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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 • • Distribution of mutational effects Recombination • • Drift Selection • • • within and among populations (metapop. structure) within and among species clinal variation – Origin of variation – Maintenance of variation – Distribution of variation Major questions • Molecular evolution • Character evolution • Process of population differentiation • Process of speciation – rate of neutral and selected sequence changes – gene and genome structure – rate of evolution – predicted or reconstructed direction of – evolutionary constraints – genotype-phenotype relationship (development) – outbreeding depression and hybrid inviability – 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 Freq. before selection AA Aa p2 2pq Total aa q2 Relative fitness wAA wAa waa After selection p2 wAA 2pq wAa q2 waa p2 wAA 2pq wAa q2 waa w w w Normalized 1 = p2 + 2pq + q2 w p2w AA 2pqwAa q 2waa Evolution Allele frequencies in the next generation p2w AA pqwAa p w • • pqwAa q 2waa q w 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 Change in allele frequency across generations p - p pq p(w AA w Aa ) q(w Aa waa ) p w Equilibrium frequency ( pˆ ) reached when p = 0 0 pˆ (w AA w Aa ) qˆ (w Aa waa ) Equilibrium pˆ w Aa waa 2w Aa w AA waa For our example: pˆ w Aa waa 1 0.8 2 2w Aa w AA waa 2 0.9 0.8 3 Stability of equilibrium can be assessed by a Taylor series expansion about pˆ 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) AA p2 + pqF • Aa 2pq - 2pqF aa q2 + pqF Can rederive evolutionary equations using these new genotype frequencies 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): 2 N i 2 N 1 Pr( i ) p q i i p 2N 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 Phenotypic variance Genetic variance Environmental variance VP VG VE Additive Genetic variance Dominance variance Epistatic variance VP VA VD VI VE Selection • Statistical association between a trait and fitness expressed as a covariance (Price Phenotypic 1970) value s cov( z,w ) • This covariance gives the change in the trait mean within a generation s z - z 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 2cov(zP ,zO ) VA Evolution 2cov(zP ,zO ) VA h VP(P) VP(P) 2 • Change in trait mean r zt 1 zt h s 2 Questions • Evolution of multiple traits • Testing validity of assumptions – genetic relationship between traits – non-independent evolution – genetic constraints – 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 course Modeling 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