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Conflict between alleles and modifiers in the evolution of genetic polymorphismsPowerPoint Presentation

Conflict between alleles and modifiers in the evolution of genetic polymorphisms

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on the Hardy-Weinberg the mutation probability per haplotype per birth, the covariances of the mutational effects of modifiers.

### The end a stop only if

Conflict between alleles and modifiersin the evolution of genetic polymorphisms

Hans Metz

&

Mathematical Institute,

Leiden University

(formerly

ADN)

IIASA

VEOLIA-

Ecole

Poly-

technique

NCBnaturalis

the tool

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

the canonical equation of adaptive dynamics

with Mendelian reproduction:

evolutionary

stop

= 0

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.

selection

phenotype

genotype

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

reading direction

coding region

DNA

regulatory regions

evolutionary constraintsthe canonical equation of adaptive dynamics

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

will lead to a phenotypic C that changes

with the genetic changes underlying the change in X.

additional (biologically unwaranted) assumption

I only showed (and use)

the canonical equation for the case of

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.

R0 : average life-time offspring number

Ts : average age at death

: effective variance of life-time offspring number

of the residents

of the residents

Tr : average age at reproduction

the canonical equation of adaptive dynamicsCE is derived via two subsequent limits

individual-based stochastic process

t

trait value

mutational step size

0

system size ∞

successful mutations/time 0

limit type:

this talk: evolution of genetic polymorphisms

individual-based stochastic process

t

trait value

mutational step size

0

system size ∞

successful mutations/time 0

limit type:

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

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

- (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.

the equilibria can be calculated by

Organism with a potentially polymorphic locus with two segregating alleles, leading to the phenotype vector , with .

: instantaneous ecological environment

: expected expected per capita lifetime macrogametic output

(= average number of kids mothered)

: expected per capita lifetime microgametic output times

fertilisation propensity

(average number of kids fathered)

Abbreviations: , etc. (and similar abbreviations later on).

the eco-genetic model: total birth rate density (C: total population density, )

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

( and )

: genotype birth rate densities (C: genotype densities, , etc)

random union of gametes:

Point equilibria:

with

, etc.

example ecological feedback loop:

the eco-genetic modelC = classical discrete time model

the evolutionary play (

Assumptions: no parental effects on gene expressions

(mutation limitation, mutations have small effect)

I. Evolution through allelic substitutions (

allelic trait vectors

genotype to phenotype map: etc.

II. Evolution through modifier substitutions

Abbreviations: etc.

b: original allele on generic modifier locus,

B: mutant, changing into

long term evolutionTwo models

Model I ((allelic evolution)

If

then

Model II (modifier evolution)

then

If

smooth genotype to phenotype mapswith (

with

the mutation probabilities per allele per birth,

and

the mutational covariance matrices,

Model I: phenotypic change in the CE limitModel I: phenotypic change in the CE limit (

Convention:

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

matrix

and (the allelic coevolution equations)

with

Model I: phenotypic change in the CE limitin matrix notation:

Model I: phenotypic change in the CE limit (

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

with

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

with (

effect of the resulting phenotypic change in the

aa-homozygotes

heterozygotes

AA-homozygotes

and

Model I: phenotypic change in the CE limiton the Hardy-Weinberg manifold (pA = qA)

summary of Model I (allelic trait substitution) (

on the Hardy-Weinberg manifold:

with , the mutation probability per haplotype per birth, the covariances of the mutational effects of modifiers.

with

Model II: phenotypic change in the CE limiton the Hardy-Weinberg manifold:

summary: model comparison the mutation probability per haplotype per birth, the covariances of the mutational effects of modifiers.

Model I (allelic substitutions):

Model II (modifier substitutions):

summary: model comparison the mutation probability per haplotype per birth, the covariances of the mutational effects of modifiers.

Model I (allelic substitutions):

Model II (modifier substitutions):

on the Hardy-Weinberg the mutation probability per haplotype per birth, the covariances of the mutational effects of modifiers.

manifold

summary: model comparisonModel I (allelic substitutions):

Model II (modifier substitutions):

on the Hardy-Weinberg the mutation probability per haplotype per birth, the covariances of the mutational effects of modifiers.

manifold

summary: model comparisonon the Hardy-Weinberg the mutation probability per haplotype per birth, the covariances of the mutational effects of modifiers.

manifold

summary: model comparisonmanifold

summary: model comparisonA the mutation probability per haplotype per birth, the covariances of the mutational effects of modifiers.

B

on the Hardy-Weinberg

manifold

summary: model comparisonsummary: model comparison the mutation probability per haplotype per birth, the covariances of the mutational effects of modifiers.

Model I (allelic substitutions):

Model II (modifier substitutions):

in reality alleles and modifiers will both evolve the mutation probability per haplotype per birth, the covariances of the mutational effects of modifiers.

combining Models I and II:

evolutionary statics the mutation probability per haplotype per birth, the covariances of the mutational effects of modifiers.

uniformly has full rank and uniformly has maximal rank.

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.

genetical and developmental assumptionsIn 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.

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).

Evolutionary stops satisfy uniformly has maximal rank.

I:

II:

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

I:

respectively

II:

evolutionary stopsHence at the stops: uniformly has maximal rank.

or equivalently,

evolutionary stopsAllelic evolution for model I:

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

I

and

II

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 do the alleles and modifiers agree?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.

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

Hence, generically there is never agreement.

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

exceptions to the generic case a stop only if

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).

Phenotype space can be decomposed (at least locally near the ESS) into a component that influences only , and one that only influences (as is the case in organisms with separate sexes),

and moreover the Ideal Free assumption applies.

In that case at ESSes aa =aA =AA =1 and aa =aA =A., Hence (HW) applies, and therefore pA =qA.

inverse problem: find all the exceptions a stop only if

Assumption:4m≥n

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

Gcommon = 0.

or (b) in their neighbourhood: a stop only if

(i) or

or

(ii) or

Examples: A priori Hardy Weinberg: .

Ecological effect only through one sex: either or .

Sex determining loci: for AA females and aA males:

inverse problem: find all the exceptionsFor 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,

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

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.

Olof Leimar a stop only if

biological conclusionsWhen 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.

This arms race can be interpreted as

a tug of war between trait evolution and sex ratio evolution.

Generically there is disagreement,

with one biologically supported exception:

the case where the sexes are separate.

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

Prediction

Hermaphroditic species have a higher turn-over rate of their genome than species with separate sexes.

Carolien de Kovel

basic ideas and first derivation (1996) a stop only if

hard proofs (2003)

extensions (2008)

Ulf Dieckmann & Richard Law

Nicolas Champagnat & Sylvie Méléard

Michel Durinx & me

hard proof for pure age dependence

Chi Tran

(2006)

not yet published

non-rigorous

historyMendelian

diploids

general

life histories

discrete generations

with Poisson

# offspring

so far only for

community equilibria

non-rigorous

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

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.

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.

rarely will be constant as and generically change with changes inX.

in reality alleles and modifiers will both evolvein “reality”:

the canonical equation of adaptive dynamics equations are incomplete dynamical descriptions as , and may still change as a result of the evolutionary process.

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.

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