The effect of gene interactions on the long-term response to - PowerPoint PPT Presentation

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 = α 1 X 1 + α 2 X 2 + ǫ 12 X 1 X 2 Assuming “linkage equilibrium” E [ X i X j ] = E [ X i ] E [ X j ]. Mean: z = α 1 p 1 + α 2 p 2 + ǫ 12 p 1 p 2 where p i = E [ X i ] Tiago Paix˜ ao, Nick Barton Epistasis

Variance Decomposition Trait: z = α 1 X 1 + α 2 X 2 + ǫ 12 X 1 X 2 Assuming “linkage equilibrium” E [ X i X j ] = E [ X i ] E [ X j ]. Mean: z = α 1 p 1 + α 2 p 2 + ǫ 12 p 1 p 2 where p i = E [ X i ] Tiago Paix˜ ao, Nick Barton Epistasis

Variance Decomposition Trait: z = α 1 X 1 + α 2 X 2 + ǫ 12 X 1 X 2 Variance in the population: Var [ z ] = α 2 1 V 1 + α 2 2 V 2 + ǫ 2 12 V 1 V 2 + α 1 α 2 Cov [ X 1 , X 2 ] � �� � =0 + α 1 ǫ 12 Cov [ X 1 , X 1 X 2 ] + α 2 ǫ 12 Cov [ X 2 , X 1 X 2 ] � �� � � �� � V 1 p 2 V 2 p 1 = ( α 1 + ǫ 12 p 2 ) 2 V 1 + ( α 1 + ǫ 12 p 1 ) 2 + ǫ 2 V 2 12 V 1 V 2 � �� � � �� � � �� � A 2 A 2 V AA 1 2 � �� � V A where V i = Var [ X i ] = E [ X 2 i ] − E [ X i ] 2 Tiago Paix˜ ao, Nick Barton Epistasis

Variance Decomposition Trait: z = α 1 X 1 + α 2 X 2 + ǫ 12 X 1 X 2 Variance in the population: Var [ z ] = α 2 1 V 1 + α 2 2 V 2 + ǫ 2 12 V 1 V 2 + α 1 α 2 Cov [ X 1 , X 2 ] � �� � =0 + α 1 ǫ 12 Cov [ X 1 , X 1 X 2 ] + α 2 ǫ 12 Cov [ X 2 , X 1 X 2 ] � �� � � �� � V 1 p 2 V 2 p 1 = ( α 1 + ǫ 12 p 2 ) 2 V 1 + ( α 1 + ǫ 12 p 1 ) 2 + ǫ 2 V 2 12 V 1 V 2 � �� � � �� � � �� � A 2 A 2 V AA 1 2 � �� � V A where V i = Var [ X i ] = E [ X 2 i ] − E [ X i ] 2 Tiago Paix˜ ao, Nick Barton Epistasis

Variance Decomposition Trait: z = α 1 X 1 + α 2 X 2 + ǫ 12 X 1 X 2 Variance in the population: Var [ z ] = α 2 1 V 1 + α 2 2 V 2 + ǫ 2 12 V 1 V 2 + α 1 α 2 Cov [ X 1 , X 2 ] � �� � =0 + α 1 ǫ 12 Cov [ X 1 , X 1 X 2 ] + α 2 ǫ 12 Cov [ X 2 , X 1 X 2 ] � �� � � �� � V 1 p 2 V 2 p 1 = ( α 1 + ǫ 12 p 2 ) 2 V 1 + ( α 1 + ǫ 12 p 1 ) 2 + ǫ 2 V 2 12 V 1 V 2 � �� � � �� � � �� � A 2 A 2 V AA 1 2 � �� � V A where V i = Var [ X i ] = E [ X 2 i ] − E [ X i ] 2 Tiago Paix˜ ao, Nick Barton Epistasis

Variance Decomposition More generally V z = V A + V AA + V AAA + . . . � ∂ z � 2 � V A = V i ∂ p i i � ∂ 2 z � 2 V AA = 1 � V i V j 2 ∂ p i ∂ p j ij � � 2 ∂ 3 z V AAA = 1 � V i V j V k 3! ∂ p i ∂ p j ∂ p k ijk . . . Tiago Paix˜ ao, Nick Barton Epistasis

Response to selection z t = α 1 p t 1 + α 2 p t 2 + ǫ 12 p t 1 p t 2 ∆ z = A i ∆ p i if you remember that, if fitness is w = 1 + β z ∆ p i = p i (1 − p i ) ∂ w = β A i V i ∂ p i and so: � A i ∆ p i = β A 2 ∆ z = i V i i = β V A Tiago Paix˜ ao, Nick Barton Epistasis

Response to selection z t = α 1 p t 1 + α 2 p t 2 + ǫ 12 p t 1 p t 2 ∆ z = A i ∆ p i if you remember that, if fitness is w = 1 + β z ∆ p i = p i (1 − p i ) ∂ w = β A i V i ∂ p i and so: � A i ∆ p i = β A 2 ∆ z = i V i i = β V A 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. of fixation can be approximated by u ( p 0 ) = 1 − exp ( − 4 N e sp 0 ) 1 − exp ( − 4 N e s ) � ( N e s ) 2 � = p 0 + 2 p 0 (1 − p 0 ) N e s + O and the expected total response is (summing the response at each generation): � � R ∞ = � � � � � � − p 0 p 0 − p 0 p ∞ α i = α i u i i i i i i � � � α 2 i p 0 1 − p 0 β N e = 2 β N e V 0 A = 2 N e R 0 ≈ 2 i i i Tiago Paix˜ ao, Nick Barton Epistasis

