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1An important component of plant water transport is the design of the vascular network, including the size and shape of water-conducting elements or xylem conduits.
2For over 100 years, foresters and plant physiologists have recognized that these conduits are consistently smaller near branch tips compared with major branches and the main stem. Empirical data, however, have rarely been assembled to assess the whole-plant hydraulic architecture of woody plants as they age and grow.
3In this paper, we analyse vessels of Fraxinus americana (White Ash) within a single tree. Vessels are measured from cross-sections that span 12 m in height and 18 years’ growth.
4We show that vessel radii are determined by distance from the top of the tree, as well as by stem size, independently of tree height or age.
5The qualitative form for the scaling of vessel radii agrees remarkably well with simple power laws, suggesting the existence of an ontogenetically stable hydraulic design that scales in the same manner as a tree grows in height and diameter.
6We discuss the implications of the present findings for optimal theories of hydraulic design.
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Plant hydraulic architecture must satisfy a number of functions and constraints, including transport of water, nutrients, and hormones from roots to shoots (Zwieniecki et al. 2001a, 2001b; Choat et al. 2003; Enquist 2003); maintenance of a continuous water column while minimizing the risk of cavitation (Tyree & Sperry 1989; Melcher et al. 2003; Sperry et al. 2003); and providing structural support for the above-ground tissues (Tyree & Zimmerman 2002). Little is known, however, about how the transport system is designed to simultaneously satisfy these functions and constraints across branches (space) and ontogeny (time). Regardless of the optimality principle governing hydraulic design, the flow of water from roots to leaves is partially limited by the size and arrangement of the conducting elements. Empirical studies have repeatedly shown that the hydraulic conducting elements of woody plants are smaller near branch tips compared with major branches and basal stems (Bailey & Shepard 1915; Carlquist 1975; Zimmermann & Potter 1982; Aloni 1987). For example, woody plants exhibit a continuous basipetal increase in the size of individual vascular elements within the above-ground structure within a given year, as well as a monotonic decrease in size from the most recent to the oldest growth rings at a given height (Iqbal & Gouse 1977; Ewers & Fischer 1989; Gartner et al. 1997; Nijsse et al. 2001). However, there is evidence that these monotonic trends approach an asymptote in the main trunk (Becker et al. 2003) and even show non-monotonic behaviour possibly attributed to ageing (Spicer & Gartner 2001). These age- and height-dependent trends are conjectured to reflect underlying trade-offs between maximizing hydraulic conductance and minimizing the likelihood of hydraulic failure due to catastrophic embolism (xylem cavitation; Comstock & Sperry 2000).
Complementing long-standing empirical work, theoretical models have been proposed to explain the structure and allometry of plant vascular systems. These formulations include, but are not limited to, the pipe model (Shinozaki et al. 1964); Murray's law (Murray 1926; McCulloh et al. 2003); and a theory of fractal branching networks (West et al. 1999). The highly cited pipe model predicts that a unit of leaf area is supported by a continuous ‘pipe’, the cross-section of which is constant (Shinozaki et al. 1964). Of particular importance to the latter two models is the manner in which the conducting elements change along the transport pathway (note that the pipe model makes no claim about the properties of individual vessels or tracheids within the conducting tissue). The dimensions of the conducting elements are important because the resistance to fluid transport in a pipe at low Reynolds number (in the laminar flow regime) increases with inverse radius to the fourth power and with length to the first power (Tritton 1988). Thus changes in conduit diameter have a disproportionate effect on resistance compared with changes in conduit length. Murray's law states that the sum of radii cubed is conserved before and after branch junctions (Murray 1926; McCulloh et al. 2003, 2004), but does not make any assumptions about changes in vessel length and so cannot be used to predict the tapering of radii with path length. Fractal branching network (FBN) theory has been used to derive a more comprehensive set of predictions concerning hydraulic and structural design in woody plants (West et al. 1999). In particular, the FBN theory developed by West et al. (1999), hereafter the WBE model, incorporates energy minimization as well as path-length independence of hydraulic resistance. Murray's law predicts that tapering can be more or less than that of the WBE model depending on how the number of conduits changes between branch points. For example, Murray's law requires no taper if the number of conduits is constant (as assumed in the WBE model), but a steeper taper than the WBE model if the number of conduits doubles at each branching. An extension of the WBE model to incorporate pit resistance (Becker et al. 2003) predicts that tapering should vary between that of Murray's law for bifurcating conduit number and the WBE model, depending on the scaling relationship between pit and lumen resistance.
