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Hydraulic conductivity (K) in the soil and xylem declines as water potential (Ψ) declines. This results in a maximum rate of steady-state transpiration (Ecrit) and corresponding minimum leaf Ψ (Ψcrit) at which K has approached zero somewhere in the soil–leaf continuum. Exceeding these limits causes water transport to cease. A model determined whether the point of hydraulic failure (where K = 0) occurred in the rhizosphere or xylem components of the continuum. Below a threshold of root:leaf area (AR:AL), the loss of rhizosphere K limited Ecrit and Ψcrit. Above the threshold, loss of xylem K from cavitation was limiting. The AR:AL threshold ranged from > 40 for coarse soils and/or cavitation-resistant xylem to < 0·20 in fine soils and/or cavitation-susceptible xylem. Comparison of model results with drought experiments in sunflower and water birch indicated that stomatal regulation of E reflected the species’ hydraulic potential for extracting soil water, and that the more sensitive stomatal response of water birch to drought was necessary to avoid hydraulic failure. The results suggest that plants should be xylem-limited and near their AR:AL threshold. Corollary predictions are (1) within a soil type the AR:AL should increase with increasing cavitation resistance and drought tolerance, and (2) across soil types from fine to coarse the AR:AL should increase and maximum cavitation resistance should decrease.
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As postulated by the cohesion-tension theory, the flow of water from soil to leaf represents a ‘tug-of-war’ on a hydraulic rope. If the hydraulic continuum breaks, the plant cannot access atmospheric CO2 without desiccating to death. There are two weak spots in the continuum: at the rhizosphere where steep water potential gradients may create dry non-conductive zones (Newman 1969), and in the xylem where cavitation can eliminate water transport (Zimmermann 1983). While earlier studies have considered the limitation of water uptake by one or the other of these processes (Newman 1969; Bristow, Campbell & Calissendorff 1984; Tyree & Sperry 1988), it is an open question how rhizosphere and xylem properties interact to limit water uptake. In this paper, we answer this question with a model.
The theory of hydraulic limits on water uptake begins with Darcy's law, which can be applied to steady-state flow through the soil–plant hydraulic continuum:
where E is the transpiration rate (per leaf area), dΨ/dx is the water potential gradient driving flow, and K is the hydraulic conductivity expressed per leaf area (Table 1 lists symbols, definitions, and units). Figure 1 shows the steady-state relationship between E and leaf Ψ for a constant bulk soil Ψ (Ψs = the Ψ intercept). If K is a constant, E is directly proportional to the decrease in leaf Ψ and there is no hydraulic limit to E or leaf Ψ (dashed line 4 in Fig. 1).
Table 1. . List of major symbols and their definitions. Units are those used in equations. Values cited in text or figures may have different units
Hydraulic limits arise because K is not constant, but instead decreases in xylem and soil as a function of decreasing Ψ. In the xylem, the decline in K is caused by cavitation, and the K (Ψ) function is described by a ‘vulnerability curve’ (e.g. Fig. 3). In the soil, the decrease in K occurs by the same mechanism causing cavitation in xylem: the displacement of water-filled pore space by air as capillary forces fail (Hillel 1980; Pockman, Sperry & O’Leary 1995). The K(Ψ) function for soil depends largely on soil texture, with more sensitive functions for coarser soils (Hillel 1980).
When Ψ-dependent K is incorporated into Darcy's law, there is no longer a directly proportional relationship between E and Ψ (Fig. 1, curves 1–3). Instead, increases in E are associated with progressively disproportionate decreases in Ψ because of declining K. The E reaches a maximum (Ecrit) at the corresponding minimum leaf Ψ (Ψcrit). At these critical values, K (Ψ) has approached zero somewhere in the hydraulic continuum (Appendix). As Ψs decreases, Ecrit declines (Fig. 1, compare curves 1–3). When Ψs = Ψcrit, the plant cannot transport water.
If stomata allow E to exceed Ecrit long enough for steady-state conditions to develop, the positive feedback between decreasing K and Ψ becomes unstable: a phenomenon dubbed ‘runaway cavitation’ when it occurs in xylem (Tyree & Sperry 1988). Runaway cavitation breaks the hydraulic rope and eliminates water transport by driving K to zero. A model by Tyree & Sperry (1988) predicted an Ecrit that was only slightly greater than actual maximum E in four diverse tree species, suggesting stomatal regulation of E was adaptive in avoiding hydraulic failure of the xylem. The gas exchange capacity of plants may have hydraulic constraints.
The Tyree and Sperry model, however, did not incorporate the K (Ψ) relationship for the soil. Transpiration-driven decreases in Ψ soil are accentuated in the rhizosphere because of the cylindrical geometry of water uptake (Cowan 1965; Newman 1969; Bristow, Campbell & Calissendorff 1984), and ‘runaway cavitation’ can potentially occur at the soil–root interface. Rhizosphere limitations should be especially important for coarse soils and plants with less absorbing root area relative to their transpiring leaf area (Newman 1969). Although variable rhizosphere conductance has been incorporated in water uptake models (Cowan 1965; Bristow et al. 1984), none have incorporated variable xylem conductance. It is not clear whether below-ground hydraulic constraints are more or less important than those of the xylem.
