Appendix S1. Derivations of model predictions.
Leaf mass-to-area ratio
Consider a section of leaf with total dry mass M, volume V, and total projected area A. This mass
and volume can be partitioned into contributions from terminal veins (MV, VV) and from the
lamina (ML, VL). Recall that the mass density of terminal veins is V and the mass density of the
lamina is L. Then the leaf mass-to-area ratio is:
(A1)
The volume of the veins is the product of the total vein length and the vein cross-sectional area:
(A2)
The volume of the lamina is the volume of the leaf minus the volume of the veins:
(A3)
Substituting Equations A2 and A3 into A1, then using Equation 2 gives
(A4)
This equation explicitly ignores the mass contributions of primary and secondary veins, though
they could be incorporated as additional mass and volume in the lamina. A fuller treatment of
LMA can be found in Niklas et al. (Niklas et al. 2009).
Peak photosynthetic rate
Peak carbon assimilation rate per mass (Am, mol CO2 g-1 s-1) is defined as:
(A5)
where, E (mol H2O m-2 s-1) is the maximum per-area transpiration rate, WUE is the water use
efficiency (mol CO2 mol-1 H2O), a species-specific constant that characterizes the biochemical
efficiency of the leaf, and LMA is leaf mass per area (g m-2) that converts between the per-area
basis of E and the per-mass basis of Am. We can detail how E relates to the venation network by
extending a recently developed model for optimal leaf vein spacing that assumes diffusion
directly limits water flux from veins to the atmosphere (Noblin et al. 2008). Noblin et al. (2008)
considered the case of regularly spaced channels in a porous medium and confirmed predictions
by measuring water fluxes in artificial leaves. We extend their modeling approach to include
more complex venation networks and stomatal limitations to transpiration. Where boundarylayer effects are negligible, extensions of the Noblin et al. (2008) model predict the conductance
of a whole leaf to water vapor.
We assume that the leaf transpires water to the atmosphere only if that water diffuses
from the veins through intercellular spaces and then through stomata. Using the concentration of
water vapor immediately outside the veins (saturation vapor concentration c0) and the
concentration of water vapor immediately outside the stomata corrected for relative humidity, h,
where c1 = h c0, we define the total per-area transpiration rate, E, as a linear function of the
whole-leaf conductance gL:
(A6)
With the previous assumption, there are three potential limits to water conductance: bulk fluid
flow in the xylem, gX, vapor diffusion through the air space away from the veins, gV, and vapor
diffusion through the stomata, gS. Since water must pass first through the xylem, then through the
air spaces, and finally through the stomata, these conductances add as series resistances, or
(A7)
Here we assume that the xylem does not limit conductance, so gX = ∞. Controversy exists over
the partitioning of resistance within and outside the venation network, but it is clear that
resistances outside the venation are important (Sack & Holbrook 2006). We acknowledge that
the venation network influences conductance because of the resistance of viscous fluids to
transport through conduits (Murray 1926). While a full treatment of the total conductance of the
venation network is beyond the scope of this paper, such a treatment could be included with
additional detailed knowledge of the connectivity of the venation network and the packing
function for conduits within veins.
We next show how the venation conductance gv is given by steady-state diffusion. Our
model assumes that veins are packed into bundles of radius, rV, and are distributed spatially on
the z=0 plane. We characterize the leaf plane as having an upper and lower surface at z=±/2.
Water moves from the veins toward the leaf surface as governed by the diffusion constant, D.
The steady-state diffusion of water is governed by Laplace's equation, 2c(x,y,z) = 0, where
c(x,y,z) is the spatially-dependent concentration of water within the leaf. To uniquely determine
c(x,y,z), we must know boundary conditions. Specifically, we start by assuming that immediately
outside the veins, c = c0, and at the surface of the leaf, c = cs (a reasonable assumption when
stomatal density is low). Then the conductance of water vapor due to the venation network is:
(A8)
where R is an arbitrary region of leaf area. Note that the diffusion constant D will be influenced
by the environment –humidity and temperature changes alter the value for transpiration. For
arbitrary venation geometries this integral has no simple analytic solution. As such, we provide
an approximate solution for rectangular geometry.
Suppose that veins are periodically spaced in the x-direction with interval kx and in the
y-direction with interval x (k ≥ 1 without loss of generality). Here,  = (k+1)/(kx) and d =x .
Because water can move in parallel through venation in both the x-direction and the y-direction,
the total venation conductance is the sum of the conductance of x-direction veins (gVx), and ydirection veins (gVy):
(A9)
Next, consider the situation where we consider only veins in the y-direction (Noblin et al. 2008).
