New insights into carbon allocation by trees from the hypothesis that annual wood
production is maximised
Ross E. McMurtrie1, Roderick C. Dewar2
Supporting Information Table S1 & Notes S1–S3
Table S1 Optimal values of plant traits, and stand structural and functional variables
predicted by MaxW with Umax = 0.012 and 0.008 kg N m-2 year-1
Foliage/root property
Modelled
Modelled
Umax = 12
Umax = 8
Canopy-average needle N:C ratio, nf (kg N kg-1 C)
0.033
0.028
Canopy-average leaf N content = Ntot / Ltot (kg N m-2)
0.0040
0.0028
Plant traits
Canopy-average leaf mass per unit area = Ftot / ( Ltot) (kg DW 0.25
0.20
m-2)
Average root-C density = Rtot / Dmax (kg C m-3 soil)
0.11
0.08
C payback by basal leaves per unit N invested, εf(Dmax) (kg C 446
812
kg-1 N)
N payback by basal roots per unit C invested, εr(Dmax) (kg N kg- 0.00224
1
0.00123
C)
Bottom line, εf(Ltot)  εr(Dmax)
1
1
Total canopy LAI, Ltot (m2 leaf m-2 land)
4.5
2.5
Total leaf biomass, Ftot (kg C m-2 land)
0.54
0.25
Maximum rooting depth, Dmax (m)
1.31
1.11
Stand structure
Total fine root C, Rtot (kg C m-2 land)
0.14
0.09
Total canopy N content, Ntot (kg N m-2 land)
0.0177
0.0072
Annual gross canopy photosynthesis, Atot (kg C m-2 land year-1)
2.03
1.07
Annual foliage production, Cf (kg C m-2 land year-1)
0.06
0.032
Annual root production, Cr (kg C m-2 land year-1)
0.14
0.09
Annual wood production, Cw (kg C m-2 land year-1)
0.71
0.36
NPP (kg C m-2 year-1)
0.91
0.48
Foliage C allocation, Cf/NPP
0.07
0.07
Root C allocation, Cr/NPP
0.15
0.19
Wood C allocation, Cw/NPP
0.77
0.74
Annual root N uptake, Utot (kg N m-2 land year-1)
0.0053
0.0029
Root N–uptake fraction = Utot / Umax (dimensionless)
0.44
0.36
Stand function
Notes S1 Solution of the MaxGPP optimisation hypothesis
The MaxGPP optimisation hypothesis states that annual gross canopy photosynthesis (Atot, kg
C m-2 land area year-1, Eqn 5) is maximised with respect to the vertical distribution of leaf N
content (Na(L), kg N m-2 leaf area) and total leaf-area index (Ltot) subject to a fixed total
canopy–N content (Ntot, kg N m-2 land area, Eqn 4). We follow Dewar et al. (2012) in
assuming there is a lower limit to leaf mass per unit area (ma*, kg DW m-2), and that leaf N:C
ratio (nf) is independent of L. If leaf mass per unit area (LMA) is equal to ma* at the canopy
base (L = Ltot), it is appropriate to consider a solution for optimal Na(L) for which this
condition applies to an extended region of the lower canopy:
for Lcrit L  Ltot,
Na(L) = N a*
Eqn S1.1
where leaf-N content at the canopy base is N a* = nf ma* , where  (kg C kg-1 DW) is the
carbon content of biomass. The values of nf and Lcrit, which may depend on Ltot, are yet to be
determined.
This constrained optimisation problem can be solved by the Lagrange–multiplier method as
follows. We introduce a Lagrange multiplier 1 and maximise the goal function
1  Atot  1 N tot  
Ltot
0

Lcrit ( Ltot )
0
Aa ( N a ( L), L) dL  λ 1 
L tot
0
 Aa ( N a ( L), L)  λ1 N a ( L)  dL  L
Ltot
crit ( Ltot )
N a ( L) dL
A ( N
a
*
a
, L)  λ 1 N a*  dL
Eqn S1.2
Eqn S1.3
The change 1 associated with small variations Ltot and Na(L) for 0  L < Lcrit is
1  
Lcrit
0
 Aa


