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The effect of gene interactions on the long-term response to selection

Tiago Paix˜ ao Nick Barton

Institute for Science and Technology Austria

Dagsthul 2016

Tiago Paix˜ ao, Nick Barton Epistasis

adaptation from standing variation

No new mutations Recombining population

Tiago Paix˜ ao, Nick Barton Epistasis

Variance Decomposition

Trait: z = α1X1 + α2X2 + ǫ12X1X2 Assuming “linkage equilibrium” E[XiXj] = E[Xi]E[Xj]. Mean: z = α1p1 + α2p2 + ǫ12p1p2 where pi = E[Xi]

Tiago Paix˜ ao, Nick Barton Epistasis

Variance Decomposition

Trait: z = α1X1 + α2X2 + ǫ12X1X2 Assuming “linkage equilibrium” E[XiXj] = E[Xi]E[Xj]. Mean: z = α1p1 + α2p2 + ǫ12p1p2 where pi = E[Xi]

Tiago Paix˜ ao, Nick Barton Epistasis

Variance Decomposition

Trait: z = α1X1 + α2X2 + ǫ12X1X2 Variance in the population: Var[z] =α2

1V1 + α2 2V2 + ǫ2 12V1V2 + α1α2Cov[X1, X2]

• =0

+ α1ǫ12 Cov[X1, X1X2]

• V1p2

+α2ǫ12 Cov[X2, X1X2]

• V2p1

= (α1 + ǫ12p2)2

• A2

1

V1 + (α1 + ǫ12p1)2

• A2

2

V2

• VA

+ ǫ2

12V1V2

• VAA

where Vi = Var[Xi] = E[X 2

i ] − E[Xi]2

Tiago Paix˜ ao, Nick Barton Epistasis

Variance Decomposition

Trait: z = α1X1 + α2X2 + ǫ12X1X2 Variance in the population: Var[z] =α2

1V1 + α2 2V2 + ǫ2 12V1V2 + α1α2Cov[X1, X2]

• =0

+ α1ǫ12 Cov[X1, X1X2]

• V1p2

+α2ǫ12 Cov[X2, X1X2]

• V2p1

= (α1 + ǫ12p2)2

• A2

1

V1 + (α1 + ǫ12p1)2

• A2

2

V2

• VA

+ ǫ2

12V1V2

• VAA

where Vi = Var[Xi] = E[X 2

i ] − E[Xi]2

Tiago Paix˜ ao, Nick Barton Epistasis

Variance Decomposition

Trait: z = α1X1 + α2X2 + ǫ12X1X2 Variance in the population: Var[z] =α2

1V1 + α2 2V2 + ǫ2 12V1V2 + α1α2Cov[X1, X2]

• =0

+ α1ǫ12 Cov[X1, X1X2]

• V1p2

+α2ǫ12 Cov[X2, X1X2]

• V2p1

= (α1 + ǫ12p2)2

• A2

1

V1 + (α1 + ǫ12p1)2

• A2

2

V2

• VA

+ ǫ2

12V1V2

• VAA

where Vi = Var[Xi] = E[X 2

i ] − E[Xi]2

Tiago Paix˜ ao, Nick Barton Epistasis

Variance Decomposition

More generally Vz = VA + VAA + VAAA + . . . VA =

• i

∂z ∂pi 2 Vi VAA = 1 2

• ij

∂2z ∂pi∂pj 2 ViVj VAAA = 1 3!

• ijk
• ∂3z

∂pi∂pj∂pk 2 ViVjVk . . .

Tiago Paix˜ ao, Nick Barton Epistasis

Response to selection

zt = α1pt

1 + α2pt 2 + ǫ12pt 1pt 2

∆z = Ai∆pi if you remember that, if fitness is w = 1 + βz ∆pi =pi(1 − pi)∂w ∂pi = βAiVi and so: ∆z =

• i

Ai∆pi = βA2

i Vi

= βVA

Tiago Paix˜ ao, Nick Barton Epistasis

Response to selection

zt = α1pt

1 + α2pt 2 + ǫ12pt 1pt 2

∆z = Ai∆pi if you remember that, if fitness is w = 1 + βz ∆pi =pi(1 − pi)∂w ∂pi = βAiVi and so: ∆z =

• i

Ai∆pi = βA2

i Vi

= βVA

Tiago Paix˜ ao, Nick Barton Epistasis

adaptation from standing variation

No new mutations Recombining population

Tiago Paix˜ ao, Nick Barton Epistasis

Limits on Additive functions I: a fixation probability derivation

Robertson (1960) Linear Functions: assuming weak selection, prob.

