Population Genetics I. Bio
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1 Population Genetics I. Bio Don Conrad
2 Why study population genetics? Functional Inference Demographic inference: History of mankind is written in our DNA. We can learn about our species population size changes, migrations, etc. Complex disease: What approaches for analysis make sense? Molecular biology: Measure rates of biological processes like mutation and recombination, learn about gene regulation, speciation Sequencing era. Framework for understanding these sequences. You will have your own genome sequence
3 Outline for Part I and Part II Theory Hardy-Weinberg Forward Models: Wright Fisher Model Backward Models: Coalescent Data Mutation, mutation rates Global diversity, serial bottleneck model Recombination, LD blocks, recombination hotspots Natural Selection
4 Hardy-Weinberg What is the fate of a neutral genetic variant at a biallelic locus in an infinite population? Udney Yule: individuals with dominant traits will increase in the population over time Hardy: Yule is wrong, and that expected genotype frequencies are simply the product of underlying allele frequencies assuming independence
5 Hardy s Argument: Generation 1 Males are 100% AA Females are 100% aa What will be the genotype composition of generation 2? Males A (100%) a (0%) Females A (0%) AA (0%) Aa (0%) a (100%) Aa (100%) aa (0%)
6 Hardy s Argument: Generation 2 Now males are 100% Aa and females are 100% Aa What will Gen 3 look like? Males A (50%) a (50%) Females A (50%) AA (25%) Aa (25%) a (50%) Aa (25%) aa (25%) Freq of a = p = ( 2*25+1*50 ) / 200 = 0.5 Freq of A = q = 1-p = 0.5 Allele frequencies are the same in Gen 3 as in Gen 2 So we have reached an equilibrium
7 Hardy s Argument: Generation 3 Males A (50%) a (50%) Females A (50%) AA (25%) Aa (25%) a (50%) Aa (25%) aa (25%) p = ( 2*25+1*50 ) / 200 = 0.5 q = 1-p = 0.5 The famous equation describes genotype frequencies as a function of allele frequencies at equilibrium: p 2 + 2pq + q 2 = 1
8 Gcbias.org
9 Modern Synthesis Reconciliation of Mendelian genetics with observations of the Biometrists Reconciliation of Mendelian genetics with Darwinian evolution R.A. Fisher Sewall Wright J.B.S Haldane
10 Wright-Fisher Model Assumptions: Two allele system N diploid individuals in each generation 2N gametes Random mating, no selection Discrete generations Generation t A a Gamete pool A a a A A A a a A a t + 1
11 Let s play a round of this game
12 The game is faster by computer I = 400 A = 200 R = 100 G = 100 I = Number of Generations A = Population size (gametes) G = Count of the G allele R = Count of the R allele
13 I = 400 A = 200 R = 100 G = 100
14 I = 400 A = 200 R = 100 G = 100
15 Let s investigate this phenomenon Change Population Size Change allele frequencies
16 I = 40 A = 20 R = 10 G = 10
17 I = 40 A = 20 R = 10 G = 10
18 I = 1000 A = 2000 R = 1000 G = 1000
19 I = 1000 A = 2000 R = 1000 G = 1000
20 I = 400 A = 200 R = 150 G = 50
21 I = 400 A = 200 R = 150 G = 50
22 I = 400 A = 200 R = 150 G = 50
23 Each generation, the new population is made by sampling with replacement from the previous generation A a a A A A a a A a AA Aa aa Let: P t = freq (A) among gametes P t+1 =. In the next generation n t+1 = count of (A).. Aa Then: n t+1 ~ Binomial (P t, 2N) Pr( n t+1 = m)! = # 2 N " m $ & % p 1 pt tm ( ) 2 N m E( p t+1 ) = P t Var( p t+1 ) = p t (1-p t ) 2N Implications: sampling variance ( genetic drift ) is dependent on population size. Allele frequency is a random sequence of numbers: p 1, p 2, p 3, Eventually p = 1 or p = 0. Stay fixed until new mutation.
24 Thinking like a (classical) population geneticist Simulations versus analysis Classical population geneticists tried to prove features of the Mendelian system Lets explore this hypothesis: One allele must always win.
