Body Size, Metabolic Rate and Allometric Scaling
No isometric relationship – doubling body mass rate does not double metabolic rate
(ie. = an allometric relationship)
JBS Haldane – “higher animals are not larger than lower because they are more
complicated. They are more complicated because they are larger” – need new
adaptations that enable the metabolic rate of all cells to be maintained (account for
problems associated with surface area)
(Rubner, 1883) – Surface area to volume ratio
- Measured a metabolic rate of 7 “surface hypothesis”: metabolic rate of birds
and mammals maintaining a steady body temperature is roughly proportional to
body surface area
- Rate at which dogs lose heat associated with surface area ( for endotherms)
- Gradient of line refers to s/a and not volume
Max Kleiber’s “mouse to elephant” curve – metabolic rate does not conform to
either s/a (exponent 0.67) or mass (exponent 1) but to scaling exponent 0.75
(3/4 scaling rule – reasons unknown)
Is there a single allometric exponent?
Dodds et al., reanalysed dataset available for 391 species of mammals and 398
species of birds – insufficient basis for rejecting slope of 2/3 in favour of ¾
Though White and Seymour (2003) used data from 619 species of mammals
correcting for temperature differences excluding taxa not likely to have yielded
true BMR values found that mammalian BMR scaling exponent is 2/3 and ¾
There is far from a universal acceptance of a universal scaling exponent – why is it
worthy of debate
Concept of allometric scaling
Metabolic rate increases with body size but
does not increase as much as would be
expected on basis of either mass or surface
area alone
Metabolic rate per kg (or per cell) falls with
body mass – each individual cell consumes less
oxygen to sustain life in a human compared to a rat
Rat vs. human liver – resting metabolic rate of rat liver cells is 7x greater than that
of humans even though tissue/cell density appears to be the same
Hemmingsen’s 1960 plot – Energy metabolism as related to body size and
respiratory surfaces and its evolution. Extended the ¾ scaling rule to single cells
W=aMb , parallel lines reflect phylum-specific values of a
, - Universal scaling law with an exponent 0.75
- Correct for temperature – all metabolic rates
appear to line up on a single line (methodology has
been criticised)
Metabolic theory of ecology
Fact that the relationship is allometric – has profound effect in terms of
fertility(amount of offspring), evolution
- “Elephant sized pile of mice would have metabolic rate 21x the elephant – ie.
mice consume 21x as much oxygen and food per min cf elephant”
- Profound effects on growth rates, behaviour (foraging etc.), fertility, number of
offspring, lifespan, population density, mobility, distribution, speed of evolution
- Small animals tend to grow fast, breed early, die young they are “r-selected”
- If metabolic rate scaled with an exponenet of 1 – would be impossible to sustain
viable populations of animals much larger than sheep
Speed of evolution – molecular clocks to aid estimates of divergence time – if rate
of sequence change is based on metabolic rates , molecular clock data could
potentially be inaccurate – have to account for metabolic rate (assumes metabolic
clock does not tick at a steady rate – depends on rate of organism)
- Effect of body mass on silent rates on nucleotide substitution
- Correcting for body size gives estimates of divergence dates that agree more
closely with the fossil records
- Model suggests that there is a single molecular clock originally proposed by
Zuckerkandl and Pauling but it ticks at a constant substitution rate per unit of
mass-specific energy rather than per unit of time
Model therefore links energy flux and genetic change
More generally, model suggests that body size and temperature combine to
control the overall rate of evolution through effects on metabolism (Gillooly et
al., 2005)
WEB (West, Enquist, Brown) postulates…
Or down to fractal geometry of supply networks (capillary diameter of a mouse is
the same as an elephant, but the branching that leads onto that capillary diameter
is a fractal network)
- Predict the ¾ power scaling on basis of fractal geometry from “first principles”
- Based on that arrived at the exponent of 0.75 (but mathematics is inaccurate)
(West and Brown, 2002) Use framework of a general model of a fractal-like
distribution networks together with data on energy transformation in mammals to
, analyse and predict allometric scaling of aerobic metabolism over 27 orders of
