molecular population genetics of adaptation from recurrent beneficial mutation

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Molecular population genetics of adaptation from recurrent

beneficial mutation

Joachim Hermisson and Pleuni Pennings,

LMU Munich

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How can genetic variation be maintained in a population in the face of positive selection?

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Selective sweepwith recombination

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Selective sweep with recombination

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Selective sweepwith recombination

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Recurrent mutation

Classical view:Adaptive substitutions occur from a single

mutational origin

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Recurrent mutation

Classical view:Adaptive substitutions occur from a single

mutational origin

What happens if the same beneficial allele

occurs recurrently in a population?

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Soft sweepfrom recurrent mutation

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Soft sweepfrom recurrent mutation

time →

freq

uenc

y →

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

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Is recurrent mutation relevant?

• What is the probability of a soft sweep under recurrent mutation?

• What is the impact on patterns of neutral polymorphism?

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Model

• Haploid population of constant size Ne

• At selected locus: recurrent mutation of rate u to a beneficial allele (or a class of equivalent alleles) with selective advantage s

• Scaled values: = 2Ne u , = 2Ne s, R = 2Ne r• Generation update: Wright-Fisher model (fitness

weighted multinomial sampling)

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Coalescent viewGenealogy of a sample from a linked locus

• What can happen one generation back in time?

time

freq

uen

cy

x1- x

n lines

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Coalescent viewCoalescence of two lines

• Rate per generation:

time

freq

uen

cy

x1- x

xN

n

e

nc1

2)(,

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Coalescent viewRecombination


time

freq

uen

cy

x1- x

e

nrN

xnR

1

)(,

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Coalescent viewNew mutation at selected site


time

freq

uen

cy

x1- x

xN

xn

e

nm

1

2)(,

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Coalescent view

Problem: Rates for

• coalescence

• recombination

• beneficial mutation

depend on the frequency x of the selected allele:stochastic path

xN

n

e

nc1

2,

xN

xn

e

nm

1

2,

e

nrN

xnR

1,

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Coalescent viewClassic case: Coalescence and recombination

• Probability for multiple haplotypes in a sample after a sweep due to recombination:

(Higher orders: Etheridge, Pfaffelhuber, Wakolbinger)• small for large strong selection makes broad sweep patterns)

)Log(

PrR

reco

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Coalescent view Coalescence and mutation, sample of size 2

Probability for coalescence before mutation (single haplotype)

xNe

c1

)(2,

xN

x

e

m)1(

)(2,

xi

cmchard iiP

1

1

1

2,2,2,2, )()(1)(

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xi ie

i

e

hardxN

x

xNP

1

1

1

2,)1(1

11

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xi ie

i

e

hardxN

x

xNP

1

1

1

2,)1(1

11

1

1

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x

τ

i ie

i

τe

τ

xτ

τ

i ie

i

τe

τhard,

xN

xθ

xN

xθ

xN

xθ

xN

xθ

θP

1

1

1

1

1

1

2

)1(11

)1(11

)1(1

1

1

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x

τ

i ie

i

e

xτ

τ

i ie

i

τe

τhard,

xN

xθ

N

θ

xN

xθ

xN

xθ

θP

1

1

1

1

1

1

2

)1(11

)1(11

)1(1

1

1

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Probability for single or multiple haplotypes:

12 1

1

1T

N

θ

θP

e

hard,

e

soft,N

T

θP 1

2 11

T1: average time to the first coalescence or mutation-event

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Sampling at time of fixation: 0 < T1 < Tfix

θP

N

T

θsoft,

e

fix

11

12

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General: sampling Tobs generations after fixation:

obs

e

soft,

obs

ee

fixT

NθP

T

NN

T

θ

11

1

111

12

extra factor can be ignored for Tobs << Ne

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Sampling at time of fixation: 0 < T1 < Tfix

θP

N

T

θsoft,

e

fix

11

12

Tfix / Ne ≈ 4 log() / = 2Ne s (scaled selection strength)

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100 1000 100002Nes

0.05

0.1

0.15

0.2

0.25

0.3

θP

θsoft,

1

)log(41

12

Simulation results (θ = 0.4)

2soft,P

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For > 500 : Tfix / Ne << 1, thus

θN

T

θP

e

soft,

11

1

12

Corresponds to approximation:

xNxN

x

ee

m

)1(

)(2,

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Coalescent view Coalescence and mutation, sample of size n

xN

n

e

nc1

2)(,

xN

nx

e

nm 12

)(,

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xN

n

e

nc1

2)(,

xN

nx

e

nm 12

)(,

Continuous time and time rescaling:

)/(~ xNe

2~

,n

nc2

~,

nnm

Neutral coalescent !

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• Problem independent of the path x and all selection parameters

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• Problem independent of the path x and all selection parameters• Coalescent of the infinite alleles model• Forward in time: “Hoppe urn” or Yule process with immigration

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• Problem independent of the path x and all selection parameters• Coalescent of the infinite alleles model• Forward in time: “Hoppe urn” or Yule process with immigration

The sampling distribution of ancestral haplotypes

can be approximated by the distribution of family sizes

in a Hoppe urn or a Yule process with immigration

Solved problem

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ResultsEwens sampling formula

• Probability for k haplotypes, occurring n1,…, nk times

in a sample of size n:

)1()1(!

