Murray's law, the ‘Yarrum’ optimum, and the hydraulic architecture of compound leaves


Author for correspondence:
Katherine McCulloh
Tel: +1 541 737 4208


  • • There are two optima for maximizing hydraulic conductance per vasculature volume in plants. Murray's law (ML) predicts the optimal conduit taper for a fixed change in conduit number across branch ranks. The opposite, the Yarrum optimum (YO), predicts the optimal change in conduit number for a fixed taper.
  • • We derived the solution for YO and then evaluated compliance with both optima within the xylem of compound leaves, where conduits should have a minimal mechanical role. We sampled leaves from temperate ferns, and tropical and temperate angiosperms.
  • • Leaf vasculature exhibited greater agreement with ML than YO. Of the 14 comparisons in 13 species, 12 conformed to ML. The clear tendency towards ML indicates that taper is optimized for a constrained conduit number. Conduit number may be constrained by leaflet number, safety requirements, and the fact that the number of conduits is established before their diameter during development.
  • • Within a leaf, ML compliance requires leaf-specific conductivity to decrease from petiole to petiolule with the decrease in leaf area supplied. A similar scaling applied across species, indicating lower leaf-specific petiole conductivity in smaller leaves. Small leaf size should offset lower conductivity, and petiole conductance (conductivity/length) may be independent of leaf size.


Interest in the link between carbon uptake and water loss has yielded numerous efforts to quantify the efficiency of the water transport network of plants (Banavar et al., 1999; McCulloh et al., 2003, 2004; McCulloh & Sperry, 2005a,b; Anfodillo et al., 2006; Weitz et al., 2006; Atala & Lusk, 2008). Much of this work has focused on the xylem tissue of woody stems, where the xylem conduits (vessels or tracheids) may be providing structural support to the plant in addition to transporting water. The additional support task compromises the potential hydraulic efficiency of water transport (McCulloh et al., 2004). Within leaves, though, substantial structural support is provided by hydrostatic pressure and specialized nonvascular tissues such as collenchyma and sclerenchyma. To the extent that this nonxylary structural support holds up the leaf, this would theoretically leave the xylem conduits free to achieve their maximum transport efficiency. Assessing transport efficiency within the lamina is complicated by the typically reticulate networks that are losing water to transpiration (Canny, 1993; McCulloh & Sperry, 2005b). By contrast, it is relatively easy to quantify network efficiency in compound leaves because the vascular tissue can be compared at discrete branching levels, the petiole and petiolule. Using these distinct levels in 13 species, we compared compliance with two theoretical optima that maximize the hydraulic conductance of this simple two-ranked vascular network.

The first optimum, Murray's law (ML), predicts how the conduit diameters should change across branching points of a vascular network so as to minimize the power (work/time) driving a given flow rate through a network of fixed total volume (Murray, 1926). As Sherman (1981) demonstrated, a Murray law network also maximizes the hydraulic conductance (volume flow rate per pressure drop). Maximizing conductance (rather than minimizing power at one flow rate) is undoubtedly the real adaptive significance of Murray's law, because a maximum-conductance network is optimal at all flow rates (Sherman, 1981). Murray's law was derived for the specific case of the cardiovascular system which delivers blood through a single branching tube. The law states that conductance is maximized when the sum of the conduit radii cubed (Σr3) is conserved at all points along the flow path from aorta to the beginning of the capillaries. Conservation of Σr3 requires an increase in the cross-sectional area (Σr2) from aorta to capillaries.

Unlike the cardiovascular system for which it was developed, Murray's law does not define the sole hydraulic optimum for the plant vascular system. In the cardiovascular system the branching of the vascular network equals the branching of a single tube. The ratio of the number of daughter-to-mother tubes (NR) cannot be varied independently of the network branching structure. Murray's law gives the ‘best’ daughter-to-mother tube taper ratio (DR) for the NR dictated by branching architecture. The evolution of NR is constrained by the need to deliver fluids spatially, and Murray's law DR evolved to maximize the conductance of this mammalian vascular topography.

Xylem networks are more complex because they are composed of many conduits in parallel and in series at every branch level, including the trunk. That means that NR is potentially independent of the branching network. When two daughter arteries join at the aorta, NR = 2. When two petiolules each containing hundreds of conduits join at the petiole, NR does not have to be 2. NR can be any number greater than zero or less than the theoretical maximum for a petiole with just one conduit. Optimization of the network conductance is not just a matter of DR, because NR can also vary independently of the network branching topography.

The fact that there are at least two variables that are potentially free to vary across the branch junctions of plants means that there are at least two theoretical optima for maximizing conductance. The distinction between the two optima is shown graphically in Fig. 1. The contour lines on this figure are hydraulic conductances of a simple network with one petiole and two petiolules. The tubes within each branch rank are of constant diameter, and the total number of petiolule tubes is constant. All other parameters, including the volume of the network, are constant except for NR and DR.

Figure 1.

