Steen on mathematics and biology

Here's a fascinating article on what math is good for in biology: The "Gift" Of Mathematics in the Era of Biology, by Lynn Arthur Steen. Steen gives lots of examples about what math is good for in biology. Somewhat surprisingly to me, he doesn't really mention one of the first things that came to mind, namely the use of combinatorial techniques to study the genome, which is nothing but a word on a four-letter alphabet. It's possible that he subsumes this in "statistics", though; to take a simple example, one might want to know how many times a certain sequence of bases would appear in a "random" genome in order to determine whether the fact that such a pattern appears often is signal or noise. Still, he makes the point that while the traditional mathematics curriculum (with lots of calculus and differential equations) takes its scientific inspiration from physics, biology is ascending.

A shorter version of this article is available at The Chronicle of Higher Education.

(How did I find this? Steen was one of the authors of Counterexamples in Topology, which I mentioned yesterday, so I went over to his web site.)

College kids are better than monkeys

A misleading headline from MSN: College Kids and Monkeys About Equal on Math.

This is in reference to the work of Elizabeth Brannon, who actually claims that certain monkeys have an intuitive "number sense" which is as good as humans; see for example this article. Despite what those of us who teach college students may occasionally think, they are better at mathematics than monkeys.

Nautilus shells aren't golden spirals

You know how everybody says that the nautilus shell grows in a golden spiral, i. e. a logarithmic spiral that grows at a rate of φ = 1.618... per quarter turn? (Formally, it's the graph of r = φ^4θ in polar coordinates.)

Turns out it's not true. (This is also why I'm not worrying about saying who "everybody" is.)

I'm not surprised; a lot of the places where φ shows up in nature seem to be related to the fact that it's the "most irrational number" (i. e. the one that's hardest to approximate accurately by rational numbers, which follows from the fact that its continued fraction expansion consists entirely of 1s). This is useful for, say, growing leaves on a tree; if the angle between successive leaves is φ^-1 revolutions then leaves end up not on top of each other, and therefore not blocking the sun. I can't see any reason why a shell should grow like that, though I could be suffering from a failure of imagination, and of course I am not a biologist. A logarithmic spiral with any growth rate is self-similar, though, and self-similarity is everywhere in biology.

Of course, technically I should go out and find my own shells and do my own measurements, and not just trust Some Guy On The Internet. But it's Sunday morning, and I really shouldn't even be awake yet, so cut me some slack.

Ivars Peterson at Science News wrote about this a few years ago, it seems. A more reputable source seems to be The Golden Ratio: A Contrary Viewpoint, Clement Falbo, The College Mathematics Journal, March 2005, pp. 123-134; among other things Falbo measured some actual shells, and the growth rate for a typical shell appears to be about 1.33 per quarter-turn, although there's a pretty wide range, but the growth rate never seems to get as large as φ. The mathematical study of mollusk shells from the AMS web site might also be of interest

Variance in Olympic events

It's often claimed that the reason that there are many more men than women in certain academic disciplines (mathematics is one, but that's not the point of this post) is not that men and women have different mean abilities, but rather that the standard deviation of male ability is larger than the standard deviation of female ability. (Of course, it is unwise to espouse these views publicly, for political reasons; that's what got Larry Summers in a lot of trouble.)

It occurs to me, having watched lots of the Olympics in the last few days, that something similar might be true in athletic events. I'm not claiming that men and women are physically identical (I'm not blind), or that their average performance in physical feats is the same. But it may be the case that the difference between the very best men and the very best women in physical feats (say, times in some sort of race, because these are the most easily quantified) is larger than the difference between the average man and the average woman, because there could be more variance among men than women.

Is there any evidence for this? I'm obviously not a student of this sort of thing (in fact, I don't even know what "this sort of thing" is called, although it's clearly some subfield of biology or medicine).

