Modeling evolutionary genetics

Jason Wolf

Department of ecology and evolutionary biology

University of Tennessee

Goals of evolutionary genetics– Basis of genetic and phenotypic variation

• # and effects of genes• gene interactions• pleiotropic effects of genes• genotype-phenotype relationship

– Origin of variation• Distribution of mutational effects• Recombination

– Maintenance of variation• Drift• Selection

– Distribution of variation• within and among populations (metapop. structure)• within and among species• clinal variation

Major questions

• Molecular evolution– rate of neutral and selected sequence changes– gene and genome structure

• Character evolution– rate of evolution– predicted or reconstructed direction of – evolutionary constraints– genotype-phenotype relationship (development)

• Process of population differentiation– outbreeding depression and hybrid inviability

• Process of speciation– genetic differentiation– reproductive isolation

Approaches

• Traditionally two major approaches have been used–Mendelian population genetics

• examine dynamics of a limited # of alleles at a limited # of loci

–quantitative genetics• assume a large # of genes of small effect

• continuous variation

• statistical description of genetics and evolution

Population genetic example

• Example captures basic approach to evolutionary models–evolution proceeds by changes in the

frequencies of alleles

–basic processes underlie almost all other approaches to modeling

• Conclusions from simple pop-gen models can be a useful first approach

A population genetic model

• Assumptions– a single locus with two alleles (A and a)

–diploid population

–random mating

–discrete generations

– large population size

The population

• With random mating the frequencies of the three genotypes are the product of the individual allele frequencies

• This is the “Hardy-Weinberg equilibrium”

• F(A) = p F(a) = q

AA Aa aa

p2 2pq q2

Selection

Genotype Total

AA Aa aa

Freq. before selection p2 2pq q2 1 = p2 + 2pq + q2

Relative fitness wAA wAa waa

After selection p2 wAA 2pq wAa q2 waa

Normalized p2 wAA 2pq wAa q2 waa

aa2

AaAA2 wq2pqwwp w

w w w

Evolution

w

pqwwp p AaAA

2

w

wqpqw q aa

2Aa

Allele frequencies in the next generation

• Selection biases probability of sampling the two alleles when constructing the next generation

• Genotype frequencies are still in H-W equilibrium at the frequencies defined by p and q

Selection

• Can define any mode of selection–frequency dependence

–overdominance

–diversifying

–sexual

–kin

An example

• Assume overdominance (heterozygote superiority)

• fitness of Aa is greater than the fitness of AA or aa

wAA = 0.9 wAa = 1 waa = 0.8

• What is the equilibrium allele frequency?

Equilibrium

w

wwwwpq p aaAaAaAA )()(

qp

Change in allele frequency across generations p - p

Equilibrium frequency ( ) reached when p = 0

)(ˆ)(ˆ0 aaAaAaAA wwqwwp

p̂

Equilibrium

For our example:

3

2

8.09.02

8.01ˆ

aaAAAa

aaAa

ww2w

ww p

aaAAAa

aaAa

ww2w

ww p

ˆ

Stability of equilibrium can be assessed by a Taylor series expansion about p̂

Other factors to consider

• Lots of questions remain and can be addressed in this framework• effects of non-random mating

• inbreeding

• limited migration

• metapopulation structure

• other modes of assortative mating

• effects of sampling variance (drift)

• behavior of non-selected alleles

• interaction between drift and selection

Inbreeding

• Non-random mating (between related individuals)

• Leads to correlation between genotypes of mates

• Frequencies are no longer products of allele frequencies

• Leads to reduction in heterozygosity (measured by F)

• Can rederive evolutionary equations using these new genotype frequencies AA Aa aa

p2 + pqF 2pq - 2pqF q2 + pqF

Drift

• Is random variation in allele frequencies due to sampling error of gametes

– sampling probabilities are given by the binomial probability function

• Sampling variance depends on population size (N)

• The probability of a population having i alleles of type A (where i has a value between 0 and 2N):

122)Pr(

Niqp

i

Ni

N

ip

2

Drift

• Can model probability of fixation (p = 0 or 1)– rate of molecular evolution

– neutral theory

– molecular clocks

• Can combine with selection– deterministic versus stochastic dynamics

• Can introduce mutations– balance of mutation and drift

• Changes through time can be modeled with differential equations and a diffusion approximation

