1. Conflict between alleles and modifiers in the evolution of genetic polymorphisms (formerly ADN ) IIASA Hans Metz VEOLIA- Ecole Poly- technique & Mathematical Institute, Leiden University NCB naturalis

2. the tool (Assumptions: mutation limitation, mutations have small effect.)

3. the canonical equation of adaptive dynamics X : value of trait vector predominant in the population N e : effective population size,  : mutation probability per birth C : mutational covariance matrix, s : invasion fitness, i.e., initial relative growth rate of a potential Y mutant population. with Mendelian reproduction: = 0 evolutionary stop

4. evolutionary constraints phenotype genotype directional selection coding region regulatory regions DNA reading direction Most phenotypic evolution is probably regulatory, and hence quantitative on the level of gene expressions.

5. the canonical equation of adaptive dynamics The canonical equation is not dynamically sufficient as there is no need for C to stay constant. Even if at the genotype level the covariance matrix stays constant, the non-linearity of the genotype to phenotype map  will lead to a phenotypic C that changes with the genetic changes underlying the change in X.

6. additional (biologically unwaranted) assumption symmetric phenotypic mutation distributions saving grace? I have reasons to expect that my final conclusions are independent of this symmetry assumption, but I still have to do the hard calculations to check this. I only showed (and use) the canonical equation for the case of

7. the canonical equation of adaptive dynamics R 0 : average life-time offspring number T s : average age at death : effective variance of life-time offspring number of the residents T r : average age at reproduction

8. t CE is derived via two subsequent limits system size  ∞ successful mutations / time  0 trait value individual-based stochastic process mutational step size  0 branching limit type:

9. t this talk: evolution of genetic polymorphisms system size  ∞ successful mutations / time  0 trait value individual-based stochastic process branching limit type: mutational step size  0

10. the ecological theatre Assumptions: but for genetic differences individuals are born equal, random mating, ecology converges to an equilibrium.

11. equilibria for general eco-genetic models  (1) setting the average life-time offspring number over the phenotypes equal to 1,  (2) calculating the genetic composition of the birth stream from equations similar to the classical (discrete time) population genetical ones,  with those life-time offspring numbers as fitnesses. For a physiologically structured population with all individuals born in the same physiolocal state, mating randomly with respect to genetic differences, the equilibria can be calculated by

12. the eco-genetic model Organism with a potentially polymorphic locus with two segregating alleles, leading to the phenotype vector, with. Abbreviations :, etc. (and similar abbreviations later on). : expected per capita lifetime microgametic output times fertilisation propensity (  average number of kids fathered) : instantaneous ecological environment : expected expected per capita lifetime macrogametic output (= average number of kids mothered)

13. the eco-genetic model C = classical discrete time model random union of gametes : Point equilibria: with, : allelic frequencies in the micro- resp. macro-gametic outputs ( and ) : total birth rate density ( C : total population density, ) : genotype birth rate densities ( C : genotype densities,, etc), etc. example ecological feedback loop :

14. the evolutionary play Assumptions: no parental effects on gene expressions (mutation limitation, mutations have small effect)

15. long term evolution Two models I. Evolution through allelic substitutions allelic trait vectors genotype to phenotype map: etc. Abbreviations : etc. II. Evolution through modifier substitutions b : original allele on generic modifier locus, B : mutant, changing into

16. smooth genotype to phenotype maps If Model I (allelic evolution) Model II (modifier evolution) If then

17. Model I: phenotypic change in the CE limit with the mutation probabilities per allele per birth, the mutational covariance matrices, and

18. Model I: phenotypic change in the CE limit Convention : Differentiation is only with respect to the regular arguments, not the indices.

19. notation I the identity matrix of any required size and denotes the Kronecker product:

20. Model I: phenotypic change in the CE limit in matrix notation: and (the allelic coevolution equations) with structure matrix

21. Model I: phenotypic change in the CE limit combining the previous results gives: with and

22. Model I: phenotypic change in the CE limit with

23. Model I: phenotypic change in the CE limit an explicit expression for the allelic (proxy) selection gradient: with on the Hardy-Weinberg manifold (p A = q A ) :

24. Model I: phenotypic change in the CE limit with and effect a mutation in the a --allele A -allele

25. Model I: phenotypic change in the CE limit with and effect of the resulting phenotypic change in the aa -homozygotes heterozygotes AA -homozygotes on the Hardy-Weinberg manifold (p A = q A )

26. summary of Model I (allelic trait substitution) on the Hardy-Weinberg manifold:

27. Model II: phenotypic change in the CE limit with, the mutation probability per haplotype per birth, the covariances of the mutational effects of modifiers. with on the Hardy-Weinberg manifold:

28. summary: model comparison Model I (allelic substitutions) : Model II (modifier substitutions) :

29. summary: model comparison Model I (allelic substitutions) : Model II (modifier substitutions) :