In this case, if each vein segment is independent of all other segments (kx >> ), then a solution
for c(x,y,z) is the periodic function c(x,y,z) = c(x+nkx,y,z) = A log√(x2+z2) + B, where n is an
integer. For the boundary conditions c(x,y,) = cs and c(0,0,rV) = c0 we then can show that A = (c0 - cs)/log(/rV) and B = c0 – A log(rV). The conductance can be found by dividing the total
water flow rate through the region -kx/2 < x < kx/2, yL < y < yu, z=, where yL and yu are
arbitrary positions along the y-axis. Because we assumed kx >> , only one vein contributes to
this region. Thus, we can solve for gVx where:
(A10)
As the size of an areole decreases (as x and d approach 0), the value of arctan(kx/2)/kx in
Equation A6 reaches a finite asymptotic value. Thus, we find, as was pointed out by Noblin et
al. 2008, that large decreases in vein distance only produce a diminishing increase in
conductance. Now because we assumed kx >> , this equation can be linearized by performing
a first-order series expansion around kx/2 = ∞. Equation A10 then reduces to
(A11)
A similar expression can be obtained for gVy by substituting k=1 into Equation A11:
(A12)
Thus, the total venation conductance can be calculated by combining Equations A9, A11, and
A12:
(A13)
Equation A13 can be rewritten in terms of the venation traits  and d as:
(A14)
Equation A14 shows that the approximate venation conductance increases with vein density, but
decreases with vein distance. However the overall magnitude of these changes is controlled by
the diffusion constant for water vapor.
We next link vein geometry with the stomatal conductance, gs. The number density of
open stomata is ns, and the average area of a stomatal pore is as. If the thickness of a stomatal
pore is ts,, then the effective thickness, taking into account boundary effects, is ts + √(as/π) (Nobel
1999). A straightforward application of Fick's first law for diffusion gives
(A15)
Equation A15 shows that stomatal conductance is mainly governed by the dimensions and
number of stomata. Maximization of Equation A7 implies that Equations A14 and A15 should be
of roughly equal magnitudes, requiring coordination of the venation network and stomata.
The peak per-mass carbon assimilation rate, Am, can now be determined by combining
Equations A5, A6, A7, A14, and A15:
(A16)
Equation A16 simplifies to
(A17)
We have written Equation A17 to emphasize how Am depends on the venation traits  and d. The
numerator of this equation sets the overall normalization of Am, while the two groups of terms in
the denominator set the scale of the relationship between  and d. This equation demonstrates the
complex dependence of photosynthetic rate on different venation geometries, as reflected in the
multiple co-occurrences of  and d terms (Box 1).
Nitrogen concentration
We assume that nitrogen (Nm) is partitioned into two pools: photosynthetic (Nm,p), and nonphotosynthetic (Nm,n):
(A18)
Photosynthetic nitrogen is proportional to the peak per-mass photosynthetic rate:
(A19)
Non-photosynthetic nitrogen is the mass of non-photosynthetic nitrogen per mass of leaf. If the
leaf has mass M (of which m is nitrogen) and projected area A, then
(A20)
The mass of nitrogen per unit area (m/A) is proportional to the volume of nitrogen in the lamina
(Vn,L), which is proportional to the volume of the lamina, via an undetermined constant k3:
(A21)
As in the calculations for LMA, the volume of the lamina is the volume of the leaf minus the
volume of the veins:
(A22)
Combining equations A18, A19, A20, A21, and A22 yields
(A23)
Using Equation 2 to rewrite  in terms of the venation trait d, we obtain
(A24)
as a final result. This result explicitly ignores the effects of primary and secondary venation,
which could contribute a large amount of mass and a small amount of nitrogen to the leaf. As
such, Equation A24 may overestimate the actual leaf nitrogen concentration.
Appendix S2. Abbreviations, definitions, units, ranges, mean values and references for all
parameters used in the model involving venation traits. Rounded values near the mean of each
reported distribution were chosen.
Symbol
Description
Units
Range
Value used
Reference
Constant in Figure 3
rV
Vein bundle
radius
Leaf peak
carbon
assimilation
rate, per mass
Stomatal pore
area
Saturation
vapor
concentration
of water in air
Diffusion
constant of
water in air
Atmospheric
pressure
m
7.010-6 4.5 10-5
5.010-9 6.610-7
2.010-5
(Altus et al. 1985)
x
Constant in Figure
4/A3
x
-
(Wright et al. 2004)
m2
7.810-11
1.010-10
(Nobel 1999)
x
x
mol
H2O
m-3
0.3 - 2.8
(Buck 1981)
x
m2 s-1
2.010-5 3.510-5
(Nobel 1999)
x
kg m1 -2
s
3.0104 1.01105
10000.61365exp(1
7.502Tc/(240.97+T
c)) / (8.314472
(273.15+Tc))
2.12106
(1+Tc/273)1.81.01
105/P
1.0105
x
d
Vein distance
m
5.010-5 2.510-3
2/ , 1/
h
-
0.0 - 1.0
0.5
k0
k1
Relative
humidity
Constant
Constant
Value chosen to
approximately match
atmospheric pressure
at sea level
(Noblin et al. 2008) Values chosen to
represent reticulate
and open venation
respectively.