What, then, is the current state of comparison between empirical analysis of conduit dimensions and that of theory? Empirical data on the tapering of xylem conduits in non-structural supporting tissue show clear evidence in support of Murray's law (McCulloh et al. 2003). However, xylem conduits in the branches and main trunk can act as structural support in addition to pipes for fluid transport, and the applicability of Murray's law diminishes in tree species and tissue types of which the wood plays an increasingly greater role in support (McCulloh et al. 2004). Limited, indirect evidence based on calculations of the contribution of different branch orders to total pathway resistance (Yang & Tyree 1993) supports the WBE model (West et al. 1999); while analysis of the hydraulic conductance of pathways from segments to whole trees suggests a range of possible scalings, of which only some agree with the WBE model (Mencuccini 2002). Many predictions of the WBE model have not been tested, and the model remains controversial because of its dependence on unrealistic assumptions such as the presence of a space-filling network of branches (Hacke & Sperry 2001), and requirement that all path-lengths from trunk to leaves are identical (McCulloh & Sperry 2005). As indicated by Robinson (2004), there is a clear need for direct empirical tests to provide a basis for testing the assumptions and predictions of present (and future) theories of hydraulic design in whole plants and trees.
The aim of this paper is to demonstrate a method for measuring large numbers of xylem conduits, then to apply this method to determine the scaling of conduit dimensions throughout the ontogenetic history of a tree. The methodology for analysing conduit cross-sections has three components: (i) a simple means of preparing trunk and branch cross-sections for analysis; (ii) a semi-automated algorithm for measuring numerous conduit dimensions within the cross-sections; and (iii) scaling techniques to compare dimensions of conduits formed at different times and positions within the tree. We use the method to measure the geometrical properties of over 8900 vessels in growth rings of cross-sections obtained at different trunk heights for a single tree of Fraxinus americana (White Ash). White Ash is an ideal species for this study because it has large vessels that facilitate the semi-automated measurement techniques and also pose a significant risk of cavitation (Zwieniecki et al. 2001a). We use the data to answer the following questions: (i) does an ontogenetically stable hydraulic design exist, i.e. do vessel radii scale in a manner that is independent of tree height and age? (ii) what is the quantitative form of vessel tapering, and how does it compare with current models of hydraulic design?
Materials and methods
A mature White Ash tree at the Stony Ford Farm Center for Ecological Studies near Princeton, NJ, USA was cut down in the summer of 1995. Trunk cross-sections (‘cookies’) were obtained at various heights and stored indoors until analysis. White Ash is a deciduous, ring-porous, early- to mid-successional species widespread throughout the eastern USA (Schlesinger 1990). This species typically has a long, straight main stem or trunk (Griffith 1991; Nesom 2000) and pinnately compound leaves with very short stalks (petiolules) (Nesom 2000).
preparation of sections
Cookies were collected from the trunk at 3·2, 3·6, 4·8, 5·6, 6·0, 6·6, 9·2, 12·6, 15·2 and 16·0 m above ground, and were prepared for measuring vessel dimensions. The objective of the preparation is to stain vessel lumens black without staining the non-conducting tissue. To accomplish this, individual cookies were prepared by first cutting a fresh surface with a carbide-tipped circular saw, then hand-sanding, followed by application of paste-wax shoe polish with a leather pad, then resanding, followed by a second application of shoe polish, then hand buffing with a soft cotton cloth, power buffing with a soft cloth on an orbital sander, and finally hand buffing with a hard linen cloth. Examples of untreated and treated cookies are shown in Fig. 1.
