Ecology Letters, (2007) 10: 1084–1093
doi: 10.1111/j.1461-0248.2007.01104.x
LETTER
SanioÕs laws revisited. Size-dependent changes in the
xylem architecture of trees
Maurizio Mencuccini,1* Teemu
Hölttä,1 Giai Petit2 and Federico
Magnani3
1
School of GeoSciences,
University of Edinburgh, Crew
Building, West Mains Road, EH9
3JN Edinburgh, UK
2
TeSAF, Treeline Ecology
Research Unit, University of
Padova, Viale dellÕUniversità, 16
I-35020 Legnaro, Italy
3
Dip. Colture Arboree,
Università di Bologna, via Fanin
46, I-40127 Bologna, Italy
*Correspondence: E-mail:
[email protected]
Abstract
Early observations led Sanio [Wissen. Bot., 8, (1872) 401] to state that xylem conduit
diameters and lengths in a coniferous tree increase from the apex down to a height below
which they begin to decrease towards the tree base. SanioÕs law of vertical tapering has
been repeatedly tested with contradictory results and the debate over the scaling of
conduit diameters with distance from the apex has not been settled. The debate has
recently acquired new vigour, as an accurate knowledge of the vertical changes in wood
anatomy has been shown to be crucial to scaling metabolic properties to plant and
ecosystem levels. Contrary to SanioÕs hypothesis, a well known model (MST, metabolic
scaling theory) assumes that xylem conduits monotonically increase in diameter with
distance from the apex following a power law. This has been proposed to explain the
three-fourth power scaling between size and metabolism seen across plants. Here, we (i)
summarized available data on conduit tapering in trees and (ii) propose a new numerical
model that could explain the observed patterns. Data from 101 datasets grouped into 48
independent profiles supported the notions that phylogenetic group (angiosperms versus
gymnosperms) and tree size strongly affected the vertical tapering of conduit diameter.
For both angiosperms and gymnosperms, within-tree tapering also varied with distance
from the apex. The model (based on the concept that optimal conduit tapering occurs
when the difference between photosynthetic gains and wall construction costs is
maximal) successfully predicted all three major empirical patterns. Our results are
consistent with SanioÕs law only for large trees and reject the MST assumptions that
vertical tapering in conduit diameter is universal and independent of rank number.
Keywords
Allometry, meta-analysis, metabolic scaling theory, MurrayÕs law, scaling, WBE, wood
anatomy.
Ecology Letters (2007) 10: 1084–1093
INTRODUCTION
Drawing conclusions from the analysis of a sample of Scots
pine (Pinus sylvestris L.) trees, Karl Sanio (Sanio 1872)
formulated five general laws describing the changes in
tracheid length and diameter occurring along a radius in a
stem from the pith outwards (first law), along a tree stem
from the apex downwards (second law) and between
branches, stems and roots (third, fourth and fifth laws).
SanioÕs second law described vertical conduit tapering along
a plant axis and stated that conduit diameters gradually
increase from the apex downward until a maximum is
reached, below which a rapid decline starts towards the stem
base. When initial tests of SanioÕs propositions on other
conifers appeared to support his statements (e.g. Mell 1910),
2007 Blackwell Publishing Ltd/CNRS
the general conclusion was reached that tree species were
similar with regard to their anatomical configuration.
However, further investigations (e.g. Gerry 1915) showed
several examples of discrepancies from these proposed laws
and without a solid theoretical model, the discussion was
progressively abandoned.
More recently, the tapering of conduit dimensions from
the apex downward (SanioÕs second law) has been used as
the cornerstone of a novel theory of organismal architecture
as a fractal-like network of transport conduits connecting
peripheral volume-filling tissues (West et al. 1997, 1999) and
the century-old discussion of whether and how conduits
taper along the stem of plants has resurfaced again. The
metabolic scaling theory (MST) proposes that the hydraulic
architecture of plants can be described as a series of
Letter
continuous parallel tubes running from the base of the trunk
to the petiole arranged in a self-similar fractal-like structure,
i.e. such that the ratio of properties such as diameter and
length across ranks is independent of rank number.
Provided that the lumen radii of xylem conduits increase
with distance from the apex (or rank number, cf. eqn. 11 in
Appendix S1) according to a power function with exponent
of at least a = 1 ⁄ 4 (or a = 1 ⁄ 6, respectively, i.e. quarterpower scaling), the hydraulic resistance of a Ôcontinuous
tubeÕ running from the base of the tree trunk to the petiole
will remain constant (West et al. 1999) or, more precisely,
increase rather little during height growth (Mäkelä &
Valentine 2006a).
Based on this model, the inter-specific allometry of some
properties in plants has been successfully predicted. For
instance, there is some evidence that allometry of plant
biomass fractions follows universal rules that are almost
entirely independent of the size of the organism (Enquist &
Niklas 2002; Niklas & Enquist 2002), although other pieces
of evidence show that this is not always the case (Mäkelä &
Valentine 2006b; Reich et al. 2007). Interspecific anatomical
scaling in trees also follows MST (Anfodillo et al. 2006).