‘Contour map’ showing how the conductance of a branch network with fixed vascular volume is influenced by the daughter-to-mother ratio of tube diameters (DR) and numbers (NR; log–log plot). Conductance contours (upper x-axis, arbitrary units) were calculated for a network with one mother-daughter furcation using the reciprocal of Eqn 1. Volume (Eqn 2), lengths (l0 and l1), and number of daughter conduits (n1) were held constant. Murray's law (solid ML diagonal) gives the DR that maximizes network conductance when NR= constant. The Yarrum optimum (solid YO diagonal) gives the NR that maximizes conductance for DR= constant. The very best conductance is given by the Murray law DR for the theoretical maximum NR, which corresponds to a single tube in the mother axis (asterisk on ML line at NR max). Conduit area preservation (dashed AP diagonal) gives the DR that results in constant conduit cross-sectional area across ranks.

The contours in Fig. 1 describe a diagonal ridge that ascends to greater conductance as NR increases and DR decreases. The ascent of the ridge corresponds to fewer, larger tubes in the petiole. The ridge cannot be ascended indefinitely because of limits on permissible values of DR and NR. The NR reaches its maximum theoretical value when the petiole has just a single tube (NR max on ‘y’ axis; McCulloh et al., 2003). At NR=NR max, the greatest conductance is achieved at the horizontal tangent of the conductance contour where DR conforms to Murray's law (asterisk on ML diagonal in Fig. 1). This global optimum is represented by the cardiovascular system which is a single branched tube that approximates Murray law taper (Sherman, 1981).

Plants are never plumbed with a single branched tube and so are never at their NR max. Negative sap pressures make a single-tube network vulnerable to complete failure in the event of a single air leak or cavitation event (Ewers et al., 2007). Greater NR also results in a greater departure of the xylem network from area preservation (Fig. 1; AP diagonal) towards area-increasing branching at Murray's law. Extreme increases in the cross-sectional area of conduits would result in top-heavy networks that could be mechanically unstable. Hydraulic and mechanical safety considerations combine to keep the NR of plant xylem well below its theoretical maximum and potentially free to vary independently of the plant's branching system.

The alternative scenario to Murray's law for plant xylem is that DR is more constrained than NR for developmental or functional reasons. Such a constraint could result from a developmental link between vessel diameter and branch rank, or a limit to maximum vessel diameter. In this case, the ‘best’ network would have the minimum permissible DR, and the greatest conductance would be achieved at the vertical tangent of the conductance contour where NR conforms to the ‘YO’ diagonal. Because this optimum is the opposite of Murray's law, we call it the ‘Yarrum’ optimum (YO; Sperry et al., 2008).

Here, we derive both Murray's law and the Yarrum optimum for a simple network composed of the conduits within a petiolule and two daughter petiolules of equal length. The derivations make explicit the distinction between the two optima. Following Sherman's example (1981), the optima are derived from a maximum conductance criterion because this is more biologically relevant than the equivalent minimum power derivation. The series conductance of the network is represented as the reciprocal resistances (R), because they are additive in series. The Hagen–Poiseuille equation predicts:

image(Eqn 1)

where l is the axis length, n is the number of conduits in parallel, and r is the conduit radius, which is assumed constant within a rank. The subscript ‘0’ denotes petiole values, and subscript ‘1’ denotes the totals for the daughter petiolules. The DR is r1/r0. The mean radiuses of petiole or petiolule vessels are assumed to substitute for r0 or r1, and mean petiolule vessel diameters are not expected to differ within a compound leaf when leaflets are of approximately similar size. Constant η is the fluid viscosity and k is the factor by which the resistivity exceeds the Hagen–Poiseuille prediction because of end-walls and other obstructions (k ≥ 1). The volume, V, of this network is

image(Eqn 2)

Network volume is proportional to the conduit wall volume if wall thickness is proportional to conduit diameter (McCulloh & Sperry, 2006); the latter is approximately true for xylem of a given cavitation resistance (Hacke et al., 2001). In both derivations, we hold the vascular investment, V, constant and solve for the minimum R under either Murray's law or Yarrum conditions. To derive Murray's law, we hold the numbers of conduits constant (n0 and n1) and vary the conduit taper (DR) to find the value that minimizes R. Because volume (V) is constant, we must also allow r0 to vary with DR. We use the Lagrange multiplier method to find where R (DR, r0) (Eqn 1) is minimized subject to the constraint that V(DR, r0) = 0, where we have subtracted a constant from Eqn 2 to set V = 0. At the minimum R, the Lagrange multiplier, λ, defines the two equalities (Edwards & Penney, 1998, p. 864):

image(Eqn 3a)
image(Eqn 3b)

From Eqns 1 and 2 the partial derivatives in Eqn 3 are:

image(Eqn 4a)
image(Eqn 4b)
image(Eqn 4c)
image(Eqn 4d)

Substituting 4a and 4b into 3a yields

image(Eqn 5)

Substituting λ, 4c and 4d into 3b, and simplifying, yields

DR= (n0/n1)1/3


DR = NR−1/3(Eqn 6)

for NR=n1/n0. Equation 6 is Murray's law for the special case where tubes within a rank are of equal diameter. However, we have previously shown that conservation of Σr3 across branch points is within 3% whether computed from actual vessel distributions or their mean values (McCulloh & Sperry, 2006). Note that the lengths cancel out, indicating that the optimal taper is independent of branch lengths.

