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Population Genetics using Trees
Peter BeerliGenome SciencesUniversity of WashingtonSeattle WA
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Outline
1. Introduction to the basic coalescent
Population modelsThe coalescentLikelihood estimation of parameters of interestWhy do we need Markov chain Monte Carlo
2. Extensions and examples
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Population genetics can help us to find answers
the PCR revolution allows us to generate lots of data from manyindividuals and many loci
We are still interested in questions like
– Where are we or other species coming from?– How big are populations?– Are these populations species?– What is the recombination rate in species x?
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Population genetics in the age of genomics
Why do we need theoretical population genetics when we can have thecomplete sequences of our favorite organism?
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Basics: Wright-Fisher population model
All individuals release many gametes and new individuals for the nextgeneration are formed randomly from these.
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Basics: Wright-Fisher population model
All individuals release many gametes and new individuals for the nextgeneration are formed randomly from these.
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Basics: Wright-Fisher population model
All individuals release many gametes and new individuals for the nextgeneration are formed randomly from these.
![Page 8: Peter Beerli Genome Sciences University of Washington ...evolution.genetics.washington.edu/PBhtmls/courses/course-I.pdf · Seattle WA. Outline 1. Introduction to the basic coalescent](https://reader034.vdocument.in/reader034/viewer/2022050106/5f44b48d6377366cc4264435/html5/thumbnails/8.jpg)
Basics: Wright-Fisher population model
All individuals release many gametes and new individuals for the nextgeneration are formed randomly from these.
![Page 9: Peter Beerli Genome Sciences University of Washington ...evolution.genetics.washington.edu/PBhtmls/courses/course-I.pdf · Seattle WA. Outline 1. Introduction to the basic coalescent](https://reader034.vdocument.in/reader034/viewer/2022050106/5f44b48d6377366cc4264435/html5/thumbnails/9.jpg)
Basics: Wright-Fisher population model
All individuals release many gametes and new individuals for the nextgeneration are formed randomly from these.
![Page 10: Peter Beerli Genome Sciences University of Washington ...evolution.genetics.washington.edu/PBhtmls/courses/course-I.pdf · Seattle WA. Outline 1. Introduction to the basic coalescent](https://reader034.vdocument.in/reader034/viewer/2022050106/5f44b48d6377366cc4264435/html5/thumbnails/10.jpg)
Basics: Wright-Fisher population model
All individuals release many gametes and new individuals for the nextgeneration are formed randomly from these.
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Wright-Fisher population model
Population size N is constant through time.
Each individual gets replaced every generation.
Next generation is drawn randomly from a large gamete pool.
Only genetic drift is manipulating the allele frequencies.
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The Coalescent
Sewall Wright showed that the probabilitythat 2 gene copies come from the samegene copy in the preceding generation is
Prob (two genes share a parent) =1
2N
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The Coalescent
Present
Past
In every generation, there is a chance of 1/2N to coalesce. Following thesampled lineages through generations backwards in time we realize that itfollows a geometric distribution with
E(u) = 2N [the expectation of the time of coalescence u of two tips is 2N ]
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The Coalescent
JFC Kingman generalized this for k genecopies.
Prob (k copies are reduced to k − 1 copies) =k(k − 1)
4N
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Kingman’s n-coalescent
Present
Past
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Kingman’s n-coalescent
Present
Past
u4
u3
u2
p(G|N) =∏
i exp(−uik(k−1)
4N ) 12N
The expectation for the timeinterval uk is
E(uk) =4N
k(k − 1)
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BUT, what’s this good for????????????????????
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Naively we could estimate: 1.Time of the most recentcommon ancestor
For a given population size we can calculate the time of the most recentcommon ancestor [MRCA].
1. Get a TRUE genealogy (topology and branch lengths) from an infallibleoracle.
2. Get the population size from the same oracle.
3. Calculate the time of the MRCA by summing over all time intervals.
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1. Time of the most recent common ancestor [Shortcut]
1. Get the population size from another oracle
2. Use the expectation for your data type to get an estimate of the time ofthe MRCA
The expectation for the time of the MRCA is
E(u) = 4N for diploid organisms
E(u) = 2N for haploid organisms
E(u) = N for maternally transmitted mtDNA,
paternally transmitted Y-chromosome
[assumption: sex-ratio is 1:1]
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2. Calculate the size of the population
1. We get THE genealogy from our oracle
2. We know that we can calculate p(Genealogy|N)
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2. Calculate the size of the population
1. We get THE genealogy from our oracle
2. We remember the probability calculation
p(G|N) = p(u1|N, k)1
2N× p(u2|N, k− 1)
12N
× .....
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2. Calculate the size of the population
1. We get THE genealogy from our oracle
2. We remember the probability calculation
p(Genealogy|N) =T∏j
e−ujkj(kj−1)
4N1
2N
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2. Calculate the size of the population
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2. Calculate the size of the population
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2. Calculate the size of the population
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2. Calculate the size of the population
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2. Calculate the size of the population
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2. Calculate the size of the population
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2. Calculate the size of the population
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2. Calculate the size of the population
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2. Calculate the size of the population
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2. Calculate the size of the population
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2. Calculate the size of the population
N = 2270
N = 12286
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Problems with these very naive approaches
We assume we knowthe TRUE genealogy:topology and branchlength.