Some of these discrepancies can be explained by noting that
differences in species-specific allometric constants can elicit
scaling changes (Niklas & Spatz 2004). In contrast to
morphological properties, the scaling of physiological
properties appears to be more plastic, although these are
less well known. For instance, the scaling of hydraulic
conductance (Mencuccini 2002) and of water fluxes
(Meinzer et al. 2005) varied with plant size while respiratory
rates scaled with nutrient status (Reich et al. 2006). In this
last paper, the reported scaling against size varied from
study to study and implied steeper scaling in smaller plants
(Enquist et al. 2007). It is not known why size-dependent
physiological scaling occurs during growth. For instance, it
has been argued (Enquist et al. 2007) that a departure from
the one of the assumptions of the MST model (e.g. volumefilling network, minimization of resistances, biomechanical
adaptations to gravity, etc.) can mathematically account for
the observed differences from one-fourth power scaling,
however, one also needs to explain why such departures
occurred in the first place. Similarly, while the allometry of
morphological properties is remarkably general and holds
true across monocots, dicots and conifers (Enquist & Niklas
2002; Niklas & Enquist 2002), differences across life forms
or phylogenetic groups occur for the scaling of physiological
properties such as xylem sap flow against stem diameter or
mass (Meinzer et al. 2005).
Here, we (i) summarize the published literature on the
scaling of xylem conduit diameters with distance from the
apex and use meta-analysis to determine whether significant
differences generally occur across two tree phylogenetic
groups and two tree groups of different sizes, (ii) use a
Anatomical scaling during ontogeny in trees 1085
numerical model to show how these differences can be
explained based on known physiological principles and (iii)
use our findings to answer the question of whether SanioÕs
Laws or, in alternative, MST anatomical scaling generally
apply to trees.
In our numerical model, we will assume that plants
attempt to maximize not the hydraulic transport properties
per se, but rather the photosynthetic return minus the
construction costs of the biomass invested in the xylem
conduit walls. Large conduits are beneficial for a plant,
because an efficient transport system minimises hydraulic
limitations to photosynthesis. On the other hand, walls of
large conduits are costly to build because the risk of
implosion caused by tension in the transpiration stream
would be greater if wall thickness did not increase in
proportion to conduit diameter (Hacke et al. 2001). This is
similar to earlier treatments that optimized the hydraulic
return of the invested biomass (McCulloh et al. 2003;
McCulloh & Sperry 2005), but the focus here is not in
optimizing the distribution of a fixed amount of carbon
allocated to transport tissues per se, but in maximizing overall
net photosynthetic gains by the plant under a carbon cost
constraint.
MATERIALS AND METHODS
Literature meta-analysis
We summarized the available literature documenting vertical
changes in anatomical properties along tree stems. The
literature was found either in wood technological studies or
in biological studies of tree hydraulics. The primary criterion
was to select studies reporting values of conduit diameters
of the outer rings measured at known distances from the top
of the tree. Among these datasets, a further selection was
made of those studies in which measurements were taken
for at least nine distances from the tree apex, to provide
enough degrees of freedom to carry out statistical analyses
(see below). We accepted papers in which either mean or
hydraulically weighted conduit diameters were measured,
because they have been found to scale linearly with each
other (e.g. Mencuccini & Comstock 1997; Mencuccini et al.
1997; Coomes et al. 2007). We also accepted ten papers in
which tracheid lengths but not diameters were measured,
because published data for gymnosperms (e.g. Sanio 1872;
Bannan 1965; Atmer & Thörnqvist 1982; Pittermann et al.
2006a,b) support the notion that tracheid diameter scales as
diameter lengthb. For those studies tracheid diameter was
estimated from length using published interspecific scaling
relationships (Pittermann et al. 2006a). If a study contained
multiple trees sampled as replicates from within the same
group (e.g. experimental treatment, site, etc.), the values
were pooled together into a single composite profile.
2007 Blackwell Publishing Ltd/CNRS
1086 M. Mencuccini et al.
Following these criteria, 101 vertical tree analyses were
grouped into 48 independent profiles from 24 studies
(containing 994 average anatomical data points), which are
summarized in Table S2. Of these 48 profiles, 30 contained
at least nine data points and were selected for statistical
analysis. To allow plotting all data on a common axis (given
the differences in absolute conduit size across species) all
values were normalized by calculating indices of conduit
tapering T (i.e. the ratio of conduit diameter at each distance
to the estimated conduit diameter at a fixed distance from
the apex). Two normalizations were carried out. The first
one was performed by calculating T at a reference distance
of 10 m from the apex (T10), whereas the second calculated
T at a reference distance of 0.05 m from the apex (T0.05),
assumed to be the petiole length. Conduit diameter at the
petiole was estimated by inverting eqn. 5 in Weitz et al.
