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June 2004

The Pythagorean Theorem of Baseball has just been simplified. This news from the web-based Science Daily for March 30, 2004. The original PTOB is due to the baseball statistician and conoisseur Bill James. It estimates a team's winning chances in terms of two numbers: Rs, the number of runs scored, and Ra, the number of runs allowed. The formula is

  Rs2 P = ------------ . Rs2 + Ra2

Suppose that in 12 games your team scored 72 hits and allowed 64 hits. The PTOB gives P = .56 so it should have won 7 games and lost 5. If it won more than 7, it is "overperforming," if less, then "underperforming." This is supposed to help in predicting future performance.
The simplification is due to Michael Jones and Linda Tappin (Montclair State University). Their formula runs

  P = 0.5 + β(Rs - Ra),

where β is a constant, "chosen to give best results" for each season, ranging between .0053 and .0078 and averaging 0.0065. For your hypothetical team the streamlined formula with β = .0065 gives P = .55 and leads, after round-off, to the same prediction as the PTOB. (Science Daily's βs were off by a factor of 10). [Linearizing the PTOB about the equilibrium Rs = Ra gives P ~ 0.5 + (Rs - Ra)/(Rs + Ra). TP]

DNA does the twist. And the writhe. A "News and Views" item in the May 13 2004 Nature picked up a preprint posted by Maria Barbi, Julien Mozziconacci and Jean-Marc Victor, all with the CNRS. "In the cells of higher eukaryotes, e.g. animals or plants, meters of DNA are packaged by means of proteins into a nucleus of a few micrometer diameter, providing an extreme level of compaction." As we know, the nuclear DNA contains a library with all the instructions for making and maintaining a cell. But how does one access an item in a library where all the text is on a single line miles long bunched up into a volume inches in diameter? We know there are enzymes (topoisomerases) that allow one strand of DNA to pass through another, so there is no topological obstruction to moving any particular segment of DNA to where it may be copied. But transcription can take place without topoisomerases. How? Barbi and collaborators studied the way that DNA is coiled. The first two levels of packing result in a chromatin fiber, with structure given schematically in the following figure.

 A segment of chromatin fiber and its detailed structure, representing the first two levels of DNA folding. The DNA double helix (yellow tube) is wound around spools of proteins. The nucleosomes (spools plus linkers) are stacked to form the fiber. Spool radius R ~ 5.3nm, DNA helix radius r ~ 1 nm, linker ~ 20 nm, each spool has 1 and 3/4 turns of a left-handed superhelix. The packing is characterized by the angles: αi between the incoming and outgoing strands on the i-th spool, and βi between the axes of the (i-1)st spool and the i-th. Images courtesy Jean-Marc Victor, reprinted with permission.

"In order to provide the transcription machinery with access to specific genomic regions, the corresponding [chromatin] loop has to be selectively decondensed, via a reversible unwinding process that elongates the fiber." The CNRS team analyzed the way the differential-geometric quantities "twist" and "writhe" vary in terms of the angles and discovered that there is a unique way to simultaneously vary the αs and the βs so that the fiber elongates isotopically: without changing the linking number of the DNA. The unfolding process is illustrated in the following picture, where it is compared with the non-isotopic stretchings that come from changing the αs and the βs separately. These are frames from movies illustrating the three stretchings, and where the top nucleosome is colored blue for reference.

 Initial and final configurations for the three stretchings of the chromatin fiber segment illustrated as 1. In 1-2 the βs are kept constant; in 1-3 the αs are kept constant; in 1-4 αs and βs vary so that the linking number stays constant.

Understanding the ununderstandable. There's an essay about the nature of mathematical understanding in the May 25 2004 New York Times Science section. Susan Kruglinski interviewed four prominent popularizers of mathematics to find out how much of "the inconceivable, undetectable, nonexistent and impossible" described by mathematics can possibly be explained to a general audience.

• Ian Stewart, asked if there exist mathematical concepts that cannot be explained to a general audience: "Oh, yes -- possibly most of them."

• Keith Devlin, speaking of the Hodge Conjecture: "Those equations ... are beyond visualization."

• Brian Greene defends imperfect metaphors: "The equations that govern a violin string are pretty close to the equations that govern the strings we talk about in string theory. So although the notion of strings is metaphorical, it's pretty close." And adds: "I suspect that the overarching aim of every mathematical study can be described, even if you can't get to the guts."
In what sense do scientists, including mathematicians, understand their own work?
• John Casti: "Mathematics is an intellectual activity --at a linguistic level, you might say-- whose oputput is very useful in the natural sciences."
This approach sidesteps the question of math's connection to reality, so understanding may well be besides the point. Brian Greene has the last word: "Our brains evolved so that we could survive out there in the jungle. Why in the world should a brain develop for the purpose of being at all good at grasping the true underlying nature of reality?"

-Tony Phillips
Stony Brook