POLYGENIC ADAPTATION: SWEEPS & SHIFTS Adaptation in polygenic traits Criteria for sweeps and shifts Joachim Hermisson Mathematics & Biology, University of Vienna
POLYGENIC ADAPTATION: SWEEPS & SHIFTS Adaptation in polygenic traits Ilse Hölliger Pleuni Pennings University of San Francisco State Vienna University
POLYGENIC ADAPTATION: SWEEPS & SHIFTS Adaptive scenarios quantitative molecular genetics popgen “ Sweeps “ “ Shifts “ • • adaptation due to adaptation due to independent large changes small collective shifts at single loci at many loci clear molecular footprint no clear sweep pattern Pritchard et al. 2010
POLYGENIC ADAPTATION: SWEEPS & SHIFTS Adaptive scenarios quantitative molecular genetics popgen “ Sweeps “ “ Shifts “ Which • • adaptation due to scenario adaptation due to independent large changes small collective shifts should at single loci at many loci clear molecular footprint no clear sweep pattern we expect ? Pritchard et al. 2010
POLYGENIC ADAPTATION: SWEEPS & SHIFTS Adaptive scenarios • panmictic population, new selection pressure Assume: • adaptation from mutation-selection-drift balance few loci many loci standing genetic variation new mutation weak selection strong selection weak mutation strong mutation
POLYGENIC ADAPTATION: SWEEPS & SHIFTS Adaptive scenarios • panmictic population, new selection pressure Assume: • adaptation from mutation-selection-drift balance few loci many loci standing genetic variation new mutation weak selection strong selection weak mutation strong mutation Which scenario is favored under which conditions ?
POLYGENIC ADAPTATION: SWEEPS & SHIFTS Additive quantitative trait under stabilizing selection • 𝑂 haploids, 𝑀 biallelic loci 𝑋 𝑎 = exp − 𝜏 2 (𝑎 − 𝑎 opt ) 2 • recurrent mutation 𝜄 𝑗 = 2𝑂𝑣 𝑗 𝑀 𝛿𝑞 𝑗 fitness W 𝑎 = 𝑗=1 trait Z Z 0 Z opt
POLYGENIC ADAPTATION: SWEEPS & SHIFTS Additive quantitative trait under stabilizing selection • 𝑂 haploids, 𝑀 biallelic loci 𝑋 𝑎 = exp − 𝜏 2 (𝑎 − 𝑎 opt ) 2 • recurrent mutation 𝜄 𝑗 = 2𝑂𝑣 𝑗 𝑀 𝛿𝑞 𝑗 fitness W 𝑎 = 𝑗=1 trait Z Z 0 Z opt
POLYGENIC ADAPTATION: SWEEPS & SHIFTS Additive quantitative trait under stabilizing selection • 𝑂 haploids, 𝑀 biallelic loci 𝑋 𝑎 = exp − 𝜏 2 (𝑎 − 𝑎 opt ) 2 • recurrent mutation 𝜄 𝑗 = 2𝑂𝑣 𝑗 𝑀 𝛿𝑞 𝑗 fitness W 𝑎 = 𝑗=1 𝑎 trait Z Z 0 Z 1 Z opt
POLYGENIC ADAPTATION: SWEEPS & SHIFTS Additive quantitative trait under stabilizing selection • 𝑂 haploids, 𝑀 biallelic loci 𝑋 𝑎 = exp − 𝜏 2 (𝑎 − 𝑎 opt ) 2 • recurrent mutation 𝜄 𝑗 = 2𝑂𝑣 𝑗 𝑀 𝛿𝑞 𝑗 fitness W 𝑎 = 𝑗=1 𝑎 trait Z Z 0 Z 1 Z opt “Architecture of joint distribution polygenic adaptation” of allele frequencies 𝑞 𝑗 :
