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Lecture 6 - Evolution in finite populations. II. Ne and the Neutral Theory

# Lecture 6 - Evolution in finite populations. II. Ne and the Neutral Theory

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1 Lecture 6 - Evolution in Finite Populations, continued III. Loss of genetic variation ("decay of heterozygosity") under genetic drift Imagine a random mating population of one self-fertile hermaphrodite, initially heterozygous at the A locus, so fr(A1) = fr(A2) = 1/2. What is the probability that the single surviving offspring is heterozygous? 1/2. What is the probability the grandchild is heterozygous (1/2)(1/2). How about the great grandchild (1/2) 3 . What about after t generations? (1/2) t Now, let us introduce the concept of "identity by descent". Two gene copies are "identical by descent" if they are descended from the same gene copy in a common ancestor. For example, if you and your brother got the same gene copy from your mother, those gene copies are identical by descent . See Figure In a population of N diploid individuals, what is the probability that two randomly drawn genes are identical by descent? 1/2N. We call the probability that two genes are identical by descent f . The way to think about this is imagine the diploid N individuals produce an effectively infinite number of gametes. Draw one gamete. The probability that any particular individual is the parent is 1/N. Now, what is the probability the gene copy in the gamete is the one the individual inherited from (say) its mother? 1/2. So, f = (1/N)(1/2) = 1/2N Let F = the probability that two gene copies are the same allele, call this "identical in state", i.e., both copies are A1 or both copies are A2. You can think of F as the probability of being homozygous, or "homozygosity". There are two ways that two gene copies can be identical in state. They can be I by D, in which case they have to be I in S (barring mutation, which we will bar), with probability 1/2N, or they can be not I by D, with probability 1 - 1/2N, and identical in state, with probability F. So, the probability that two gene copies are identical in state after one generation of random mating is: F' = 1/2N + (1 - 1/2N)F Now, define heterozygosity H = 1 - F (i.e., the probability that two randomly drawn gene copies are different in state, i.e., A1 and A2) So, H' = 1 - F'

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2 Some algebra yields H' = (1 - 1/2N)H, [Convince yourself of this, substitute 1-H for F in the equation for F'] H t = (1-1/2N) t H 0 so Δ N H = H' - H = (-1/2N)H = - f H The upshot is that heterozygosity decays at rate f (= 1/2N) per generation. Thus, the smaller the population, the faster genetic variation is lost due to drift. SUMMARY: Recall the main points about evolution in finite populations from last time 1) Random Genetic Drift (i.e., random sampling of gene copies) reduces genetic variation within populations. After a long enough time, one allele will be fixed (frequency = 1) and all other alleles will be lost (frequency = 0).
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