The model presented in this paper shows how three causal factors – (1) cavitation resistance, (2) root:leaf area ratio (AR:AL), and (3) soil type (specifically, soil texture) – interact to set hydraulic limits on water transport. The analysis of cavitation resistance includes the influence of root xylem, which is more vulnerable than canopy xylem in many species (Sperry & Saliendra 1994; Alder, Sperry & Pockman 1996; Hacke & Sauter 1996; Mencuccini & Comstock 1997). The purpose of the model is to obtain a better understanding of biophysical limits on water uptake and their relevance to physiological responses of plants to water availability.
Flux balance equations and hydraulic conductance functions
The model uses the standard finite-difference approach to solving Darcy's law for the soil–plant hydraulic pathway (Campbell 1985). The soil–leaf continuum was divided into ‘nodes’ and connecting ‘elements’ (Fig. 2; see below). The soil elements defined a cylindrical rhizosphere adjacent to the absorbing roots. For each node i (ascending from i = 1 at the leaves, Fig. 2), we wrote the flux balance, or Richards’, equation (Ross & Bristow 1990) in which the difference in flux of water leaving versus entering node i equals the change in water content at node i (ΔW = Wt = 1–Wt = 2) over time step Δt (t2–t1). These equations assume the driving forces (ΔΨi = Ψi+ 1–Ψi) are differences in water pressure (i.e. osmotic effects are ignored):
where ki is the hydraulic conductance of the element subtending node i, and Fi is the mass balance for node i which equals zero for the correct values of Ψi, Ψi + 1, and Ψi– 1. Note that in formulating Eqn 2, hydraulic conductivity (K, Eqn 1), which is a length- and/or area-independent parameter, is converted to conductance (k) which incorporates the specific geometry of element i (Table 1).
The water content (W, moles of water per volume of tissue or soil) and hydraulic conductance in Eqn 2 are both functions of Ψ. In the soil we used Campbell's (1985) equation,
where Ws is the water content at maximum hydration where Ψi = Ψe, and b is a soil-texture parameter that increases with finer texture (Table 2). The value for Ψe in soil was taken as the ‘air-entry’ value, which is also a function of soil texture (Table 2; Campbell 1985). In the plant, ΔWi for Eqn 2 was calculated as:
Table 2. . Soil parameters. GMD, geometric mean particle diameter; GSD, geometric standard deviation of particle size; Ψe, air entry potential (Ψe = –0·5 GMD–0·5; Campbell 1985); b, soil texture parameter (b=–2 Ψe + 0·2 GSD, Campbell 1985); Ks, saturated (maximum) hydraulic conductivity. All soils were assumed to have a bulk density of 1·3 Mg m–3
where C is the whole-plant capacitance or change in water content per change in Ψ per leaf area, AL is the leaf area, ΔΨi is the change in Ψ at node i over the time step, and ai is the fraction of the total plant volume with which node i exchanges water.
The hydraulic conductance function for the rhizosphere elements was taken from Campbell (1985):
where Ks is the maximum soil hydraulic conductivity at saturation (at Ψi = Ψe), b is the soil texture parameter in Eqn 3, and Xi is a factor that converts hydraulic conductivity to hydraulic conductance. We assumed a cylindrical geometry for water uptake by roots so that nodes were at progressively greater radial distances from the root centre (Fig. 2). The ‘conductance factor’ for the cylindrical flow geometry of the rhizosphere elements is:
where l is the total length of absorbing roots, and ri is the radius of node i (Campbell 1985).
We used a log transformation (Passioura & Cowan 1968) to set rhizosphere nodes exponentially closer together nearer to the root where Ψ gradients are largest. This transformation equates to:
where ri is the radius of node i, rs is the radius of the root surface, rmax is the radius of the outermost rhizosphere node, s is the node number of the root surface, and m is the number of rhizosphere elements. We assumed the absorbing roots were uniformly aligned in the soil volume such that their associated soil cylinders exhibited closest packing in the soil space. Under these conditions, rmax is related to root length density of absorbing roots (L; length per soil volume):
The hydraulic conductance function for the plant elements was based on a Weibull model fit (Rawlings & Cure 1985; Neufeld et al. 1992) to empirical vulnerability curves (Fig. 3). The Weibull equation includes two curve-fitting parameters (d and c):
where ki is the hydraulic conductance of element i, Ψi is the xylem pressure at node i and ks is the element's maximum conductance in the absence of cavitation. Values for d and c were obtained using a standard curve-fitting routine. There is no conductance factor in Eqn 9 because ks was inputted as conductance rather than conductivity.
A preliminary model solved the nodal Richards’ equations (Eqn 2) by converting them to ordinary differential equations (for dΨ/dt) and integrating with the Runge-Kutta procedure (Press et al. 1989). This proved inordinately time-consuming on the computer because the non-linearity of the k functions in the equation required extraordinarily small time steps (< 0·001 s) under wet soil conditions. The present version linearized the Richards’ equation using the ‘Kirchhoff transform,’ an integral transform (Ross & Bristow 1990) that substitutes ‘matric flux potential’ (Φ) for ΔΨ as the driving force for flow (Campbell 1985). Matrix flux potential is the integral of hydraulic conductance from Ψ = Ψi to –∞:
Integrating Eqn 5 for soil element conductance gives:
Equation 9, the Weibull function for plant hydraulic conductance, can only be integrated using numeric methods. The equation was converted to the complement of an incomplete gamma function for which a numerical routine was available (Press et al. 1989):
where z = (–Ψi/d)c, and h = 1/c.
