Adaptation dynamics of a polygenic trait Kavita Jain J. Nehru - PowerPoint PPT Presentation

Adaptation dynamics of a polygenic trait Kavita Jain J. Nehru Centre, Bangalore K. Jain & W. Stephan G3 (2015); Genetics (2017); MBE (2017) K. Jain & A. Devi, EPL (2018)

� ✄ ✷ ✂ ✁ t Modes of polygenic adaptation AlleleFrequency AlleleFrequency 1. 0.7 0.8 0.65 0.6 0.6 0.4 0.55 0.2 time 10 1 10 3 10 4 10 5 10 1 10 3 10 4 10 5 10 6 10 10 Subtle shift in allele frequencies Sweeps in allele frequencies → Quantitative-genetic picture → Population-genetic picture Integrate these approaches into a single framework

Phenotype-Genotype map How selection acts on phenotype Phenotypic dynamics ↓ ↑ Phenotype-to-Genotype map Genotype-to-Phenotype map ↓ ↑ Work out allele frequency dynamics But P-G maps are too complicated, not much known Make simple and reasonable assumptions (e.g., additive map)

Cumulant hierarchy (B¨ urger 1991) Mean trait dynamics depend on genetic variance (and higher cumulants) Variance dynamics depend on skewness (and higher cumulants) ... • assume variance is roughly constant (Lande 1983, Turelli & Barton 1990, Rattray & Shapiro 2001,...) but not always a good approximation • come up with effective models for specific situations (Chevin & Hospital 2008) but not general/detailed enough • numerical simulations of full model but restricted to few loci (Pavlidis et al. 2012, Franssen et al. 2017, ...)

Today’s talk - consider a detailed model that accommodates both sweeps and shifts (de Vladar & Barton 2014) - develop new approximations that allow detailed understanding of dynamics (Jain & Stephan 2015, 2017)

Model (de Vladar & Barton, 2014) • Infinitely large population of diploids • Linkage equilibrium • Additive phenotype-genotype map • Quantitative trait determined by finite number of biallelic loci • Effect sizes γ are locus-depn but do not depend on ℓ • Trait is under stabilising selection w ( z ) = 1 − ( s/ 2)( z − z 0 ) 2

Allele frequency dynamics (Wright 1935, Barton 1986) • Initially population is equilibrated to z 0 • Then phenotypic optimum moves from z 0 to z f p i q i ∂ ln ¯ w p i ˙ = + Mutation term 2 ∂p i + sγ 2 i = − sγ i (¯ z − z f ) p i q i 2 p i q i (2 p i − 1) + µ ( q i − p i ) � �� � � �� � � �� � MUTATION DIRECTIONAL FIXATION ℓ � Mean trait, ¯ z = γ i p i − γ i q i i =1

At large times, fixation and mutation matter (de Vladar & Barton 2014) If the population is well adapted, both polymorphism and fixation are possible  1 / 2 , γ i < ˆ γ  (small effects) p ∗ i ≈ 0 or 1 , γ i > ˆ γ  (large effects) 1 Equilibrium allele frequency Large effects Small effects 0.5 0 0 1 2 3 ^ / γ i Relative effect, γ

At short times, selection dominates (Jain & Stephan 2015) sγ 2 ❳❳❳❳❳❳❳❳❳❳❳❳❳❳ ✘ ✘✘✘✘✘✘✘✘✘✘✘✘✘✘ i p i ˙ = − sγ i (¯ z − z f ) p i q i + 2 p i q i (2 p i − 1) + µ ( q i − p i ) � �� � � �� � � �� � MUTATION DIRECTIONAL FIXATION ❳ p i ˙ ≈ S i p i q i , S i = − sγ i (¯ z − z f )

Directional selection model 1.0 MeanDeviation 0. 0.8 - 2. Variance 0.6 - 4. 2.5 2. 0.4 1.5 - 6. 1. 0.2 0.5 - 8. time 10 100 1000 10 6 time 1000 time 10 4 10 5 10 100 1000 1 10 100 captures bulk of adaptation solvable even when variance (almost always) changes dramatically

☎ I. Response to a sudden shift in phenotypic optimum e.g., sudden outbreak of disease Shift in optimum 0.4 MeanDeviation 0.35 0. 0.3 Phenotypic Fitness - 0.5 0.25 0.2 - 1. 0.15 - 1.5 0.1 10 6 time 0.05 10 1 10 3 10 4 10 5 10 0 -4 -2 0 2 4 6 Distance from the optimum

Most effects are small Variance AlleleFreq 1.00 0.8 0.55 0.53 0.95 0.7 FULL � 2 � , � 4 � 0.51 DIR SEL � 2 � , � 10 � 0.6 FULL � 4 � 0.90 10 1 10 3 10 5 DIR SEL � 15 � , � 16 � DIR SEL � 10 � APPROX � A3.14 � 0.5 0.85 � 4 � , � 20 � 0.4 10 6 time 10 6 time 10 4 10 5 1 10 100 1000 10 4 10 5 10 100 1000 • Initial genetic variance is large • Subtle shifts in minor allele freq • Remains roughly constant • Sweeps may occur at major loci → Behavior is essentially same as that in infinitesimal model

Most effects are small • Initial genetic variance drives bulk of adaptation g t = − z f e − sℓ ¯ Mean deviation ≈ − z f e − sσ 2 γ 2 t • At long times, true stationary state reached MeanDeviation 0.0 � 0.026 � 0.027 � 0.5 � 0.028 FULL � 1 � , � 4 � 10 4 10 5 10 6 DIR SEL � 1 � , � 10 � � 1.0 DIR SEL � 13 � , � 15 � APPROX � 18 � � 1.5 10 6 time 10 4 10 5 10 100 1000 Numerical solution for ˆ γ = 0 . 13 , ¯ γ = 0 . 04 , z f = 2 , ℓ = 200