To find the Yarrum optimum, we hold the taper constant (DR) and vary the relative numbers of conduits across ranks (varying n0 for a constant n1) to find the value that minimizes R. Because volume (V) is constant, we must allow r0 to vary with n0. As before, we use the Lagrange multiplier method to find where R(n0, r0) (Eqn 1) is minimized subject to the constraint that V(n0, r0) = 0. At the minimum R, the Lagrange multiplier, λ, defines the two equalities:

image(Eqn 7a)
image(Eqn 7b)

From Eqns 1 and 2 the partial derivatives in 7a are:

image(Eqn 8a)
image(Eqn 8b)

The derivatives for 7b are 4c and 4d. Substituting 8a and 8b into 7a yields:

image(Eqn 9)

Substituting λ, 4c and 4d into 7b, and simplifying, gives:

image(Eqn 10a)


image(Eqn 10b)

where the branch length ratio LR = l1/l0. Using the quadratic formula to solve for NR yields the following relevant root (the other root gives negative NR; signs in the valid root have been simplified):

image(Eqn 11)

which is the Yarrum optimum. Note that the branch lengths do not cancel out, indicating that the optimal NR depends on the relative lengths of the petiole and petiolule. The shorter the petiolules relative to the petiole (smaller the LR), the fewer conduits they need in parallel (relative to the petiole) to minimize the total resistance (the smaller the optimal NR). As LR approaches zero, the lowest optimal NR converges on 1.41 inline image. For typical DR < 1, Yarrum's NR will always exceed 1.4, meaning that petiolule conduits will always be more numerous than petiole conduits at the Yarrum optimum.

The existence of two hydraulic optima raises the question of what variable is most constrained during evolution and development. If the evolution of NR is more constrained (as in the single-tube mammalian cardiovascular system where it is at NR max), then the vascular conductance through the branching system can only be optimized by evolving the value of DR predicted by Murray law. Alternatively, if the DR and LR are more constrained, then the vascular conductance through the branch system would be maximized by evolving the Yarrum optimum NR. To date, the Yarrum optimum has not been recognized, or evaluated, in any system. To address the question of whether leaf vasculature follows Murray's law, or the Yarrum optimum, or conduit area preservation, or none of these patterns, we measured DR, LR, and NR across the petiole vs petiolule ranks of compound leaves and compared them with the predictions. Leaves were studied in six tropical and five temperate angiosperm species, and in two temperate fern species. In addition to evaluating these two optima, we examined the evidence for size- and species-independent scaling patterns in the hydraulic architecture of leaves.

Materials and Methods

Plant material

Compound leaves were collected in the Republic of Panama in April 2005 and in the USA in September 2006 and July 2007. The six tropical species (Table 1) were collected in two seasonally dry tropical forest sites (9°N, 79°W). Three species with leaves exposed in the upper canopy, Schefflera morototoni (Aubl.) Maguire, Steyerm. & Frodin (Araliaceae), Arrabidaea candicans Dc. (Bignoniaceae), and Serjania cornigera Turcz. (Sapindaceae), were accessed using the Smithsonian Tropical Research Institute (STRI) Canopy Crane in the Parque Metropolitano, near Panama City. Leaves from three species growing in the understory, Dalbergia retusa Hemsl. (Fabaceae), Paullinia pterocarpa Triana & Planch. (Sapindaceae), and Stizophyllum riparium (H.B. & K.) Sandwith (Bignoniaceae), were collected in Soberania National Park near Gamboa, Panama. The temperate angiosperms were Daucus carota L. (Apiaceae), Rubus discolor Weihe & Nees (Rosaceae), Fragaria×ananassa Duchesne (Rosaceae), Sambucus caerulea Raf. (Caprifoliaceae), and Clematis armandii Franch. (Ranunculaceae), and the temperate ferns used were Adiantum pedatum L. (Adiantaceae) and Pteridium aquilinum (L.) Kuhn (Dennstaedtiaceae). All temperate species were collected in and around Corvallis, OR, USA (45°N, 123°W; Table 1). When names were in conflict, International Plant Names Index conventions were followed (IPNI;

Table 1.  Abbreviations used in figures and characteristics of the species examined (tropical angiosperms, temperate angiosperms and temperate ferns are grouped separately)
SpeciesAbbr.Habit and habitatbPetiole vessel D (µm) (± SD)cPetiolule vessel D (µm) (± SD) LR (± SD)dNR (± SD)eYO predicted inline imageDR (± SD)ML predicted inline imageMLhLeaf typei
  • a

    The asterisk here indicates that, for Adiantum pedatum, one rank distal to the petiolule rank was measured in addition to the petiole and petiolule ranks. For this comparison, the petiolule value for vessel diameter is listed under ‘petiole,’ because it is more basal than the third rank, which is treated as the ‘petiolule’.

  • b

    Abbreviations for the habit and habitat: L, liana; T, tree; S, shrub; H, herbaceous plant; F, fern; Tr, tropical; Te, temperate.

  • c

    Petiole and petiolule vessel diameters (D) are the grand mean of the average diameters of the n = 5 (n = 4 for A. pedatum) leaves from each species.

  • d

    LR is the median ratio for each species of the length of the petiole:petiolule shoots.

  • e

    NR, the ratio of the number of conduits in the petiolule: petiole, and DR, the ratio of the mean conduit diameter in the petiolule: petiole, are the median of the ratios from n = 5 (n = 4 for A. pedatum) leaves.

