Hein, Schierup & Wiuf: Genealogies, Variation & Evolution.www.stats.ox.ac.uk/hein/lectures www.coalescent.dk (mheide@birc.dk)

Contents

H: The Basic Coalescent

H: The Coalescent with Recombination

S: The Coalescent with History, Geography & Selection

S: The Coalescent & Gene Mapping

H: The Coalescent & Combinatorics

W: The Coalescent & Ancestral Analysis

W: Parameter Estimation & Hypothesis Testing

H: The Coalescent & Human Evolution

Book & Tutorials

The Coalescent with Geography, History, & Selectionwww.stats.ox.ac.uk/hein/lectures www.coalescent.dk (mheide@birc.dk)

Review: Coalescence/Recombination

Geography

History

Selection

Scenarios Detection

Continuous-Time Coalescent

1 24 56 3 0.0

1.0

1.0 corresponds to 2N generations

0

2N

Discrete Continuous

Recombination-Coalescence IllustrationCopied from Hudson 1990 Intensities

Coales. Recomb.

1 2

3 2

6 2

3 (2+b)

1 (1+b)

0

b

History Scenarios.

Logistic growthBottleneck Size jump Exponential growth

Stochastic Fluctuations of Population Size.

Simple Deterministic Models of Population Size.

LB

Bottleneck Severity

LB/Ne

Bottleneck Age

T/Ne

teNtN 0)( )1()('

N

KtN

T

The Coalescent & Population Growth

Growth will elongate leaf edges relative to deep edges!

If the population size is known as function of time N(t), time can be

scaled as for exponential growth et, this givesdssN

Nt

t

0 )(

)0()(

]1[1

)( tet

Tests of History.

Distortion of branch lengths towards the present.

Tajima (1989) - Fu and Li (1993)

Mismatch Distribution Pairwise Distances

Rogers & Harpending,1992.

Likelihood Models

Beerli & Felsenstein,1999

Mismatch distributionsRogers & Harpending,1992, Slatkin and Hudson, 1991

Watterson’s Estimator

1 2 3 54

ACCTGAACGTAGTTCGAAG

ACCTGAACGTAGTTCGAAT

ACCTGACCGTAGTACGAAT

ACATGAACGTAGTACGAAT

ACATGAACGTAGTACGAAT

* * * *

Expected Number Segregating Sites:

*(1+1/2+ +1/(k-1))

W := Segr/(1+1/2+ +1/(k-1))= 4/ [11/6]=24/11=2.1818

Var(segr) =

1j

i2

n1

2 1j2

i1

n1

AA

B

B C

C

D

D

Pairwise Distance Estimator

1 2 3 54

PD := Average Pairwise Distance (above 2.166)

VarPD) = (n+1)/3(n-1) + 2 2(n2+n+3)/9n(n-1)

44

4 44

6

6

A mutation on a (n,n-k) branch will be counted n*(n-k) times. I.e. deep branches have higher weights.

A

ACCTGAACGTAGTTCGAAG

ACCTGAACGTAGTTCGAAT

ACCTGACCGTAGTACGAAT

ACATGAACGTAGTACGAAT

ACATGAACGTAGTACGAAT

* * * *

BC

D

A B C D

D = (PD -W)/Sd(PD -W)

A large value indicates shortened tips

A small value indicates shortened deep branches.

Tajima’s Test

Mitochondria (Ingman et al. 2000) Remade from McVean

52 complete molecules

521 segregating sites

PD = 44.2 W = 115.3

V(D) =31.8 D = -2.23

Geography Scenarios.2 Demes

Same Size

Continent & Island

N Demes with structure

stepping stone: 1- dimension

stepping stone: 2 dimensions

Continuous Geography

1 Dimension

2 Dimensions

Two Demes:

Symmetric:

Island/Continent:

M1

M2

Distribution of MRCA in d-deme model.

2 alleles. T(2,0) time to coalesce if in different demes

T(0,1) time to coalesce if in the same demes

Recursions Solutions

)]0,2([)1(4

)1(

)1(2

1)]1,0([

TEdM

dM

ddMTE

)]1,0([)1/(

1)]0,2([ TE

dmTE

MTE 2/11)]1,0([

1)]0,2([ TE

d=5

m

1/d

From Hudson, 1990

Distribution of coalescence times within/between demes

Population subdivision - two demes

Stepping stone models

Line (1-Dim)

Plane (2-Dim)

Continuous Geography(Wright 43, Malecot 48, Felsenstein 75, Barton 96 02)

1 dimensional 2 dimensional - plane or torus (wrapped rectangle)

These models can be obtained by

A. A limit of stepping stone models

B. Directly by a Brownian Motion model of movement.

Sequences can only find common ancestors when at the same place. This doesn’t happen in the continuous models for dim >1. A neighborhood has to be defined.

Continuous Geography(Barton & Wilson 96)

Several artifacts in this model.

Increasing lumping over time.

Testing Geography.

Hudson, Boos & Slatkin’s (1990) Permutation test

Maddison & Slatkin (1989) Assignment of ancestral geography.

Likelihood Bahlo & Griffiths (2000) Pritchard’s Structure (2000) Kuhner et al. (2000)

Selection Scenarios.

Haploid

Selection Directional

Frequency Dependent

Diploid

Balancing Selection

Directional Selection

Allelic Types

Alleles: A a

Fitnesses 1 1-s

Fitnesses 1-pA 1-pa

Genotypes AA AA aa

Fitnesses 1 1+s 1

Fitnesses 1 1+hs 1+s

sNs In the coalescent scaling

The 1983 Kreitman Data(M. Kreitman 1983 Nature) from Hartl & Clark, 1997

11 alleles 3200 bp long.

