Molecular population genetics of adaptation from recurrent beneficial mutation

1
Molecular population genetics of adaptation from
recurrent beneficial mutation

• Joachim Hermisson and Pleuni Pennings,
• LMU Munich

2

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

3
Selective sweepwith recombination
4
Selective sweep with recombination
5
Selective sweepwith recombination
6
Selective sweepwith recombination
7
Selective sweepwith recombination
8
Recurrent mutation

• Classical view
• Adaptive substitutions occur from a single
  mutational origin

9
Recurrent mutation

• Classical view
• Adaptive substitutions occur from a single
  mutational origin
• What happens if the same beneficial allele
• occurs recurrently in a population?

10
Soft sweepfrom recurrent mutation
11
Soft sweepfrom recurrent mutation
12
Soft sweepfrom recurrent mutation
13
Soft sweepfrom recurrent mutation
14
Soft sweepfrom recurrent mutation
frequency ?
time ?
15
Is recurrent mutation relevant?

• What is the probability of a soft sweep under
  recurrent mutation?
• What is the impact on patterns of neutral
  polymorphism?

16
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 q 2Ne u , a 2Ne s, R 2Ne r
• Generation update Wright-Fisher model
  (fitness weighted multinomial sampling)

17
Coalescent viewGenealogy of a sample from a
linked locus

• What can happen one generation back in time?

t 1 t
1- xt
xt
n lines
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Coalescent viewCoalescence of two lines
t 1 t

• Rate per generation

1- xt
xt
19
Coalescent viewRecombination
t 1 t

• Rate per generation

1- xt
xt
20
Coalescent viewNew mutation at selected site
t 1 t

• Rate per generation

1- xt
xt
21
Coalescent view

• Problem Rates for
• coalescence
• recombination
• beneficial mutation
• depend on the frequency x of the selected allele
• stochastic path

22
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 a (strong selection makes broad
  sweep patterns)

23
Coalescent view Coalescence and mutation, sample
of size 2

• Probability for coalescence before mutation
  (single haplotype)

24
Coalescent view Coalescence and mutation, sample
of size 2

• Probability for coalescence before mutation
  (single haplotype)

25
Coalescent view Coalescence and mutation, sample
of size 2

• Probability for coalescence before mutation
  (single haplotype)

26
Coalescent view Coalescence and mutation, sample
of size 2

• Probability for coalescence before mutation
  (single haplotype)

27
Coalescent view Coalescence and mutation, sample
of size 2

• Probability for coalescence before mutation
  (single haplotype)

28
Coalescent view Coalescence and mutation, sample
of size 2

• Probability for single or multiple haplotypes

T1 average time to the first coalescence or
mutation-event
29
Coalescent view Coalescence and mutation, sample
of size 2

• Sampling at time of fixation 0 lt T1 lt Tfix

30
Coalescent view Coalescence and mutation, sample
of size 2

• General sampling Tobs generations after fixation

extra factor can be ignored for Tobs ltlt Ne
31
Coalescent view Coalescence and mutation, sample
of size 2

• Sampling at time of fixation 0 lt T1 lt Tfix

Tfix / Ne 4 log(a) / a , a 2Ne s (scaled
selection strength)
32
Coalescent view Coalescence and mutation, sample
of size 2
Simulation results (? 0.4)
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Coalescent view Coalescence and mutation, sample
of size 2

• For a gt 500 Tfix / Ne ltlt 1, thus

• Corresponds to approximation

34
Coalescent view Coalescence and mutation, sample
of size n
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Coalescent view Coalescence and mutation, sample
of size n
36
Coalescent view Coalescence and mutation, sample
of size n
Continuous time and time rescaling
Neutral coalescent !
37
Coalescent view Coalescence and mutation, sample
of size n

• Problem independent of the path xt and all
  selection parameters

38
Coalescent view Coalescence and mutation, sample
of size n

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

39
Coalescent view Coalescence and mutation, sample
of size n

• Problem independent of the path xt 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

40
ResultsEwens sampling formula

• Probability for k haplotypes, occurring n1,, nk
  times
• in a sample of size n

41
ResultsEwens sampling formula

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

42
ResultsProbability of a soft sweep
43
ResultsProbability of a soft sweep
Simulation (2Ne s 10 000, n 20)
gt4 haplos
100
4 haplos
80
3 haplos
60
2 haplos
40
1 haplo
20
0
q 1
q 4
q 0.4
q 0.04
q 0.004
44
ResultsProbability of a soft sweep
Simulation (2Ne s 10 000, n 20)
gt4 haplos
100
4 haplos
80
3 haplos
60
2 haplos
40
1 haplo
20
0
q 1
q 4
q 0.4
q 0.04
q 0.004
Probability for multiple haplotypes gt 5 for q gt
0.01 gt95 for q gt 1
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ResultsFrequency of major haplotype
0.5
Sample size 10
0.4
a 100
a 1000
0.3
a 10000
prediction
0.2
0.1
0
5/10
6/10
7/10
8/10
9/10
46
When should we expect soft sweeps?Multiple
haplotypes due to recurrent beneficial mutations

