Hydraulic trade-offs and space filling enable better
predictions of vascular structure and function
in plants
V. M. Savagea,b,1, L. P. Bentleyc, B. J. Enquistb,c, J. S. Sperryd, D. D. Smithd, P. B. Reiche, and E. I. von Allmend
a
Department of Biomathematics, David Geffen School of Medicine, and Department of Ecology and Evolutionary Biology, University of California, Los
Angeles, CA 90095; bSanta Fe Institute, Santa Fe, NM 87501; cDepartment of Ecology and Evolutionary Biology, University of Arizona, Tucson, AZ 85721;
Department of Biology, University of Utah, Salt Lake City, UT 84112; and eDepartment of Forest Resources, University of Minnesota, St. Paul, MN 55108
d
Edited by Simon A. Levin, Princeton University, Princeton, NJ, and approved November 9, 2010 (received for review August 17, 2010)
Plant vascular networks are central to botanical form, function,
and diversity. Here, we develop a theory for plant network scaling
that is based on optimal space filling by the vascular system along
with trade-offs between hydraulic safety and efficiency. Including
these evolutionary drivers leads to predictions for sap flow, the
taper of the radii of xylem conduits from trunk to terminal twig,
and how the frequency of xylem conduits varies with conduit
radius. To test our predictions, we use comprehensive empirical
measurements of maple, oak, and pine trees and complementary
literature data that we obtained for a wide range of tree species.
This robust intra- and interspecific assessment of our botanical
network model indicates that the central tendency of observed
scaling properties supports our predictions much better than the
West, Brown, and Enquist (WBE) or pipe models. Consequently,
our model is a more accurate description of vascular architecture
than what is given by existing network models and should be used
as a baseline to understand and to predict the scaling of individual
plants to whole forests. In addition, our model is flexible enough
to allow the quantification of species variation around rules for
network design. These results suggest that the evolutionary drivers that we propose have been fundamental in determining how
physiological processes scale within and across plant species.
U
nderstanding the coevolution of plant internal vascular
networks and external branching networks is essential to
predict botanical form and function (1–4). Seminal studies have
attempted to unite these internal and external networks (3, 5, 6).
A decade ago West, Brown, and Enquist (3) proposed a model
(WBE) that focuses on the primacy of vascular networks, predicts myriad aspects of plant form and function (3, 7), and has
subsequently been tested by the collection of new data for vascular networks (8, 9) and analyses of fluxes through plants (10,
11), forests, and ecosystems (12, 13). Since the publication of the
WBE model, several criticisms have been published that question its basic framework, assumptions, and generality (14–16).
Indeed, focusing on plant models, several studies have: (i) highlighted how hydraulic safety and efficiency may have shaped the
evolution of vascular networks (2, 17), (ii) questioned whether
vascular safety and efficiency are adequately described by the WBE
model (8, 18, 19), and (iii) revealed empirical patterns that contradict parts of the WBE model (9, 20–23). For example, building
on earlier work (24, 25), Sperry and colleagues (2) compiled data
for the xylem conduits that transport water in plants, and they
documented a general principle termed the “packing rule”—the
frequency of xylem conduits varies approximately inversely with the
square of conduit radius. This packing rule contradicts the WBE
model’s assumption that conduit frequency remains unchanged as
conduit radii taper, decreasing in size from trunk to terminal twig.
Safety and efficiency considerations have been proposed to underlie the packing rule (2), suggesting new theory is needed to accurately describe vascular architecture (2, 19, 26, 27).
Here, we construct a plant network model that allows for
a flexible and realistic representation of xylem vascular networks.
We show how the internal vascular and external branching networks are coupled and jointly optimized by general evolutionary
22722–22727 | PNAS | December 28, 2010 | vol. 107 | no. 52
principles (2, 3, 22, 27–29). We argue that the evolution of plant
networks is primarily guided by (i) space-filling geometries to
maximize carbon uptake by leaves and sap flow through conduits;
(ii) increasing hydraulic conductance and resource supply to
leaves; (iii) protection against embolism and associated decreases
in vascular conductance; (iv) enforcement of biomechanical constraints uniformly across a plant; and (v) independence of terminal
twig size, flow rate, and internal architecture with plant size.
Principles ii and iii are more central in our model than in the WBE
model (3, 5). Furthermore, we apply space filling not only to the
external network (like WBE) but also to the internal network,
allowing us to relate conduit radius to conduit frequency. These
principles together enable us to predict and incorporate the
packing rule and other vital plant properties that better match real
plant networks and empirical data.
Theoretical Framework
Geometric Model for Internal and External Networks in Plants. Plants
are characterized as a symmetric, hierarchical branching network
within our model (Fig. 1). Each branching junction is considered to
be symmetric because the daughter branches have identical properties to each other for both the external branch and the internal
conduits (30). The overall network is hierarchical because each
branch segment can be labeled as being within level k, according to
the number of branching points from the trunk to that branch segment (Fig. 1). We place the external and internal networks on the
same footing by using three parallel scaling ratios that describe the
changes in the radius, length, and number throughout the plant
network. Letting N•,k be the number of xylem conduits (branches) at
level k, where • represents either int or ext, we define the ratios
next;k ¼
Next;kþ1
Nint;kþ1
and nint;k ¼
:
Next;k
Nint;k
[1a]
The ratios for radii, r•,k, and lengths, l•,k, across branching levels are
βext;k ¼
rext;kþ1
rint;kþ1
and βint;k ¼
rext;k
rint;k
[1b]
γext;k ¼
lext;kþ1
lint;kþ1
and γint;k ¼
:
lext;k
lint;k
[1c]
The WBE model uses ak instead of rint,k for xylem conduit radius.