The Richards’ equation (Eqn 2) written in the form of Φ gives:
which results in element conductance of unity, linearizes the steady flow equation, and allows it to be integrated for each element using practical time steps of 1 h (the default setting). However, the transient flow problem remains non-linear because of the dependence of W on Ψ (Eqn 3). To solve the set of i + 1 simultaneous equations for Fi = 0 at each time step, we used the Newton–Raphson method (Campbell 1985). This routine iteratively adjusts each Ψi until Σi=1i=n|Fi| converged on zero (where n = total nodes in model). The model was written in visual basic within the Excel 5·0 spreadsheet application (Microsoft, Inc.).
Discretizing the soil–plant continuum
An important advantage of the Kirchhoff transform (Eqn 13) is that it minimizes the number of nodes required to obtain the steady-state solution of Ecrit and associated parameters. With the Richards’ equation (Eqn 2), Ecrit estimates converge on the proper solution as the continuum is more finely discretized, and preliminary tests are necessary to determine an acceptable number of nodes. The Kirchhoff transform gives the correct Ecrit and Ψcrit whether the continuum is represented by 1 or 100 nodes (Appendix), as long as the k (Ψ) function is the same along the flow path. To solve for steady-state Ecrit, discretizing is only necessary if the k (Ψ) functions change through the continuum, as they do within soil because of the changing conductance factor Xi (Eqns 5 & 6), and within the plant because of possibly different values of d and c for the Weibull function (Eqn 9) along the flow path. We applied the same procedures developed by Ross & Bristow (1990) for applying the Kirchhoff transform to a hydraulic continuum across media with different k (Ψ) and W (Ψ) functions.
We divided the soil–leaf continuum into 16 nodes with 15 connecting elements (Fig. 2). The plant consists of four elements: absorbing roots, transporting roots, stems, and leaves. This allowed us to vary the k(Ψ) function between these compartments if desired. In organizing the plant elements we applied the pipe-model principle (Shinozaki et al. 1964) and assumed that the entire hydraulic pathway of a plant from fine roots to evaporating surfaces in the leaf can be represented as a bundle of parallel pipes with equal hydraulic conductance. This means, for example, that the hydraulic conductance from the root collar to the base of the lamina of each leaf is the same, and similarly, the conductance from the base of the lamina to every evaporating surface within the leaf is equivalent. This allowed us to represent the branch system of roots and shoots as a catena of conductance elements and nodes in series, where each element represents the collective (series and parallel) conductance of morphologically equivalent units of the plant (Fig. 2).
Node 1 is the evaporating surface of the leaves, and the subtending conductance (k1) is the collective conductance of all laminae from the junction with the petiole to the evaporating surface. This flow path includes both xylem and mesophyll components. Node 2 is the petiole–lamina junction, and its subtending conductance (k2) is the conductance of the xylem of the entire branched stem system, including the petioles. Node 3 is the stem–root junction, and the subtending conductance (k3) is the conductance of the xylem of the non-absorbing root system. Node 4 is the junction between water-absorbing and non-absorbing roots. The associated conductance (k4) is the collective conductance of all absorbing roots from the root surface to node 4. Although the flow path in these roots includes both xylem (axial flow) and non-xylem (radial flow) components, we assume that the axial conductance is infinite, and that the water uptake along the length of an absorbing root will be equal. This is a reasonable assumption for the presumably short (≈ 100–200 mm) absorbing zones behind root tips (Steudle 1994).
The rhizosphere was more finely discretized than the plant because of the changing k (Ψ) function with distance from the root surface (Eqn 5). The total rhizosphere volume was divided into 10 nodes. However, tests showed that, under conditions when soil properties were determining Ecrit and Ψcrit (see ‘Results’), as few as three nodes were required for these parameters to be within 2% of their value when using a 50 node model. The reason for such efficient discretizing is the log transformation of the nodes which concentrates them close to the root where the limiting conductance develops (Fig. 2; Eqn 7; Passioura & Cowan 1968).
Associated with each node is a volume of plant tissue or soil with which water was exchanged as nodal pressures changed. The total nodal volume was taken as the adjacent half of the element volume above (downstream from) the node plus the adjacent half of the element volume below (upstream from) the node.
Determination of hydraulic limits
The model was applied to finding Ecrit and Ψcrit for the continuum. Boundary conditions for the equation set (Eqn 13) were that Ψ at the outermost soil node (i = 16) was set to a constant (Φ16 = constant), as was the product E AL, which in terms of Eqn 13 was the flux of element i = 0 (substituted for Φ1–Φ0). Initial conditions were that Ψ at all nodes equalled Ψ16.
The model was verified by setting E to permissible values and testing for flux balance. Under these conditions, fewer than 20 iterations were required to make Σi=1i=16|Fi| < 0·001 mmol s–1, and cumulative water depletion was within 0·0001% of cumulative transpiration.
To determine Ecrit, E was incremented in steps of 0·01 mmol s–1 m–2 from permissible values until Σi=1i=16|Fi| failed to converge on zero after a maximum of 30 iterations. At each increment, the model was run for enough time steps to achieve steady-state flow (Σi=1i=16|ΔΨi|/Δt < 0·001 MPa h–1). We deliberately set liberal requirements for model failure by adjusting the criteria for flux balance to Σi=1i=16|Fi|≤ 1 mmol s–1. The Ecrit and Ψcrit were the last permissible values before model failure.
