Yes .... well. Speaking in general terms, Israel and/or Zionism considers you Jewish if you were born of a Jewish mother and (in the case of males) have been circumcised. Males who've already been circumcised and want to convert to Judaism must be ritually re-circumcised by drawing blood from the penis in a mock circumcision formality. However, converts are not eligible for Zionist indulgences unless they have been converted by an
Orthodox rabbi. There may be one or two exceptions but any road up, a Jew need not have any
"Semitic DNA" 
what-so-ever. Judaism is a religion - duh - not a race.
You're confused and projecting modern Western terminology on an ancient culture.
There's no such thing as "Semitic DNA".
'Semitic' is a linguistic cultural term.
Religion, also a Western term under which the West categorized Jewish culture.
As well as 'Judaism', but nowhere is it mentioned in Torah.
So is 'race'.
You folks in the West have religions and races,
and here you're simply projecting your theories of blood purity on Judaism,
which is not there. King David A"H was a grandson of the first convert, our royal line.
If for you that's not "clean" enough...who's problem is that?
For all I care since Khan all your Czars are basically Mongols, and that changes what?