Because the WBE model does not include the internal branching
Author contributions: V.M.S., B.J.E., J.S.S., and P.B.R. designed research; V.M.S., L.P.B.,
B.J.E., J.S.S., D.D.S., P.B.R., and E.I.v.A. performed research; V.M.S. and J.S.S. contributed
new reagents/analytic tools; L.P.B. and D.D.S. analyzed data; and V.M.S., L.P.B., B.J.E., and
J.S.S. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
1
To whom correspondence should be addressed. E-mail: [email protected].
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.
1073/pnas.1012194108/-/DCSupplemental.
www.pnas.org/cgi/doi/10.1073/pnas.1012194108
Hydraulic Trade-Offs, Space Filling, and Scaling in Internal Vascular
Structure of Plants. Our characterization of the internal network
differs from WBE’s because we invoke the full generality of
biomechanical stability, space filling, and hydraulic safety while
also allowing the number of conduits to vary across branching
levels. Our model’s flexibility allows the full incorporation of
principles i–v that together imply an optimal vascular network
should (a) have sufficient nonconducting tissue in each branch to
provide biomechanical strength, (b) pack as many conduits as
possible into the remaining wood tissue area to increase conductance and buffer against embolism, (c) taper conduit radii to
increase conductance and the number of parallel hydraulic
pathways, and (d) minimize conduit taper, subject to the previous constraints, to decrease chances of embolism. We show
how selection “drivers” a–d are constrained by the external
network (Eqs. 2–4) and have shaped form and function within
and across plants.
ratio, nint,k, or length ratio, γint,k, it tacitly assumes that nint,k = 1,
preventing conduit number from increasing as conduits taper.
In our model, because nint,k differs from 1, we can assess how
selection for increased conductance and protection against
embolism through redundancy can shape the internal xylem
network. We assume that the interconduit pits contribute a constant fraction to branch resistance (31), so within a branch,
conduits connected end-to-end by pits have a resistance that
scales like a single conduit that is the length of the external
branch (i.e., lint,k = lext,k and γint,k = γext,k), as in the WBE model.
The conduit radius, rint,k, represents the average over all conduits
within a level k, and we assume that a constant fraction of xylem
cross-sectional area is actively conducting.
Biomechanical Trade-Offs, Space Filling, and Scaling in External
Branching Networks of Plants. Eqs. 1a–1c can describe a variety
of networks and diverse plant types because the scaling ratios can
differ at each level k. However, many networks in nature are selfsimilar: n•,k = n•, β•,k = β•, and γ•,k = γ•, where • represents
either int or ext. Our principles i–v predict self-similarity for both
the internal and the external networks. Principle i gives
− 1=3
−b
γext;k ¼ γext ¼ γint;k ¼ γint ¼ next
¼ next ;
[2]
2=3
[3]
because summing across levels predicts that plant path length,
2=3
lTOT, scales with trunk radius as lTOT ∝ rext;0 for large plants. This
relationship is observed for large plants and results in biomechanical stability with constant safety margins from Euler
buckling—the deformation or buckling of a column (plant stem or
tree trunk) under its own weight (32, 33). Combining Eqs. 2 and 3
predicts da Vinci’s rule of area preservation (21, 34, 35),
− a=2
βext;k ¼ βext ¼ next
− 1=2
¼ next ;
[4]
where constant a indicates self-similarity and a = 1 is area
preservation. Eqs. 2–4 are equally prioritized optimality constraints with any two implying the third.
Savage et al.
stant fraction of the total wood area does not conduct water
because it is required to maintain mechanical strength by driver
a. The remainder of wood area is potentially available for
conducting water (Fig. 1), and by driver b, the total water2
conducting area in each branch, ATOT
= Nint,kπrint;k
, must fill
int
this entire remainder portion of wood area to maximize conductance. Because total wood area is constant across levels
(Eq. 4), the total branch conducting area, which is a constant
fraction of the total, must also be constant across branching
levels. Rearranging the expression for the water-conducting
area gives the packing rule
2
−2
Nint;k ¼ ATOT
int =πrint;k ∝ rint;k :
[5]
Second, we derive the maximum conductance (driver c). This conductance calculation explicitly couples the internal geometry, via each
conduit’s resistance, and the external geometry, via number of terminal twigs, Next;N ¼ nN
ext , where N is the number of levels. Together,
this coupling determines the total number of conduits at level k,
seg
seg
−k
−k
N
N −k
Nint;k ¼ Nint;N = nN
¼ ðNint;N = nN
int
int Þ next ¼ ðNint;N = nint ÞNext;N ,
seg
where Nint;N is the number of conduits in a single terminal twig.
Further, we define p to be the unique exponent such that nint ¼ npext .
Using these relationships, the whole-plant conductance is
b−p
2=a
−
1
κN
n
ext
rext;0
κplant ¼
;
[6]
2ðb − pÞ=a
rext;N
a=2
−1
next ðrext;0 = rext;N Þ
seg
where constant b indicates self-similarity and b = 1/3 is space
filling (3). Principle iv yields
lext;k ∝ rext;k
Joint Optimization of Internal Vascular and External Branching
Networks As Guided by Core Evolutionary Principles. First, a con-
4
= 8μlN is the conductance for laminar flow
where κN ¼ Nint;N πrint;N
through all conduits in a terminal twig, invariant by principle v,
and μ is the viscosity. As plants increase in size (rext,0/rext,N) >> 1,
there are two possibilities dictated by the difference in the
scaling exponent, b, for branch length ratios and the exponent, p,
related to the taper of radii of xylem conduits: (i) If b − p ≥ 0,
conductance scales with stem radius, rext,0, to an effective exponent, q, that is <2/a; or (ii) if b − p < 0, conductance scales
2=a
maximally with plant size as κplant ∝ rext;0 . The latter result is
approached asymptotically and holds only for plants of infinite
size (SI Text) (36). For realistic size ranges, conductance always
scales less than the maximal value (q < 2/a; Fig. 2).
