Pharyngula

Basics: Allometry

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.

i-3d5180877073514077e0595c411d7cf2-fetal_growth.gif

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.

i-8a5eda4dfcd875ff584084648ed3a500-fetal_proportions.gif

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.

i-234eb40497d665f21f7dde5450b9ac5f-organ_growth.gif

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.

i-a480e097de2a81469bd686d846f038c1-chimp_growth.jpg

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:

i-103063279736bd2d74d76603a4edc866-baboon_skulls.gif

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 = axb

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.

Comments

  1. #1 Blake Stacey, OM
    June 3, 2007

    It looks like you’ve got an unclosed font tag, which is upsetting the formatting of the text following the pretty math.

  2. #2 Caledonian
    June 3, 2007

    It would be an interesting statistical exercise to sort human faces by their allometric properties, and then see if the most babyish-looking people tended to have higher IQs.

  3. #3 PZ Myers
    June 3, 2007

    That’s trivial math, not pretty math, and I’ll have you know I never use the <font> tag — that’s been deprecated. There was a dangling <sup>, though — I’ve fixed it.

  4. #4 Callandor
    June 3, 2007

    See, this is what I never get. I can only describe processes and information like this in two words: fucking awesome.

    And yet people can not be interested in science at the same time. I’ll never understand it.

  5. #5 Richard
    June 3, 2007

    This is a wonderful post. I’m glad that you never get caught up in the whole God/bullshit stuff enough to forget about explaining some biology to us all. Thanks!

  6. #6 RamblinDude
    June 3, 2007

    These are laymen’s questions, but I don’t think the answers are common knowledge.

    Does brain growth take priority in the early years of development? I mean, if there is not enough proper nutrition during the formative early years, does the body shunt its limited resources to brain development, to the detriment of skeletal size, etc?

    Also, does any thing else influence brain size? Like cognitive interaction, more complex stimuli? Etc.

  7. #7 Paguroidea
    June 3, 2007

    Thanks for the post, PZ. This is really fascinating. I’m amazed how human-like that baby chimp is.

  8. #8 Mike Saelim
    June 3, 2007

    I love how you still preface these articles with what amounts to an abstract. It just kills me.

    I’m wondering, though – does puberty upset the nice exponential relation between two body parts? I can understand that two parts may experience the growth spurt together and keep the relation valid, but what about, say, comparing the body and the brain? Does puberty upset the exponential relation, and if so, is the deviance from the model small enough that you can simply use a best-fit line to determine b?

    I’d try to use Excel to figure it out myself, but I’m lazy and I have a feeling you have the answer.

  9. #9 Colugo
    June 3, 2007

    Caledonian: “It would be an interesting statistical exercise to sort human faces by their allometric properties, and then see if the most babyish-looking people tended to have higher IQs.”

    Well, maybe. However, without digging up a bunch of cites or getting into details, the emerging view (with some dissenters) is that humans are not neotenic apes.

    There is, however, a negative correlation within humans between paedomorphic facial features and violence. How so? Women – who are more facially paedomorphic – tend to be less violent than men. Testosterone is a common causal factor in both traits. (Insert long spiel about overlapping distributions, plasticity, cultural context…)

  10. #10 Colugo
    June 3, 2007

    Just to be clear, I am merely observing the male-female differences in these traits, not stating nor predicting the relationship between these traits within each sex.

  11. #11 David Marjanovi?
    June 3, 2007

    I love how you still preface these articles with what amounts to an abstract. It just kills me.

    I bet that comes with practice… :-)

  12. #12 David Marjanovi?
    June 3, 2007

    I love how you still preface these articles with what amounts to an abstract. It just kills me.

    I bet that comes with practice… :-)

  13. #13 Andre
    June 3, 2007

    Are there underlying reasons to pick this kind of scaling for the growth, or is it an empirical thing?

  14. #14 No One of Consequence
    June 4, 2007

    What I find interesting is the trade off between brain and facial growth — in the article pz wrote about blind cave fish, part of the cave fish’s face evolved larger to allow better sensing with it’s mouth, but a byproduct was that the eyes evolved smaller and eventually disappeared in the adult form.

    I wonder if a similar evolutionary change occurred between humans and chimps. One group had evolutionary pressure for bigger brains, and as a consequence the faces got smaller, while the other group had the converse evolutionary pressures which made the large face a more important feature than increasing brain size.

  15. #15 CCP
    June 4, 2007

    Didn’t Gould have a column illustrating principles of mammalian ontogentetic allometry with the appearance of Mickey Mouse over the decades?

    Allometry is extremely important not just in studying relative growth rates, but in understanding patterns of variation in functional and physiological traits. Nearly all rates (running speed, heart rate, metabolic rate, lifespan, etc.) scale to body size allometrically, both intraspecifically (ontogeny) and interspecifically (phylogeny).

  16. #16 Torbjörn Larsson, OM
    June 4, 2007

    If it is of any comfort, his global infamy as a low-brow, jingoistic schmuck now extends to Uppsala, Sweden, as well.

    Make that extended in spades. (Hi, forsen, the world is getting both larger and smaller for each blog, if you get my drift.)

    Btw, speaking of Sweden, I would guess national (I think) regulations for burial sites prevents undue noises here. At least I haven’t heard anything I can remember, so perhaps even wind chimes are unusual or forbidden.