30. summary: model comparison Model I (allelic substitutions) : Model II (modifier substitutions) : on the Hardy-Weinberg manifold

31. summary: model comparison on the Hardy-Weinberg manifold

32. summary: model comparison on the Hardy-Weinberg manifold

33. summary: model comparison on the Hardy-Weinberg manifold

34. summary: model comparison A B on the Hardy-Weinberg manifold

35. summary: model comparison Model I (allelic substitutions) : Model II (modifier substitutions) :

36. in reality alleles and modifiers will both evolve combining Models I and II:

37. evolutionary statics

38. genetical and developmental assumptions In biological terms: there are no local developmental or physiological constraints. So-called genetic constraints are rooted more deeply than in the physiology or developmental mechanics. Example: some phenotypes can only be realised by heterozygotes. When there are developmental or physiological constraints, we can usually define a new coordinate system on any constraint manifold that the phenotypes run into, and proceed as in the case without constraints. IF : There are no constraints whatsoever, that is, any combination of phenotypes may be realised by a mutant in its various heterozygotes. (known in the literature as the “ I deal F ree” assumption). uniformly has full rank and uniformly has maximal rank.

39. evolutionary stops Evolutionary stops satisfy I: II: that is, G common should lie in the null-space of I: respectively II:

40. evolutionary stops Allelic evolution for model I: Hence at the stops: or equivalently,

41. when do the alleles and modifiers agree? The alleles on the focal locus and the modifiers agree about a stop only if I and II The seemingly simpler G common = 0, amounts to 4 n equations. If the dimensions of phenotypic and allelic spaces are n resp. m, then I is a system of min{4n, 2m}, II a system of 3n equations. Hence, generically there is never agreement. In the case of modifier evolution, these have to be satisfied by 3n, in the case of allelic evolution by min{2m, 3n} unknowns (since the act only through the ). (When 2m > 4n, the alleles cannot even agree among themselves!)

42. exceptions to the generic case We have already seen a case where the alleles and modifiers agree: if p A = q A. This can happen for two very different reasons: 1. When (HW) (the standard assumption of population genetics). 2.Phenotype space can be decomposed (at least locally near the ESS) into a component that influences only, and one that only influences  (as is the case in organisms with separate sexes), and moreover the Ideal Free assumption applies. In that case at ESSes aa = aA = AA = 1 and  aa =  aA =  A., Hence (HW) applies, and therefore p A = q A.

43. inverse problem: find all the exceptions Assumption: 4m ≥ n In that case there is only agreement at evolutionary stops iff at those stops G common = 0.

44. inverse problem: find all the exceptions If not (a), any individual-based restriction doing the same job implies (b). Examples: A priori Hardy Weinberg:. Ecological effect only through one sex: either or. Sex determining loci: for AA females and aA males: The conditions for higher dimensional phenotype spaces are that after a diffeomorphism the space can be decomposed into components in which one or more of the above conditions hold true. For one dimensional phenotype spaces the individual-based restrictions on the ecological model that robustly guarantee that G common = 0 are that (a) at evolutionary stops (HW) holds true, or (b) in their neighbourhood: (i) or or (ii) or

45. biological conclusions When the focal alleles and modifiers fail to agree the result will be an evolutionary arms race between the alleles and the rest of the genome. Generically there is disagreement, Prediction Hermaphroditic species have a higher turn-over rate of their genome than species with separate sexes. This arms race can be interpreted as a tug of war between trait evolution and sex ratio evolution. (Even though in all the usual models there is agreement ! ) Olof Leimar with one biologically supported exception: the case where the sexes are separate.

46. The end Carolien de Kovel

47. history with Poisson # offspring discrete generations Assumptions still rather unbiological (corresponding to a Lotka- Volterra type ODE model): individuals reproduce clonally, have exponentially distributed lifetimes and give birth at constant rate from birth onwards general life histories Mendelian diploids Michel Durinx & me extensions (2008) Ulf Dieckmann & Richard Law basic ideas and first derivation (1996) Nicolas Champagnat & Sylvie M é l é ard hard proofs (2003) so far only for community equilibria non-rigorous not yet published non-rigorous hard proof for pure age dependence Chi Tran (2006)

48. in reality alleles and modifiers will both evolve in “reality”: Generically in the genotype to phenotype map all three equations are incomplete dynamical descriptions as, and may still change as a result of the evolutionary process. and are constant when is linear and and resp. the are constant (two commonly made assumptions!). Otherwise constancy of and requires that changes in the various composing terms precisely compensate each other. rarely will be constant as and generically change with changes in X.

49. the canonical equation of adaptive dynamics X : value of trait vector predominant in the population n e : effective population size,  : mutation probability per birth C : mutational covariance matrix, s : invasion fitness, i.e., initial relative growth rate of a potential Y mutant population.