How physics stops animals getting really really big in 285 words
April 12, 2022
Consider a small sauropod of length x, as shown on the left above. Its mass is proportional to x cubed, it stands on leg bones whose cross-sectional area is x squared, and it ingests food through a gullet whose cross-sectional area is x squared. Now consider a larger sauropod of length 2x, as shown on the right above. Its mass is proportional to 2x cubed = 8x, it stands on leg bones whose cross-sectional area is 2x squared = 4x, and it ingests food through a gullet whose cross-sectional area is 2x squared = 4x. The bigger sauropod has to carry proportionally twice as much mass on its leg bones, and ingest proportionally twice as much food through its gullet. (Similarly, a 104-foot tall gorilla, 20 times as tall as a real one, is only 400 times as strong but 8000 times as heavy — which is why we can’t have Skull Island.)
In practice, big animals tend to have adaptations such as thicker limb bones that mean the numbers aren’t quite as bad as this, but the principal holds: the bigger an animal gets, the worse the problems imposed by scaling. It’s not possible to “solve” this problem because so many biological properties scale this way. Something is always the limiting factor. Suppose it were leg-bones or gullet. If somehow a hypothetical ultra-sauropod evolved extra thick leg-bones and gullet, scaling of respiration would suffocate it, or scaling of digestion would starve it, or scaling of heat-loss through the skin would boil it. The fundamental reason that you can’t just scale an animal up is that some parts of its function scale with volume while most — respiration, digestion, etc. — scale with surface area.
Sauropod Vertebra Picture of the Week 
April 12, 2022 at 12:03 pm
One of the best little articles I ever read on these biological fundamentals. Breathing is indeed another important one of those surface area limitations, and the main reason why the by far most successful animal class on Earth, the insects of course, cannot grow bigger.
April 12, 2022 at 3:44 pm
This is one I have always pondered too – you see the giant sauropods with their tiny (by proportion) heads and therefore mouths (and by your extension, their gullets), and you feel like they would need to be eating *constantly* just to get enough nutrition through their mouth/gullet to stay alive
April 12, 2022 at 4:12 pm
True, Pete, and it’s widely thought that sauropods did spend pretty much all their time eating. But then, elephants do the same, despite their negligible magnitude. That’s because they waste so much time chewing what they ingest. Sauropods skipped that part: it was just crop and ingest, and let the hindgut do the rest. That seems to have been part of the special sauce that let them get so big.
April 12, 2022 at 5:51 pm
But precisely that hindgut scaled volume-wise and not area-wise, which is one of your stated reasons fermenting herbivores want to scale big – though your point is likely that microvilli of the intestines are area limited. Also, what’s to prevent the lungs from maintaining constant-sized alveoli-like microstructures that would allow the lungs themselves to also scale volume-wise? Especially with flow thru breathing, you don’t have to worry about dead spots like we mammals have as a partial limit. And since bone tends to add to itself in response to stress, I’d expect the structural bones to keep pace with the body – and if pneumatic penetration of bones is more or less automatic, that needn’t be such a massive weight penalty. Your post on pelican bones shows that in an extreme, you could almost get away with area growth instead of full volume effects. Actually, when you double a beam width, iirc, you make it 8x more resistant to deflection – so, linear scaling of the horizontal vertebral column automagically keeps up, and the limbs keep pace stiffness-wise if not entirely compressive-wise (if weight alone on the limb is truly the limit, then the pelican would need to thicken that shell, not just expand its circumference).
But, I’ve never looked closely at dinosaur remains outside a few minutes peering at museum displays, and I recognize that I’m hand waving away details that, in fact, probably contribute to the real limits.
April 12, 2022 at 7:12 pm
Brad, well, maybe lung capacity does scale with volume: the flow-through lung processes according to the cross-sectional area of a bundle of tubes and perhaps the lengths of those tubes — though I imagine that past a certain tube length you’re not going to be extracting much more oxygen. But the broader point stands: whatever limiting factor is defeated by a clever evolutionary innovation, all it does it make something else become the limiting factor.
