A house mouse weighs about 20 grams. An African elephant weighs around 4,000 kilograms. That is 200,000 times more animal. So does the elephant burn 200,000 times more energy than the mouse? It does not. It burns roughly 9,000 times more. Nature charges big animals a steep discount per kilogram, and the discount has a name.
In 1932 the Swiss biologist Max Kleiber measured how much oxygen resting animals used, from small ones to large ones. He found a tidy pattern: an animal's resting metabolic rate grows as its body mass raised to the power of about three quarters. Double the mass and the energy bill rises by only about 68 percent, not 100.
That was a surprise because there is an obvious competing guess. Heat leaves through skin, and skin area grows as mass to the power of two thirds. If animals simply had to dump their heat, you would expect the exponent to be 2/3. Kleiber's measurements kept landing nearer 3/4. The gap between 0.67 and 0.75 sounds tiny. Across a mouse and an elephant it is the difference between a factor of about 3,400 and a factor of about 9,400.
The chart above is drawn on logarithmic axes, so a straight line means a power law. All three lines are pinned to a 70 kilogram human using Kleiber's classic figure of about 70 kilocalories per day per kilogram to the 3/4, which is roughly 80 watts. Slide the mass down to a mouse and watch how little power it needs in total, yet how much per gram. Then change the exponent. At 1.00 the elephant's bill balloons; at 0.67 it shrinks.
Look at the energy per kilogram. A mouse burns around 20 times more per kilogram than an elephant. Every gram of mouse is running a furnace. That has a visible consequence, and it is the next thing to try.
If tissue needs more fuel per kilogram, the blood has to deliver it faster. Heart rates fall as body mass rises, roughly as mass to the power of minus 1/4. A mouse heart can beat around 550 times a minute, and some figures go as high as 700. An elephant's beats about 30 to 35 times a minute. You are in between at roughly 70.
The little heart above beats at the model's rate, slowed to a tenth so you can see it. Drag from mouse to elephant and it settles from a flutter into a slow thump.
Lifespan scales the other way, roughly as mass to the power of plus 1/4. Small animals live fast and die young; large ones live slowly for longer. Multiply the two quarter-powers together and the masses cancel. Heart rate times lifespan is about the same for a mouse and an elephant. A mouse living two to two and a half years at about 550 beats a minute gets roughly 600 to 700 million beats. An elephant at 30 to 35 beats a minute for about 30 years gets a similar amount. Across non-human mammals the average works out to roughly 700 million, which people round to “a billion”, though the spread between species is wide.
Humans are the famous exception. At about 70 beats a minute for 65 years or more, we use something like 2.4 to 3 billion beats, a few times the usual mammal total. Our life expectancy has mostly risen since the late 19th century, largely through preventing and treating disease, so the extra beats are better read as medicine than as biology. The “model” lifespan on the chart is the one the rule predicts for a wild mammal of that mass, which is why the human setting reads low.
This is still argued over. In 1997 Geoffrey West, James Brown and Brian Enquist proposed that the answer lies in branching delivery networks, the arteries, veins and capillaries (or the branching of a tree), that must reach every cell. In their model the exponent comes out near 3/4. Other researchers have since shown that simple networks can give 2/3, and several analyses of mammal data find the exponent shifts with body size or that neither value fits perfectly.
One 2014 proposal, from physicists at the University of Maryland and the University of Padua, adds a missing ingredient: how fast the fluid flows. If blood velocity itself creeps up with body mass, as mass to the power of 1/12, a 3/4 exponent can emerge without a perfectly fractal network. In their models, a constant velocity gives exactly 2/3, and 3/4 is the upper limit.
So the headline is robust and the explanation is not: bigger animals are cheaper to run per kilogram, with a heartbeat that slows as they grow. How exactly nature pays that discount is still an open question, and it is yours to poke at with the exponent slider.