Counting ancestors

I have been puzzling in reference 1 about why Elizabeth II probably has no DNA from William the Conqueror, despite the latter being a reasonably well-attested ancestor of the former.

The Reich story seems to be that the number of people from whom we can take genes goes up linearly as we go back in time, while the number of people from whom we can claim ancestry goes up geometrically, so the former is always going to get small relative to the latter, making it improbable that anyone has genes from anyone in particular from a long time ago.

We note that 2 raised to the power of 20 is around 1,000,000. So, neglecting royal inbreeding, any royal has about 1,000,000 ancestors of the twentieth generation, which takes us maybe half way back to the time of William the Conqueror. So given that the fictional population of the United Kingdom, the United Kingdom not having been invented at that time, everyone has as much a share in the present monarch as anyone else.

In what follows we neglect the facts that some genes are only passed down the male line and that some genes are only passed down the female line. We also neglect the fact that most of the genes of most of us are all the same, and many of those are the same as those of the potato. But we can, nevertheless, still say that all our genes have come to us from someone, even if the result is likely to be just the same as it would have been had they had come to us from someone else.

So suppose now we each have just one copy of one chromosome and sex has not yet been invented. Then everyone in any one person’s line of descent has the same genes, barring mutation - which might be considerable if we were able to breed as fast as bacteria.

Suppose we now invent sex.

Suppose we still have just one copy of one chromosome and each new person, each child, simply takes their chromosome at random from either their mother or their father. Then all children will have either their mother’s chromosome or their father’s. But grandchildren will have four grandparents to chose their chromosome from, great-grandchildren eight great-grandparents and so on. In this scenario, Elizabeth II has just the one genetic ancestor and it is indeed most unlikely that that ancestor is William the Conqueror – or, indeed, anyone else in particular.

Suppose now that we all have one copy of each of ten chromosomes and that each new person picks their chromosomes at random from their mother and their father. These ten chromosomes are all different, so, for example, the person’s ninth chromosome must be the ninth chromosome from either their mother or their father. In this scenario Elizabeth II has at most ten genetic ancestors and it remains most unlikely that any of those ancestors are going to be William the Conqueror. Allowing a hundred chromosomes would not much change things.

We now move a bit closer to the real world and build new chromosomes by mix n’match, rather than by just choosing one or the other. Thinking of these chromosomes as long strings of beads, we build our new chromosomes by splicing together segments of beads taken from each of the two parental chromosomes in turn. This is done in some random way, but in such a way that the n-th bead of the new chromosome is the n-th bead from one of the two parental chromosomes corresponding. With this random alternation resulting, say, in an average of 50 such segments to each new chromosome. If each person has ten chromosomes, it is almost certain that all those chromosomes will contain a good mixture of genetic material from both parents. So the question is, do we now have enough mixing for everyone to have at least some genetic material from all their ancestors?

Note that we are assuming that every chromosome of a given type has the same number of beads and that corresponding beads from different chromosomes of that given type are usually more or less the same. The fun lies in the small differences.

At the limit, the answer to the question is no. We are presently thought to have around 20,000 protein coding genes, but even if we allow a total of fifty thousand beads, we still only have room for at most fifty thousand ancestors – against the suggested half way total of a million – so where are they all going, given the splendid mixing of the paragraph before last?

The answer is that our chromosomes are finite and they cannot be chopped into infinitely small pieces. If the pieces get small enough they just vanish, just as William the Conqueror has probably vanished from the Royal genome.

And so Reich seems to be right, even if I do still fail to understand his explanation.

PS: cautionary note: real chromosomes do not work quite in the way suggested above. For a better story the interested reader can start with reference 2.

References

Reference 1: Who We Are and How We Got Here: Ancient DNA and the new science of the human past - David Reich – 2018.

Reference 2: https://en.wikipedia.org/wiki/Human_genome.