Isometry

Geometrically similar objects exhibit what is called **isometric
scaling** (iso = same, metric = measure); the relationships
between surface area, volume, and length follow the simple rules
we noted earlier. That is, if we consider a species of snail that
exhibits a constant shape regardless of size, then the volume
enclosed by its shell will be proportional to its (height)** ^{3
}**. To test this, you could collect 100 shells of different
sizes, measure the height and volume of each, and plot log(volume)
vs. log(height). If scaling is isometric in the shells of this
species, then the data should fall along a straight line with
slope

When an animal (or organ or tissue) changes shape in response
to size changes (*i.e.,* does not maintain geometric similarity),
we say that it scales **allometrically** (allo = different,
metric = measure). Allometric scaling is common in nature, both
when comparing two animals of different sizes and when comparing
the same animal at two different sizes (*i.e.,* growth).
The reasons, and there are many, underlying this rule will keep
us occupied for much of this course.

**Example: How do lungs scale with body size in mammals?**
The primary function of lungs, gas exchange, is accomplished in
the alveoli, thin sacks filled with air and lined with blood capillaries.
Oxygen diffuses from air into the blood across the lining of the
alveoli as carbon dioxide diffuses in the opposite direction.
How would you expect the surface area of the alveoli to scale
with body mass in a series of mammals from mice to elephants?

Hypothesis 1: Lungs scale isometrically.If large mammals were simply scaled-up versions of small ones (i.e.,geometrically similar), then what relationship would we predict between alveolar surface and body mass?

If you set

y=alveolar surfaceandx=body mass, the power function isalveolar surface(mass)^{b }or alveolar surface=a(mass)^{b }The question now becomes what scaling coefficient (

b) corresponds to isometry?To figure this out, convert each variable into a length raised to some power. Alveolar surface is an area, and for isometric scaling,

areaL2^{}.Body mass is proportional to volume (as most animals have similar densities), and

volumeL3^{}The predicted scaling coefficient for isometry can be calculated as the exponent on the y-axis divided by the exponent on the x-axis, so in this case

b= 2/3 or 0.67.

Hypothesis 2: Lungs scale allometrically.Any scaling coefficient other than 2/3 would indicate an allometric relationship between lung size and body mass, but you might consider what coefficient is appropriate to the function of the lungs. If the tissues of large animals have the same need for gas exchange surface as those of small animals, then alveolar surface area should be proportional to body mass, oralveolar surface(mass)1^{}which would mean a scaling exponent of 1.

Testing your hypotheses:If you actually measure body mass and alveolar surface area for a variety of mammals and perform a linear regression of log(alveolar surface) against log(mass), the slope is about 1.02-1.06. Thus scaling of mammalian alveoli is allometric, and the specific hypothesis that alveolar surface is proportional to body mass is supported. Elephants have more alveolar surface than would be the case if their lungs were scaled up isometrically from those of mice. If the slope had turned out to be 0.4, then we would still have rejected the hypothesis of isometry and accepted the hypothesis of allometry, but not the specific form of allometry we had predicted.

**Structural Support: **Structures break when the load (force)
per unit cross sectional area exceeds the strength of the material
from which the structure is built. A tree trunk, for instance,
must be strong enough to support the tree's branches and leaves
against the force of gravity and to withstand the forces exerted
by winds, but must not be so large that it crushes under its own
weight. Similarly, the limbs of terrestrial animals, whether they
be elephants or insects, must support the weight of the animal's
body without being so large that moving the legs during locomotion
becomes too costly. Within most groups of organisms, density is
relatively constant, so the load on the support structure is proportional
to the volume of the organism. The strength of the support material
is generally proportional to its cross-sectional area (e.g. the
area of wood exposed when you cut through a tree trunk). Think
about the implications of this for support of large organisms.
Will isometrically scaling organisms be relatively robust or fragile
as they get bigger?

**Practice Questions **(click answer to test your response)

For each example, pick the scaling exponent, **b**, that
would signify an isometric relationship . Assume that the log
of the first variable would be plotted on the y-axis and the log
of the second variable on the x-axis.

1. Trunk diameter vs. height in oak trees.

b= 0.33b= 0.5b= 1.0b= 2.0

2. Trunk cross-sectional area vs. total above-gound mass in oak trees.

b= 0.33b= 0.67b= 1.0b= 1.5

3. Total leaf area vs. height in oak trees.

b= 0.5b= 1.0b= 2.0b= 3.0

Question 2 on the worksheet
asks that you calculate the dimensions of a sugar cube that would
crush under its own weight, given it's density, its crushing strength
(which is equivalent to about 200 lb. for an average sugar cube
- if you don't believe this, try standing on one!), and the acceleration
due to gravity. **Hint: **What you need to figure out is the
length of one side of a cube whose weight (mass times acceleration
due to gravity) divided by the area of the bottom face is equal
to the crushing strength. The answer will surprise you. The point
is that there are limits to size for anything - sugar cubes, mountains,
or organisms.

For question 3 on the worksheet, use the discussion of scaling of mammalian alveoli above as an example of how to tackle the four cases in the question. It's helpful to think of the graphs you would make and the power function appropriate for that case.

The scaling principles we've presented can be used with certain other properties of organisms besides the three spatial dimensions, for instance heartbeat frequency or metabolic rate. After completing question 5 on the worksheet, examine the following statistics and contemplate the significance of scaling of metabolic rate with body size for thermoregulation for large animals:

heat output of a resting 409 kg. camel: 240 watts heat output of a resting 20g mouse: 0.2 watts heat output of 409 kg of resting mice: 3840 watts

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