Introduction
 Top of page
 Summary
 Introduction
 Materials and methods
 Results
 Discussion
 Acknowledgements
 References
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 aboveground 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 aboveground 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 nonmonotonic behaviour possibly attributed to ageing (Spicer & Gartner 2001). These age and heightdependent trends are conjectured to reflect underlying tradeoffs between maximizing hydraulic conductance and minimizing the likelihood of hydraulic failure due to catastrophic embolism (xylem cavitation; Comstock & Sperry 2000).
Complementing longstanding 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 crosssection 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 pathlength 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 nonstructural 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 spacefilling network of branches (Hacke & Sperry 2001), and requirement that all pathlengths 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 crosssections has three components: (i) a simple means of preparing trunk and branch crosssections for analysis; (ii) a semiautomated algorithm for measuring numerous conduit dimensions within the crosssections; 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 crosssections 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 semiautomated 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?
Results
 Top of page
 Summary
 Introduction
 Materials and methods
 Results
 Discussion
 Acknowledgements
 References
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, r̄_{e}, and the average hydraulically weighted effective radius, r̄_{h}, 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 cookieheight 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 r̄_{e} (or r̄_{h}) 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 powerlaw 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 r_{e} 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 r_{e} and resampling of r_{h}, 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 r_{e} (effective conduit radius) or r_{h} (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) Method  X  Y  a  b  95% CI b 


1  h − z  r_{e}  4·02  0·27  [0·26, 0·29] 
2  h − z  r_{e}  4·03  0·27  [0·25, 0·29] 
3  h − z  r_{h}  4·36  0·25  [0·23, 0·27] 
1  R  r_{e}  4·21  0·25  [0·24, 0·26] 
2  R  r_{e}  4·21  0·25  [0·24, 0·26] 
3  R  r_{h}  4·54  0·23  [0·22, 0·24] 
Discussion
 Top of page
 Summary
 Introduction
 Materials and methods
 Results
 Discussion
 Acknowledgements
 References
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 powerlaw 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 tradeoffs 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 crosssection 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 freezinginduced embolism (Hacke & Sperry 2001)). Even if a powerlaw 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 powerlaw 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 powerlaw 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 powerlaw 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 semiautomated 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 sidebranching networks, not deterministic fractal networks, although methods are available to relate wholeorganism scaling for both types (Turcotte et al. 1998). Additional tests using data from multiple individuals, spanning a variety of species, including ringporous and diffuseporous hardwoods and tracheidbearing 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 limitcase 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.