conflict between alleles and modifiers in the evolution of genetic polymorphisms

Conflict between alleles and modifiers in the evolution of genetic polymorphisms

(formerly ADN) IIASA

Hans Metz

QuickTime™ en eenTIFF (ongecomprimeerd)-decompressorzijn vereist om deze afbeelding weer te geven.VEOLIA-Ecole Poly-technique

QuickTime™ en eenTIFF (ongecomprimeerd)-decompressor

zijn vereist om deze afbeelding weer te geven.

&Mathematical Institute, Leiden University

QuickTime™ en eenTIFF (ongecomprimeerd)-decompressor

NCB naturalis

the tool

(Assumptions: mutation limitation, mutations have small effect.)

the canonical equation of adaptive dynamics

X: value of trait vector predominant in the population Ne: effective population size, : mutation probability per birth C: mutational covariance matrix, s: invasion fitness, i.e., initial relative growth rate of a potential Y mutant population.

= 2 Ne C ∂sX Y( )∂Y Y=X

⎡⎣⎢

⎤⎦⎥T

with Mendelian reproduction:

= 0evolutionary

evolutionary constraints

phenotype

genotype

directional selection

coding region

regulatory regions

reading direction

Most phenotypic evolution is probably regulatory, and hence quantitative on the level of gene expressions.

The canonical equation is not dynamically sufficient as there is no need for C to stay constant.

Even if at the genotype level the covariance matrix stays constant, the non-linearity of the genotype to phenotype map Φ

will lead to a phenotypic C that changes with the genetic changes underlying the change in X.

additional (biologically unwaranted) assumption

symmetric phenotypic mutation distributions

saving grace?

I have reasons to expect that my final conclusions are independent of this symmetry assumption,

but I still have to do the hard calculations to check this.

I only showed (and use)the canonical equation for the case of

= 2 Ne C ∂sX Y( )∂Y Y=X

⎡⎣⎢

⎤⎦⎥T

sX Y( ) ≈ln R0X Y( )⎡⎣ ⎤⎦

TrNe =

TrNTsse

Tss e2 C

∂R0X Y( )∂Y Y=X

⎡⎣⎢

⎤⎦⎥T

R0 : average life-time offspring number

Ts : average age at death

: effective variance of life-time offspring numbers e2

of the residents of the residents

Tr : average age at reproduction

CE is derived via two subsequent limits

system size ∞ successful mutations / time 0

trait valuex

individual-basedsimulation

adaptive dynamicslimit

individual-based stochastic process

mutational step size 0

canonical equationlimit

branching

limit type:

this talk: evolution of genetic polymorphisms

system size ∞ successful mutations / time 0

trait valuex

individual-basedsimulation

adaptive dynamicslimit

individual-based stochastic process

canonical equationlimit

branching

limit type:mutational step size

the ecological theatre

Assumptions: but for genetic differences individuals are born equal,random mating, ecology converges to an equilibrium.

equilibria for general eco-genetic models

(1) setting the average life-time offspring number over the phenotypes equal to 1,

(2) calculating the genetic composition of the birth stream from equations similar to the classical (discrete time) population genetical ones,

with those life-time offspring numbers as fitnesses.

For a physiologically structured population with all individuals born in the same physiolocal state, mating randomly with respect to genetic differences,

the equilibria can be calculated by

the eco-genetic model

Organism with a potentially polymorphic locus with two segregating alleles, leading to the phenotype vector , with .XG G ∈ aa, aA, AA{ }

X := Xaa XaA XAA( )T

Abbreviations: , etc. (and similar abbreviations later on). μG := μ (XG;E)

: expected per capita lifetime microgametic output times fertilisation propensity ( average number of kids fathered)

μ(XG;E)

: instantaneous ecological environmentE

l(XG ;E) : expected expected per capita lifetime macrogametic output (= average number of kids mothered)

the eco-genetic model C = classical discrete time model

baa =paaB,random union of gametes:

baA =(pAa + paA)B, bAA =pAAB.

