- Top of page
- Materials and methods
Murray's law defines the vascular structure that should maximize conductance per investment for blood circulation in animals (Murray 1926). The optimal structure minimizes the two costs associated with cardiovascular function (Murray 1926; Sherman 1981). The first is the cost of moving the blood through an organism, which is inversely proportional to hydraulic conductance. The second is the cost of constructing and maintaining the vascular system, which is proportional to the vascular tissue volume. These costs are in conflict because, for the generally laminar flow in blood vessels, the length-independent conductivity follows the Hagen–Poiseuille equation and is proportional to the fourth power of the conduit radius, while volume per length increases with the second power of the radius. Murray (1926) demonstrated that the point at which the hydraulic conductivity per volume is maximized is when the volume flow rate (Q) at any point is proportional to the sum of the conduit radii cubed at that point (Σ r3). Q does not change in animals, so Murray's law predicts that the optimal transport system will conserve Σ r3 at any cross-section of the system. This solution is independent of vessel length and branching pattern. The validity of Murray's law has been shown in a variety of animal vascular systems (Sherman 1981; LaBarbera 1990).
We have shown that Murray's law applies equally well to the xylem of plants (McCulloh et al. 2003). Like blood flow, water flow in xylem is laminar and conductivity is proportional to the conduit radius raised to the fourth power. Also like blood flow, investment in xylem conduits should be proportional to the conduit radius squared. In animals the major investment is in the blood volume itself, and the cost of the thin vessel walls can be ignored. In contrast, in xylem the water is cheap but the conduit walls must be thick. Thickness must be proportional to the conduit diameter to withstand implosion caused by negative pressures in the conduits (Hacke et al. 2001). Therefore investment in conduits should still scale with the radius squared, as in animals, and Murray's law should apply.
Although the Murray's law optimum of maximum conductance per vascular volume is always achieved by Σ r3 conservation across branch points, the actual value of the maximum conductance depends on how the number of conduits changes from mother to daughter branches. In animals one mother vessel usually branches into two daughters, and the ratio of conduits between daughter and mother ranks is always ≥2 (Fig. 1a). We refer to this ratio of daughter (distal) to mother (proximal) ranks as the conduit furcation number (F). In plants no such rule applies because the conduits are not continuous. The flow path is made up of multiple conduits in series and in parallel, and the furcation number can be any positive value. For example, the number of conduits can remain the same across ranks (Fig. 1b), or it can increase distally (Fig. 1c). To obtain the greatest conductance per fixed vascular volume, not only must Σ r3 be conserved, but the furcation number should be maximized (McCulloh et al. 2003). Higher furcation numbers with Σ r3 conservation mean fewer, wide conduits proximally and many, narrow ones distally.
Figure 1. Vascular networks in animals (a) vs plant xylem (b, c). The most proximal rank (0) branches to form more distal ranks (e.g. rank 1). In the animal cardiovascular system (a) a single continuous tube in rank 0 branches to form two daughter tubes in rank 1. The number of tubes increases by a factor ≥ 2 (the conduit furcation number, F) between ranks. In the plant system (b, c) the tubes are not continuous, but divided into individual conduits. (b) The number of conduits is identical at every rank and F = 1; (c) F = 2 and the number of conduits doubles from rank 0 to rank 1, as in the animal network (a). The value of F in (b, c) represents points in a continuum because, with hundreds of conduits in a single branch (a fraction is shown for clarity), F can be any positive value.
Download figure to PowerPoint
Furcation numbers in xylem reach a maximum of 1·4 in compound leaves (McCulloh et al. 2003), which is less than the minimum of 2 in animals. Higher furcation numbers in xylem would be more efficient, but would compromise safety by reducing the number of conduits in parallel in the major branches. Multiple pathways are necessary to maintain transport under negative pressure, where a single cavitation or wound embolizes the entire conduit.
Mechanical support requirements may also limit F in plants. An F > 1 in combination with Murray's Σ r3 conservation means the cross-sectional area of the conduits increases with height. If the conduit area is proportional to the wood area, a tree with this area profile would fall over. This area profile of an inverted cone was found in our previous study, but the xylem conduits were not involved in structural support because we examined leaves and vines (McCulloh et al. 2003). Trunk and branch wood may require an F ≤ 1 to avoid becoming top-heavy. Moreover, Murray's Σ r3 law may not apply if conduit walls are contributing to support of the shoot. Mechanical support is not considered in the derivation of Murray's law (Murray 1926).