measurement of xylem conduit dimensions
The prepared cookies were scanned on an Epson Photosmart 2450 scanner at 2400 dpi in photo mode and stored as raw greyscale images (pixel intensity values 0–255). A growth ring was selected using the freehand tool in Adobe photoshop and saved as a separate file. Then individual vessels within the ring, including earlywood and latewood vessels, were selected using the freehand tool and exported as a new file. The grey-scale images were converted to black-and-white images (black is conducting tissue) by manually adjusting the threshold to retain contrast while simultaneously admitting minimal noise; automatic thresholding techniques were found to be unreliable. Occasionally cell walls separating neighbouring vessels were missing or undetected. In such cases, we used the single-pixel white pencil tool to manually separate fused vessels. We visually compared the original grey-scale image with the final prepared image to ensure accuracy of the method. The final product is a Boolean image file for which the conduit dimensions can be extracted automatically.
Conduit detection and measurement algorithm
We developed an automated image analysis program that takes a Boolean image as input. Connected clusters of black pixels were grouped automatically using a recursive search. We applied a non-linear least-squares ellipse-fitting procedure (Fitzgibbon et al. 1999) that estimates the major axis a, minor axis b, orientation, and geometrical centre of each cluster. Examples of the results produced by the ellipse-fitting procedure are shown in Fig. 2. For each elliptical vessel we determined an effective radius, re. There are several ways to calculate re: we chose to define it as the radius of a circular tube that yields the same Hagen–Poiseuille conductance as an elliptical conduit with major and minor axis dimensions, a and b, respectively (Lewis & Boose 1995). This results in the following solution for effective radius:
( eqn 1)
Measurements in units of pixels were converted to microns.
analysis of vessel data
FBN scaling model
Rather than fitting arbitrarily chosen regression functions to the vessel radii data, we derive a fairly simple model for analysing the data that is based on FBN theory. The use of a deterministic FBN model of tree architecture to derive optimal hydraulic design was first applied to woody plants by West et al. (1999). Numerous objections have been raised regarding the assumptions of this model: see Hacke & Sperry (2001); Becker et al. (2003); McCulloh & Sperry (2005) for a more extensive discussion of these issues. Here we make no assumptions regarding the optimality of the shape of elements (vessels and branches) within the network. Rather, we derive the scaling of conduit cross-section dimensions for a deterministic branching network in which conduit dimensions are constant within a branch and change at each branch junction. Although we also invoke several simplifying and somewhat biologically unrealistic assumptions to arrive at the final model, it is still preferred over other fitting models that lack a biophysical basis.
Consider a deterministic branching network where k = 0 denotes the non-branching trunk portion below the crown, and k = N denotes the terminal units (leaves/petioles). The conduit radius at level k = 0 ... N is given by
rk = r0n−āk/2,( eqn 2)
where n is the number of branches at each bifurcation; r0 is the xylem conduit radius at the base; and ā is the conduit-width tapering exponent. Note that r0 = rNnāN/2, where N is the number of branch nodes and rN is the radius of xylem conduits in the petiole, which is assumed to be statistically invariant with respect to tree height and age. Although the xylem network extends into the petioles and leaves, the hydraulic properties of these terminal units do not enter into the scaling behaviour of hydraulic design except for possible modification of prefactors.
All pathways are considered equivalent in an FBN model, so the height of a tree, h, is equal to the sum of branch lengths, lk, along a path:
( eqn 3)
where lN is petiole length and is a scaling exponent related to the taper of branch lengths. We invert equation 3 to find N as a function of h:
( eqn 4)
where log denotes natural logarithms. Note that for a distance Δh = h − z away from the petiole, the radius of a xylem conduit is the same as that of a conduit at the base of a tree of the same height (h − z), and therefore:
( eqn 5)
For saplings and mature trees, h and h − z are in the order of metres for all but the most distal internodes, while lN is usually in the order of mm or cm; for example, for F. americana the rachis length is approximately 5 cm. Thus (h − z) >> lN and equation 5 can be approximated as:
( eqn 6)
So long as the tapering parameters remain constant (which we assume to be the case within a single tree), the prefactors in equation 6 can be absorbed into a single constant leading to the scaling prediction:
r(h, z) = C(h − z)α,( eqn 7)
where C is a constant and α = ā/(2). Similar reasoning leads to a related prediction of how xylem conduit radius scales with stem radius, R:
r(R) = DRω,( eqn 8)
where D is a constant, ω = ā/(2), and is the scaling exponent related to the tapering of branch diameters. Thus a broad class of FBN theories should predict power-law scaling of vessels when measured along a single pathway from the tip or among branches of different sizes. The trunk or main stem supports a crown and thus at some point branches originate from the trunk, resulting in a trunk that is essentially comprised of a series of nodes and internodes. Thus we assume that equations 7 and 8 can be used to evaluate the vessel radius data obtained from the White Ash tree.