Most effects are large AlleleFreq MeanDeviation 1.0 0. 0.8 - 2. Variance 0.6 - 4. 2.5 2. 0.4 1.5 - 6. 1. 0.2 0.5 - 8. time 10 100 1000 10 6 time 1000 time 10 4 10 5 10 100 1000 1 10 100 • Initial genetic variance is small • Sweeps at major allele freq • Increases dramatically • At both short and long times

When most effects are large γ (ln ℓ )2 Mean deviation decreases as ∼ e − sz f ¯ ln α t • Effect size (not initial variance) is the driving force • Only a few large-effect loci are important MeanDeviation 0 � 2 FULL � 1 � , � 4 � � 4 DIR SEL � 1 � , � 10 � DIR SEL � 15 � , � 23 � � 6 APPROX � A5.14 � � 8 time 1 5 10 50 100 500 1000 Numerical solution for ˆ γ = 0 . 03 , ¯ γ = 0 . 3 , z f = 10 , ℓ = 200

When do selective sweeps occur at major loci? when major allele freq at end of directional selection > 1 / 2 z − z f ) p i q i + sγ 2 ❤❤❤❤❤❤❤❤ ✭ i p i = ✭✭✭✭✭✭✭✭ 2 p i q i (2 p i − 1) + ✘✘✘✘✘ ❳❳❳❳❳ ✘ ˙ − sγ i (¯ µ ( q i − p i ) ❳ ❤ AlleleFreq 1.0 0.8 0.6 0.4 0.2 10 6 time 10 4 10 5 10 100 1000

When do selective sweeps occur? (Jain & Stephan 2017) For exponentially distributed effects: • when most effects are large, selective sweeps can occur at short times since moderately large effect sizes needed � ℓ ¯ � � 2 γ i � 2¯ γ γ γ i > � ln ln � z f γ ˆ γ 2¯ ln γ ˆ • when most effects are small, sweeps are prevented since quite large effect sizes needed � 2 γ i � γ ℓ ¯ γ γ i > 2¯ ln z f γ ˆ Simpler model (Lande 1983) gives same results (Chevin & Hospital 2008)

II. Response to a slowly moving phenotypic optimum e.g., global warming Moving optimum 0.4 0.35 0.4 0.3 Phenotypic Fitness 0.3 0.25 c 1 ( t )/ z τ 0.2 0.2 0.15 0.1 0.1 0.05 0 0 0 0.2 0.4 0.6 0.8 1 1.2 -4 -2 0 2 4 6 8 Distance from the optimum t / τ

Most effects are small Is the behavior same as that in infinitesimal model in which mean trait moves with speed of optimum and maintains constant lag? z ( t ) = vt − v ¯ (B¨ urger & Lynch 1995) sσ 2 g

Most effects are small (Jain & Devi 2018) As before, genetic variance remains constant But mean trait moves slower than optimum 0.6 Variance c 2 ( t ) Full model captured 0.4 0.6 FULL 0.2 by directional selec- DIR SEL 0 10 0 10 2 10 4 tion+ mutation c 1 ( t )/ z τ 0.4 Time t l = 50 u 1 0.2 l = 100 v = � � 2 l = 200 γ ˆ 1 + 1 ℓ γ 2¯ OPTIMUM 0 0 0.2 0.4 0.6 0.8 1 1.2 t / τ

Most effects are small (Devi & Jain, unpubl.) Unlike infinitesimal model, finite loci and recurrent mutations included Finite pop. size? Speed is slow for stochastic populations also 1 0.9 N = 100 0.6 u/v 0.8 0.3 0 0.6 0 10 20 30 u/v ˆ γ/ ¯ γ 0.4 0.2 N = 1000 deterministic ( u/v ) 0 10 20 30 40 50 60 γ/ ¯ γ ˆ

Most effects are large • Negligible mutations → mean trait moves with the speed of optimum • Changing genetic variance → lag larger than in infinitesimal model 10 2 10 0 Variance c 2 ( t ) 10 - 1 10 - 2 10 0 Mean c 1 ( t ) 10 - 3 10 0 10 1 10 2 10 3 10 4 10 5 Time t 10 - 2 FULL DIR SEL OPTIMUM 10 - 4 10 0 10 1 10 2 10 3 10 4 10 5 Time t

Key results • When effects are small, recurrent mutations can change the behavior from that of infinitesimal model • When effects are large, genetic variance does not remain constant unlike in infinitesimal model; it increases dramatically and is an indicator of sweeps

Open questions • Linkage (although QLE terms can be added) • Asymmetric mutations (but equilibria not known) • Periodic, fluctuating optimum • Random genetic drift • Multiple traits (pleiotropic effects)

Directional selection model is solvable (Jain & Stephan, 2017) • p i ˙ = − sγ i p i q i ∆ c 1 ( { p 1 , ..., p ℓ } ) p j ˙ = − sγ j p j q j ∆ c 1 ( { p 1 , ..., p ℓ } ) Allows to express the allele frequencies in terms of just one of them p j = F j ( p i ) , j � = i p i ˙ = − sγ i p i q i ∆ c 1 ( F ( p i )) • Variance, skewness... can be found using mean c 1 ˙ = − s ∆ c 1 c 2 c 2 ˙ = − s ∆ c 1 c 3 No arbitrary truncation needed !

A general result for the allele frequency p i = − sγ i p i q i ( c 1 − z f ) ˙ At short times, all + alleles increase in frequency Suggestion: consider many loci simultaneously instead of individual SNPs