Oh, and Jordan Ellenberg wrote an explanation of why the new gymnastics scoring system is good. I'm glad he did, because I'd had a feeling it was better than the old system but was having trouble articulating why.

Worms doing calculus?

Worms do calculus to find food. (Um, not really.)

But apparently worms use salt concentration to find food, and tend to head in the direction of the gradient of salt concentration. That is, they go where there's more salt. This is due to neuroscientist Shawn Lockery and his students at the University of Oregon. I think the paper is the following:

Suzuki H, Thiele TR, Faumont S, Ezcurra M, Lockery SR, Schafer WR (2008). "Circuit motifs for spatial orientation behaviors identified by neural network optimization." Nature 454:114-117.

but I can't be 100 percent sure -- Penn's libraries don't allow access to the electronic version of papers from Nature until twelve months have passed, and I'm not on campus right now. (This is, however, the only paper on Lockery's list with a title fitting the description.)

Saying "worms do calculus to find food" seems a bit disingenuous to me, though. It seems like saying that baseball players do calculus to catch fly balls. The larger point, though, is that neural processes -- of worms or of humans -- can be modeled using mathematical techniques, which may be of use to people trying to develop artificial systems that do these things.

(From John Scalzi, via 360. Apparently this first appeared in blogs a couple weeks ago, but I'm posting it here anyway, because it's new to me, which means it's probably also new to a lot of you.)

Can a biologist fix a radio?

Can a biologist fix a radio?, by Yuri Labeznik, via Anarchaia. This article asks a question: say biologists decided to research radios in the same way that they research things like how cells work. Then they would buy a lot of radios, classify and dissect them, eventually conclude that there was some sort of evolutionary explanation for why the antenna is really long, and so on. Much work would be duplicated, because the biologists do not have a particularly good language for communicating to each other how complex systems work. (The author compares the language used by biologists to that of stock market analysts.) Engineers, the author claims, have this problem less, because they have found standardized ways to describe such systems, simulate their workings in computers, and so on. I found the following quote interesting:

In biology, we use several arguments to convince ourselves that problems that require calculus can be solved with arithmetic if one tries hard enough and does
another series of experiments.

Yes, but if the biologists figure this out, and they make their students take calculus, how do I feel about that? (I actually think I feel good about it; if I'm not mistaken the biology undergrads already take calculus, but they think it's unnecessary for them.)

mapping functions and genes "crossing over"

In genetics they have a unit called the centimorgan. This unit is a unit of what is called recombinant frequency, and it doesn't seem to be well-defined. For those of you who don't remember your biology (and I'll admit I'm one of them), recall that almost all cells contain chromosomes in pairs (23 pairs in humans). in the process of meiosis, cells are produced which contain a copy of one member of each pair. When fertilization occurs, these come together to form a new pair of chromosomes. However, this new pair mixes up or "recombines" parts of the old pair, as can be seen in the image.

The result is that two genes which are physically close together on the same chromosome will be inherited together, but two genes which are physically far apart might not be inherited together. When one learns about this in an introductory biology class, I think that the fact that two cross-overs is, in a sense, the same as no cross-over at all is ignored. That is, if the chromosomes cross over twice, or four times, or six times, or any even number of times between two genes, then those genes will end up on the same copy of the chromosome even after crossing over. (A more quotidian analogy: you walk down a street, arbitrarily crossing it "when the mood strikes"; the probability that at some given moment in the future you are on the opposite side from where you started is not the same as the probability that you have ever crossed the street, because you might have crossed back.)

Certainly, I don't remember hearing it in high school biology, and it's not mentioned in Time, Love, Memory: A Great Biologist and His Quest for the Origins of Behavior, which is the book I'm reading right now. It's nominally a biography of Seymour Benzer (who is still an active researcher) but is also something of a history of molecular biology.

