Thank everybody. It seems that Rollof method don't work well when there is a complex admixture stories. That 40% ( or 60% from other side ) is not real.
It gives good results when the event is simple like in the case of Turks.
Nevertheless I found some historic events that could bring some Near Eastern admixtures in the Middle Ages.
* It's the invasions of Saracens and their activities. But assessing it's real impact is difficult.
* The appearance of
Cathars in North Italy and South France.
Of course religions and ideas can move without mass movement, but from historic sources we know that Paulicians and related Manichean sects were wiped out by the Byzantine emperors from Anatolia and Armenia. They were mostly relocated to Bulgary. Did they move from there to Italy is unknown.
Before one speculates about the genetic impact of the Saracens or any other ethnic group, it might make sense to know who they were ethnically and what areas they actually impacted. As has been pointed out by another poster, Saracen is just a name for the predominantly Berber populations of North Africa after their conversion to Islam, and later it was also applied to their compatriots in Al Andalus. Another name sometimes applied to them is "Moor". Their only contact with Toscana was through some coastal raiding, the kind of coastal raiding that took place along all the southern European Mediterranean coastlines.
The genetic impact of this coastal raiding was not on Toscana or any of the coastal areas, it was on North Africa, through the thousands of slaves that were kidnapped and sold in the slave markets of North Africa. The only place where there was actual settlement of North African Berbers was Sicily and a few places in southern Italy, and, of course Iberia. At any rate, if you take a look at Hellenthal, the signal they're talking about is "Cypriot like" (which is virtually the same as EEF like), not Berber or North African like, which is a separate component even in Hellenthal.
As for the Cathars, there is absolutely
no indication in the historical record that the Cathar movement had anything to do with any actual migration from the Near East, other than the initial settlement of some heretics in Bulgaria.
"The Cathars were largely a homegrown, Western European/Latin Christian phenomenon, springing up in the Rhineland cities (particularly Cologne) in the mid-12th century, northern France around the same time, and particularly southern France — the
Languedoc — and the northern Italian cities in the mid-late 12th century. In the Languedoc and
northern Italy, the Cathars attained their greatest popularity, surviving in the Languedoc, in much reduced form, up to around 1325 and in the Italian cities until the
Inquisitions of the 1260s–1300s finally rooted them out."
https://en.wikipedia.org/wiki/Catharism
If all devotees of religious cults with roots in the East were held to have ancestry from there, then Hellenthal should show an admixture event for Swedes with Palestine around the time that they adopted Christianity.
Also, unfortunately, as was pointed out in the above article, most Cathar groups were in
northern Italy, not Toscana, yet the same kind of "admixture" event is not posited for northern Italy.
http://catharslideshow.blogspot.com/2010/11/italy-and-beyond.html
Plus, I don't quite see how the Cathars could have had a genetic impact on anyone since a primary tenet of their religion involved forswearing intercourse.
The kind of situation you raise where people from different ethnic groups can provide organs for one another is not at all uncommon. It's a coincidence of immunity profiles. Plus, even if you had to share broad ethnic ancestry for such matches to be possible, Armenians and southern Europeans share ancestry since the Neolithic and this was reinforced by shared ancestry from Yamnaya incursions. Why would the admixture event show up only in Toscana, and only at this time?