Metabolic scaling

Why does a gram of elephant burn far less energy each day than a gram of mouse?

A mouse and an elephant are both mammals, built of cells that look much the same under a microscope and run the same biochemistry. Yet a gram of mouse consumes roughly twenty times the energy per day that a gram of elephant does. The mouse must eat something like its own body weight in a few days; the elephant, on the same reckoning, eats a small fraction of itself.

If the cells are the same, why does putting more of them in one animal make each of them slower? That is a strange thing for arithmetic to do — and the first explanation almost everyone offers turns out to be measurably wrong.

Start with the fact itself. Across mammals, basal metabolic rate is described tolerably well by about 70 times body mass in kilograms raised to the power three-quarters, in kilocalories per day. Max Kleiber established that fit in 1932. Test it before trusting it: for a 70-kilogram human, 70 times 70 to the power 0.75 gives about 1,700 kilocalories a day, which is close to measured human basal rates. That agreement between a fitted rule and an independent measurement is the reason to keep using the rule.

Now divide through by mass. If the whole animal scales as mass to the three-quarters, then per gram it scales as mass to the minus one-quarter. Take a 25-gram mouse and a four-tonne elephant: a mass ratio of 160,000, and the fourth root of that is about twenty. The observed twentyfold difference is not a separate fact — it is the exponent, restated.

So the whole question reduces to: why three-quarters, rather than one?

Here is the explanation almost everyone reaches for. Animals produce heat throughout their volume and lose it across their surface, and surface goes as mass to the two-thirds. A large animal, with proportionally less skin, would overheat unless it ran slower. Max Rubner proposed exactly this in 1883, and the reasoning is impeccable.

It also makes a specific prediction: the exponent should be two-thirds. Measured, it is closer to three-quarters. That is a small difference on paper and a large one across a millionfold range of body mass. Worse for the heat-loss story, similar sublinear scaling shows up in animals that do not regulate their temperature at all, in plants, and even across unicellular organisms — none of which have a heat-shedding problem. And a mammal measured in thermally neutral conditions, where heat balance is not the binding constraint, still shows it. Whatever the cause is, it is not principally about staying cool.

So look for a constraint that scales differently from a surface. Consider what limits a cell's rate of work: not the cell itself, but the delivery of oxygen and fuel to it. That delivery runs through a branching network — arteries to arterioles to capillaries — and the capillary is the same size in a mouse as in an elephant, because it is sized by red blood cells and diffusion distances, not by the animal. So a bigger animal is not a scaled-up version of a smaller one; it is the same terminal units, fed by a deeper hierarchy of branching.

West, Brown and Enquist argued in 1997 that a space-filling branching network with size-invariant end points, tuned to minimise the energy lost in pumping, yields exactly the three-quarters exponent. This is elegant and it is contested. Critics have challenged the derivation's assumptions, and careful re-analyses of mammal data — notably by White and Seymour in 2003, controlling for body temperature and digestive state — recovered something much closer to two-thirds. Larger compilations across wider mass ranges find genuine curvature, meaning no single exponent fits everything.

What is not in dispute is that the exponent is well below one. Per gram, big animals are slower, and that fact carries consequences whatever explains it: heart rate falls roughly as mass to the minus a quarter, lifespan rises roughly as mass to the plus a quarter, and the product — lifetime heartbeats — comes out roughly constant, on the order of a billion, across mammals. Humans are a conspicuous exception, living several times longer than our mass predicts.

It is worth separating this from the other famous scaling argument in biology. The square-cube law genuinely governs structural support: bone resists compression in proportion to cross-sectional area, which grows as the square of length while weight grows as the cube, and that really does set limits on how large a land animal can be. The same geometric reasoning applied to metabolism gives two-thirds and does not match. Same starting move, two different answers — because a skeleton's limit really is a surface in the ordinary Euclidean sense, whereas a metabolic limit runs through a branching network whose relevant geometry is not.

One smaller clarification: these are relations across species, not within one — a fat human does not obey Kleiber's law relative to a thin one, because the added tissue is not metabolically like the rest.

• Why the energetic equivalence rule makes population density scale as mass to the minus three-quarters, so each species uses roughly equal energy per unit of habitat.
• How allometric scaling is used, and misused, to set drug doses across species.

Every term the collection defines is gathered in the glossary.