  • f

    The optimal NR predicted by the Yarrum optimum (YO) based on the observed DR.

  • g

    The optimal DR predicted by Murray's law (ML) based on the observed NR.

  • h

    Compliance with ML is based on whether the sum of the conduit radii cubed in petiolules was different from that in petioles as indicated by P-values from paired t-tests greater than 0.05. Asterisks indicate the two species that deviated from the ML optimum.

  • i

    The compound leaves were separated into pinnate (Pinn), which have a rachis, and palmate (Palm), which do not.

Arrabidaea candicansACL, Tr18.2 (1.7)14.6 (0.9)0.47 (0.07)2.9 (0.3)4.50.83 (0.05)0.70P = 0.70Palm
Dalbergia retusaDRT, Tr21.0 (1.2)10.5 (0.8)0.13 (0.03)3.7 (0.1)35.90.50 (0.01)0.65P < 0.001*Pinn
Paullinia pterocarpaPPL, Tr17.4 (1.6)14.5 (1.1)0.12 (0.01)1.8 (0.2)12.60.83 (0.04)0.83P = 0.13Pinn
Schefflera morototoniSMT, Tr53.9 (2.3)33.8 (1.7)0.09 (0.01)6.0 (0.3)28.40.63 (0.01)0.55P < 0.01*Palm
Serjania cornigeraSCL, Tr22.4 (2.5)16.8 (1.0)0.15 (0.01)2.4 (0.1)13.10.74 (0.02)0.74P = 0.18Palm
Stizophyllum ripariumSRL, Tr21.7 (1.2)15.2 (1.2)0.70 (0.03)2.8 (0.2)5.50.72 (0.02)0.71P = 0.10Palm
Clematis armandiiCAL, Te17.2 (1.4)12.8 (1.1)0.36 (0.08)1.8 (0.2)7.20.73 (0.04)0.83P = 0.34Palm
Daucus carotaDCH, Te15.1 (0.4)12.3 (1.7)0.13 (0.04)1.3 (0.1)11.40.86 (0.02)0.91P = 0.41Palm
Fragaria×ananassaFAH, Te7.7 (0.9)6.1 (0.5)0.03 (0.01)2.3 (0.3)46.90.79 (0.04)0.76P = 0.08Palm
Rubus discolorRDS, Te18.8 (1.0)11.6 (0.5)0.18 (0.04)3.2 (0.6)15.90.63 (0.03)0.68P = 0.17Pinn
Sambucus caeruleaSaCT, Te21.9 (4.0)13.1 (2.0)0.13 (0.02)3.3 (0.4)24.30.59 (0.02)0.67P = 0.49Pinn
Adiantum pedatumAPF, Te22.4 (1.4)16.7 (1.8)0.06 (0.02)1.5 (0.3)26.30.79 (0.06)0.87P = 0.14Palm
Adiantum pedatum*aAP*aF, Te16.7 (1.8)15.4 (1.5)0.70 (0.12)1.3 (0.1)2.90.92 (0.01)0.92P = 0.46Palm
Pteridium aquilinumPAF, Te31.2 (5.3)25.1 (2.3)0.09 (0.06)1.7 (0.1)18.00.80 (0.07)0.84P = 0.84Pinn

Five leaves from one individual were sampled for each species. For all species, the ranks of petiolule and petiole were compared, except for A. pedatum, for which a third, more distal rank was also compared with the petiolule rank. For this species, only four leaves were measured.

For the tropical species, petioles and petiolules were stored in vials containing 50% ethanol and shipped to Oregon State University for sectioning and image analysis. Temperate species were sectioned in fresh condition. Total leaflet area was measured for each leaf using either a leaf area meter (Li-Cor 3100C; Li-Cor, Lincoln, NE, USA), or a flatbed scanner (Hewlett Packard Scanjet 6200C). Scanned leaf areas were measured using ImageJ (NIH, USA,

Anatomical measurements

Cross-sections were hand-cut near the midpoints of the petioles and petiolules using a sharp razor blade and stained with 0.5% toluidine blue or 0.5% safranin O to clearly distinguish the xylem vessels from surrounding tissues. Images of cross-sections were taken with a Photometrics Coolsnap camera (Tucson, AZ, USA) mounted on a Nikon Eclipse E400 microscope connected to a PC using metavue software (Universal Imaging Corp., Downington, PA, USA). For all species except S. morototoni, all vessels in the petioles and petiolules from all five leaves were measured. Vessel cross-sectional area measurements were made on the images using ImageJ, and from these measurements the radius (r) of each vessel was calculated as the radius of a circle with the same area.

The large number of vessels in the petioles and petiolules of S. morototoni leaves made it necessary to determine the average conduit number and diameter per vascular bundle area on a subsample and scale up to the whole cross-section from its total bundle area. Every fifth bundle around the circumference of the cross-section was measured to obtain the average number and diameter of conduits per vascular bundle area.

Length ratio measurements

The ratio of petiolule:petiole lengths (LR) was determined in order to estimate the Yarrum optimum NR (Eqn 11). Species-specific length ratios were estimated from measurements on live material, herbarium sheets of pressed plants, or both. The lengths of petioles and petiolules from four to eight leaves were measured for each species.