Other questions

• Can look at dynamics through time to examine common ancestry

• Can be used to examine relationships of genes, populations and species

• Coalescent models examine the probability that two alleles were derived from the same common ancestor– looks back in time until a common ancestor is found – this is

a coalescent event

– various models are used to calculate these probabilities

• Coalescent events are nodes in a tree of diversification

More complex genetic systems

• Dynamics of the 1 locus system are easily expanded to a 2 locus system– allows for consideration of linkage between loci

and interactions between loci (epistasis)

– can model more complex modes of selection (e.g., sexual selection)

– can examine dynamics of simultaneous selection at two loci (interference)

• Dynamics of a 3 locus system start to become too cumbersome to work with analytically (27 genotypes)

Quantitative genetics

• More complex genetic systems are too complex to model using the algebra of pop. gen. models

• Potentially very large number of genes contribute to trait variation– human genome contains 40-70,000 genes

• Effect of each locus is likely to be very small

• Most traits have continuous variation anyway (e.g., body size, seed production)

From genes to distributions

• Number of genotype classes increases exponentially as # of loci increases

• Distribution becomes increasingly smooth as # of classes increases

• Continuous random variation smoothes distribution

• Genotype classes vanish and a continuous distribution emerges

• This distribution can be described by statistical parameters (mean, variance, covariance etc.)

• Parameters can be used to model aggregate behavior of genes

Evolution

• Evolution occurs when moments of the trait distribution change– usually focus on changes in the mean

• Most models based on the “infinitesimal model” (Fisher 1918)– infinite # of loci, each with an infinitesimal effect on

the trait

– allele frequency changes at any single locus are negligible, but sum of changes significant

– higher moments remain constant if selection is weak

Trait variation

• Variation can be partitioned into additive components

EGP VVV

EIDAP VVVVV

Phenotypic variance

Genetic variance

Environmental variance

Additive Genetic variance

Dominance variance

Epistatic variance

Selection

• Statistical association between a trait and fitness expressed as a covariance (Price 1970)

• This covariance gives the change in the trait mean within a generation

),cov( wzs

Phenotypic value

z-zs

Evolution

• Within generational changes transformed into cross generational changes

• Degree to which changes within a generation are maintained across generations is determined by the heritability of traits

• Heritability measures resemblance of parents and offspring (measured as a covariance)

• Resemblance is primarily due to additive effects of genes

AOP Vz,z )(2cov

Evolution

• Change in trait mean

shzzr t1t2

P(P)

A

P(P)

OP

V

V

V

z,zh

)(2 2cov

Questions

• Evolution of multiple traits– genetic relationship between traits

– non-independent evolution

– genetic constraints

• Testing validity of assumptions– Approaches to examining genetic architecture of

these types of traits

• Violation of assumptions– fewer genes of larger effect

– strong selection

Other approaches

• Models can be used as tools to define dynamics of a system in computer-based approaches– define dynamics of Monte-Carlo simulation

– move through search space in a genetic algorithm similation

– define transition probabilities in an iterative model

• Models can be made spatially explicit– cellular automata

– individual based models

NIH short courseModeling evolutionary genetics of complex traits

• Hierarchical approach– genes RNA proteins developmental

modules phenotypes populations metapopulations

• Focused on genotype –phenotype relationship and its impacts on evolutionary processes

• Grant support available

• Summer 2003 – Date TBA

Course on quantitative genetics

• NC State Summer Institute in Statistical Genetics– Quantitative Genetics

– Genomics

– Molecular Evolution

• http://sun01pt2-1523.statgen.ncsu.edu/sisg/

Recommended texts

• Principles of Population Genetics – D. L. Hartl and A. G. Clark – Sinauer

• An Introduction to Population Genetics Theory – J. F. Crow and M. Kimura – Burgess Publishing (Alpha Editions)

• Evolutionary Quantitative genetics – D. A. Roff – Chapman and Hall

• Introduction to Quantitative Genetics – D. S. Falconer and T. F. C. Mackay - Longman