    Cemeteries are supposed to be handled with decorum, and be dead calm. ;-) I’m a bit surprised about, I assume, local regulations failing that simple respect.

    Otherwise I could contemplate making, oh, say as a first attempt this fake coffin lid for my grave site which would come off by motion sensors with a skeleton hand going out, with much moaning and groaning, and the sign it would hold would say “Gimme some rest, finally!”

  17. #17 Torbjörn Larsson, OM
    June 4, 2007

    If it is of any comfort, his global infamy as a low-brow, jingoistic schmuck now extends to Uppsala, Sweden, as well.

    Make that extended in spades. (Hi, forsen, the world is getting both larger and smaller for each blog, if you get my drift.)

    Btw, speaking of Sweden, I would guess national (I think) regulations for burial sites prevents undue noises here. At least I haven’t heard anything I can remember, so perhaps even wind chimes are unusual or forbidden.

    Cemeteries are supposed to be handled with decorum, and be dead calm. ;-) I’m a bit surprised about, I assume, local regulations failing that simple respect.

    Otherwise I could contemplate making, oh, say as a first attempt this fake coffin lid for my grave site which would come off by motion sensors with a skeleton hand going out, with much moaning and groaning, and the sign it would hold would say “Gimme some rest, finally!”

  18. #18 Torbjörn Larsson, OM
    June 4, 2007

    Please disregard previous comment, posted in the wrong thread.

  19. #19 Torbjörn Larsson, OM
    June 4, 2007

    Please disregard previous comment, posted in the wrong thread.

  20. #20 Torbjörn Larsson, OM
    June 4, 2007

    messy biology always sticks its nose into our pretty mathematics

    Well, it is a model. But if I would introduce it, or any other math, I would try to motivate it. (At least in a basics post and/or as long as the given references are books.)

    Are there underlying reasons to pick this kind of scaling for the growth, or is it an empirical thing?

    I am wondering the same thing. The motivation behind could be that if we want to compare different dimensioned characteristics, such as sizes to areas, or areas to volumes (mass), we would expect power laws.

    This is AFAIK what motivates BMI, modeling body fat as dominantly being proportional to skin area (in reasonably healthy individuals).

    “Nearly all rates (running speed, heart rate, metabolic rate, lifespan, etc.) scale to body size allometrically”, and that is what Wikipedia motivates it with as well.

    Otherwise, physically I guess you would naively expect a linear model for a single trait, dx/dt = a*x(t), from cells (volume, mass) multiplying at a certain rate constrained by equal resources (forgetting unequal blood supply et cetera). This gives exponential growth in mass terms as the basic growth.

    So I presume comparisons sets the preferred equation form here.

    an exponential rate b relative to the other

    Hm. An exponential model y = a*exp(c*t) would have a growth or decay constant c. I would call the equation a description of (absolute) exponential rate.

    I would call b in y = a*x^b a power or an exponent, and indeed Wikipedia calls it a scaling exponent ( http://en.wikipedia.org/wiki/Allometric_law ).

    The above is not exactly wrong, but it doesn’t feel unambiguous either. Perhaps it is my understanding of english which is at fault. But I would like to call it “a power rate b relative to the other”. Alternatively “exponent rate” if such a beast exist.

    My reason to keep the distinction between exponential laws and power laws clear is that exponential functions have Taylor expansions of all powers, not just one (b). Which can give some interesting differences. :-)

  21. #21 Torbjörn Larsson, OM
    June 4, 2007

    messy biology always sticks its nose into our pretty mathematics

    Well, it is a model. But if I would introduce it, or any other math, I would try to motivate it. (At least in a basics post and/or as long as the given references are books.)

    Are there underlying reasons to pick this kind of scaling for the growth, or is it an empirical thing?

    I am wondering the same thing. The motivation behind could be that if we want to compare different dimensioned characteristics, such as sizes to areas, or areas to volumes (mass), we would expect power laws.

    This is AFAIK what motivates BMI, modeling body fat as dominantly being proportional to skin area (in reasonably healthy individuals).

    “Nearly all rates (running speed, heart rate, metabolic rate, lifespan, etc.) scale to body size allometrically”, and that is what Wikipedia motivates it with as well.

    Otherwise, physically I guess you would naively expect a linear model for a single trait, dx/dt = a*x(t), from cells (volume, mass) multiplying at a certain rate constrained by equal resources (forgetting unequal blood supply et cetera). This gives exponential growth in mass terms as the basic growth.

    So I presume comparisons sets the preferred equation form here.

    an exponential rate b relative to the other

    Hm. An exponential model y = a*exp(c*t) would have a growth or decay constant c. I would call the equation a description of (absolute) exponential rate.

    I would call b in y = a*x^b a power or an exponent, and indeed Wikipedia calls it a scaling exponent ( http://en.wikipedia.org/wiki/Allometric_law ).

    The above is not exactly wrong, but it doesn’t feel unambiguous either. Perhaps it is my understanding of english which is at fault. But I would like to call it “a power rate b relative to the other”. Alternatively “exponent rate” if such a beast exist.

    My reason to keep the distinction between exponential laws and power laws clear is that exponential functions have Taylor expansions of all powers, not just one (b). Which can give some interesting differences. :-)

  22. #22 Steven
    June 5, 2007

    Fantastic post.

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