Point equilibria: lqA = λ AA pAqA + 12 λ aA (pAqa + paqA ),

l := pAqAλ AA + (pAqa + paqA )λ aA + paqaλ aa = 1,

μpA = μ AA pAqA + 12 μ aA (pAqa + paqA ),

μ := pAqAμ AA + (pAqa + paqA )μ aA + paqaμ aa .

pa =1−pA

, : allelic frequencies in the micro- resp. macro-gametic outputs ( and )

pa , pA qa ,qA

qa =1−A

: total birth rate density (C: total population density, )B N

baa ,baA ,bAA : genotype birth rate densities (C: genotype densities, , etc)naa

naa =Ts,aabaa , etc.

example ecological feedback loop: E =Φ φ1(E, XG )nGG∑ , ... , φk(E, XG )nG

G∑⎛

⎝⎜⎞⎠⎟

the evolutionary play

Assumptions: no parental effects on gene expressions(mutation limitation, mutations have small effect)

long term evolution

Two modelsI. Evolution through allelic substitutions

Xa , XA : allelic trait vectorsXaA =Φ(Xa, XA),Φ : genotype to phenotype map: etc.

Abbreviations: etc.∂Φ(Xa , XA )∂Xa

=: ∂1Φ(Xa , XA ) =: ∂1ΦaA ,

II. Evolution through modifier substitutions b : original allele on generic modifier locus,

B: mutant, changing into X =Xaa

⎝⎜⎜

⎠⎟⎟ =

Xbb,aa

Xbb,aA

Xbb,AA

⎝⎜⎜⎜

⎠⎟⎟⎟=: Xb

XbB,aa

XbB,aA

XbB,AA

⎝⎜⎜⎜

⎠⎟⎟⎟=: XB.

smooth genotype to phenotype maps

IfModel I (allelic evolution)

Xa =XA + eZA

Xaa =XAA + e ∂1ΦAAZA + e ∂2ΦAAZA + O(e2 ) =XAA + 2 e ∂2ΦAAZA + O(e

Model II (modifier evolution)

XbB,aa

XbB,aA

XbB,AA

⎝⎜⎜⎜

⎠⎟⎟⎟=

Xbb,aa

Xbb,aA

Xbb,AA

⎝⎜⎜⎜

⎠⎟⎟⎟+ e

⎝⎜⎜

⎠⎟⎟If

Xaa = XaA + e ∂2ΦaAZA + O(e2 ), XAa = XAA + e ∂2ΦAAZA + O(e

XBB,aa

XBB,aA

XBB,AA

⎝⎜⎜⎜

⎠⎟⎟⎟=

Xbb,aa

Xbb,aA

Xbb,AA

⎝⎜⎜⎜

⎠⎟⎟⎟+ 2e

⎝⎜⎜

⎠⎟⎟ +O(e

2 )then

Model I: phenotypic change in the CE limit

with dXa

dt = ψ aN pa2aCa

∂R0,XaXA(Xa )

⎛⎝⎜

⎞⎠⎟a=a

dt = ψ AN pA2ACA

∂R0,XaXA(Xa )

⎛⎝⎜

⎞⎠⎟a=A

ψ a = 2 (Ts,aσ a2 ), ψ A = 2 (Ts,Aσ A

2 ),witha , θ A the mutation probabilities per allele per birth,

the mutational covariance matrices,Ca , CA

p a := 12 (pa + qa ), π A := 1

2 (pA + qA ).and

⎛⎝⎜

⎞⎠⎟a

⎛⎝⎜

⎞⎠⎟A

2∂1ΦaadXa

∂1ΦaAdXa

⎜⎜⎜⎜⎜⎜⎜

⎟⎟⎟⎟⎟⎟⎟

+ ∂2ΦaAdXA

2∂2ΦAAdXA

⎜⎜⎜⎜⎜⎜⎜

⎟⎟⎟⎟⎟⎟⎟

∂R0.Xa XA(Xa )

=∂R0, Xaa , XaA , XAA

(Xaa , XaA )∂Xaa

∂1Φaa +∂R0, Xaa , XaA , XAA

(Xaa , XaA )∂XaA

∂1ΦaA

Convention:

Differentiation is only with respect to the regular arguments, not the indices.

