- An exact isometric relationship between two animals occurs when when all linear body dimensions scale up or down by the same multiplier. When height doubles, arm length doubles, distance between the eyes doubles - All linear dimensions double. But, volume will increase to 8 times the original volume and surface area will increase to 4 times the original value. (Eckert, Fig. 16-6a)
Let's put the isometric relationship of body dimensions with scale in terms of the formula for the volume of a cube or volume of a sphere.

- With allometric change the proportions vary. Note the size of the head and eyes as this zebrafish grows. (Eckert, Fig. 16-6b)
- And, most groups of animals show allometric relationships between adults of different species. This is shown very clearly when considering metabolic rates. Metabolic rate as a function of body mass isn't a simple relationship. (Eckert, Fig. 16-7b)
- There are many interesting examples of scaling. In general, large animals live longer than small animals. Warm blooded animals tend to die after a fixed number of heart beats, somewhere in the neighborhood of 1-2 billion heart beats. Chicken heart rate = 300 beats/minute. Elephant heart rate = 30 beats/minute. Chickens live 5 years, at the most, but elephants live 50 to 70 years. But, note that an elephant's mass = 10,000 times chicken mass. Note that heart rate isn't proportional (or inversely proportional) to mass of the animal. Is the heart rate proportional or inversely proportional to the linear dimension of the animal.
Do you think that the heart rate comparison is an example of isometric or allometric scaling? Let's do the math.

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Question: Suppose that we multiply by 10 each of the linear dimensions of an animal. So, the nose becomes 10 times as long, the arm becomes 10 times as wide, etc. As a result, the larger animal's surface area to volume ratio, S/V, becomes`

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- The basal metabolic rate is the stable rate of energy metabolism measured in mammals and birds under conditions of minimum enviromental and physiological stress (at rest). Why "birds and mammals? Well, all birds and mammals and
*some*other animals maintain their temperature above the ambient enviromental temperature, using metabolic energy, and thus can be grouped together as having similar metabolic mechanisms. - Metabolic rate (R) can be described as a power function of body mass (M):
R = aM ^{b}Mass specific metabolic rate (R/M) is then:

R/M = aM ^{b}/M = aM^{b-1}The two relationships above are what we saw in (Eckert, Fig. 16-7b). We can look at the relationship between mass-specific metabolic rate and animal mass for a range of mammals - (Eckert, Fig. 16-7a)

- It is easier to visualize the relationships between metabolic rate and mass by taking the log of both sides of the equations:
log(R) = log(a) + (b)log(M) and ...

log(R/M) = log(a) + (b-1)log(M) These two relationships are straight lines if we plot log(R) or log(R/M) vs. log(M). (Eckert, Fig. 16-7c) The slopes of these straight lines are b and (b-1), respectively. IMPORTANT - Equation 16-6 on pg. 677 is wrong!

- Measurements on actual animals show that the value of b is very close to 0.75 (and thus the value of (1-b) is close to 0.25.) (Eckert, Fig. 16-8) . Organisms seem to follow a rule, known as Kleiber's law, that b = 0.75, at least within groups of unicellular organisms and ectothermic or endothermic animals.

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Question: Assume that the proportions of different animals scale isometrically with length of the animal. If we plot log of each animal's mass against log of the the animal's length, we obtain a straight line. What is the slope of that straight line?
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Question: If we plot log of the mass specific metabolic rate vs. log of the mass, measurements on actual animals yield a straight line. What is the slope of this line?

- Let's go back to isometric relationships, where we scale up or down an animal, keeping the same increase or decrease in dimension for length, width and depth. Surface (S) is proportional to linear dimension (l) in the following way:
S = c l ^{2}If we are comparing two spherical bodies, l is the radius and surface area = 4 Pi l

^{2}(c = 4 Pi). If we double the radius, the surface area increases by 4 times. If we are comparing two cubes, l is the width and height of each one of the faces of the cube and surface area = 6 l^{2}(c = 6). If we double the linear dimension of each face, the surface area of each face, and thus of the cube, increases by 4 times.Volume (V) is proportional to linear dimension (l) in the following way:

V = d l ^{3}If we are comparing two spherical bodies, volume = (4/3) Pi l

^{3}(d = (4/3) Pi). If we double the radius, the volume increases by 8 times. If we are comparing two cubes, volume = l^{3}(d = 1). If we double the width of one face, the volume increases by 8 times. Mass is also proportional to the cube of linear dimension since if we have more volume and the density of material remains the same, so, M = e l^{3}, where e is just another proportionality constant.Now, let's take the log of the surface and the mass relationships:

log(S) = log(c) + 2 log(l) and ...

log(M) = log(e) + 3 log(l) So, if we plot log of the animal's surface area vs. log of the animal's mass for different animals of different sizes (different l's), as we increase l (the animal becomes longer, higher, and deeper in the same proportion), the log(S) increases by 2 units while at the same time log(M) increases by 3 units. In other words, the slope of the log(S) vs. log(M) relationship is 2/3 or 0.67. This is for an isometric relationship. Only adults of the same species obey this relationship.

When we compare many different species, the log(S) vs. log(M) relationship has a slope of 0.63 instead of 0.67, because of the thickening of bones and other body structures in larger animals (

*allometric*scaling - proportions change as overall size changes). (Eckert, Fig. 16-9) - In 1883, Max Rubner proposed that log of the metabolic rate would vary with log of body mass in just the same way that the log of surface area does, BECAUSE heat is lost through the external surface of the animal. If the animal is to balance the heat loss across its external surface, then it must increase heat production in proportion to surface area. So, Rubner predicted that the slope of a plot of log(R) vs. log(M) would be 0.67. This is true comparing different sizes of adult animals within a single species - (Eckert, Fig. 16-10a). But, it is not true, as we saw before, when comparing animals of different species - (Eckert, Fig. 16-10b) - the slope is really 0.75.
- One problem with Rubner's argument is that we still get 0.75 when comparing animals that don't control their body temperature independently of the environment. (Eckert, Fig. 16-8)
- We will discuss reasons for why the 0.75 slope seems to be a universal constant of nature.

All text and images, not attributed to others, including course examinations and sample questions, are Copyright, 2012, Thomas J. Herbert and may not be used for any commercial purpose without the express written permission of Thomas J. Herbert.