Friday Feature: A natural curve

July 2, 2010 § Leave a comment

As your Friday treat, here is an amazing image Felice Frankel made for Walter Fontana’s group, as a cover suggestion for their recent paper on curvature in metabolic scaling (Kolokotrones T, Van Savage, Deeds EJ, Fontana W. 2010. Curvature in metabolic scaling. Nature 464 753-6.PMID: 20360740). There’s a lot of substance to this image, and it’s really too bad that the issue of Nature the paper ended up in happened also to be the one that celebrated the 10th anniversary of completing the human genome.  Some things are hard to compete with.

So what’s going on here?  In this paper, Kolokotrones et al. take on the ancient assertion that basal metabolic rate scales as body mass to the pth power, and the huge debate over what, in that case, p might be.  Some say 3/4; others 2/3; others, that it all depends. To all of these, Kolokotrones et al. respond “well… kinda”.  Using a new data set that (among other things) permits better correction for temperature, they show that the best fit for a log/log plot is not a straight line, but a curve.  This, for the first time, makes sense of both the smallest animals and the largest: the orca (top right) has always been an outlier before.  It now becomes clear why it has been hard to identify a single value for p — it depends where you take the tangent to the curve.

Why should there be any relationship between mass and metabolism?  An influential idea, suggested by West, Brown and Enquist in 1997 (who were trying to explain why p should be 3/4) is that the relationship is the result of the need to distribute resources (in this case blood) to all the cells of an organism.  An optimal oxygen-distribution network should maximize the space filled by the network of blood vessels (which leads to a fractal kind of branching structure), and minimize the effort needed to push the blood through the tubes. If evolution has built a vascular network with these properties, then (they argue) the resultant supply of oxygen causes metabolic rate to scale as mass to the 3/4 power.

West et al. are cautious enough to point out that their model should be viewed as an “idealized zeroth-order approximation”.  On examining their proof carefully, Kolokotrones et al. realized that “idealized” in this case includes the fact that 3/4 scaling only really happens in the limit, at infinite mass.  If you use the same equations to look at animals of real mass, you get a curve — but it’s the other way up from the curve you see in real data.  What went wrong?

A key issue in modeling blood flow is how to handle the pulses that result from heartbeats.  Near the heart, blood flow is pulsatile; in the capillaries, it’s smooth.  Where does this transition happen?  West et al. assumed that no matter how large the organism, the transition happens a fixed number of levels up from the capillaries.  This doesn’t seem very plausible, when you think about it. The network in large animals must have many layers — large arteries branching into smaller arteries, dividing further until finally they get to the size of capillaries — whereas small animals must have fewer layers.   It makes little biophysical sense for the transition from pulses to smooth flow to happen at a uniform point.  Kolokotrones et al. show that if instead you assume that the pulses go away at a network level that is proportional to the overall size of the network — say, halfway down — then you recover a curve that looks just like the experimental data.

Now then, be honest.  Would you have guessed that you could get all that information into a single image? Felice wants to show us that with visual imagination all things are possible — even communicating results from systems biology.  She’s very interested in what’s going on in the Department and is willing to work with you if you have an interesting challenge.  Send her an e-mail at felice_frankel[at]harvard.edu and let us know what happens.

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