperture (R_a) could not be estimated from Eq. 15 because the aperture cannot be regarded as being infinitely short in length. Dagan et al. (1982) provide an approximate solution for the hydraulic resistance of circular pores of finite length that combines Eq. 15 with the Hagen-Poiseuille equation:

(20)

where t_a is the length of one aperture (Fig. 1C). We used Eq. 20 to calculate the R_a for the cavitation data set, calculating t_a from the double wall thickness (t_w):

(21)

where y_l was the distance from the inner aperture edge to the membrane at zero deflection (Fig. 1D). The y_l was calculated from radius of curvature of the chamber wall (r_ya; Eq. 11).

The total pit resistance (R_p) was equal to R_a and R_m in series, ignoring the resistance of the pit chamber:

(22)

To represent the hydraulic efficiency of individual pits, we converted the pit resistance to a conductance per membrane area (pit K_sp = 4/(R_pD_m²).

The accuracy of Eqs. 17, 20, and 22 was tested by applying them to physical models of pits with well-characterized shape, pore sizes, and aperture configurations (Lancashire and Ennos, 2002 ). The agreement with measured values was extremely close (Fig. 5), suggesting that the largest source of error will be the estimates of pore size and number in Eqs. 18 and 19.

View larger version (20K): [in this window] [in a new window]   Fig. 5. Measured vs. calculated hydraulic resistance of physical models of circular bordered pits (Rp). Measured values are from Ennos and Lancashire (2002) . Calculated values are from Eqs. 17, 20, and 22 based on the pit model dimensions. Aperture length (ta) was not reported, but the aperture was constructed from a hose gasket, so assumed ta = 2 mm. This is equivalent to ta = 1.1 µm according to the scale factor of 1830

(23)

where R_l is the contribution of the conduit lumen and R_w the contribution of the conduit wall pitting to the total conduit resistance. The R_l is the Hagen-Poiseuille resistance of the lumen for half of the conduit length, because water on average traverses just half of a conduit's total length:

(24)

where L is the total length of the vessel.

Similarly, R_w is the parallel resistance of just half of the vessel's pits:

(25)

The number of pits per vessel (n_pi) was given by

(26)

where int[number] returns the largest integer less than or equal to the number and LX is the cumulative length of pitted walls, with the "pitting coefficient" X being the proportionality factor between the vessel length and the cumulative pitted length. For example, if a vessel had pitted walls on two sides for all of its length, X = 2. Vessel length (L) and pitting coefficient (X) were not measured, and we present a sensitivity analysis of these variables in the results. Equation 26 makes the simplifying assumption that the minimum vertical spacing between horizontal rows of pits equals the minimum spacing within a single row. The n_pi' term in Eq. 26 is the number of pits fitting side-by-side across the conduit wall of width b in an opposite pitting arrangement (Fig. 1):

(27)

The R_c is the resistance of a vessel—the hydraulic pressure drop across the vessel divided by the flow rate. To assess the hydraulic efficiency of a conduit, we expressed the hydraulic conductance of a single vessel on a per-conduit length and per-unit cross-sectional wall area basis (K_sc):

(28)

where A_w is the cross-sectional area of the wall of a single conduit. To account for the wall investment in the conduit over its entire length, the A_w was calculated from the total wall volume of the conduit (V_t) divided by the conduit length (A_w = V_h/L). The V_t was

(29)

where V_s is the volume of the wall assuming no pits, V_c is the volume of one pit chamber, and V_a the volume of one pit aperture. Volume components were computed as:

	RESULTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS MODEL DESCRIPTION RESULTS DISCUSSION LITERATURE CITED

 
Wall implosion pressure (P_i) and conduit dimensions
The conduit implosion pressure was positively correlated with the measured air-seed pressure with an average safety factor (P_i/P_a) greater than 1 as predicted (Fig. 6A). The stem and root regressions were not significantly different in slope or intercept and were pooled. From the regression line (P_i = 0.90P_a + 1.0), the safety factor from implosion ranged from 2.1 at P_a = 1 MPa to 1.0 at P_a = 10 MPa, with an average of P_i/P_a = 1.8. One outlier (Laurus nobilis) had an implosion pressure more than three standardized residuals greater than the mean (Fig. 6A, open symbol).