magnitude in mass encompassing 4 levels of organisation: individual organisms, single
cells, intact mitochondria, enzyme molecules
Observation suggests that aerobic energy transformation at all levels of biological
organisation is limited by transport of materials through hierarchical fractal-like
networks with properties specified by the model
Data is inaccurate
Hemmingsen’s data, when eliminating non-single cell data, revealed an exponent of
0.61 (Prothero, 1987)
Scaling exponent varies – giant bacteria eg. Thiomargerita excluded from data
(exponent of ~1) So there is no universal scaling law
- If it did depend on fractal scaling – single-celled cells do not possess such
networks – easier to explain if the exponents did differ
Scaling exponent varies – large-scale analysis shows different scaling exponenets
for prokaryotes, protists, metazoans (DeLong, 2010)
In mammals and birds – scaling exponent of 0.67
- Insectivores have different metabolic rates (does not seem to be congruent
with other vertebrates)
- Exponent is never statistically equal to 0.75 corrected to 37 oC
Fractal Geometry Implications/Predictions
There is a lot of uncertainty about the exponent itself, but this is still controversial
and varies a lot between studies. Giving WEB the benefit of the doubt… what does
fractal scaling of supply networks predict?
1. Fractal scaling is a CONSTRAINT. Supply networks restrict metabolic rate of
tissues, therefore scaling cannot be above 0.75 even in specific organs.
2. Basal metabolic rate is ‘idling rate’, and unlikely to be constrained by network supply;
maximal metabolic rate should be. If MMR is constrained by fractal geometry it should
also scale with predicted fractal exponent of 0.75.
Maximal metabolic rate – (0.88) – closer to an isometric relationship
Rate of nutrient consumption is dependent on the network, then the maximal metabolic
rate should be constrained and not the minimal metabolic rate
3. Supply and demand. Angiogenesis should be determined by the properties of the
branching network and not the tissues they serve
Criticisms of Fractal Scaling
No isometric relationship – doubling body mass rate does not double metabolic rate
(ie. = an allometric relationship)
JBS Haldane – “higher animals are not larger than lower because they are more
complicated. They are more complicated because they are larger” – need new
adaptations that enable the metabolic rate of all cells to be maintained (account for
problems associated with surface area)
(Rubner, 1883) – Surface area to volume ratio
- Measured a metabolic rate of 7 “surface hypothesis”: metabolic rate of birds
and mammals maintaining a steady body temperature is roughly proportional to
body surface area
- Rate at which dogs lose heat associated with surface area ( for endotherms)
- Gradient of line refers to s/a and not volume
Max Kleiber’s “mouse to elephant” curve – metabolic rate does not conform to
either s/a (exponent 0.67) or mass (exponent 1) but to scaling exponent 0.75
(3/4 scaling rule – reasons unknown)
Is there a single allometric exponent?
Dodds et al., reanalysed dataset available for 391 species of mammals and 398
species of birds – insufficient basis for rejecting slope of 2/3 in favour of ¾
Though White and Seymour (2003) used data from 619 species of mammals
correcting for temperature differences excluding taxa not likely to have yielded
true BMR values found that mammalian BMR scaling exponent is 2/3 and ¾
There is far from a universal acceptance of a universal scaling exponent – why is it
worthy of debate
Concept of allometric scaling
Metabolic rate increases with body size but
does not increase as much as would be
expected on basis of either mass or surface
area alone
Metabolic rate per kg (or per cell) falls with
body mass – each individual cell consumes less
oxygen to sustain life in a human compared to a rat
Rat vs. human liver – resting metabolic rate of rat liver cells is 7x greater than that
of humans even though tissue/cell density appears to be the same
Hemmingsen’s 1960 plot – Energy metabolism as related to body size and
respiratory surfaces and its evolution. Extended the ¾ scaling rule to single cells
W=aMb , parallel lines reflect phylum-specific values of a
, - Universal scaling law with an exponent 0.75
- Correct for temperature – all metabolic rates
appear to line up on a single line (methodology has
been criticised)
Metabolic theory of ecology
Fact that the relationship is allometric – has profound effect in terms of
fertility(amount of offspring), evolution
- “Elephant sized pile of mice would have metabolic rate 21x the elephant – ie.