∂R0, Xa XA(XA )

=∂R0, Xaa , XaA , XAA

(XAA , XaA )∂XAA

∂2ΦAA +∂R0, Xaa , XaA , XAA

(XAA , XaA )∂XaA

∂2ΦaA

notation

A ⊗B=

a11B L a1nBM M

aμ1B L aμnB

⎝⎜⎜

⎠⎟⎟

I the identity matrix of any required size

denotes the Kronecker product:⊗

= 2 0 0 00 1 1 00 0 0 2

⎝⎜⎜

⎠⎟⎟⊗I

∂1Φaa

∂1ΦaA

∂2ΦaA

∂2ΦAA

⎜⎜⎜⎜

⎟⎟⎟⎟

⎜⎜⎜

⎟⎟⎟

in matrix notation:

⎜⎜⎜

⎟⎟⎟=ψ allelicN 2allelic

%ψ a%apaCa 00 %ψ A

%ApACA

⎛⎝⎜

⎞⎠⎟

∂1ΦaaT

0∂1ΦaA

0∂2ΦaA

0∂2ΦAA

⎛⎝⎜

⎞⎠⎟

∂R0,Xaa , XaA, XAA(Xaa, XaA)

∂Xaa

⎡⎣⎢

⎤⎦⎥T

∂R0,Xaa , XaA, XAA(Xaa, XaA)

∂XaA

⎡⎣⎢

⎤⎦⎥T

∂R0,Xaa , XaA , XAA(XAA, XaA)

∂XaA

⎡⎣⎢

⎤⎦⎥T

∂XAA

⎡⎣⎢

⎤⎦⎥T

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

and (the allelic coevolution equations)

withψ allelic = π aψ a + π Aψ A , allelic = π aθa + π AθA ,

%ψ a = ψ a ψ allelic , %ψ A = ψ A ψ allelic , %a = θa θallelic , %A = θA θallelic .

structurematrix

= ψ allelicN 2 0 0 00 1 1 00 0 0 2

⎜⎜⎜

⎟⎟⎟⊗I 2allelicYCallelicGallelic

combining the previous results gives:

%ψ a%θaπ a 0 0 00 %ψ a

%θaπ a 0 00 0 %ψ A

%θAπ A 00 0 0 %ψ A

%θAπ A

⎜⎜⎜⎜

⎟⎟⎟⎟⊗I

Callelic :=

∂1Φaa

∂1ΦaA

∂2ΦaA

∂2ΦAA

⎜⎜⎜⎜

⎟⎟⎟⎟

Ca 00 CA

⎛⎝⎜

⎞⎠⎟

∂1ΦaaT

0∂1ΦaA

0∂2ΦaA

0∂2ΦAA

⎛⎝⎜

⎞⎠⎟.

Gallelic =

∂R0,Xaa ,XaA , XAA(Xaa, XaA)

∂Xaa

⎡⎣⎢

⎤⎦⎥T

∂R0,Xaa ,XaA , XAA(Xaa, XaA)

∂XaA

⎡⎣⎢

⎤⎦⎥T

∂XaA

⎡⎣⎢

⎤⎦⎥T

∂XAA

⎡⎣⎢

⎤⎦⎥T

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

ba = 1−(pa −a)

laaμaA

4lμ−laAμaa

4lμ⎛⎝⎜

⎞⎠⎟, bA = 1−(pA−A)

lAAμaA

4lμ−laAμAA

4lμ⎛⎝⎜

⎞⎠⎟

Gallelic =B−1 Gcoμ μ on

an explicit expression for the allelic (proxy) selection gradient:

ba 0 0 00 ba 0 00 0 bA 00 0 0 bA

⎜⎜⎜⎜

⎟⎟⎟⎟⊗I

on the Hardy-Weinberg manifold (pA = qA) :

with μaa = μ (Xaa;EXaa XaA XAA), etc.

l =paqaλ aa + (paqA + pAqa )λ aA + pAqAλ AA = 1and

effect a mutation in the

a--allele A-allele

Gcommon =

pa +μaA

2μpA−A( )

⎛⎝⎜

⎞⎠⎟

∂laa

2l∂Xaa

⎡⎣⎢

⎤⎦⎥T

+ a +laA

2lA−pA( )

⎛⎝⎜

⎞⎠⎟

∂μaa

2μ∂Xaa

⎡⎣⎢

⎤⎦⎥T

pA +μaa

2μpa −a( )

