How long animals live mathematically linked to size, shark study shows

New 3D modelling techniques show that shark size follows a “two-thirds scaling law”, helping reshape understanding of biology.

The law refers to the ratio between the surface area of an organism and its volume (or mass). It is a geometric formula which says that the surface area of the organism increases with the volume raised to the power of ⅔ (or 0.67).

This is not just an interesting numerical or geometric relationship but relates directly to an organism’s metabolism. Previous studies into the scaling law in organisms has focused on individual cells or tissues, according to the writers of the new study published in the Royal Society Open Science journal.

“This law helps explain how animals exchange heat, energy, and oxygen with their environment – so confirming it in full-sized animals, not just cells, is a big deal,” says lead author Joel Grayford, a PhD student at Australia’s James Cook University (JCU).

“We found that sharks follow what’s known as the ‘two-thirds scaling law’ almost perfectly.”

Studies in the 19th and early 20th centuries showed mathematically that a two-thirds scaling is expected based on how energy dissipates from an organism’s surface area. This was first shown in respiration trials of dogs by German physiologist Max Rubner in 1883. His theory was further developed in his book, The Rate of Living, published in 1928.

This “rate of living” scale is believed to be related to observations like the shorter and faster lives of smaller animals compared to larger ones.

The theory is largely accepted, but the exact scaling factor has been subject to debate. Max Kleiber, a Swiss biologist, found that metabolic rates could be predicted by taking an organism’s mass to the power of ¾.

Grayford and colleagues tested the ratio by modelling the body shapes of 54 shark species using high-resolution 3D scans created with the help of New York-based computer graphics artist Johnson Martin.

“This ratio is fundamental,” says co-author Jodie Rummer, also from JCU. “It underpins how animals breathe, regulate temperature, and process waste. And now, for the first time, we’ve shown it holds true in animals as complex and diverse as sharks.”

The team used a statistical model to consider evolutionary relationships between the shark species and relate them to the animals’ shape and size.

They found that shark surface area is proportional to volume raised to the power of 0.64, just 3% off the theoretical prediction of 0.67.

“It’s remarkable,” says Rummer. “This suggests sharks have evolved to stick to this ratio, possibly because deviating from it is too costly or constrained by early development.”

Understanding these evolutionary and developmental constraints could explain why sharks living in vastly different ways and environments all follow the same scaling rule.

“Changing the way tissue is distributed throughout the body might require major changes during early embryonic development – and that’s expensive, energetically speaking,” Gayford explains.

The findings could be used to answer energy and lifespan questions in other animals.

“Surface area-to-volume ratios are key inputs in equations used to model how animals respond to climate change, like how fast they regulate their body temperatures or how efficiently they use oxygen,” Gayford says. “Now, we can use those equations with much greater confidence in sharks and other large animals.”