Data analysis and statistics

To test whether plants more closely complied with the Murray's law or the Yarrum optimum, the mean predicted error was calculated for each relationship. The mean predicted error from the YO optimum was calculated from the absolute value of the difference: (measured NR/optimal NR) – 1 summed across all species. The optimal NR was calculated from the median DR and LR for each species from Eqn 11. The median ratios were used rather than the mean to avoid bias created by the averaging of ratios (Lawren Sack, pers. comm.; Packard & Boardman, 1988; Jasienski & Bazzaz, 1999). The analogous mean square error was calculated for the ML optimum from the difference: (measured DR/optimal DR) – 1. The optimal DR was calculated from the median NR for each species from Eqn 6. A mean predicted error was also calculated for the deviation of median DR from the value required for area-preserving branching (‘AP’, conserved conduit lumen area). In this case the difference was the following: (measured DR/DR_AP) – 1, where DR_AP is the diameter ratio that conserves lumen area for a particular inline image.

Species-specific compliance with the Murray's law optimum was also evaluated by testing the extent of Σr3 conservation (r= conduit radius) between the ranks of the petiole and the petiolule (McCulloh et al., 2003, 2004). For each leaf and rank, the Σr3/distal leaf area was multiplied by the total leaf area of all five leaves for that species. This provided five estimates of the combined Σr3 for each rank in each species. Cubing the vessel radii strongly skewed the data, so the Σr3 values were log transformed for statistical analyses. The similarity between the total rank Σr3 values was assessed using a paired t-test. If the P-value was > 0.05, Murray's law could not be rejected for that species. The equivalent test for compliance with YO was not possible because of the complexity of the YO optimum (Eqn 11).

Reduced major axis (RMA) regressions were used to compare leaf traits. The slopes were calculated from ordinary least squares (OLS) regression slopes as: RMA slope = OLS slope/R, where R is the correlation coefficient. RMA intercepts were calculated as Y– RMA slope X, where Y is the mean log y-axis value and X is the mean log x-axis value (Quinn & Keough, 2002). Slope values and y-intercepts from intraspecific comparisons were determined to be different from the pooled, interspecific comparisons when the 95% confidence intervals did not overlap.


On the landscape of possible DR vs NR combinations, the species clustered more closely around the ML (Murray's law; Fig. 2a) optimum than the YO one (Yarrum optimum; Fig. 2b). The RMA regression of the observed DR vs the Murray predicted DR was not significantly different from the 1 : 1 line (slope = 1.08). The corresponding regression for observed NR vs the Yarrum predicted NR was significantly different from the 1 : 1 line (slope = 0.12), with measured NR falling far short of the YO optimum. Greater agreement with ML was also supported by the much smaller mean predicted error for the ML optimum than the YO alternative (1.1 vs 10.9, respectively). For the individual species, Murray's law could not be rejected for 11 of the 13 species measured (Table 1). The comparison of the petiolule versus the third rank in A. pedatum also was consistent with ML. The two species that deviated from ML were tropical tree species (S. morototoni and D. retusa). The former had less conduit taper than predicted by ML, and the latter exhibited more taper than the optimum and was quite close to the area-preserving line (Fig. 2c).

Figure 2.

(a) The median measured DR, which is the ratio of the mean vessel diameter in petiolules: petiole, versus the DR predicted by Murray's law (ML). The solid line indicates pooled reduced major axis (RMA) regression (slope = 1.08, R2 = 0.62) with dash-dotted 95% confidence intervals. The gray dashed line is the 1 : 1 ML optimum. The asterisks designate the two species that deviated significantly from ML (Table 1). (b) The median measured NR, which is the ratio of the number of vessels in petiolules: petiole, vs the optimum NR predicted by the Yarrum optimum (YO). The solid line indicates the pooled RMA regression (slope = 0.12, R2 = 0.52) with dash-dotted 95% confidence intervals, and the dashed line is the 1 : 1 YO optimum. (c) The median measured DR vs the DR that would result in conduit area preservation between the petiolule and petiole. The solid line shows the pooled RMA regression (slope = 0.83, R2 = 0.64) with dash-dotted 95% confidence intervals. The dashed gray line is for area-preserved branching of the vascular network. Error bars on symbols represent standard deviations from the medians among n = 5 leaves (or n = 4 for Adiantum pedatum). Species abbreviations are defined in Table 1.

Consistent with the tendency towards Murray's law, the data generally fell above the area-preserving line (Fig. 2c). Although the slope of the RMA regression in Fig. 2(c) was not significantly different from the area-preserving slope of 1, the y-intercept was marginally greater than zero, which indicated that the lumen cross-sectional area summed across the petiolule rank was greater than in the petiole for most species. The mean predicted deviation from the AP line was also greater than for the ML optimum (2.0 vs 1.0, respectively). This increase in area occurred despite the decline in vessel diameters from the petiole to petiolule ranks, because the number of vessels increased (Table 1). Area-increasing conduit branching is predicted for Murray's law when NR > 1.

Of most consequence to the plant is how much a given deviation from ML or YO anatomy costs in terms of lost conductivity per investment. Again, ML appears to define the more relevant optimum. Species averaged 97 ± 5% of the peak conductivity at ML (Fig. 3) vs 47 ± 26% of the YO value (not shown). Of the two species that deviated statistically from ML anatomy (S. morototoni and D. retusa), in only one of them did this result in a sizable deviation from ML conductance (Fig. 3; 82% of optimum). Interestingly, this species (D. retusa) fell close to the area-preserving condition (Fig. 2c). Excluding D. retusa, species averaged 98 ± 2% of their ML conductivity.

