albatross

D. exulans -- up to 12 kG and 3.6 m wingspan

Hummingbird Bee hummingbird Mellisuga helena

0.6 g and 0.06 m wingspan

Source: http://www.birdlife.org/images/sized/450/b_bee_hummingbird.jpg.jpg

How do we understand the effects of differences in size on the structures and processes of organisms?

In the previous two slides, body mass differed by:

12,000/0.6 = 2000X

This is over 3 orders of magnitude!

Scaling

• A technique for determining how two phenotypic variables change with respect to each other.

• Most useful when the variables are considered over wide ranges of values.

• The variables can be morphological or process.

• Therefore, useful in answering large scale questions about biological design.

Isometric and Allometric Scaling

When two variables have a linear relationship with respect to each other, we say that they scale isometrically. With respect to size this means that they are scale models of each other.

However, if the relationship is non-linear it means that the two organisms are not exact scale models of each other with respect to the variables of interest. We call this allometric scaling.

Graphically

Equations

Isometric: -- if similarity is maintained, a = 1

Y = mX1.0 + b

Allometric: -- here a has any value not close to 1.0

Y = mXa+ b

Generalized power function:

Y = mXa+ b

Double log Plots of Scaling Relationships

log(Y) = slope log(X) + log (y-intercept)

log(Y) = b * log(X) + log(a)

Metabolism and Body Size

Isometric:

log(Q) = 1.0*log(M) + log(a)

Allometric -- many possible versions differentiated from each other by their value of the coefficient of log(M).

Let’s use one that presupposes that metabolism is constrained by surface area. Why SA?

Derivation of a SA Model Where SA is Expressed in Terms of Mass (volume)0.67

Volume L3 or V = k1*L3

L V0.334 or L= k1* V0.334

Volume1.0 Mass1.0 or V1.0 = k2* M1.0

L M0.334 or L = k3 * M0.334

SA L2 or SA = k3*L2

SA (M0.334)2 or SA = k4 * M0.67

Double Log Plots of the Models

Hemmingsen

Explanations

Compromise between need to maintain a per unit tissue metabolic rate and the surface constraint.

What happens to the metabolism of a unit mass of tissue with change in size?

Given the Allometry, How Is it Best To Compare Animals to See if they Conform to the Allometry?

See how close they come to overall regression

“% expected”

Divide metabolic rate by mass raised to the expected scaling coefficient for the group -- thus, for most groups, divide by M0.75 or the “known” exponent for that group (known to the extent that different animals have been measured)