⎛⎝⎜

⎞⎠⎟

∂laA

2l∂XaA

⎡⎣⎢

⎤⎦⎥T

+ A +laa

2la −pa( )

⎛⎝⎜

⎞⎠⎟

∂μaA

2μ∂XaA

⎡⎣⎢

⎤⎦⎥T

pa +μAA

2μpA−A( )

⎛⎝⎜

⎞⎠⎟

∂laA

2l∂XaA

⎡⎣⎢

⎤⎦⎥T

+ a +lAA

2lA−pA( )

⎛⎝⎜

⎞⎠⎟

∂μaA

2μ∂XaA

⎡⎣⎢

⎤⎦⎥T

pA +μaA

2μpa −a( )

⎛⎝⎜

⎞⎠⎟

∂lAA

2l∂XAA

⎡⎣⎢

⎤⎦⎥T

+ A +laA

2la −pa( )

⎛⎝⎜

⎞⎠⎟

∂μAA

2μ∂XAA

⎡⎣⎢

⎤⎦⎥T

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

with μaa = μ (Xaa;EXaa XaA XAA), etc.

l =paqaλ aa + (paqA + pAqa )λ aA + pAqAλ AA = 1and

effect of the resulting phenotypic change in the

aa-homozygotes heterozygotes AA-homozygotes

Gcommon =

pa +μaA

2μpA−A( )

⎛⎝⎜

⎞⎠⎟

∂laa

2l∂Xaa

⎡⎣⎢

⎤⎦⎥T

+ a +laA

2lA−pA( )

⎛⎝⎜

⎞⎠⎟

∂μaa

2μ∂Xaa

⎡⎣⎢

⎤⎦⎥T

pA +μaa

2μpa −a( )

⎛⎝⎜

⎞⎠⎟

∂laA

2l∂XaA

⎡⎣⎢

⎤⎦⎥T

+ A +laa

2la −pa( )

⎛⎝⎜

⎞⎠⎟

∂μaA

2μ∂XaA

⎡⎣⎢

⎤⎦⎥T

pa +μAA

2μpA−A( )

⎛⎝⎜

⎞⎠⎟

∂laA

2l∂XaA

⎡⎣⎢

⎤⎦⎥T

+ a +lAA

2lA−pA( )

⎛⎝⎜

⎞⎠⎟

∂μaA

2μ∂XaA

⎡⎣⎢

⎤⎦⎥T

pA +μaA

2μpa −a( )

⎛⎝⎜

⎞⎠⎟

∂lAA

2l∂XAA

⎡⎣⎢

⎤⎦⎥T

+ A +laA

2la −pa( )

⎛⎝⎜

⎞⎠⎟

∂μAA

2μ∂XAA

⎡⎣⎢

⎤⎦⎥T

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

l =pa2λ aa + 2 pa pAλ aA + pA

2λ AA = 1

on the Hardy-Weinberg manifold (pA = qA)

summary of Model I (allelic trait substitution)

= ψ allelicN

2 0 0 00 1 1 00 0 0 2

⎜⎜⎜

⎟⎟⎟⊗I 2allelicYCallelic B

−1 G coμ μ on

Gcommon =

pa +μaA

2μpA−A( )

⎛⎝⎜

⎞⎠⎟

∂laa

2l∂Xaa

⎡⎣⎢

⎤⎦⎥T

+ a +laA

2lA−pA( )

⎛⎝⎜

⎞⎠⎟

∂μaa

2μ∂Xaa

⎡⎣⎢

⎤⎦⎥T

pA +μaa

2μpa −a( )

⎛⎝⎜

⎞⎠⎟

∂laA

2l∂XaA

⎡⎣⎢

⎤⎦⎥T

+ A +laa

2la −pa( )

⎛⎝⎜

⎞⎠⎟

∂μaA

2μ∂XaA

⎡⎣⎢

⎤⎦⎥T

pa +μAA

2μpA−A( )

⎛⎝⎜

⎞⎠⎟

∂laA

2l∂XaA

⎡⎣⎢

⎤⎦⎥T

+ a +lAA

2lA−pA( )

⎛⎝⎜

⎞⎠⎟

∂μaA

2μ∂XaA

⎡⎣⎢

⎤⎦⎥T

pA +μaA

2μpa −a( )

