IIt's a universal fact: since any three-dimensional object, from Plato's sphere to a cage to an elephant, grows in all directions, its total surface area will increase more slowly than the space it occupies (its volume). If the geometry and shape of an object remains the same as it grows larger, then its surface area will increase at about the same rate as its volume, to the power of two-thirds. For centuries, biologists have wondered whether life forms obey this two-thirds scaling law, even though they come in a bewildering variety of shapes and sizes. If this is true, it means that there are fundamental constraints fundamental to evolution that can influence how life interacts with the world around it.
Researchers recently used CT scans and digital tools to calculate the surface areas and volumes of an ancient and diverse lineage of animals: sharks. Team analysis, published in Royal Society Open Scienceincluded more than 50 species of sharks and provides some of the best empirical evidence to date for the existence of a hard-and-fast scaling rule in zoology. The team found that, as with the sphere, the surface area and body mass of sharks do indeed follow the two-thirds scaling law. If this is true for other groups of animals, it likely reflects basic rules of heat, metabolism, or development that constrain evolution.
If you're looking for a group of animals to study biological scaling, it's hard to do better than sharks, according to Joel Gayford, a shark biologist at James Cook University in Australia who led the new study. They share a general shape, but come in different sizes, occupy many niches, and have huge differences in body shape. In his study of the morphological evolution of sharks, Gayford noticed what appeared to be large-scale relationships between their body parts, such as the size of their fins. This led him to wonder whether there might be more fundamental rules limiting the forms that sharks can take.
However, he was able to find few high-quality studies on scale formation in large animals. Research in single cells discovered many deviations from expected rules; rare studies on smaller animals such as insects And snakes found some evidence of two-thirds scaling. But few studies have included larger animals, and most were conducted decades ago. Additionally, Gayford found that existing data on animal scaling was somewhat confusing. Due to technological limitations in the 19th and 20th centuries, attempts to accurately measure the surface area and volume of animals were “prone to error and also ethically questionable,” he said.
He wasn't the only one who thought so. “One of the big limitations—especially if you read these early biology studies—is how do you measure the surface area of a cow?” said Brian Enquist, an evolutionary biologist at the University of Arizona who was not involved in the study.
Until recently, options were limited. Researchers could run a measuring wheel over the animal's skin and mark the units with chalk, or skin the creature and measure its surface area by hand. To calculate its volume, they could throw an animal into a bathtub filled with water and see how much liquid it displaces; some went even further and poured water directly onto the newly freed skins.
Gayford's team had much more advanced technology. They measured the surface area and volume of 54 different species of sharks, from the 9-inch pygmy shark, one of the smallest in the world, to the whale shark, the largest living fish. But instead of skinning them, they took CT scans of high-quality museum exhibits to create detailed virtual reconstructions. For species too large to fit in a CT scanner, they used photogrammetry software, which combines many photographs of an object's surface to create a 3D model. (In one case, the object in question was a 37-foot-long whale shark that lives at the Georgia Aquarium.) They then loaded the models into 3D rendering software called Blender, which was originally designed for rendering objects in video games. To calculate the shark's surface area, Gayford simply had to press a button.
In addition to the huge range of animal sizes, the dataset also represents sharks that occupy a variety of ecological niches, from reef dwellers to bottom predators to open ocean predators. According to Gayford, they had “many different unique morphologies”, including several species with elongated hammerheads; an ordinary thresher, the tail fin of which is almost the same length as the rest of the body; and the flat and frilly wobbegong, as well as the more standard shark-shaped wobbegongs. And while most sharks are cold-blooded or ectothermic, some species (including great white sharks) can generate their own heat. Gayford's team included one of these regionally endothermic sharks, the thresher shark, in the data set.
Despite such diversity in size, shape, lifestyle, and metabolism, sharks conform almost perfectly to the two-thirds scaling rule. “They showed that there’s not a lot of variety in this, so that’s really cool,” Enquist said.
Analysis suggests that this two-thirds scaling rule may be universal across animals. To be sure, more research is needed in other animal groups, including terrestrial animals that may have complex external geometries such as feathers and hair, and warm-blooded, or endothermic, animals such as mammals and birds. To this end, Gayford's team is collecting more data; he hopes other researchers will continue to test biological scaling in the animals they study.
However, surface area measurements can still be considered incomplete since they only include the external features of sharks. Although structures such as gills are hidden inside animals' bodies, their surfaces are actually external from a topological perspective, says Carl Niklas, professor emeritus of biomechanics at Cornell University. If the researchers had also analyzed the sharks' gills, Niklas speculated that they would have found a scaling factor closer to three-quarters. However, the consistency in numbers of many different shark species suggests that this rule is no coincidence. “We have to think of this as a kind of reflection of adaptive evolution,” Niklas said.
Scientists are not sure what fundamental mechanisms may limit the size and shape of sharks and other animals, but they have hypotheses. One of them relates to tissue distribution during early growth. To visualize this, imagine the developing animal as a lump of clay. “There are so many ways to stretch clay to create different shapes without using energy,” Gayford said. In this case, scaling relationships are important to the embryo and limit the potential shapes that the adult organism can take.
Alternatively, this relationship may reflect a fundamental limitation of heat transfer. In animals that can absorb external heat and generate it through metabolism or movement, the scaling principle, in which surface area grows more slowly than volume, will create an insulating effect when moving from small to larger species. “The surface area to volume ratio is really important for heat transfer,” said Van Savage, a computational biologist at the University of California, Los Angeles, who was not involved in the study. This may explain why arctic species tend to be large and bulky, while species living in tropical climates can be slender: larger bodies are more difficult to cool than smaller ones. This applies even to ectothermic animals, which may have to conserve heat when moving between warm and cool environments—for example, when whale sharks dive into deeper, colder water.
The research provides insight into the mathematical limits of evolution and may help identify the fundamental mechanisms limiting the topology of life. But calculating how organisms scale also has practical value. This can help veterinarians figure out what dose of anesthesia should be given to a cat versus a Great Dane, for example, or help doctors determine drug dosages for infants versus adults.
According to Gayford, this highlights the need for continued empirical research into biological scaling. “It’s really important that people actually check these laws,” he said. “Because often they are just assumed to be correct.”
This article was originally published on Quanta.
Main image: Samantha Mash for Quanta Magazine.
