Derivation of scaling theory for wood decomposition rates. Appendix S1 for Traits drive global wood decomposition rates more than climate

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Appendix S1. Derivation of scaling theory for wood decomposition rates. 1
Quantification of wood decomposition rates 2
Wood decomposition rate constants k (yr-1) are quantified as the first-order reaction 3
(Atkins & de Paula, 2010) 4
( )
0
ln[]/[]A A kt= −
(S1) 5
or 6
0
[] []
kt
A Ae
−
=
(S2) 7
where A (mass or density; g or g cm-3, respectively) is the amount of wood at time t (yr) 8
and A0 (g or g cm-3) is the initial amount of wood at t = 0. 9
10
Scaling theory linking climate and wood traits to decomposition rates 11
Building from metabolic scaling theory for decomposition (Brown et al., 2004; Allen 12
et al., 2005), we hypothesize that decomposition rate constants will exhibit a 13
Boltzmann-Arrhenius temperature dependence 14
B
/
0EkT
k ke
−
=
(S3) 15
where k0 (yr-1) is a decay normalization constant, E (eV) is an effective activation 16
energy characterizing the kinetics of decomposition, kB is the Boltzmann constant 17
(8.617 × 10-5 eV K-1), and T (K) is temperature. 18

The decomposition normalization constant k0 can be unpacked to account for several 19
biotic and abiotic variables that may influence wood decomposition rates. Specifically, 20
we can identify the potential influences of precipitation, relative humidity, active season 21
length, wood nitrogen content, and wood particle size on decomposition rates. We 22
hypothesize that k0 is characterized by: 1) a power-law dependence on precipitation 23
(Table S1), 2) a power-law dependence on relative humidity (Table S1) a direct 24
proportionality to active season length (Eq. (S1)), 4) an exponential dependence on 25
wood mass density (Table S1), 5) a power-law dependence on wood nitrogen 26
(Anderson-Teixeira & Vitousek, 2012), and 6) a power-law dependence on wood 27
particle diameter (section d, below), so that 28
01
l
h as N
PD
a
as
k kP h l e N D
α
αα
αα
ρ
=⋅ ⋅⋅ ⋅⋅ ⋅
(S4) 29
where k1 (
1
yr mm d cm g N g M
ll
as as N N
PD
αα αα
αα
+− −
−−
) is a decomposition normalization 30
constant, P (mm) is precipitation, αP (dimensionless) is a precipitation scaling 31
exponent, h (dimensionless) is relative humidity, αh (dimensionless) is a relative 32
humidity scaling exponent, las (d yr-1) is the active season length,
as
l
α
(dimensionless) 33
is an active season length scaling exponent, ρ (g cm-3) is wood mass density, a is a wood 34
density coefficient (cm3 g-1), N (g N g M-1) is wood nitrogen content, αN (dimensionless) 35
is a wood nitrogen content scaling exponent, D (cm) is wood diameter, and αD 36
(dimensionless) is a wood diameter scaling exponent. Substituting Eq. (S4) into Eq. 37
(S3) and rearranging gives 38

/
1
l
h as N
BP D
EkT a
as
k ke P h l e N D
α
αα
αα
ρ
−
=
(S5) 39
Eq. (S5) expresses several hypotheses regarding the dependence of wood 40
decomposition rate on climate and wood trait variables, including temperature, 41
precipitation, relative humidity, active season length, wood mass density, wood 42
nitrogen content, and wood diameter. 43
Since decomposition rate constants k are quantified at an annual time resolution, 44
their magnitudes are confounded by variation in active season length among study sites. 45
By definition, when all else is equal, sites with longer active seasons will have larger 46
values of k. In order to remove the confounding influence of active season length, we 47
recast Eq. (S5) to yield a more instantaneous daily decomposition rate k/las (d-1) as 48
B
/
2
hN
PD
EkT a
as
kke P h e N D
l
αα
αα
ρ
−
=
(S6) 49
where k2 (
1
d mm g N g M cm
NN
PD
αα
αα
−
−−
−
) is another decomposition normalization 50
constant. 51
Eqs. (S5) and (S6) can be linearized, respectively, as 52
1
B
ln( ) ln( ) ln( ) ln( ) ln( ) ln( ) ln( )
as
P h l as N D
E
k k P h la N D
kT
α α α ρα α
= − + + + ++ +
(S7)53
and 54
2
B
ln ln( ) ln( ) ln( ) ln( ) ln( )
Ph N D
as
kE
k P ha N D
l kT
α α ρα α

= − + + ++ +


(S8) 55
56

Boltzmann-Arrhenius temperature dependence of wood decomposition rates 57
Relationships between temperature and decomposition rates can be revealed using 58
Arrhenius plots. Specifically, Eqs. (S5) and (S6) can be loge-transformed to give the 59
linear forms 60
( )
( )
1
B
1
ln ln
l
h as N
PD
a
as
k E kP h l e N D
kT
α
αα
αα
ρ