(2006; cf. eqn. 12 in Appendix S1) for the point measured
closest to the apex. This gives a T of 1.00 at a distance of
0.05 m from the petiole tip. T10 was employed to
demonstrate the common shape of the scaling curves for
the tall trees. T0.05 was employed to test for specific values
of tapering compared with the slope of a theoretical power
law. Notice also that a differs from T, as a represents the
tapering of conduit diameters across two successive ranks (a
is positive if the distal diameter is smaller than the proximal
diameter, cf. eqn. 11 in Appendix S1).
For each of the 30 independent vertical profiles, values of
T0.05 were plotted as function of the distance from the tree
apex and we determined whether the best statistical model
corresponded to a power or to an alternative function. The
alternative models were either a sigmoid function leading to
a plateau in conduit T with distance from the apex, a
parabolic model or a modified power function with a
negative linear term dependent on distance from the apex,
to allow for a peak and a subsequent decline in T with
distance. An alternative analysis tested the distribution of
model residuals but is not reported here as the results were
entirely coincident. All models were fitted using the
nonlinear procedure in SPSS. Competing models were
compared using the Akaike Information Criterion, corrected
for small sample sizes (AICc; Hurvich & Tsai 1998;
Burnham & Anderson 2002). The differences in AICc
between each model and the best performing model
(DAICc) were then calculated. Values equal to or greater
than four were taken to indicate that considerably less
evidence was present for a given model compared with the
best alternative (Hurvich & Tsai 1998; Meinzer et al. 2005).
If values of DAICc for the MST model were found to be
greater than zero but less than four, this was taken as
evidence against the alternative models, i.e. we treated the
MST power model as our ÔcontrolÕ hypothesis, against which
we tested the alternative hypothesis that either a plateau or a
declining function performed considerably better. After
2007 Blackwell Publishing Ltd/CNRS
Letter
determining DAICc, all profiles were classed into an
angiosperm or a gymnosperm group and into two classes
of tree size, small (shorter than 10 m) or large (taller than
10 m) trees. This procedure generated a 2 · 2 · 2 contingency table which was analysed using a chi-square test.
A FisherÕs exact test for small sample sizes (Fisher 1922) was
used to test the hypothesis that the frequencies in each class
were not significantly different from those predicted by
chance and did not suggest an association between the
classification variables and the best supported fitting
function. Hence for this analysis, agreement between data
and MST predictions was tested with regard to the shape of
the curves, not the absolute value of the slope coefficients.
Finally, for the 21 (of 30) datasets that reported measurements taken within the top 3 m of distance from the apex,
we tested whether T close to the apex occurred at the same
rate as predicted by MST, by checking whether the
confidence intervals for the reduced major axis regression
slope of a log–log function through the T0.05 data also
included the slope of the power-law approximation for eqn.
5 (Weitz et al. 2006). An additional analysis based on residual
distribution was also carried out here but it is not reported,
since the results were coincident.
Numerical modelling
We modelled the optimal hydraulic tapering allowing
maximization of whole-plant net photosynthetic gains (i.e.
the increased gross gains because of greater hydraulic
conductance minus the costs of building wider conduits). A
numerical model (Supplementary materials) was used to test
whether this approach might explain the reported patterns.
A numerical model was chosen, in alternative to an
analytical solution, to allow for greater flexibility in relaxing
many of the assumptions that must be made to achieve a
closed solution. Carbon gains were represented using a
photosynthesis model incorporating a stomatal conductance
term, a linear response function to air carbon dioxide
concentrations and a saturating response function to
irradiance. The stomatal conductance term was set to be
directly dependent on the hydraulic conductance of the
xylem system (which depended on conduit tapering), and a
set minimum water potential term, while it was inversely
dependent on leaf to air vapour pressure deficit. This
approach is based on the observation that minimum water
potential is frequently strongly conserved in plants and it is
consistent with the idea that plants will tend to maximize the
driving force for water flow through the xylem without
incurring in substantial penalties due to turgor loss and
cavitation (e.g. Saliendra et al. 1995). Total modelled gains
were obtained by setting constant values for the relevant
environmental conditions (air CO2 concentration, vapour
pressure deficit and irradiance) and by integrating over time.
Letter
The construction costs were calculated from the biomass
used in building xylem conduit walls. Evidence shows that
the ratio between wall thickness at the point of contact
between two adjacent conduits and conduit radius is largely
a function of the prevailing water stress conditions
experienced by the plant and that walls are reinforced to
prevent implosion caused by negative water potentials
(Hacke et al. 2001). Hence building larger conduits incurs
the penalty of having to build proportionally thicker walls to
resist implosion. Costs related to additional tree mechanical
support (e.g. fibre construction) were not considered, since
they do not necessarily scale with conduit diameter and do
not therefore influence optimal T. Additional tree metabolic
costs such as leaf, rays and fine root respiration were also
not considered for the same reason. For instance, if one
treats maintenance respiration as a carbon ÔtaxÕ on photosynthesis, its inclusion in the model will lower gross gains
but will not affect optimal conduit T. Roots and leaves
extra-xylary hydraulic resistances were modelled as giving
fixed water potential losses. While conduit numbers doubled
with each rank following MST, conduit tapering was
allowed to vary to maximize net canopy carbon gain.
