Metabolic theory and ecological scaling

Geoffrey West James Brown Brian Enquist

rK2LK

rK

LKrK2LK

rK

LK

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.

an n

K K

V rV r

b

n n

K K

A rA r

1aV c r 2

bA c r

M A

W Vx

n n

K K

M WM W

3/ 4M BKleiber’s rule

2/3M B

In the Euclidean case A V2/3

Metabolic rate is proportional to tissue surface area

Animal or plant body weight is proportional to body volume

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).

34M W

32

vaWBE proposition

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

aEkTk Ae

Ea = Activation energyk = 8.314 Jmol-1K-1 T = absolute temperatureA = Proportionality factor

The Arrhenius equation holds approximatily for most enzymatic processes.

Ea is often not a constant but related to the square root of T

aE c T

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

273 293 313 333 353 373 393 413 433

Temperature [K]

k

The basic equation of metabolic theory

3/ 40

aEkTM M W e

M: metabolic rate

M0: basal metabolic rate

W: body weigth

Ea: Mean activation energy for biochemical reactions

k: Boltzmann factor: 8.314 Jmol-1K-1 = 0.0000862eVK-1

Andrew Allen

Adding the concentration of an assumed limiting resource gives

3/ 40[ ] ( ) ( ) ( )

aEkTM M R W e f R f W f T

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

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

175413/ 40[ ]

TM M R W e

James Gillooly

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

0.760/ E kT

M M We

The temperature corrected metabolism – weight relation

From Brown et al. (2004), Ecology 85: 1771-1789

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

/3 / 4

/3/ 4

[ ]

[ ]

a

a

E kT

E kT

cNR M e

R M e

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

/3 / 40

aE kTN M M e 3/ 40

aEkTM M W e

3/ 4 3/ 40 0

a aE EkT kTNM M W e M W e 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.

totaldNM NM Edt

cell

wNw

N is the total number of cells at time t, M is energy demand. M total is the total energy needed

w is the ontogenetic mass and wcell the mean cell mass.

3/ 4 3/ 4cell 0 celltotal 0

cell

wdw M ww dw MM M w M E w w

w dt dt E E

EkT

totalM e

3/ 4 1/ 43/ 40 cell 0 cell

wd M w M w Ww M w dw M wW wdt E W E W dt E E W

3/ 4dww W Cwdt

E

3/ 4 kTdw Cw edt

E3/ 4 kTdw Cw e

dt

E E E3/ 4 1/ 4kT kT kT

3/ 4

dw Ce dt w dw 4w Ce dt Ce tw

EkT

E1/ 4kT

t 4 4 ew C

Ce

1/ 4E

kT

t w4 eC

Body weight corrected developmental times should exponentially decrease with

increasing environmental or body temperature

Temperature corrected developmental times should scale to body weight to a

power of ¼.

0

1

2

3

0 5 10 15 20

x

y

y=w1/4

0

1

2

3

0 5 10 15 20

x

y

y=exp(a/T)

0

1

2

3

0 5 10 15 20

x

y

y=exp(a/T)

0

1

2

3

0 5 10 15 20

x

y

y=w1/4

0

1

2

3

0 5 10 15 20

x

y

y=exp(a/T)

0

1

2

3

0 5 10 15 20

x

y

y=exp(a/T)

Biological times should scale to body weight to the quarter power

Examples: Generation time, lifespan, age of maturation, average lifetime

of a species

E1/ 4 kTt w e

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

E1/ 4 kT1r w e

t

The slope –E/k is predicted to be about -7500.

For a slope –E/k of about -11000 E should take a value of 0.95eV.

0

20

40

60

80

100

270 280 290 300 310

T/(1+T/273)

t/w1/

4 Marine fish

-E/k = -10000

050

100150200250300

0 10 20 30 40

T/(1+T/273)

t/w1/

4 Aquatic ectotherms

Birds

-E/k = -12000

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.

0

5

10

15

20

25

30

35

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

Slope

Obs

erva

tions

z

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.

Cell size theory (Kozłowski et al. 2003)

cell

z z0 cell

W nW

M M n W

Total body weight is the product of mean cell size Wcell and its number nCell metabolic rate scales allometrically to cell size

Upper boundary: If body size increases solely due to the increase in the number of cells we get

z z 10 cell 0 cellM nM W nM W W

Lower boundary: If body size increases due to the increase in cell size we get

z z z0 cell 0M n M W M W 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.

Brown, J. H., Gillooly J. H., Allen A. P., Savage V. M., West G. B., 2004, Towards a metabolic theory of ecology. Ecology 85: 1771-1789. www.nceas.ucsb.edu/~drewa/pubs/brown_jh_2004_e85_1771.pdf Ecology’s big, hot idea: http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=535575

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