View larger version (27K): [in this window] [in a new window]   Fig. 6. (A) Implosion pressure (Pi) vs. air-seed pressure (Pa) for xylem samples in the data set. Symbols are measured values, dashed line is regression through data excluding one outlier (open circle, >3 standardized residuals from mean), and solid line is 1 : 1. (B) Components of implosion pressure vs. air-seed pressure. Left ordinate and solid symbols is the thickness-to-span ratio (tw/b)2. Right ordinate is the ligament efficiency (Le; open circles) or the moment ratio (Ih/Is; open triangles)

For most species, the spacing between pits in the vessel wall (s, Eq. 2) was negligible, meaning that the L_e was approximated by 1 – D_a/D_m. The increase in L_e with air-seed pressure was primarily a result of a decrease in D_a (Fig. 7A, solid symbols) rather than any increase in D_m with air-seed pressure (Fig. 7A, open symbols). Although pit membranes varied from a D_m of 2.4–7.5 µm, the D_a was correlated with D_m (Fig. 7B), keeping L_e relatively constant with pit size.

View larger version (20K): [in this window] [in a new window]   Fig. 7. (A) Pit aperture diameter (Da; solid circles) and membrane diameter (Dm; open circles) vs. air-seed pressure for data set. (B) Pit aperture diameter vs. pit membrane diameter

Mode of air-seeding and the mechanical properties of membrane microfibrils (E, F)
Whether or not a pit rupture-seeded or capillary-seeded and whether pits aspirated or not before seeding depended on the strength (F) and elastic modulus (E) of the radial microfibril-based spokes (Fig. 8). The ambiguity of spoke structure—whether the spokes are continuous microfibrils or aggregates of overlapping ones—made it difficult to assign F and E values. For individual cellulose microfibrils, estimates of F range from an upper limit of 25 GPa to a lower value near 1 GPa and E ranges from 250 to 3 GPa (Mark, 1967 ; Petty, 1972 ; Jeronimidis, 1980 ; Ashby et al., 1995 ; Hepworth and Vincent, 1998a , b ). Estimates for F and E of primary cell walls fall within the lower end of this range and below (Vincent, 1999 ).

View larger version (24K): [in this window] [in a new window]   Fig. 8. Influence of microfibril spoke strength (F) and elastic modulus (E) on pit aspiration and mode of air-seeding. Pits were of average Dm (4.9 µm), Da (1.6 µm), and air-seed pressure (Pa = 3.6 MPa). Type 1 pits capillary-seeded after aspiration. Type 2 pits capillary-seeded without aspiration. Type 3 pits rupture-seeded without aspiration. Type 4 pits rupture-seeded after aspiration. The solid "rupture boundary" divides rupture-seeding pits (below) from capillary-seeding pits (above). The dashed "aspiration boundary" divides aspirating pits (above and to left) from nonaspirating pits (below and to right). Decreasing the pit size, or increasing the microfibril spacing, caused the circled four-way intersection between pit types to move down the aspiration diagonal (arrow); and vice versa. The dotted lines show the boundary shift associated with increasing sf to 3.0 µm from the default of 1.5

Type 1 pits showed aspiration and capillary-seeding. These pits had high F and low E—strong, flexible membranes (Fig. 8, upper left).

Type 2 pits showed no aspiration and capillary-seeding. These were associated with high F and high E—strong, stiff membranes (Fig. 8, upper right).

Type 3 pits showed no aspiration and rupture-seeding. These were associated with relatively low F and high E—weak, stiff membranes (Fig. 8, lower right).

Type 4 pits showed aspiration and rupture-seeding. These pits occupied a thin wedge between type 1 and 3 pits (Fig. 8).

The boundaries between pit types were essentially independent of the air-seed pressure. Although increasing the air-seed pressure increased the force on the pit membrane, it also required an increase in the number of sheets making up the membrane and hence, an increase in the number of load-bearing microfibril spokes. The result was that the force per spoke at air-seeding did not vary substantially, keeping the boundaries between pit types relatively constant.

The boundaries between pit types did depend on the aspiration strain (e_ya), pit size, and the choice of microfibril spacing, s_f. The aspiration strain setting of e_ya = 0.03 demarcated the diagonal portion of the "aspiration boundary" (Fig. 8, dashed line). The ratio F/E is the membrane strain at rupture and so no pit could aspirate without rupturing first at an F/E below 0.03. The vertical portion of the aspiration boundary was set by the E threshold above which aspiration could not occur because it was preceded by capillary seeding.

Decreasing the pit size (while maintaining observed scaling between D_a and D_m; Fig. 7B) or increasing the microfibril spacing (s_f) shifted the four-way intersection of pit types (Fig. 8, circled point) to a lower point on the F/E = 0.03 diagonal (Fig. 8, arrow showing shift for s_f changed from 1.5 to 3 µm). This reduced the range of type 1 and type 4 pits (Fig. 8, dot-dash boundaries between pit types). In both cases, the force per spoke at air-seeding was reduced, meaning that aspiration occurred for a smaller range of E. A small membrane size reduced the force on each spoke according to Eq. 5, and a higher s_f required more membrane sheets (and hence, spokes) to achieve the same membrane porosity, leading to less force per spoke.

