# What are some examples of scaling laws in biology?

I've seen that metabolic rate scales logarithmically as function of mass for many animals over an extremely large span of parameters. What other scaling laws exist at the individual level?

• Obviously, one has surface-area to volume issues (e.g., it is speculated that huge dragonflies were possible in the past due to increase O2 despite lack of lungs) and cross-sectional area to volume/mass issues (e.g., bigger animals tend to require disproportionately thicker legs). May 15 '13 at 13:01

Here are some off the top of my head.

• The height an animal can jump depends on the muscle cross-sectional area ($l^2$) and its mass ($l^3$). Mass grows faster with body size ($l$) so small animals can therefore jump higher relative to their body size ($l$) than large animals. Very similar scaling exists for strength of limbs vs. mass, ability to fly vs. mass, etc.

• In pinhole eyes (as found in clams and nautilus) sensitivity to light depends on the size of the pupil ($l^2$), the bigger the hole the more light gets in. However the ability to focus depends on the reciprocal of the size of the hole (but I don't know the scaling, so $l^{-n}$). Therefore there is a trade off between sensitivity and ability to focus.

• There are some recent models about foraging behaviour in a 2D terrestrial environment (with the front of the animal being a length $l$) vs. foraging in a 3D marine environment (with the front of the animal being a surface $l^2$). The model predicts different behaviours for the size of prey animals should aim for.

• The rate at which sound attenuates scales with the frequency of the sound as $1/f^2$. The frequency of a resonator scales with the reciprocal of volume $1/l^3$. So small animals make high pitched sounds which attenuate quickly and don't travel very far.

• Finally, the properties of fluids change with size. The Reynolds number describes viscosity and is dependent on linear scale $l$. Therefore small animals experience fluids as viscous and can "crawl" through water whereas large animals have to "swim" through water.

However, any directly physical or chemical-physical property of biology will experience scaling laws.

• This is great! Do you have any references for some of these so I can read more, especially the first three? May 16 '13 at 2:22
• Are you looking for academic paper style references or something a bit softer? Here's a few papers anyway. Hope you don't get stuck with paywalls. sciencedirect.com/science/article/pii/S0003347271801343 <- Bats calls damtp.cam.ac.uk/user/gold/pdfs/purcell.pdf <- 70's paper about Reynolds jeb.biologists.org/content/208/18/3581.full <- allometry of flight May 16 '13 at 2:46
• Just for the record, jumping height depends also on the distance ($l$) along which a muscle's force delivers energy. The net result is independence ($l^0$) between jumping height and body size, but smaller animals still jump higher relative to size (by a factor of $1/l$). Feb 11 '16 at 1:57