Figure 3.

The fraction of the maximum possible network conductivity for Murray's law (ML) vs the ratio of the measured DR, which is the ratio of the mean vessel diameter in the petiolule: petiole, to the ML predicted optimum DR. The solid curve shows the shape of the optimum, calculated for the two-rank network depicted in Fig. 1 for NR= constant. NR is the ratio of the number of conduits in petiolules: petiole. The curve is essentially independent of the particular NR value. Species conductivities for the same two-rank (petiole–petiolule) network were calculated using their measured NR value, and comparing conductivity for the observed vs the Murray law DR. Species abbreviations are defined in Table 1.

The general agreement with ML suggests that, within a species, DR adjusts to a ‘preset’ value of NR rather than vice versa as would be the case for the Yarrum optimum. In all species, NR exceeded the often assumed value of 1 (West et al., 1997), ranging from 1.3 in the more distal comparison of A. pedatum to ∼6 in S. morototoni, and with a median of 2.4 ± 0.4 (Figs 2b, 4). An advantage of NR > 1 is that, for a given leaf structure and vascular volume, the larger the NR the greater the conducting efficiency at Murray's law (Fig. 1; McCulloh et al., 2003). Variation in NR between species was at least in part related to differences in leaf structure: species with more leaflets per leaf tended to have greater NR (Fig. 4).

Figure 4.

The median ratio of the number of vessels in petiolules: petiole for each species (NR) versus the average number of leaflets. The dashed line shows the 1 : 1 relationship and the solid line shows the regression of the data. Regression analysis showed that the slope was different from 0 (slope = 0.23; P-value < 0.001; R2 = 0.51). Symbol markings are defined in Fig. 2 for each species, and symbol shapes are: circles, tropical species; squares, temperate angiosperm species; and triangles, temperate fern species. Error bars are not shown for simplicity, but standard deviations in the number ratio are the same as in Fig. 2 and deviation in the average leaflet number ranged from 0 to 9%.

Conduit number also scaled significantly with leaf and leaflet area within most species, and also across species (Fig. 5a). In order for NR to be > 1 across petiole to petiolule ranks, the number of conduits must decrease less than the drop in leaf area moving from whole leaf to leaflet. Hence, the species-specific regressions of conduit number vs leaf and leaflet area supplied followed power functions with an exponent of < 1 (Fig. 5a; species log–log regressions, all slopes < 1), which is the value required for NR = 1 (Appendix, Eqn A2). Exponents varied from −0.7 to 0.8 (Table 2), with the variation in part reflecting the different numbers of leaflets in the different species (Fig. 4). The negative exponents, for example, corresponded to the species with the fewest leaflets per leaf (A. candicans and S. riparium). There was also significant variation between species in the number of conduits for a given leaf area, as indicated by significant differences in the intercepts of the log–log relationships (Table 2). Pooling across all species indicated a significant increase in conduit number with area supplied with an exponent of 0.53. The low R2 of this pooled regression (0.35) reflects the variation in slopes and intercepts between species: four of the 13 species had slopes significantly different from the pooled regression, and nine species had different intercepts (Table 2).

Figure 5.

Intra- and interspecific log–log scaling of petiole and petiolule vasculature with the area of the leaf or leaflet (A) supplied. (a) Number of vessels, (b) the mean diameter of vessels, and (c) the theoretical hydraulic conductivity of petioles and petiolules. Petiolule and leaflet data are averages per leaf. Conductivity was calculated from mean vessel diameter and number using the Hagen Poiseuille equation. Solid black lines show intraspecific regressions across petiole (open symbols) and petiolule (closed symbols) ranks. Statistics of these intraspecific regressions are shown in Table 2. Solid gray lines show pooled regressions with 95% confidence intervals (dashed lines). In (b) and (c), the dotted lines show the scaling required for Murray's law (‘ML’; see Appendix). In (c), the scaling is also shown for invariant leaf-specific conductivity across leaf size (‘KL’).

Table 2.  The slope, R2 values and y-intercepts from the reduced major axis regressions shown in Fig. 5
SpeciesVessel number vs leaf areaVessel diameter vs leaf areaTheoretical conductivity vs leaf area
Slope (± CI)aR2y-interceptSlope (± CI)R2y-interceptSlope (± CI)R2y-intercept
  • a

    The 95% confidence intervals (CIs) of the slope are indicated in parentheses.

  • b

    Bold typeface indicates that the slope or y-intercept deviated from the pooled values.