!)Pr(

1

1

nnnk

nnn

k

k

k

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ResultsEwens sampling formula

• Probability for more than one ancestral haplotype in a sample (“soft sweep”):

1

1

, 1n

i

nsofti

iP

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ResultsProbability of a soft sweep

Ewens approximation, sample size n = 20

0%

20%

40%

60%

80%

100%

= 0.

004

= 0.

04

= 0

.4

=

1

= 4

>4 haplos

4 haplos

3 haplos

2 haplos

1 haplo

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Simulation (2Ne s = 10 000, n = 20)

0%

20%

40%

60%

80%

100%

= 0.

004

= 0.

04

= 0

.4

=

1

= 4

>4 haplos

4 haplos

3 haplos

2 haplos

1 haplo

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Simulation (2Ne s = 10 000, n = 20)

0%

20%

40%

60%

80%

100%

= 0.

004

= 0.

04

= 0

.4

=

1

= 4

>4 haplos

4 haplos

3 haplos

2 haplos

1 haplo

Probability for multiple haplotypes > 5% for > 0.01 >95% for > 1

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0

0.1

0.2

0.3

0.4

0.5

5/10 6/10 7/10 8/10 9/10

α =100

α =1000

α =10000

prediction

ResultsFrequency of major haplotype

Sample size 10

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• Strong dependence on the mutation rate– More than 5% for > 0.01– E.g. African D. melanogaster: ≈ 0.05 (Li / Stephan 2006)About 16% of all single-site adaptations “soft”

• Particularly relevant for – Large populations (e.g. bacteria)– Adaptive (partial) loss-of-function mutations

When should we expect soft sweeps?Multiple haplotypes due to recurrent beneficial mutations

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Soft sweeps in data?

• Drosophila – Schlenke and Begun (Genetics 2005): LD pattern at 3 immunity

receptor genes in Californian D. simulans

• Humans– Multiple origin of FY-0 Duffy allele (loss of function)

• Plasmodium– Multiple origins of pyrimethamine resistance mutations

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Generality of the resultmigration instead of mutation

• Beneficial alleles enter by recurrent migration at rate M = 2Ne m from a genetically diverged source population

• Coalescent analysis with migration rate

xN

nM

e

nm2

)(,

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• Beneficial alleles enter by recurrent migration at rate M = 2Ne m from a genetically diverged source population

• Coalescent analysis with migration rate

• Directly proportional to coalescence rate (no factor 1- x) Approximation holds exactly in this case

xN

nM

e

nm2

)(,

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100 1000 10000

0.05

0.1

0.15

0.2

0.25

0.3

Psoft, 2

Selection strength 2Nes

M = 0.4

= 0.4

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Generality of the resulttime or frequency-dependent selection

• Results independent of the stochastic path x of the frequency of the beneficial allele

Independent of any form of time or frequency dependence of the selection strength

In particular: Independent of the level of dominance In particular: Holds also for adaptation from standing

genetic variation (number of independent origins)

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Generality of the resultvariance in selection coefficients

If beneficial allele corresponds to a class of alleles: some fitness differences among variants likely

Assume: 2 classes of alleles with selective advantage

(D = coefficient of variation)

D 1

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0

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%θθ = 0.01 θ = 1

>4

4

3

2

1

= 0.1

0.0

1

0.0

50

.1

0.2 0

0.0

1

0.0

50

.1

0.2 0

0.0

1

0.0

50

.1

0.2

Nu

mb

er

of h

ap

loty

pes

D

10000

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D=0

D=0.01

D=0.05

D=0.1

D=0.2

5/10 6/10 7/10 8/10 9/10

0.4

0.3

0.2

0.1

0

= 0.1

Fre

que

ncy

of m

ajo

r ha

plo

type

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Ewens + neutral coalescent prior to the sweep:

Derive frequency distribution of ancestral variation that survives the sweep

Skew toward intermediate allele frequencies (singleton frequency lower than neutral)

In contrast:

Recombination haplotypes are most likely at low frequency

Footprint of selectionFrequency spectrum of polymorphic sites

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coalescence

recombination

mutation


Pro

bab

ility

of e

ven

t

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time

freq

uen

cy x1-x

recombination

mutation

coalescence

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Footprint of selectionIncluding recombination

• Analytical results– E.g. Probability for a single haplotype in sample of two:

– General: “Marked Yule process with immigration” For now …• Simulation results

– Add recurrent mutation to simulation program by Yuseob Kim

log2exp

1

1Prsing

R

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Footprint of selectionPower of Tajima’s D test at the selected gene

Neutral locus at recombination distance R to selected site:

• Recombination width of the neutral locus Rn = 10

• Neutral mutational input n = 10

• = 2Ne s = 10000

• Sample size 20

Power of Tajima’ D for various recombination distances and

sampling times after fixation of the beneficial allele

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0 10 20 100 200 600

Footprint of selectionPower of Tajima’s D test: single origin

0

1

0.5

0.2

0.1

0.05

0.01

Tim

e si

nce

fixat

ion

in 2

Ne

gene

ratio

ns

Distance in unitsof R = 2Ne r

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0 10 20 100 200 600

Footprint of selectionPower of Tajima’s D test: = 0.1

0

1

0.5

0.2

0.1

0.05

0.01

Tim

e si

nce

fixat

ion

in 2

Ne

gene

ratio

ns


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0 10 20 100 200 600

Footprint of selectionPower of Tajima’s D test: = 0.4

0

1

0.5

0.2

0.1

0.05

0.01

Tim

e si

nce

fixat

ion

in 2

Ne

gene

ratio

ns


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0 10 20 100 200 600

Footprint of selectionPower of Tajima’s D test: = 1

0

1

0.5

0.2

0.1

0.05

0.01

Tim

e si

nce

fixat

ion

in 2

Ne

gene

ratio

ns


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0 10 20 100 200 600

Footprint of selectionCondition on soft sweeps: negative D

0

1

0.5

0.2

0.1

0.05

0.01

Tim

e si

nce

fixat

ion

in 2

Ne

gene

ratio

ns


= 0.1

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0 10 20 100 200 600

Footprint of selectionCondition on soft sweeps: positive D

0

1

0.5

0.2

0.1

0.05

0.01

Tim

e si

nce

fixat

ion

in 2

Ne

gene

ratio

ns


= 0.1

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Footprint of selectionTests based on linkage disequilibrium

E.g. number-of-haplotypes test (K-test) by Depaulis and Veuille

• Conditioned on number of segregating sites• Zero recombination assumed for neutral comparison• Other values as before

Power of K for various recombination distances and

sampling times after fixation of the beneficial allele

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INTRODUCTION

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0 10 20 100 200 600

Footprint of selectionPower of haplotype test: single origin

0

1

0.5

0.2

0.1

0.05

0.01

Tim

e si

nce

fixat

ion

in 2

Ne

gene

ratio

ns


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INTRODUCTION

COALESCENT VIEW

RESULTS

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0 10 20 100 200 600

Footprint of selectionPower of haplotype test: = 0.1

0

1

0.5

0.2

0.1

0.05

0.01

Tim

e si

nce

fixat

ion

in 2

Ne

gene

ratio

ns


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INTRODUCTION

COALESCENT VIEW

RESULTS

SUMMARY

0 10 20 100 200 600

Footprint of selectionPower of haplotype test: = 0.4

0

1

0.5

0.2

0.1

0.05

0.01

Tim

e si

nce

fixat

ion

in 2

Ne

gene

ratio

ns


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INTRODUCTION

COALESCENT VIEW

RESULTS

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0 10 20 100 200 600

Footprint of selectionPower of haplotype test: = 1

0

1

0.5

0.2

0.1

0.05

0.01

Tim

e si

nce

fixat

ion

in 2

Ne

gene

ratio

ns


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INTRODUCTION

COALESCENT VIEW

RESULTS

SUMMARY

0 10 20 100 200 600

Footprint of selectionPower of haplotype test: = 4

0

1

0.5

0.2

0.1

0.05

0.01

Tim

e si

nce

fixat

ion

in 2

Ne

gene

ratio

ns


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INTRODUCTION

COALESCENT VIEW

RESULTS

SUMMARY

0 10 20 100 200 600

Footprint of selectionCondition on soft sweeps: number of haplotypes

0

1

0.5

0.2

0.1

0.05

0.01

Tim

e si

nce

fixat

ion

in 2

Ne

gene

ratio

ns


= 0.1

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SUMMARY

Can we extend high power to a longer time after fixation?

Idea:• Use only ancestral variation

– E.g. local adaptation to an “island”: use only shared polymorphisms with the continental founder population

• Adapt neutral standard of the test accordingly

Footprint of selectionTests based on linkage disequilibrium

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INTRODUCTION

COALESCENT VIEW

RESULTS

SUMMARY

0 10 20 100 200 600

Footprint of selectionCondition on soft sweeps: ancestral haplotypes

0

1

0.5

0.2

0.1

0.05

0.01

Tim

e si

nce

fixat

ion

in 2

Ne

gene

ratio

ns


= 0.1

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INTRODUCTION

COALESCENT VIEW

RESULTS

SUMMARY

0 10 20 100 200 600

Footprint of selectionCondition on soft sweeps: “ancestral ZnS”

0

1

0.5

0.2

0.1

0.05

0.01

Tim

e si

nce

fixat

ion

in 2

Ne

gene

ratio

ns


= 0.1

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RESULTS

SUMMARY

Summary

• Soft sweeps from recurrent mutation likely for biologically realistic parameter values

• Pattern described by Ewens sampling distribution• Result very stable with respect to the selection scenario• May be detected by LD tests, in particular if recent

mutations can be sieved out

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INTRODUCTION

COALESCENT VIEW

RESULTS

SUMMARY

Open Issues

• Unified Yule process (?) theory of coalescence, recombination, and mutation

• Description of LD patterns after soft (or hard) sweeps: Which aspect lasts the longest?

molecular population genetics of adaptation from recurrent beneficial mutation

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