-
mo m
1.08
Unknown
1.0
1.0105
x
x
x
k2
Constant
gs
mol-1
C
Unknown
1.0105
x
x
k3
Constant
-
Unknown
1.0103
x
x
LL
Leaf life span
mo
-
LMA
Leaf mass to
area ratio
Number
density, open
g m-2
9.010-1 2.9102
1.4100 1.5103
2.5108
(Noblin et al. 2008)
Chosen to produce
LL10 mo for 104
m-1 (Wright et al.
2004)
Chosen to match the
orders of magnitude
for photosynthetic
and nonphotosynthetic
nitrogen content;
overall magnitude
chosen to produce Nm
 1% for 104 m-1
(Niinemets 1999)
Chosen to match the
orders of magnitude
for photosynthetic
and nonphotosynthetic
nitrogen content;
overall magnitude
chosen to produce Nm
 1% for 104 m-1.
(Niinemets 1999)
(Wright et al. 2004)
-
(Wright et al. 2004)
1.0108
(Nobel 1999)
x
x
Am
as
c0
D
P
ns
mol
CO2
g-1 s-1
m-2
x
Nm
ts
WUE



L
V

Tc
stomata
Leaf nitrogen,
per mass
Stomatal pore
thickness
Leaf water use
efficiency
Distance
between vein
and leaf
evaporative
surface (halfthickness)
Stoichiometric
constant
Vein
loopiness
Mass density,
lamina
Mass density,
veins
Vein density
Air
temperature
gN
g-1
m
2.010-3 6.410-2
2.010-5
-
(Wright et al. 2004)
2.010-5
(Nobel 1999)
x
mol
CO2
mol-1
H2O
m
4.010-4 1.610-2
1.010-3
(Nobel 1999)
x
5.0 10-5 2.5 10-3
d/k0
(Niinemets 1999;
Noblin et al. 2008)
mol
C g-1
m-2
Unknown
Not used in text
-
Unknown
(d-1)/d2
-
g m-3
9.2104 1.3106
Unknown
3.0105
(Niinemets 1999)
x
x
1.0106
x
x
5.0102 2.5104
0.0 - 4.0
101
5.0102 - 2.5104
Chosen to match the
density of water
(Brodribb et al.
2007)
Chosen to fall within
the temperature range
for maximum carbon
assimilation (Nobel
1999)
g m-3
m-1
°C
3.0 101
x
x
Appendix S3. Experimental methods.
Field and laboratory measurements to measure leaf functional traits
From August 31 - September 3, 2010 we measured 3 leaves from 25 woody species located on
the University of Arizona main campus (Tucson, AZ). First, we measured peak net carbon
assimilation rate using a portable gas exchange system (Li-6400XT, Li-COR, Lincoln, NE,
USA). Measurements were taken on watered plants between 6AM - 11AM at ambient air
temperature (27 - 38 °C), ambient relative humidity (21-52%), saturating light conditions (2000
µmol m-2 s-1 photon flux density) and ambient CO2 concentrations (400 ppm). Gas exchange
measurements were logged multiple times within a 2-3 minutes interval for a given leaf and
values were averaged to obtain a single observation. After gas exchange measurements, petioles
were removed from all leaves and placed on ice to stop biochemical reactions. Upon returning to
the laboratory, each leaf was immediately digitally scanned (Epson, Expression 1680). Surface
area was determined by binary thresholding using the software program, ImageJ (NIH,
http://rsbweb.nih.gov/ij/). To determine leaf mass per area (LMA), each leaf was pressed flat and
dried at 65°C for 48 hours, after which dry mass was measured with an analytical balance. LMA
(g/m²) was then calculated as the dry mass/leaf area. A 1-cm2 section of each leaf was cut and
saved for venation trait analysis. Remaining tissue was used for analyses of leaf nitrogen content
per unit mass (Nm, %; 100 x g N g-1), determined by standard elemental analysis of ground and
combusted leaf tissue (Europa Scientific 20/20 mass spectrometer, acetanilide standard).
