The definition of a turing machine is mathematically perfect. No threading, no IO, no error correction, no errors, no asynchronous events, no processes fighting over shared resources, no resources that might or might not disappear at the blink of an eye, in short no nothing. In that it is equivalent to a spherical brain, any complexity relevant to the problem at hand removed.
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.