To complete our optimization calculation, we assume that the
probability of embolism within the xylem increases with conduit
radius (31). Because terminal twig size is assumed to be independent of plant size, increased tapering leads to larger conduit radii in the base. Thus, by driver d selection to maximize
hydraulic safety will minimize taper relative to other constraints.
Using Eqs. 5 and 6, the taper of xylem conduit radii is
PNAS | December 28, 2010 | vol. 107 | no. 52 | 22723
PLANT BIOLOGY
Fig. 1. Branching structures depicting the difference in internal network
structure for our model compared with the WBE model. Trees are labeled
from the base (level k = 0) to the terminal twigs (level k = N). The left and
right columns represent simplified versions of the models. Both models
predict conduit taper, but our model also allows the number of conduits to
increase and potentially fill a constant fraction of available wood area
(shown to the right).
(31) is necessary for species- and habitat-specific predictions and
rigorous evaluation and extensions of our model.
Results
Tests of Predictions for Internal Vascular Structure and Hydraulic
Function Within Plants. We use the central predictions of our model
Fig. 2. (A) Plots of the logarithm of conductance [kg/(Pa·s)] vs. the logarithm
of the ratio of base radius to terminal twig radius. We chose taper exponents
of p = 0, 1/6 (WBE model), 1/3 (our model), and 1. We used next = 2, rext,N = 0.5
seg
¼ 200,
mm, lext,N = 5 mm, μ = 10−6 kg/(mm·s), ρ = 1 g/cm3, rint,N = 0.01 mm, Nint;N
and a size range of rext,0 = 1 cm to rext,0 = 4 m for our parameters. The corresponding effective conductance exponent, q, is given beside each line.
(B) Plot of the conductance exponent, q, vs. the taper exponent p. For plants
of infinite size, the asymptotic relation is q = 4/3 + 2p for p ≤ 1/3 and q = 2 for
p ≥ 1/3. For a realistic size range of rext,0 = 1 cm to rext,0 = 4 m (finite-size
plants), q varies continuously with p. Nevertheless, the point of maximum
curvature (i.e., the magnitude of the second derivative) occurs at p = 1/3,
seemingly always an important transition point in the scaling of whole-plant
conductance.
rint;kþ1
¼
rint;k
Nint;k
Nint;kþ1
1=2
− 1=2
¼ nint
− p=2
¼ next
¼
rext;kþ1 p=a
:
rext;k
[7]
Eqs. 6 and 7 reveal a trade-off between safety and conductance
(Fig. 2). Specifically, as the taper exponent, p, increases, the
scaling exponent of plant conductance with trunk radius, q, saturates to its maximum of 2 when a = 1 (Fig. 2). For extremely
large trees, the scaling of conductance is maximized when b −
p ≤ 0 or p ≥ b = 1/3 (SI Text; Fig. S1). Relative to this constraint,
xylem taper is minimized when the tapering exponent p = 1/3,
predicting that the ratio of radii across levels and thus the degree
of taper follows rint,k+1/rint,k = ðrext;kþ1= rext;k Þ1=3 when a = 1. For
realistic size ranges of plants, the scaling of conductance, q,
varies with the scaling of taper, p, more continuously (Fig. 2B).
Nevertheless, irrespective of plant size, the taper exponent p = 1/3
still represents an important transition in the trade-off curve
between conductance and safety. The maximum curvature in the
q vs. p plot (Fig. 2B) occurs at p = 1/3, consistent with a breakeven point in the diminishing gain for the scaling of conductance,
q, as the exponent for taper, p, increases beyond 1/3. Thus, selection to maximize rates of return for conductance while tapering less for safety leads to p being close to 1/3. Elucidating the
mechanisms underlying this trade-off between taper and safety
22724 | www.pnas.org/cgi/doi/10.1073/pnas.1012194108
—the packing rule and taper scaling exponent p—to determine
a suite of branch, stem, and whole-plant scaling relationships
(Table 1 and Table S1). We test model predictions that differ
substantially from the WBE and pipe models by compiling intraand interspecific data from the literature (SI Text; Table S2) and
our own detailed measurements of trees with diverse vascular
anatomy: oak (Quercus gambelii), maple (Acer grandidentatum),
and pine (Pinus edulis). In Table 1 and Fig. 3, scaling exponents
were calculated separately for each species using standardized
major axis (SMA) regression. These intraspecific exponents were
then used to calculate cross-species (interspecific) weighted
averages of exponents and 95% confidence intervals (CI) (Methods). Fig. 3 A and B shows conduit frequency changes with conduit
radius with an average cross-species exponent of −2.04 (−2.74,
−1.34) for literature data and −2.16 (−3.35, −0.97) for our
measurements (Table 1). These average cross-species exponents
agree well with our prediction of −2 for the packing rule and have
95% CIs that easily exclude the WBE model prediction of 0. Fig. 3
C and D displays that conduit taper, both across level k (axial) and
within levels of k (radial) (SI Text), has cross-species average
scaling exponents of 0.27 (0.20, 0.34) for literature data and 0.29
(0.08, 0.50) across measured trees (Table 1). These 95% CIs include our model prediction (p = 1/3 ≈ 0.33) and exclude the WBE
model’s prediction (p = 1/6) for literature data. We also predict
4=3
seg
Nint;k ∝ rext;k for the scaling of conduit number with branch radius.
