Statistical population genetics

Transcription

1 Statistical population genetics Lecture 7: Infinite alleles model Xavier Didelot Dept of Statistics, Univ of Oxford Slide 111 of 161

2 Infinite alleles model We now discuss the effect of mutations. Kimura and Crow (1964) proposed the following mutational model: Definition (The infinite alleles model). Each mutation creates a new allele. Slide 112 of 161

3 Infinite alleles model A A B C D C Slide 113 of 161

4 Infinite alleles model Data from the infinite alleles model can be represented as a vector a = (a 1,...,a n ) where a i is the number of alleles for which i copies exist in the sample of sizen. a is called the allelic partition of the data. n = n i=1 ia i and K n = n i=1 a i is the number of allele types For example, in the previous slide, we have n = 6,K n = 4 and a = (2,2,0,0,0,0). Slide 114 of 161

5 Number of alleles Theorem (Number of alleles). P(K n = k) = S(n,k) θ k n 1 i=0 (θ +i) wheres(n,k) is the Stirling number of the first kind. Slide 115 of 161

6 Number of alleles Proof. If the last event was a coalescent, then just before that we hadn 1 lineages and k distinct alleles. If the last event was a mutation, then the mutating lineage is a unique allele, and then 1 other lineages contained k 1 distinct alleles. It follows that: P(K n = k) = θ n 1+θ P(K n 1 = k 1)+ n 1 n 1+θ P(K n 1 = k) with initial condition P(K 1 = 1) = 1. Solving this recursive equation gives the result. Slide 116 of 161

7 Number of alleles Slide 117 of 161

8 Ewens sampling formula Theorem (Ewens sampling formula). The probability of an allelic partitionain a sample of size n is equal to: P n (a) = n! n 1 i=0 (θ +i) n j=1 ( ) aj θ 1 j a j! This formula is called Ewens sampling formula (ESF) because it was discovered by Ewens (1972). The ESF has since been found to have many applications, and is thus an important result in theoretical probability. Slide 118 of 161

9 Ewens sampling formula Proof. Let e i be the vector of size n filled with zeros except for a one at the i-th position. We decompose P n (a) according to whether the last event was a coalescence (C) or a mutation (M): P n (a) = P(a C)P(C)+P(a M)P(M) = n 1 n 1+θ P(a C)+ θ n 1+θ P(a M) If the last event was a mutation, then the mutating lineage has a unique allelic type and then 1 other lineages need to generate the rest of the profile, ie. a e 1 so that P(a M) = P n 1 (a e 1 ). If a 1 = 0 then this probability is of course equal to zero. Slide 119 of 161

10 Ewens sampling formula If the last event was a coalescence, we decompose P(a C) according to all the profiles of size n 1 that could be observed just before the coalescence: P(a C) = a P n 1 (a )P(a C,a ) The coalescence may have happened between any two genes that share the same allele ina. Let j denote the number of copies inaof the allele of the genes that coalesced. Given j, we have a = a e j +e j 1. Thus: P(a C) = n P n 1 (a e j +e j 1 )P(a C,a e j +e j 1 ) j=2 Slide 120 of 161

11 Ewens sampling formula The last term is the probability that a coalescence event happens to one of the (j 1)(a j 1 +1) genes for which there are a j 1 copies ina e j e j 1. Since there are n 1 genes ina e j e j 1, we have: Putting this altogether we get: P(a C,a e j +e j 1 ) = (j 1)(a j 1 +1) n 1 P n (a) = θ n 1+θ P n 1(a e 1 ) n + n 1 n 1+θ j=2 (j 1)(a j 1 +1) P n 1 (a e j +e j 1 ) n 1 with boundary condition P 1 (1) = 1 and P n (a) = 0 if any of thea j < 0. Solving this recursion equation leads to the ESF. Slide 121 of 161

12 Example For a sample of size n = 3, there are three possible allelic profiles: (3,0,0), (1, 1, 0) and (0, 0, 1) with respective probabilities: P 3 (3,0,0) = θ θ +2 P 2(2,0) = θ 2 (θ +1)(θ +2) P 3 (1,1,0) = θ θ +2 P 2(0,1)+ 2 θ +2 P 2(2,0) = θ 1 θ+2θ θ +2 3θ = (θ+1)(θ+2) P 3 (0,0,1) = 2 θ +2 P 2(0,1) = θ θ +1 2 (θ +1)(θ +2) Slide 122 of 161

13 Sufficiency of number of alleles Definition (Sufficiency of a statistic). A statistict(x) is sufficient for underlying parameter θ if the conditional distribution of the datax given the statistict(x) is independent of θ, ie: P(X T(X),θ) = P(X T(X)) Theorem (Sufficiency of the number of alleles). The number of alleles is a sufficient statistic for parameter θ. Slide 123 of 161

14 Sufficiency of number of alleles Proof. Since the number of alleles K n is completely determined by the allelic profilea, the distribution ofagiven K n reduces to: P(a K n = k,θ) = P n(a) P(K n = k) = n! S(n, k) n j=1 1 j a j aj! This distribution does not depend onθ, therefore K n is sufficient for parameter θ. Slide 124 of 161

15 Example Coyne (1976) studied the xanthine dehydrogenase gene (Xdh) of Drosophila persimilis by electrophoresis. This method reveals whether two genes are identical, but not how closely related they are. The infinite alleles model is therefore particularly well suited for the analysis of such data. They found K n = 23 alleles in a sample of n = 60 individuals with the following allelic profile: a 1 = 18, a 2 = 3,a 4 = 1, a 32 = 1 What is the maximum likelihood estimator ofθ based on this data? Slide 125 of 161

16 Example Since K n is sufficient for θ, we estimate θ based on K n only. The likelihood ofθ is: L(θ) = P(K 60 = 23 θ) = S(60,23) Taking the logarithm and deriving byθ gives: θ i=0 (θ +i) dl(θ) dθ This is equal to zero when: = θ i=0 1 θ +i 23 = 59 i=0 θ θ +i Solving gives a maximum likelihood estimator for θ of Slide 126 of 161

17 Summary In the infinite alleles model, each mutation creates a new allele The Ewens sampling formula gives the probability of a dataset occurring under this mutational model We derived an equation for the number of alleles The number of alleles is a sufficient statistic in this model, making it very useful to draw inference from genetic data The infinite alleles model is particularly well suited to Analise data from electrophoresis Slide 127 of 161