Conduit T was allowed to vary during tree growth in two
different ways, i.e. by either finding the optimal constant
a or by finding the optimal variable a within an individual.
The model assumed constant leaf water potential throughout tree growth. A list of parameter values is given in
Table S1, where the implications of the relaxation of
assumptions of the MST model and the role of environmental factors are also discussed. Because of variable
conduit a during growth and variable construction costs,
optimal T was not predicted to be the same for different
plant sizes or different taxa.
RESULTS AND DISCUSSION
Empirical data
A summary of the 48 profiles is given in Fig. 1, together
with reference MST power curves and the numerical model
output, for the four combinations of tree size and
phylogenetic group. To highlight the overall trend, values
of conduit T were normalized across studies such that for all
curves T10 = 1 at a distance from the apex D = 10 m, with
the same normalization carried out also for the reference
theoretical curves (variability in T across studies can be seen
more clearly when values are normalized such that T0.05 = 1
at a distance D = 0.05 m from the apex, cf. Fig. S5). The
first statistical analysis determined whether a power model
(black lines in Fig. 1) adequately described the vertical
tapering profiles across phylogenetic and tree size groups,
i.e. whether overall tapering along the entire length of a tree
followed a power function or showed evidence of a gradual
Anatomical scaling during ontogeny in trees 1087
change in curve shape with plant height to a plateau or to a
declining trend close to the stem base. Of the 30 profiles for
which the analysis could be carried out, 18 profiles (i.e. 60%)
significantly diverged from a power model (i.e. DAICc > 4
compared with a plateauing or declining model, see
Table S2). A chi-square test showed a significant association
across studies between lack of support for a power model
and plant size (v2 = 5.93, Fisher P = 0.024, n = 30),
suggesting a departure of the large plants from power
scaling. Similarly, a chi-square test showed a significant
association across studies between lack of support for a
power model and phylogenetic group (gymnosperms versus
angiosperms; v2 = 7.75, Fisher P = 0.009, n = 30), suggesting a departure of the gymnosperms from power scaling.
When the comparison of size classes was carried out by
blocking for phylogenetic group, v2 = 0.24 and v2 = 10.98
(Fisher P = 1.00 and P = 0.004 for angiosperms and
gymnosperms, respectively, n = 30), suggesting that size
differences were very significant for gymnosperms but not
for angiosperms. Finally, when the comparison of phylogenetic group was carried out by blocking for size differences,
v2 = 13.37 and v2 = 0.00 (Fisher P = 0.002 and P = 1.000
for large and small trees, respectively, n = 30), suggesting
that differences between gymnosperms and angiosperms in
the best supported model were only apparent for trees taller
than 10 m. Maximum height across the angiosperm dataset
seldom exceeded 25 m, i.e. only about half of the maximum
height of the gymnosperm dataset.
Most datasets showed rapid conduit T at the top. This
was investigated in more detail for the 21 datasets in which
measurements were reported for the first 3 m of distance
from the apex (Fig. 2, cf. Fig. S5) and required using T0.05
instead of T10. A log–log function was fitted to the pooled
T0.05 data from all 21 datasets and its slope compared to the
expected value of 0.1861 (Eqn.5 of Weitz et al. 2006;
corresponding to a = 1 ⁄ 6 over the first 3 m). Gymnosperm
trees showed a regression slope above 0.2 in both size
classes (which was significantly larger than MST predictions,
Table 1), suggesting that both tree size classes tapered faster
than predicted by MST close to the apex. Angiosperm trees
showed a slope steeper than the predicted value in only one
case (Table 1), suggesting that tapering close to the apex was
faster than MST in small but not in large plants (cf. Fig. 2).
The small angiosperms had a regression slope for the first
3 m slightly below 0.3, but the slope within the first 0.5 m
approached one-third.
