Ahhh, but you can see that it does "write", at least! A calculator doesn't write back to the same tape that it read from, either, but it still performs calculations and computations, and can (for most of them) still qualify as a Turing Machine.
Moreover, the entire theory of Evolution is dependent upon there being mistakes mutating their way into the "written" output of DNA replication! If DNA systems (ooops, for Sentis "cellular systems") truly didn't "write" anything, as you stated above in this thread, then that would put quite a damper on things, wouldn't it?!
Not to worry, however, as DNA systems do indeed qualify as Turing Machines because such systems do indeed read commands and data, process said commands and data, as well as generate output (i.e. the "written" aspect of the Turing test) from those commands and data.