⎛⎝⎜

⎞⎠⎟

∂lAA

2l∂XAA

⎡⎣⎢

⎤⎦⎥T

+ A +laA

2la −pa( )

⎛⎝⎜

⎞⎠⎟

∂μAA

2μ∂XAA

⎡⎣⎢

⎤⎦⎥T

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

on the Hardy-Weinberg manifold:

Model II: phenotypic change in the CE limit

=2 ψ μ odifierN 2haplotψpe Cμ odifiersGμ odifier

ψ modifier = 2 (Ts,modifierσ modifier2 )with , the mutation probability per

haplotype per birth, the covariances of the mutational effects of modifiers.

haplotype

Cmodifiers

γa = a

2μ+ pa

2l−1+ bA, γA = A

2μ+ pA

2l−1+ ba

G = γabA +γAba( )−1

0 0 γA

0 0 0 γA

⎜⎜⎜⎜

⎟⎟⎟⎟⊗I

with Gmodifier =

⎜⎜⎜

⎟⎟⎟⊗I G Gcoμ μ on

G = Π :=

0 0 pA

0 0 0 pA

⎜⎜⎜⎜

⎟⎟⎟⎟⊗I

on the Hardy-Weinberg manifold:

summary: model comparison

= ψ allelicN 2 0 0 00 1 1 00 0 0 2

⎜⎜⎜

−1G coμ μ on

= 2ψ μ odifierN 2haplotψpeCμ odifiers 100

⎜⎜⎜

⎟⎟⎟⊗I G G coμ μ on

Model I (allelic substitutions):

Gcommon =

a +laA

2lA−pA( )

⎛⎝⎜

⎞⎠⎟

∂μaa

2μ∂Xaa

⎡⎣⎢

⎤⎦⎥T

+ pa +μaA

2μpA−A( )

⎛⎝⎜

⎞⎠⎟

∂laa

2l∂Xaa

⎡⎣⎢

⎤⎦⎥T

A +laa

2la −pa( )

⎛⎝⎜

⎞⎠⎟

∂μaA

2μ∂XaA

⎡⎣⎢

⎤⎦⎥T

+ pA +μaa

2μpa −a( )

⎛⎝⎜

⎞⎠⎟

∂laA

2l∂XaA

⎡⎣⎢

⎤⎦⎥T

a +lAA

2lA−pA( )

⎛⎝⎜

⎞⎠⎟

∂μaA

2μ∂XaA

⎡⎣⎢

⎤⎦⎥T

+ pa +μAA

2μpA−A( )

⎛⎝⎜

⎞⎠⎟

∂laA

2l∂XaA

⎡⎣⎢

⎤⎦⎥T

A +laA

2la −pa( )

⎛⎝⎜

⎞⎠⎟

∂μAA

2μ∂XAA

⎡⎣⎢

⎤⎦⎥T

+ pA +μaA

2μpa −a( )

⎛⎝⎜

⎞⎠⎟

∂lAA

2l∂XAA

⎡⎣⎢

⎤⎦⎥T

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

Model II (modifier substitutions):

summary: model comparisonModel I (allelic substitutions):

Gcommon =

a +laA

2lA−pA( )

⎛⎝⎜

⎞⎠⎟

∂μaa

2μ∂Xaa

⎡⎣⎢

⎤⎦⎥T

+ pa +μaA

2μpA−A( )

⎛⎝⎜

⎞⎠⎟

∂laa

2l∂Xaa

⎡⎣⎢

⎤⎦⎥T

A +laa

2la −pa( )

⎛⎝⎜

⎞⎠⎟

∂μaA

2μ∂XaA

⎡⎣⎢

⎤⎦⎥T

+ pA +μaa

2μpa −a( )

⎛⎝⎜

⎞⎠⎟

∂laA

2l∂XaA

⎡⎣⎢

⎤⎦⎥T

a +lAA

2lA−pA( )

⎛⎝⎜

⎞⎠⎟

∂μaA

2μ∂XaA

⎡⎣⎢

⎤⎦⎥T

+ pa +μAA

2μpA−A( )

