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Michaletz et al. 2014 Supplement

Authors:
Sean Michaletz at University of British Columbia
Sean Michaletz
Dongliang Cheng at Fujian Normal University
Dongliang Cheng
Andrew J Kerkhoff at Kenyon College
Andrew J Kerkhoff
Brian J. Enquist at University of Arizona
Brian J. Enquist
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SUPPLEMENTARY INFORMATION doi:10.1038/nature13470
Supplementary Information
Derivation of metabolic scaling theory for terrestrial net primary production
We characterize the net stand biomass production rate Gtot (g yr-1) in terms of the average
individual growth rate G(r) (g yr-1) and the stand size distribution f(r) (m-1). Assuming that
growth rate is directly proportional to metabolic rate, we build from Enquist et al.1 so that
tot 0 0 m 0 0 m
dd
nn n
G Gr
f
rr
g
cr
g
cr r
g
cr  

(S1)
Here, G(r) = g0r2, where g0 is a growth normalization constant (g m-2 yr-1), and
f(r) = dn/dr = cnr-2 (m), where cn is a size-corrected measure of the number of individuals of a
given size. If the radius of the smallest tree stem r0 (m) is much smaller than the radius of the
largest tree stem rm (m), then the net stand biomass production rate is approximately proportional
to the radius of the largest tree2, which is given by1
8/3
m tot
5
3
m
n
c
rM
c
 
 
 
(S2)
where cm is a normalization constant relating stem radius to tree mass (r = cmm3/8; m g-3/8), Mtot is
the total forest biomass (g), and α is a scaling exponent (dimensionless). The value of α can be
shown to be related to the allometric scaling of individual metabolism with size (see derivation
in ref. 1).
Next, we can substitute Eq. (S2) into Eq. (S1) to show explicitly how the production rate
tot
G scales with total stand biomass tot
M
8/3 8/3
tot 0 tot 0 tot
55
33
mm
nn
nn
cc
G
g
cM
g
cM
cc

 
 
 
 
(S3)
Note that the units of the normalization constant g0 must change (i.e., g m-1-α(5/3) yr-1) if the value
of the scaling exponent varies from the theoretical value1 of α = 3/5. The corresponding
autotrophic or whole-stand annual net primary productivity NPP (g m-2 yr-1) is
8/3
tot
0 tot
5
NPP 3
nm
n
G cc
g
M
A Ac

  

(S4)
where A is the stand area under consideration (m2) and cn/A is the size-corrected density of
individuals per unit area of a given size (cn/A = [dn/dr][1/A]r2).
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Next, we show that the normalization constant g0 can be partitioned into a number of key
abiotic components that influence NPP. Specifically, we identify the potential influences of
precipitation, growing season length, plant age, and temperature on NPP. We hypothesize that g0
is characterized by: (i) a power-law dependence on precipitation (Extended Data Fig. 1), (ii) a
direct proportionality to growing season length3,4, (iii) a power-law dependence on plant age
(Extended Data Fig. 1), and (iv) an Arrhenius temperature dependence5,6, so that
gs /
0 1 gs
la
PE kT
g gP l a e
 
(S5)
where g1 is a growth normalization constant ( gs gs
1
1 (5/3)
g m yr mm mo
la l
P

 
 ), P is
precipitation (mm),
P
is a precipitation scaling exponent, lgs is the growing season length (mo
yr-1), gs
l
is a growing season length scaling exponent, a
is a plant age scaling exponent, E is an
activation energy (eV), k is the Boltzmann constant (8.617 x 10-5eV K-1), and T is temperature
(K). Plant age a (yr) is that of the individuals constituting most of the stand biomass; it will equal
the stand age for even-aged stands and the age of the dominant size classes for secondary and
old-growth forests. Substituting Eq. (S5) into Eq. (S4) and rearranging gives
gs
8/3
/
gs 1 tot
5
NPP 3
la
PE kT nm
n
cc
Pl ae
g
M
Ac



(S6)
Eq. (S6) expresses a number of hypotheses regarding the dependence of NPP on climatic and
biotic variables, including precipitation, growing season length, plant age, temperature, and total
stand biomass. We can recast Eq. (S6) to yield a more instantaneous monthly net primary
production (NPP/lgs; g m-2 mo-1) as
8/3
/
2 tot
gs
5
NPP
3
aP E kT nm
n
cc
P
ae g M
l Ac



