### Metabolic theory

```Metabolic theory and ecological scaling
Geoffrey West
James Brown
Brian Enquist
The grand idea of scaling theory is that animal and plant branching systems and
tissues are self similar. They are assumed to have a fractal structure.
Vn
VK
V  c1 r
a
 rn 


 rK 
a
A  c2 r
b
 rn 


 rK 
An
AK
b
rK
r K 2 L K
Metabolic rate is proportional to tissue
surface area
LK
M  A
Animal or plant body weight is proportional
to body volume
W V
In the Euclidean case A  V2/3
M B
2/3
M B
3/ 4
Kleiber’s rule
Mn
MK
 Wn 


 WK 
x
A textbook example for Kleiber’s rule
Hemmingson classic plot of metabolic rate against body size.
Each regression line has a slope of 3/4
How to derive Kleiber’s rule?
The second grand idea of Brown, West and Eberhard is that life during
evolution optimized the relation between tissue surface and volumen to get
maximum energetic efficiency at a given body volume (size).
2
av
3 
WBE proposition
3
M W
4
The optimization argument is still heavility disputed.
Recent analysis rather point to rather complex relationships between
tissue surface and volume.
The ¾ power function is then only a rough approximation.
Metabolism and temperature
0.14
kT
0.1
0.12
k
k  Ae

Ea
0.08
0.06
Ea = Activation energy
k = 8.314 Jmol-1K-1
T = absolute temperature
A = Proportionality factor
0.04
0.02
0
273
293 313
333
353 373
393 413
Temperature [K]
The Arrhenius equation holds approximatily for most enzymatic processes.
Ea is often not a constant but related to the square root of T
Ea  c T
433
The basic equation of metabolic theory
M  M 0W
3/4

e
Ea
kT
Andrew Allen
James Gillooly
M: metabolic rate
Ea: Mean activation energy for biochemical reactions
M0: basal metabolic rate
k: Boltzmann factor: 8.314 Jmol-1K-1 = 0.0000862eVK-1
W: body weigth
Adding the concentration of an assumed limiting resource gives
M  M 0 [ R ]W
3/4

e
Ea
kT
The activation energy can be estimated from
plots of M against W (at constant
temperature) or from plots of M against 1/T
for species of similar body size
M  M 0 [ R ]W
 f ( R ) f (W ) f (T )
Ea takes values from 0.6 to 0.7 eV with a
mean of 0.65 eV = 62693Jmol-1
-Ea/k = 0.65/0.0000862 = 7541K
3/4
7541
e
1
T
The basic assumption of metabolic theory
•
Living organsims are composed of fractal networks
•
Evolution operated as to optimize energetic efficiency
•
Limiting constraints operate in a multiplicative manner
•
Parameter values are whole organism means are fairly
constant
•
Body weight is independent of temperature
•
Parameters are not functions of the amounts of resources
available.
•
The theory cannot explain why animals of the same size can
have strikingly different metabolic rates and lifespan.
•
Kleiber’s law holds even for organisms without fractal networks.
Empirical support
The temperature corrected metabolism
– weight relation
M
e
From Brown et al. (2004), Ecology 85: 1771-1789
 E / kT
 M 0W
0 .7 6
It is now easy to derive other relationships that involve basic ecological variables:
Abundance: Take the the expression MN as the total metabolism of of a
population of size N. At equilibrium (dN/dt = 0) the product NM must be
constant (NM = c) that is total resource use remains stable. Hence
c
N 
[ R ]M
 [ R ]M
3/4
e
3 / 4
 E a / kT
e
E a / kT
Hence at stable temperature
metabolic theory predicts that
abundance scales to body weight to
a power of -3/4.
From Brown et al. (2004), Ecology 851771-1789
The energy equivalence rule
N  M 0M
3 / 4
e
M  M 0W
E a / kT
N M  M 0W
3 / 4
Ea
e kT M 0W
3/4

e
3/4

e
Ea
kT
Ea
kT
 const
The product NM is the total amount of energy use of the whole
population of size N.
In poikilothermes and plants NM is a proxi to population biomass.
The equal biomass hypothesis
Ontogenetic growth
Assume multicellular animals (or plants) where energy is transported through a network of
branches. The total energy can be expressed as the sum of the energy need to maintain
the existing cells plus the energy needed to create new cells.
