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 in
environmental 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

genetic Gaussian discrete phenotypic
polymorphism noise morphs plasticity


Genetic polymorphism VS environmental decanalisation
phenotypic variance


genetic Gaussian
polymorphism noise


phenotypic variance


genetic Gaussian
polymorphism noise

genetic contribu- degree of canali-
genetically controlled via
tion to a trait sation of the trait


phenotypic variance


genetic Gaussian
polymorphism noise


frequency depen- as bet-hedging
why adaptive?
dent selection strategy

if both are adaptive:


phenotypic variance


genetic Gaussian
polymorphism noise


frequency depen- as bet-hedging
why adaptive?
dent selection strategy

if both are adaptive:
what
? evolves? ?

Genetics
Clonal reproduction
Phenotype is a quantitative trait x
Phenotype is determined by genetic component µx and random
2
environmental eﬀects (Gaussian noise with variance σx )
Amount of environmental canalisation genetically controlled:
σx heritable

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

σx heritable canalisation

µx x
heritable genetic component


a lot of noise

canalised


The question

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

x
We compare selection for genetic polymorphisms in µx to selection for
increased σx :

σx1 σx2
σx
VS

µx1 µx2 x µx x

In models where both – genetic polymorphism and environmental
decanalisation – are adaptive: What does evolve?

The Lottery model (Chesson and Warner 1981): Temporal
variation 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
2
(but has stationary distribution with mean µθ , variance σθ )
2
Gaussian selection of strength 1/σs on distance |x − θ|
surviving juveniles randomly compete to ﬁll up the population size
back to K
(equivalent to the seed bank model)


Model ingredients

occurrence probability external environment:
σθ optima distribution with
2
mean µθ and variance σθ

optimal phenotype θ


Model ingredients

p special case 1−p
occurrence probability
σθ

optimal phenotype θ θ1 µθ θ2


Model ingredients

2

optimal phenotype θ
heritable
frequency

genotypic values:
σx µx and σx determine gene-
tic contribution and noise
µx phenotype x
level


Model ingredients

2

θt optimal phenotype θ
heritable
frequency

genotypic values:
σx µx and σx determine gene-
tic contribution and noise
µx x phenotype x
level
survival

σs selection: depends on diﬀe-
rence optimum⇔phenotype
0 |x − θt |

The two possibilities independently
selected if
2
σθ
decanalisation (σx > 0) 2 2
σθ > σs
2
σs noise
2
γσθ
genetic polymporphism 2 2
γσθ > σs
(disruptive selection in µx ) 2 genetic p.
σs


The two possibilities independently
selected if
2
σθ
decanalisation (σx > 0) 2 2
σθ > σs
2
σs noise
2
γσθ
genetic polymporphism 2 2
γσθ > σs
(disruptive selection in µx ) 2 genetic p.
σs

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

µθ µx

Adaptive dynamics of the genotypic values µx , σx

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

λ(µxm , σxm , µxr , σxr ) =

(θ−µxr )2 (θ−µxm )2
 
2 2
σs + σxr exp −
2 2
2(σs +σxr ) 2 2
2(σs +σxm )
1 − (1 − γ) 1 − 
2 2
σs + σxm

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

2 2 2
Noise will evolve to its optimum: σx = σθ − σs
Additional genetic polymorphism (branching) are selected if:
√4
γ>
gθ +4+ 8˜2 +gθ
µ3θ 2

µ3θ ... skewness of optima distribution
˜
gθ ... kurtosis of optima distribution
optima distribution sufficiently asymmetric
optima distribution has fatter tails than Gaussian (extremes more likely)
⇒ If noise can evolve, genetic polymorphisms are only selected if the
optima distribution is sufficiently different from Gaussian


Examples of optima distributions
optima distribution example branching branching in
sum of small
never -
eﬀects

number of
4λ
predation γ> √
1+4λ+ 1+8λ
µx , σx
events

? γ > 2/5 σx

p 1−p

occurence of 2p(1−p)
γ> 1−2p(1−p) µx , σx
thunderstorm
µθ


Two possible optima

evolutionary convergence if asymmetric:
to optimal noise level further genetic branching

σx
σx

µθ µx µθ µx
θ1 θ2
p = 0.8 1 − p = 0.2

If genetic polymorphism evolve, mostly both, µx AND σx , diverge
between 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:
γ= polymorphism often
0.75 unstable

γ=
parametres: p = 0.8, σs = 0.1,
0.95
∗
µθ = −0.3, σx = 0.39, γ = 0.47


Conclusion

Under temporally fluctuating selection noise evolves easier than
genetic polymorphisms
Genetic branching at optimal noise level if
optima distribution sufficiently asymmetric
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 of
fluctuating selection could be made


Thanks for your attention!


Hannes Svardal - The role of environmental variance as adaptive response to fluctuating selection

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