Key Concepts: Scaling

Anyone who has dealt with puppies, kittens, or human babies has probably noticed their freakish (“cute”) body proportions relative to adults. The heads are too big and the limbs too small! Fortunately, most of us grow out of this condition as we mature. The change (or lack of) in the proportions of an organism as its overall size changes is called scaling.

Scaling relationships can be compared between species, within species, or even within the same individual over the course of its development. In the present round of the ODP, we’ll be focusing on the first comparison, although many of the examples that follow will draw on within-species or within-individual scaling.

Scaling, as a whole, can be divided into two general forms: isometry and allometry. The key difference between these terms is how the different parts of an organism change in proportion to each other with increasing or decreasing overall size. Isometry (a.k.a., isometric scaling) means that everything stays the same. Here, imagine a condition in which a baby maintains its proportionately large head size into adulthood. Contrast this with the actual condition, allometry. In humans (and most other vertebrates), the head becomes proportionately smaller to the rest of the body. When different parts of the body scale at different rates, this is called an allometric scaling relationship. You can see this in the puppy and adult golden retrievers at right.

Allometry is divided into two classes – negative and positive. Our oft-cited baby’s head is a classic example of negative allometry—the head becomes relatively smaller with increasing body size. Other features might show positive allometry—for instance, the horns of male bighorn sheep grow at a much faster rate than the rest of the body. In a lamb, the horns are tiny compared to the skull. In the adult, the horns are huge!

Scaling comparisons are pretty straight-forward for linear measurements (e.g., comparing scaling of femur length relative to humerus length). Isometry is indicated by a one-for-one change in size between the two elements. But, the complications of mathematics mean that things are a little less intuitive when considering areas or volumes. Consider a cube measuring 1 mm along each side. Its volume (length x width x height) is 1 cubic mm. If we double the size to 2 mm along each side, its volume doesn’t just double, but leaps to 8 cubic mm (2 mm x 2 mm x 2 mm). Bumping the size up to 3 mm along each side, the volume is now 27 cubic mm (3 mm x 3 mm x 3 mm). Length along any given dimension has changed by a factor of 3, but volume has changed by a factor of 27 (3 cubed). Similarly, the cross-sectional area has changed by a factor of 8 (2 cubed). We won’t be doing much with areas or volumes for the present study, but this is an important attribute of scaling to keep in mind.

So, one important analysis to be considered in ODP 1.0 is how limb bone lengths scale against each other. For instance, does the humerus increase its length at the same rate as the ulna, or metacarpals, or phalanges? Or, does the humerus get relatively shorter in larger animals? The departure from simple isometry might be for any number of reasons – perhaps larger animals need relatively more surface area for muscle attachment. Or perhaps weight limitations mean that certain bones can’t increase their size too quickly. We’ll just have to analyze the data and see what the trends are!

Image credits: Puppy image licensed under Creative Commons License 2.0; adult golden retriever in the public domain.

Key Concepts: Scaling
Anyone who has dealt with puppies, kittens, or human babies has probably noticed their freakish (“cute”) body proportions relative to adults. The heads are too big and the limbs too small! Fortunately, most of us grow out of this condition as we mature. The change (or lack of) in the proportions of an organism as its overall size changes is called scaling.

Scaling relationships can be compared between species, within species, or even within the same individual over the course of its development. In the present round of the ODP, we'll be focusing on the first comparison, although many of the examples that follow will draw on within-species or within-individual scaling.

Scaling, as a whole, can be divided into two general forms: isometry and allometry. The key difference between these terms is how the different parts of an organism change in proportion to each other with increasing or decreasing overall size. Isometry (a.k.a., isometric scaling) means that everything stays the same. Here, imagine a condition in which a baby maintains its proportionately large head size into adulthood. Contrast this with the actual condition, allometry. In humans (and most other vertebrates), the head becomes proportionately smaller to the rest of the body. This is called an allometric scaling relationship.

Allometry is divided into two classes – negative and positive. Our oft-cited baby's head is a classic example of negative allometry—the head becomes relatively smaller with increasing body size. Other features might show positive allometry—for instance, the horns of male bighorn sheep grow at a much faster rate than the rest of the body. In a lamb, the horns are tiny compared to the skull. In the adult, the horns are huge!

Scaling comparisons are pretty straight-forward for linear measurements (e.g., comparing scaling of femur length relative to humerus length). Isometry is indicated by a one-for-one change in size between the two elements. But, the complications of mathematics mean that things are a little less intuitive when considering areas or volumes. Consider a cube measuring 1 mm along each side. Its volume (length x width x height) is 1 cubic mm. If we double the size to 2 mm along each side, its volume doesn't just double, but is cubed (2 mm x 2 mm x 2 mm) to 8 cubic mm. Bumping the size up to 3 mm along each side, the volume is now 27 cubic mm (3 mm x 3 mm x 3 mm). Length along any given dimension has changed by a factor of 3, but volume has changed by a factor of 27 (3 cubed). Similarly, the cross-sectional area has changed by a factor of 8 (2 cubed). We won't be doing much with areas or volumes for the present study, but this is an important attribute of scaling to keep in mind.

So, one important analysis to be considered in ODP 1.0 is how limb bone lengths scale against each other. For instance, does the humerus increase its length at the same rate as the ulna, or metacarpals, or phalanges? Or, does the humerus get relatively shorter in larger animals? The departure from simple isometry might be for any number of reasons – perhaps larger animals need relatively more surface area for muscle attachment. Or perhaps weight limitations mean that certain bones can't increase their size too quickly. We'll just have to analyze the data and see what the trends are!