25 The Decay of Heterozygosity Define G t, the homozygosity at generation t. = probability of randomly picking two chromosomes from population and they are the same allele Then the heterozygosity H t = 1- G t. What happens to G t over time, under the assumptions of Wright-Fisher?
26 What is G 0 A A B BB B B B 1. Pick A then A = number of A s / 2N * number A s-1 / (2N-1) Generation 0 2. Pick B then B = number of B s / 2N) * (number B s-1) / (2N-1)
27 What is G 1? Probability = 1/2N Generation 0 Generation 1 Probability (1-1/2N)*G 0 Generation 0 Generation 1
28 Proof of decay of heterozygosity
29 What is the half life of H? H 0 /2 = H 0 (1-1/2N) t t = 2Nln2 N = 10^4, t = 1.1e5 generations
30 What does this mean? In a large population, eventually, every allele will have descended from a single allele in the founding population! All but 1 allele will have died off.
31 -Genealogical Analysis of all 131K Icelanders born after 1972
32 Drift versus Darwin How can we add selection to our game? We need to account for dominant and recessive alleles!
33 The Wright Fisher Game v0.2 Define relative fitness for each possible individual Fitness RR = 1 Fitness RG = 1.1 Fitness GG = 2 Modify rules. Pick an individual with probability proportional to the fitness of her genotype. A given GG individual is twice as likely to be picked. Now choose one chromosome and put into the next generation.
34 What relative fitness should we select? Conserved elements <0.01% increase in fitness
35 Drift versus Darwin I = 100 A = 100 R = 99 G = 1 fg = 2*fR
36 I = 100 A = 100 R = 99 G = 1 fg = 3*fR
37 I = 100 A = 100 R = 99 G = 1 fg = 3*fR
38 I = 100 A = 100 R = 99 G = 1 fg = 3*fR
39 I = 100 A = 2000 R = 1999 G = 1 fg = 3*fR
40 Some startling results! Survival of the fittest luckiest. Sometimes drift can overcome selection. Depends on allele frequency, population size. Most new advantageous mutations are not fixed!
41 Mutation Infinite alleles model Assumptions
42 I = 5000 U = Start as Homozygous At allele A U=mutation rate
43 Summary thus far Chance can play a large role in determining which polymorphisms are fixed in a population. These findings are/were not obvious. Amount of a variation at a locus, and the fate of individual alleles, is the product of Mutation-selection-drift balance
44 Recursion equations Genotype Total AA Aa aa Freq in generation t q 2 2pq p 2 1 = q 2 + 2pq + p 2 Fitness w 11 w 12 w 22 Freq (after selection) q 2 w 11 2pqw 12 p 2 w 22 ŵ = q 2 w pqw 12 +p 2 w 22 p t+1 = p 2 w 22 +pqw 12 ŵ q t+1 = q 2 w 11 +pqw 12 ŵ Recursion equations for analysis of selection Assumptions in this example: no drift or mutation, discrete generations, random mating
45 Evolutionary dynamics in a simplex for a biallelic locus aa AA Analysis of recursion equations, and later, differential equations, was the central tool for classical popgen Aa 2010 by National Academy of Sciences Modified from Gokhale C S, Traulsen A PNAS 2010;107:
46 Dynamics:Topics covered Selection (additive, balancing, frequency-dependent) Altruism, kin selection Structural variation (inversions) Multiple loci (recombination, epistatic selection) Population structure (island model, stepping stone model, isolation by distance, metapopulation models) Assortative mating Sex-specific effects (migration, selection) Variable environments, etc
47 Sampling with Replacement Past Some alleles pass on no copies to the next generation, while some pass on more than one. Present
48 The Coalescent Process ACTT T G C G ACGT ACGT ACTT ACTT AGTT Backward in time process Discovered by JFC Kingman, F. Tajima, R. R. Hudson c DNA sequence diversity is shaped by genealogical history Genealogies are unobserved but can be estimated Conceptual framework for population genetic inference: mutation, recombination, demographic history
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