The scaling function in equation 7 suggests that vessel radii depend on distance from the top of the tree, h −z, but we cannot measure h − z directly because of the limited number of cookies available for our analysis. However, we can estimate h − z from cookie height and growth-ring data. For example, for a tree of final age A, each growth ring (g) was at one time the outermost ring of a tree of age A − g + 1 (see Fig. 3 for illustration). Thus if a cookie collected at height z has G total growth rings, then the tree at age A − G was at most z m tall. We assume that the height of a tree (h) at any age can be described by the following function:
( eqn 9)
where H is the final height of the tree at age A; β is a parameter that describes the growth rate of the tree. Equation 9 ensures that h is bounded between 0 (at G = A) and H (at G = 0). The height–age relationship of the White Ash tree was estimated by fitting equation 9 to observed cookie height and number of growth rings by conducting a non-linear least-squares regression. The fitting procedure yielded parameter estimates for H, A and β. The tree's final age and height were estimated to be A ≈ 37·3 years and H ≈ 17·5 m. The exponent was β ≈ 1·05, which does not differ significantly (P = 0·33) from linear height growth with age. A comparison between data and the fitted model is shown in Fig. 4.
We used several different approaches to estimate the tapering relationship between re and (h − z), and between re and R. We calculated the height of the tree when ring g from a cookie at height z was the outermost ring, using h(G) in equation 9 where G = g − 1, and incorporating the parameter estimates of H, A and β from the height–age non-linear regression. Then we conducted least-squares linear regressions of log re vs log[h(G) − z] and logre vs logR to estimate the parameters C, α, D and ω in equations 7 and 8.
The above analysis assumes that all radii are ‘equally important’. However, large water-filled vessels contribute substantially more than smaller-diameter vessels to water transport. Accordingly, we also considered a distribution where each vessel's radius, re, is weighted by its Hagen–Poiseuille conductance, which is proportional to . This weighting scheme yields a hydraulically meaningful empirical distribution of radii, rh ~ fh(re), the mean hydraulic effective radius of which is given by:
( eqn 10)
where V is the number of vessels in the growth ring. Note that the mean of the actual effective radii, r̄e, is estimated by the sample average of re.
We implemented a resampling routine to account for uncertainty in the reconstructed tree height–age chronology and to obtain samples of vessel radii from both effective and hydraulically weighted distributions. For the effective radii, we first drew a triplet of A, H and β from the multivariate distribution estimated by the height–age regression associated with equation 9. Then effective vessel radii were drawn at random, with replacement, from the empirical distribution, with sample sizes equal to the original number of vessels. Then a linear regression of the resampled log re vs log[h(G) − z] as well as log re vs log R was conducted. The resampling procedure was repeated 500 times to obtain a distribution of scaling parameter values, from which the mean and empirical 95% confidence intervals (CIs) were obtained. We also used this resampling method to estimate the scaling parameters for the hydraulically weighted distribution of radii; the only difference being that rh values were drawn at random from fh(re).
Although reduced major axis regression and similar methods are often employed for estimating allometric or structural relationships (Niklas & Buchman 1994; Herrera 2005), such approaches are not appropriate here because: (i) h(G) − z is not directly measured; and (ii) the measurement error or variation in vessel radii is much greater, on a relative scale, than the error associated with measuring cookie height or radius (Rayner 1985). However, the resampling routines do allow us to incorporate uncertainty in h(G) into estimates of the scaling parameters C, α, D and ω.