The choice of the E increment can influence the Ecrit and Ψcrit estimates. The prediction of Ecrit is less sensitive to the E increment than Ψcrit because E approaches Ecrit in an asymptotic manner, while Ψ decreases abruptly to Ψcrit (Fig. 1). Mathematically, Ψcrit = –∞ because our soil and xylem K(Ψ) functions never reach zero (Eqns 5 & 9; Appendix). Thus, decreasing the E increment caused the model to converge on Ecrit as Ψcrit became increasingly negative. Our choice of a 0·01 mmol s–1 m–2E increment was based on a very conservative estimate of the control sensitivity of stomata. Using this increment, Ψcrit corresponded to > 98% loss of conductance in the limiting element of the continuum (e.g. Fig. 7).
Plant parameters were based largely on data from woody plants.
Secondary parameters were held constant for all simulations except when varied for a sensitivity analysis (Table 3).
Table 3. . Sensitivity analysis. Parameters tested were capacitance, whole-plant ks/AL, percentage of plant resistance in root elements 3 and 4 (% R 3+4; with equal resistance in elements 3 and 4), percentage of resistance in root element 4 (% R 4; with equal resistance for 3+4 and 1+2), percentage of resistance in leaf (% R 1, equal resistance for 3+4 and 1+2), root length (l), distance to outermost soil node (rmax), and k4 = ks4[vs. the same k (Ψ) function as k3]. Default and test values of each parameter are shown; %Δ=percentage of increase in test value vs. default; %ΔEcrit=percentage change in Ecrit; %ΔΨcrit=percentage change Ψcrit. Simulations were for xylem-limited conditions in sagebrush xylem at AR:AL=1·9, soil volume=0·3 m3, loam soil, with Ecrit evaluated at soil Ψ=–2·2 MPa (=1/3 Ψcrit)
Maximum hydraulic conductance in the absence of cavitation (ks in Eqn 9) was equal for all plant elements and scaled with leaf area to give a leaf-specific conductance of 5 mmol s–1 MPa–1 m–2 for the whole plant. This is a typical value for woody plants (Meinzer et al. 1995; Saliendra, Sperry & Comstock 1995). The equal conductance for shoot and root systems was a realistic approximation based on measurements in a variety of herbaceous and woody plants (Hellkvist, Richards & Jarvis 1974; Saliendra & Meinzer 1989; Meinzer et al. 1992; Sperry & Pockman 1992; Yang & Tyree 1993; Mencuccini & Comstock 1997). While there is evidence that leaves (k1) and absorbing roots (k4) have lower conductance than stems and transporting roots (Yang & Tyree 1994; Lopez & Nobel 1991), we set them equal for the sake of simplicity, while conducting a sensitivity analysis to determine the influence of lower conductance in the k1 and k4 components.
For plant capacitance (C in Eqn 4), we used the whole-plant, leaf-specific value of 5 mol Mpa–1 m–2 as estimated for apple trees by Landsberg, Blanchard & Warrit (1976). The choice of a in Eqn 4 divided whole-plant capacitance into leaf (a = 0·4), stem (a = 0·5), transporting root (a = 0·05), and absorbing root (a = 0·05) components. The choice of a was based on biomass fractions of roots, stems, and leaves reported in Givnish (1995) for ≈ 2–6 m trees. The radius of absorbing roots (rs, in Eqn 7), was 0·1 mm; a typical value for fine roots (Caldwell & Richards 1986).
Primary parameters were the hypothetical causal factors underlying the hydraulic limits: cavitation resistance, root:leaf area (AR:AL), and soil texture.
Four cavitation resistances were chosen to represent the span of known values (Fig. 3) and to establish the d and c values for the k(Ψ) functions of the plant elements (Eqn 9). On the vulnerable end was water birch (Betula occidentalis, water birch; Alder et al. 1997), representative of obligate riparian trees in the Western United states; on the resistant end was ceanothus (Ceanothus crassifolius, hoary leaf ceanothus; Portwood et al. 1997), a shrub of the California chaparral. Intermediate vulnerabilities were represented by sagebrush, a mesic-adapted Artemisia species (Artemisia tridentata ssp. vaseyana, mountain sagebrush; K.J. Kolb and J.S. Sperry, in review), and boxelder (Acer negundo; U. Hacke and J. S. Sperry, unpublished). All vulnerability data were collected using either the air-injection method (Cochard, Cruiziat & Tyree 1992; Sperry & Saliendra 1994) or the centrifugal force technique (Pockman et al. 1995; Alder et al. 1997). Data were from stem segments of between 5 and 10 mm diameter. In water birch and boxelder there were additional data from similar-sized root segments (Fig. 3; dashed curves). No data were available for smaller absorbing roots or for leaf xylem.
For sagebrush and ceanothus simulations where we had no root vulnerability data, all four conductance elements were given the same k(Ψ) function. In water birch and boxelder, the two root elements were given the k(Ψ) function for root xylem and the two shoot elements were given the corresponding function for stem xylem. Although the absorbing root and leaf elements include non-xylary flow paths, without any data on the k(Ψ) function we applied the xylem functions by default.
We ran simulations using d and c values that resulted in no cavitation and constant plant k over physiological values of Ψ (d > 50, c > 100). Under these circumstances, the only hydraulic constraint on flux and pressure was in the soil and rhizosphere components of the continuum.
