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