Evolution of quantitative traits under a migration–selection balance: When does skew matter?

Evolution of quantitative traits
under a migration–selection balance:
When does skew matter?
S. Yeaman,F. Débarre, F. Guillaume.
@flodebarre @sam_yeaman @fred_guillaume
Débarre, Yeaman & Guillaume Skew matters? SMBE – -- /

Quantitative genetics and Gaussian distributions
(c)Blakeslee()
Débarre, Yeaman Guillaume Skew matters? SMBE – -- /

Quantitative genetics and Gaussian distributions
(c)Blakeslee()
(e.g., Turelli Barton )

Divergent selection and trait distributions
Migration

Migration
Optimum ✓ Optimum ✓

Migration
Traits
0
Traits
0
?Are the local distributions Gaussian?

Migration
Traits
0
Traits
0
?Are the local distributions Gaussian?
If not, can we nevertheless approximate them as Gaussian
to predict phenotypic divergence between demes?

Outline

Analytics: include the eect of skew in predictions
of phenotypic divergence;

Simulations: estimate the amount of skew for
dierent genetic architectures.

Life cycle
Reproduction
MigrationSelection
Regulation
Census

Life cycle
Reproduction
MigrationSelection
Regulation
Census
Reproduction
Random mating with each deme,
unbiased mutation.
¯z
(r)
i = ¯z
(t)
i

Life cycle
Reproduction
MigrationSelection
Regulation
Census
Migration
Fraction m of dispersers.
pm
i (z) = m pr
j (z) + ( m) pr
i (z).

Life cycle
Reproduction
MigrationSelection
Regulation
Census
Selection
Selection towards an optimum value ✓i
in each deme,
wi(⇣) = exp
✓
(⇣ ✓i)
!
◆
,
so that
ps
i (z) =
Wi(z)
W
m
i
pm
i (z).

Life cycle
Reproduction
MigrationSelection
Regulation
Census
Regulation
Brings densities back to carrying
capacity (large)
p
(t+)
i (z) = ps
i (z).

Life cycle
Reproduction
MigrationSelection
Regulation
Census z
(t)
i = z
(t+)
i z
(t)
i .

Life cycle
Reproduction
MigrationSelection
Regulation
Census z
(t)
i = z
(t+)
i z
(t)
i .
Trick: assume weak selection (large !).
z
(t)
i = f
⇣
z
(t)
i , z
(t)
j , vm
i , m
i
⌘
.

The eect of skew on phenotypic dierentiation
D⇤
= z⇤
z⇤
.

D⇤
= z⇤
z⇤
.
Without specifying the distribution
D⇤
=
vm⇤ ✓ + m⇤
vm⇤ ( m) + m !
(with m⇤
).

D⇤
= z⇤
z⇤
.
Without specifying the distribution
D⇤
=
vm⇤ ✓ + m⇤
vm⇤ ( m) + m !
(with m⇤
).
Assuming a Gaussian distribution, and constant variance
D⇤
G =
v⇤ ✓
v⇤ ( m) + m !
.
(Hendry et al., )

Outline

Analytics: include the eect of skew in predictions
of phenotypic divergence;

Simulations: estimate the amount of skew for
dierent genetic architectures.
a Diallelic model, influence of eect size;
b Continuum of alleles model, evolvable eects.

Diallelic model
unlinked loci
alleles at each locus: + and .
loci of eect .,
locus of eect a.

Diallelic model: Results
a = .
m = ⇤
−Θ 0 Θ
Trait
Density

a = .
m = ⇤
−Θ 0 Θ
Trait
Density
a = .
m = ⇤
−Θ 0 Θ
Trait
Density

a = .
m = ⇤
−Θ 0 Θ
Trait
Density
0
∆Θ 2
∆Θ
10−6
10−5
10−4
10−3
10−2
10−1
Migration m
DifferentiationD*
D⇤
, D⇤
G, simulations.
a = .
m = ⇤
−Θ 0 Θ
Trait
Density

a = .
m = ⇤
−Θ 0 Θ
Trait
Density
0
∆Θ 2
∆Θ
10−6
10−5
10−4
10−3
10−2
10−1
Migration m
DifferentiationD*
D⇤
, D⇤
G, simulations.
a = .
m = ⇤
−Θ 0 Θ
Trait
Density
0
∆Θ 2
∆Θ
10−6
10−5
10−4
10−3
10−2
10−1
Migration m
DifferentiationD*
D⇤
, D⇤
G, simulations.