Assuming that pits aspirate prior to air-seeding, in agreement with limited observations (Petty, 1972 ; Thomas, 1972 ), the F and E must lie within the aspiration boundary that includes both type 1 and type 4 pits (Fig. 8). Making the further assumption that pits do not rupture-seed, given that this could cause irreparable damage to the membrane, we can narrow down the F and E values further as those leading to type 1 pits. From the boundary between type 1 and 4 pits this means F/E must exceed approximately 0.1. Except where noted, the default setting for all subsequent analyses was F = 2.2 and E = 5 GPa, which was substantially within the type 1 domain for all pit dimensions.

Pit conductivity (pit K_sp) vs. air-seed pressure (P_a)
The model predicted a significant decline in pit K_sp with increasing P_a (Fig. 9A, r² = 0.88). Pit K_sp dropped by a factor of 30 for a 10-fold increase in P_a from 1 to 10 MPa. These calculations were for pits and vessels of measured diameters and of a wall thickness giving the P_i predicted from the regression in Fig. 6A.

View larger version (23K): [in this window] [in a new window]   Fig. 9. (A) Pit conductivity per membrane area (pit Ksp) vs. air-seed pressure (Pa). Solid symbols represent pits of measured dimensions from the data set. Dashed line ("narrow") is for pits of average dimension in a conduit of minimum diameter (Dc = 23 µm); solid line ("wide") is for pits of average dimension in a conduit of maximum diameter (Dc = 102 µm). The r2 for the regression was 0.88. (B) Components of pit Ksp vs. air-seed pressure. Solid circles represent membrane conductivity per membrane area; open circles show aperture conductivity per membrane area for pits of measured dimensions. Dashed ("narrow") and solid ("wide") lines represent the aperture conductivities for narrowest and widest vessels in the data set as in panel (A)

The scatter in the aperture conductivities was a result of the variation in conduit diameter (D_c). A higher D_c required a thicker wall to maintain a given implosion pressure and hence a lower aperture conductivity. As a result, the wider the conduit, the lower the pit conductivity. This effect is illustrated by the "wide" vs. "narrow" curves for aperture conductivity (Fig. 9B) and pit conductivity (Fig. 9A). These curves were calculated for the widest (102 µm) and narrowest (23 µm) conduits in the data set (using average pit dimensions). This disadvantage of wider conduits had the most influence on pit K_sp at low P_a (Fig. 9A) where aperture conductivity was lower than membrane conductivity (Fig. 9B) and hence more limiting. Thus, although a wider conduit has a much greater lumen conductivity according to the Hagen-Poiseuille equation, its pits are necessarily less efficient conductors on a per-membrane-area basis.

Pit conductivity predictions were insensitive to F as long as these settings allowed type 1 pits. This was because membrane strength does not determine the air-seed pressure for capillary-seeding pits. Pit conductivity declined slightly with increasing E, because more microfibrils were needed to compensate for greater stretch. The more microfibrils, the narrower the pores and the lower the conductivity of the relaxed membrane. This effect was rather minor, however, resulting in a 13% decline in average pit K_sp for an 80% reduction in E (from 5 to 1 GPa).

Pit conductivities were insensitive to the spacing of microfibrils in a membrane sheet (s_f) for type 1 pits. Changing the s_f from the default of 1.5 µm to 0.5 or 3.0 µm only altered the number of membrane sheets and radial spokes required to obtain an air-seed pressure; the membrane porosity and thus conductivity was essentially constant regardless of s_f.

Vessel conductivity (K_sc) vs. air-seed pressure
The conducting unit in angiosperm xylem is not one pit but the vessel that consists of multiple pits in series with the vessel lumen. The vessel K_sc reflects the contribution of pit and lumen components to the hydraulic conductivity of a single conduit. The hydraulic conductivity was expressed per conduit wall area per unit length. A greater vessel K_sc means the plant can move more water with less drop in water potential per unit length and less investment in wall material per unit length.

To calculate vessel K_sc we needed to know the conduit length (L) and the "pitting coefficient," X, which determined the cumulative length of pitted wall (FX; Eq. 26). Neither parameter was measured, and we present a sensitivity analysis in which all other parameters were constant at values for the average vessel.

The vessel K_sc increased in sigmoidal fashion with L (Fig. 10). Short conduits had low K_sc because their conductivity was dominated by the pits with their narrow channels and high resistance to flow (Fig. 10, "pit-limited"). Long conduits had higher K_sc because their conductivity was dominated by the conduit lumen with its very low resistance to flow (Fig. 10, "lumen-limited"). At a certain length, the vessel K_sc became saturated at a value set by the Hagen-Poiseuille conductivity of the lumen and the cross-sectional area of the conduit wall. This "saturated vessel K_sc" was calculated directly from the model, but its dependence on conduit diameter and wall thickness can be represented in simplified form:

(33)

where the proportionality depends in part on the volume of the pits.