Fits to empirical data for this relationship yield a scaling exponent
of 1.19 (0.86, 1.52) (Table 1), again with 95% CIs that include our
prediction of 4/3 and exclude the WBE model’s prediction of 2.
Moreover, in our model the total cross-sectional area of conducting tissue and fluid flow rate are conserved across levels, so the
sap velocity is approximately constant within and across trees.
Conversely, the WBE model predicts that velocity increases toward
the terminal twigs (Fig. 1). We find no clear interspecific trend that
maximum velocity changes with plant size or external branch radius
(Fig. 4) or that conducting-to-nonconducting ratio changes with
branch radius (Table 1). These results show that our model better
matches available data, both within and across species, than the
WBE or pipe models (3, 5).
Predictions for Allometric Scaling with Plant Mass. Much of the field
of allometric scaling, as applied to animals and plants, focuses on
how physiological and anatomical properties scale with wholebody or whole-plant mass (3, 6, 37, 38). To facilitate comparisons
across allometric scaling studies and promote future tests of our
model, we list the predictions for both the WBE model and our
new model in the SI Text (Table S1). For whole-plant properties,
the WBE model and our model agree (by default) for those
parameters defined by the external network. For properties of
the basal stem (level k = 0), the models again share predictions
of the external network but differ for many properties tied to the
internal network, such as area of conductive tissue, number of
conduits, conduit radius, and fluid velocity. These differences are
measurable and provide an ideal test for empirically distinguishing between our model, the WBE model, and other models.
The predicted scaling exponents in Table S1 ignore finite-size
effects that result from restricting the size range of plants to what
is observed in nature (30). These finite-size effects are typically
small but nonnegligible and can occur in a variety of ways (30).
As an example, stem radius scales with plant mass as rext,0 ∝ M3/8
(Table S1). Therefore, plant conductance scales with plant mass
q
∝ M 3q=8 . For optimal plants with a taper expoas κplant ∝ rext;0
nent of p = 1/3, Fig. 2 shows that q ≈ 1.85 for a realistic range
(finite size) of tree sizes. Thus, if the pressure difference driving
water transport is independent of tree size, we predict that
conductance scales with plant mass as κplant ∝ M0.69 for finiteSavage et al.
Table 1. Predicted scaling exponents for physiological and anatomical variables of plant internal networks as a function of branch
radius (rext,k) for our model and the WBE model
Internal
network property
Packing (conduit frequency
vs. conduit radius, rint,k,
not branch radius)
Conduit radius taper (rint,k)
Conduits in branch segment
seg
(Nint;k
)
Fluid velocity (uk)
Conducting-tononconducting ratio
Network conductance (κk)
Branch segment
conductivity (Kk)
Leaf-specific conductivity
(Kk/Nleaves)
Volume flow rate (Qk)
Pressure gradient along
branch segment (ΔPk =lk )
Branch segment conductance
(Zk/Nint,k)
Observed average
cross-species
exponent for all
measured trees
for rext,k (axial
data only)
WBE model
exponent for
rext,k
Our model
exponent for
rext,k
Observed average crossspecies exponent from the
literature for rext,k
Observed average cross-species
exponent for all
measured trees for rext,k
(radial and axial
data combined)
n.s.
−2
−2.04 (−2.74, −1.34)
−2.16 (−3.35, −0.97)
−1.86 (−2.91, −0.81)
1/6 ≈ 0.17
2
1/3 ≈ 0.33
4/3 ≈ 1.33
0.27 (0.20, 0.34)
n.d.
0.29 (0.08, 0.50)
1.19 (0.86, 1.52)
0.34 (0.04, 0.64)
1.03 (0.11, 1.95)
−1/3
1/3
0 or n.s.
0 or n.s.
n.s.
n.d.
n.m.
0.00 (−0.88, 0.88)
n.m.
0.13 (−0.66, 0.92)
2
1.44 (ref. 12)
n.m.
n.m.
8/3 ≈ 2.67
1.84 (finite)
2 (infinite)
8/3 ≈ 2.67
2.78 (ref. 12)
n.m.
n.m.
2/3 ≈ 0.67
2/3 ≈ 0.67
2.12 (−1.38, 5.62)
n.m.
n.m.
2
−2/3
2
−2/3
1.77 (1.38, 2.16)
n.d.
n.m.
n.m.
n.m.
n.m.
2
2
n.d.
n.m.
n.m.
sized trees, as opposed to the prediction of κplant ∝ M3/4 for
infinite-size trees. Although the WBE value of 3/4 is a useful
reference point, there is scope within our theory for species- and
environment-specific variation that corresponds well with empirical data from this and prior studies (22, 23).
Discussion
Our predictions are for a snapshot of an allometrically ideal plant
(Table S1). However, plant architecture varies, likely reflecting
phenotypic plasticity, the relative strength of differing selection
pressures, phylogenetic histories, and differing size ranges (Fig. 2)
(35). For example, if biomechanical constraints are not uniformly
adhered to, the ratio of water-conducting area to total wood area
may vary across branching levels, changing the space available to
be filled by xylem conduits and allowing plant architecture to explore more of phenotypic space. We expect measured exponents
for plants to cluster around the predicted exponents in Table 1.