The AR:AL was varied from a minimum of 0·24 to 40, a range that includes most measured values (Rendig & Taylor 1989; Glinski & Lipiec 1990; Tyree, Velez & Dalling 1997). This ratio also represented the ratio of hydraulic conductance in rhizosphere versus plant because these were proportional to their respective root and leaf areas. The ratio was varied by changing AL and/or AR. The AR = 2πrsLV, where V = soil volume = 0·3 m3. We varied AR by changing L. This influenced rhizosphere conductance via changes in l (Eqn 6) and rmax (Eqn 7). We also analysed the independent effect of changing l versus rmax.
Several soil types were chosen to span the range of textures from clay through loam to coarse sand (Table 2).
Controlled drought experiments
A comprehensive comparison between predictions of Ecrit and Ψcrit with empirical data awaits experiments designed explicitly for that purpose. However, we were able to make a preliminary comparison using data from controlled drought experiments on sunflower (Helianthus annuus) and water birch (Betula occidentalis).
Potted plants were grown from seed in fritted clay under well-watered greenhouse conditions until they were ≈ 0·5–1·5 m tall. Cylindrical pots were ≈ 0·15 m in diameter and 0·76 m tall, holding ≈ 0·014 m3 of soil. Values for soil parameter b were obtained by best fit of Eqn 3 to moisture release data for fritted clay (van Bavel, Lascano & Wilson 1978: b = 5·12). The saturated hydraulic conductivity (Ks, Eqn 5) was set to 11·1 mol s–1 MPa–1 m–1 based on data from van Bavel et al. (1978) and assuming their water content for a drained pot of 0·39 m height.
Vulnerability curve parameters c and d (Eqn 9) were the same as reported in Fig. 3 for water birch. In sunflower, they were obtained from the best fit of Eqn 9 to data obtained for mature stems (c = 3, d = 2·3; J. S. Sperry, unpublished results) using the centrifugal force method (Pockman et al. 1995; Alder et al. 1997). The sunflower vulnerability curve was very similar to that for boxelder stem xylem (Fig. 3).
Values for ks of the plant elements (Eqn 9) were based on k measurements of root and shoot systems of well-watered plants using the vacuum canister method of Kolb, Sperry & Lamont (1996). This method gave approximate ks values for the two root elements in series and the two shoot elements in series. Conductance was divided equally among the two elements within the root and shoot system.
The AR:AL was not measured. A range was estimated based on published values of root length per leaf area which range from 3900 to 14 000 m m–2 (Rendig & Taylor 1989). These correspond to AR:AL = 2·4–9·1, assuming rs = 0·1 mm. Measurements of AL and a soil volume of 0·014 m3 allowed us to set a corresponding range of L (Table 4).
Table 4. . Plant parameter settings for sunflower and water birch simulations shown in Fig. 8. Soil settings were for fritted clay (Table 2) using a soil volume of 0·014 m3. Means are based on n=10 plants for sunflower and n=9 plants for water birch. Plant parameters were whole-plant ks (all four elements in series), percentage of total resistance in elements 3 and 4 (% R 3+4, with equal resistances per element), leaf area (AL) and root length density (L). Settings for L corresponded to a range of 2·4–9·1 for AR/AL. The d and c values for the Weibull function (Eqn 9) are given in Fig. 3 for water birch, and were 2·3 and 3, respectively, for sunflower
Given the uncertainty of AR:AL, and of using ks values measured on nonintact plants to represent the in situ values, we used the model to estimate a liberal range of Ecrit values as a function of soil Ψ (i.e. Ψ16). The high end was based on maximum AR:AL in combination with a 20% increase in ks of each plant element over the measured value; the low end used minimum AR:AL and a 20% decrease of ks.
To obtain data on how E varied with soil Ψ (Ψs), water was withheld and periodic measurements of E and Ψs were made during the drought. The E was measured in a whole-canopy open gas exchange system as described in Saliendra et al. (1995). The E measurements were destructive because leaf area was measured by defoliating the plant and using a bench-top leaf area meter (LiCor, Lincoln, NE, USA). The Ψs was measured psychrometrically by removing soil samples from ports in the sides of the pots at 300 mm depth (n = 3 samples per pot) and sealing the samples in psychrometer chambers (Merrill Scientific, Logan, UT, USA).
Analysis of Ecrit and Ψcrit
The dependence of Ecrit on Ψs is shown in Fig. 4 for loam soil (Table 2). The five xylem types are shown, four with varying cavitation resistance (open symbols) and one with no cavitation (solid symbols). Where the non-cavitating curve shows higher Ecrit and lower Ψcrit than the cavitating curves, the xylem was limiting. Where the two curves are the same, the rhizosphere was limiting. For all curves, Ecrit decreased to zero as Ψs decreased to Ψcrit (Ψcrit is shown by arrows on the Ψs axis in Fig. 4a).
At the relatively high AR:AL of 10 in 4Fig. 4a, xylem cavitation was limiting (compare open versus solid symbols). The more vulnerable the xylem to cavitation, the lower was Ecrit at a given Ψs, and the higher (less negative) was Ψcrit. The Ψcrit was sufficient to cause ≈ 98–99% loss of xylem conductance based on the xylem vulnerability curve (Fig. 3).
Decreasing AR:AL from 10 to 5 (Fig. 4b) and 1 (Fig. 4c) caused a gradual transition from xylem cavitation to rhizosphere conductance as the limiting factor for Ecrit and Ψcrit. This is evident from the identical Ecrit for ceanothus xylem versus non-cavitating xylem at AR:AL = 1 (Fig. 4c). The same was nearly true for sagebrush. The more cavitation-susceptible xylem types (boxelder, water birch) were still xylem-limited (Fig. 4c). However, as AR:AL was decreased further (data not shown), the rhizosphere became limiting for all xylem types.