To conclude, for both phylogenetic groups, we found
strong evidence that scaling of conduit diameter against
distance from the apex depended on individual size and
vertical position (i.e. rank number). Scaling was faster than
MST at the top of all trees (except large angiosperms) and it
either declined towards MST power scaling (at the base of
both small angiosperms and small gymnosperms) or it
2007 Blackwell Publishing Ltd/CNRS
1088 M. Mencuccini et al.
1.4
Small angiosperms
1.2
1.2
1.0
1.0
0.8
0.8
Pi.sy. S (Coomes et al 2007)
Pi.ab. S (Coomes et al 2007)
Pi.ce. T (Petit et al 2007b)
Pi.ce. H (Petit et al 2007b)
La.de. T (Petit et al 2007b)
La.de. H (Petit et al 2007b)
Ts.ca. (Ewers & Zimmermann 1984)
MST
No.so. H (Coomes et al 2007)
Ac.ps. 1 (Petit et al 2007a)
Ac.ps. g1 (Petit et al 2007a)
Ac.ps. g4 (Petit et al 2007a)
Po.gr. (Zimmermann 1978)
Be.pa. (Zimmermann 1978)
MST
0.6
0.4
0.2
1.8
0
MST
10
Numerical model
1.6
20
30
distance from the apex, m
1.4
Conduit tapering, T10
Pi.sy. L (Coomes et al 2007)
Pi.ab. L (Coomes et al 2007)
Pi.sy. (Atmer & Thörnqvist 1982)
Pi.ab. (Atmer & Thörnqvist 1982)
Pi.ta. (Jackson 1959)
La.la. (Yang et al 1986)
Ps.me. (Becker et al 2003)
La.de. (Anfodillo et al 2006)
Pi.ab. (Anfodillo et al 2006)
Pi.sy. (Sanio 1872)
40
Pi.ta. (Bethel 1941)
Pi.si. (Dinwoodie 1963)
Se.se. (Resch & Arganbright 1968)
Pi.sy. (Trendelenburg & Mayer 1955)
Ab.al. (Trendelenburg & Mayer 1955)
Pi.ab. (Trendelenburg & Mayer 1955)
Pi.ru. (Bailey & Shephard 1915)
Ps.me. (Bailey & Tupper 1918)
Ab.co. (Anderson 1951)
Ab.pr. (Anderson 1951)
Ps.me. (Anderson 1951)
Ps.me. (Lee & Smith 1916)
Pi.co. OSG (Kienholz 1931)
Pi.co. SB (Kienholz 1931)
1.2
1.0
0.8
0.4
0.2
Large angiosperms
0.4
0.2
2.0
1.8
1.6
1.4
1.2
1.0
No.so. M (Coomes et al 2007)
No.so. L (Coomes et al 2007)
Fr.ex. (Anfodillo et al 2006)
Fr.am. Weitz et al 2006)
Ac.ps. 4 (Petit et al 2007a)
Fr.ex. (Trendelenburg & Mayer 1955)
Eu.gl. (Leitch 2001)
Sc.mo. (James et 2003)
Co.al. (James et 2003)
An.ex. (James et 2003)
Fi.in. (James et 2003)
Numerical model
MST
0.6
0.6
Conduit tapering, T10
Small gymnosperms
Conduit tapering,T10
Conduit tapering, T10
1.4
Letter
0.8
0.6
0.4
Large gymnosperms
0.0
0
10
20
30
40
0
Distance from the apex, m
20
40
60
80
Distance from the apex, m
Figure 1 Tapering profiles for two phylogenetic groups (angiosperms and gymnosperms) and two tree size classes (large and small plants).
The Y-axis represents a tapering coefficient (i.e. T10 (h) = r (h) ⁄ r(h = 10), where r(h) is conduit radius at distance h from the apex, and
r(h = 10) is conduit radius estimated at a distance of 10 m from the apex). All profiles and theoretical curves go through the point T = 1 at
h = 10 m, so that across-species variability in absolute conduit size is eliminated. The black curve gives the theoretical MST scaling (Weitz
et al. 2006), while the thick red curves in the two panels at the bottom give an output from the numerical model. Notice that most series for
the gymnosperms (particularly in large trees) taper more slowly than predicted by the MST power curve after the first few meters, with many
showing even a decline towards the tree base. Most angiosperm series taper faster than predicted by MST in the small plants, but follow MST
more closely in the large plants. Species abbreviations can be found in Table S2. Letters and numbers next to a species code refer to different
treatments, age classes or sampling sites within a study.
developed into a saturating trajectory, not consistent with a
power function (in the much taller gymnosperms).
Numerical modelling
Maintaining maximum net photosynthetic gains during tree
growth required varying conduit a. The model predicted
steeper conduit tapering in short trees and slower or even
reverse tapering in tall trees, reflecting the empirical trend
noticed above. When the model was run assuming fixed a
within a tree, optimal T was found to gradually decline during
growth at rates dependent on apical conduit diameter (cf.