⎛⎝⎜

⎞⎠⎟

∂laA

2l∂XaA

⎡⎣⎢

⎤⎦⎥T

A +laA

2la −pa( )

⎛⎝⎜

⎞⎠⎟

∂μAA

2μ∂XAA

⎡⎣⎢

⎤⎦⎥T

+ pA +μaA

2μpa −a( )

⎛⎝⎜

⎞⎠⎟

∂lAA

2l∂XAA

⎡⎣⎢

⎤⎦⎥T

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

= ψ allelicN 2 0 0 00 1 1 00 0 0 2

⎜⎜⎜

−1G coμ μ on

= ψ μ odifierN 2haplotψpeCμ odifiers 200

⎜⎜⎜

Gcommon =

a +laA

2lA−pA( )

⎛⎝⎜

⎞⎠⎟

∂μaa

2μ∂Xaa

⎡⎣⎢

⎤⎦⎥T

+ pa +μaA

2μpA−A( )

⎛⎝⎜

⎞⎠⎟

∂laa

2l∂Xaa

⎡⎣⎢

⎤⎦⎥T

A +laa

2la −pa( )

⎛⎝⎜

⎞⎠⎟

∂μaA

2μ∂XaA

⎡⎣⎢

⎤⎦⎥T

+ pA +μaa

2μpa −a( )

⎛⎝⎜

⎞⎠⎟

∂laA

2l∂XaA

⎡⎣⎢

⎤⎦⎥T

a +lAA

2lA−pA( )

⎛⎝⎜

⎞⎠⎟

∂μaA

2μ∂XaA

⎡⎣⎢

⎤⎦⎥T

+ pa +μAA

2μpA−A( )

⎛⎝⎜

⎞⎠⎟

∂laA

2l∂XaA

⎡⎣⎢

⎤⎦⎥T

A +laA

2la −pa( )

⎛⎝⎜

⎞⎠⎟

∂μAA

2μ∂XAA

⎡⎣⎢

⎤⎦⎥T

+ pA +μaA

2μpa −a( )

⎛⎝⎜

⎞⎠⎟

∂lAA

2l∂XAA

⎡⎣⎢

⎤⎦⎥T

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟

Model I (allelic substitutions):

= ψ allelicN 2 0 0 00 1 1 00 0 0 2

⎜⎜⎜

−1G coμ μ on

on the Hardy-Weinberg manifold

⎜⎜⎜

⎟⎟⎟⊗I Π G coμ μ on

summary: model comparison on the Hardy-Weinberg manifold

2haplotψpeCμ odifiers 200

⎜⎜⎜

⎟⎟⎟⊗I Π

2 0 0 00 1 1 00 0 0 2

⎜⎜⎜

⎟⎟⎟⊗I 2allelicYCallelic

⎜⎜⎜

⎟⎟⎟⊗I Π

2 0 0 00 1 1 00 0 0 2

⎜⎜⎜

2 0 0 00 1 1 00 0 0 2

⎜⎜⎜

%ψ a%θaπ a 0 0 00 %ψ a

%θaπ a 0 00 0 %ψ A

%θAπ A 00 0 0 %ψ A

%θAπ A

⎜⎜⎜⎜

⎟⎟⎟⎟⊗ I = Π Ξ

⎜⎜⎜

⎟⎟⎟⊗I Π

%ψ a%θaπ a 0 0 00 %ψ a

%θaπ a 0 00 0 %ψ A

%θAπ A 00 0 0 %ψ A

%θAπ A

⎜⎜⎜⎜

⎟⎟⎟⎟⊗ I = Π Ξ

⎜⎜⎜

⎟⎟⎟⊗I Π

2 0 0 00 1 1 00 0 0 2

⎜⎜⎜

⎟⎟⎟⊗I Π 2allelicXCallelic

summary: model comparisonModel I (allelic substitutions):