Figure 5 extends the findings in Fig. 4 to different soils. It shows Ψcrit as a function of AR:AL for cavitating (solid lines) versus non-cavitating (dashed lines) xylem. 5Figure 5a is for sandy loam (Table 2). At lower AR:AL, the Ψcrit was the same with or without cavitation (overlapping dashed and solid lines) and the rhizosphere was limiting. At higher AR:AL, the Ψcrit for cavitating xylem converged on the Ψ causing 98–99% loss in xylem conductance (Fig. 3) while the Ψcrit for non-cavitating xylem continued to decrease. The threshold AR:AL marking the transition between rhizosphere versus xylem limitation is where Ψcrit values for cavitating versus non-cavitating xylem depart.
It is evident from Figs 4 and 55a that each xylem type had a threshold AR:AL above which xylem cavitation was limiting and below which the rhizosphere conductance was limiting. The AR:AL threshold increased as cavitation resistance increased.
5Figure 5b generalizes the results in 5Fig. 5a across five soil types. The dashed lines again represent Ψcrit in the absence of cavitation, with lines 1–5 indicating soils of increasingly finer texture from sand to loam (3 is sandy loam from Fig. 5a). The horizontal solid lines represent Ψcrit = Ψ at 99% loss of xylem conductance. The AR:AL threshold for a given soil and xylem type is approximated by the intersection of dashed and solid lines. The actual transition was smooth (Fig. 5a) indicating a range of AR:AL over which the rhizosphere exerted a gradually diminishing influence (compare Figs 5a & b).
The coarser the soil type, the more limiting the soil was to plant fluxes and pressures relative to the xylem, and the higher was the AR:AL threshold for xylem limitation. For example, the coarsest soil (sand, dashed line 1), limited Ψcrit in all but the most cavitation-susceptible type (water birch), and then only at the highest AR:AL. In contrast, in the finest soil analysed (loam, dashed line 5), the xylem determined Ψcrit even in the most cavitation-resistant species. Simulations for soils finer than loam were all xylem-limited.
Increasing AR:AL above the xylem-limiting threshold continued to influence Ecrit up to a second threshold (Fig. 6). The influence is evident in the sagebrush results in Fig. 4: increasing AR:AL caused Ecrit to increase for intermediate Ψs (e.g. Ψcrit > Ψs < 0). The Ecrit was maximized by progressively higher AR:AL in more cavitation-resistant species (Fig. 6).
The results in Fig. 5 show conditions under which xylem versus rhizosphere conductance is more hydraulically limiting, but they do not indicate in which conducting element k approaches zero at hydraulic failure. When the xylem was limiting, the element with k≅ 0 was in the leaf element, because it has the lowest Ψ, and also in the transporting root element, if the root elements were more vulnerable to cavitation than the stem elements (as for water birch and boxelder). Figure 7 shows an example from boxelder xylem under xylem-limiting conditions (triangles, sandy loam soil, AR:AL = 27). The leaf and transporting root elements (1 and 3) had near-zero hydraulic conductance at Ecrit (Fig. 7, solid triangles). Although a substantial reduction in hydraulic conductance also occurred at the root–soil interface, rhizosphere element conductance (e.g. 5, 6 and 7) was still above that in xylem elements 1 and 3. (Note: elements were exponentially shorter as they approached the root; Fig. 2).
When the soil conductance was limiting Ecrit, the hydraulic bottleneck at the root–soil interface was responsible. Figure 7 also shows boxelder under soil-limiting conditions (circles, sandy loam soil, AR:AL = 1). At Ecrit, the elements with the lowest conductance were adjacent to the root surface (Fig. 7, solid circles, elements 5, 6 and 7) rather than in the plant.
It should be emphasized that, while Ecrit and Ψcrit are associated with k reaching zero somewhere in the continuum, the total conductance in the continuum may still be substantial. In Fig. 7, for example, the total loss of conductance in the continuum at Ecrit was 58% and 86% for xylem- and rhizosphere-limited cases, respectively.
Decreasing AR:AL transferred the hydraulic limitation from xylem to rhizosphere because it reduced the ‘resting’ (i.e. E near 0) rhizosphere conductance (Fig. 7; compare open triangles with open circles), therefore causing the rhizosphere bottleneck to create the lowest k values in the continuum as E approached Ecrit. At high AR:AL, the resting conductance in the rhizosphere was much greater than in the plant (Fig. 7, open triangles); thus, even though a rhizosphere bottleneck developed as E increased, it did not result in the limiting conductance for the continuum. Lower AR:AL brought the resting conductance closer to those in the plant (Fig. 7, open circles) with the result that rhizosphere conductance became limiting as flux increased. At intermediate AR:AL, the relative limitation of rhizosphere versus xylem was less pronounced, and both components exerted influence, as evidenced by the smooth transition from rhizosphere to xylem limits in 5Fig. 5a, and the influence of AR:AL on Ecrit under xylem-limited conditions (Fig. 6).
Table 3 summarizes a sensitivity analysis for secondary model parameters under xylem-limited conditions. As expected, Ψcrit = Ψ at 98–99% loss of xylem conductance regardless of parameter settings.