=−+


(S9) 61
and 62
( )
2
B
1
ln ln
hN
PD
a
as
kE kP h e N D
l kT
αα
αα
ρ
 
=−+
 


(S10) 63
These loge-transformed decomposition rates are then regressed over ambient 64
temperature 1/kBT, yielding a slope –E that is equal in magnitude but opposite in 65
direction to the effective activation energy E. It has been hypothesized that the 66
temperature-dependency of wood decomposition rates may reflect activation energies 67
for the biochemical reactions of aerobic metabolism (~0.65 eV; see Allen et al., 2004; 68
Gillooly et al., 2001) or microbial ecoenzymes kinetics (~0.31 – 0.56 eV; see 69
Sinsabaugh & Follstad Shah, 2012; Wang et al., 2012; Follstad Shah et al., 2017). 70
71
Scaling of wood decomposition rates with wood diameter 72
Decomposition of wood mass/volume occurs across its surface area. Thus, we 73
hypothesize that decomposition rates k (yr-1) and k/las (d-1) are proportional to the 74

surface-area-to-volume ratio of the wood particle (Schmidt-Nielsen, 1984; Niklas, 75
1994) 76
as
kA
klV
∝∝
(S11) 77
where A (cm2) is the surface area and V (cm3) is the volume. Approximating wood 78
particles (stems and branches) as cylinders, the decomposition rate constant and the 79
surface-area-to-volume ratio can be expressed in terms of wood particle diameter D 80
(cm) as 81
211
2
2 ( / 2) 2 ( / 2) 42
as
kA Dl D
k Dl
l V rl
ππ
π
−−
+
∝ ∝= = +
(S12) 82
where l (cm) is the length of the wood particle. Eqs. (S11) and (S12) show that 83
surface-area-to-volume ratios and, ultimately, decomposition rates will vary with the 84
unique length-to-diameter geometry of each wood particle in question. 85
To make progress, we consider three idealized wood particle geometries that allow 86
us to express Eq. (S12) in terms of wood diameter only. These geometries originate 87
from different theoretical optimality theories for length-to-diameter geometry in woody 88
plant stems and branches. It is important to consider these various idealizations since 89
length-to-diameter geometries vary spatially (Bertram, 1989) and temporally (Niklas, 90
1995) within individuals, reflecting spatial and temporal variation in constraints on 91
scaling of plant form and function. 92

First, we consider the geometric similarity theory (Niklas, 1995), which predicts 93
isometric scaling such that 94
1
lD∝
(S13) 95
Substituting Eq. (S13) into Eq. (S12) and simplifying gives 96
1
6
as
kA
kD
lV
−
∝ ∝=
(S14) 97
Second, we consider the elastic similarity (McMahon & Kronauer, 1976; Niklas, 98
1995) and the West-Brown-Enquist (West et al., 1999) (WBE) network theories, both 99
of which predict allometric scaling according to 100
2/3
lD∝
(S15) 101
Substitution of Eq. (S15) into Eq. (S12) and simplifying yields 102
1 2/3
42
as
kA
k DD
lV
−−
∝ ∝= +
(S16) 103
This Eq. (S16) can be solved numerically to show that the surface-area-to-volume 104
ratio scales with diameter as 105
0.76
as
kA
kD
lV
−
∝ ∝∝
(S17) 106
Third, we consider the stress similarity theory (Niklas, 1995), which predicts 107
allometric scaling such that 108
1/2
lD∝
(S18) 109

Substituting Eq. (S18) into Eq. (S12) and simplifying gives 110
1 1/2
42
as
kA
k DD
lV
−−
∝ ∝= +
(S19) 111
Eq. (S19) can also be solved numerically to show that the surface-area-to-volume 112
ratio scales with diameter as 113
0.60
as
kA
kD
lV
−
∝ ∝∝
(S20) 114
These optimality models predict that surface-area-to-volume ratios and 115
decomposition constants (Eq. (S11)) will scale with diameter with a scaling exponent 116
that depends on the unique geometry of the wood particle in question. While the 117
different optimality models yield different numerical values for these exponents, all are 118
consistent with our surface-area-to-volume hypothesis for scaling of decomposition 119
with diameter. 120
Importantly, deviations from these predicted exponents reflect departures from the 121
optimality assumptions. Decomposing wood particles will almost always be broken 122
and/or partially decomposed, so their lengths will generally be shorter than expected 123
for idealized geometries and the scaling of length with diameter will thus depart from 124
the optimality assumptions. For example, if wood particle length scales with diameter 125
as 126
1/4
lD∝
(S21) 127

then 128
0.34
as
kA
D
lV
−
∝∝
(S22) 129
where the scaling exponent of -0.34 is approximately equal to the empirically observed 130
value of αD = -0.35 (Table S3). 131
132
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