POLYGENIC ADAPTATION: SWEEPS & SHIFTS Binary trait with complete redundancy • 𝑂 haploids, 𝑀 biallelic loci • recurrent mutation 𝜄 𝑗 = 2𝑂𝑣 𝑗 • fitness function (e.g. resistance): before env. change after env. change fit. fit. 1+ s b 1 1 1- s d # mut. # mut. 0 2 3 0 2 3 1 1 wt mutant phenotype
POLYGENIC ADAPTATION: SWEEPS & SHIFTS Binary trait with complete redundancy • 𝑂 haploids, 𝑀 biallelic loci 𝑋 𝑎 = 1 ± 𝑡 𝑐,𝑒 𝑎 • recurrent mutation 𝜄 𝑗 = 2𝑂𝑣 𝑗 𝑀 (1 − 𝑞 𝑗 ) 𝑎 = 1 − 𝑗=1 • fitness function (e.g. resistance): freq. of mutant phenotype before env. change after env. change fit. fit. 1+ s b 1 1 1- s d # mut. # mut. 0 2 3 0 2 3 1 1 wt mutant phenotype
POLYGENIC ADAPTATION: SWEEPS & SHIFTS Maths of polygenic adaptation Evolutionary trajectories (2 loci, schematic): sampling rapid phenotypic adaptation slow change (neutral) 1 st locus (SGV or) 2 nd locus time
POLYGENIC ADAPTATION: SWEEPS & SHIFTS Maths of polygenic adaptation Evolutionary trajectories (2 loci, schematic): sampling rapid phenotypic adaptation slow change (neutral) 1 st locus (SGV or) 2 nd locus time establishment phase competition phase stochastic: deterministic: mutation & drift selection & epistasis
POLYGENIC ADAPTATION: SWEEPS & SHIFTS Maths of polygenic adaptation Establishment phase (both models): Yule branching process track only mutant copies destined for establishment, prob. 𝑞 est (𝑡 𝑐 , 𝑡 𝑒 ; 𝜏, 𝛿) per locus new lines (mutation) • ~ 𝜄 𝑗 ∙ 𝑞 est time split rate (reproduction) ~ 𝑞 est per line • copies at all loci 𝑜 1 , 𝑜 2 , … mutation and drift during establishment create stochastic differences among loci locus 1 locus 2
POLYGENIC ADAPTATION: SWEEPS & SHIFTS Maths of polygenic adaptation Establishment phase (both models): Yule branching process track only mutant copies destined for establishment, prob. 𝑞 est (𝑡 𝑐 , 𝑡 𝑒 ; 𝜏, 𝛿) per locus new lines (mutation) • ~ 𝜄 𝑗 ∙ 𝑞 est time split rate (reproduction) ~ 𝑞 est per line • copies at all loci 𝑜 1 , 𝑜 2 , … mutation and drift during establishment create stochastic differences among loci ratios independent of 𝑡 𝑐/𝑒 ; 𝜏, 𝛿 𝑦 𝑗 = 𝑜 𝑗 /𝑜 1 locus 1 locus 2
POLYGENIC ADAPTATION: SWEEPS & SHIFTS Maths of polygenic adaptation Establishment phase (both models): Yule branching process track only mutant copies destined for establishment, prob. 𝑞 est (𝑡 𝑐 , 𝑡 𝑒 ; 𝜏, 𝛿) per locus new lines (mutation) • ~ 𝜄 𝑗 ∙ 𝑞 est time split rate (reproduction) ~ 𝑞 est per line • copies at all loci 𝑜 1 , 𝑜 2 , … mutation and drift during establishment create stochastic differences among loci ratios independent of 𝑡 𝑐/𝑒 ; 𝜏, 𝛿 𝑦 𝑗 = 𝑜 𝑗 /𝑜 1 joint distribution of frequency ratios 𝑦 𝑗 depends only on mutation rates 𝜄 𝑗 : locus 1 locus 2 (inverted Dirichlet distribution)
