Lecture 6-7: Scaling Flashcards
What is scaling (allometry)?
study of the relationship between body size and shape, anatomy, physiology, and behaviour – how morphology and biological processes scale with body size
What is the most important factor in biology?
size
What is influenced by size?
almost everything:
- most physiological functions
- most anatomical features
- most ecological characteristics
- many behavioural traits
What is isometry?
two variables scale in direct proportion with one another (scale with a factor of 1)
What is allometry?
non-equal scaling (scale with a factor < 1 or > 1)What does
What does scaling allow us to do? (3)
- understand how a structure works
- differentiate between differences due to size and differences due to adaptation
- examine how changes in shape might be necessary to maintain functional equivalence
What is positive allometry?
Y changes rapidly relative to X
What is negative allometry?
Y changes slowly relative to X
What is the power law?
Y = aM^b
variable Y changes in proportion with mass to the power b
where a = variable-specific coefficient
where b = scaling factor (power)
What is the log equation of the power law?
log(Y) = log(a) + b log(M)
What is the square-cube law?
- area is proportional to L^2
- volume is proportional to L^3
What is uniform scaling?
increase all linear dimensions of an object by the same factor
How does absolute surface area of a cube change as its volume (or mass) increases?
- larger objects have less surface area per unit volume
- for every increase in an object’s linear dimensions, volume increases with the cube of L (L^3) and area increases with the square of L (L^2)
What does it mean for two objects to be geometrically similar?
same shape, different size
What do the scaling relationships for length, area, and volume (or mass) form the basis for?
for testing whether biological structures deviate from expected geometric principles as they change in size – allows us to remove the effect of size and only compare shapes
When is mass proportional to volume
when the density of the object doesn’t change as it increases in size
What happens to the SA/V ratio as objects are scaled up uniformly?
decreases (assuming mass is proportional to volume)
What is uniform (isotropic) scaling?
all linear dimensions have increased by the same factor
What is non-uniform (anisotropic) scaling?
some linear dimensions have increased by different factors – object is disproportionately increasing
What is the SA/V ratio of an animal?
ratio of an animal’s inside to its outside
What is an animal’s SA?
interface between an animal and its environment, which determines how it exchanges oxygen, carbon dioxide, heat, nutrients, wastes, etc.
What would happen if every aspect of an animal were to increase geometrically with its body mass (which is proportional to volume)?
functions requiring large SA would progressively become limited
How can you avoid decreasing SA/V?
get bigger in only one dimension
What do linear dimensions scale with?
M^0.33
What does surface area scale with?
M^0.67
What do independents (constants) scale with?
M^0
What do geometric exponents form the basis of?
form the basis of any scaling argument where we assume uniform scaling (geometric similarity) of an organism
using empirical measurements, we can then see if animals deviate from these ‘expected’ geometric exponents
What are the uses of allometry (4)?
- primary information – how does the variable (Y) change with body size
- compare Y from a single species to the overall trend seen in a group
- compare two different groups for differences in slope (b) and elevation (a)
- make an estimate of Y for an unmeasurable species
What is the technique of allometric substitution?
how to calculate an exponent for an unknown scaling relationship with M
same process as dimensional analysis – but only considering how a variable (Y) changes in relation to one of the organism’s dimensions, mass (M)
- think of a relationship (equation containing variables)
- substitute allometric equations into it (replace each variable with some value of Mx related to the variable
- solve for the missing exponent (b) or elevation (a)
Exponents (b) substituted into equations are either…
- based on the assumption of uniform geometric scaling (ie. the exponents show in the graph)
- derived from measurement (ie. lines fitted ito collected biological data with ‘b’ determined empirically)
Exponents (b) substituted into equations are either…
- based on the assumption of uniform geometric scaling (ie. the exponents show in the graph)
- derived from measurement (ie. lines fitted ito collected biological data with ‘b’ determined empirically)
What does volume scale with?
M^1
What does length scale with?
M^0.33
How does mass-specific surface area (surface area per unit mass) scale with body mass?
surface area (SA) / mass (M) = surface area per unit mass (m2/kg)
M^0.67 / M^1 = M^-0.33
What are the 2 hypotheses for how skeletons scale with body mass?
hypothesis 1: isometric scaling – mass of skeleton increases uniformly with body mass (b = 1) to maintain constant shape
hypothesis 2: allometric scaling – skeleton scales in some other allometric fashion (b > 1, 0 < b < 1) to maintain functional equivalence
How does a skeleton work? What must it do?
- skeleton is a stiff support system for the animal’s muscles and organs
- must be able to bear the weight of the animal – ie. be able to withstand a certain amount of pressure or stress (force/area)
What is the force on the skeleton proportional to?
mass of the animal
Skeleton Scaling
READ LECTURE NOTES
What is resting metabolic rate (RMR)?
rate at which an animal consumes fuel at rest
Do smaller or bigger animals use less energy per unit mass?
bigger animals
Does RMR scale isometrically or allometrically with mass?
allometrically – but exponent is unknown
different researchers have suggested different exponents (b)
MR Scaling Exponent – Rubner
- measured RMR of different sized dogs
- MR = aM^0.67
- suggests that MR scaled in proportion to dog’s surface area
argued that this was due to heat loss:
- for body temperature to be constant, MR (heat produced) = heat flux out
- heat flux = conductance (Ta -Tb)
- conductance ∝ surface area
MR Scaling Exponent – Kleiber (and Kleiber’s Rule)
- measured MR of 13 animals belonging to 8 species
- decided that MR had to be measured under standardized conditions to be comparable (basal MR – animal had to be adult, resting, non-reproductive, post-absorptive, and at a thermo-neutral temperature)
- BMR = M^0.75
- could not explain the origin of this exponent (it isn’t based on geometric principles)
MR Scaling Exponent – Brody and Hemmingsen
- RMR scales with b = 0.75
- this established Kleiber’s work as a ‘rule’
What did fractal distribution networks do in the interest of scaling exponents?
created a sensation and renewal of interest in scaling exponent
- even tried to apply it to single celled animals and trees
- eventually applied it to entire ecosystems: “Metabolic Theory of Ecology”
What were problems with fractal distribution networks?
- unicells have no fractal circulation, but their MR still scales approximately as M^0.7
- BMR depends mainly on gut metabolism, but maximum MR depends on muscles
- maximum MR scales to about 0.89
- the mathematics may be flawed
Is there a universal SMR exponent?
NO – no dichotomy between 0.67 and 0.75 if there is no ‘universal’ scaling exponent to describe the relationship between SMR and mass
Universal Scaling of MR – Summary
- no single scaling exponent describes the relationship between basal MR and body mass for all life
most likely multiple scaling exponents exist for different organisms - allometric relationship between MR and mass need not have a single underlying factor
- RMR scales with mass with b usually somewhere between 0.67 and 0.75