(S7)
where g2 is another growth normalization constant ( gs
1
1 (5/3)
g m mo mm yr
la
P

 ). Thus,
Eqs. (S6) and (S7) provide different perspectives on the interpretation of production rates in
woody plant communities.
Relationships between temperature and production rates can be revealed using Arrhenius
plots. Specifically, Eqs. (S6) and (S7) can first be loge-transformed to give the linear forms

gs
8/3
gs 1 tot
5
1
ln NPP ln 3
la
Pnm
n
cc
E Pl ag M
kT A c




 


 

(S8)
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8/3
2 tot
gs
5
NPP 1
ln ln 3
a
Pnm
n
cc
E Pag M
l kT A c

 



 



 


(S9)
These loge-scaled production measures are then regressed over ambient temperature 1/kT,
providing a slope –E that is equal in magnitude but opposite in direction to the activation energy
E. It has been hypothesized that the temperature-dependency of plant metabolism and growth
might be characterized by E = 0.32 eV, reflecting the activation energy of photosynthesis7.
Importantly, the same mass-scaling relationships predicted for community-level Gtot,
NPP, and NPP/lgs (Eqs. S3, S4, S6, and S7) are also predicted for rates of individual-level growth
and production by component (root, aboveground woody, and foliage), because rates of root
growth GR(r), aboveground woody growth GAGW(r), and foliage growth GF(r) are predicted to be
isometric8 within individual plants, and these components sum to give both whole-plant growth
rates (i.e. G(r) = GR(r) + GAGW(r) + GF(r)) and community-level production rates (e.g. NPP/lgs =
[NPPR + NPPAGW + NPPF]/lgs). Thus, for example, monthly net primary production NPP/lgs is
predicted to scale isometrically between components; this is marginally supported by our data as
NPPR/lgs scales as the 1.22 power (95% CI = 1.18 to 1.27) of NPPAGW/lgs, NPPF/lgs scales with
the 1.12 power (95% CI = 1.06 to 1.18) of NPPAGW/lgs, and NPPR/lgs scales as the 1.10 power
(95% CI = 1.04 to 1.16) of NPPF/lgs. Further, monthly net primary production for each
component is predicted to scale with total stand biomass as
R gs AGW gs F gs tot
NPP / NPP / NPP /l l lM
 