The Ecrit increased 7·1% for a 40% increase in whole-plant ks/AL (maximum leaf-specific conductance). The effect of increasing L (and AR) was primarily due to the increase in l (28·6% response) rather than a decrease in rmax (10·7% response), meaning that total root length was more important than density. The Ecrit was not influenced by the allotment of ks among elements; however, it did increase by 9·6% when the absorbing root k was held constant (Table 3).
The Ecrit values were for steady-state conditions where Ψ of outermost soil node was held constant. As a result, capacitance had no influence on Ecrit (Table 3). In reality, soil Ψ will decrease as water is withdrawn. However, when we modelled the non-steady-state case we found no significant change in the Ecrit versus Ψs relationships shown in Fig. 4 (simulations not shown).
Controlled drought experiments
Figure 8 shows the comparison between model predictions of Ecrit versus Ψs, and data from controlled drought experiments. A range of Ecrit is shown based on a ±20% deviation in plant ks from measured values and a 2·4–9·1 range in AR:AL (see Table 4 for parameters). Xylem cavitation was limiting for both sunflower and water birch. The greater resistance of sunflower to cavitation than water birch is reflected in the lower Ψcrit and greater Ecrit of sunflower than water birch (Fig. 8).
In both species, stomatal closure during drought was necessary and sufficient to keep E below maximum Ecrit and so avoid a predicted hydraulic failure. However, safety margins from the Ecrit range decreased substantially as drought progressed. As predicted because of its greater cavitation resistance, sunflower maintained higher E relative to water birch at all soil Ψ. If water birch had not restricted E below that of sunflower, the model would predict hydraulic failure, even under well-watered conditions.
Both rhizosphere and xylem constraints were important for setting the range of possible flux and pressure in plants, but their relative importance depended on conditions. The rhizosphere was limiting for low AR:AL, coarse textured soils, and species resistant to cavitation. Xylem cavitation was limiting for higher AR:AL, fine textured soils, and species vulnerable to cavitation (Fig. 5b). The incorporation of the rhizosphere component is a significant improvement over previous attempts to estimate hydraulic limits (Tyree & Sperry 1988; Jones & Sutherland 1991; Alder et al. 1996).
The method of analysis also improves over earlier attempts. Previous models did not employ the Kirchhoff transform (Ross & Bristow 1990), and their accuracy depended on how finely the continuum was discretized (Appendix). Insufficient discretizing resulted in predictions of Ψcrit much less negative than that which caused 100% loss of xylem conductance (Jones & Sutherland 1991; Alder et al. 1996; Mencuccini & Comstock 1997). While Jones & Sutherland (1991) emphasized that stomatal conductance may be maximized at the expense of some xylem conductance, our model predicts that stomatal conductance (a proxy of E) will be maximized at the expense of all conductance at the limiting point in the continuum. According to Fig. 7 this will be in the root and/or leaf xylem under xylem-limiting conditions, or in the rhizosphere. At Ecrit the loss of conductance for the entire continuum (soil-to-leaf) may be substantially below 100% (e.g. 59–86% for simulations in Fig. 7).
Assuming a benefit from maximizing leaf area and stomatal conductance while minimizing root biomass and cavitation resistance, plants should operate as close as they can to their hydraulic limits without risking failure. This is consistent with the drought experiments (Fig. 8), and the analysis of Tyree & Sperry (1988). Although the Tyree & Sperry study did not incorporate the Kirchhoff transform or rhizosphere resistances, it was discretized and would match our model predictions under xylem-limiting conditions. A large body of empirical work also supports the existence of small safety margins in plants that are relatively vulnerable to cavitation and likely to be xylem-limited (Sperry & Pockman 1993; Sperry, Alder & Eastlack 1993; Tyree et al. 1993, 1994; Cochard et al. 1996; Saliendra et al. 1995; Alder et al. 1996; Lu et al. 1996).
Plants with very cavitation-resistant xylem (sagebrush, ceanothus) may only approach their hydraulic limits when Ψs drops during drought. At high Ψs, the Ecrit for these xylem types (Fig. 4) was far above typical maximum values for plants (≈ 10 mmol s–1 m–2), and E would probably be limited by other factors such as the maximum diffusive conductance of the leaf-to-water vapour. A decrease in Ψs during drought would reduce Ecrit well within the physiological range and constrain gas exchange, a prediction borne out by application of the model to field data from sagebrush (Kolb & Sperry, manuscript in preparation) and ceanothus (Portwood et al. 1997).
Being able to predict hydraulic limitations based on interactions between soil type, water availability, AR:AL, and xylem type (Fig. 5b) leads naturally to hypotheses about adaptive combinations of these properties. A set of these hypotheses can be formulated if we assume at the outset that plants operate near their hydraulic limits, at least on a seasonal basis.
The most general hypothesis is that plants will be near the AR:AL threshold where xylem is limiting. The AR:AL should be at least as high as the threshold because this minimizes Ψcrit. Increases in AR:AL beyond the threshold confer progressively diminishing returns with respect to water uptake (Figs 5 & 6), and would represent a waste of root biomass with respect to water uptake.