 λ1  N a ( L) dL  Aa ( N a* , Ltot )  λ1 N a* δLtot
 N a



Eqn S1.4
where we have assumed continuity of Na and Aa at L = Lcrit (i.e. N a ( Lcrit )  N a* ) so that the
variation in 1 due to variation of Lcrit vanishes. Then setting 1 to zero, with Na(L) and
Ltot treated as independent variations, gives
Aa
 λ1
N a
for 0  L < Lcrit ,
Aa ( N a* , Ltot )  λ1 N a* .
Eqn S1.5
Eqn S1.6
Combining Eqns S1.5 and S1.6 to eliminate 1 gives
Aa
A (N * , L )
 a a * tot
N a
Na
for 0  L < Lcrit.
Eqn S1.7
The solution for L  Lcrit is obtained from Eqn S1.1:
Aa  Aa ( N a* , L)
for Lcrit  L  Ltot.
Eqn S1.8
Note: if Lcrit = Ltot, i.e. if ma(L)  m a* throughout the canopy, then photosynthetic N-use
efficiency (PNUE) of basal leaves, Aa / Na, is maximized at the canopy base (L = Ltot), but not
elsewhere in the canopy (cf. McMurtrie & Dewar, 2011). However, if Lcrit < Ltot so that
ma(Ltot) = m a* , then PNUE is not maximised at the canopy base.
Following Dewar et al. (2012), we assume that gross leaf photosynthesis is a rectangular
hyperbolic function of light intensity:
Aa ( L)   mc
Asat ( L) α I ( L)
,
Asat ( L)  α I ( L)
Eqn S1.9
where  (s year-1) = growing–season length, mc = a constant for unit conversion from mol
CO2 to kg C,  (mol CO2 mol -1 quanta) = quantum yield of photosynthesis, Asat(L) (mol CO2
m-2 s-1) = light-saturated photosynthetic rate, which is a linear function of leaf N content:
Asat ( L)  a N N a L   N o  ,
Eqn S1.10
where aN (mol kg-1 N s-1) = slope of the relationship, No (kg N m-2) = threshold leaf N content
for photosynthesis. Photosynthetic photon-flux density (PPFD) incident on leaves at
cumulative LAI L (I(L), mol quanta m-2 s-1) is
I ( L)  k L I in e  k L L
Eqn S1.11
where Iin (mol quanta m-2 land s-1) = PPFD incident at the top of the canopy, and kL = lightextinction coefficient. It follows from Eqn S1.9 that
Aa
 mc a N

N a  a N L   N   2
o
1  N a

α
I
(
L
)


for 0  L < Lcrit
Eqn S1.12
for 0  L < Lcrit.
Eqn S1.13
and from Eqn S1.7 that
Aa

N a
 mc a N
a N N a*
1

1  N o N a* α I ( Ltot )
Combining Eqns S1.12 and S1.13 leads to the following equation for Na(L):
N a ( L)  N o 
where  =

N a* e  kL L 
1
k L Ltot

ζ
e

1


*
ζ

 1  N o N a
for 0  L < Lcrit , Eqn S1.14
a N N a*
. Lcrit is determined as a function of Ltot from Eqns S1.1 and S1.14,
α k L I in
assuming continuity at L = Lcrit:
 1


 
1
1
k L Ltot

  .
Lcrit Ltot   max 0 ,
log e 

ζ
e

1
* 
*
 

k
ζ
(
1

N
N
)
1

N
N

L
o
a 
o
a
 

Eqn S1.15
Total canopy N and canopy photosynthesis are evaluated as functions of Ltot by substituting
Eqns S1.1, S1.9, S1.14 and S1.15 into Eqns 4 and 5, respectively:
N tot Ltot   N o Lcrit  N a* Ltot  Lcrit  
α I in
aN