POLYGENIC ADAPTATION: SWEEPS & SHIFTS Maths of polygenic adaptation Competition phase: binary trait model • deterministic dynamics maintains ratios • 𝑞 𝑗 𝑒 𝑞 𝑗 = 𝑞 𝑗 𝑡 𝑐 1 − 𝑎 ⟹ = 0 zooms up differences 𝑒𝑢 𝑞 𝑘
POLYGENIC ADAPTATION: SWEEPS & SHIFTS Maths of polygenic adaptation Competition phase: binary trait model • deterministic dynamics maintains ratios • 𝑞 𝑗 𝑒 𝑞 𝑗 = 𝑞 𝑗 𝑡 𝑐 1 − 𝑎 ⟹ = 0 zooms up differences 𝑒𝑢 𝑞 𝑘 • joint distribution of mutant frequencies 𝑞 𝑗 at 𝑥 : 𝑎 = 1 − 𝑔
POLYGENIC ADAPTATION: SWEEPS & SHIFTS Maths of polygenic adaptation Competition phase: binary trait model • deterministic dynamics maintains ratios • 𝑞 𝑗 𝑒 𝑞 𝑗 = 𝑞 𝑗 𝑡 𝑐 1 − 𝑎 ⟹ = 0 zooms up differences 𝑒𝑢 𝑞 𝑘 • joint distribution of mutant frequencies 𝑞 𝑗 at 𝑥 : 𝑎 = 1 − 𝑔 • depends only on mutation rates 𝜄 𝑗 • independent of selection strength 𝑡 𝑐 , 𝑡 𝑒
POLYGENIC ADAPTATION: SWEEPS & SHIFTS Maths of polygenic adaptation Competition phase: quantitative trait [DeVladar/Barton 2014 • deterministic dynamics (LE and weak selection) Jain/Stephan 2017] 𝑎 + 𝜏𝛿 2 (2𝑞 𝑗 − 1) 𝑞 𝑗 = 𝑞 𝑗 1 − 𝑞 𝑗 𝜏𝛿 𝑎 opt − disruptive directional selection selection
POLYGENIC ADAPTATION: SWEEPS & SHIFTS Maths of polygenic adaptation Competition phase: quantitative trait [DeVladar/Barton 2014 • deterministic dynamics (LE and weak selection) Jain/Stephan 2017] 𝑞 𝑗 = 𝑞 𝑗 1 − 𝑞 𝑗 𝜏𝛿 𝑎 opt − 𝑎 directional selection
POLYGENIC ADAPTATION: SWEEPS & SHIFTS Maths of polygenic adaptation Competition phase: quantitative trait [DeVladar/Barton 2014 • deterministic dynamics (LE and weak selection) Jain/Stephan 2017] 𝑞 𝑗 𝑧 𝑗 𝑒 𝑧 𝑗 ≔ 𝑞 𝑗 = 𝑞 𝑗 1 − 𝑞 𝑗 𝜏𝛿 𝑎 opt − 𝑎 ⟹ = 0 1−𝑞 𝑗 𝑒𝑢 𝑧 𝑘
POLYGENIC ADAPTATION: SWEEPS & SHIFTS Maths of polygenic adaptation Competition phase: quantitative trait [DeVladar/Barton 2014 • deterministic dynamics (LE and weak selection) Jain/Stephan 2017] 𝑞 𝑗 𝑧 𝑗 𝑒 𝑧 𝑗 ≔ 𝑞 𝑗 = 𝑞 𝑗 1 − 𝑞 𝑗 𝜏𝛿 𝑎 opt − 𝑎 ⟹ = 0 1−𝑞 𝑗 𝑒𝑢 𝑧 𝑘 • joint distribution of mutant frequencies 𝑞 𝑗 at 𝑎 = 𝑎 1 = 𝛿𝑑 𝑎 : • depends only on mutation rates 𝜄 𝑗 • independent of locus effect and selection strength 𝛿, 𝜏
POLYGENIC ADAPTATION: SWEEPS & SHIFTS Results: binary trait • equal loci, 𝜄 𝑗 = 𝜄 • start in mutation-selection-drift balance • adaptation until 95% mt phenotypes ( 𝑎 = 1 − 𝑔 𝑥 = 0.95) • loci ordered according to their contribution to the adaptive response: – locus with largest frequency: major locus – all other loci: minor loci
POLYGENIC ADAPTATION: SWEEPS & SHIFTS Relative adaptive response (2 loci) 𝑞 < 𝑞 > 𝜄
POLYGENIC ADAPTATION: SWEEPS & SHIFTS Relative adaptive response (2 loci) 𝑞 < 𝑞 > N = 10000 , sampling at 95% mt. phenotype s b N = s d N = 1000, LE 𝜄
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