Now on to this week's topic. Recently my wife Ann and daughter Sonia were visiting the Newport Aquarium on the Oregon coast, and came across a large tank of crabs. Recognizing her mom's favorite food, Sonia remarked, "They look tasty! But don't tell them I said so." Ann and the other spectators began laughing. But they were laughing for somewhat different reasons. I

think most of the visitors were amused by Sonia's implication that the crabs could be offended. Ann, however, recognized that the crabs in the tank were spider crabs, with long legs and roundish shells, while everyone knows that the tastiest crabs are the oblong-shelled Oregon Dungeness crabs.

This got me thinking about the many different types of crab shells. How is it that from the same basic type of creature, so many different forms could evolve? How did random genetic mutation continually rewrite the complex blueprints of the crab shell, so each cell knew exactly the right place to grow to complete these complex shapes?

Actually, about 100 years ago, British mathematical biologist D'Arcy Thompson investigated this issue. He asked the question: is there a simple mathematical relationship between the shapes of different species of related animals? He began looking at groups of creatures such as crabs, fish, and primates, and made an amazing discovery. By drawing a coordinate grid over the shapes and making simple mathematical transformations, he could change

one species into another.

For example, he started with a fish species called Argyropelecus. He then showed that by applying simple transformations to each (x,y) coordinate, where x'=ax+by, and y'=cx+dy, he could distort the fish into other species known as Sternoptyx or Scarus. Basically this transformation is the equivalent of stretching the fish at different rates in the x and y directions, and then tilting one axis with respect to the other. This change, called an affine transformation, is a subset of what is known as "rubber sheet" transformations; more complex conformal transformations, which you may be familiar with from flattened maps of the Earth, allow curved aspects and lead to a greater variety of possible forms. The show notes contain a link to a nice web tool that lets you stretch a fish of your own, mimicking some of Thompson's thought experiments. Thompson showed with a series of diagrams how the shapes of fish, shells of various crab species, and skulls of apes and humans were mathematically related to each other.

What does this all mean? Well, for one thing, it makes the process of evolution much easier to understand. When we talk about stretching, slanting, or compressing a part of a living creature, what we're really talking about is increasing or reducing the rate of growth of various parts. So what Thompson showed was that to produce the various forms we see in nature, it's not necessary for some mutation to radically change a

detailed blueprint, but simply to tweak the uniform mathematical formula that governs the rates of cell growth in a creature. This seems like a much more plausible avenue for a random mutation to create a new creature, as opposed to the total rewrite that would be necessary if DNA was working like a literal blueprint of a building.

Thompson went on to discover many other amazing mathematical relationships in biology, such as the relationship of the Fibonacci Sequence to spiral structures and the relation between jellyfish forms and random dispersal of viscous fluids, that we might discuss in future podcasts. I'm not sure if he ever found the true formula for which crab is tastiest though.

And this has been your Math Mutation for today.

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