One of the common principles used to characterize the rules of growth in developing systems is the idea of *allometry*. Here, I’ll briefly summarize the concept with a few clear illustrations and a tiny amount of simple math.

“Allometry” means “different growth or measurement”, and it refers to the fact that different body parts grow at different rates during development—we aren’t simply linearly scaled up versions of our younger selves, but instead we have different proportions. One vivid and familiar example is the growth of our heads. Look at a baby, and their heads and eyes seem *huge* relative to the rest of their body; there is going to be a shift in proportions over time, with the body growing faster than the head so that by adulthood they have adult proportions. Here, for instance, are scale diagrams of fetal growth that show you how much change there is just during pregnancy.

There is an overall pattern of growth of everything—we could draw a line along the tops of those heads to get a nice curve illustrating growth rate—but the different parts grow allometrically, at different rates. This next picture is a little freaky: all of the fetuses are scaled to the same size to illustrate the change in proportions.

At nine weeks, about *half* the fetus’s body length is taken up with just its head; if those proportions were retained into adulthood, your chin would be down around your bellybutton and you’d fit right in in a Mardi Gras parade. By 38 weeks, the fetus’s head is reduced to about a quarter of its length, and even that would look weird and rather creepy in an adult.

This pattern of differential growth continues after birth. This lovely graph from D’Arcy Thompson shows the relative sizes of the whole body, the brain, and the heart over time. You keep growing through childhood and adolescence (you can also see the growth spurt in the increasing steepness of the body curve, starting in the teens), and the heart also grows continuously through that period to keep up, but look at the brain—it grows for a while, and then it just stops. You know all that stuff you’re trying to manage now, as an adult? You’re doing it with the same sized brain you had at four years old, and with fewer neurons than you had then.

Here’s another example from one of Gould’s books. We aren’t the only organisms to experience this differential growth in our body parts; the chimpanzee’s brain also stops growing shortly after birth, but the rest doesn’t. In particular, you can see that there is massive growth of the face and jaws that greatly exceeds that of the cranium, turning the dome-skulled, almost human looking baby into the beetle-browed adult with its protruding muzzle.

Those photos give you clue about why allometric growth is of interest in evolution, as well. A set of mutations that slowed the rate of growth of the face and increased the rate of growth in the cranium of our ancient common ancestor would have produced something that looked like us, just by retaining the proportions of the younger form. One way we can smoothly shift from one form to another over evolutionary time is for the regulation of the genes that control the timing and rate of growth of these different body parts to be tweaked in quantitative ways.

One more example of the flexibility of the regulation of allometric growth:

These are baboon skulls. The top one is an infant, the second a juvenile, and the bottom two are adults. The same pattern of allometric growth seen in the chimpanzee is present here—the cranium grows relatively little, but the face and jaws just keep going and going. The other interesting twist is that (c) is a female and (d) is a male; differential regulation of allometric growth can be used to generate sexual dimorphism. It’s also seen in, for instance, the different morphological forms of ant castes.

Differential growth can be modeled with a simple equation. If we’re comparing two different structures, for instance brain mass vs. total body mass, or brain mass vs. jaw mass, the two are related by the allometric equation, where the size of one feature is equal to an empirically determined constant times the size of the second feature to the power of another constant, or, if our two measurements are *x* and *y*,

*y* = **a** • *x*^{b}

The growth of one is at an exponential rate **b** relative to the other. A log transformation of this equation produces

Log(*y*) = Log(**a**) + **b** • Log(*x*)

This is an easy equation to work with. All you have to do is measure the items you want to compare, plot them on log-log paper, and then fit a straight line to them. The slope of the line gives you the relative rate of growth. If the slope **b** is less than one, it’s hypometric growth; that is, one organ (the brain, for instance) is growing at a slower rate than the other (whole body size, for instance). If the slope **b** is greater than one, then the organ is growing hypermetrically, or faster relative to the other—male baboon jaws probably grow disproportionately faster than the rest of the body. If **b** is exactly 1, then growth is *isometric*—that is, the two organs are always in the same relative proportion throughout development.

The power of this approach and the root of its appeal is that it reduces details of morphology to two parameters, **a** and **b**, that are modified by evolutionary processes. The question then becomes one of figuring out what a and **b** actually are (messy biology always sticks its nose into our pretty mathematics.) The parameter **a**, the intercept of the line, could be something like the time in development at which growth of the organ begins, or the fraction of cells in the embryo that are initially allocated to the tissue; this leads us to look for timing regulators or patterning elements in the genome. Rates of growth, the **b** parameter, might be controlled by levels of signals, for instance hormone titers (insulin-like growth factors are candidates here), or by clocklike internal regulators of the mitotic machinery, such as the ASPM gene.

The utility of allometry, the scaling relationship between two characters in development, is that detecting it is a clue that there is some independent regulation of the two characters; it is an indication that there is an interesting growth relationship between the two, and that there may be a simple underlying differential regulator of their growth. In phylogenetic comparisons, it’s a way to begin looking for the molecular mechanisms that underlie morphological differences.

Gould SJ (1977) *Ontogeny and Phylogeny*^{(amzn/b&n/abe/pwll)}. Harvard University Press, Cambridge, MA.

Kalthoff KO (2001) *Analysis of Biological Development*^{(amzn/b&n/abe/pwll)}. McGraw-Hill, New York.

Moore KL, Persaud TVN (1993) *Before We Are Born*^{(amzn/b&n/abe/pwll)}. Saunders, Philadelphia

Thompson DW (1942) *On Growth and Form*^{(amzn/b&n/abe/pwll)}. Dover, New York.