In the SI Text (Table S3), we show that scaling exponents
cluster around our predictions while exhibiting significant variation among species. Species-level variation may be due to differences in xylem anatomy that correspond to differences in
functional types (e.g., ring porous, diffuse porous, or coniferous)
(39) or differing growth environments (e.g., light and soil moisture). On the basis of size range of plants in Fig. 2 and measured
tapering exponents for pine, maple, and oak [p = 0.20, 0.23, and
0.46 (Table S3)], we use Fig. 2B to obtain species-specific predictions for the conductance exponent of q = 1.66, 1.70, and
1.92, respectively. These values are in reasonable agreement with
previously reported values for pine and maple (17) and bracket
our interspecific prediction of 1.84 (Table 1; Fig. 2). Our model
predicts that whole-plant hydraulic conductance scales with plant
mass to the exponent 3q/8 for large plants, yielding speciesspecific predictions of 0.62, 0.64, and 0.72 for pine, maple, and
oak. Because hydraulic conductance can limit rates of photosynthesis, the scaling of metabolic rate with plant mass is likely
influenced by these values of the scaling exponent for conductance, q, potentially leading to scaling exponents lower than the
Savage et al.
WBE value of 0.75 for large plants. This argument is in contrast
to studies that focus on carbon, rather than water use, and find
metabolic scaling exponents for saplings that are >3/4 (40, 41).
More intra- and interspecific data are needed to further test our
model, evaluate species differences, and understand environmental and selective pressures.
Our model predictions are well matched by the cross-species
average and interspecific trends across maple, oak, and pine as
well as across our compiled literature data. However, closer
analysis of our detailed empirical data (Table S3) for maple, oak,
and pine reveals significant differences among these three species
(39). The largest deviations away from our model are observed
for oak. These deviations among different scaling exponents appear to be systematically interrelated in a way that characterizes
hydraulic trade-offs specific to oak architecture. Specifically, the
scaling exponent for the packing rule is significantly shallower
than −2, and the scaling exponent for taper is significantly steeper
than 1/3 (Table S3) (18, 42). Moreover, oak is the only tree for
which the conducting-to-nonconducting ratio shows a significant
trend with external branch radius. As discussed above, changes in
the conducting-to-nonconducting ratio are reflected in the packing rule. When the fraction of conducting tissue increases toward
the base, as for oak, more space is available for packing conduits.
This additional space within the branch increases the total area
packed by xylem conduits with larger radii and effectively flattens
the relationships for the packing rule, as indicated by the exponent for oak of −1.44. This exponent for the packing rule influences the calculation for maximizing conductivity and thus the
taper of conduit radii, with flatter packing rules leading to steeper
taper, again as observed for oak (18). Consequently, our model
predictions can be used as a baseline comparison for species
differences among tree architectures and internal networks.
Extensions of our model will help further illuminate the interconnections among these deviations in scaling exponents (23, 42).
We constructed a plant network model for the evolution of form
and function within the xylem network of Tracheophytes. We tested
our predictions using intra- and interspecific data compiled from
PNAS | December 28, 2010 | vol. 107 | no. 52 | 22725
PLANT BIOLOGY
Observed values for average cross-species scaling exponents [mean, 95% confidence intervals using standardized major axis (SMA) regression] are shown
for literature data and our measurements for oak, maple, and pine. n.d., no data found; n.m., not measured; n.s., nonsignificant.
xylem hydraulics, canopy conductance, sapwood-to-leaf area ratio, and canopy height (45). Elaboration of our model to include
hydraulic limitation arising from finite-size effects may help to
explain some of these relationships. To make realistic predictions
for whole forests and land–atmosphere interactions, researchers
need models that account for across-species differences and can
translate across the water hydraulics–carbon metabolism gap, yet
are simple enough to include in large-scale simulations (13, 35, 38,
45–47). Our model represents an important step in this direction.
Our model allows xylem conduit frequency to vary across levels,
includes additional principles for protection from embolism, and
more generally applies the principle of space filling than prior
models. Indeed, the packing rule arises because the total cross-sectional area of conduits fills the space of a constant fraction of wood
area. Thus, the principle of space filling now spans from conduits (2,
25) to branches (3) to canopies to forests (46), suggesting that spacefilling geometries for the capture and delivery of resources may be
one of the most pervasive principles in plant biology.
Methods
Fig. 3. Plots of the packing rule and the conduit taper. For the packing rule A is
reproduced from ref. 2, and B shows our measurements. For the taper C shows
literature data (SI Text; Table S2) and D shows our measurements. Each colored
symbol represents a different species. The packing limits in both plots (upper lines)
represent wood tissue composed entirely of closest-packed conduits (i.e., the
closest packing of both square-shaped conduits in conifers and circular-shaped
conduits in angiosperms).
the literature and our own detailed measurements from maple, oak,
and pine. These data match our predictions better than those of the
WBE or pipe models. Our model does not consider how differences
in leaf form and function influence scaling relationships. Thus,
further improvements of our theoretical framework may arise from
an emerging literature on venation patterns and allometric scaling
in leaves (27, 43, 44). Also, apart from the xylem tissue that delivers
water, future work may add more realism to models by including the
other transport tissue, phloem, that vascular plants use to deliver
sucrose and other organic nutrients to cells.
Hydraulic architecture regulates the flow of water and resources
to leaves and ultimately influences the fate of carbon in the canopy. Recent findings and models indicate relationships among
Fig. 4. Plot of maximum sap velocity (m·h−1) as a function of branch radius (m). Data were compiled from the literature (SI Text; Table S2) and each
colored symbol represents a different species. The interspecific trend is nonsignificant (n.s.), matching the predictions of our model and contradicting the
WBE model prediction. Intraspecific trends were nonsignificant (n.s. for eight
species), significant but measured over a small radius range [much less than
a factor of 2 (three species)], or significant for one species, Acer saccharum.
Thus, these intraspecific trends support the predictions of our model.