• f fixation can be approximated by

u (p0) = 1 − exp (−4Nesp0) 1 − exp (−4Nes) = p0 + 2p0 (1 − p0) Nes + O

• (Nes)2

and the expected total response is (summing the response at each generation): R∞ =

• i

αi

• p∞

i

− p0

i

• =
• i

αi

• u
• p0

i

• − p0

i

• ≈ 2
• i

α2

i p0 i

• 1 − p0

i

• βNe = 2βNeV 0

A = 2NeR0

Tiago Paix˜ ao, Nick Barton Epistasis

Limits on Additive functions I: a fixation probability derivation

Robertson (1960) Linear Functions: assuming weak selection, prob.

• f fixation can be approximated by

u (p0) = 1 − exp (−4Nesp0) 1 − exp (−4Nes) = p0 + 2p0 (1 − p0) Nes + O

• (Nes)2

and the expected total response is (summing the response at each generation): R∞ =

• i

αi

• p∞

i

− p0

i

• =
• i

αi

• u
• p0

i

• − p0

i

• ≈ 2
• i

α2

i p0 i

• 1 − p0

i

• βNe = 2βNeV 0

A = 2NeR0

Tiago Paix˜ ao, Nick Barton Epistasis

Limits on Additive functions I: a fixation probability derivation

Robertson (1960) Linear Functions: assuming weak selection, prob.

• f fixation can be approximated by

u (p0) = 1 − exp (−4Nesp0) 1 − exp (−4Nes) = p0 + 2p0 (1 − p0) Nes + O

• (Nes)2

and the expected total response is (summing the response at each generation): R∞ =

• i

αi

• p∞

i

− p0

i

• =
• i

αi

• u
• p0

i

• − p0

i

• ≈ 2
• i

α2

i p0 i

• 1 − p0

i

• βNe = 2βNeV 0

A = 2NeR0

Tiago Paix˜ ao, Nick Barton Epistasis

Limits on Additive functions I: a fixation probability derivation

Robertson (1960) Linear Functions: assuming weak selection, prob.

• f fixation can be approximated by

u (p0) = 1 − exp (−4Nesp0) 1 − exp (−4Nes) = p0 + 2p0 (1 − p0) Nes + O

• (Nes)2

and the expected total response is (summing the response at each generation): R∞ =

• i

αi

• p∞

i

− p0

i

• =
• i

αi

• u
• p0

i

• − p0

i

• ≈ 2
• i

α2

i p0 i

• 1 − p0

i

• βNe = 2βNeV 0

A = 2NeR0

Tiago Paix˜ ao, Nick Barton Epistasis

Limits on Additive functions II: a QG derivation

Assume that selection is weak such that at any individual locus, the dynamics are mostly determined by the sampling noise (genetic drift) Infinitesimal Model E[pt+1] = pt V t+1

i

= (1 − F)V t

i

Rt = βV t

A

For a randomly mating population F = 1

N .

This would be the case if an infinite number of genes contributed infinitesimally to the trait. But as long as the expected change in allele frequency is small compared to its variance (∆pi = pi(1 − pi) ∂w

∂pi ≪ pi(1−pi) N

) it should approximately apply.

Tiago Paix˜ ao, Nick Barton Epistasis

Limits on Additive functions II: a QG derivation

Additive variance then evolves as V t

A =

• i

α2

i V t i =

• i

α2

i (1 − F)V t−1 i

=

• i

α2

i (1 − F)tV 0 i = (1 − F)tV 0 A

and the total response is: R∞ = β

∞

• t=0

V t

A = β ∞

• t=0

V 0

A

• 1 − 1

N t = βNV 0

A

Tiago Paix˜ ao, Nick Barton Epistasis

Limits on Additive functions II: a QG derivation

Additive variance then evolves as V t

A =

• i

α2

i V t i =

• i

α2

i (1 − F)V t−1 i

=

• i

α2

i (1 − F)tV 0 i = (1 − F)tV 0 A

and the total response is: R∞ = β

∞

• t=0

V t

A = β ∞

• t=0

V 0

A

• 1 − 1

N t = βNV 0

A

Tiago Paix˜ ao, Nick Barton Epistasis

What about arbitrary GP maps?

Tiago Paix˜ ao, Nick Barton Epistasis

An arbitrary trait z =

• i

αiXi + 1 2

• ij

αijXiXj + 1 3!αijkXiXjXk + . . . VG = VA + VA(2) + VA(3) + . . . Infinitesimal Model E[pt+1] = pt V t+1

i

= (1 − F)V t

i

Rt = βV t

A

dynamics of variance components V t+1

A(k) = (1 − F) ∞

• k′=k

k′F k′−1V t

A(k′)

Tiago Paix˜ ao, Nick Barton Epistasis

An arbitrary trait z =

• i

αiXi + 1 2

• ij

αijXiXj + 1 3!αijkXiXjXk + . . . VG = VA + VA(2) + VA(3) + . . . Infinitesimal Model E[pt+1] = pt V t+1

i

= (1 − F)V t

i

Rt = βV t

A

dynamics of variance components V t+1

A(k) = (1 − F) ∞

• k′=k

k′F k′−1V t

A(k′)

Tiago Paix˜ ao, Nick Barton Epistasis

An arbitrary trait z =

• i

αiXi + 1 2

• ij

αijXiXj + 1 3!αijkXiXjXk + . . . VG = VA + VA(2) + VA(3) + . . . Infinitesimal Model E[pt+1] = pt V t+1

i

= (1 − F)V t

i

Rt = βV t

A

dynamics of variance components V t+1

A(k) = (1 − F) ∞

• k′=k

k′F k′−1V t

A(k′)

Tiago Paix˜ ao, Nick Barton Epistasis

Response to selection

In particular, for VA VA =(1 − F)

• V 0

A + 2FV 0 AA + 3F 2V 0 AAA . . .

