The Ancestry of Genetic Segments

Table 1

Bounds on identity probabilities for a genetic segment.

𝑟𝑁	2𝑁	MRCA		Asymptotic ancestor		Asymptotic pool size
		Lower	Upper	Lower	Upper	Lower	Upper	1.28𝑅/ln(1+𝑅)
.001	200	.98	1.000	.996	1.00	1.00	1.004	1.28
.01	2000	.77	.998	.96	.98	1.02	1.04	1.29
.1	2×10^4	.03	.977	.6	.83	1.19	1.4	1.40
1	2×10^5	<10^−19	.795		.20	2.32	5	2.33
10	2×10^6	<10^−234	.100		.00018	6.59	41	8.41
100	2×10^7	“0”	1.048×10^−10		10^−15	20.25	401	48.27

The diploid population size is 𝑁, and 𝑟 is the probability of recombination within a segment. The value 𝑟=10⁻⁵ is used for the columns which bound the probability that the MRCA of a base pair is the MRCA of the entire segment. The bounds on the probability that an asymptotic ancestor of a base pair is an asymptotic ancestor of the entire segment and the asymptotic expected size of the ancestral pool of a segment are functions of 𝑟𝑁. The last column is the estimate from Wiuf and Hein [8] which was obtained for a limited range of parameter values (𝑅=2𝑟𝑁).