22726 | www.pnas.org/cgi/doi/10.1073/pnas.1012194108
Empirical Measurements. Study species. A. grandidentatum Nutt. (bigtooth
maple) and Q. gambelii Nutt. (Gambel oak) were chosen as representatives of
diffuse- and ring-porous trees, respectively. Three mature trees from each species
were obtained from Red Butte Canyon Natural Research Area [1,640 m above sea
level (m.a.s.l.), 40° 46′ N, 111° 48′ W] in Salt Lake County, Utah. A. grandidentatum trees ranged from 72 to 90 y old and had an average height of 5.0 m
(range: 4.7–5.2 m) and an average root-collar diameter of 11.8 cm (range: 11.1–
12.5 cm). Q. gambelii trees ranged from 26 to 30 y old and had an average height
of 5.8 m (range: 5.0–6.8 m) and an average root-collar diameter of 10.3 cm
(range: 9.2–11.1 cm). P. edulis Engelm. (piñon pine) was used as a representative
of tracheid-bearing species. Three reproductively mature trees were obtained
from a ranch near Ojitos Frios, NM (35.5177° N, 105.3337° W, 2,050 m.a.s.l.). P.
edulis trees ranged from 20 to 25 y old and had an average height of 1.7 m (range:
1.3–2.4 m tall) and an average root-collar diameter of 6.5 cm (range: 3.5–11 cm).
Xylem anatomy. A subset of segments representing a wide distribution of
diameters was selected from throughout each of the trees for xylem
measurements (n = 36 for maple, n = 35 for oak, n = 25 for pine). Anatomical
measurements were made to ensure data characterized xylem properties
both axially (from the base to the tip of the tree) and radially (from the
center pith to the outer bark). For the maple and oak, each stem segment
was cut transversely and photographed using a digital camera (SPOT RT KE;
Diagnostic Instruments) mounted to a stereo microscope (SZH; Olympus).
Segments of pine were also cut transversely and photographed using
a digital camera (Optronics Microfire) mounted to a stereo microscope (SZH;
Olympus). From all photographs, stem, cambium, and pith areas were
measured with image analysis software (Image-Pro Plus; Media Cybernetics)
and converted to equivalent diameter.
To determine individual conduit size and number, transverse sections were
made from multiple locations around each cross-sectional stem segment [using
a razor blade for maple and oak and a sliding microtome (Spencer Lens Co.) for
pine]. For maple and oak, in all young stems (<ca. 15 y) and some older stems, all
rings were examined, whereas in other, older stems only the six most recent rings
followed by every fifth ring were examined. For each examined growth ring, at
least three areas of interest (AOI) were selected that occupied the entire thickness of the ring. For pine, all rings in each stem segment were examined, but only
one AOI was measured within each ring. In maple and oak, AOIs were photographed at 40, 100, or 200× magnification using a digital camera (SPOT RT KE;
Diagnostic Instruments) mounted to a light microscope (Eclipse E600W; Nikon).
In pine, AOIs were photographed at 40, 100, or 200× magnification using
a digital camera (Optronics Microfire) mounted to a light microscope (BX-51;
Olympus). Using Image-Pro Plus, ring thickness and all conduit lumen areas were
measured within each AOI. Lumen areas were converted to equivalent diameter
and ring thicknesses were averaged within each ring. These data were used to
calculate growth ring area, conduit density, and mean area-weighted lumen
diameter for each ring. Using these calculations, we evaluated the number of
conduits per branch segment across all trees (n = 3 trees/species). Because of
questionable image quality for a single pine tree, data were excluded from
calculations that required lumen area. Therefore, the packing rule, conduit taper, and conducting-to-nonconducting area were evaluated using n = 3 trees for
maple and oak and n = 2 trees for pine. These differences in the number of trees
and number of data points per tree were accounted for using our weighted
cross-species averages and 95% confidence intervals, as explained below.
Savage et al.
calculated for the residuals of the data around the fitted SMA regression line.
For the interspecific variance, we computed the difference between the
weighted cross-species average slope and each intraspecific slope, squared this
difference, multiplied by the weight for each species, and summed these values.
Statistical Analyses. Bivariate scaling relationships between branch radius and
internal network properties from both empirical and literature data were analyzed by fitting standardized major axis (SMA) regression to log-scaled variables. This technique is recommended for fitting allometric relationships because
it yields an unbiased estimate of the scaling exponent (48). Using the statistical
program (S)MATR (http://www.bio.mq.edu.au/ecology/SMATR/), SMA regression was used to estimate species-level slopes for intraspecific relationships.
To obtain cross-species measures, we calculated a weighted average of the intraspecific slopes with weights (wi = ni/N) determined by the sample size, ni, for
each species, i, relative to the total number of data points across all species, N. To
determine the 95% CIs, we summed the total intraspecific and interspecific
variance to give the total variance, took the square root of this total variance to
yield the SD, and multiplied by 1.96, corresponding to the number of SDs that
contain 95% of the data. The total weighted intraspecific variance is the sum of
the products of the weight and the variance for each species, with the latter
ACKNOWLEDGMENTS. We are grateful for having literature data provided to
us by Yehezkel Cohen, Joshua Weitz, Kelly Caylor, Danilo Dragoni, Rick
Meinzer, David Coomes, and Karen Christensen-Dalsgaard. We benefited
from help from and conversations with Vanessa Buzzard, Philippe Gregoire,
Ashley Wiede, Evan Sommer, Henry Adams, Travis Huxman and the research
staff of B2Earthscience, David Killick, Rebecca Franklin, Anna Tyler, Larry
Venable, and Kate McCulloh. V.M.S. and J.S.S. acknowledge the Australian
Research Council (ARC) and Landcare Research New Zealand (NZ) Research
Network for Vegetation Function (working group 2, vascular design: comparison of theory strands) for their meeting and for initiating our collaboration.
