Conflict between alleles and modifiers in the evolution of genetic polymorphisms

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

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

3. the canonical equation of adaptive dynamics with Mendelian reproduction: evolutionary stop = 0 X: value of trait vector predominant in the population Ne: 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.

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

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 I only showed (and use) the canonical equation for the case of 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.

7. R0 : average life-time offspring number Ts : average age at death : effective variance of life-time offspring number of the residents of the residents Tr : average age at reproduction the canonical equation of adaptive dynamics

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

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

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 For a physiologically structured population with all individuals born in the same physiolocal state, mating randomly with respect to genetic differences, • (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. the equilibria can be calculated by

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

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

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

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

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

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

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

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

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

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

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 (pA = qA):

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

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

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

27. with , the mutation probability per haplotype per birth, the covariances of the mutational effects of modifiers. with Model II: phenotypic change in the CE limit 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. on the Hardy-Weinberg manifold summary: model comparison Model I (allelic substitutions): Model II (modifier substitutions):

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

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

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

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

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. uniformly has full rank and uniformly has maximal rank. 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. 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. 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 “Ideal Free” assumption).

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

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

41. The alleles on the focal locus and the modifiers agree about a stop only if I and II 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 do the alleles and modifiers agree? 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. The seemingly simpler Gcommon = 0, amounts to 4n equations. Hence, generically there is never agreement. (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 pA = qA. This can happen for two very different reasons: 1. When (HW) (the standard assumption of population genetics). 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 pA =qA.

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

44. or (b) in their neighbourhood: (i) or or (ii) or Examples: A priori Hardy Weinberg: . Ecological effect only through one sex: either or . Sex determining loci: for AA females and aA males: inverse problem: find all the exceptions For one dimensional phenotype spaces the individual-based restrictions on the ecological model that robustly guarantee that Gcommon = 0 are that (a) at evolutionary stops (HW) holds true, If not (a), any individual-based restriction doing the same job implies (b). 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.

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

46. The end Carolien de Kovel

47. basic ideas and first derivation (1996) hard proofs (2003) extensions (2008) Ulf Dieckmann & Richard Law Nicolas Champagnat & Sylvie Méléard Michel Durinx & me hard proof for pure age dependence Chi Tran (2006) not yet published non-rigorous history Mendelian diploids general life histories discrete generations with Poisson # offspring so far only for community equilibria non-rigorous 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

48. 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 inX. in reality alleles and modifiers will both evolve in “reality”:

49. the canonical equation of adaptive dynamics X: value of trait vector predominant in the population ne: 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.