Anyway, two genes are said to be one centimorgan apart if the probability of a crossover occurring between them is 0.01 -- or if the probability of an odd number of crossovers occurring between them is 0.01 -- or if the average number of crossovers between them is 0.01 -- I can't determine which. From what I can gather, molecular biologists seem to think of centimorgans as additive, which seems to require the third definition. (It looks like sometimes they use the other definitions and use something called a mapping function to correct for this, but I'm not entirely sure I'm reading this correctly.)

Now, a first guess would be that crossovers occur basically at random over the entire chromosome, and are a Poisson process. For the sake of simplicity assume that crossovers form a Poisson process with rate 1 -- that is, in a piece of the chromosome of length λ, the number of crossovers is a Poisson distribution with mean λ, and non-overlapping pieces have independent numbers of crossovers. What is the probability of an odd number of crossovers occuring in a segment of length λ? Let X be a Poisson(λ) random variable; then it's f(λ) P(X = 1) + P(X = 3) + P(X = 5) + ... The logical question to ask is: is this an increasing function of λ? That is, as we consider points further and further apart on the chromosome, does the linkage between them actually become less strong? You could imagine that the function might not be increasing. For example, say that after one crossover, the next crossover always occurred between 9 and 11 space-units down the line. Then two genes between 11 and 18 units apart would always end up on opposite chromosomes, and two genes between 22 and 27 units apart would always end up on the same chromosome, and in general you'd have some sort of oscillatory behavior.

Under the Poisson assumption, though, the answer is yes. In fact, we have
P(X = 1) + P(X = 3) + P(X = 5) + ...
= λ e^-λ + (λ³ e^-λ)/3! + (λ⁵ e^-λ)/5! + ...
= e^-λ (λ + λ³/3! + λ⁵/5! + ...)
= e^-λ sinh λ
= (1 - e^-2λ)/2
which is known as Haldane's mapping function. It's hard to find a clear derivation of this online, because most of what's available online is course notes that are intended for people who will be using this in their work and don't particularly need to know the derivation.

What this tells us is that two genes which are separated by λ "units of space" will recombine with frequency (1 - e^-2λ)/2. Note that if λ is small, this is only very slightly smaller than λ, since cases when there is more than one crossover in the space between the genes are vanishingly rare. But it also tells us that if two genes A and B recombine with frequency p, then they are not p of these "natural units" apart, but rather they are a distance λ apart with (1-e^-2λ)/2 = p, so λ = -log(1-2p)/2. So, for example, if two genes A and B recombine with frequency .20, the average number of recombinations between them is not .20, but -log(.6)/2 = .255. And if another two genes B and C recombine with that frequency, and they are arranged on the chromosome in the order A, B, C, then the distance between them is -log(.6), and the recombination frequency is (1-e^{log .6})/2 = .32, not .40. In general, if A and B recombine with frequency p, and B and C recombine with frequency q, then A and C recombine with frequency p+q-2pq. This can be derived from the Haldane mapping function, but the following argument is nicer. In order for A and C to recombine, exactly one of the pairs (A, B) and (B, C) must recombine. With probability p(1-q), A and B recombine while B and C don't; with probability q(1-p) the reverse happens. Again, this formula seems to recur without justification in notes that I can find online.

If you know anything about special relativity, this sort of reminds me of how rapidities add (while velocities don't). The rapidity of a particle with velocity v is is tanh^-1 v/c, which is approximately v/c (or v, if you like natural units); relative rapidities are additive, while relative velocities are only approximately additive, and then only for small velocities. Something similar is going on in the genetic situation, where the usual measure of "distance" is only additive for things that are close together and a correction has to be used when they get far apart.

And how would this be different if chromosomes came in, say, triplets instead of pairs? Maybe I should be a mad scientist in my next life. Then I could find out. (Or I could just do the calculation now, but I've got better things to do.)

(I suspect there are places here where I'm not using correct biological terminology; here I follow in the footsteps of Feynman, who when he learned about zoology once went to a library and asked them for a "map of the cat".)