Elliptical dimensions were measured for over 8900 xylem conduits taken from the first (outermost), sixth, 12th and 18th growth rings of the White Ash cookies, spanning the latter half of the tree's life span and nearly all of its height. The average effective radius, r̄e, and the average hydraulically weighted effective radius, r̄h, for each combination of growth ring and cookie height are shown in Fig. 5. Clearly, each growth ring is characterized by a different conduit radius vs cookie-height trajectory (Fig. 5a,b). However, an ontogenetically stable profile emerges when average radius is plotted against the distance from the top of the tree (h − z) (Fig. 5c,d). That is, the relationship between r̄e (or r̄h) and (h − z) is independent of tree age or total tree height. This scaling fit is superior to that found by alternative indicators such as relative tree height (data not shown). The vessel tapering trajectory, in a sense, follows the apical meristem up the tree during ontogeny.
The power-law function derived from a simple FBN calculation (equation 7) fits the White Ash vessel data exceptionally well (solid line, Fig. 5c,d). A linear regression of log re vs log(h − z) yields a scaling exponent of α = 0·27, and the resampling procedure generated α = 0·27 for the effective radius and α = 0·25 for the hydraulically weighted effective radius. Similar analysis for the quantitative scaling of vessel radii with stem radii (equation 8) yields scaling exponents of ω = 0·25, 0·25 and 0·23 for the cases of linear regression, resampling of re and resampling of rh, respectively. The data for scaling of vessel radii with stem radii along with model fits are shown in Fig. 6 and the statistics are summarized in Table 1. These estimates are consistent with FBN calculations that satisfy ā/(2) = 1/4 and ā/(2) = 1/4.
Table 1. Summary statistics of the linear regression, log Y = a + b log X, where Y is either re (effective conduit radius) or rh (hydraulically weighted effective conduit radius); X is either h − z (distance from top) or R (stem width); a is a prefactor (C or D); and b is the scaling exponent (α or ω)
95% CI b
Estimates are given for a and b; 95% CI also given for b. The three methods used to obtain estimates of a and b are: (1) simple linear regression; (2) resampling of re; (3) resampling of rh from the hydraulically weighted distribution of re.
h − z
h − z
h − z
Here we provide the first direct empirical evidence of an ontogenetically stable conduit tapering profile in woody plants. That is, conduit radii of White Ash vary systematically with distance from the distal node as well as with stem width, regardless of tree height or age. Conduit radii are observed to increase with distance from the distal node to the α = 1/4 power, and with stem radius to the ω = 1/4 power. Such power-law scaling is a natural outcome of a broad class of models involving hierarchical branching (West et al. 1999; Becker et al. 2003), although the predicted scaling exponents are not necessarily universal. The scaling exponents α and ω depend on the quantitative form in which conduit radii, stem radii and stem length, change or taper along the transport path. Changes in conduit radii are expected to reflect energy minimization of construction (Murray 1926) as well as the equalization of hydraulic resistance across alternative paths (West et al. 1999; Tyree & Zimmerman 2002), while changes in stem radii and lengths are expected to reflect the structural design of limbs associated with biomechanical constraints (McMahon 1973). Thus the scaling exponents we measured are probably the product of the combined effect of hydraulic and structural trade-offs on scaled hydraulic design.
The observation of an ontogenetically stable pattern is more interesting than the quantitative details of the tapering exponents. In retrospect, a stable hydraulic design appears to be necessary to maintain efficient water transport that scales across the life span and height range of a tree of which the terminal units are statistically and operationally equivalent. We observe the striking result that conduit radii collapse along a single curve when plotted against distance from the top of the tree (Fig. 5) as well as when plotted against stem radii (Fig. 6). This ontogenetically stable relationship suggests that hydraulic design may be controlled by the terminal units of trees, which implicates a hormonal signal originating in the apices that affects cell growth and development, as might be expected of auxin (Aloni & Zimmerman 1983; Fukuda 1996). The stable relationship may also reflect possible effects of cambial age on xylogenesis, a conclusion also reached by Spicer & Gartner (2001). We hypothesize that a qualitatively similar stable design should be common to all woody plants. The scaling of conduit cross-section reflects a combination of minimizing resistance (that would tend to make vessels larger (Tyree & Zimmerman 2002)) and minimizing the likelihood of cavitation (that would tend to make vessels smaller, as in the case of freezing-induced embolism (Hacke & Sperry 2001)). Even if a power-law scaling relationship is common to many species, the relationship between conduit number, conduit dimensions (including tapering of length and radius), and properties of interconduit pits may result in a wide range of scaling exponents and prefactors.
