Why an elephant's heart beats slowly

In a big table of mammal measurements, an Asian elephant weighs about one and a half million times as much as an Etruscan shrew, yet at rest it burns energy only about 37,000 times as fast. Per gram of body, the shrew burns about forty times more (our arithmetic, on Savage and colleagues' table). Big animals live slower, and their hearts beat slower. The pattern is not in dispute. What remains in dispute is the number that describes it.

What is being claimed

Biologists write the relation as a power law: a rate Y equals a constant times body mass M raised to an exponent b. West, Brown and Enquist (1997) put the usual claim this way: whole-organism metabolic rate scales as M^3/4, while heartbeat and the rate of cellular metabolism scale as M^−1/4, and blood circulation time and life span scale as M^1/4. On a log–log plot a power law is a straight line, and b is its slope.

Why the exponent matters

The obvious guess is geometry. Heat leaves through the skin, skin area grows like M^2/3, so a body in balance should burn energy like M^2/3. West and colleagues note that if geometry were the whole story, b “should be a simple multiple of one-third. However, most biological phenomena scale as quarter rather than third powers”. So the difference between 2/3 and 3/4 is not rounding. It decides whether surface area or something else sets the pace of life. The history runs both ways: White and Seymour (2003) recount that Max Rubner reported M^2/3 in the 1880s, that Max Kleiber's 1932 monograph moved the exponent above it, and that Brody's “famous mouse-to-elephant curve” supported 3/4.

Where the 3/4 comes from

The 3/4 in West, Brown and Enquist is not fitted to data. It comes out of a model of the plumbing. Blood reaches every cell through a branching tree of tubes. The model makes three assumptions. The tree fills the whole body. Its last branches, the capillaries, are the same size in every mammal. And the network is built to spend as little energy as possible pumping. Solving for that network gives an exponent of D/(D+1), where D is the number of dimensions the body fills. For a three-dimensional animal that is 3/4. One consequence is that the tree grows mostly by adding levels, and the number of levels grows only with the logarithm of mass: they write that “a whale is 10⁷ times heavier than a mouse but has only about 70% more branchings from aorta to capillary.” The same model predicts a heart-rate exponent of −1/4, and their table lists the observed value as −0.25.

What the mammal data give

Savage and six co-authors (2004) pooled published basal metabolic rates for 626 mammal species. Fitting every species directly gives a slope of 0.712, with a 95% confidence interval of 0.699 to 0.724, which excludes both 2/3 and 3/4. They argue that fit is skewed, because 477 of the species weigh under a kilogram and only 149 weigh more. So they grouped the species into 52 bins, each 0.1 wide in log mass, averaged each bin, and fitted the bins. That gives b = 0.737, interval 0.711 to 0.762, which includes 3/4 and excludes 2/3. Binning is a choice, and it is part of what the two sides argue about. For heart rate, in 26 species, their binned slope is −0.251.

The case for 2/3

White and Seymour (2003) assembled 619 species from 19 orders. Uncorrected, their exponent was 0.69, and its interval included neither 3/4 nor 2/3. They then corrected every rate to a common body temperature, and left out groups they judged could not be measured at a true resting, fasted baseline, such as even-toed hoofed mammals, kangaroos, rabbits and their relatives, and shrews. What remained gave BMR ∝ M^2/3. Savage and colleagues reply that those exclusions remove most of the smallest and largest mammals, and they write that “values of b ranging from 2/3 to 3/4 can be obtained from data on mammalian BMR, depending on which data are included and how they are analysed,” before arguing that their own analyses strongly support 3/4. White and Seymour's analysis covers only mammals. Savage and colleagues also compile exponents across many other groups, including plants and single-celled organisms, and find values near 3/4, though whether 3/4 holds for all organisms, as West and colleagues claim, is part of the same argument.

The figure shows why the argument lasts (our arithmetic, on Savage and colleagues' 52 bins). The best 3/4 line and the best 2/3 line differ by a factor of 1.66 at the shrew end and 1.97 at the elephant end, while the typical (root-mean-square) bin sits a factor of 1.38 off the 3/4 line and 1.50 off the 2/3 line. Keep only the bins from 10 g to 10 kg, a cut of our own to show how much the range matters, and the slope falls to 0.687, with an interval that holds 2/3 and excludes 3/4.

The elephant's slow heart

Take the quarter-power rules at face value. Heart rate goes as M^−1/4, so ten thousand times the mass gives one tenth the beat rate. From the shrew bin to the elephant, mass rises about 1.5 million times, so the rule predicts a heart about 35 times slower (our arithmetic). Life span goes as M^1/4, so beats per lifetime go as M^−1/4 × M^1/4 = M⁰, the same for every size. That is our deduction from the two exponents, not a measurement in either source. These are fits across species, not rules for individual animals.

In short

Bigger mammals burn more energy in total but less per gram, and their hearts beat slower. The fractal-network model predicts a slope of exactly 3/4. Binned mammal data give 0.737, and a temperature-corrected analysis that leaves out some groups gives 2/3. Over the masses measured, the two lines are hard to tell apart, and the answer turns on which animals are counted and how.