Fig. S1), as empirically observed (G. Petit, T. Anfodillo &
M. Mencuccini, unpublished data). When the model was run
with a within each tree dependent on rank number, it
2007 Blackwell Publishing Ltd/CNRS
reproduced the major trends found in the empirical datasets,
i.e. it predicted higher a close to the tops, lower a in the centre
and finally negative a close to the base of tall trees (see thick
red lines in the bottom panels of Fig. 1). The model predicted
a plateau to occur earlier during vertical growth at the base of
gymnosperm trees, as a consequence of higher tracheid,
compared to vessel, constructions costs per unit conduit
diameter (Hacke et al. 2001) and larger numbers of smaller
tracheids per leaf. In addition to SanioÕs second law, our
model also reproduced the observation (Bailey & Shephard
1915) that the distance from the base below which conduit
diameter and length begin to decline increases with tree size,
as a result of construction costs accumulating faster in tall
trees. Finally, the model predicted steeper anatomical scaling
in the roots compared to stems (see Fig. S3), because of low
Anatomical scaling during ontogeny in trees 1089
Small angiosperms
Large angiosperms
3.0
No.so. M (Coomes et al 2007)
No.so. L (Coomes et al 2007)
Fr.ex. (Anfodillo et al 2006)
Fr.am. (Weitz et al 2006)
Ac.ps. 4 (Petit et al 2007a)
MST
2.5
2.5
2.0
2.0
1.5
1.5
No.so. H (Coomes et al 2007)
Ac.ps. 1 (Petit et al 2007a)
Ac.ps. g1 (Petit et al 2007a)
Ac.ps. g4 (Petit et al 2007a)
Po.gr. (Zimmermann 1978)
Be.pa. (Zimmermann 1978)
MST
1.0
1.0
0.5
Conduit tapering, T0.05
3.0
0.5
Small gymnosperms
Large gymnosperms
3.0
2.5
2.5
2.0
2.0
1.5
1.5
Pi.sy. S (Coomes et al 2007)
Pi.ab. S (Coomes et al 2007)
Pi.ce. T (Petit et al 2007b)
Pi.ce. H (Petit 2007b)
La.de. T (Petit 2007b)
La.de. H (Petit 2007b)
MST
1.0
Pi.sy. L (Coomes et al 2007)
Pi.ab. L (Coomes et al 2007)
La.de. (Anfodillo et al 2006)
Pi.ab. (Anfodillo et al 2006)
MST
0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Conduit tapering, T0.05
Conduit tapering, T0.05
3.0
Conduit tapering, T0.05
Letter
1.0
0.5
3.5
Distance from the apex, m
Distance from the apex, m
Figure 2 Tapering profiles for the first 3 m from the apex for two phylogenetic groups (angiosperms and gymnosperms) and two tree size
classes (large and small plants). The Y-axis represents the tapering coefficient T0.05 = ri ⁄ rapex, where rapex is conduit radius estimated at a
distance of 0.05 m from the apex. The gymnosperms (both large and small plants) and the small angiosperms taper faster than predicted by
the MST power curve in the first 3 m (all P < 0.005), whereas the series for the large angiosperms did not significantly depart from MST
(P > 0.05). The black curve gives the theoretical MST scaling calculated following (Weitz et al. 2006) for a = 1 ⁄ 6.
Sample size, RMA regression coefficients, R2, confidence intervals of the slopes and significance levels of the tests on the
relationships shown in Fig. 2 (i.e. limited to the first 3 m of distance from the apex), for two phylogenetic groups (gymnosperms and
angiosperms) and two size groups (small and large). The significance levels refer to a t-test comparing the observed slopes for each group with
the expected value of 0.1861 (Eqn. 5, Weitz et al. 2006; calculated for a = 1 ⁄ 6)
Table 1
Model
Small gymnosperms
Large gymnosperms
Small angiosperms
Large angiosperms
N
RMA slope
R2
Confidence intervals of the slopes
Significance level (P-value)
155
0.235
0.93
0.225–0.245
< 0.001
73
0.215
0.90
0.200–0.231
< 0.001
178
0.292
0.84
0.275–0.310
< 0.001
81
0.199
0.84
0.181–0.216
> 0.05
construction costs of the comparatively thinner walls of root
conduits, due to lower mechanical requirements and lower
risks of implosion (i.e. less negative water potentials). This is
also similar to available observations (e.g. Oliveras et al. 2003;
McElrone et al. 2004).
While MST predicts a universal curve, our numerical
modelling produced different curves depending on parameter values. Although the absolute values of T changed from
one model run to another, the trends mentioned above were
very robust to changes in parameter values. A sensitivity
2007 Blackwell Publishing Ltd/CNRS
1090 M. Mencuccini et al.
analysis of the major model parameters is given in the
Supplementary materials.
Impact of tree size
That conduit a should depend on individual size and vertical
position seems functional to optimal plant growth and a
simple power-law with constant a should have limited value
in describing tapering throughout a treeÕs life cycle. In small
plants, a = 1 ⁄ 6 is insufficient to prevent losses in conductance (Mencuccini et al. 2005; Mäkelä & Valentine 2006a)
while the costs of tapering are small. In fact, in the first
0.5 m of the small angiosperms, most datasets showed
a fi 1 ⁄ 3. In larger trees, a = 1 ⁄ 6 is too expensive to retain
and its decline allows achieving maximum net gains.