1
kL Ltot

 1  e kL Lcrit

ζ
e

1
 1 N N*

o
a



if Lcrit > 0

Eqn S1.16a
 N a* Ltot
if Lcrit = 0
Eqn S1.16b
and






Atot Ltot 
1
 α I in 1  e  k L Lcrit 1 

ξ mc
1
k L Lto t 

 ζe


1  N o N a*



 1  ζ 1  N o N a* e kL Ltot
1
*

 a N N a  N o Ltot  Lcrit  log e 
k L Lcrit
*

kL
1  ζ 1  N o N a e










 

if Lcrit > 0


Eqn S1.17a

1  ζ 1  N o N a* e kL Ltot  
1
 a N N a*  N o  Ltot 
log e 
 
*

k
1

ζ
1

N
N
L
o
a







if Lcrit = 0.
Eqn S1.17b
Atot – Ntot relationship
MaxGPP thus predicts Na(L) (Fig. 1a), Aa(L), Ltot and Atot as functions of the constraint Ntot.
The maximised value of Atot can be evaluated as a function of Ntot by eliminating Ltot between
Eqns S1.16 and S1.17. The resulting Atot-Ntot relationship is illustrated in Fig. 3(a), in which
the red and blue curves correspond to varying Ntot with the leaf N:C ratio fixed at its optimal
value predicted by MaxW at, respectively, high and low soil N availability Umax (see Notes
S3, step 2). The Atot-Ntot curve in Fig. 3(a) differs slightly in response to altered Umax because
the optimal value of nf changes with Umax, which affects the value of N a* in Eqns S1.16 and
S1.17.
A generic equation for the slope dAtot/dNtot of this relationship, valid irrespective of the leaf
photosynthesis model, can be obtained from Eqns 4 and 5 written in the form:
Atot  
Lcrit
0
Aa ( N a ( L), L) dL  
Ltot
Lcrit
Aa ( N a* , L) dL ,
Eqn S1.18
N tot  
Lcrit
0
N a ( L) dL  N a* Ltot  Lcrit  ,
Eqn S1.19
Under a small variation Ntot, the associated variations Atot, Ltot and Na(L) for 0  L < Lcrit,
are related via Eqns S1.18 and S1.19, giving
Atot  
Lcrit
0
Aa
N a ( L)dL  Aa ( N a* , Ltot )δLtot
N a
N tot  0Lcrit N a ( L) dL  N a* δLtot
Eqn S1.20
Eqn S1.21
where we have used continuity of Na and Aa at L = Lcrit. Using Eqn S1.7, Eqn S1.20 simplifies
to
 1
Atot  Aa ( N a* , Ltot )  *
 Na

Lcrit
0

N a ( L) dL  δLtot  ,

Eqn S1.22
and combining this with Eqn S1.21 then gives
Aa ( N a* , Ltot )
Atot 
δN tot .
N a*
Eqn S1.23
In the limit Ntot0 this becomes
dAtot
Aa ( N a* , Ltot )

,
dN tot
N a*
Eqn S1.24
Thus, the canopy-scale sensitivity of annual GPP to an increase in canopy N content is equal
to the PNUE of basal leaves.
Notes S2 Solution of the MaxNup optimisation hypothesis
The MaxNup optimisation hypothesis states that annual N uptake (Utot, kg N m-2 land area
year-1, Eqn 9) is maximised with respect to the distribution of root-C density (R(z), kg C m-3
soil volume) as a function of soil depth (z, m) and maximum rooting depth (Dmax, m) subject
to a fixed total root C (Rtot, kg C m-2 land area, Eqn 6). This hypothesis was considered by
McMurtrie et al. (2012), who assumed that Ur(z) is a saturating function of R(z):
U r ( z) 
U o ( z)
,
1  Ro R( z )
Eqn S2.1
where Uo(z) is potential annual plant N uptake, which is the asymptotic N-uptake rate in the
limit R(z)   and Ro (kg C m-3) is the root-C density yielding half the potential N-uptake
rate. Uo(z) is greatest at the soil surface and decreases exponentially with depth (Jackson et
al., 2000; Jobbágy & Jackson, 2001; McMurtrie et al., 2012):
U o ( z) 
U max  z Do
e
Do
Eqn S2.2
where Umax (kg N m-2 land area year-1) is total potential annual N uptake integrated over all
soil depths, and Do (m) is the length scale for the exponential decrease of available soil N with
soil depth.
The optimal profile of Ur(z) and optimal Dmax can be obtained by solving the equation
U r
 2
R
for 0  z  Dmax
Eqn S2.3
(McMurtrie et al., 2012), where the Lagrange multiplier 2 is
2 
U r ( R( Dmax ), Dmax ) U o ( Dmax )
.

