Contribution of ancestors in a person's genome (David Reich's "Who we are and how we got here")

Question

I was reading the book "Who we are and how we got here", written by David Reich. There it has the following explanation:

Females create an average of about forty-five new splices when producing eggs, while males create about twenty-six splices when producing sperm, for a total of about seventy-one new splices per generation. So it is that as we trace each generation back further into the past, a person’s genome is derived from an ever-increasing number of spliced-together ancestral fragments.

Any person’s genome is derived from 47 stretches of DNA corresponding to the chromosomes transmitted by mother and father plus mitochondrial DNA. One generation back, a person’s genome is derived from about 118 (47 plus 71) stretches of DNA transmitted by his or her parents. Two generations back, the number of ancestry stretches of DNA grows to around 189 (47 plus 71 plus another 71) transmitted by four grandparents. Look even further back in time, and the additional increase in ancestral stretches of DNA every generation is rapidly overtaken by the doubling of ancestors. Ten generations back, for example, the number of ancestral stretches of DNA is around 757 but the number of ancestors is 1,024, guaranteeing that each person has several hundred ancestors from whom he or she has received no DNA whatsoever. Twenty generations in the past, the number of ancestors is almost a thousand times greater than the number of ancestral stretches of DNA in a person’s genome, so it is a certainty that each person has not inherited any DNA from the great majority of his or her actual ancestors.

These calculations mean that a person’s genealogy, as reconstructed from historical records, is not the same as his or her genetic inheritance… Yet even if the genealogies are accurate, Queen Elizabeth II of England almost certainly inherited no DNA from William of Normandy, who conquered England in 1066 and who is believed to be her ancestor twenty-four generations back in time.

So, when he talks about how most of the ancestors of a person are not really genetic ancestors, he justifies that at each generation, 71 new splices are generated on average. I got that those splices are the crossovers between mother's and father's chromosomes in a germ cell. So he calculates that a person's genome is derived from the 47 stretches of DNA (46 chromosomes + mtDNA), plus 71 by generation.

To be honest, I just didn't get the logic behind it. The 71 new splices are splices for specific chromosomes, or each splice is a different combination for each of the chromosomes present in the germ cells? And since the splices are combinations of the chromosomes, in any scenario, why would be guaranteed that after 10 generations (1024 ancestors and 757 splices), some ancestors don't contribute to a person's genome? I mean, I can get that the chance is reduced, but I don't get the guarantee, since the stretches are random and uneven, parts of the chromosome could still contain DNA from any ancestor (I think!).

So, to resume:

1. Considering N generation, why the calculation for where the genome derives would be 71*N + 47?
2. Considering the point 1 is clarified, why would that be a guarantee that some ancestors don't contribute to a person's genome?

Answer 1

Point #2 is a bit easier to understand (at least to me):

Forget about "stretches of DNA", and focus on individual base pairs. There are around 3 billion (3x10^9) base pairs in a human genome. Each base pair arrived from some ancestor. If we look 1,000 years back, around 40 generations ago (assuming a generation is 25 years), then you have 2^40 ~= 10^12 (1 trillion) ancestors. It's impossible that you inherit at least 1 base pair from each of 1 trillion ancestors, if you only have 3 billion base pairs.

About #1:

You can get an intuition for it by looking at the "extreme case". If you assume that you have only 1 chromosome pair, and there are no recombinations whatsoever, only 2 individuals contributed to your genome: the DNA of only 1 grandmother and 1 grandfather would be present in your genome.

If you allow 1 recombination per generation, then you can have some DNA of all of your grandparents, but you no matter how you recombine the chromosomes, DNA of at least 2 of your great-grandparents will not survive:

Hypothetical stretches of DNA of great-grandparents that end up in next generation. Each letter represents a "DNA fragment":
#1: aaaa
#2: bbbb
#3: cccc
#4: dddd
#5: eeee
#6: ffff
#7: gggg
#8: ffff

Grandparents:
#1+#2: aabb
#3+#4: ccdd
#5+#6: eeff
#7+#7: gghh

Parents:
#1+#2+#3+#4: acdd (we're losing one great-grandparent's DNA here, since only 1 recombination is allowed)
#5+#6+#7+#8: eefh

You:
acdd & eefh (but no traces of "b" and "g" left in your genome)

So by allowing 1 recombination per parent, we got 2 extra individuals possibly contributing to the genome with each generation:

• You: max 2 people (your parents)
• Including your parents (N = 1): 2 + 2 = 4
• Including your grandparents (N = 2): 2 + 2 * 2 = 6

And so on, using 2 + 2*N formula. It works the same way for more recombinations, and more chromosome pairs, but it's a bit more difficult to get the intuition for.

About the "guarantee" that some ancestors didn't contribute to your genome:

The math works out nicely, but of course no one had 1 trillion distinct ancestors 40 generations ago, because there's some intermarrying going on. To say with certainty that:

Yet even if the genealogies are accurate, Queen Elizabeth II of England almost certainly inherited no DNA from William of Normandy, who conquered England in 1066 and who is believed to be her ancestor twenty-four generations back in time.

we would need to know how often royal families intermarry.

If we use 1 recombination per generation, and 1 chromosome pair example, and assume that instead of having 4 distinct great-grandmothers, and 4 distinct great-grandfathers, one has in fact 4 distinct great-grandmothers, but only 2 distinct great-grandfathers, that's only 6 distinct ancestors at N=2. Even though 2 + 2*N < 2^(N+1), one could still inherit some DNA from all ancestors in such a case.