Our numerical model realistically reproduced the patterns
observed in the empirical data with regards to the levelling
off of lumen dimensions at the base of large trees as well as
the prediction of faster-than-MST a at the top of trees.
However, without additional constraints on the rate of
maximum initial tapering, our model would predict very fast
rates of conduit enlargement, much larger than empirically
observed (i.e. a > 1). We therefore restricted the initial rate
of tapering below a certain threshold. While accepting that
this is not theoretically justified, we defend it on the ground
that the equivalent choice of assuming a constant a = 1 ⁄ 6
throughout is more arbitrary. There is no special physiological reason for selecting a = 1 ⁄ 6, except the mathematically appealing argument that the resistance of each pipe
increases by a constant amount every time a new rank is
added at the bottom, such that its percentage increase will
gradually decline towards zero at infinity. However, in the
range of realistic heights, even with scaling above a = 1 ⁄ 6,
resistance will continue to increase significantly. Therefore,
in both cases the initial tapering is chosen arbitrarily. Our
approach however predicts the gradual decline in a as plants
get bigger and quantifies the limitations imposed by the
plantÕs carbon balance.
MurrayÕs law (West et al. 2000) also attempts to predict
the maximum efficiency of hydraulic conductance for blood
circulation in animals. As applied to the xylem (McCulloh
et al. 2003), MurrayÕs law is based on the principle of
minimizing hydraulic resistance while keeping wall investment constant, with the optimal tapering dependent on
whether conduits furcate. However, existing data do not
generally support the presence of conduit furcation in the
stems of trees (McCulloh & Sperry 2005; G. Petit,
T. Anfodillo & M. Mencuccini, unpublished data). In the
absence of furcation, MurrayÕs law predicts that maximum
conductance for a constant amount of invested biomass
occurs for untapered pipes (i.e. a = 0). We removed the
constraint of a fixed investment in conduit walls in our
model by directly coupling the costs of xylem construction
2007 Blackwell Publishing Ltd/CNRS
Letter
to the benefits of photosynthesis, thereby overcoming the
limitations imposed by MurrayÕs law. In a small plant, wall
construction costs were trivial, therefore a larger investment
in conduit walls was beneficial and led to fast rates of
tapering (a >> 1 ⁄ 3, if the rate of initial tapering was not
restricted, as opposed to MurrayÕs law prediction of a = 0)
and increased whole-tree photosynthesis. However, as soon
as large conduits were obtained after a few generations,
hydraulic limitations to photosynthesis were greatly reduced
and a fi 0, as in MurrayÕs law. Therefore, while a = 0
(MurrayÕs law for non-furcated conduits) and a = 1 ⁄ 6
(MST) are too slow a slope for the tapering observed close
to the apex, a >> 1 ⁄ 3 (our carbon balance model) is instead
too high. Hence, none of the existing theories can currently
predict the observed rates of tapering close to the apex. The
most likely candidates are a developmental constraint
preventing a very rapid increase in conduit diameter (and
length) at short distances from the crown, the interactions
between conduit diameters, pit resistance and vulnerability
to cavitation and mechanical constraints in rapidly developing shoots.
Our model did not incorporate inter-conduit pits, on the
ground that the log–log scaling of end-wall and conduit
lumen resistance is typically linear across-species (Wheeler
et al. 2005), i.e. its inclusion would not change optimal
T. However, depending on the assumptions made, inclusion
of pits may only increase the calculated a (Becker et al.
2003), and hence it would not help our model predict
a < 1 ⁄ 3 below the apex, once again highlighting the need to
incorporate effects linked to resistance to cavitation directly.
Also, inclusion of pits would explain nor the plateau
effect, neither the differences between angiosperms and
gymnosperms.
Impact of phylogenetic group
Current empirical and theoretical work (Pittermann et al.
2006a,b; Sperry et al. 2006) suggests that the differences
between the homoxylous wood of gymnosperms and the
heteroxylous wood of angiosperms has significant implications for the functionality and ecology of these two groups
of taxa. Conifers appear to achieve efficiency in water
transport by packing large numbers of small conduits per
unit of volume and by an efficient pit pore structure.
However, because tracheids must also carry out a mechanical supporting function, their wall thickness per unit of
conduit diameter must be larger for a given vulnerability to
cavitation, thereby leading to larger carbon costs, further
enhanced by the large number of conduits per leaf. As
hypothesized here, it is possible that the maximum
achievable wall thickness varies during growth as a result
of variable construction costs and that this sets the limit to
maximum tracheid diameter in tall trees. In contrast, conduit
Letter
construction costs are lower for angiosperms and these taxa
appear capable of obtaining steeper a along the main stem
even in relatively tall trees, although at heights greater than
those available for this analysis, carbon costs should become
limiting also for angiosperms. Only a few datasets were
available for very tall angiosperm trees. Four of them (James
et al. 2003) suggested a plateauing trend for two species and
a declining trend for the remaining two, although each
vertical profile was only composed by four to five points
and could not be tested statistically.
