4 minute read
Jan. 19, 2021

The Square-Cube Law in Chemistry and Biology

Drug Hunter Team

Assume a cow is an evenly distributed sphere... sometimes.

I happened to take an animal physiology course in college because my girlfriend (now wife) needed it for her biology major and taking a class together about bird metabolism was my idea of a nice date.  I had no idea it would end up being one of the most useful courses in my career.

The most memorable takeaway for me was the influence of the square-cube law on biology.  The square-cube law simply points out that volume grows exponentially faster than surface area with increasing size, since area is a square function and volume is a cubic function.

The magic behind flow chemistry, black holes, and probably why dinosaurs died (vide infra).

The square-cube scaling of the surface area to volume ratio is why stuff dries faster in a pan than a bowl, wings bake faster than chicken breast, and why one does not simply increase the scale of a reaction 10x and expect the same results.  This probably sounds obvious, but how many times have you heard someone say, “all I did was run the reaction on a larger scale,” or “I just used a different stir bar or vial”?  Square-cube scaling is also why flow processes are preferable to batch processes in manufacturing – by changing the scaling function to time rather than volume, flow processes remove a lot of the variability that comes with scaling processes.

Guilty of all three.

Animal physiologists also realized empirically that mammals’ basal metabolic rates scale non-linearly with size.  That is, smaller animals expend more energy per kilogram of body weight than larger animals.  You’ve likely noticed this when holding hamsters that can’t seem to keep still, and parents know that children have much faster resting heart rates than adults.  Whereas a mouse burns over 160 kcal/day per kilo just being alive, an adult human female only consumes about 20 kcal/day per kilo. 

Smaller critters burn more energy per kg. How many kcal/day does a woodchuck chuck ...?

The rate at which metabolic rate scales with weight is actually surprisingly consistent across animals.  If you do a log/log plot of Metabolic Rate vs. Weight, you find that the scaling factor between the two is remarkably close to 2/3rds.  In other words, metabolic rate is proportional to weight to the 2/3rds power, or proportional to the cube root of weight, squared).

Metabolic rate scales across species roughly with the cube root of weight, squared, as if cows really were spheres. Blue lines represent what you'd predict for a mouse or cow if there were no 2/3rds correction factor in metabolic scaling.

Why would that be the case?  It makes sense if you assume a cow is an evenly distributed sphere.  We pointed out earlier that heat transfer scales with the square-cube law.  Metabolic rate per kilo, or energy consumption per kilo, is a measure of energy efficiency.  Smaller animals are less efficient, and therefore generate more heat per kilo.  They also have a larger surface area than larger animals, are so they cool down faster.  If larger animals didn’t have slower metabolic rates, they would overheat and die if their energy consumption scaled linearly from mice (see blue lines on the chart above).  Similarly, if small animals didn’t have higher metabolic rates, they would freeze to death.  “Cold-blooded” large animals like dinosaurs were doomed from the start – evolution didn’t need an asteroid to figure out that depending on external heating to maintain the temperature of a brontosaurus-sized creature is bad engineering.

Small mammals evolved to burn energy inefficiently to keep themselves warm, whereas larger mammals evolved to burn energy efficiently to avoid overheating.

Essentially, animals evolved an allosteric scaling relationship between basal metabolic rate and animal size to correct for the square-cube law.  This is also why, as drug hunters are keenly aware, drug clearance rates and fluid flows scale non-linearly with size as well.

If that were the end of it, then dose prediction wouldn’t be so bad – we’d assume a spherical cow, scale by the 2/3rds power law, and have our estimated drug clearance rate and predicted human dose.  But as every drug hunter knows, there’s a ton of other confounding factors for scaling drug doses since organisms did not evolve to metabolize new chemical entities, including those fun factors that nobody expects to find until you’re already late in development (sweat glands and lung excretion, anyone?).

Other fun variables: sweat, spit, exhalation, and involvement of other mucous membranes...

Though the art and science of DMPK is constantly improving, cross-species PK will probably always be a little frustrating.  I’ll never forget a troubleshooting meeting when we were trying to figure out how to do a tox. study when our compound had great exposure in dog/cyno but didn’t in smaller critters.  Our DMPK colleague asked, “did you try putting a fluorine on it?” to which my French chemistry lead quipped, “did you try putting it in a spider?”

Hope this was useful, or at least entertaining.  I’m off to Google whether spiders have livers.  Maybe our readers from Corteva can help me with that one…

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