Limits on Additive functions I: a fixation probability derivation Robertson (1960) Linear Functions: assuming weak selection, prob. of fixation can be approximated by u ( p 0 ) = 1 − exp ( − 4 N e sp 0 ) 1 − exp ( − 4 N e s ) � ( N e s ) 2 � = p 0 + 2 p 0 (1 − p 0 ) N e s + O and the expected total response is (summing the response at each generation): � � R ∞ = � � � � � � − p 0 p 0 − p 0 p ∞ α i = α i u i i i i i i � � � α 2 i p 0 1 − p 0 β N e = 2 β N e V 0 A = 2 N e R 0 ≈ 2 i i i Tiago Paix˜ ao, Nick Barton Epistasis

Limits on Additive functions I: a fixation probability derivation Robertson (1960) Linear Functions: assuming weak selection, prob. of fixation can be approximated by u ( p 0 ) = 1 − exp ( − 4 N e sp 0 ) 1 − exp ( − 4 N e s ) � ( N e s ) 2 � = p 0 + 2 p 0 (1 − p 0 ) N e s + O and the expected total response is (summing the response at each generation): � � R ∞ = � � � � � � − p 0 p 0 − p 0 p ∞ α i = α i u i i i i i i � � � α 2 i p 0 1 − p 0 β N e = 2 β N e V 0 A = 2 N e R 0 ≈ 2 i i i Tiago Paix˜ ao, Nick Barton Epistasis

Limits on Additive functions I: a fixation probability derivation Robertson (1960) Linear Functions: assuming weak selection, prob. of fixation can be approximated by u ( p 0 ) = 1 − exp ( − 4 N e sp 0 ) 1 − exp ( − 4 N e s ) � ( N e s ) 2 � = p 0 + 2 p 0 (1 − p 0 ) N e s + O and the expected total response is (summing the response at each generation): � � R ∞ = � � � � � � − p 0 p 0 − p 0 p ∞ α i = α i u i i i i i i � � � α 2 i p 0 1 − p 0 β N e = 2 β N e V 0 A = 2 N e R 0 ≈ 2 i i i 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 [ p t +1 ] = p t V t +1 = (1 − F ) V t i i R t = β V 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 ∂ p i ≪ p i (1 − p i ) (∆ p i = p i (1 − p i ) ∂ w ) it should approximately apply. N Tiago Paix˜ ao, Nick Barton Epistasis

Limits on Additive functions II: a QG derivation Additive variance then evolves as � � V t α 2 i V t α 2 i (1 − F ) V t − 1 A = i = i i i � α 2 i (1 − F ) t V 0 i = (1 − F ) t V 0 = A i and the total response is: � � t ∞ ∞ 1 − 1 � � V t V 0 R ∞ = A = β β A N t =0 t =0 β NV 0 = A Tiago Paix˜ ao, Nick Barton Epistasis

Limits on Additive functions II: a QG derivation Additive variance then evolves as � � V t α 2 i V t α 2 i (1 − F ) V t − 1 A = i = i i i � α 2 i (1 − F ) t V 0 i = (1 − F ) t V 0 = A i and the total response is: � � t ∞ ∞ 1 − 1 � � V t V 0 R ∞ = A = β β A N t =0 t =0 β NV 0 = A Tiago Paix˜ ao, Nick Barton Epistasis

What about arbitrary GP maps? Tiago Paix˜ ao, Nick Barton Epistasis

An arbitrary trait α i X i + 1 α ij X i X j + 1 � � z = 3! α ijk X i X j X k + . . . 2 i ij V G = V A + V A (2) + V A (3) + . . . Infinitesimal Model E [ p t +1 ] = p t V t +1 = (1 − F ) V t i i R t = β V t A dynamics of variance components ∞ � V t +1 k ′ F k ′ − 1 V t A ( k ) = (1 − F ) A ( k ′ ) k ′ = k Tiago Paix˜ ao, Nick Barton Epistasis

An arbitrary trait α i X i + 1 α ij X i X j + 1 � � z = 3! α ijk X i X j X k + . . . 2 i ij V G = V A + V A (2) + V A (3) + . . . Infinitesimal Model E [ p t +1 ] = p t V t +1 = (1 − F ) V t i i R t = β V t A dynamics of variance components ∞ � V t +1 k ′ F k ′ − 1 V t A ( k ) = (1 − F ) A ( k ′ ) k ′ = k Tiago Paix˜ ao, Nick Barton Epistasis

An arbitrary trait α i X i + 1 α ij X i X j + 1 � � z = 3! α ijk X i X j X k + . . . 2 i ij V G = V A + V A (2) + V A (3) + . . . Infinitesimal Model E [ p t +1 ] = p t V t +1 = (1 − F ) V t i i R t = β V t A dynamics of variance components ∞ � V t +1 k ′ F k ′ − 1 V t A ( k ) = (1 − F ) A ( k ′ ) k ′ = k Tiago Paix˜ ao, Nick Barton Epistasis

Response to selection In particular, for V A � � V 0 A + 2 FV 0 AA + 3 F 2 V 0 V A =(1 − F ) AAA . . . ∞ � kF k − 1 V 0 =(1 − F ) A ( k ) k =1 Putting it all together: ∞ ∞ ∞ � � � R ∞ = β V t kF k − 1 V 0 A = β (1 − F t ) t A ( k ) t =0 t =0 k =1 ∞ � V 0 A ( k ) = β N e V 0 = β N e G k =1 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