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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 • Sequence era. Framework for understanding these sequences.• You will have your own genome sequence
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, hotspots PRDM9 – Natural Selection
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
A (100%) a (0%)
A (0%) AA (0%) Aa (0%)
a (100%) Aa (100%)
aa (0%)
Hardy’s Argument: Generation 1
Males
Females
A (50%) a (50%)
A (50%) AA (25%) Aa (25%)
a (50%) Aa (25%)
aa (25%)
p2 + 2pq + q2 = 1
Hardy’s Argument: Generation 2
Males
Females
p = ( 2*25+1*50 ) / 200 = 0.5q = 1-p = 0.5
Gcbias.org
Modern Synthesis • Reconciliation of Mendelian genetics with
observations of the Biometrists • Reconciliation of Mendelian genetics with Darwinian
evolution
R.A. Fisher Sewell Wright J.B.S Haldane
Wright-Fisher Model Assumptions: • Two allele system • N diploid individuals in each generation • 2N gametes • Random mating, no selection • Discrete generations
Aa
Generationt
t + 1
A a
a A
a
A
A A
a a Gamete pool
Let’s play a round of this game
The game is faster by computer
I = 400 A = 200 R = 100 G = 100
I = Number of GenerationsA = Population size (gametes)�G = Count of the G alleleR = Count of the R allele
I = 400 A = 200 R = 100 G = 100
I = 400 A = 200 R = 100 G = 100
Let’s investigate this phenomenon
• Change Population Size
• Change allele frequencies
I = 40 A = 20 R = 10 G = 10
I = 40 A = 20 R = 10 G = 10
I = 1000 A = 2000 R = 1000 G = 1000
I = 1000 A = 2000 R = 1000 G = 1000
I = 400 A = 200 R = 150 G = 50
I = 400 A = 200 R = 150 G = 50
I = 400 A = 200 R = 150 G = 50
Can we deduce general rules?
• Larger population size = alleles stick around longer. Less susceptibility to “random walk”
• Probability of winning seems related to initial frequencies. At 50/50 50% chance of either allele winning. Hypothesize: probability of winning is proportional to initial frequency.
• Hypothesis: One allele must always win.
• 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
Aa
Let: Pt = freq (A) among gametesPt+1 = …. In the next generationnt+1 = count of (A) …..
Then: nt+1 ~ Binomial (Pt, 2N)
Pr( nt+1 = m) E( pt+1) = Pt Var( pt+1) = pt (1-pt)
2N
= 2Nm
!
"##
$
%&& pt
m1−pt( )2N−m
Implications: sampling variance (“genetic drift”) is dependent on population size. Allele frequency is a random sequence of numbers: p1, p2, p3,… Eventually p = 1 or p = 0. Stay “fixed”until new mutation.
An important concept: Drift
• Drift – stochastic fluctuations in allele frequency due to random sampling in a finite population.
Drift versus Darwin
• How can we add selection to our game?
• We need to account for dominant and recessive alleles!
The Wright Fisher Game v0.2 • Define relative fitness for each possible
individual Fitness RR = 1 Fitness RB = 1.1 Fitness BB = 2 Modify rules. Pick an individual with probability
proprotional to the fitness of her genotype. A given BB individual is twice as likely to be picked. Now choose one chromosome and put into the next generation.
What relative fitness should we select?
• Conserved elements <0.01% increase
in fitness
Drift versus Darwin
I = 100 A = 100 R = 99 G = 1 fG = 2*fR
I = 100 A = 100 R = 99 G = 1 fG = 3*fR
I = 100 A = 100 R = 99 G = 1 fG = 3*fR
I = 100 A = 100 R = 99 G = 1 fG = 3*fR
I = 100 A = 2000 R = 1999 G = 1 fG = 3*fR
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!
Mutation
• Infinite alleles model – Assumptions
I = 5000 U = 0.0001 Start as Homozygous At allele A
U=mutation rate
Summary thus far • Chance can play a large role in determining which
polymorphisms are fixed in a population. • The fittest don’t always survive. • These findings are/were not obvious. • They become (more) obvious with quantitative
investigation.
• And we’ve only scratched the surface.
Further explorations of this model
• To date our approach has been based on observations of simulations. But the model is simple – analytic approach may prove fruitful.
• Our hypotheses: – Can we prove them?
– Can we quantify them?
• Lets explore this hypothesis: One allele must always win.
The Decay of Heterozygosity
• Define Gt, the homozygosity at generation t.
= probability of picking two genomes from population and they are the same allele
• Then the heterozygosity Ht = 1- Gt .
What is G0 R
B R B B B
B B
Generation 0
1. Pick R then R
= number of R’s / 2N * number Rs-1 / (2N-1) 2. Pick B then B
= number of B’s / 2N) * (number B’s-1) / (2N-1)
What is G1?
Probability = 1/2N
Probability (1-1/2N)*G0
Generation 0 Generation 1
Generation 0 Generation 1
Proof of decay of heterozygosity
What is the half life of H?
• H0 /2 = H0(1-1/2N)t
• t = 2Nln2
• N = 10^4, t = 1.1e5 generations
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”.
• Drift-Mutation-Selection balance.
-Genealogical Analysis of all 131K Icelanders born after 1972
Analysis of selection
Genotype Total AA Aa aa
Freq in generation t q2 2pq p2 1 = q2 + 2pq + p2 Fitness w11 w12 w22 Freq (after selection) q2w11 2pqw12 p2w22 ŵ = q2w11 + 2pqw12+p2w22
pt+1 = p2w22 +pqw12ŵ
qt+1 = q2w11 +pqw12ŵ
“Recursion equations”
Assumptions in this example: no drift or mutation, discrete generations, random mating
Evolutionary dynamics in a simplex for a biallelic locus
Modified from Gokhale C S , Traulsen A PNAS 2010;107:5500-5504 ©2010 by National Academy of Sciences
AA
Aa
aa
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…
Sampling with Replacement
• Some alleles pass on no copies to the next generation, while some pass on more than one.
Present
Past
The Coalescent Process • “Backward in time process” • Discovered by JFC Kingman,
F. Tajima, R. R. Hudson c. 1980
• 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
ACTT
ACGT ACGT ACTT ACTT AGTT
T
G
C G
