Modeling evolutionary genetics

1. Modeling evolutionary genetics Jason Wolf Department of ecology and evolutionary biology University of Tennessee

2. 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

3. 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

4. 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

5. 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

6. A population genetic model • Assumptions • a single locus with two alleles (A and a) • diploid population • random mating • discrete generations • large population size

7. 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) = pF(a) = q AA Aa aa p2 2pq q2

8. Selection Genotype Total AA Aa aa Freq. before selectionp2 2pq q2 1 = p2 + 2pq + q2 Relative fitness wAA wAa waa After selection p2 wAA 2pq wAaq2 waa Normalized p2 wAA 2pq wAaq2 waa

9. Evolution 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andq

10. Selection • Can define any mode of selection • frequency dependence • overdominance • diversifying • sexual • kin

11. 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?

12. Equilibrium Change in allele frequency across generationsp - p Equilibrium frequency ( ) reached when Dp = 0

13. Equilibrium For our example: Stability of equilibrium can be assessed by a Taylor series expansion about

14. 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

15. 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

16. 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):

17. 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

18. 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

19. 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)

20. 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)

21. 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

22. 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

23. Trait variation • Variation can be partitioned into additive components Phenotypic variance Genetic variance Environmental variance Additive Genetic variance Dominance variance Epistatic variance

24. 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 Phenotypic value

25. 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

26. Evolution • Change in trait mean

27. 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

28. 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

29. 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

30. Course on quantitative genetics • NC State Summer Institute in Statistical Genetics • Quantitative Genetics • Genomics • Molecular Evolution • http://sun01pt2-1523.statgen.ncsu.edu/sisg/

31. 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