We appreciate the hospitality and accommodations of the Santa Fe Institute,
Biosphere 2, and the Centre d’Ecologie Fonctionnelle et Evolutive (CNRS),
Montpelier, France. The authors were supported by National Science Foundation Advancing Theory in Biology Award 0742800, specifically encouraging
our integration of disparate viewpoints to arrive at a common framework.
J.S.S., D.D.S., and E.I.v.A. were partially funded by National Science Foundation
Grant IBN-0743148 (to J.S.S.).
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PNAS | December 28, 2010 | vol. 107 | no. 52 | 22727
PLANT BIOLOGY
Literature Data Summary. To test the predictions of our model, intra- and
interspecific data were gathered from the available literature. This compilation results in an impressively large number of data points for plants that
span a wide range of masses and represent diverse taxa. Table S2 lists the
references and taxa used to evaluate each scaling relationship.
Supporting Information
Savage et al. 10.1073/pnas.1012194108
SI Text
Resistance per Path Length. Whole-plant resistance, Zplant, is the
inverse of plant conductance, κplant, given by Eq. 6 in the main
text. The whole-plant resistance is calculated to be
b − p
b
b
lTOT
−
1
þ
1
n
−1
ext
lN
ZN
;
nN
nbext− p − 1
ext
Zplant ¼
seg
4
Þ is the total resistance for
where ZN ¼ ð1=Nint;N Þð8μlN =πrint;N
laminar flow through all conduits in a terminal twig, which
is invariant by principle v from the main text. The total path
length from stem to terminal twig for the plant is
bðNþ1Þ
lTOT ¼ lN ðnext
− 1Þ=ðnbext − 1Þ. Our expression for whole-plant
resistance separates the dependence on the number of terminal
twigs and girth of the tree given by ZN =nN
ext , from the dependence on path length (lTOT/lN) given by the remainder of
the expression. As plants increase in height and total path length
lTOT/lN >> 1, there are two possibilities: (i) If b − p > 0, the
resistance per path length will increase with path length, lTOT,
or (ii) if b − p < 0, the total resistance is ZN =nN
ext , and the resistance per path length is independent of path length lTOT.
When b − p = 0, l’Hospital’s rule requires that resistance depends weakly on path length [∼ln(lTOT)].
Consequently, minimizing the resistance per path length,
analogous to the calculation in the WBE model, demands that
b − p ≥ 0. Minimizing taper relative to this constraint leads to
b = p = 1/3. This result emphasizes the importance of p = 1/3 as
a transition point. For allometrically optimal plants, our prediction of p = 1/3 differs from the WBE prediction of 1/6 for
taper, obtained by holding the conduit frequency constant and
not including the packing rule (1, 2).
It is important to emphasize, as in the discussion following Eq. 6
of the main text, that for realistic size ranges across plants, the
resistance per path length may still increase significantly as path
1. West GB, Brown JH, Enquist BJ (1999) A general model for the structure and allometry
of plant vascular systems. Nature 400:664–667.
2. West GB, Brown JH, Enquist BJ (1997) A general model for the origin of allometric
scaling laws in biology. Science 276:122–126.
3. Weitz JS, Ogle K, Horn HS (2006) Ontogenetically stable hydraulic design in woody
plants. Funct Ecol 20:191–199.
4. Fan ZX, Cao KF, Becker P (2009) Axial and radial variations in xylem anatomy of
angiosperm and conifer trees in Yunnan, China. IAWA J 30:1–13.
Savage et al. www.pnas.org/cgi/content/short/1012194108
length increases and may differ dramatically from the asymptotic
value. Fig. S1A shows that when the taper exponent p ≥ 1/3 the
remainder term of the resistance (dependence on path length)
appears to quickly approach an asymptote. In contrast, this is not
the case when p < 1/3. However, this interpretation is slightly
misleading because of the range of resistance considered. For
example, by restricting the scale of resistance (Fig. S1B) and
focusing only on the cases where the taper exponent p ≥ 1/3, it is
evident that resistance per path length does still increase when
p = 1/3, although at a slower rate than for p < 1/3. This result is
because for p = 1/3 the resistance per path length is predicted to
change logarithmically with path length and will not appear
constant until trees are many orders of magnitude larger than the
tallest trees.
Axial and Radial Taper Data. Our model is designed to describe
changes in conduit radii across levels k—proceeding axially, from
base to twig through the tree. However, variation in conduit radii
also occurs across growth rings—outer cambium to the pith—
radially within a single branch. The packing rule and tapering
seem to exhibit similar patterns both across the tree (axially) and
within branches (radially) (3–7), reflecting the fact that the inner
xylem of a large branch originated when the branch was a narrow
terminal twig. This correspondence has been previously used by
others to increase the amount of data for regression to perform
tests of the WBE model predictions, even though the WBE and
our model both effectively ignore variation in conduit radii
within branches (radial) (3–7). Because of our extensive data
within single trees as well as within and across species, we are able
to calculate scaling exponents both for axial measurements, the
most direct test of our and the WBE model, and for radial and
axial measurements combined, linking our results most directly to
previous studies. As shown in Table 1, the measured scaling exponents are in good agreement with each other (axial compared
with axial plus radial) and with the predictions of our model.
5. Coomes DA, Heathcote S, Godfrey ER, Shepherd JJ, Sack L (2008) Scaling of xylem
vessels and veins within the leaves of oak species. Biol Lett 4:302–306.
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along gradients of nutrient supply and altitude. Biol Lett 3:86–89.