, where α = 3/5 = 0.6 (ref. 1); results of this analysis
are discussed in the main text and Extended Data Table 3.
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Effects of leaf type, soil fertility, and biome on scaling of NPP with age and biomass
The normalization constant c of the simpler model ( tot
NPP a
ca M
) characterizes the
combined influence of climate variables and plant functional traits on NPP9,10, and is thus
expected to vary with morphological, physiological and environmental stand characteristics. The
effects of leaf type, soil fertility, and biome were evaluated using standardized major axis (SMA)
regression fits11 of NPP on tot
a
aM
(Fig. 3, Extended Data Fig. 4). SMA slopes did not vary with
leaf type (broadleaf, needle-leaf, and mixed-leaf; p = 0.271; Fig. 3b), but leaf type did influence
normalization constants as needle-leaf stands had a significantly lower elevation than broad-leaf
(p < 2.22 x 10-16) and mixed-leaf stands (p = 8.026 x 10-6). Further, broad-leaf and mixed-leaf
stands were shifted along their common axis (p = 9.811 x 10-7). Slopes for SMA fits did not vary
among soil fertility classes (p = 0.144; Fig. 3c), but the high fertility class had significantly
higher elevation than low (p = 2.072 x 10-8) and medium (p = 8.880 x 10-6) fertility classes, and
all fertility classes were shifted along their common axis (p < 2.22 x 10-16). Total stand biomass
decreased with soil fertility, with low fertility soils having greater stand biomass than medium
fertility soils (ANOVA; p = 3.87 x 10-4), and medium fertility soils having greater stand biomass
than high fertility soils (ANOVA; p = 3.3 x 10-14). This negative relationship between soil
fertility and total stand biomass likely reflects both the inverse relationship between precipitation
and soil fertility (p < 2.22 x 10-16) and the role of precipitation in limiting maximum plant size12
and total stand biomass1,13. Thus, these results show that while high fertility soils have lower
total biomass, they are more productive. Finally, across biomes, SMA slopes for temperate forest
stands varied from desert (p = 0.043; Extended Data Fig. 4), taiga (p = 0.038), and
woodland/shrubland (p = 0.005) stands; woodland/shrubland stands had higher NPP than taiga
stands (p = 2.589 x 10-6), which in turn had higher NPP than desert stands (p = 0.011).
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1 Enquist, B. J., West, G. B. & Brown, J. H. Extensions and evaluations of a general
quantitative theory of forest structure and dynamics. Proceedings of the National
Academy of Sciences 106, 7046-7051, doi:10.1073/pnas.0812303106 (2009).
2 Stephenson, N. L. et al. Rate of tree carbon accumulation increases continuously with
tree size. Nature 507, 90-93, doi:10.1038/nature12914 (2014).
3 Enquist, B. J., Kerkhoff, A. J., Huxman, T. E. & Economo, E. P. Adaptive differences in
plant physiology and ecosystem paradoxes: insights from metabolic scaling theory.
Global Change Biology 13, 591-609, doi:10.1111/j.1365-2486.2006.01222.x (2007).
4 Kerkhoff, A. J., Enquist, B. J., Elser, J. J. & Fagan, W. F. Plant allometry, stoichiometry
and the temperature-dependence of primary productivity. Global Ecology and
Biogeography 14, 585-598, doi:10.1111/j.1466-822X.2005.00187.x (2005).
5 Gillooly, J. F., Brown, J. H., West, G. B., Savage, V. M. & Charnov, E. L. Effects of Size
and Temperature on Metabolic Rate. Science 293, 2248-2251,
doi:10.1126/science.1061967 (2001).
6 Gillooly, J. F., Charnov, E. L., West, G. B., Savage, V. M. & Brown, J. H. Effects of size
and temperature on developmental time. Nature 417, 70-73 (2002).
7 Allen, A. P., Gillooly, J. F. & Brown, J. H. Linking the global carbon cycle to individual
metabolism. Functional Ecology 19, 202-213, doi:10.1111/j.1365-2435.2005.00952.x
(2005).
8 Niklas, K. J. & Enquist, B. J. On the Vegetative Biomass Partitioning of Seed Plant
Leaves, Stems, and Roots. The American Naturalist 159, 482-497, doi:10.1086/339459
(2002).
9 Enquist, B. J. & Bentley, L. P. in Metabolic Ecology: A Scaling Approach (eds R. M.
Sibly, Sandra Brown, & A. Kodric-Brown) 164-187 (John Wiley & Sons, Ltd., 2012).
10 Enquist, B. J. et al. A general integrative model for scaling plant growth, carbon flux, and
functional trait spectra. Nature 449, 218-222,
doi:http://www.nature.com/nature/journal/v449/n7159/suppinfo/nature06061_S1.html
(2007).
11 Warton, D. I., Duursma, R. A., Falster, D. S. & Taskinen, S. smatr 3– an R package for
estimation and inference about allometric lines. Methods in Ecology and Evolution 3,
257-259, doi:10.1111/j.2041-210X.2011.00153.x (2012).
12 Kempes, C. P., West, G. B., Crowell, K. & Girvan, M. Predicting Maximum Tree
Heights and Other Traits from Allometric Scaling and Resource Limitations. PLoS ONE
6, e20551, doi:10.1371/journal.pone.0020551 (2011).
13 West, G. B., Enquist, B. J. & Brown, J. H. A general quantitative theory of forest
structure and dynamics. Proceedings of the National Academy of Sciences 106, 7040-
7045, doi:10.1073/pnas.0812294106 (2009).
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A general integrative model for scaling plant growth, carbon flux, and functional trait spectra
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Kerkhoff, A. J., Enquist, B. J., Elser, J. J. & Fagan, W. F. Plant allometry, stoichiometry and the temperature-dependence of primary productivity. Glob. Ecol. Biogeogr. 14, 585-598
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