= ψ allelicN 2 0 0 00 1 1 00 0 0 2

⎜⎜⎜

−1G coμ μ on

ba 0 0 00 ba 0 00 0 bA 00 0 0 bA

⎜⎜⎜⎜

⎟⎟⎟⎟⊗I

ba = 1−(pa −a)

laaμaA

4lμ−laAμaa

4lμ⎛⎝⎜

⎞⎠⎟, bA = 1−(pA−A)

lAAμaA

4lμ−laAμAA

4lμ⎛⎝⎜

⎞⎠⎟

G = γaβ A + γ Aβa( )−1

γ a 0 0 00 γ a 0 00 0 γ A 00 0 0 γ A

⎜⎜⎜⎜

⎟⎟⎟⎟

γa = a

2μ+ pa

2l−1+ bA, γA = A

2μ+ pA

2l−1+ ba

⎜⎜⎜

in reality alleles and modifiers will both evolve

= dXdt

⎛⎝⎜

⎞⎠⎟μ odifier

⎛⎝⎜

⎞⎠⎟a

⎛⎝⎜

⎞⎠⎟A

= dXdt

⎛⎝⎜

⎞⎠⎟μ odifier

⎛⎝⎜

⎞⎠⎟allelic

combining Models I and II:

evolutionary statics

genetical and developmental assumptions

In biological terms: there are no local developmental or physiological constraints.

So-called genetic constraints are rooted more deeply than in the physiology or developmental mechanics.Example: some phenotypes can only be realised by heterozygotes.

When there are developmental or physiological constraints, we can usually define a new coordinate system on any constraint manifold that the phenotypes run into, and proceed as in the case without constraints.

IF: There are no constraints whatsoever, that is, any combination of phenotypes may be realised by a mutant in its various heterozygotes.

(known in the literature as the “Ideal Free” assumption).

uniformly has full rank and uniformly has maximal rank.Cmodifiers Callelic

evolutionary stops

Evolutionary stops satisfy

2 0 0 00 1 1 00 0 0 2

⎜⎜⎜⎜

⎟⎟⎟⎟⊗I YCallelicB

−1G coμ μ on =0

⎜⎜⎜⎜

⎟⎟⎟⎟⊗I G G coμ μ on = 0

& dCallelic

dt=0 ⎡

⎣⎢⎤⎦⎥

that is, Gcommon should lie in the null-space of

respectively

2 0 0 00 1 1 00 0 0 2

⎜⎜⎜⎜

⎟⎟⎟⎟⊗I YCallelicB

⎜⎜⎜⎜

⎟⎟⎟⎟⊗I G

evolutionary stops

Allelic evolution for model I:

⎜⎜⎜

⎟⎟⎟=ψ allelic2N allelicY

Ca 00 CA

⎛⎝⎜

⎞⎠⎟

∂1ΦaaT

0∂1ΦaA

0∂2ΦaA

0∂2ΦAA

⎛⎝⎜

⎞⎠⎟Gallelic

Gallelic =B−1 Gcoμ μ on

Ca 00 CA

⎛⎝⎜

⎞⎠⎟

∂1ΦaaT

0∂1ΦaA

0∂2ΦaA

0∂2ΦAA

⎛⎝⎜

⎞⎠⎟B

−1 G coμ μ on =0

Hence at the stops:

CallelicB−1G coμ μ on =0

or equivalently,

when do the alleles and modifiers agree?

The alleles on the focal locus and the modifiers agree about a stop only if

⎜⎜⎜⎜

⎟⎟⎟⎟⊗I G G coμ μ on =0

CallelicB−1 G coμ μ on =0

The seemingly simpler Gcommon = 0, amounts to 4n equations.

If the dimensions of phenotypic and allelic spaces are n resp. m, then I is a system of min{4n , 2m}, II a system of 3n equations.

Hence, generically there is never agreement.

Xaa , XaA , XAA( )Xa , XA( )

In the case of modifier evolution, these have to be satisfied by 3n, in the case of allelic evolution by min{2m , 3n} unknowns(since the act only through the ).

(When 2m > 4n, the alleles cannot even agree among themselves!)

exceptions to the generic case

We have already seen a case where the alleles and modifiers agree:

if pA = qA.

This can happen for two very different reasons:

1. When (HW)

(the standard assumption of population genetics).

2. Phenotype space can be decomposed (at least locally near the ESS) into a component that influences only l, and one that only influences μ (as is the case in organisms with separate sexes), and moreover the Ideal Free assumption applies. In that case at ESSes laa = laA = lAA = 1 and μaa = μaA = μA., Hence (HW) applies, and therefore pA = qA.

