Population Genetics 7: Genetic Drift

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Daniella Walters
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1 Population Genetics 7: Genetic Drift Sampling error Assume a fair coin with p = ½: If you sample many times the most likely single outcome = ½ heads. The overall most likely outcome ½ heads n P = 2 k ( ) k ( ) n / 2 / k n = n! k k!( n k )! Combinations Formula: n is the number of flips k is the number of successes k heads from n flips Probability k =5 from n =.246 k =6 from n =.25

2 Sampling error The long term average value for p H is.5; let s call that E(p H ). How do we improve our changes of getting something close E(p H )? N flips p H <.35 p H = p H = p H = p H <.65 variance If we flipped the coin times: we get very close to E(p H ) in a single try, but not exactly. A note about HWE. N = 2 flips Probability of 5:5 heads : tails =.2256 Probability of 5:5 heads : tails =.796 N = flips 2

3 Genetic drift Consider a diploid population: Ideal population: no sampling errors because infinite population size Natural population: finite size and finite sample of gametes [sampling errors] Example: Let s assume: A = p =.75; a = q =.25; N = 5 This generation: 2 individuals reproduce [4 gametes] This is a binomial sampling problem: The probability of getting p =.75 and q =.25 in next generation is: 4 P = 3 (.75) 3 (. 25) P =.46 Genetic drift Generation Generation Generation 2 Generation 3 Draw 4:6 Draw 7:3 Draw 8:2 Restock Restock Restock white =.5 white =.4 white =.7 white =.8 Genetic drift is the accumulation of random sampling fluctuations in allele frequencies over generations. 3

4 Genetic drift The magnitude of change in allele frequencies is inversely proportional to the sample size: N Ideal population with finite size and finite gamete sample per generation. See last slide for example Remember that natural populations are less than ideal in many more ways! In most natural populations the effective size (N e ) will be less than the census size. The magnitude of drift in natural populations is: N e Drift and inbreeding effects are not independent! Genetic drift N e = N e = N e = N e = 5 4

5 Genetic drift If we run this simulation long enough it will go to fixation or loss; it just takes much longer rate to fixation [under drift] slows with increasing in N e ultimate fate is fixation or loss ( if f(a) =.5, P(fixed) =.5 ) Genetic drift What is the fate (on average) of a new mutant? The probability of fixation of a new mutant is its frequency (p or q) in the population: N e This is al low as it gets. The fate of most new mutations is LOSS due to drift. W AA =.5; W Aa =.5; W aa = : ideal population: probability of fixation = population with N e = 5: probability of fixation ~.25 Probability of fixation actually declines as N e decreases! 5

.8.6.4.2 3 5 7 9 3 5 7 9 2 23 25.8.6.4.2.8.6.4.2 3 5 7 9 3 5 7 9 2 23 25 3 5 7 9 3 5 7 9 2 23 25.8.6.4.2.8.6.4.2 3 5 7 9 3 5 7 9 2 23 25 3 5 7 9 3 5 7 9 2 23 25.9.8.7.6.5.4.3.2..8.6.4.2 3 5 7 9 3 5 7 9 2 23 25 3 5 7 9 3 5 7 9 2 23 25.8.6.4.2.8.6.4.2 3 5 7 9 3 5 7 9 2 23 25 3 5 7 9 3 5 7 9 2 23 25 Genetic drift Independent populations; each started with p = q =.

6 Genetic drift Independent populations; each started with p = q = * Allele frequency * = fixation N e = 5; generations = 5 * Generation Changes in allele frequency due to drift are unpredictable! Note if we ran more generations, more popns would go to fixation Genetic drift number of populations allele frequency allele frequency initial distribution; t = generations distribution after t = 5 generations 6

7 Genetic drift The effects of drift are cumulative over time. The effects of drift are predictable as averaged over time and populations:. loss of variation within populations 2. gain in variation between populations Does genetic drift affects heterozygosity? Let x be the amount of change in p and q in a population due to drift. As we have seen the long term average, E(x), due to drift will be zero because changes in p and q are equally likely to be positive or negative. Given E(x) =, what happens to heterozygosity? Does heterozygosity change at all? Let s start with HW at generation t: H t = 2pq The allele frequencies, p and q, will change from generation to generation by the amount x: H t+ = 2(p + x)(q x) H t+ = 2pq + 2x(q p) 2x 2 Although E(x) =, the expected value of x-squared, E(x 2 ), is always positive. E(2pq + 2x(q p) 2x 2 ) 2pq 2x 2 Heterozygosity is expected reduced by genetic drift. Nice, eh? 7

8 Genetic drift and inbreeding are not independent. Unequal numbers in successive generations N = g N N N e N g (approx.) 2. Different numbers of males and females N e = 4N m + 4N f (approx.) 3. Variance in reproductive success (other than male verse female) N ( v) e 4N 2 = V + 2 k Bottlenecks and founder effects Bottleneck: is a single, extraordinarily large, reduction in population size. Change in allele frequencies, as compared with pre-bottleneck population 2. Reduction in diversity 8

population census size 2,, 8, 6, 4, 2, Population crash Population recovered to historical high Ave N Ne 2 3 4 5 6 7 8 9 2 3 4 5

9 Bottlenecks and founder effects Effective population size is dominated by historical lows and can be very much lower than current census size. population census size 2,, 8, 6, 4, 2, Population crash Population recovered to historical high Ave N Ne Time = Ne g N N N N g (approx.) Two species that have suffered extreme bottlenecks due to commercial harvesting Northern elephant seal Northern right whale Excellent population recovery Poor population recovery 9

Ellis-van Creveld syndrome Other symptoms of this disease include dwarfisms,

10 Parental population Dispersal event to a neighbouring island New population Island Island 2 Polydactyly caused by the homozygous recessive disease Ellis-van Creveld syndrome Other symptoms of this disease include dwarfisms, abnormalities of the nails and teeth, and a hole between the two upper chambers of the heart.

Picture wing Drosophila Direction of colonization Direction of archipelago growth Keynotes: Genetic drift influences both allele frequency and genotype frequency.

In specific cases the outcome of genetic drift is unpredictable. The effects of drift are predictable as an average over populations.

11 Picture wing Drosophila Direction of colonization Direction of archipelago growth Keynotes: Genetic drift influences both allele frequency and genotype frequency. Drift decreases diversity within populations and increases diversity between populations. Under genetic drift, the rate to fixation is determined by Ne and the probability of fixation by p. In specific cases the outcome of genetic drift is unpredictable. The effects of drift are predictable as an average over populations. Because drift reduces genetic variation in populations, a population s ability to evolve in response to new selective pressures might be reduced (remember Trudy MacKay s experiments). Alternatively, some believe that drift could actually increase the rate of speciation (e.g., Hawaiian Drosophila). Because the effect of drift is inversely proportional to the effective population size, its affects are particularly important in rare and endangered species. Founder effects may play an important role in some speciation events

Lecture 14 Chapter 11 Biology 5865 Conservation Biology. Problems of Small Populations Population Viability Analysis

Lecture 14 Chapter 11 Biology 5865 Conservation Biology Problems of Small Populations Population Viability Analysis Minimum Viable Population (MVP) Schaffer (1981) MVP- A minimum viable population for