<­­­­­­­­­­­­ QUIZ #2 material ended here (Friday's class) ­­­­­­­­­­­­­­­­­­­>
This week (all week):
Scaling, Allometry, Estimation
(these notes and practice problems available on the web; italics and bold are important
take­home messages for studying.)
I. Size
Size is an important property of living organisms.
Because organisms vary in size, they experience the world in very different ways. Most
organisms are much much SMALLER than we are. For example, the surface tension of
water becomes a significant force for them (water striders); we barely notice it.
Some organisms change in size over their lifetimes. A tuna larva hatches from a 2mm
egg; an adult tuna can be 14 ft long (1000X difference). Since people are about 50 cm
long at birth, this would be equivalent to growing to about half a km tall!
Also, many taxonomic groups have a wide range of size. If we consider mammals,
an adult shrew is about 2g, but an adult blue whale is 100 tonnes. The largest mammal
on land is now extinct but was the Baluchitherium, which weighed 30 million grams. It
looked like a rhinoceros, but if you are 6 ft tall and you reach, you could just reach its
underbelly.
Measuring Size
Depending on what we're interested in, we can measure size in different ways. We might
measure the length (of a tuna, or a wingspan). This would be in “linear units”, eg. cm, m,
inches. Keep in mind that linear units have dimension one.
We might measure area. For instance the surface area of the lungs might be important
physiologically, or likewise the surface area of the cortex of the brain. This would be in
square units, eg. square cm, m2, square miles. These all have dimension two.
We might also measure volume, in cubic units (m3, cubic mm, mL, etc.). Volume has
dimension three.
Finally, we might measure related quantities like the total mass (lbs, grams, kg) of a
shrew or a Baluchitherium. We'll talk later about the dimensions of these quantities.
II. Scaling
Scaling is: the study of the consequences of changing size.
As you grow, some body parts stay at the same ratio. For example, the ratio of the length
of baby's fingernail to length of baby's finger is about the same as for my fingers. But the
ratio of length of baby's arms to size of her head is very different; baby can't reach around
her own head. Imagine having a head so big that you couldn't reach the top of it.
We say fingernail length grows “isometrically” with finger length. Isometric growth is
when the ratio of two measurements stays the same. If lengths grow isometrically, you
get an exact scaled version as the organism grows. Baby's fingers look exactly like little
miniature versions of my fingers. We say arm length grows “allometrically” with head circumference. Allometric growth
is when the ratio of two measurements changes as they grow. If the lengths of parts of an
organism grow allometrically, the body changes overall shape as it grows. When we look
at the whole person, not just fingers, we grow allometrically; we are not shaped like giant
babies.
As an organism grows allometrically, some features will scale with the length of the
organism, while others might scale with surface area or volume.
Some examples of both types of growth:
The length of the forearm probably grows isometrically with the length of the organism.
The length of a dog's paws probably grow allometrically with the height of the dog,
because puppies have big paws compared to full­grown dogs.
The total number of hairs on the back of a fern leaf might grow isometrically with the
total area of the fern leaf.
The rate of oxygen exchange might scale isometrically with the surface area of the lungs.
The length of a leaf on a tree grows allometrically with the age of the tree; hundred year
old trees do not have leaves that are 100 times as long as 1 year old trees!
III. Power Models
How do we look at this mathematically?
A. Isometric Growth
When measurements scale isometrically, we can write:
Y = kX
or, equivalently,
Y X "Y is proportional to X"
where Y is one measurement, X is another, and k is just a constant. For example: my little finger, X, is 7 cm long, and my fingernail, Y, is 1 cm long. What
is k?
1 = k (7)
1/7 = k
And then if I told you that fingernails grow isometrically with fingers, that means
k stays constant for people of all sizes. So if baby's finger is 14mm long, how long is
baby's fingernail? (2mm)
We can use the idea of dimension, and other information, to make intelligent guesses
about which quantities we expect to scale isometrically. •
If the shape stays the same, then length measures (dimension 1) should grow
isometrically with other length measures; dimension 2 measures will grow
isometrically with others of dimension 2; likewise dimension 3
•
We expect that mass grows isometrically with volume
•
We expect that rates of exchange (oxygen in the lungs, CO2 in the gills, etc.) will
scale isometrically with surface area
•
We expect that maximum weight supported scales isometrically with cross­sectional
area (this is because a skeletal bone can only support a certain number of lbs/square
foot; thus if we have more lbs, we need more square feet). This is an important point
for giant spiders (see below).
B. Allometric Growth
When measurements scale allometrically, we can write:
Y = kXp
or, Y Xp
Here k is a constant again, but p is the "power" of the power law.
Note that when p=1 we have the special case of isometric growth.
For example, let X be the radius of a spherical cell, and let Y be the volume of the cell.
What are k and p?
volume = 4/3 π r3
Y = 4/3 π X3
So we see that k = 4/3 π , and p = 3.
We can also make intelligent guesses of what p should be, just by considering units. In
the example above, we were comparing radius (dimension 1) with volume (dimension 3).
The "best guess" for p is the ratio of the dimension of Y over the dimension of X. We call
this a best guess, rather than the truth, because it's the best estimate we can make if we
don't have data. It's what we expect as a default.
eg. If Y is area and X is length, our guess for p would be 2.
What if you don't know the dimension of one quantity? Use the dimension of something
that is isometric to that quantity.
eg. If Y is mass and X is area, we would use dimension 3 for mass (isometric to
volume), and expect that p is 3/2.
Now we can also answer questions like the following:
eg. If the length of a bacterium triples, by how much will its mass increase?
answer: Let M = mass and L = length. We expect
M = k L3
Now what happens if we triple L, that is, what do we get on the LHS if we substitute 3L
for L on the RHS?
new mass = k(3L) 3 = k3 3 L3 = 27 (kL3) = 27 M
The new mass will be 27 times the old mass if we triple the length.
Finally, we can consider giant spiders in horror films, that are scaled isometrically (just
huge versions of real spiders). Suppose a spider is scaled isometrically to be 1000 times
taller than a real spider. By how much does its mass change? By how much does the
surface area of its legs change? From your answers to these questions, why are giant
spiders impossible?