μ(XG;E) ∝ λ (XG;E)

inverse problem: find all the exceptions

Assumption: 4m ≥ n

In that case there is only agreement at evolutionary stops iff at those stops

Gcommon = 0.

inverse problem: find all the exceptions

If not (a), any individual-based restriction doing the same job implies (b).

Examples: A priori Hardy Weinberg: .Ecological effect only through one sex: either or .Sex determining loci: for AA females and aA males:

μ ∝ λ∂l ∂x = 0 ∂μ ∂x = 0

μAA = 0, λ aA = 0.

The conditions for higher dimensional phenotype spaces are that after a diffeomorphism the space can be decomposed into components in which one or more of the above conditions hold true.

For one dimensional phenotype spaces the individual-based restrictions on the ecological model that robustly guarantee that Gcommon = 0 are that (a) at evolutionary stops (HW) holds true,

∂μaA ∂xaA =0 ⇒ ∂laA ∂xaA =0 ∂laA ∂xaA =0 ⇒ ∂μaA ∂xaA =0

∂μaa ∂xaa =0 & ∂laa ∂xaa =0 ∂μAA ∂xAA =0 & ∂lAA ∂xAA =0

or (b) in their neighbourhood:

(i) oror(ii) or

biological conclusions

When the focal alleles and modifiers fail to agree the result will be an evolutionary arms race

between the alleles and the rest of the genome.

Generically there is disagreement,

PredictionHermaphroditic species have a higher turn-over rate of their genome

than species with separate sexes.

This arms race can be interpreted as a tug of war between trait evolution and sex ratio evolution.

(Even though in all the usual models there is agreement !)

QuickTime™ en eenTIFF (LZW)-decompressorzijn vereist om deze afbeelding weer te geven.

Olof Leimar

with one biologically supported exception: the case where the sexes are separate.

The end

QuickTime™ and a decompressor

are needed to see this picture.

Carolien de Kovel

history

= 12 N e C ∂sY X( )∂Y

⎣⎢⎢

⎦⎥⎥

with Poisson # offspring

discrete generations

Assumptions still rather unbiological (corresponding to a Lotka- Volterra type ODE model): individuals reproduce clonally, have exponentially distributed lifetimes and give birth at constant rate from birth onwards

general life histories

Mendelian diploids

Michel Durinx & me

extensions (2008)

QuickTime™ en eenTIFF (LZW)-decompressor

Ulf Dieckmann & Richard Law

basic ideas and first derivation (1996)

QuickTime™ en eenTIFF (ongecomprimeerd)-decompressorzijn vereist om deze afbeelding weer te geven.

Nicolas Champagnat & Sylvie Méléard

hard proofs (2003)

QuickTime™ en een-decompressorzijn vereist om deze afbeelding weer te geven.

so far only for community equilibria

non-rigorous

not yet publishednon-rigorous

hard proof for pure age dependence

Chi Tran(2006)

in reality alleles and modifiers will both evolve

= dXdt

⎛⎝⎜

⎞⎠⎟μ odifier

⎛⎝⎜

⎞⎠⎟a

⎛⎝⎜

⎞⎠⎟A

= dXdt

⎛⎝⎜

⎞⎠⎟μ odifier

⎛⎝⎜

⎞⎠⎟allelic

in “reality”:

Generically in the genotype to phenotype map all three equations are incomplete dynamical descriptions as , and may still change as a result of the evolutionary process.

ΦY Callelic Cmodifier

and are constant when is linear and and resp. the are constant (two commonly made assumptions!). Otherwise constancy of and requires that changes in the various composing terms precisely compensate each other.

Callelic Cmodifier Φ Ca CA

Cmodifier Callelic

rarely will be constant as and generically change with changes in X.Y p a p A

X: value of trait vector predominant in the population ne: effective population size, e: mutation probability per birth C: mutational covariance matrix, s: invasion fitness, i.e., initial relative growth rate of a potential Y mutant population.

= 2 ne e C ∂sY X( )∂Y

⎣⎢⎢

⎦⎥⎥

conflict between alleles and modifiers in the evolution of genetic polymorphisms

phenotypic change

genetic changes

genetic composition

ce limitmodel

ce limitcombining

ce limitconvention

ce limitwithmodel

effective population

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