R( Dmax )
Ro
Eqn S2.4
Here Ur(z) is given by Eqn S2.1, and the last equality is obtained by taking the limit R(Dmax)
 0, which is the case for the particular N uptake model described by Eqn S2.1. Optimal
values of Rtot and Utot are related to Dmax by the equations

 Dmax




2
D
o
Rtot Dmax   Ro  2 Do  e
 1  Dmax









and






Eqn S2.5
D

 max

U tot Dmax   U max 1  e 2 D o



2


 .



Eqn S2.6
Utot – Rtot relationship
MaxNup thus predicts R(z) (Fig. 2a), Ur(z), Dmax and Utot as functions of the constraint Rtot.
The maximised value of Utot can be expressed as a function of Rtot by eliminating Dmax
between Eqns S2.5 and S2.6:



Rtot

,
 log e 1    
1 
 2R D
o
o




Eqn S2.7
where  (= Utot/Umax) is the fraction of available N taken up annually (Fig. 2b of McMurtrie et
al., 2012). This relationship is illustrated in Fig. 3(b). A generic equation for the slope
dUtot/dRtot, valid irrespective of the root N uptake model, can be obtained from Eqns 6 and 9,
analogous to the derivation of the slope dAtot/dNtot in Notes S1, to give
dU tot U r ( R( Dmax ), Dmax )

.
dRtot
R( Dmax )
Eqn S2.8
Thus the sensitivity of annual root-system N uptake to an increase in total root biomass is
equal to annual N uptake per unit C of basal roots.
Notes S3 Solution of the MaxW optimisation hypothesis
The MaxW optimisation hypothesis states that annual wood production (Cw, kg C m-2 land
area year-1, Eqn 3) is maximised with respect to the vertical profiles of leaf N content at
cumulative LAI L (Na (L), kg N m-2 leaf area) and root-C per unit soil volume at soil depth z
(R(z), kg C m-3 soil volume), total leaf-area index (Ltot), maximum rooting depth (Dmax, m)
and leaf N:C ratio (nf), under constraints that C and N are conserved (Eqn 10).
It is convenient to solve the problem in two steps (Table 1):
(1) optimise Na(L), R(z), Ltot , and Dmax for a given nf;
(2) optimise nf.
The motivation here is that step 1 leads to generic results that are independent of the specific
details of the models for leaf photosynthesis and root N uptake, while the result of step 2 is
more dependent on those details.
Step 1: Maximisation of Cw with respect to Na(L), R(z), Ltot, Dmax
In contrast to MaxGPP and MaxNup where, respectively, Ntot and Rtot are constrained to be
fixed, the key constraint in MaxW is C and N conservation. Cw is given from C conservation
by Eqn 3, while we introduce a Lagrange multiplier λ for the constraint of N conservation
(Eqn 8). We then maximise the goal function


N (1  r ) Rtot nr
Ψ  C w  λ U tot  tot

 C w nw  .
τf
τr


Eqn S3.1
From Eqn 3, this is equivalent to

N
1
  1  λ nw CUE Atot  λ U tot   1  λ nw  - λ (1  r )  tot
nf

 τf
 1  λ nw    nr 
Rtot
,
τr
Eqn S3.2
where Ntot, Atot, Rtot and Utot are given by Eqns 4, 5, 6 and 9, respectively. We assume there is
a lower limit to LMA as in Notes S1.  is then maximised independently with respect to
Na(L), Ltot, R(z) and Dmax. Leaf N:C ratio (nf) is a constant, whose value will be determined
below in step 2.
From Eqn S3.2, the change  associated with small variations Na(L), Ltot, R(z), Dmax is
  
L crit
0



A
 CUE a 1  λ n w    1  λ n w  1 - λ (1  r )  1

τ

N a
nf

 f


 N a ( L) dL




 N a* 
1
*

 Ltot
  CUE Aa ( N a , Ltot )1  λ n w    1  λ n w  - λ (1  r ) 

n
τ
f

 f 


Dmax
0
 U r
1
 λ
 1  λ nw   λ nr 
τr
 R

δR( z ) dz


R( Dmax ) 
 δDmax ,
  λU r ( R( Dmax ), Dmax )  1  λ nw   λnr 
τr


Eqn S3.3
where the variation in  due to variation of Lcrit vanishes as in Notes S1. Setting  to zero
gives the following four optimisation criteria for Na(L), Ltot, R(z) and Dmax:
CUE


