1. Evolution of functional traits (Joel Kingsolver, Biology) Traits as functions: functional, function- valued, infinite-dimensional A primer in evolutionary models: –Variation, inheritance, selection, evolution Approaches to analysing functional traits: –understanding genetic variation –Estimating selection –Predicting evolutionary responses & constraints

2. Examples of functional traits Functions of age: –growth trajectories; life history; aging Functions of environmental state: –Physiological ‘reaction norms’ Descriptions of 2-D or 3-D shape, etc

3. Age-specific mortality rates (Drosophila) Pletcher et al, Genetics (1999)

6. Evolution of quantitative traits: some basics Individual organism: –Phenotype: observable trait with value z –Genotype: genetic ‘type’ (usually inferred) Population: –Phenotypic variance, P = G + E –Genetic variance, G Evolution = change in mean trait value per generation,  z

7. Evolution, in 3 easy steps P Trait value z Frequency z

8. Evolution, in 3 easy steps (2) Trait value z Frequency z z’

9. Evolution, in 3 easy steps (3) Trait value z Frequency z z’ zz

10. A simple evolutionary model Variation and inheritance –Variance: P = G + E Selection –Selection gradient:  = P -1 ( z’ - z ) –Also:  = d[ln(W)]/dz, where W = mean population fitness Evolutionary response –  z = G 

11. Evolution on an adaptive landscape Mean phenotype, z Mean fitness, W 

12.  z = G  z may be: a scalar a vector a function  z(t) = G(t,s)  s  ds

13. Analysing genetics of functional traits Estimating G: an example Biological hypotheses about G: eigenfunction analysis G and the response to selection

14. Temperature & caterpillar growth rates: Thermal performance curves (TPCs) z(t), where t = temperature

16. GeneticVar-Cov for RGR

17. Basis = orthogonal polynomials

18. Analysing genetics of functional traits Estimating G: an example Biological hypotheses about G: eigenfunction analysis G and the response to selection

19. Faster- slower Hotter- colder Generalist- specialist

20. Eigenfunction Faster- slower Hotter- colder Generalist- specialist

21. Caterpillar growth rates

22. Temperature Performance Hotter-Colder II

23. G-covariance function: Variation in TPC position

24. First eigenfunction (65%)

25. Second eigenfunction (25%)

26. Analysing genetics of functional traits Estimating G: an example Biological hypotheses about G: eigenfunction analysis G and the response to selection

27. Eigenfunction Faster- slower Hotter- colder Generalist- specialist

29. Evolutionary Constraints: Identifying zero eigenfunctions

30. Selection and evolutionary response Estimating selection,  (s): an example Predicting evolutionary responses

31. Selection on caterpillar growth rate TPCs Measure z(t) for a sample of individuals in the lab --> estimate P(s,t) Measure fitness of those individuals in the field Estimate  (s) (cubic splines)

33. Evolutionary Response to Selection Selection Evolutionary response Evolutionary change in one generation

34. Challenges Estimation methods for G Hypothesis testing of eigenfunctions Estimating zero eigenfunctions Estimation methods for  Predicting evolutionary responses