The mechanical limitation on furcation number and Σ r3 conservation should be greater in conifer wood than in angiosperm wood. Conifer wood is 90–95% tracheids by volume (Gartner 1995). These tracheids function simultaneously in transport and support, and one function cannot be optimized at the expense of the other. Angiosperm wood is <55% conduits by volume, the rest being largely fibre cells (Gartner 1995). This more complex xylem has the potential independently to vary vessel structure to optimize transport without sacrificing the mechanical integrity of the fibres. The vessel fraction is especially low in ring-porous trees with their wide-diameter early wood vessels.
Here we test the hypothesis that branch wood of conifers will not follow Murray's Σ r3 law, whereas angiosperm wood will more closely approach Σ r3 conservation, particularly ring-porous wood. We also examine the values of the conduit furcation number (F) exhibited and the degree of the conduit taper to evaluate the area profile. We tested our hypothesis in some of the same species used in our previous study of leaf xylem to compare results between leaf and branch. We also include results from our previous study on vine wood to compare with the results from free-standing trees. As structural parasites, vines provide a natural control to free-standing trees because the xylem of vines should be freed of any role in structural support. We used two methods to test these hypotheses. First, we determined the degree of Σ r3 conservation by making time-consuming anatomical measurements of conduit radii. Second, we developed a more efficient functional test that determined the validity of Murray's law by comparing the ratios of hydraulic conductivity between branch ranks.
- Top of page
- Materials and methods
As hypothesized, the results revealed a progressive departure from Murray's law as xylem conduits play an increasing role in mechanical support of the plant. Compound leaves, which are supported hydrostatically, showed Σ r3 conservation (Figs 2 and 3; McCulloh et al. 2003). Similarly vines, which are structural parasites, also showed agreement with Murray's law (Figs 2 and 3). In both these cases F was relatively high, between 1·12 and 1·45, and the collective conduit area increased distally, giving a conduit-area profile of an inverted cone (Fig. 3, McCulloh et al. 2003). The xylem conduits appear to be optimized for their sole function in transport.
In stark contrast to leaf and vine stem xylem, conifer stem xylem deviated unambiguously from Murray's law according to both the anatomical and conductivity measurements. This deviation was expected because tracheids provide both structural support and water conduction. The conifer deviated from Murray's law because the proximal Σ r3 far exceeded the distal Σ r3 (Fig. 3). The large basal Σ r3 resulted from slightly more conduits proximally than distally, and the conduits narrowed distally (Table 1). This conduit configuration resulted in an area profile of an upright cone, which is consistent with the mechanical requirements of conifer wood, which is built primarily of conduits.
The stem xylem in the diffuse-porous and ring-porous trees was closer to Murray's law than the xylem in the conifer stems. This is consistent with their conduits occupying less wood area and presumably playing less of a role in mechanical support than conifer tracheids. The ring-porous species had the lowest fraction of stem area devoted to conduits and showed the least deviation from Murray's law (Fig. 4). It also exhibited a significantly higher F than the diffuse-porous or coniferous species (Fig. 3). The conduit area declined with rank, but not as sharply as the diffuse-porous or coniferous species (Table 1).
Compared with the ring-porous species, the diffuse-porous species had a higher conduit-area fraction, and the anatomical data indicated greater deviation from Murray's law (Figs 2–4). The deviation was in the same direction as in the conifer, with more Σ r3 basally than distally. The same combination of low F with conduit taper that was found in the conifer resulted in a conduit-area profile of an upright cone in A. negundo (Table 1). The results from the conductivity measurements in A. negundo contradicted the anatomical data. The discrepancy between methods may have arisen for several reasons. First, the branches used for conductivity measurements were from adult trees, while the anatomical measurements were made on young saplings. Second, the most distal rank for comparison was the petiole in the hydraulic measurements instead of the petiolule rank. Third, the higher level of variability in the hydraulic measurements probably reduced the sensitivity for detecting deviation from Murray's law.
The patterns we observed in conduit-area profiles within a tree are consistent with patterns of sap velocity within trees. If the sap at the top of a tree is moving faster than in the trunk, then the total conduit area must decrease at the top of the tree. Measurements on diffuse porous trees and conifers show increased sap velocity from the trunk to the branch, which is consistent with the upright cone profile for conduit area that we measured for these wood types (Huber & Schmidt 1936; McDonald, Zimmermann & Kimball 2002). In contrast, ring-porous trees tend to show nearly equal velocities in the branches and in the trunk, indicating a constant area profile (Huber & Schmidt 1936). Although the conduit-area fraction declined distally in the ring-porous species examined, it did not decrease as steeply as in the conifer or diffuse-porous species (Table 1), which is consistent with a smaller change in velocity.