Given the controversy surrounding a recent FBN theory applied to plant vascular systems (the WBE model: West et al. 1999; Hacke & Sperry 2001; McCulloh & Sperry 2005), it is worthwhile to consider how the results found here can be compared to other theories of the plant vascular system. Our finding of a power-law tapering is in accord with what one would expect based on theoretical considerations of a continuum of networks whose hydraulic and structural tapering obey certain branching properties, that is, they must satisfy ā ≈ /2 as well as ā ≈ /2 (see FBN scaling model for definitions). As we have used only a single trunk from a single tree, we do not have the necessary branching or comparative data to evaluate the comprehensive predictions of any particular FBN theory. However, recent work suggests that the taper of vessel diameter as a function of distance from the top in the most recent growth rings from multiple individuals of birch and poplar obey similar qualitative forms (McCulloh & Sperry 2005). Additionally, although the power-law scaling functions derived from the FBN theory (equations 7 and 8) fit the data exceptionally well, this does not prove that the tree is a fractal branching network that possesses the properties assumed in the derivation of these power laws. For example, it is likely that the number of conduits in any particular branching level is correlated with its branching order. It is entirely possible that the hydraulic architecture of other trees are not well approximated by an FBN, in which case identifying these scaling exponents may prove to be a fruitless endeavour. Furthermore, we do not know if the predictions of power-law scaling of vessel radii with distance from top, as well as with branch radii, are unique to FBN theory, or if other network structures or alternative tapering scenarios (continuous vs discrete) would result in a similar prediction.
What is the next step toward testing whether or not the hydraulic design of woody plants conforms to optimality principles, and if so what those principles are, particularly at the level of whole organisms? Given the inexpensive and semi-automated nature of the methods described here, we believe it is feasible for future experiments to quantify directly and independently the tapering of xylem conduit radii as well as branch lengths and radii. Such a test would permit a statistical disentangling of the related hydraulic scaling exponents α and ω, and the degree to which they depend on the structural morphology of the tree. Actual trees have side-branching networks, not deterministic fractal networks, although methods are available to relate whole-organism scaling for both types (Turcotte et al. 1998). Additional tests using data from multiple individuals, spanning a variety of species, including ring-porous and diffuse-porous hardwoods and tracheid-bearing gymnosperms, are critical for discerning the generality of the hydraulic design observed in the present study. Such data will be particularly valuable for understanding the most distal reaches of the vascular network where a transition between different tapering regimes is possible, such as shifts away from Murray's law (McCulloh et al. 2003, 2004) as the fractional contribution of woody support tissue increases. Finally, the dimensions of conducting elements affect the likelihood of hydraulic failure, for example, conduit radius affects the probability of cavitation nucleated by the formation of gas bubbles during freeze–thaw events (Hacke & Sperry 2001). The impact and relevance of cavitation resistance to the scaling of hydraulic design of whole trees remains an open and fascinating question.
The genetic, physiological and biomechanical controls on a plant's hydraulic design, including the tapering of conduit dimensions, are not fully understood, and deserve considerable experimental and theoretical evaluation. It is of utmost importance to develop limit-case models of optimal hydraulic design under well controlled settings (e.g. plants with negligible lumen resistance and large pit resistance; plants with high risk of cavitation, etc.). Such case studies will be essential in developing a comprehensive theory of plant hydraulic architecture that can be tested against measurements from real trees for which multiple factors and design constraints are simultaneously balanced.
This material is based on work supported by the National Science Foundation under a grant awarded in 2003. The authors would like to thank Xiaoyu Peng for her assistance in preparing computerized images, Ben Ouyang for his work on the best-fit ellipse algorithm, Megan McGroddy, Jeremy Lichstein, Drew Purves and Christian Wirth for many helpful conversations, and N. Michelle Holbrook and two anonymous reviewers for comments on the manuscript.