Genetic Inference

Over at the Genetic Genealogist, Blaine Bettinger has a Q&A post up about the difference between a genetic tree and a genealogical tree. The destinction is that your genealogical tree is the family tree of all your ancestors, but your genetic tree only contains those ancestors that actually left DNA to you. Just by chance, an individual may not leave any DNA to a distance descendant (like a great-great-great-grandchild), and as a result they would not appear on their descendant’s genetic tree, even though they are definitely their genealogical ancestor.

At the end of his post, Blaine asks a couple of questions that he would like to be able to answer in the future;

• At 10 generations, I have approximately 1024 ancestors (although I know there is some overlap). How many of these ancestors are part of my Genetic Tree? Is it a very small number? A surprisingly large number?
• What percentage, on average, of an individual’s genealogical tree at X generations is part of their genetic tree?

I think that I can answer those questions, or at least predict what the answers will be, using what we already know about sexual reproduction.

Simulating Sex with Recombination

We can give a simple answer by assuming that each chromosome is passed on intact, with a 50% chance of getting either one from a pair. This gives us a maximum of 44 genetic ancestors, and means that the probability of being related to any particular ancestor N generations ago is 1 - (1 - 0.5^{N - 1})²². We’d have about 43 genetic ancestors out of 1024 genealogical ancestors after 10 generations.

This is an underestimate; outside of the Y chromosome and the mitochondria, DNA is not passed down as whole chromosomes; recombination occurs, in which chromosomes come together and swap DNA. This mixes up DNA and stop is getting lost, and lets you have DNA from more than 44 genetic ancestors.

To answer Blaine’s two questions, we need to take recombination into account. To do this, I put together a computer simulation of recombination, using data from Chowdhury et al’s large population study of recombination; they found that recombination rates vary from person to person, and especially between genders, with significantly more recombination in women than in men (see this graph for the data I used). I simulation individuals with the 22 non-sex chromosomes, and each person had their own recombination rate, chosen at random, dependent on their gender. I simulated sexual reproduction with unrelated individuals, and checked whether their DNA was present in their descendants N generations in the future.

The results

Here are the results of the simulation:

This graph shows the probability that an ancestor of mine from N generations ago will pass DNA onto me, i.e. the probability that a genealogical ancestor is also a genetic ancestor. The black line is the simulation with recombination, the red line is our prediction without recombination; as we guessed, recombination increases your chance of being genetically related to your ancestors, though it still drops of pretty dramatically; after 10 generations, only about 12% of your genealogical tree is in your genetic tree.

The probability of having DNA from all of your genealogical ancestors at a particular generation becomes vanishingly small very rapidly; there is a 99.6% chance that you will have DNA from all of your 16 great-great grandparents, only a 54% of sharing DNA with all 32 of your G-G-G grandparents, and a 0.01% chance for your 64 G-G-G-G grandparents. You only have to go back 5 generations for genealogical relatives to start dropping off your DNA tree.

We also care about how many genetic ancestors we have after a certain number of generations, shown below:

The number of genetic ancestors starts off growing exponentially, but eventually flattens out to around 125 (at 10 generations, 120 of your 1024 genealogical ancestors are genetic ancestors).

As a final note, there is an interesting effect of the larger recombination rate in women; you are, on average, slightly more closely related to your maternal line (your maternal grandmother, your mother’s maternal grandmother, etc) than you are to you paternal line (your father’s paternal grandfather, etc). We can see the sharing probability for the maternal line and the paternal line show:

You are about 30% more likely to be genetically related to your maternal-line ancestor 10 generations ago than you are the corresponding paternal-line ancestor (14% vs 11%).

My not-very-well-documented R code for this simulation can be found here

Tags: genealogy, genetics

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