You're making things far too complex, and confusing the issue, and yourself, as a result. Let's turn to the first sentence from Wikipedia:
"A Turing machine ... manipulates symbols on a strip of tape according to a table of rules"
Whichever programming language you are fond of ultimately reduces to this mode of computation. However, with DNA, RNA, and Proteins, that is not the case. The way that we compute is simplistic compared with the way that biology computes. Thus: the crude analogy in fact hinders understanding, and should be discarded.
"A Turing machine ... manipulates symbols on a strip of tape according to a table of rules"
Whichever programming language you are fond of ultimately reduces to this mode of computation. However, with DNA, RNA, and Proteins, that is not the case. The way that we compute is simplistic compared with the way that biology computes. Thus: the crude analogy in fact hinders understanding, and should be discarded.