• =(1 − F)

∞

• k=1

kF k−1V 0

A(k)

Putting it all together: R∞ =β

∞

• t=0

V t

A = β ∞

• t=0

(1 − Ft)

∞

• k=1

kF k−1

t

V 0

A(k)

=βNe

∞

• k=1

V 0

A(k) = βNeV 0 G

The long-term response to selection depends only on the initial genetic variance in the population, regardless of genetic architecture.

Tiago Paix˜ ao, Nick Barton Epistasis

Response to selection

In particular, for VA VA =(1 − F)

• V 0

A + 2FV 0 AA + 3F 2V 0 AAA . . .

• =(1 − F)

∞

• k=1

kF k−1V 0

A(k)

Putting it all together: R∞ =β

∞

• t=0

V t

A = β ∞

• t=0

(1 − Ft)

∞

• k=1

kF k−1

t

V 0

A(k)

=βNe

∞

• k=1

V 0

A(k) = βNeV 0 G

The long-term response to selection depends only on the initial genetic variance in the population, regardless of genetic architecture.

Tiago Paix˜ ao, Nick Barton Epistasis

Response to selection

In particular, for VA VA =(1 − F)

• V 0

A + 2FV 0 AA + 3F 2V 0 AAA . . .

• =(1 − F)

∞

• k=1

kF k−1V 0

A(k)

Putting it all together: R∞ =β

∞

• t=0

V t

A = β ∞

• t=0

(1 − Ft)

∞

• k=1

kF k−1

t

V 0

A(k)

=βNe

∞

• k=1

V 0

A(k) = βNeV 0 G

The long-term response to selection depends only on the initial genetic variance in the population, regardless of genetic architecture.

Tiago Paix˜ ao, Nick Barton Epistasis

Response to selection

In particular, for VA VA =(1 − F)

• V 0

A + 2FV 0 AA + 3F 2V 0 AAA . . .

• =(1 − F)

∞

• k=1

kF k−1V 0

A(k)

Putting it all together: R∞ =β

∞

• t=0

V t

A = β ∞

• t=0

(1 − Ft)

∞

• k=1

kF k−1

t

V 0

A(k)

=βNe

∞

• k=1

V 0

A(k) = βNeV 0 G

The long-term response to selection depends only on the initial genetic variance in the population, regardless of genetic architecture.

Tiago Paix˜ ao, Nick Barton Epistasis

Long-term response as a function of epistatic variance

13 0. 0.5 1. 1.5 2. 2.5 VAA

0 VG

R RΒNe VG Ne750 Ne500 Ne250

Tiago Paix˜ ao, Nick Barton Epistasis

The infinitesimal approach holds for large numbers of loci

0.01 1 100 104 0.1 0.5 1.0 5.0 10.0 50.0 100.0

NeΒ R VG

Tiago Paix˜ ao, Nick Barton Epistasis

Conclusions Simple predictions for the long-term response Genetic architecture is irrelevant under weak selection Infinitesimal model holds for epistatic architectures and for strong selection on the trait (provided there are many genes contributing) Under strong selection, all that matters is the shape of the fitness landscapes (in particular, if sign reversal of alleles happens ) Open issues / Future stuff It might be useful to analyse GAs at the early stages of adaptation, taking into account the initial variation in the population.a Make this analysis rigorous (is this an upper bound for the response?)

aPaix˜

ao, T., and Barton, N. A Variance Decomposition Approach to the Analysis of Genetic Algorithms. (GECCO 2013), 845–852.

Tiago Paix˜ ao, Nick Barton Epistasis

Conclusions Simple predictions for the long-term response Genetic architecture is irrelevant under weak selection Infinitesimal model holds for epistatic architectures and for strong selection on the trait (provided there are many genes contributing) Under strong selection, all that matters is the shape of the fitness landscapes (in particular, if sign reversal of alleles happens ) Open issues / Future stuff It might be useful to analyse GAs at the early stages of adaptation, taking into account the initial variation in the population.a Make this analysis rigorous (is this an upper bound for the response?)

aPaix˜

ao, T., and Barton, N. A Variance Decomposition Approach to the Analysis of Genetic Algorithms. (GECCO 2013), 845–852.

Tiago Paix˜ ao, Nick Barton Epistasis