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As you know DNA is composed of strings of 4 letters. I am wondering if the number 4 here has any significance? Any property of 4 that makes using 4 letters more advantageous over more (or less) letters to encode information?

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I'm no expert, but I would expect the answer to be chemistry or biology and not mathematics. The four letters correspond to certain acids, and so it's not simply an encoding, but also has a functional part. –  Helge Jul 18 '10 at 10:49
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I'm no expert either, but to me this looks a lot more like numerology than biology or chemistry, let alone mathematics. –  Franz Lemmermeyer Jul 18 '10 at 12:02
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Also, there are technically more than four bases - there are less common modified bases, such as m5C. See en.wikipedia.org/wiki/Nucleobase. –  Thomas Bloom Jul 18 '10 at 12:10
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If you believe in the RNA world, this probably originated as some sort of compromise between having a minimally complex chemistry with sufficiently interesting enzymatic/catalytic behavior. –  Rob Grey Jul 18 '10 at 12:49
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There might also be a genome-compactness/efficiency vs. chemical complexity/difficulty-of-error-correction argument. –  Rob Grey Jul 18 '10 at 12:52
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A short (and not entirely correct) answer is: redundancy. There is a very nice article on this in Brian Hayes' book, Group Theory in the Bedroom. The precursor of the chapter appeared in the American Scientist: "The Invention of the Genetic Code". One interesting aspect of the story is that George Gamov proposed several clever triplet genetic codes of great abstract beauty. Other physicists and mathematicians (Feynman, Teller, Golomob) also made suggestions. But nature chose its own path.

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I'm pretty sure his name is Brian, not Barry. :) I read his blog bit-player.org quite often. In fact, since this is CW, I'm just going to in and edit it. BTW I highly, highly recommend Group Theory in the Bedroom. –  Willie Wong Jul 18 '10 at 14:24
    
@Willie: Thanks for the correction! –  Joseph O'Rourke Jul 18 '10 at 14:57
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A 1993 PRL by Hornos and Hornos suggested that the evolution of the genetic code manifests itself through symmetry breaking. According to them the basic symmetry is supposed to be $SU(2)^{\otimes 3}$ (three nucleotides per codon).

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It's an interesting idea, but the paper reads more like pattern fitting than actual a priori derivation. Then again, I am no expert in biophysics. –  Willie Wong Jul 18 '10 at 14:21
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