I am no biologist, but as a physicist, a spontaneous mutation (seen as a chemical transformation) should lower the energy of the system, at least locally. So I wonder if any research has been done along these lines for the DNA.
a spontaneous mutation (seen as a chemical transformation) should lower the energy of the system
Why do you think that? Because it’s an endothermic reaction?
Consider that mutations don’t happen “just like that” (at least to my knowledge) –they are triggered by external energy influx from things like reactive oxygen species or ionising radiation. So yes, they cost energy but that energy doesn’t come energetically unstable chemical bonds; it is supplemented from external sources.
So it’s not necessarily true that mutations produce lower-energy chemical bonds.
I'm going to attempt an answer to this at the level of undergraduate biochemistry: apologies in advance if this is not sufficiently sophisticated.
The most frequent spontaneous mutation is deamination of cytosine to uracil (hydrolysis with loss of ammonia). Strictly in terms of the double helical structure of the DNA molecule this, it seems to me, would be destabilizing because of the change from a GC base pair (3 hydrogen bonds) to a GU base pair (2 hydrogen bonds).
The reaction kinetics of DNA mutations are complicated by the fact that the DNA doesn't exist in isolation, and in fact is actively processed and maintained by the cellular machinery.
For example, as Alan Boyd notes, the most common spontaneous mutation is the deamination of cytosine to uracil: Cyt + H2O → NH3 + Ura. On its own, the reverse of this reaction, i.e. spontaneous amination of uracil to cytosine, would be very unlikely, if only because the low concentration of ammonia compared to water in cellular fluid.
However, in a living cell, there are DNA repair enzymes that actively look for uracil in DNA and remove it, allowing other enzymes to come along and replace it with the original cytosine. This keeps the effective Cyt → Ura mutation rate far below what it would be in the absence of active repair.