Like, the connectome for C. elegans has been mapped out; it's can be written down as a 2 megabyte ascii text file. Just the connectivity is not enough to actually reproduce the behavior of the worm, you would also need data about the weight of each connection, but it's still a lot less data than the worm genome (about 25 megabytes---I hope I got the number right this time!). The worm genes also need to contain a lot of additional stuff to build functioning cells internals, etc, stuff which hopefully is irrelevant to the actual cognition.
I cannot adequately put the insane laugh required as response to that into text form. So I will only write this and be just as right: going by physics the brain of a mouse can be adequately approximated by a perfect sphere.
"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.