43 segregating sites (columns with variation).

1 amino replacement event

1 insertion-deletion (indel)

P(A)=p, P(B)=q, we sample k1 A alleles and k2 B alleles

Two locus balancing selection modelHudson, Darden & Kaplan,88-89

q p

Two cases:

Strong selection with fixed ancestral frequencies

Weaker selection with fluctuating ancestral frequencies.

k1 k2

Directional Selection/ Bottlenecks

Geography Selection

Local Global

Heterogenisation Balancing Selection Geo.Subdivision

Homogenisation Selective Sweeps Bottlenecks

Two alleles, A and a, A has an advantage of sMutation rate between types = u

The ancestral selection graphKrone & Neuhauser, 1997

Construction of the ancestral selection graph

Recovery of the

coalescent tree with

directional selection

Krone & Neuhauser, 97

A Coalescent relating Allele Classes.(Takahata,1990)

AiAj AiAi

1 1-s

2

3

2)

16ln(

2

M

S

M

sfS

NM NsS

Examples: Major Histocompatibility Genes

Self-incompatibility Alleles in plants.

Allelic genealogy in

the self-incompatibility system of Solanaceae

From Vekemans, 1998

Tests of Selection

Tajima’s D (1989) (Fu’s test)

Hudson, Kreitman & Aquade (1987) HKA

Kreitman-MacDonald test (1990)

Likelihood tests

HKA-Test (Hudson, Kreitman & Aquade)Hudson,Kreitman & Aquade,1987

Gene 1

Specie 1 Specie 2

1

Speciation, T:

MRCA1

Gene 2

Specie 1 Specie 2

2

Speciation, T:

MRCA2

Are the 2 loci linked or unlinked?

The original data set ADH & 5’ prime region, D. sechellia & D.melanogaster

d=210 (MRCA1) d=18 (MRCA2)

S=9 (Lk1*1) S=8 (Lk2*2)

1 = 2.7 2=0.7 T=13.4Ne

Rejection.

Kreitman/McDonald-Test Kreitman/McDonald,1990 + Eanes, 1994

Specie 1 Specie 2 Within Specie

Between Specie

Replacement Synonymous

21 BR

2 WR 36 WS

26 BS

Tested by 2*2 indendence in contingency table.

Parameter estimation not necessary.

Rejection

Summary of Coalescent with HGS

History

Geography

Selection

Scenarios Detection

References.(Balding,D. et al. (2000) “Handbook of Statistical Genetics” Wiley Articles by Rousset, Nordborg, Stephens, Hudson,

Barton,NH, Depaulis & Etheridge (2002) Neutral Evolution in Spatially Continuous Populations Theor.Pop.Biol. 61.31-48.Barton, N. & I.Wilson (1996) “Genealogies and Geography” in New uses for New Phylogenies eds. Harvey et al. OUPDonnelly,P., Nordborg, M. & Joyce,P. (2001) Likelihoods and Simulation Methods for Classes of Non-neutral Population Genetics Models. Genetics 159.853-867.

Golding,B. (ed.) (1994) “Non-Neutral Evolution” Chapman & Hall articles by Eanes, Aquadro,.Hudson, McDonald,

Hein,JJ (2002) Slides: www.stats.ox.ac.uk/hein/lectures

Hudson, Boos & Kaplan (1992) A Statistical Test for Detecting Geographical Subdivision” Mol.Biol.Evol. 9.1.138-151

Hudson, Kreitman & Aquade (1987) A test of Neutral Molecular Evolution Based on Nuclear Data. Genetics 116.153-9

Hudson, Darden & Kaplan (1988)

Hudson and Kaplan (1988) “The Coalescent Process in Models with Selection and Recombination” Genetics 120.831-840.

Krone & Neuhauser (1997)

McVean,G. (2002) course: www.stats.ox.ac.uk/mcvean

Neuhauser & Krone (1997) The Genealogy of Samples in Models with Selection Genetics 145.519-534.

Nordborg, M.(1997) “Structured Coalescent Processes on Different Time Scales” Genetics 146.1501-1514.

Pybus,OG et al(2000) An Integrated Framework for the Inference of Viral Population History from Reconstructed Genealogies. Genetics 155.1429-1437.

Schierup, M. et al.(2002) Coalescent Simulator: www.coalescent.dk

Slatkin (1991) Inbreeding Coeffecients and Coalescence Times. Genet. Res. Camb. 58.167-175.

Slatkin & Hudson (1992)

Slade (2000) “Simulation of Selected Genealogies” Theor.Pop.Biol. 57.35-49

Takahata,N.(1990) “A simple genealogical structure of strongly balanced allelic lines of transpecies evolution of polymorphism. PNAS 87.2419-23.

Wiuf, C. and J.Hein (2000) “The Coalescent with Gene Conversion” Genetics 155.451-462.

FST’s & Geography & Coalescent.

History of Coalescent with HGS

Stepping Stone Model introduced by Wright

Krone Neuhauser introduces Selection Graph

Evidence against background selection

From Aquadro, 1994

Evidence for genetic hitch-hiking

From Aquadro, 1994

From Aquadro, 1994

Variation in recombination rate in Drosophila, chromosome 3

Selective Sweeps/Background Selection.

Excoffier, 2000

Mismatch distributions in human populations

External () versus Internal Branches. e and i the number of mutations in external and internal branches.

E() = 2 E() = Lk-2 E(e )= V(e )= E(i )= (Lk-2) V(i )= (Lk-2)/(n-1) + c]

1 if n =2c = b= 2[nLk - 2(n-1)]/(n-1)(n-2)

Fu’s TestFrom Li,1997

1i2

i1

k 1

Waiting intensities in general model of population subdivision