mice consume 21x as much oxygen and food per min cf elephant”
- Profound effects on growth rates, behaviour (foraging etc.), fertility, number of
offspring, lifespan, population density, mobility, distribution, speed of evolution
- Small animals tend to grow fast, breed early, die young they are “r-selected”
- If metabolic rate scaled with an exponenet of 1 – would be impossible to sustain
viable populations of animals much larger than sheep
Speed of evolution – molecular clocks to aid estimates of divergence time – if rate
of sequence change is based on metabolic rates , molecular clock data could
potentially be inaccurate – have to account for metabolic rate (assumes metabolic
clock does not tick at a steady rate – depends on rate of organism)
- Effect of body mass on silent rates on nucleotide substitution
- Correcting for body size gives estimates of divergence dates that agree more
closely with the fossil records
- Model suggests that there is a single molecular clock originally proposed by
Zuckerkandl and Pauling but it ticks at a constant substitution rate per unit of
mass-specific energy rather than per unit of time
Model therefore links energy flux and genetic change
More generally, model suggests that body size and temperature combine to
control the overall rate of evolution through effects on metabolism (Gillooly et
al., 2005)
WEB (West, Enquist, Brown) postulates…
Or down to fractal geometry of supply networks (capillary diameter of a mouse is
the same as an elephant, but the branching that leads onto that capillary diameter
is a fractal network)
- Predict the ¾ power scaling on basis of fractal geometry from “first principles”
- Based on that arrived at the exponent of 0.75 (but mathematics is inaccurate)
(West and Brown, 2002) Use framework of a general model of a fractal-like
distribution networks together with data on energy transformation in mammals to
, analyse and predict allometric scaling of aerobic metabolism over 27 orders of
magnitude in mass encompassing 4 levels of organisation: individual organisms, single
cells, intact mitochondria, enzyme molecules
Observation suggests that aerobic energy transformation at all levels of biological
organisation is limited by transport of materials through hierarchical fractal-like
networks with properties specified by the model
Data is inaccurate
Hemmingsen’s data, when eliminating non-single cell data, revealed an exponent of
0.61 (Prothero, 1987)
Scaling exponent varies – giant bacteria eg. Thiomargerita excluded from data
(exponent of ~1) So there is no universal scaling law
- If it did depend on fractal scaling – single-celled cells do not possess such
networks – easier to explain if the exponents did differ
Scaling exponent varies – large-scale analysis shows different scaling exponenets
for prokaryotes, protists, metazoans (DeLong, 2010)
In mammals and birds – scaling exponent of 0.67
- Insectivores have different metabolic rates (does not seem to be congruent
with other vertebrates)
- Exponent is never statistically equal to 0.75 corrected to 37 oC
Fractal Geometry Implications/Predictions
There is a lot of uncertainty about the exponent itself, but this is still controversial
and varies a lot between studies. Giving WEB the benefit of the doubt… what does
fractal scaling of supply networks predict?
1. Fractal scaling is a CONSTRAINT. Supply networks restrict metabolic rate of
tissues, therefore scaling cannot be above 0.75 even in specific organs.
2. Basal metabolic rate is ‘idling rate’, and unlikely to be constrained by network supply;
maximal metabolic rate should be. If MMR is constrained by fractal geometry it should
also scale with predicted fractal exponent of 0.75.
Maximal metabolic rate – (0.88) – closer to an isometric relationship
Rate of nutrient consumption is dependent on the network, then the maximal metabolic
rate should be constrained and not the minimal metabolic rate
3. Supply and demand. Angiogenesis should be determined by the properties of the
branching network and not the tissues they serve
Criticisms of Fractal Scaling
