The Coalescent Evolution backward in time Joachim Hermisson - - PowerPoint PPT Presentation

The Coalescent

Evolution backward in time

Joachim Hermisson Mathematics and Biosciences Group Mathematics & MFPL, University of Vienna

Massive accumulation of DNA sequence data

• 1980’s:

3-4 years PhD projects to sequence a single gene (some 1000 base pairs)

• 1990 – 2003: Human Genome Project (~ 3 109 (3 billion) bases)

expected: 3 billion $, final: ~ 300 Mio $

• since 2010:

1000 Genome Project 4000 $ – 10000 $ per genome, soon < 1000 $

• today: extended to 2500 (25 x 100), completed May 2013

1000 genomes also for Drosophila, Arabidopsis …

Introduction to the Coalescent

data, data, data, …

A C A T T A A G C G T A G A C T T A G G T G T T G C A C A T T A A G C C T A G A C A T A G G T G T T G C A G A T T C A G C C T A G A C T T A G G T G A T G C A G A T T C A G C C T A G A C T T A G G T G T T G C A C A T T A A G C C T A G A C A T A G G T G T T G C A C A T T C A G C C T A G A C T T A G T T G T T G C

Patterns of Evolution

”Summary Statistics”

Sample size (n = 6) Sequence alignment (length m = 26) 4(6×26) = 8.3 ×1093

A C A T T A A G C G T A G A C T T A G G T G T T G C A C A T T A A G C C T A G A C A T A G G T G T T G C A G A T T C A G C C T A G A C T T A G G T G A T G C A G A T T C A G C C T A G A C T T A G G T G T T G C A C A T T A A G C C T A G A C A T A G G T G T T G C A C A T T C A G C C T A G A C T T A G T T G T T G C

Patterns of Evolution

”Summary Statistics”

• nly polymorphic sites …

C

Patterns of Evolution

”Summary Statistics”

C A G T C C C C C A A G G G G G G G A A A T T T T T T T T T C C C C C G T G A T C

• utgroup

compare with outgroup …

Patterns of Evolution

”Summary Statistics”

forget about molecular state … (assumes infinite sites mutation model)

Patterns of Evolution

Summary statistics based on segregating sites

• number of segregating sites and allele frequencies

4 3 1 2 1 1 mutation “size“:

Patterns of Evolution

Summary statistics based on segregating sites

• number of segregating sites and allele frequencies
• associations not important (“molecular bean bag“)

4 3 1 2 1 1 mutation “size“:

Patterns of Evolution

Summary statistics based on segregating sites

• number of segregating sites and allele frequencies
• associations not important (“molecular bean bag“)

4 3 1 2 1 1 mutation “size“:

• genome position

does not matter

Patterns of Evolution

Summary statistics based on segregating sites

Site Frequency Spectrum

4 3 1 2 1 1

1 2 3

1 2 3 4 5

Reconstruction of evolutionary history

selection and demographic events distributions for summary statistics (S, p) Statistical Reconstruction

Process Pattern

Patterns of Evolution

• bserved

patterns (S, p from data) estimated parameters

Reconstruction of evolutionary history

standard neutral model Distributions ?

Process Pattern

Patterns of Evolution

How does pure randomness look like ?

• Null-model of the evolutionary theory

Neutral genetic variation

• single locus, multiple alleles

Drift:

• random sampling of parents
• k types: multinomial offspring distribution

Mutation:

• probability u for each offspring
• infinite alleles model: every mutation leads

to a new allele (“new color”) 1.

• 2. generation

population (size 2N)

Patterns of Evolution

Wright-Fisher model

Patterns of Evolution

Wright-Fisher model

sample generation

Patterns of Evolution

Wright-Fisher model

Patterns of Evolution

Wright-Fisher model

Patterns of Evolution

coalescence process

All information about the genetic variation pattern is contained in the sample genealogy.

Patterns of Evolution

coalescence process

Construct a process to generate genealogies: „coalescence-process“ continuous time All information about the genetic variation pattern is contained in the sample genealogy.

Coalescent Theory

The standard neutral model

Haploid Wright-Fisher population of size 2N :

• Genetic differences have

no consequences on fitness

• No population subdivision
• Constant population size

Individuals are equivalent with respect to descent Exchangable offspring distribution, independent of any state label (genotype, location, age, …) `State´ and `Descent´ are decoupled

• 1. Construct genealogy independently of the state
• 2. Decide on the state only afterwards

2 steps:

• Wright-Fisher: multinomial sampling

Coalescent Theory

Construction of the Genealogy: Sample Size 2

2N Coalescence probability … in a single generation:

N pc 2 1

1 , 

Coalescent Theory

Construction of the Genealogy: Sample Size 2

Coalescence probability … in a single generation: 2N

N pc 2 1

1 , 

N N p

t t c

2 1 2 1 1

1 , 

       