Aa
1  λ nw    1  λnw  1 - λ (1  r )  1
N a
nf

 τf
for 0  L < Lcrit ,

 N*
1
CUE Aa ( N a* , Ltot )1  λ nw    1  λ nw  - λ (1  r )  a ,
nf

 τf
λ
U r
1
  1  λ nw   λ nr 
R
τr
for 0  z  Dmax ,
λU r ( R( Dmax ), Dmax )   1  λ nw   λ nr 
R( Dmax )
.
τr
Eqn S3.4
Eqn S3.5
Eqn S3.6
Eqn S3.7
Combining Eqns S3.4 and S3.5 gives
Aa
Aa ( N a* , Ltot )

N a
N a*
for 0  L < Lcrit ,
Eqn S3.8
which is identical to Eqn S1.7 obtained from MaxGPP. Therefore the solution for optimal
Na(L) and Aa(Na(L),L) is as presented in Notes S1. Combining Eqns S3.6 and S3.7 gives
U r U n ( R( Dmax ), Dmax )

R
R( Dmax )
for 0  z  Dmax ,
Eqn S3.9
which is equivalent to Eqns S2.3, S2.4 obtained from MaxNup (McMurtrie et al. 2012).
MaxW thus gives the same solutions for Na(L) and Aa(L) as MaxGPP, and the same solutions
for R(z) and Ur(z) as MaxNup. It follows that the optimal values of Ltot and Atot obtained from
MaxW are the same functions of Ntot and nf as those given by Eqns S1.16 and S1.17, even
though in MaxW the value of Ntot is no longer constrained to be fixed but is an emergent
outcome of the optimisation. Similarly the optimal values of Dmax and Utot obtained from
MaxW are the same functions of Rtot as those given by Eqns S2.5 and S2.7, even though Rtot is
no longer fixed. In other words, the vertical profiles of leaf N content and root-C density
predicted by MaxW are just those that would be predicted by MaxGPP and MaxNup if Ntot
and Rtot had been fixed at their MaxW values.
Combining Eqns S3.5 and S3.7 leads to the bottom line condition
 CUE Aa ( Ltot )

1   U (D )

τ f    r max τ r  nr   1  r ,
nf   R( Dmax )
 N a ( Ltot )

Eqn S3.10
which, using Eqns 13 and 14, is equivalent to Eqn 15.
For the specific models of leaf photosynthesis and root N uptake described in Notes S1 and
S2, the bottom line Eqn S3.10 becomes



 aN τ f
1


*
a n  ma kL Ltot  nf
1
 

 N f
e
*


 1 N N

k
I
L
in
o
a





D
 1 U max e max


Ro Do
 1  r  


Do
τr

 nr   1 .

Eqn S3.11
where  = CUE  mc. Eqn S3.11 leads to the following equation for Dmax as a function of
Ltot:
Dmax




U τ
 Do log e  max r
 Ro Do













1  r 
  log e 
 nr  .
 aN τ f
1




*


a N nf  ma kL Ltot nf
1

e


*
 k L I in
 1 No Na

Eqn S3.12
A second relationship between Dmax and Ltot is obtained when Eqns S1.16, S1.17, S2.5 and
S2.6 are substituted into the C-N conservation constraint (Eqn 10). This second equation
relates Dmax, Ltot, nf and nr. Dmax is then eliminated from these two equations to obtain an
equation for Ltot as a function of nf and nr, which can be solved numerically. Given this
solution for Ltot, we then determine Dmax (Eqn S3.12), Ntot (Eqn S1.16), Atot (Eqn S1.17), Rtot
(Eqn S2.5), Utot (Eqn S2.6), Cf (Eqn 2), Cr (Eqn 2) and Cw (Eqn 1).
Step 2: Maximisation of Cw with respect to leaf N:C ratio
The solution for maximal Cw in step 1 depends on the value of nf, which is assumed to be
constant through the canopy. Cw has a maximum with respect to nf, which can be determined
numerically. Optimal solutions for nf and other plant traits and stand variables are listed in
Table S1.