View larger version (20K): [in this window] [in a new window]   Fig. 10. Vessel conductivity per length and per wall area (Ksc) vs. vessel length (L) for the average vessel (see text). Symbols represent vessels with pitting on 50% of their wall area (X = 2, Eq. 26). Dashed lines indicate 10% and 100% pitting as indicated. Short vessels had low Ksc determined by the dominating influence of pit resistance. In long vessels, the relative contribution of pit resistance diminished, and the Ksc saturated at a maximum value determined by the conductivity of the lumen and the transverse wall area. The "saturating vessel length" was the length required to achieve 95% of the maximum Ksc (asterisked point for 50% pitting; L = 0.02 m)

The saturating vessel length depended on how much of the wall was occupied by pits, as determined by the pitting coefficient, X. The maximum X was 4 when all vessel walls were pitted over their entire length (100% pitting). The higher the pitting percentage, the higher was the pit conductivity, and the shorter the vessel length required to saturate the K_sc (Fig. 10; 100% pitting line, X = 4). The lower the pitting percentage, the more limiting was the pit conductivity, and the longer the vessels had to be to maximize K_sc (Fig. 10; 10% pitting line, X = 0.4). For the average vessel, a 50% pitting percentage (X = 2) corresponded with a saturating vessel length of 2 cm (Fig. 10, asterisk). The 50% pitting percentage was adopted as the default for subsequent analyses.

The saturating vessel length also depended on the vessel diameter (Fig. 11A). The wider the vessel, the higher the lumen conductivity, and the greater must be the length for lumen conductivity to become limiting. Saturating lengths varied from 2.3 mm for D_c = 20 µm to 4.4 cm for D_c = 100 µm (Fig. 11A, solid line). While these may seem like short vessel lengths, they compare favorably to measurements of median hydraulic diameter (smallest diameter class achieving 50% or more of the cumulative Hagen-Poiseuille conductivity) vs. median vessel length from previous work on woody temperate trees and shrubs, including some of the same species used in the present study (Fig. 11A, open symbols). These species were: Alnus crispa, A. incana, Populus tremuloides, Betula occidentalis, B. papyrifera, Artemisia tridentata, Acer rubrum, and Quercus gambelii (Zimmermann and Potter, 1982 ; Sperry and Sullivan, 1992 ; Sperry et al., 1994 ; Kolb and Sperry, 1999a ). Medians were used because vessel length distributions are strongly skewed to shorter length classes (Zimmermann and Jeje, 1981 ). An exception was made for A. rubrum for which only mean vessel diameter was reported (Zimmermann and Potter, 1982 ).

View larger version (22K): [in this window] [in a new window]   Fig. 11. (A) Saturating vessel length vs. hydraulic mean conduit diameter (Dc) modeled from the data set (solid circles). The median vessel length and corresponding median hydraulic diameter from the literature on temperate woody plants of the same genera (and species, in some cases, Zimmermann and Potter, 1982 ; Sperry and Sullivan, 1992 ; Sperry et al., 1994 ; Kolb and Sperry, 1999a ) are shown as open circles. (B) Saturated vessel Ksc vs. mean hydraulic conduit diameter (Dc). Curve fit based on Eq. 33

The saturated vessel K_sc increased as a power function of vessel diameter according to Eq. 33 (Fig. 11B). The scatter results from variation in air-seed pressure that was independent of conduit diameter. Air-seed pressure influenced the saturated K_sc by changing the wall thickness (t_w) required to maintain the necessary implosion resistance (Eq. 33).

There was a significant negative relationship between the saturated vessel K_sc and increasing air-seed pressure, but with considerably more scatter (Fig. 12, r² = 0.33) than seen for the pit K_sp vs. P_a relationship in Fig. 9A (r² = 0.88). As for pit K_sp, the scatter was related to conduit diameter. Unlike the pit K_sp, however, wider conduits had much higher vessel K_sc than narrow ones (Fig. 12, compare wide vs. narrow lines). Although the pits in wide conduits are less conductive (Fig. 9), in long vessels where pit conductivity is less important, this disadvantage is masked by the much greater conductivity of a wide lumen. The scatter results from the fact that vessel diameter was not correlated with air-seed pressure.

View larger version (17K): [in this window] [in a new window]   Fig. 12. Saturated vessel conductivity per length and per wall area (Ksc) vs. air-seed pressure (Pa) for each xylem sample in the data set. The r2 for the regression (not shown) was 0.33. The correlation was weak, because vessel diameter was not correlated with Pa. The solid line was for average pit dimensions and the widest vessel diameter (Dc = 102 µm); the dashed line was for average pit dimensions and the narrowest vessel diameter (Dc = 23 µm)