… for exactly t generations:

Coalescent Theory

Construction of the Genealogy: Sample Size n

Multiple (e.g. triple) mergers: 2N

 

2 2

4 1



   N N ptriple

Coalescent Theory

Construction of the Genealogy: Sample Size n

Multiple (e.g. triple) mergers: 2N

 

2 2

4 1



   N N ptriple

 

2 2 ,

Pr



   N p t

c

Multiple coalescences:

Coalescent Theory

Construction of the Genealogy: Sample Size n

Multiple (e.g. triple) mergers: 2N

 

2 2

4 1



   N N ptriple

 

2 2 ,

Pr



   N p t

c

Multiple coalescences: can be ignored if N >> n :

• nly binary mergers for

  N

“Kingman coalescent“

Coalescent Theory

Construction of the Genealogy: Sample Size n

Coalescence probability (single binary merger) 2N

N n n n N p n

c

4 ) 1 ( 2 2 1

) ( 1 ,

          

… in a single generation:

N n n N n n p

t n t c

4 ) 1 ( 4 ) 1 ( 1

1 ) ( ,

         



… for exactly t generations:

Coalescent Theory

Distribution of Coalescence Times

Coalescence time T2 for sample size 2: coalescence time

   

 



             

 

exp 2 1 1 Pr

2 2 N N

N T

Define coalescence time scale:

N t 2  

T2

Exponential distribution with parameter 1:



  1

E

2 

T

(2N generations)

Coalescent Theory

Distribution of Coalescence Times

with sample size n:

 

 

 

                  

             

2 2 2

exp 2 1 1 Pr

n N N n n

N T

Exponential distribution with parameter :

 

) 1 ( 2 E   n n Tn

T4

2 ) 1 ( 2           n n n

iterate until most recent common ancestor (MRCA): T3 T2 coalescence time



Coalescent Theory

Tree Topologies

• pick two random individuals

from the sample and merge

• sample size n → n-1 and

iterate until n = 1 (MRCA) “random bifurcating tree“ coalescence time



• all individuals exchangable
• topology invariant under

permutation of “leaves“

Coalescent Theory

Tree Topologies

• pick two random individuals

from the sample and merge

• sample size n → n-1 and

iterate until n = 1 (MRCA) “random bifurcating tree“ coalescence time



• all individuals exchangable
• topology invariant under

permutation of “leaves“ same topology

Coalescent Theory

Tree Topologies

• pick two random individuals

from the sample and merge

• sample size n → n-1 and

iterate until n = 1 (MRCA) “random bifurcating tree“

• all individuals exchangable
• topology invariant under

permutation of “leaves“ different topology coalescence time



Coalescent Theory

Tree Topologies

• pick two random individuals

from the sample and merge

• sample size n → n-1 and

iterate until n = 1 (MRCA) “random bifurcating tree“ coalescence time



• all individuals exchangable
• topology invariant under

permutation of “leaves“ Distribution of tree topologies

• independent of coalescence times
• depends only on the separation of

state and descent and on the “no multiple merger“ condition

T4 T3 T2

Coalescent Theory

Mutation “Dropping”

• nly number of mutations
• n each branch matters

Infinite sites mutation model: mutation rate u, all mutations

• n the genealogy are visible as polymorphisms on different sites
• Poisson distributed with

parameter , 2 2 L L Nu    







k j i i

T L

branch length

• f branch from state j through k

state 4 3 2

(also other mutation schemes possible)

Coalescent Theory

Basic Properties

Three independent stochastic factors determine the polymorphism pattern:

• 1. coalescent times
• 2. tree topology
• 3. mutation

(very easy to implement in simulations)

Coalescent Theory

Basic Properties

Time to the most recent common ancestor:

 

 

  

n k n k k MRCA

k k T E T E

2 2

) 1 ( 2 ] [ ] [                  



n k k

n k

1 1 2 1 1 1 2

2

T4 T3 T2

1 ] [ 2  T E

Compare: More than half for the last two branches!

Coalescent Theory

Basic Properties

Total length of the tree and expected number

• f polymorphic sites:

T4 T3 T2

n n k

a k S E    



 

 

1 1

1 ] [

with:

 

  

 

1 1 2

1 2 ] [ ] [

n k n k k tree

k T kE L E

577 . log 1

1 1

  

 

n k a

n k n

(logarithmic dependence on sample size)

Coalescent Theory

Basic Properties

Expected site frequency spectrum:

 

 

   

 

1 1 1 1

1 ] [

n k k n k

E k S E  

indeed:

0,2 0,4 0,6 0,8 1

1 2 3 4 5 6

k



  / ] [

k

E

 

k E

k

  

Number of mutations that appear k times in the sample (= of size k) in particular:

  

 

1

E

k

n = 7