[pallada92]

1) The answer is both yes and no. Any particular digital circuit can be implemented, but not universal Turing machine, simply because there is no way to deal with arrays.

There is mind-blowing paper, that gives exhaustive anwser to this question: https://hal.inria.fr/hal-01519828/document

This paper is about chemical reaction networks, but in 2010 it was shown, that any chemical reaction network is realizable in DNA: https://www.pnas.org/content/pnas/107/12/5393.full.pdf

2) Yes, it was figured out in 2006, how to do logic gates in DNA: https://science.sciencemag.org/content/314/5805/1585

[jacquesm]

> simply because there is no way to deal with arrays.

I wouldn't be so quick to make that call. That we don't know how to do it does not mean it can't be done and there is some weak evidence (innate knowledge) that nature has indeed found a way to do this, just that we don't understand it.