ED. 19 allometry and scaling laws

How did different sized organisms evolve

• rules of geometric similarity, and what it means for living organisms

• Greek philosophers (300-200 BCE), Euclid Archimedes codified the geometric ‘laws’ of increasing size - surface area increases by the square of linear dimension, volume increases by the cube of linear dimension

• Galileo discussed growth in size - maximum size limits vs changes in relative proportions and strength of materials

Allometry

• the study of the relationship between size and shape, how biological processes scale with body size and with each other and the impact of this relationship on ecology and evolution

• allometry is best understood as a result of differential growth (explained by developmental biology)

Can be studied at three levels:

• static - size relationship of traits among individuals of the same age - typically adults

• evolutionary (infraspecific) - size relationship among species

• ontogenetic (intraspecific) - growth relationships during development between two traits or between one trait and the whole organism

Allometric Equation

Y = aX^b (power function)

• Y = body part being measured in relationship to the size of the organism 

• X = measure of size used for basis of comparison (usually a measure of body size e.g. mass or length) 

• a = initial growth index (size of Y when X = 1) 

• b = allometric or scaling exponent (proportional change in Y per unit X) 

Scaling exponent → if b=1

• defines the type of scaling relationship

• if b=1, there is no differential growth:

• the relative size of Y to X is the same at all values of X 

• isometry (maintenance of geometric similarity)

Scaling exponent → if b<1

• if b<1, Y increases at a slower rate than X:

• as X increases, Y becomes relatively smaller 

• negative allometry 

Scaling exponent → if b>1 

• if b>1, Y increases at a faster rate than X 

• as X increases, Y becomes relatively larger 

• positive allometry

Allometric data expressed as linear functions of log-transformed data

Y=aX^b

to

log Y = log a + b log X

• the slope of the line (b) indicates the type of scaling relationship

Types of scaling relationships

• if b = 1, isometry (geometric similarity)

• if b<1, negative allometry 

• if b>1, positive allometry

• but the above is only true when we compare like dimensions such as length vs. length, mass to mass

Isometry for different dimensions

e.g. head length to body length

• linear dimension (m¹) vs. linear dimension (m²)

e.g. head length vs. body mass

• linear dimension (m¹) vs. cubic dimension (m_³)

• isometry: m¹/m³, b=1/3 =0.333

e.g. surface area vs. body mass

• square dimension (m²) vs. cubic dimension (m³)

• isometry: m²/m³, b = 2/3=0.667

Scaling of skeleton to body

• we expect b=1 for skeleton mass/body mass isometry

• however, because of the increased weight loading, may expect b>1:

• → skeleton becomes relatively more massive with increased body size

• remember that skeleton (dense bone material) itself disproportionately adds to body weight

• considering the properties of the skeleton, we observe the scaling of 1.33 for isometry for skeletal weight loading

Implications of scaling and importance of body size and mass

• a male African elephant weighs 11,000 kg, whereas the piebald shrew weighs 11g

• these animals differ in body mass by 3 orders of magnitude

Why is size and scaling important

• living organisms vary hugely in body mass and size

• the magnitude of many biological/physiological processes depends on the body size and mass e.g.:

• → it determines the surface/volume ratio of an organism

• → it affects the metabolic rate, respiration, digestion, water balance, thermoregulation etc.

Geometrically similar objects have predictable properties

in geometrically similar objects:

• the ratio of two linear dimensions is equal and independent of the size of the objects

• if L = length (or a linear dimension) then:

• area is proportional to L² - areas increase with the square (L²) of linear dimensions

• mass and volume are proportional to L³ - volumes (and hence masses) increase with the cubic (L³) of linear dimension

SA: vol decreases with an organisms size 

• the exchange surfaces (epithelia) tend to increase their areas of contact by folding, flattening, and branching

Scaling relationships of size/mass and metabolism

• the amount of energy used by animals per unit mass decreases with body size - per gram, a shrew uses a lot more energy than an elephant

How does differential scaling arise

• ontogenetic growth - growth during development

• allometric growth in humans - juveniles are not isometrically scaled models of adults (the brain and heart grow at different rates relative to the body

• growth of the heart is more or less isometric to body size

• in contrast, growth of the brain is initially hypoallometric to body size, before growth stops once the body reaches a certain size, at about age 6

• consequently, head size becomes proportionally smaller as individuals grow to their final body size

Using allometry in biology

• quantify similarities and differences between taxa (populations, individuals etc.) - differences in the intercept of the allometry between species indicate differences in the proportionate size of the wing, irrespective of body size

• points to operation of similar or differing processes - in contrast, differences in the slope of the allometry between species indicate differences in how the relative size of the wing changes with body size within a species

• can use propose and test theory - the slope and the intercept for morphological static allometries captures the relationship between size and form within and between species

How we use allometry in biology

• organismal growth occurs in a way to preserve correct size relations between all traits results - so there must be general mechanisms of development that govern trait value and these mechanisms must be remarkably robust

• one of the great challenges, then, is to discover how these size relations are generated

Allometry matters - the sad story of an elephant and LSD

• a previous study found that a dose of 0.1mg was safe for 2.6kg cats but sufficient for a psychotic effect - Tusko weighed 7722 kg, and hence Jolly and co. decided to scale up the dose by 2970

• body mass is the determinant of the magnitude of most physiological processes such as metabolic rate, but these processes often do not vary in direct proportion with body mass