Laboratory measurements to measure leaf venation traits
Leaf sections were chemically prepared to expose the venation network using standard methods
(Ellis et al. 2009). To remove extraneous laminar tissue, each leaf section was soaked in a warm
5% w/v NaOH/H2O bath for 2-3 days. Bath solution was changed every 24 hours. Leaves were
then rinsed in deionized water and soaked for fifteen minutes in commercial bleach (6% w/v
NaOCl/H2O). After a second rinse in deionized water, leaves were stained in a 1% w/v
safranin/ethanol solution for 10 minutes before being destained for 20 minutes in 100% ethanol.
Leaves were sequentially transferred to 50% ethanol / 50% toluene, 100% toluene, and 100%
immersion oil (Cargille, Type B) before being mounted in immersion oil on glass slides.
To visualize venation, slides were trans-illuminated and photographed with a digital
camera (Canon, Rebel T2i) attached to a dissecting microscope at 25x magnification (SZX12,
Olympus, SZX12; MM-DSLR adapter, Martin Microscope). Images were edited to prepare for
venation trait analysis (MATLAB, The MathWorks). To improve contrast, only the green
channel of the resulting digital image was used. Image contrast was further improved using
contrast limited adaptive histogram equalization. We then hand-traced the venation network for
each image and stored the resulting binary images for further analyses (GIMP;
http://www.gimp.org/).
We used MATLAB to calculate venation traits. We determined vein density as the total
number of pixels in a skeletonized transformation of the image divided by the number of pixels
in the image. We determined vein distance as twice the mean of the regional maxima of the
Euclidean distance transformation of the complement of the skeletonized image. We determined
loopiness as the number of connected components of the complement of the skeletonized image
divided by the area of the image. All statistics were corrected for scale. These codes are available
as a supplementary download, venation.zip.
Data analysis
Of the 75 leaves we collected from 25 species, we were unable to obtain venation trait
values for 15 leaves: a malfunctioning hotplate caused an unexpected explosion of our
experimental apparatus. All statistics and model predictions were calculated in R
(http:/www.r-project.org/). Model II regressions were determined with the 'smatr'
package (http://www.bio.mq.edu.au/ecology/SMATR/).
Figure S1. Structure of a model leaf including terminal veins. The leaf is approximated
as a uniformly thick lamina with one order of embedded cylindrical terminal veins. Water
is assumed to diffuse from veins through intercellular spaces until it reaches the
atmosphere by way of open stomatal pores. a. A leaf cross-section shows periodically
spaced veins. In the horizontal direction, the vein spacing is kx. The leaf has thickness 2
and the veins have radius rv. The density of venation is V and the density of lamina is L.
Water has a saturation vapor concentration c0 immediately outside of the vein and an
environmental concentration c1=c0(1-h) beyond the leaf surface, where h is relative
humidity. b. A transverse view of a surface of the leaf shows open stomata (gray dots) and
rectangular venation geometry (black lines). Each stomate has area as and depth ts; the
number density of open stomata is ns.
Figure S2. Vein spacing and leaf thickness. Empirical data demonstrate an
approximately linear relationship between the distance between veins (d) and the distance
from vein to leaf abaxial surface (). Triangles: individual leaves from this study;
thicknesses were measured with calipers. SMA slope = 0.90; OLS r2 = 0.22, p < 10-4. Circles:
data reproduced from Noblin et al. (2008); each point represents a species-mean value
obtained through analysis of multiple thin tissue sections. SMA slope = 1.08; OLS r2 = 0.979,
p < 10-15. These data sets both support the form of Eqn 2 (main text) used in our model.
Figure S3. Empirically measured venation and functional traits. Observed and
predicted values for a) maximum leaf carbon assimilation rate (Am), b) leaf life span (LL), c)
leaf mass per area (LMA), and d) leaf nitrogen content (Nm). Points represent
measurements on individual leaves for 25 species in Tucson, AZ. These data are identical to
those shown in Fig. 4. The dashed line is the 1:1 line. The solid line is the OLS regression;
r²- and P-values are given. There is a significant relationship between observed and
predicted values for all traits, but explained variation remains limited. This may be because
we did not directly measure all parameters in the model, relying instead on some speciesmean values that could bias our results (Appendix B). More information on leaf structure
(e.g. leaf area, secondary vein density, stomatal density) is necessary to explain residual
variation. Additionally, more extensive studies with a wider range of trait values and
species are necessary to fully test the limits of the model.
Supporting references
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Brodribb T., Feild T. & Jordan G. (2007). Leaf maximum photosynthetic rate and venation
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Buck A.L. (1981). New equations for computing vapor pressure and enhancement factor. J.
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Ellis B., Daly D. & Hickey L. (2009). Manual of Leaf Architecture. New York Botanical Garden.
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