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Variability of the coalescent
10 coalescent trees generated with the same population size, N = 10, 000
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Variability of mutations
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How many samples do we need?
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Summary of the basic Coalescent
Mathematically tractable way to calculate probabilities of genealogies ina population.
The coalescent is a very noisy distribution of times on a tree.
Variability because of mutation increases the uncertainty of these times.
The population size is correlated with the depth of the tree.
Estimations of population size or the time of the MRCA from a singletree are very error-prone.
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Variable population size
In a small population lineages coalesce quickly
In a large population lineages coalesce slowly
This leaves a signature in the data. We can exploit this and estimate thepopulation growth rate g jointly with the population size Θ.
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Exponential population size expansion or shrinkage
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Grow a frog
-50 0 50 100
1
0.01
0.1
Θ
gMutation Rate Population sizes
-10000 generations Present10−8 8, 300, 000 8, 360, 00010−7 780, 000 836, 00010−6 40, 500 83, 600
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Parameter estimation using maximum likelihood
Mutation model: Nucleotide mutation model, ...
Population genetics model: the Coalescent
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Parameter estimation using maximum likelihood
Mutation model: Nucleotide mutation model, ...
Population genetics model: the Coalescent
Prob (N, µ|data)
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Parameter estimation using maximum likelihood
Mutation model: Nucleotide mutation model, ...
Population genetics model: the Coalescent
Prob (N, µ|data) =Prob (data|N, µ) Prob (N, µ)
Prob (data)
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Parameter estimation using maximum likelihood
Mutation model: Nucleotide mutation model, ...
Population genetics model: the Coalescent
Prob (N, µ|data) =Prob (data|N, µ) Prob (N, µ)
Prob (data)
L(N, µ) = Prob (data|N, µ) = c Prob (N, µ|data)
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Parameter estimation using maximum likelihood
L(N, µ) = Prob (data|N, µ)
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Parameter estimation using maximum likelihood
L(N, µ) = Prob (data|N, µ) =∫
G
p(G|N, µ) Prob (data|G, µ)
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Parameter estimation using maximum likelihood
L(N, µ) = Prob (data|N, µ) =∫
G
p(G|N, µ) Prob (data|G, µ)
We cannot observe the mutation events. Instead of estimating N and µ weestimate the product Θ = 4Nµ and scale G with µ
L(Θ) =∫
G∗p(G∗|Θ) Prob (data|G∗)
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Parameter estimation using maximum likelihood
L(N, µ) = Prob (data|N, µ) =∫
G
p(G|N, µ) Prob (data|G, µ)
We cannot observe the mutation events. Instead of estimating N, and µwe estimate the product Θ = 4Nµ and scale G with µ
L(Θ) =∫
G∗p(G∗|Θ) Prob (data|G∗)
Problem: We need to integrate over all genealogies: all different labelledhistories, all different branchlengths
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Can we calculate this sum over all genealogies?
We need to integrate over all genealogies: all different topologies, alldifferent branchlengths
Tips Topologies
3 3
4 18
5 180
6 2700
7 56700
8 1587600
9 57153600
10 2571912000
15 6958057668962400000
20 564480989588730591336960000000
30 4368466613103069512464680198620763891440640000000000000
40 30273338299480073565463033645514572000429394320538625017078887219200000000000000000
50 3.28632 × 10112
100 1.37416 × 10284
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A solution:Markov chain Monte Carlo
Metropolis recipe
0. first state
1. perturb old state and calculateprobability of new state
2. test if new state is better thanold state: accept if ratio of newand old is larger than a randomnumber between 0 and 1.
3. move to new state if acceptedotherwise stay at old state
4. go to 1
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How do we change a genealogy
zA B
C D
1
2j
k
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Markov chain Monte Carlo
1
L(G1|Θ)
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Markov chain Monte Carlo
1
create a new tree
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Markov chain Monte Carlo
12
L(G2|Θ)
Evaluate
r <p(G2|Θ)P(D|G2)P(G1|G2)p(G1|Θ)P(D|G1)P(G2|G1)
luckily reduces most often to:
r <P(D|G2)P(D|G1)
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Markov chain Monte Carlo
12
Store G1
Make another change to the tree
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Markov chain Monte Carlo
12
3
L(G3|Θ)
r <P(D|G3)P(D|G2)
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Markov chain Monte Carlo
12
3
Store G2
Make another change to the tree
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Markov chain Monte Carlo
12
3
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Markov chain Monte Carlo
12
3
4
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Markov chain Monte Carlo
12
3
4
5
67
8
9
1110
121314
15
16
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MCMC walk I
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MCMC walk result
Tree space Tree space
0.002
0.005
0.01
Pro
babi
lity
0.002
0.005
0.01
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MCMC walk result
Tree space Tree space
0.002
0.005
0.01
Pro
babi
lity
0.002
0.005
0.01
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Improving our MCMC walker: MCMCMC or MC3
Metropolis Coupled Markov chain Monte Carlo
Run several independent parallel chains: each has a different temperature
After some sampling of genealogies, swap the genealogies of a pair ofchains if the ratio between probabilities in the cold and the hot chain islarger than a random number drawn between 0 and 1.
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MCMC walk II
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better MCMC walk result