M total
 dN 
 NM  E 

 dt 
N 
w
w cell
N is the total number of cells at time t, M is energy demand. Mtotal is the total energy
needed
w is the ontogenetic mass and wcell the mean cell mass.
M total  M 0 w
d
3/4

w
w cell
w
W  M 0 w cell  w 


dt
E
W 
w  W 
dw
dt
 Cw
w

d
 w
cell
M E
 dt


3/4

M w
E W
3/4



M 0 w cell 3 / 4 M
dw


w

w
dt
E
E



dw

dt
M total  e
M 0 w cell W
1/ 4
w
E

E
3/4

M w
E W
dw
kT
dt
 Cw
3/4

e
E
kT
dw
 Cw
3/4

E
kT
e
dt
dw
w
3/4
t
w
 Ce
1/ 4

E

kT
4
Ce

E

dt 
kT
4
w
3 / 4
dw  4 w
1/ 4
E

 Ce
kT
E
e
 dt  C e
t
kT
C
4
E
E
kT
t
1/ 4
e kT
C
33
33
 w

1 /4
1 /4
y=
y=eexxpp(a(a/T/T) )
y=
y=ww
22
yy
yy
22
11
11
00
00
00
55
1100
1155
2200
xx
Body weight corrected developmental
times should exponentially decrease with
increasing environmental or body
temperature
00
55
1100
1155
2200
xx
Temperature corrected developmental
times should scale to body weight to a
power of ¼.
The slope –E/k is predicted to be about -7500.
100
300
-E /k = -10000
1/4
60
-E /k = -12000
250
M a rine fish
t/w
t/w
1/4
80
40
200
A q ua tic e cto the rm s
150
100
20
50
0
0
270
280
290
300
310
B ird s
0
10
20
30
40
T /(1 + T /2 7 3 )
T /(1 + T /2 7 3 )
For a slope –E/k of about -11000 E should take a value of 0.95eV.
Biological times should scale to body
weight to the quarter power
Examples: Generation time, lifespan,
of a species
E
t w
1/ 4
e
kT
The inverse of time are rates.
Examples: Growth rates, mutation rates,
species turnover rates, migration rates
Hence biological rates should scale to
body weight and temperature by
r
1
t
 w
1 / 4

e
E
kT
Criticisms
Living organims are not or at least not in total fractals.
The derivation of the ¾ scaling rule is mathematically flawed.
The ¾ scaling rule has no empirical justification.
The Arrhenius temperature term is much too simplified and the
parameters are poorly defined.
More important than universal scaling is the variability in metabolism.
The theory predicts average scaling rates but does not account for the
oberserved variance in scaling relationships.
The variablility of living organisms prohibits any useful predictions from
the theory. In other words, even if the thory is correct it does not tell
much about the functioning of real world ecological systems.
z
30
O b s e rv a tio n s
35
25
20
15
10
5
0
0
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
1
1 .1
S lo p e
Slopes of metabolic rate – body weight relationships of different taxonomic
groups scatter around 0.75. There is no single slope for all taxa.
Data from Peters 1983.
1 .2
Cell size theory (Kozłowski et al. 2003)
Total body weight is the product of mean cell size Wcell
and its number n
Cell metabolic rate scales allometrically to cell size
W  nW cell
M  M 0 n W cell
z
z
Upper boundary: If body size increases solely due to the increase in the
number of cells we get
z 1
M  nM 0 W cell  nM 0 W cell W
z
Lower boundary: If body size increases due to the increase in cell size we get
M  n M 0 W cell  M 0 W
z
z
z
z<1
The theory predicts
•
Metabolic rate should scale allomtrically to body weight
•
The scaling exponent should be less than 1
•
If cell metabolic rate is proportional to cell surface then z = 2/3
•
2/3 < z < 1
Major criticism
•
The theory uses species as as major driver in evolution.
•
Cells have a fractal nature. Hence the lower slope boundary
should be 0.75.
•
The theory applies only to taxa with large differences in cell size
like mammals.
•
The theory does not explain scaling intercepts.
•
The theory does not explain scaling in plants and invertebrates.