Most studies of hydraulic architecture show a decline in leaf-specific conductivity from trunks to branches and to leaves. Accordingly, our conductivity ratios (most distal rank over proximal rank; Fig. 5) would be expected never to be >1, and to decline steadily with older ranks. However, this is not what we found. Instead, we saw a conductivity ratio >1 in the angiosperm species when the petiole was compared with the 1-year-old wood. However, the studies that have found steadily declining leaf-specific conductivities examined many more ranks than we did (Zimmermann 1978; Cochard et al. 1997; Zotz, Tyree & Patino 1997). When we compare our data with previous results made over a similarly fine scale of branching ranks, the findings are consistent (Tyree & Zimmermann 2002).
Our results indicate that when vascular tissue functions in both transport and mechanical support, it should not and does not follow Murray's law. This raises the question of what law should be followed that would simultaneously optimize conductivity and mechanics. The current dogma consists of Da Vinci's rule of constant area across branch ranks or its modern equivalent in the pipe model (Shinozaki et al. 1964; Richter 1970). The latter assumes a constant xylem area per unit leaf area, but contradicts the Da Vinci rule by predicting a monotonic increase in wood area below the canopy from the accumulation of disused pipes connected to shed leaves (Shinozaki et al. 1964). However, neither rule was derived from an analysis of hydraulics or mechanics. McMahon's important work showing that tree height scales with trunk diameter to the 2/3 power, as required to preserve elastic similarity, is independent of the distribution of area with height and holds equally well for area-preserving or area-tapering branching (McMahon 1973). Measurements of changes in stem cross-sections across branch points are limited, but show that area is generally preserved except at the petioles, where the area increases (Horn 2000). These findings are not surprising, considering that, for mechanical purposes, the total area of the wood (supporting plus transporting elements) should at the very least not increase with height to avoid a top-heavy structure. It is likely that the optimal area profile would be an upright cone, consistent with mechanical analyses of the shape of the tallest free-standing column for a given volume of material (Keller & Niordson 1966).
We found that as the conduit-area fraction increased, and wood hydraulics and mechanics came more into conflict, there was a trend for F to drop to <1 and for proximal Σ r3 to increase relative to distal Σ r3 (Fig. 3, dashed curve). This configuration achieves a mechanically desirable area profile even in the conifer where nearly all the wood is composed of conduits. However, the same area profile could be achieved at F > 1 simply by narrowing the tracheid diameter more steeply from the base to the top of the tree. Although this configuration would not reach the maximum efficiency of Murray's law, it would still exploit the efficiency of fewer, wider conduits proximally. For example, conifer wood could theoretically have maintained the same F as the leaf xylem, achieved the necessary straight area profile for mechanics by disobeying Murray's law through increased tapering of tracheid diameter, and enjoyed higher hydraulic conductivity.
If there is a more efficient option, why should conifers and diffuse-porous trees exhibit such low values of F? We suspect that the geometry of wood development prevents an F >1 in trees where conduits are a significant fraction of wood volume (Fig. 6). Wood grows in concentric rings, and the number of rings decreases with increasing branch rank. In a simple simulation we assumed that the conduit diameter is set to the width of a growth increment, and the number of conduits is set by the circumference and number of increments (Fig. 6, cross-sections). For the mechanical requirement of either a straight or upright cone-area profile, our calculations indicate this geometry of growth dictates F < 1, a tapering of conduit diameter with rank, and a Σ r3 ratio <1. These predictions are consistent with the data from the conifer. Although a very simplified representation of a complex growth process, this analysis makes the point that growth constraints might limit F where mechanical considerations also constrain the area profile.
Figure 6. A simplistic model of wood growth applied to a 3-year-old tree. Growth increments are rings of conduits around the pith, and the number of increments increases with stem age. To preserve a constant wood area from the 3-year-old trunk to the most distal 1-year-old branch rank, conduit diameter must decrease and conduit number must increase. The tree will have a conduit furcation number (F) <1, exhibit a tapering of conduit diameter distally, and deviate from Murray's law by having higher proximal Σ r3. A constraint of this nature may explain the trend for trees with high conduit-area fractions (conifer and diffuse-porous species) to have relatively low F in combination with a departure from Murray's law (Fig. 3).
Download figure to PowerPoint
We conclude that the optimal area profiles for hydraulics vs mechanics of free-standing trees are in direct conflict. Water conduction is most efficient in an area profile of an inverted top-heavy cone, while mechanical support is favoured by a straight column or upright cone. In conifers one cell type must execute these two tasks simultaneously, with opposing optima. We have described how the conducting network in A. concolor adjusts to mechanical constraints by a decrease in F to <1 and a conduit diameter taper in excess of the Murray's law optimum. However, with the evolution of vessels and fibres, angiosperms can more closely approach Murray's law while still maintaining a mechanically stable area profile.