Adaptation of a quantitative trait to a moving optimum

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Adaptation of a quantitative trait to a moving
optimum

• Michael Kopp Joachim Hermisson

Ludwig-Maximilian-University Munich
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How does adaptation work?

• How many steps (substitutions)?
• What distribution of step sizes ?
• What order of step sizes?

Old optimum
New optimum
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Classical picture of adaptive walks

• Sudden environmental change
• Low mutation rate

Old optimum
New optimum

• Small overall number of steps
• Exponential distribution of step sizes
• Large steps first

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Our approach

• Moving optimum (global warming, range expansion,
  coevolution)
• Arbitrary mutation rates
• Sexual reproduction
• Multilocus quantitative trait

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Stabilizing selection with a moving optimum
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Optimal phenotype
0
Time
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Fixation of a single beneficial mutation
Lag time TL
Waiting time TW
Fixation time TF
Frequency
Phenotype /
A
a
Time
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Expected time to fixation
Waiting time
Total time to fixation
Fixation time
Lag time
Mutational effect
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Expected time to fixation
Mutation rate
Low
High
(Nu 0.01)
(Nu 10)
Fast
(1 gen.)
Time to fixation
Environmental change
Slow
(4000 gen.)
Mutational effect
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Fastest mutation
Small mutationsare favored by
Effect of fastest mutation

• Slow env. change
• High mutation rate
• Strong selection

Duration of environmental change (in 1000 gen.)
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A race between two mutant alleles
Frequency
Phenotype /
A b
a b
a B
Phenotype /
Frequency
a b
Time
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A race between two mutant alleles
Frequency
Phenotype /
A b
a b
TWA gt TWB(high variance)
TLA lt TLB
TFA gt TFB
a B
Phenotype /
Frequency
a b
Time
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The environmentally-limited regime
Frequency
Phenotype /
A b
a b
a B
Frequency
Phenotype /
a b
Time
TL gtgt TWTF Small allele fixes earlier
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The mutation-limited regime
Frequency
Phenotype /
A b
a b
a B
Frequency
Phenotype /
a b
Time
TW gtgt TLTF Probability to fix first
proportional to (final) selection
coefficient
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The fixation time-limited regime
Frequency
Phenotype /
A b
a b
a B
Frequency
Phenotype /
a b
Time
TF gtgt TLTW Large allele is faster
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Three regimes
Fixation time-limited(s small)Large allele
first
Mutation-limited(Nus small)Large allele first
on average
Speed of environm. change
Environmentally-limited(v small)Small allele
first
Mutation rate Nu
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Simulation results and analytical prediction
1
1/500
Speed of environm. change
1/1000
1/2000
1/4000
0.01
0.1
1
10
Mutation rate Nu
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Conclusions

• Theory for constant selection does not generalize
  to cases with gradual environmental change.
• Slow environmental change favors small mutational
  steps.
• Three potentially limiting time-scales for
  adaptation.
• One-locus theory can partially explain multilocus
  results (work in progress).

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Is the fastest allele from the one-locus case
also the fastest in the two-locus case?
Mutation rate
Low
High
(Nu 0.01)
(Nu 10)
Fast
(1 gen.)
Winning rate of optimal allele
Environmental change
Slow
(4000 gen.)
Mutational effect of competitor allele
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Overview

• One-locus model
• Two-locus model
• Multilocus model Characteristics of adaptive
  walks

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Multilocus simulations

• Optimum moves from 0 (wild type) to 1
• 40 haploid, diallelic loci
• Locus effects drawn from uniform distribution
  between -2 and 2
• Summary statistics over 1000 simulations with
  different locus effects

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Mean number of steps
1
1/500
Speed of environm. change
1/1000
1/2000
1/4000
0.01
0.1
1
10
Mutation rate Nu
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Distribution of fixed alleles
Mutation rate
Low
High
(Nu 0.01)
(Nu 10)
Mutation-limited
Fixation time-limited
Fast
(1 gen.)
Frequency
Environmental change
Environment- limited
Slow
(4000 gen.)
Mutational effect
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Sequence of steps
Mutation rate
Low
High
(Nu 0.01)
(Nu 10)
Fixation time-limited
Mutation-limited
Fast
(1 gen.)
Mutational effect
Environmental change
Environment- limited
Slow
(4000 gen.)
Rank of step