hannes svardal - the role of environmental variance as adaptive response to fluctuating selection

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Hannes Svardal (Vienna) environmental vs. genetic variance 1. Juni 2010 1 / 18

Does fluctuating selection favour an increase inenvironmental or in genetic variance?

Hannes Svardal, Claus Rueffler, and Joachim Hermisson

Institute of Mathematics, University of Vienna

1. Juni 2010


Observations

Quantitative traits show considerable amounts of phenotypic variation

Variation could be adaptive (favoured by selection) or a constraint(mutation selection balance)

We are looking at adaptive sources of phenotypic variation


Sources of phenotypic variance in a quantitative trait

phenotypic variance

genetic environmental

random GxE interaction

phenotypicplasticity

discretemorphs

Gaussiannoise

geneticpolymorphism


Genetic polymorphism VS environmental decanalisation

phenotypic variance

genetic environmental

Gaussiannoise

geneticpolymorphism

genetically controlled viagenetic contribu-tion to a trait

degree of canali-sation of the trait

why adaptive?frequency depen-dent selection

as bet-hedgingstrategy

if both are adaptive:

? ?what

evolves?


Genetics

Clonal reproduction

Phenotype is a quantitative trait x

Phenotype is determined by genetic component µx and randomenvironmental effects (Gaussian noise with variance σ2x)

Amount of environmental canalisation genetically controlled:σx heritable

Probability that a genotype (µx, σx) produces a phenotype x:

µx

σx

x

prob

abili

ty

heritable canalisation

heritable genetic component


canalised

a lot of noise


The question

Most models treat fully canalised genotypes (µx, σx) = (x, 0)

x

We compare selection for genetic polymorphisms in µx to selection forincreased σx:

µx2

σx2

µx1

σx1

x

VS

µx

σx

x

In models where both – genetic polymorphism and environmental

decanalisation – are adaptive: What does evolve?Hannes Svardal (Vienna) environmental vs. genetic variance 1. Juni 2010 8 / 18

The Lottery model (Chesson and Warner 1981): Temporalvariation in selective optimum

Ecological assumptions:

discrete generations

maximum population size K

generation overlap γ⇒ ∼ (1− γ)K adults die each year, no selection on adults

Selection on juveniles:

selective optimum θ changes from year to year(but has stationary distribution with mean µθ, variance σ2θ)

Gaussian selection of strength 1/σ2s on distance |x− θ|surviving juveniles randomly compete to fill up the population sizeback to K

(equivalent to the seed bank model)


Model ingredients

optimal phenotype θ

σθ

occurrence probability

θt

external environment:optima distribution withmean µθ and variance σ2θ

µθ

p

θ1

1− p

θ2

special caseexternal environment:optima distribution withmean µθ and variance σ2θ

µx

σx

phenotype x

heritablegenotypic values:µx and σx determine gene-tic contribution and noiselevel

freq

uen

cy

x

0 |x− θt|

σs

surv

ival

selection: depends on diffe-rence optimum⇔phenotype


Model ingredients

optimal phenotype θ

σθ

occurrence probability

θt


µθ

p

θ1

1− p

θ2

special case


µx

σx

phenotype x

heritablegenotypic values:µx and σx determine gene-tic contribution and noiselevel

freq

uen

cy

x

0 |x− θt|

σs

surv

ival

selection: depends on diffe-rence optimum⇔phenotype


The two possibilities independently

selected if

decanalisation (σx > 0) σ2θ > σ2s

σ2θ

σ2s noise

genetic polymporphism(disruptive selection in µx)

γσ2θ > σ2s

γσ2θ

σ2s genetic p.

Now: analysis of evolution in the 2D”genotype-space“ (µx, σx):

σx

µxµθ


Adaptive dynamics of the genotypic values µx, σx

Growth rate of mutant (µxm, σxm) in resident population (µxr, σxr):

λ(µxm, σxm, µxr, σxr) =

1− (1− γ)

1−

√σ2s + σ2xr exp

((θ−µxr)22(σ2

s+σ2xr)− (θ−µxm)2

2(σ2s+σ

2xm)

)√σ2s + σ2xm

Invasion fitness of mutant m = (µxm, σxm):

w(m, r) =

∫ln(λ(m, r|θ))h(θ)dθ

⇒ Calculate zeros of selection gradient ∇w and investigate stability


Results

Noise will evolve to its optimum: σ2x = σ2θ − σ2sAdditional genetic polymorphism (branching) are selected if:

γ > 4

gθ+4+√

8µ23θ+g2θ

µ3θ ... skewness of optima distributiongθ ... kurtosis of optima distribution

I optima distribution sufficiently asymmetricI optima distribution has fatter tails than Gaussian (extremes more likely)

⇒ If noise can evolve, genetic polymorphisms are only selected if theoptima distribution is sufficiently different from Gaussian


Examples of optima distributions

optima distribution example branching branching in

sum of smalleffects

never -

number ofpredation

events

γ > 4λ1+4λ+

√1+8λ

µx, σx

? γ > 2/5 σx

µθ

p 1− p

occurence ofthunderstorm

γ > 2p(1−p)1−2p(1−p) µx, σx


Two possible optima

evolutionary convergenceto optimal noise level

if asymmetric:further genetic branching

σx

µxµθ

σx

µxµθθ1 θ2

p = 0.8 1− p = 0.2

If genetic polymorphism evolve, mostly both, µx AND σx, divergebetween the populations (cf. Doebeli and Ispolatov 2010)


Simulation Results: Two possible optima

γ =0.5 general observations:

↑ γ stabilises (lhs)↑ σs stabilises↑ p destabilises

conclusion:

polymorphism oftenunstable

γ =0.75

γ =0.95 parametres: p = 0.8, σs = 0.1,

µθ = −0.3, σ∗x = 0.39, γ′ = 0.47


Conclusion

Under temporally fluctuating selection noise evolves easier thangenetic polymorphisms

Genetic branching at optimal noise level ifI optima distribution sufficiently asymmetricI optima distribution has fatter tails than Gaussian

Polymorphism of divergent genotypes often unstable

In sexual populations: selection for increased genetic variance

Predictions about the heritability of traits under different forms offluctuating selection could be made


Thanks for your attention!


hannes svardal - the role of environmental variance as adaptive response to fluctuating selection

Technology

variance x

genetic variance1

genetic component x

genetic polymorphisms

increased x

x heritableprobability

genotype x

distance x