So here I am, having both paternal and maternal ancestries traced to the central Asian mountains and valleys, to the Hunzas and the Persians. How did I get here?
The 64 : 36 admixture of European and Asian polymorphic markers in my genome has evidently been preserved over many generations, because there is no European history known in either of my lineage within at least 10 generations. Were it the case that there was a single homozygous European who married a homozygous Asian, then their child would have both markers. Since there are many more Asians than Europeans in Bengal, and if all were homozygous Asians, then in every subsequent generation there is an overwhelming probability that the descendent of that lineage will breed with a homozygous Asian, thus at every subsequent generation the proportion of European markers will be halved (if the markers are all unlinked). This is akin to successive back-crosses with the Asian stock. Thus, after 10 generations, 2^(-10) or only 1 in approximately 1,000 or 0.1% of the European markers, if all are unlinked, will still exist.
The actual proportion might be somewhat more, because of linkage and linkage disequilibrium, which can be calculated, which will lead to loss of heterozygosity at the rate of (1 – r)^t, where t is the number of generations and r is the recombination frequency between marker pairs. But the frequency of retention under the above simplifying assumption will be far below 64%. This is because r for most marker pairs (~700,000 markers if randomly distributed over 23 chromosome pairs) would be approximately 0.04 for human chromosomes (~1 centiMorgan per megabase pairs). Therefore, the erosion will be approximately at a rate of (1 – 0.04)^t, which translates to 0.96^t. For 10 generations, we need to divide 0.1% by approximately 66%, which leads to a retention of 0.15% of the heterozygous markers on average. Although I arbitrarily chose 10 generations (i.e., t = 10), it is probably true that anyone in my ancestry mating with a person homozygous for European markers goes far deeper into the past because the very first Europeans in historical times came to Bengal only about 20 generations ago. The overwhelming conclusion is that my assumptions are incorrect. Where are they incorrect?
The main assumptions were that there was one rare mating between a European and an Asian, and that most people in Bengal are homozygous for Asian markers. Both are nearly certainly incorrect.
The heterozygosity of markers over many generations, in the absence of direct natural selection due to selective advantage (unlikely because it would predict an enormous selective advantage to rare heterozygous markers), is probably the result of selective breeding or kin-selection—the inevitable result of the caste system in India. It is because of this selective breeding and kin-selection that the heterozygous markers were conserved over many generations. Therefore in a moment of somewhat dampened literary inspiration, I am compelled to moderate my romantic scenarios of a Hunza couple eloping together and settling in Bengal, or a wayward Yemeni sailor marrying an Asian woman.
The reality is likely to be quite different. The most likely scenario is that my ancestors descended from individuals in south central Asia, the inhabitants of Afghanistan, Persia, and central Asian plateaus at some remote Vedic or pre-Vedic time, through selective marriages among a small number of communities who rarely married into the indigenous Asian stock. This ensured that the members of these communities are all highly heterozygous. If nearly all of these people are heterozygous at most markers, then the chance that any individual will be heterozygous at any marker is nearly 50%. The observed 64 : 36 distribution of markers is close enough to this expectation, if one assumes a slight bias towards marrying into families with more European (i.e., Brahmin) than Asian markers. Thus, for my ancestors, intermarriages largely restricted within the community in most generations with only rare breeding with non-Brahmins (having somewhat higher frequency of Asian markers) is a good explanation for my lineage.
This scenario is well attuned to an oral myth of the Vaidya or Baidya communities of Bengal. Tradition has it that the ancestor of the Vaidya caste was the result of an illegitimate union between a Vaishya woman by the name of Birabhadra and a Brahmin Galava Muni. The latter was reputed to be a Vedic Brahmin, apparently from the area currently known as northern Pakistan/Afghanistan (but see also: this). The child born became known as Dhanavantri. Since the child had no legal father, (s)he belonged to the family of his/her mother. Of course that is only one of the narratives, and there are several competing narratives. One fact is clear: the Vaidyas generally intermarried among their own communities, thus maintaining their genetic heterogeneity.
This is all a remarkable congruence of oral tradition and science, perhaps even more interesting than my romantic story of the eloping Hunza couple.