How many genetic ancestors do you have?

Short answer: you get DNA from about 2ⁿ ancestors n generations back, up to 8 generations, then about 68 additional ancestors for each generation after that.

• Background
• Tables, number of ancestors
• Tables, number ancestors you got 1/2ⁿ of your DNA from
• Miscellaneous observations

Other people have done this same analysis, see here and here. They got similar but slightly different results than I did.

The number of genealogical ancestors you have n generations is 2ⁿ: 2 parents, 4 grandparents, 8 great-grandparents, and so forth. The only way to have fewer is if some of them are the same person. (For example, I have two great-great-great-great grandparents who are also my great-great-great-great-great grandparents by another line.)

The difference between genealogical ancestors and genetic ancestors is that genetic ancestors are the ones that you actually got some DNA from. They're a subset of your genealogical ancestors. Humans have about 3 billion base-pairs of DNA, so that limits the number of genetic ancestors to about 3 billion no matter how far back you go. There are also around 46,000 hotspots (reference Genetic Crossovers Are Predicted Accurately by the Computed Human Recombination Map, figure 6). Hotspots are the places where crossovers usually happen. Each of the 46,000 segments bounded by neighboring hotspots usually has a single line of descent, so you're limited to about 46,000 ancestors.

But I'm interested in something that happens much sooner than that: after about eight generations back, the number of genetic ancestors only increases linearly with the number of generations, while the number of genealogical ancestors keeps increasing exponentially. Once you go back 20 generations, you have only 1300 or so genetic ancestors despite having over a million genealogical ancestors.

The exponential growth up to 8 generations is due to having many independent chromosome pairs. Humans have 23 pairs of chromosomes, and which side of the pair you inherit from is a random choice for each of the chromosomes. You start out with both sides of the pair yourself. For your parents, you got one side of each of the 23 pairs from each parent. Grandparents, that's about one side of 23/2=12 pairs. Great-grandparents, one side of 6 pairs each. Great-great-grandparents, one side of 3 pairs. Eventually you're down to inheriting DNA on just one side of one pair from each ancestor. At that point the exponential growth of ancestors stops.

The linear growth after that is a bit more complicated. The linear growth is due to having a limited number of crossovers per chromosome per generation. Each generation, you don't just get one side of one of your parent's chromosomes. Instead, the parent mixes their pair before splitting it into two. The process is called meiosis. The points where the pairs cross are called crossovers. There's around one or two crossovers per chromosome pair per generation.

Being more precise about crossovers (reference "Genetic Analysis of Variation in Human Meiotic Recombination"). The 23rd chromosome is special, it's the XY chromosome. If you're a male you have XY, if you're female you have XX. The male does not mix the XY, daughters get the X directly and sons get the Y directly. If you're female you have XX, and XX has about as many crossovers as any other chromosome. Other than that, there's always at least one crossover per chromosome per generation (otherwise meiosis doesn't work properly), and sometimes more than one. How many more is proportional to the length of the chromosome (reference Broad-Scale Recombination Patterns Underlying Proper Disjunction in Humans, figure 5). If you're male, you have an average of 26.4 crossovers total when producing sperm, so spread among 22 chromosomes, that's a little over 1 crossover per chromosome. If you're female, it's more like an average of 41.1 crossovers when producing eggs, so given 23 chromosomes total, that's a little under 2 crossovers per chromosome. Each generation each chromosome does these crossovers, and also decides randomly which side of each chromosome pair to give you.

If you're male, you got your Y chromosome from exactly 1 ancestor n generations back. The X chromosome (men have half a pair, women have a whole pair) is also unusual: you're most likely to get it from your mother's father's mother's father's mother etc, but it might have come from your mother's mother's etc. Reference That Unruly X ... Chromosome That Is.

Tables: number of genetic ancestors N generations back

So ... what's that add up to? How many genetic ancestors do you get n generations back? I simulated it, C++ code here, doing the crossovers at random spots, handling XY correctly, and the number of crossovers per generation for males and females correctly too. Other than XY, I treated all chromosomes as the same: same length, same chance of crossover. Instead of starting with 2ⁿ ancestors and trying to derive a person, I started with a person and worked backwards. Any ancestors contributing no DNA can be ignored, because their ancestors will also contribute no DNA, so the time and space complexity of the simulation is linear with the number of genetic ancestors for however many generations back you want to go.

Here's a table of the results, starting with a female:

And again, starting with a male:

One practical upshot of this is that those ancestors you do inherit something from, you inherit a lot from. And large stretches of DNA are passed down from generation to generation, those stretches usually aren't finely divided.

How closely related are cousins? It's pretty much a matter of the number of meioses again, except you go up to a pair of ancestors then down again. Siblings shared half their DNA due to inheriting from both parents. The same chart that shows how much DNA your ancestors contirbuted can show how much DNA cousins will shared, just about. There is the difference that the messiness around the common parents adds about one extra generation of splitting without adding one extra generation of diverging.

How many cousins do you have that share 2^-X of your DNA? That cannot be answered by these charts directly: it is strongly influenced by how many children your ancestors had. If someone shared 2^-X of your DNA, what type of cousin are they probably? Again can't tell, it depends on how many children your ancestors had. If they all had 2 children each, say for 2^-9, you take the mean generation of the 9:??? values and that would tell you (somewhere between 2nd cousin and 100th cousin, with 7th cousin once removed most likely).

How many ancestors X generations back gave you closest to 2^-Y of your DNA from (starting from a female):

For males, the constant 6:1 is from one ancestor contributing 1/46th of the DNA, namely the Y chromosome along the direct paternal line. There are a few strongly related individuals for females too, due to X having crossovers about every other generation.

How many ancestors X generations gave closest to 2^-Y of your DNA from (starting from a male):

I used 23andMe's genetic testing for me and my family, and found a cousin I share 0.31% of my DNA with. However, my son shares 0.48% of his DNA with that same cousin. How can that be? Does that mean my wife and I are closely related? Probably not. More likely, I'm a distant cousin on his mother's side, and my wife is a distant cousin on his father's side. That doesn't imply that my wife and I are related at all, yet it still allows our son to have more DNA in common with him than either of us has.

The small number of genetic ancestors implies it is much easier to not be inbred according to genetic ancestors than by genealogical ancestors. You had a trillion genealogical ancestors 40 generations back, which far exceeds the total number of people who have ever existed, so most of your ancestors appear many many times in your genealogical tree. But you had under 3000 genetic ancestors 40 generations back (and under 7000 at 100 generations back). It's quite feasible for all your genetic ancestors to have been distinct for as far back as it's practical to go.

I wonder what the mechanical process is for meiosis. It sounds you get two pairs, then you exchange the ends of one side of each of those pairs to force crossover. Maybe the other ends are twisted around, or moved independently in a fixed manner. I don't know if both ends are twisted or just one end, but the middle doesn't seem to be moved in a controlled manner.

Bob Predicts the Future

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