David E. Kephart replied to my query as follows:
From dkephart@csee.usf.edu Mon Jan 17 11:06:29 2005 To: propp@math.wisc.edu CC: Natasha Jonoska <jonoska@math.usf.edu>
James,
I was forwarded your musing about biologically relevant (or otherwise) group actions on DNA. I know you had polyhedral groups in mind, originally. Now, everybody loves groups, but semi-groups have their good points, too, you know.
I say this because single-stranded DNA molecules can be quite naturally modelled as elements of the free semigroup generated by {A,C,T,G} with the antimorphic involution $\theta$, which maps A to T, T to A, C to G, G to C. And why a semigroup? Because the nucleotide chain itself is oriented, 5'-3', just as a word over {A,C,T,G}* has a first symbol, second symbol, etc.
As you might guess, I'm not just thinking this up on the spur of the moment, but echoing some research I have been involved in. If you want to check out more about this, and it is worth it, particularly since, yes, the uracil thing has to be accounted for, there should be a way of modeling the melting temperature problem, etc., some of which you refer to, I would suggest checking out http://math.usf.edu/~jonoska, the Web site of Natasha Jonoska here at USF and hunting through the links (some of them are on her "Classes" page, some of them elsewhere) to find papers by Lila Kari, et al, Natasha, Kalpana Mahalingam, and myself. Can I put in a good word for the CodeGen DNA word-generating software available there which I wrote, based on these ideas.
Now if you only will be happy with group actions, then there is one little problem when it comes to biology...what is a meaningful inverse operation? Taking apart a DNA strand (as in $S_4$)? Dehybridizing it? The closest thing I can think of is the various splicing operations, with their obvious inverses, that are specific to the enzymes that act on double-stranded DNA. With respect to that, rather than group actions, I believe Natasha has done some work on the four-dimensional topological implications...I can give you a reference on that, too, if you like.
Anyhow, I would be interested to find out what other responses you may get regarding this, and I hope that what I mentioned is of interest to you!
Yrs, David E. Kephart graduate student at University of South Florida
Jim