April 13, 2022 at 4:36 am
Guts and lungs are still surface-area-limited. Alveoli are nice hacks, which let humans carry around the gas exchange area of a basketball court. But gas exchange in the respiratory system, and the nutrient absorption in the guts, still happen across membranes, and it’s the surface area of those membranes that Mike is discussing in the post.
April 13, 2022 at 8:46 am
In practice, of course, nothing ever scales exactly with the square or the cube of linear dimension. As noted above, limb-bones trend (though with plenty of exceptions) to being proportionally thicker in bigger animals, so that their cross-sectional area varies with a bit more than the square of linear dimension. Gas exchange in lungs is another that will be somewhere between the square and the cube: if you double the size of a sauropod lung you get four times the number of parallel gas-exchange tubes but they are also twice as long. But being twice as long won’t get you twice as much gas exchange.
April 13, 2022 at 9:49 am
I always thought that while a lot of e.g. physiological problems might be solved by evolutionary innovation, the hard limiter was that at a certain size the legs of a tetrapod would be too thick compared to body size to walk, in order to support its mass. Is that true, or would it only happen at ludicrously large sizes?
April 13, 2022 at 9:52 am
Well, it’s true; the question is, would that ever be the limiting factor, or would something else stop it getting big enough for that to be an issue?
April 13, 2022 at 10:21 am
I think it would be interesting to read a study on that- the order in which various factors restrict body size without transformative adaptations. Similarly, what the differences would be for an aquatic animal like a whale.
April 13, 2022 at 10:27 am
OK, it’s obviously time for me to do a post on my long-postponed paper WEASS. Stand by …
April 13, 2022 at 11:31 am
[…] discussed this project with Matt, usually under the acronym WEASS, and its substance has come up in the previous post, and especially Mickey Mortimer’s […]
April 13, 2022 at 11:31 am
As promised, Mickey!
April 14, 2022 at 3:48 am
The avian lung is already so efficient, I doubt that doubling the length of the tubes can double to amount of oxygen extracted from them. On the other hand, doubling the number of tubes might do the trick – you get more air in and can get most of the oxygen out of it. Now, the geometry of such a lung…?
I have my A&P students read the classic LaBarbera (https://fathom.lib.uchicago.edu/2/21701757/ ) and we discuss the scaling laws. Looking at the Incredible Shrinking Man, the assumption is that as the volume of the body (including lungs) shrinks by a cube, the surface (including the total respiratory surface of the alveoli) shrinks only by square, so all is well. But, there is a question; what exactly shrinks there? If the alveoli stay the same size, will the total surface really shrink only by square or by more (and shrinking in size increases metabolic need so need for more oxygen) – what is the geometry there? Alternatively, if the size of individual alveoli also shrinks, they will get too small, so type II cells cannot produce surfactant that can actually counteract the force of surface tension of water at that small scale and alveoli will collapse after the first expiration. Has anyone thought about this, made calculations? What about the lungs of really tiny mammals like shrews?
April 17, 2022 at 6:50 pm
I would very much doubt that doubling the length of tubes would help. Gas transfer and heat transfer are different things, of course, but I suspect there are similar principles involved. So look at the boiler tubes in a steam locomotive boiler. Late (WW II era) designs– 2-10-4 and the like — were often much bigger than earlier (WW I era) types — 2-8-2 and so on, but the boiler tubes were often no longer: extensions of the fire-box into the boiler barrel (“combustion chambers”) were used to keep the boiler tube length down. I think there were at least two factors involved. (a) The exhaust gas cools down as it transfers heat through the tube walls to the boiler water, so there is less heat left to be transferred by the time it gets to the end of a long boiler tube. Something similar should be the case for gas transfer. (b) It takes energy to force the hot gas through the relatively narrow tubes, so if you make the tubes longer you end up wasting energy (typically by using more steam pressure to induce draught rather than to move pistons in the cylinders) to keep the gas flowing. I don’t know how the analogy works on this one, but if you doubled the length of tubes in the lungs, wouldn’t it get harder to breathe? (Need more muscle power from the diaphragm if you’re a mammal, something else for other critters.)
April 17, 2022 at 9:44 pm
That’s more or less the kind of thing I had in mind, Allen.