Thursday, March 22, 2007

My big dog can outrun your little dog

kw: book reviews, nonfiction, biology, scaling

Item: A nice 4-m yacht, under full sail across a brisk wind, leaps along, dashing merrily amidst the spume. It's really fast! But should it run parallel to a 300-passenger liner, the big ship will steadily, grandly leave it in the spray.

Item: A mouse can live four to five years. In that time its heart beats at most two billion times. A medium-sized pet dog can live fifteen to eighteen years, and its heart beats a billion to 1.3 billion times. Humans are said to experience two billion heartbeats, seldom more; at 80 beats per minute, or 42 million per year, that's only 48 years, so the two billion limit is obviously off by a factor of about two, even for folks with a slower heart. But the funny thing is, how can the longevity of a mouse's heart, in total beats, be anywhere close to that of a human's, when the human is sixty times as long and weighs 2,000 to 3,000 times as much?

Item: The walking speed, for animals that walk, and the running speed, for animals that run, depends on only two things: whether two or four legs (four is faster), and how long it is from hip or shoulder to toe. Interestingly, the variation in walking speeds is much greater than the variation in running speed.

A most interesting Item: The attribute that correlates most closely to life span is the weight of the brain. A tiny shrew with a 1-gram brain lives 2-3 years; an small wildcat with a 50-gram brain lives about fifteen years (in a zoo, much less in the wild); and an elephant with a 7-kg (7000-gram) brain lives at least sixty years, and we don't know how long one could live if its teeth didn't wear out at that age. Oddly, a human with brain weight of 1.2 to 1.6 kg, is above the trend, living 80-110 years (the trend predicts 40-50). The shrew-wildcat-elephant trend, appropriately crunched, yields an allometric exponent near 0.4; Yrs Wbrain0.4, which means the brain has to be more than five times heavier for a doubling of life span. A good rough guide.

Where'd all this come from? John Tyler Bonner, Professor Emeritus at Princeton, has studied scaling in plants and animals all his life, and recently written Why Size Matters: From Bacteria to......Blue Whales. I distilled the Items from his book, all except the first, which I threw in to illustrate that scaling affects everything, not just living creatures.

The book is built around five "size rules":

  • Strength varies with size.
  • Rates of gas and food ingestion vary with size, because of lung and digestive membrane scaling.
  • Complexity varies with size.
  • Metabolism and other life process rates vary with size.
  • Abundance of individuals varies with size.

Dr. Bonner's key message is that all these trends are governed by the principle of allometry, that changes in size are accompanied by changes in proportion. Thus, a baby's head is about a third of its body length, while a typical adult's head is one-seventh or one-eighth the total height (fashion models are selected from rare cases of extra-small heads, as small as one-tenth the total height, as long as their other proportions are pleasing).

A less-obvious allometric trend is that very tall persons have thicker limbs, in proportion to their size, than very small, yet otherwise "normal-looking" midgets. If you take a photo of a basketball player and a non-dwarfed stage midget, the midget looks skinnier, with twiggy arms and legs. That's because weight increases as the cube of length, but strength (muscle or bone cross-sectional area) increases only as the square of length:

Suppose a five-foot-tall young girl weighs 90 pounds, a normal weight for that height. If every proportion is increased by 6/5 so she is six feet tall, the cube of 6/5 is 1.728, so she'd weigh 155 pounds. That's thin enough to be almost unhealthy; a more normal weight for a six-foot young woman is 175 pounds. Let's consider their relative strength, by looking at the leg just above the ankle, where it is thinnest. Suppose at five feet, the cross section is about three square inches (the leg diameter is close to two inches). Then at six feet, the diameter is just under 2.35 inches, and the cross sectional area is 4.32 sq. in. That's 1.44 times the 3 sq. in. from before, because the square of 6/5 is 1.44.

Here is the important fact here: when standing on one leg, the 90-pound 5-footer is putting a stress of 30 pounds per sq. in. (psi) on that part of the leg. When the 155-pound 6-footer stands on one leg, the stress is 36 psi. The extra factor of 6/5 showed up as extra stress. This isn't much difference, but suppose we "blow up" the poor girl to thirty feet tall, with all the same proportions. Then the stress would be 180 psi, which is nearly enough to break the leg bone.

Going back to the six-footer: I wrote above that a more normal weight would be 175 pounds. She needs stronger legs to bear her weight with the same safety factor, so we want a stress of 30 psi. That requires 5.8 sq. in., or a diameter of 2.73 inches. This is 1.4 times the ankle diameter for the five-foot girl, which explains why taller people must be extra-wide.

That's just one of the five size rules. The author develops each one, in much simpler terms than I(!), making this a very informative and enjoyable book.

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