Arrabidaea candicans−0.68 (0.38)b0.543.630.36 (0.13)0.810.53−0.85 (0.68)0.820.43
Dalbergia retusa0.53 (0.08)0.971.370.26 (0.02)0.990.671.57 (0.11)0.99−4.13
Paullinia pterocarpa0.62 (0.12)0.940.870.14 (0.06)0.710.901.14 (0.21)0.97−3.64
Schefflera morototoni0.28 (0.05)0.962.040.19 (0.02)0.981.071.09 (0.09)0.99−2.05
Serjania cornigera0.21 (0.09)0.721.900.28 (0.05)0.950.761.29 (0.18)0.97−3.20
Stizophyllum riparium−0.54 (0.42)0.113.380.43 (0.17)0.760.481.41 (0.42)0.87−3.36
Clematis armandii0.82 (0.48)0.490.680.30 (0.11)0.800.631.73 (0.47)0.89−4.50
Daucus carota0.62 (0.23)0.791.300.22 (0.13)0.520.881.42 (0.63)0.71−3.28
Fragaria ¥ananassa0.39 (0.29)0.162.370.24 (0.09)0.800.521.08 (0.32)0.87−3.40
Rubus discolor0.32 (0.16)0.642.040.29 (0.09)0.850.691.35 (0.27)0.94−3.22
Sambucus caerulea0.35 (0.16)0.701.500.33 (0.04)0.980.731.63 (0.26)0.96−3.73
Adiantum pedatum0.47 (0.24)0.741.260.29 (0.23)0.360.551.60 (1.07)0.55−4.72
Pteridium aquilinum0.70 (0.13)0.950.840.20 (0.13)0.400.881.43 (0.58)0.76−3.68
Pooled0.53 (0.08)0.351.380.31 (0.03)0.710.621.50 (1.12)0.82−3.79

Average vessel diameter also scaled significantly with leaf area supplied both within and between species (Fig. 5b). Species-specific scaling exponents ranged from 0.1 to 0.4, which corresponds well with the 0.1 to 0.6 range predicted for Murray's law from the NR scaling in Fig. 5(a) (Appendix, Eqn A7). The pooled data exhibited an exponent of 0.31 for diameter vs leaf area scaling, deviating slightly from the 0.16 value for perfect ML compliance across species (Fig. 5b, dotted ML line; Appendix, Eqn A7). The R2 of 0.71 for the pooled regression suggests greater convergence of intra- and interspecific scaling for conduit diameters than for conduit numbers (Fig. 5a). Only three of the 13 species had slopes different from the pooled value, and three species had different intercepts (Table 2).

The greatest convergence between intra- and interspecific scaling was in the calculated conductivity vs leaf area (Fig. 5c). Petiole or petiolule theoretical conductivity scaled with leaf area supplied to the 1.50 power across species (Fig. 5c; Table 2). The relatively high R2 of 0.82 for the pooled regression reflected the fact that only one of 13 species deviated from the pooled regression for slope and one for intercept. The pooled slope of 1.50 is greater than the exponent of 1 for conductivity increasing in direct proportion to leaf area (Fig. 5c; dotted ‘KL’ line for constant leaf-specific conductivity). An increase in leaf-specific conductivity with leaf area was consistent with the general compliance with ML, which predicts a conductivity vs leaf area exponent of 1.16 (Fig. 5c, ML line; Appendix Eqn A10). Computing conductivities on the basis of the actual conduit distributions rather than the average conduit diameter did not change the indication that conductivity increases disproportionately with leaf area (R2 = 0.83, slope = 1.23; not shown).


Compound leaves from a wide range of species complied more closely with Murray's law (ML) than with the Yarrum optimum (YO; Fig. 2a vs 2b). Only two species deviated statistically from ML (Table 1), but even the greatest deviant (D. retusa) was within 18% of the broad optimal ML conductivity (Fig. 3; Sherman et al., 1989). These results suggest that vessel diameter ratio (DR) ‘tunes’ to a constrained number ratio (NR) and not vice versa. This sequence makes sense for several reasons. First, the number of conduits (hence NR) is determined much earlier in development than conduit diameter (hence DR). Yarrum's NR also depends on the ratio of petiolule-to-petiole length (LR), which like DR is determined fairly late in development. Secondly, although plants cannot achieve the theoretical NR max of a maximally efficient single-tubed vascular network, they may have evolved to achieve the ‘next best’NR within species-specific constraints of leaflet number (Fig. 4) and considerations of transport safety and redundancy. Finally, achieving the Yarrum optimum requires an even more extreme area-increasing branching of conduits than Murray's law (Fig. 1), which may not be possible within the confines of petiole/petiolule architecture. Although it is known that vessel diameters increase basipetally as the vascular auxin concentration declines (Aloni & Zimmermann, 1983), how the plant develops the particular ML diameter ratio is unknown.

Conduits within woody stems tend to be closer to area preserving than the ML optimum, suggesting that the need to avoid top-heavy networks constrains the hydraulic efficiency of trees (McCulloh et al., 2004; Atala & Lusk, 2008). It is worth noting that this area preservation is not da Vinci's rule, which refers to preservation of the total cross-sectional area of branches, not their internal conducting tubes (Horn, 2000). Interestingly, the compound leaf that deviated most significantly from Murray's law (leaves of D. retusa; Fig. 2a) was fairly close to the area-preserving line (Fig. 2c). This species had 12–14 leaflets arranged pinnately on a long, rigid axis and perhaps relied more than other species on its xylem conduits for mechanical support. This reliance may have prevented area-increasing branching of vessels which otherwise would have been more hydraulically efficient.