a = .
m = ⇤
−Θ 0 Θ
Trait
Density
0
∆Θ 2
∆Θ
10−6
10−5
10−4
10−3
10−2
10−1
Migration m
DifferentiationD*
D⇤
, D⇤
G, simulations.
a = .
m = ⇤
−Θ 0 Θ
Trait
Density
0
∆Θ 2
∆Θ
10−6
10−5
10−4
10−3
10−2
10−1
Migration m
DifferentiationD*
D⇤
, D⇤
G, simulations. ⇤

Diallelic model: Explanations
a = .
0.00
0.25
0.50
0.75
1.00
10−6
10−5
10−4
10−3
10−2
10−1
Migration m
Freq.of+allele

a = .
0.00
0.25
0.50
0.75
1.00
10−6
10−5
10−4
10−3
10−2
10−1
Migration m
Freq.of+allele
a = .
0.00
0.25
0.50
0.75
1.00
10−6
10−5
10−4
10−3
10−2
10−1
Migration m
Freq.of+allele

a = .
0.00
0.25
0.50
0.75
1.00
10−6
10−5
10−4
10−3
10−2
10−1
Migration m
Freq.of+allele
● ● ● ● ● ● ●0.000
0.025
0.050
0.075
10−6
10−5
10−4
10−3
10−2
10−1
Migration m
Thirdmoment
a = .
0.00
0.25
0.50
0.75
1.00
10−6
10−5
10−4
10−3
10−2
10−1
Migration m
Freq.of+allele

a = .
0.00
0.25
0.50
0.75
1.00
10−6
10−5
10−4
10−3
10−2
10−1
Migration m
Freq.of+allele
● ● ● ● ● ● ●0.000
0.025
0.050
0.075
10−6
10−5
10−4
10−3
10−2
10−1
Migration m
Thirdmoment
a = .
0.00
0.25
0.50
0.75
1.00
10−6
10−5
10−4
10−3
10−2
10−1
Migration m
Freq.of+allele
● ●
●
●
●
● ●0.000
0.025
0.050
0.075
10−6
10−5
10−4
10−3
10−2
10−1
Migration m
Thirdmoment

Continuum-of-alleles model
unlinked loci
Continuum of alleles at each locus.
Mutation: z0
(i) = z(i) + dz,
with dz normally distributed (N(, ↵)).

Continuum-of-alleles model: Results
!
=
0
∆Θ 2
∆Θ
10−6
10−5
10−4
10−3
10−2
10−1
Migration m
DifferentiationD*
!
=
0
∆Θ 2
∆Θ
10−6
10−5
10−4
10−3
10−2
10−1
Migration m
DifferentiationD*

Continuum-of-alleles model: Results
!
=
0
∆Θ 2
∆Θ
10−6
10−5
10−4
10−3
10−2
10−1
Migration m
DifferentiationD*
0.0
0.1
0.2
0.3
0.4
10−6
10−5
10−4
10−3
10−2
10−1
Migration m
Skewproportion
!
=
0
∆Θ 2
∆Θ
10−6
10−5
10−4
10−3
10−2
10−1
Migration m
DifferentiationD*
0.0
0.1
0.2
0.3
0.4
10−6
10−5
10−4
10−3
10−2
10−1
Migration m
Skewproportion

Continuum-of-alleles model: Explanations
(c)YeamanWhitlock()Evolution

Continuum-of-alleles model: Explanations
(c)YeamanWhitlock()Evolution
See Sam Yeaman’s poster A.

Summary // Take-Home messages
I Neglecting the skew of distributions can lead to the
underestimation of phenotypic divergence D⇤
D⇤
=
vm⇤
✓ + m⇤
vm⇤ ( m) + m !
;

D⇤
=
vm⇤
✓ + m⇤
vm⇤ ( m) + m !
;
I There is virtually no eect of the timing of measurement of
variance and skew (e.g., before/aer dispersal) on D⇤;

D⇤
=
vm⇤
✓ + m⇤
vm⇤ ( m) + m !
;
I Skew is more likely to be generated when there are loci of
major eect;

D⇤
=
vm⇤
✓ + m⇤
vm⇤ ( m) + m !
;
major eect;
I But under the conditions of the infinitesimal model, little skew
is generated: the Gaussian approximation remains a good one.

D⇤
=
vm⇤
✓ + m⇤
vm⇤ ( m) + m !
;
major eect;
I For more details, see

D⇤
=
vm⇤
✓ + m⇤
vm⇤ ( m) + m !
;
major eect;
I For more details, see
Thanks for your attention!

Evolution of quantitative traits under a migration–selection balance: When does skew matter?

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Evolution of quantitative traits under a migration–selection balance: When does skew matter?