# Scaling

Scaling can be defined as the structural and functional consequences of a change in size and scale among similarly organized animals. To examine what "consequences of a change in size" means, consider what would happen if one scaled up a cockroach simply by expanding it by a factor of 100 in each of its three dimensions. Its mass, which depends on volume, would increase by a factor of 1 million (100 x 100 x 100). The ability of its legs to support that mass, however, depends on the cross-sectional area of the leg, which has only increased by a factor of ten thousand (100 x 100). Similarly, its ability to take in oxygen through its outer surface will also grow only by ten thousand, since this too is a function of surface area. This disparity between the rapid growth in volume and the slower growth in surface area means the super-sized cockroach would be completely unable to support its weight or acquire enough oxygen for its greater body mass.

The consequences of body size on the physiology, ecology, and even
behavior of animals, can be appreciated if one examines in more detail
differences in function between organisms of widely different sizes. For
example, consider that a 4-ton elephant weighs about 1 million times more
than a 4-gram shrew, and further consider that the shrew consumes enough
food daily to equal about 50 percent of its body weight. Imagine then what
the daily food consumption of 1 million shrews would be (2 tons of food),
and realize that the elephant is probably consuming instead only about 100
pounds of food. From this example it is obvious that daily food
requirements do
*
not
*
scale directly with body mass. In fact, most body processes scale to some
proportion of body mass, rarely exactly 1.0.

## Allometric Analysis

How can one determine the relationship of body processes to body mass? The
best technique for uncovering the relationship is to plot one variable
(for example, food requirements or metabolic rate) against body mass for
groups of similar animals (for example, all mammals, or even more
specifically, carnivorous mammals). Such a plot is called an X-Y
regression. Using a statistical technique called least-squares regression
gives an equation
that best fits the data. The equation for scaling of any variable to body
mass is
*
Y = aW
*
^{
b
}
*
,
*
where
*
Y
*
is the variable to be determined,
*
W
*
is the animal body mass (or weight), and
*
a
*
and
*
b
*
are empirically derived constants from the regression. The exponent
*
b
*
is of particular interest, since it gives the scaling relationship one is
looking for in nonlinear relations, such as that of
**
metabolism
**
and body mass. This mathematical technique is called allometric analysis.
Allometric analysis can be used to predict the capacity or requirements of
an unstudied animal, one that might be too rare to collect or too
difficult to maintain in captivity for study.

## Metabolism

Using this technique, several interesting relationships between animal
structure and function have been uncovered. Among the most well studied is
the relationship between animal metabolism and body mass, introduced
above, in which
*
M
*
(metabolism) scales to the 0.75 power of body weight (
*
M = aW
*
^{
0.75
}
). This means that while the total energy needs per day of a large animal
are greater than that of a small animal, the energy requirement
*
per gram
*
of animal (mass-specific metabolism) is much greater for a small animal
than for a large animal. Why should this be the case? For birds and
mammals that maintain a constant body temperature by producing heat, the
increased mass-specific metabolism of smaller animals was once thought to
be a product of their greater heat loss from their proportionately larger
surface area-to-volume ratio. However, the same mathematical relationship
between metabolism and body mass has been found to hold for all animals
studied, and even unicellular organisms as well. Therefore, the
relationship of metabolism to body size seems to represent a general
biological rule, whose basis eludes scientific explanation at this time.

Allometric analysis has shown that different body processes, involving different organs, scale with different exponents of body mass. For example, blood volume, heart weight, and lung volume all scale almost directly with body mass (exponent = 0.99–1.02). Thus, the oxygen delivery system (heart and lungs) is directly proportional to body mass, even though the metabolism, and thus oxygen requirements, of the body scale with body mass to the 0.75 power. If the hearts are proportionately the same size for large and small animals, but mass-specific oxygen requirements are higher for small animals, then this implies that hearts in small animals must pump faster to deliver the greater quantity of oxygenated blood. Similarly, lung ventilation rates of smaller animals must be higher than those of larger animals. Both predictions have been borne out by measurements that support this conclusion from the allometric analysis.

## Locomotion

The energy requirement for locomotion also scales with body size, in much
the same way that metabolism does. But here another factor comes into
play: the type of locomotion. It is obvious that locomotion is much more
energetically expensive than sitting still, but are some types of
locomotion more expensive than others? Let's compare running,
swimming, and flying. In plotting the cost of running versus body mass,
one notes that metabolic cost increases directly as a function of mass.
What about swimming and flying?
Again, cost increases with mass, but the regression lines for these
allometric analyses exhibit different slopes than the one for runners. As
might be expected the cost (per kilometer per gram of animal) is lowest
for swimmers, where the body mass is supported by buoyancy; next highest
for flyers, where body mass is partially supported by air mass; and
highest for runners, who lose energy to friction with the ground. While
water is more
**
viscous
**
to move through than air, swimmers (especially fish) have streamlined
bodies that reduce frictional drag and reduce cost.

Allometric analysis helps explain why animals can only get so large or so small. Limits placed on structural support, amount of gut surface area required to process the required energy per day, and cost of locomotion become limiting factors for large animals. High surface area-to-volume ratios, high metabolic costs of existence, and limits on the speed of diffusion and cell surface area become limiting factors for small animals. Thus, animal structural design has functional implications that determine physiological processes and ultimately the ability to exist under specific ecological constraints.

**
SEE ALSO
**
Circulatory Systems
;
Flight
;
Gas Exchange
;
Physiological Ecology
;
Temperature Regulation

*
Susan Chaplin
*

## Bibliography

Peters, Robert H.
*
The Ecological Implications of Body Size.
*
Cambridge: Cambridge University Press, 1983.

Schmidt-Nielsen, Knut.
*
How Animals Work.
*
Cambridge: Cambridge University Press, 1972.