Givnish (1978) suggested that a benefit of compound leaves was as ‘throw-away branches’ that could achieve the same area as simple leaves for a smaller investment in woody biomass. Here, we propose that, by following Murray's law more closely than stems, compound leaves are also more hydraulically efficient than the equivalent stem structure. The efficiency is achieved by a greater uncoupling of mechanical support from hydraulic supply, which allows the conduit network to be area-increasing as required for the most efficient Murray law networks.

Intraspecific scaling of the petiole–petiolule network with leaf area did not carry over precisely to a single universal interspecific scaling (Fig. 5). The most variation between species was in how the number of conduits scaled with leaf area (Fig. 5a). However, consistent with NR being > 1, all intraspecific exponents were < 1; as was the pooled exponent (Table 2). Diameter scaled more consistently across species with leaf area, but still with significant interspecific variation (Fig. 5b). This result is consistent with the strong degree of scaling between the hydraulically weighted vessel diameter and leaf area observed by Coomes et al. (2008) across multiple vein orders in leaves of various oak species. The variation in diameter scaling observed in our data partially compensated for variation in conduit number, because theoretical conductivity (calculated from diameter and number) exhibited a fairly strong interspecific relationship with leaf area (Fig. 5c). Actual conductivity tends to be proportional to the theoretical value (Sperry et al., 2006; Choat et al., 2008). These data imply that leaf-specific conductivity of petioles increases with leaf area, a trend consistent with following Murray's law. Experimental data on hydraulic conductivities measured in petioles of simple leaves are consistent with this trend (Sack et al., 2003).

Unlike petiole conductivity, petiole conductance (= conductivity/length) may be relatively invariant with leaf size because petioles tend to be longer in larger leaves (Niklas, 1994; Sack et al., 2003). A recent survey found that petiole length scaled with leaf area to approximately the 0.5 power (Arcand et al., 2008). Combination of this with the 1.5 scaling of conductivity (Fig. 5c) would predict that petiole conductance scales isometrically with leaf area as does lamina conductance (Sack et al., 2003). Variation in conductivity or conductance by leaf area scaling may be related in part to light environment and the prevailing evaporative gradient, with sunlit leaves or leaves in drier air developing greater conductivity for a given leaf area than shaded leaves or leaves in humid air (Sack et al., 2003).

The petioles and petiolules of compound leaves are discrete branching ranks that provide an opportunity to examine compliance with two distinct optima that both maximize hydraulic conductance per investment in vascular tissue volume. The Yarrum optimum does not seem to represent an achievable optimum, suggesting that NR is more constrained than DR in the evolution and development of the leaf vasculature. Most species tracked the alternative Murray law optimum. Convergence on a hydraulic optimum suggests the importance of hydraulic efficiency for maximizing plant fitness. Deviations from the ML optimum in some leaves and in most stems tend to approach the alternative area-preserving behavior, a more adaptive alternative when the xylem conduits are called upon to supply mechanical support in addition to water.


The authors are deeply indebted to Osvaldo Calderon for his identification of the Panamanian species. We thank the Smithsonian Tropical Research Institute for laboratory facilities and support, as well as the expertise of the Canopy Crane operators. KM thanks Diego Quesada and Mylthon Jimenez Castillo for field and linguistic assistance, and is grateful to Kathryn Irvine and Lisa Ganio for statistical advice. Comments of David Ackerly, Lawren Sack and two anonymous reviewers significantly improved the manuscript. Fred Adler of the University of Utah provided input on the derivations of ML and YO. Funding was provided by NSF grants 99-05012 to FM and 05-44470 to KM, JS, FM and BL. CA was funded by MECESUP UCO0214 and CONICYT.

Appendix: Relations between scaling exponents in Fig. 5

Figure 5(a) shows that the number of conduits (N) is proportional to leaf or leaflet area (A) to the ‘x’ power:

NαAx(Eqn A1)

Accordingly, the number of conduits per petiolule/number of conduits per petiole ≈ (Ai/AL)x, where Ai is the average leaflet area and AL is the whole leaf area. The approximation is because mean leaflet area is used and the proportionality in Eqn A1 is estimated from a regression. Multiplying (Ai/AL)x by the number of petiolules (≈ AL/Ai) yields

NR ≈ (Ai/AL)x (AL/Ai) (Eqn A2.)

Using AR = Ai/AL,

NR ≈ AR(x–1)(Eqn A3.)

Because AR < 1, it is clear that for NR > 1 as observed, x must be < 1.

Fig. 5(b) shows that the average diameter of conduits (D) is proportional to leaf area to the ‘y’ power:

DαAy(Eqn A4.)

This yields

image(Eqn A5)

From Eqn 6, inline image at Murray's law. Equating A3 and A5 at Murray's law gives:

image(Eqn A6)

Thus, at Murray's law:

y = (1 − x)/3 (Eqn A7)

For x = 0.53, the pooled regression exponent from Fig. 5(a), y = 0.16 for Murray's law (ML line in Fig. 5b).

Figure 5(c) shows that conductivity (K) scales with A to an exponent ‘z’:

K α Az(Eqn A8)

Assuming that KαND4, then from A1 and A4:

KαA(x+4y)(Eqn A9)

which means z = x + 4y. Plugging in y from A7 gives z as a function of x for Murray's law:

z = 1.33 − 0.33x(Eqn A10)

For x = 0.53 from Fig. 5(a), z = 1.16 at Murray's law (ML line in Fig. 5c).