While at the Joint Winter Meetings in Atlanta, I saw a rather
curious object sitting on a table: a regular tetrahedron made
of white cardboard, whose sides were labelled "A", "C", "G",
and "T". (Does anyone know who created it, and why?)
This sighting prompted a question which is perhaps more suited
to Michael Kleber than to any other person alive, since it concerns
both polyhedral models (an old interest of his) and nucleotides
(a newer interest of his), but since the answer may interest
lots of people besides the two of us, I thought I'd ask it in
this forum:
What are the biologically relevant group-actions on the set
{A,C,G,T}?
Leaving aside the symmetric group action ("they're all
necleotides, aren't they?") and the trivial group action
("yeah, but they're DIFFERENT nucleotides"), there's the
4-element group action that stabilizes the set {A,G} and
the set {C,T} ("sure, but the two purines are more like
each other than they are like the two pyrimidines, and
vice versa"), and there's the 8-element group action that
stabilizes the partition {{A,T},{C,G}} ("okay, but what
really matters is that A pairs with T and C pairs with G")
and there's the 4-element group action that stabilizes
the sets {A,T} and {C,G} ("fine, but don't forget that
the two pairs behave differently vis-a-vis transcription
of DNA into RNA (the whole uracil thing)"), and ...
I'd be surprised if there were some biologically relevant
action of the 2-element group that leaves T and G alone
but acts like Sym({A,C}) on the other two elements (to
give just one example of a group-action that probably
isn't biologically meaningful). But, like most of you,
I like to be surprised, which is why I'm asking.
Also, do any linear representations (as opposed to permutation
representations) of Sym({A,C,G,T}) play a role in genomics?
Jim Propp
