This question is more on the theoretical-encoding side. Given the known kinds of genetic mutation, transposition, deletion, and such, what can be said about genes that would be left untouched by the mutation, left invariant by the permutation, if you will? Are there any such sequences? Would they be more common in a population, if not detrimental?
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2$\begingroup$ By definition, mutations are genetic variation, so a gene cannot be “left invariant by permutation”. Are you perhaps referring to variation in the proteins encoded by protein-coding genes or the downstream phenotypes? $\endgroup$– acvillCommented Oct 10, 2021 at 23:53
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$\begingroup$ You are correct, I mislabeled what I meant. Technically not mutations, but defects than don't affect the sequence, i.e.: $(ACT)_1(ACT)_2 -> (ACT)_2(ACT)_1$ the resulting sequence is the same, but there was a change of the 'physical' bases. It occurred to me that, given the possible mechanisms for error production, some sequences would be more stable, and therefore, less prone to mutation. $\endgroup$– Elkin MontoyaCommented Oct 11, 2021 at 22:15
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2$\begingroup$ @ElkinMontoya - you can't distinguish the examples you gave, they are literally chemically the same thing. I suggest you look up a codon table and see the redundancy in the codons and think about how that affects your question. $\endgroup$– bob1Commented Oct 11, 2021 at 22:38
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$\begingroup$ @bob1 That is precisely the point, the point I'm trying to get across is that some sequences, like $CGA$ can, due to imperfect replication, translate into $AGC$ (the first base is transposed with the third), but others, like $CAC$, will still produce the same sequence even if the same physico-chemical events that made the other sequence change where to happen. And so, my reasoning goes, proteins that are encoded by sequences like $CAC$ will not mutate at the same rate as others Because even if something were to happen, it will still be the same sequence $\endgroup$– Elkin MontoyaCommented Oct 13, 2021 at 22:07
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$\begingroup$ @ElkinMontoya - I'm not sure that you can study mutations where the "error" doesn't actually produce an error, because it then isn't a mutation at all. You can examine the mutation rate for individual bases in a protein coding sequence though. $\endgroup$– bob1Commented Oct 13, 2021 at 22:26
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Not in the way you suggest: a sequence of letters with the property of being immutable by any type of permutation on its constituent bases. I would be interested if anything of the likes exists in mathematics.
Having said that, there are examples of sequences that seem to have such property but only in the light of evolution and the biological processes that come with it:
Ultra-conserved regions have been described as sequences of about 200 base pairs that appear to be exactly the same from humans to mice. They have not experience change in about ~300 million years of evolution (!) and exposure to the various forces of mutation.
The reason for this is not an underlying 'invariant to permutation' property, but rather strong negative selection that prevents individuals that acquire changes on the sequence from being able to pass the changes over to their progeny, thus effectively maintaining only the original version in the population.
The exact mechanisms, functions and properties of these regions are under research. You might want to see if the invariant properties you have in mind have anything to do with these interesting regions.
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2$\begingroup$ Enzymes' catalytic site(s) are also very often left unchanged for the same reason. If there happen to be a mutation there, the protein's function might be compromised. But sometimes it is instead enhanced.... $\endgroup$ Commented Oct 18, 2021 at 8:24
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$\begingroup$ Very interesting. Based on the criteria that their existence is due to very strong selective pressure, and not due to external mechanisms, I figure it is possible that, given they are relative few in number, the UCR sequences that are not completely essential could be something of this kind, were some to be actually inessential, but it is also possible that the UCR change with species, so it may actually take an enormous to prove/disprove. $\endgroup$ Commented Oct 18, 2021 at 20:08
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$\begingroup$ Yes, I guess UCRs are specific to particular groups of species and/or have certain upper limits to the evolutionary time within which they remain conserved. The UCR shared by humans and mice for instance were present in their common ancestor, so therefore all species derived from this common ancestor have them too. As usual, proving a certain property in this or other UCR example might well lead to some degree of generalization, $\endgroup$ Commented Oct 19, 2021 at 7:42