Inbreeding depression in corn. Inbreeding. Inbreeding depression in humans. Genotype frequencies without random mating. Example.

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1 nbreeding depression in corn nbreeding Alan R Rogers Two plants on left are from inbred homozygous strains Next: the F offspring of these strains Then offspring (F2 ) of two F s Then F3 And so on November 0, 207 (Jones 924) 2 / 30 / 30 nbreeding depression in humans Genotype frequencies without random mating Describes any bi-allelic locus Genotype A A A A2 A2 A2 Offspring of cousin marriages are less likely to survive (Bittles and Neel 994) Frequency p 2 + pqf 2pq( F ) q 2 + pqf F = 0 under random mating Reduces to Hardy-Weinberg F > 0 under inbreeding Gives excess of homozygotes F is the coefficient of inbreeding 3 / 30 Example Outline of theory Assume p = /2 Genotype A A A A2 A2 A2 4 / 30 Frequency F = 0 F = nbred population has more homozygotes Suffers if either Heterozygotes tend to have high fitness Deleterious alleles tend to be recessive nbreeding increases F, which increases homozygosity, which decreases fitness 5 / 30 6 / 30

2 Decay of heterozygosity under selfing Decay of heterozygosity under selfing Gen A A A A 2 A 2 A 2 0 N Half of heterozygosity is lost each generation 7 / 30 8 / 30 What about cousin mating, or mating between sibs? Kinds of gene identity t is extremely difficult to work this out, using the method we just used Between 903 and 95, no one could get it right Pearl (93): Only for brother-sister matings does inbreeding reduce heterozygosity [Wrong!] Solution: build theory looking backwards in time, not forwards There are two senses in which a pair of gene copies may be identical: identity in state : copies of same allele identity by descent : copies of same gene copy in an ancestor Abbreviation: BD = dentity by Descent 9 / 30 0 / 30 Gametes a and b are identical by descent dentical in state, not by descent May be BD relative to an earlier generation a b a b Not BD relative to the pedigree shown here BD is always relative to a particular generation / 30 2 / 30

3 dentical neither in state nor by descent Uniting gametes Consider the two gametes that unite to form an individual F, is the probability that they are BD What is the probability that they both carry A? a b Event Probability BD from A -bearing ancestor Fp Descend from two random ancestors who both carry A ( F )p 2 P = Fp + ( F )p 2 = p 2 + pqf 3 / 30 4 / 30 All three genotypes For blackboard Genotype Frequency A A p 2 + pqf A A 2 2pq( F ) A 2 A 2 q 2 + pqf Same formulas as before F is no longer arbitrary F is probability that uniting gametes are BD Calculating F from pedigrees 5 / 30 6 / 30 Darwin-Wedgewood Genealogy Josiah Wedgewood Sarah Wedgewood Susannah Wedgewood Josiah Wedgewood Charles Darwin Emma Wedgewood George Darwin 7 / 30 8 / 30

4 A complex pedigree Mating between full siblings A B C E D F A C B D E G 9 / / 30 nbreeding and drift nbreeding and Genetic Drift Alan R Rogers Even under random mating, there is inbreeding in any finite population This random inbreeding is the same thing as genetic drift November 0, / / 30 Number of ancestors generation year ancestors , , , , ,384 Number of ancestors: generation year ancestors , , , , , ,048, ,097, ,94, ,388, ,777, ,554, ,08, ,27, ,435, / / 30

5 Number of ancestors: generation Drift and inbreeding year ancestors 536,870,92,073,74,824 2,47,483,648 4,294,967,296 Drift After t generations of genetic drift, the expected heterozygosity is E [H(t) p0 ] = 2p0 q0 ( /2N)t nbreeding f Ft is the average inbreeding coefficient in generation t, relative to generation 0, f you were born in 994, then you had over 4 billion ancestors in 066 E [H(t) p0 ] = 2p0 q0 ( Ft ) But there were not that many people on the planet Equating these expressions gives Many of your ancestors in 066 were the same people we are all inbred Ft = ( /2N)t nbreeding is genetic drift Let us build a model of this inbreeding 25 / / 30 Genotype frequencies and fitnesses Genotype A A A A2 A2 A2 Fitness and nbreeding Frequency Fp + ( F )p 2 2pq( F ) Fq + ( F )q 2 Fitness hs s Mean fitness: Alan R Rogers w = 2pq( F )hs [Fq + ( F )q 2 ]s = a bf November 0, 207 linear func of F where a = 2pqsh + q 2 s b = 2pqs(/2 h) 27 / 30 Why is inbreeding harmful? 28 / 30 What inbreeding depression tells us We have just seen that mean fitness is nbreeding depression is widespread in Nature w = a bf where mplies that deleterious alleles tend to be partially recessive b = 2pqs(/2 h) Fitness decreases with inbreeding if b > 0, which is true if s > 0 and h < /2, or in other words, if deleterious alleles are at least partially recessive (Jones 924) 29 / / 30

6 Morton and Crow: how large is the effect? Model S = Pr[survival] L = a i b i F i= i= e a i b i F e A BF where A = a i and B = b i Estimates  = 062 ˆB = 734 Example For mating between full sibs, F = /4, and S = exp{ /4} = 085 So we expect 5% mortality in the offspring of full-sib matings 3 / 30