The evidence suggests that most plants are xylem-limited. Measurements of AR:AL vary tremendously, from 0·24 to over 10 (Tyree et al. 1998; Rendig & Taylor 1989). Even at the low end of this range, all plants would be xylem limited if growing in a loam- or finer-textured soil (Fig. 5b, dashed line 5). Excess of AR:AL beyond the threshold may reflect requirements for uptake of nutrients rather than water. For a modest AR:AL of 2, soils would have to be at least as coarse as a sandy loam (Fig. 5b, dashed line 3) to limit fluxes in relatively cavitation-susceptible plants such as boxelder, water birch, and probably most crop species (Sperry 1998). These conclusions are consistent with Newman's (1969) analysis of the rhizosphere bottleneck, wherein he estimated that, for a sandy loam, AR:AL less than 0·62 would be necessary for an appreciable rhizosphere resistance to develop. Our results also parallel those of Bristow et al. (1984) where rhizosphere resistances influenced plant water uptake for coarse soils with a b value below 3·5 (Table 2), and other factors constrained uptake in finer soils (b > 3·5).
A related hypothesis is that plants growing in drought-prone habitats should be more resistant to cavitation and have higher AR:AL than plants in wetter habitats. This is true with respect to cavitation resistance (Sperry 1998), and in many instances for root density (Glinski & Lipiec 1990) which may correspond to higher AR:AL.
Superimposed on the hypothetical trend with drought exposure are adaptive trends related to soil type. Plants should be hydraulically compatible with their soil. Plants in finer textured soils would tend to develop the lowest AR:AL and have the widest range of cavitation resistance. Plants in sandier soils would have the highest AR:AL and be more uniformly vulnerable to cavitation. Rooting density is known to be higher for certain plants in coarse versus fine soils (Glinksi & Lipiec 1990), and a recent analysis of vulnerability curves of forests in Brunei indicated rather uniformly vulnerable xylem for species of the heath and Dipterocarp forests on relatively coarse soil (Becker, Patino & Tyree 1998).
A caveat to these hypotheses is that an increase in Ecrit and/or a decrease in Ψcrit can have the disadvantage of promoting faster consumption of water and accelerating soil drought. This is the case where a fixed soil volume is available to the roots, as in potted plants. In the ground, however, the total soil volume drained by roots is more ambiguous, and could actually be dependent on the manner in which E is regulated during the drought. Drought simulations incorporating pararhizal resistances (Newman 1969) and larger soil volumes would be necessary to explore these interactions.
The use of the pipe model leads to an even draining of the soil volume and an even distribution of cavitation among morphologically equivalent units of the plant. This may approach reality for shallowly rooted plants with fibrous root systems and weak apical dominance in the shoot, but otherwise it is an oversimplification. Nevetheless, the prediction of Ψcrit should not depend on the model's representation of hydraulic architecture because under xylem-limited circumstances it is constant at near the 100% loss point of the vulnerability curve (Table 3). The Ecrit, however, may be more accurately predicted by a species-specific representation of morphology. The pipe model also predicts uniform hydraulic failure at Ecrit, whereas the branched catena approach of Tyree & Sperry (1988) demonstrated a patchwork pattern of canopy dieback with hydraulically favoured branches surviving at the expense of others.
Synthesizing the effects of rhizosphere and xylem conductance on plant water use has led to explicit hypotheses concerning the coordinated evolution of root-shoot ratios and cavitation resistance in response to soil type and water availability. Evaluation of these hypotheses should be accompanied by finer resolution of the k(Ψ) relationships in xylem of different organs (especially roots and leaves), and in non-xylary tissues of roots and leaves, as well as a more quantitative understanding of root architecture and function.
This work was supported by NSF (IBN-9319180) and USDA (9500965) grants to J.S.S. N.Z.Saliendra conducted the controlled drought experiments in context of an associated project. We thank J.B. Passioura and W.T. Pockman for helpful discussions regarding preliminary versions of the model. The comments of three anonymous reviewers were useful in improving the manuscript.
Solution of Ecrit and Ψcrit
Equation 1, Darcy's law, can be expressed in terms of hydraulic conductance between two points in the flow path, where k (Ψ) represents the Ψ-dependent leaf-specific hydraulic conductance between those points:
where xl is the distance between the points. If k (Ψ) represents the conductance between soil and leaf, Eqn A1 can be solved for Ecrit by separating variables and integrating over the soil–leaf continuum, yielding:
where Ψl = leaf Ψ and Ψs = soil Ψ. E is maximized (=Ecrit) when Ψl = Ψcrit = limit of Ψ as k goes to zero. The Ψcrit is independent of Ψs, and Ecrit decreases as Ψs decreases.
The Kirchhoff transform and the solution of Ecrit and Ψcrit
Transforming Eqn A1 in terms of matric flux potential (Φ) equates to:
Separating variables and integrating over the continuum from soil to leaf (as for Darcy's law, above) gives:
where ΦL and ΦS represent leaf and soil Φ, respectively.
From the definition of Φ in Eqn 10:
Substituting E for – (ΦL–ΦS) in Eqn A5 gives Eqn A2 as derived from Darcy's law. E = Ecrit under the same conditions as for Eqn A2.
The use of matric flux potential for a single element model (i.e. two nodes, soil and leaf) yields the correct value for steady-state Ecrit and Ψcrit as long as the k(Ψ) function is identical throughout the pathway. Discretizing is only necessary when there is a change in the k(Ψ) function. In contrast, numerically solving Darcy's law for the same parameters requires extensive discretizing even if the k(Ψ) function is constant in the continuum. For example, assuming a linear k(Ψ) function with a zero intercept, no discretizing caused a 50% underestimation of Ecrit and overestimation of Ψcrit. Over 30 elements were required to get within 5% of the correct values.