• Strong dependence on the mutation rate
• More than 5 for q gt 0.01
• E.g. African D. melanogaster q 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

47
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

48
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

49
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
• Directly proportional to coalescence rate (no
  factor 1- xt)
• Approximation holds exactly in this case

50
Generality of the resultmigration instead of
mutation
M 0.4
q 0.4
a
51
Generality of the resulttime or
frequency-dependent selection

• Results independent of the stochastic path xt 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)

52
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)

53
Generality of the resultvariance in selection
coefficients
?
?
0.01
?
1
0.1
100
90
80
70
gt4
60
4
50
Number of haplotypes
3
40
30
2
20
1
10
0
D
0
0.1
0
0.1
0
0.1
0.2
0.2
0.2
0.01
0.05
0.01
0.05
0.01
0.05
54
Generality of the resultvariance in selection
coefficients
q 0.1
0.4
0.3
D0
Frequency of major haplotype
D0.01
0.2
D0.05
D0.1
0.1
D0.2
0
5/10
6/10
7/10
8/10
9/10
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Footprint of selectionFrequency spectrum of
polymorphic sites

• 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

56
Footprint of selectionFrequency spectrum of
polymorphic sites
Probability of event
57
Footprint of selectionFrequency spectrum of
polymorphic sites
x
1-x
recombination
coalescence
mutation
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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

59
Footprint of selectionPower of Tajimas 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 qn 10
• a 2Ne s 10000
• Sample size 20
• Power of Tajima D for various recombination
  distances and
• sampling times after fixation of the beneficial
  allele

60
Footprint of selectionPower of Tajimas D test
single origin
0
0.01
0.05
Time since fixation in 2Ne
generations
0.1
0.2
0.5
1
100
0
10
20
200
600
Distance in units of R 2Ne r
61
Footprint of selectionPower of Tajimas D test
q 0.1
0
0.01
0.05
Time since fixation in 2Ne
generations
0.1
0.2
0.5
1
100
0
10
20
200
600
Distance in units of R 2Ne r
62
Footprint of selectionPower of Tajimas D test
q 0.4
0
0.01
0.05
Time since fixation in 2Ne
generations
0.1
0.2
0.5
1
100
0
10
20
200
600
Distance in units of R 2Ne r
63
Footprint of selectionPower of Tajimas D test
q 1
0
0.01
0.05
Time since fixation in 2Ne
generations
0.1
0.2
0.5
1
100
0
10
20
200
600
Distance in units of R 2Ne r
64
Footprint of selectionCondition on soft sweeps
negative D
0
q 0.1
0.01
0.05
Time since fixation in 2Ne
generations
0.1
0.2
0.5
1
100
0
10
20
200
600
Distance in units of R 2Ne r
65
Footprint of selectionCondition on soft sweeps
positive D
0
q 0.1
0.01
0.05
Time since fixation in 2Ne
generations
0.1
0.2
0.5
1
100
0
10
20
200
600
Distance in units of R 2Ne r
66
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

67
Footprint of selectionPower of haplotype test
single origin
0
0.01
0.05
Time since fixation in 2Ne
generations
0.1
0.2
0.5
1
100
0
10
20
200
600
Distance in units of R 2Ne r
68
Footprint of selectionPower of haplotype test
q 0.1
0
0.01
0.05
Time since fixation in 2Ne
generations
0.1
0.2
0.5
1
100
0
10
20
200
600
Distance in units of R 2Ne r
69
Footprint of selectionPower of haplotype test
q 0.4
0
0.01
0.05
Time since fixation in 2Ne
generations
0.1
0.2
0.5
1
100
0
10
20
200
600
Distance in units of R 2Ne r
70
Footprint of selectionPower of haplotype test
q 1
0
0.01
0.05
Time since fixation in 2Ne
generations
0.1
0.2
0.5
1
100
0
10
20
200
600
Distance in units of R 2Ne r
71
Footprint of selectionPower of haplotype test
q 4
0
0.01
0.05
Time since fixation in 2Ne
generations
0.1
0.2
0.5
1
100
0
10
20
200
600
Distance in units of R 2Ne r
72
Footprint of selectionCondition on soft sweeps
number of haplotypes
0
q 0.1
0.01
0.05
Time since fixation in 2Ne
generations
0.1
0.2
0.5
1
100
0
10
20
200
600
Distance in units of R 2Ne r
73
Footprint of selectionTests based on linkage
disequilibrium

• 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

74
Footprint of selectionCondition on soft sweeps
ancestral haplotypes
0
q 0.1
0.01
0.05
Time since fixation in 2Ne
generations
0.1
0.2
0.5
1
100
0
10
20
200
600
Distance in units of R 2Ne r
75
Footprint of selectionCondition on soft sweeps
ancestral ZnS
0
q 0.1
0.01
0.05
Time since fixation in 2Ne
generations
0.1
0.2
0.5
1
100
0
10
20
200
600
Distance in units of R 2Ne r
76
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

77
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?