BIOl 325 - Scaling

Scaling or Allometry

The study of the relationship between body, size, and shape, anatomy, physiology and behaviour

Isometry

• Two variables scale in direct proportion with each other

• 1 unit change in one creates a 1 unit change in the other

Allometry

• non-equal scaling of 2 variables

• 1 unit change in one results in less or more than 1 unit change in another 

Equation of Scaling 

Y = aM^b

b = scaling factor 

a = variable specific coefficient 

M = mass (body mass, equivalent to volume)

Positive Allometry

Y changes rapidly with respect to X

Negative Allometry

Y changes slowly with respect to X

Relationships for value of b:

1. b > 1

2. b = 1

3. 0 < b < 1

4. b = 0

5. b < 0

Answers:

1. Allometric - getting bigger disproportionately 

2. Isometric - changing proportionately 

3. Allometric - getting bigger disproportionately but more slowly

4. Independent of each other

5. Allometric - getting smaller disproportionately 

Log equation of scaling

• Log Y = Log a + b x Log M

• equation of a straight line (y = mx + c); taking log converts all curves to straight lines.

• 10^{log x} = x 

b for the following:

1. Length 

2. Area

3. Volume (Mass)

Values:

1. 0.33 (1/3)

2. 0.67 (2/3)

3. 1 (3/3)

Isotropic Scaling

Scales according to the expected scale relationships (eg: b for Linear dimension  = 0.33)

*different from Isometry, where scaling is in direct proportion (b = 1)

Anisotropic scaling

Does not scale according to expected/predicted scale relationships (b values)

Importance of SA:Vol ratio & conservation

• ratio of animal’s inside to the animal’s outside

• SA, how an animal interacts with its environment; exchange of gases, heat, nutrients, wastes etc. 

• If everything increases, functions relating to SA would diminish. 

• Ratio can be conserved by changing in only one dimension (eg: tapeworm increases in length but not thickness) OR increase surface area (alveoli in lungs; villi, microvilli in gut ) 

Uses of Allometry 

1. See how Y changes with Mass

2. compare the Y trend information of a species with the group 

3. compare b & a of 2 groups 

4. predict the Y value for species for which measurement is not possible (extinct species)

b for skeletal Mass scaling for terrestrial animals 

1.08

b for skeletal mass scaling for aquatic animals

1.02 - whales

1.03 - fishes 

RMR - Resting Metabolic Rate

Rate at which an consumes fuel at rest

b for RMR given by Rubner

0.67 = (2/3)

based on conductance, which is directly proportional to surface area 

b for BMR (Basal) given by Kleiber 

0.75 = (3/4)