It is interesting to note the differences between the two
phylogenetic groups in the rates of initial a close to the
crown. In the conifers, a close to the crown continued to be
clearly faster than one-sixth even in tall trees. As a
consequence, large conduits with thick walls rapidly
developed and costs started to increase rapidly. In the
angiosperms, the tall trees tapered more slowly than the
short trees close to the crown and this may have contributed
to maintaining rates of tapering consistent with MST.
Implications for scaling
Our meta-analysis confirms that SanioÕs second law is
generally applicable to gymnosperms and clarifies that its
applicability depends on the size of the tree. Scaling varies
within each individual from top to bottom, with a generally
declining with distance from the apex and also varying as a
function of plant size, with exponents declining with tree
size.
If conduit a is size-dependent during growth, then it
logically follows that other processes should be equally so.
Indeed available physiological data, relevant to both water
transport (Mencuccini 2002, 2003; Meinzer et al. 2005),
photosynthesis (Bond 2000; Koch et al. 2004) and respiration rates (Reich et al. 2006; Enquist et al. 2007) show that
this is the case. This creates an interesting conflict with
other sources of evidence, which on the contrary show sizeindependent allometry for much of the plant kingdom (e.g.
Enquist & Niklas 2002; Niklas & Enquist 2002). One
possible resolution of this conflict is a careful examination
of inter-individual versus inter-specific scaling of both
morphological and physiological properties and in examining the possibility that different scaling rules apply to these
two types of allometry. Another possibility is that the
analysis of morphological properties of (especially large)
plants does not have enough temporal resolution to provide
meaningful insights into plant physiological processes.
CONCLUSIONS
Conduit a was not found to be constant within trees (hence
the network is not self-similar) but varied as an inverse
function of distance from the apex, resulting in values close
Anatomical scaling during ontogeny in trees 1091
to one-third at top of (especially) young trees, values close to
zero in the centre and even negative values at the trunk base
(especially in the gymnosperms). In the angiosperms, large
trees showed lower values of a at the top compared to small
trees, whereas in the gymnosperms, large trees departed
from power law scaling at their base. A numerical model,
maximizing the photosynthetic gains minus the conduit wall
construction costs, could explain all three trends, highlighting the fundamental role of xylem construction costs in
explain anatomical patterns in trees. Despite this success,
the model overestimated the rapid, but gradual, conduit
tapering observed inside the crowns of trees. Other
constraints linked to, e.g. cavitation vulnerability, mechanical needs or developmental restrictions may exert additional
pressures that will affect a throughout the tree and
particularly close to the tree crown. Overall, we conclude
that SanioÕs second law applies to gymnosperms and
possibly also to very tall angiosperms for which current
data are very limited. Models such as MST must account for
position-dependent tapering within individuals and sizedependent tapering during growth.
ACKNOWLEDGEMENTS
T. Hölttä was supported by a grant from Helsingin sanomain
100-vuotissäätiö. The research was carried out in the
framework of a NERC-funded project to M. Mencuccini
(competitive grant NER ⁄ A ⁄ S ⁄ 2001 ⁄ 01193).
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SUPPLEMENTARY MATERIAL
The following supplementary material is available for this
article:
Appendix S1 Structure of the numerical model.
Figure S1 Optimal tapering coefficient a as a function of
distance from the apex (or total number of ranks N )
assuming constant tapering within a tree (i.e. a is independent of rank number and positive) for four different values
of conduit radius. Note that a is used in this graph, not T.
Figure S2 An example of how the model calculated gross
photosynthetic gains, total carbon construction costs and
net gains (the difference between the first two) for trees of
variable number of ranks (hence distance from the apex).
Letter
Figure S3 In this simulated coniferous tree, conduits
tapered for a few metres below the tree apex at a constant
rate (corresponding to a = 0.20).
Figure S4 An example of the impact of allowing variable
maximum tapering rates just below the apex of angiosperm
and gymnosperm trees.
Figure S5 Tapering profiles for two phylogenetic groups
(angiosperms and gymnosperms) and two ontogenetic
classes (large and small plants).
Table S1 List of parameter values employed in the
numerical model.
Table S2 Datasets employed in the meta-analysis.
Anatomical scaling during ontogeny in trees 1093
Please note: Blackwell publishing are not responsible for the
content or functionality of any supplementary materials
supplied by the authors. Any queries (other than missing
material) should be directed to the corresponding author for
the article.
Editor, Josep Penuelas
Manuscript Submitted 8 June 2007
First decision made 11 July 2007
Manuscript accepted 31 July 2007
This material is available as part of the online article from:
http://www.blackwell-synergy.com/doi/full/10.1111/j.1461.
0248.2007.01104.x.
2007 Blackwell Publishing Ltd/CNRS