A number of assumptions have been made while trying to work out the Fertile Crescent "who and how much" in the Southwestern European populations Tuscans, Northern Italians and Spaniards. I'll state them explicitly here.
First, we have only chosen three populations thus far. It is possible that other Fertile Crescent populations are the "who" in Southwestern Europe. Notably, we've thus far left out Jews, Greeks, Cretans, Jordanians and Egyptians from possible sources for a westward migration.
The reason I haven't looked at Jews is because I don't have an origin population for them. Dienekes' looks at Ashkenazi and Sephardic Jews, but Iraqi Jews and many others were dropped from Dienekes' later K=10 run that I have been using. It's clear that in their long diaspora in Europe, the European Jewry have absorbed some European genes. In the Ashkenazim and Sephardim, you can see that in their Northern European components as well as in their elevated Southern European components, compared to Iran and Iraq Jews (see Dienekes' earlier k=10 run). Without an origin population for Jews, it's difficult to infer their contribution to Southwestern European populations. It's also difficult to nail down a position in the Fertile Crescent for Jews. Iran and Iraq Jews look very Northern Fertile Crescent and Ashkenazy and Sephardic Jews look very Mediterranean Fertile Crescent.
I will say that many Jewish groups seem to bear the unmistakable marker of a small Mozabite component, but again, looking at Dienekes' earlier k=10 run, it is notably absent in Iraq Jews. So we can't say for certain that the combination of a Fertile Crescent + Mozabite components is a universal marker for Jews.
For now, the best we can do is to say that the Jewish origin population fits somewhere in the Fertile Crescent triangle, but we don't know where.
I'll note that a close look at Dienekes' new K=10 run shows that there is a small Mozabite component across Tuscans, Northern Italians, Cypriots and certainly Spaniards. That hints at a small contribution to Tuscans, Northern Italians, Cypriots and Spaniards from an Ashkenazi or Sephardic Jewish origin.
I haven't included an obvious point in the path of a westward migration: that of Crete and other points of Greece. Crete and Greece aren't in the Fertile Crescent, but based on Dienekes' latest runs with Greeks in the mix, Greeks are looking like (1) a Fertile Crescent/Northern European mashup or (2) a Fertile Crescent/Northern European/Southern European mashup. Greeks look a lot like Tuscans across four components. I'll be adding Greeks to the analysis in the next few days.
I also haven't included Jordan or Egypt.
The Fertile Crescent Signature of Jordan looks very similar to that of Syria which is why I didn't include it.
It would be improbable that the point of origin of Fertile Crescent populations was from Egypt. The Egyptian signature is tilted heavily in favor of Southwest Asia and away from West Asia. As a contributor to Tuscans, Spaniards and Northern Italians, it yields a weird result where almost all of the West Asian in Southwestern Europe would have arrived by another non Mediterranean route. Egypt is also an unlikely source because of the West African and Eastern African contribution in the Egyptians. It's likely that due to ancient movements up and down the Nile, these West and East African components have been present in Egyptians at least since the Neolithic. Therefore, if Egypt made a significant population contribution to Southwestern Europe since the Neolithic, these West And East African components would appear in Europe. They do not.
A significant result of these linear additions is that Modern Turkish and Armenian Fertile Crescent populations don't work as a source for Southwestern Europe. They've got too much West Asian compared to their Southwest Asian component to yield a solution for Tuscans. It's likely that Turkey has not only absorbed Turkic populations in the last several thousand years, but has also continued to absorb more of a West Asian component. Ancient Anatolia and certainly ancient Southern Anatolia likely had less of the West Asian component. We don't know how much less. What I've done here is set the limit on the greatest amount of the West Asian component, set against the Turkish Southwest Asian component, that can still yield an admixture solution for Tuscans. I've called that mix "Anatolian".
We can improve upon inferred results from this model by including 1-sigma standard deviation results for each component in each population. I haven't done that yet, but hope to in the future.
Finally, in terms of inferring background populations in Southwestern Europe, I've made two critical assumptions:
1. The West Asian-Southern Europe-Southwest Asian component ratio in Fertile Crescent autosomal DNA is stable over thousands of years. This key assumption allows isolating the contribution from the Fertile Crescent in Southwestern European populations on the Southwest Asia component.
2. I assume that the Southwest Asia component was not present in Southern European background populations.
The method of inference for the background populations is most sensitive to these two assumptions.
That's an exhaustive discussion of assumptions. I'll add to this and develop upon this in future posts.