Fractal Biology

I first learned about the intermediate-dimension objects called fractals in the late 1970s, from Marvin Gardner's wonderful "Mathematical Games" column in Scientific American. One of the cool and compelling things they can do is explain is how highly branched circulatory system, with an effective dimension between two and three, could have an effectively infinite surface area, abutting every cell in the body, while taking up only a fraction of the body volume.

Twenty years later, Geoffrey West and his collaborators used this fractal model to explain the well known "3/4" law of metabolism, in which different organisms' resting metabolic rate varies as the 3/4 power of their body mass. West, an erstwhile theoretical physicist from Los Alamos who recently stepped down as the head of the delightfully eclectic Santa Fe Institute (for which I've done some writing), used similar scaling analyses of things for other aspects of biology as well as resource usage in cities.

Unfortunately, according to Peter Dodds at the University of Vermont, the well known 3/4 law is also wrong. In my latest story for Physical Review Focus, I briefly describe how Dodds uses a model of the branched network to derive an exponent of 2/3.

Interestingly, this 2/3 exponent is precisely what you'd expect from a simple computation of the surface to volume ration of any simple object. A 2/3 law for metabolism was first proposed in the mid 1800s, Dodds said, at "a tobacco factory in France, trying to figure out how much to feed their workers, based on their size. They asked some scientists and they said 'we think this 2/3 rule would make sense.'" Experimental data on dogs seemed to fit this idea.

But later experiments hinted at a slightly higher exponent. "At some point it became more concretely ¾," Dodds said, based on the work of Max Kleiber published in 1932. "He'd measured some things that looked like 0.75 to him. You know, he had nine or ten organisms, and it was easier on a slide rule." At a conference in the 1960s, scientists even voted to make 3/4 the official exponent.

But in the wake of the fractal ideas, Dodds and his collaborators re-examined the data in 2001. "What really amazed me was I went back and looked at the original data and it's not what people thought. People had sort of forgotten about it by that point." Instead, Dodds, found, the data really matched 2/3 better. At the very least, the 3/4 law was not definitive. This doesn't mean that the fractal description is not useful, only that it has a different connection to the metabolic rate.

From C.R. White and R.S. Seymour, Allometric scaling of mammalian metabolism, Journal of Experimental Biology 208, 1611-1619 (2005). BMR is resting metabolic rate. The best fit line has a slope (exponent) of 0.686±0.014 (95% CI), much more consistent with 2/3 than with 3/4.

Other authors have since supported this conclusion, especially if they omit big herbivores like kangaroos, rabbits, and shrews, whose resting metabolism is hard to measure. The real biological data is messy, and perhaps it is silly to expect a simple mathematical law to apply to diverse biological systems. In any case, the difference between the two exponents is modest, amounting to a factor of about 2.5 in metabolism over the range of experimental data in the plot.

Still, some experts, such as the commentators that I interviewed for the Focus story, still think that the 3/4 law is correct. But it seems plausible that many decades of experimental observations have been colored by researchers' expectations. Science remains a human endeavor.