Conflict between alleles and modifiers in the evolution of genetic polymorphisms
Download
1 / 49

Conflict between alleles and modifiers in the evolution of genetic polymorphisms - PowerPoint PPT Presentation


  • 75 Views
  • Uploaded on

Conflict between alleles and modifiers in the evolution of genetic polymorphisms. Hans Metz. & Mathematical Institute, Leiden University. (formerly ADN ) IIASA. VEOLIA- Ecole Poly- technique. NCB naturalis. the tool.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Conflict between alleles and modifiers in the evolution of genetic polymorphisms' - chapa


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Conflict between alleles and modifiers in the evolution of genetic polymorphisms
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
the tool

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


The canonical equation of adaptive dynamics
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.


Evolutionary constraints

directional

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 constraints


The canonical equation of adaptive dynamics1
the 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
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.


The canonical equation of adaptive dynamics2

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 dynamics


Ce is derived via two subsequent limits

branching

CE 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

branching

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
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
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


The eco genetic model

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


The eco genetic model1

: 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 model

C = classical discrete time model


The evolutionary play
the evolutionary play (

Assumptions: no parental effects on gene expressions

(mutation limitation, mutations have small effect)


Long term evolution

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 evolution

Two models


Smooth genotype to phenotype maps

Model I ((allelic evolution)

If

then

Model II (modifier evolution)

then

If

smooth genotype to phenotype maps


Model i phenotypic change in the ce limit

with (

with

the mutation probabilities per allele per birth,

and

the mutational covariance matrices,

Model I: phenotypic change in the CE limit


Model i phenotypic change in the ce limit1
Model I: phenotypic change in the CE limit (

Convention:

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


Notation

denotes the Kronecker product: (

notation

and

I the identity matrix of any required size


Model i phenotypic change in the ce limit2

structure (

matrix

and (the allelic coevolution equations)

with

Model I: phenotypic change in the CE limit

in matrix notation:


Model i phenotypic change in the ce limit3

with (

and

Model I: phenotypic change in the CE limit

combining the previous results gives:



Model i phenotypic change in the ce limit5
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):


Model i phenotypic change in the ce limit6

with (

effect a mutation in the

a--allele

A-allele

and

Model I: phenotypic change in the CE limit


Model i phenotypic change in the ce limit7

with (

effect of the resulting phenotypic change in the

aa-homozygotes

heterozygotes

AA-homozygotes

and

Model I: phenotypic change in the CE limit

on the Hardy-Weinberg manifold (pA = qA)


Summary of model i allelic trait substitution
summary of Model I (allelic trait substitution) (

on the Hardy-Weinberg manifold:


Model ii phenotypic change in the ce limit

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

with

Model II: phenotypic change in the CE limit

on the Hardy-Weinberg manifold:


Summary model comparison
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 comparison1
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 comparison2

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

manifold

summary: model comparison

Model I (allelic substitutions):

Model II (modifier substitutions):


Summary model comparison3

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

manifold

summary: model comparison


Summary model comparison4

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

manifold

summary: model comparison


Summary model comparison5

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

manifold

summary: model comparison


Summary model comparison6

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

B

on the Hardy-Weinberg

manifold

summary: model comparison


Summary model comparison7
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):


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


Genetical and developmental assumptions

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 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.

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

Evolutionary stops satisfy uniformly has maximal rank.

I:

II:

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

I:

respectively

II:

evolutionary stops


Evolutionary stops1

Hence at the stops: uniformly has maximal rank.

or equivalently,

evolutionary stops

Allelic evolution for model I:


When do the alleles and modifiers agree

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
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
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.


Inverse problem find all the exceptions1

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 exceptions

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,

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.


Biological conclusions

Olof Leimar a stop only if

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.

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.


The end

The end a stop only if

Carolien de Kovel


History

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

history

Mendelian

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


In reality alleles and modifiers will both evolve1

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 evolve

in “reality”:


The canonical equation of adaptive dynamics3
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.


ad