Efforts to identify the origins of Ashkenazi Jews through DNA analysis began in the 1990s. Like most DNA studies of human migration patterns, the earliest studies focused on two segments of the human genome, the Y-chromosome (passed on only by males), and the mitochondrial genome (mtDNA, passed on only by females). Both segments are unaffected by recombination. Genome-wide association studies have also been employed to yield findings relevant to genetic origins. Genetic studies revealed that Ashkenazi Jews originated in Middle East during Bronze Age (between 2500 BC and 700 BC) spreading later to Europe. I guess the question is, is Turkey in the Middle East or do the two share a high level of genetic relatedness.
A study of haplotypes of the Y-chromosome, published in 2000, addressed the paternal origins of Ashkenazi Jews. Hammer found that the Y-chromosome of Ashkenazi and Sephardic Jews contained mutations that are also common among Middle Eastern peoples, but uncommon in the general European population. This suggested that the male ancestors of the Ashkenazi Jews could be traced mostly to the Middle East. However, unlike male lineages, the female lineages in Ashkenazi Jewish communities "did not seem to be Middle Eastern", and that each community had its own genetic pattern and even that "in some cases the mitochondrial DNA was closely related to that of the host community." In his view this suggested "that Jewish men had arrived from the Middle East, taken wives from the host population and converted them to Judaism, after which there was no further intermarriage with non-Jews.
A 2005 study by Nebel et al., based on Y chromosome polymorphic markers, showed that Ashkenazi Jews are more closely related to other Jewish and Middle Eastern groups than to the populations among whom they lived in Europe. However, 11.5% of male Ashkenazim were found to belong to Haplogroup R1a, the dominant Y chromosome haplogroup in Eastern Europeans, suggesting possible gene flow. Referencing The Thirteenth Tribe, the study's authors note that "Some authors argue that after the fall of their kingdom in the second half of the 10th century CE, the Khazar converts were absorbed by the emerging Ashkenazi Jewish community in Eastern Europe." They conclude "However, if the R-M17 chromosomes in Ashkenazi Jews do indeed represent the vestiges of the mysterious Khazars then, according to our data, this contribution was limited to either a single founder or a few closely related men, and does not exceed ~ 12% of the present-day Ashkenazim".
Writing in Science, Michael Balter states Koestler's thesis "clash[es] with several recent studies suggesting that Jewishness, including the Ashkenazi version, has deep genetic roots." He refers to a 2010 study by geneticist Harry Ostrer which found that Ashkenazi Jews "clustered more closely with Middle Eastern and Sephardic Jews, a finding the researchers say is inconsistent with the Khazar hypothesis" and concludes "that all three Jewish groups—Middle Eastern, Sephardic, and Ashkenazi—share genomewide genetic markers that distinguish them from other worldwide populations". Geneticist Noah Rosenberg asserts that although recent DNA studies "do not appear to support" the Khazar hypothesis, they do not "entirely eliminate it either.