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1 of 4
Fig. S1. (A) Plots of the remainder term of the resistance, capturing the dependence on total path length, vs. the ratio of total path length to the terminal
twig length for realistic plant sizes. For these plots we used a branching ratio of next = 2 and a terminal twig length of 5 mm and assumed that the total path
length for plants ranged from 1 cm, a small seedling, to 80 m, the approximate height of the tallest tree on earth. For values of p < 1/3 the remainder term for
the resistance increases without bound. For values of p ≥ 1/3 the remainder term for the resistance approaches an asymptotic limit and appears to be constant
as a function of path length. (B) Plots of the remainder term of the resistance, capturing the dependence on total path length, vs. the ratio of total path length
to the terminal twig length. Here, we have restricted the plot to values of p = 1/3 and p = 1 so that it is clear that the remainder term for the resistance
continues to increase substantially with path length, not reaching its asymptote until the path is much longer than for the tallest trees. Consequently, it is not
accurate to claim that resistance is truly independent of path length even for taper exponents of p ≥ 1/3.
Table S1. Predicted scaling exponents for physiological and anatomical variables of whole-plant and base properties as a function of
plant mass (M) for the West, Brown, and Enquist (WBE) model and our model
Network property
Whole plant
No. of leaves (Nleaves)
No. of branches (Nbranches)
Metabolic rate (B)
Stem-to-petiole path length (lTOT)
Total fluid volume
Stem/branch
Length (lext,0)
Radius (rext,0)
Area of conductive tissue (ATOTint,0)
Active conducting conduit area-to-nonconducting area ratio
No. of conduits (Nint,0)
Conduit radius (rint,0)
Fluid velocity (u0)
Conductivity (K0)
Leaf-specific conductivity (K0/Nleaves)
Pressure gradient ((P1 − P0)/l0)
Resistance (Z0/Nint,0)
WBE model exponent for
plant mass (M)
Our model exponent for
plant mass (M)
3/4
3/4
3/4
1/4
25/24
3/4
3/4
3/4
1/4
1
1/4
3/8
7/8
1/8
3/4
1/16
−1/8
1
1/4
−1/4
−3/4
1/4
3/8
3/4
0
1/2
1/8
0
1
1/4
−1/4
−3/4
The predictions derived from the external network (identical between models) are unchanged, whereas predictions that depend on the internal conduit
network differ. For these predictions we ignore finite-size effects, which alter these scaling exponents depending on the size ranges of the plants considered.
See also discussion connected to Fig. 2 in main text.
Savage et al. www.pnas.org/cgi/content/short/1012194108
2 of 4
Table S2. Sources for literature data
Network property
Taxa
Reference
Packing
Angiosperms/conifers
(1)
Taper
Betula pendula
Fraxinus americana
Larix decidua
Nothofagus solandi, Picea abies, Pinus sylvestris
Tsuga canadensis
(2)
(3)
(4)
(5)
(6)
Velocity
Pseudotsuga menziesii
Eucalyptus regnans
Citrus sinensis, Cupressus sempervirens, Eucalyptus camaldulensis, Malus domestica, Persea americana,
Pinus halepensis, Quercus calliprinos, Quercus ithaburensis
Acer saccharum
Liriodendron tulipifera
(7)
(8)
(9)
(10)
(11)
Sap flux
Angiosperms
Angiosperms
Conifers
Angiosperms
Angiosperms
Angiosperms
Conifers
Conifers
Conifers
Angiosperms
Angiosperms
Conifers
Angiosperms
Angiosperms
Conifers
Angiosperms
Conifers
Angiosperms
Conifers
Angiosperms
Angiosperms
Angiosperms
Angiosperms
Angiosperms
Angiosperms
(12)
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)
(36)
Hydraulic conductivity
Acer rubrum, Acer saccharum, P. sylvestris, Pinus banksiana
(37)
Leaf-specific hydraulic
conductivity
Tsuga canadensis
Ficus glabrata
Thuja occidentalis
A. saccharum, Schefflera morototoni, T. occidentalis
Sequoiadendron giganteum, Sequoia sempervirens
(6)
(38)
(39)
(40)
(41)
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Table S3. Observed values for intraspecific scaling exponents (and 95% confidence intervals and r2 values) as a function of branch radius
(rext,k) for empirical data related to internal network structure measured across three contrasting species (n = 3 trees for all values for oak
and maples; n = 2 trees for all values for pine except number of conduits with n = 3)
Network property
No. of conduits in a
branch segment
(Nsegint,k)
Conducting-tononconducting
ratio
Conduit radius
(taper, rint,k)
Packing rule (no.
frequency of
conduits vs.
conduit radius,
rint,k, not branch
radius, rext,k)
Our model
predicted exponent
for rext,k
Oak observed
exponent for rext,k
Maple observed
exponent for rext,k
Pine observed
exponent for rext,k
4/3 = 1.33. . .
1.12 (1.01, 1.24; r2 = 0.51)
1.08 (0.99, 1.18; r2 = 0.36)
1.45 (1.40, 1.49; r2 = 0.95)
0
0.56 (0.49, 0.63; r2 = 0.30)
−0.42 (−0.47, −0.38; r2 = 0.04)
0.33 (0.28, 0.40; r2 = 0.06)
1/3 = 0.33. . .
0.46 (0.42, 0.50; r2 = 0.63)
0.23 (0.21, 0.25; r2 = 0.36)
0.20 (0.17, 0.22; r2 = 0.55)
−1.44 (−1.55, −1.33; r2 = 0.72)
−2.73 (−2.90, −2.57; r2 = 0.68)
−1.67 (−1.78, −1.56; r2 = 0.86)
−2
Regression exponents were calculated using RMA regression within (S)MATR software using both radial and axial xylem data. Performing regressions on
individual trees within the same species leads to very similar results.
Savage et al. www.pnas.org/cgi/content/short/1012194108
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