Adaptive Dynamics Workshop: Budapest, 14-19 june 2004 1 Disruptive selection on a continuous multi-locus trait Carlo Matessi Istituto di Genetica Molecolare.

Presentation on theme: "Adaptive Dynamics Workshop: Budapest, 14-19 june 2004 1 Disruptive selection on a continuous multi-locus trait Carlo Matessi Istituto di Genetica Molecolare."— Presentation transcript:

1 Adaptive Dynamics Workshop: Budapest, 14-19 june 2004 1 Disruptive selection on a continuous multi-locus trait Carlo Matessi Istituto di Genetica Molecolare - Consiglio Nazionale delle Ricerche - Pavia - Italy Marco Archetti Dept. Biologie, Ecologie et Evolution - Université de Fribourg - Switzerland Alexander Gimelfarb San Francisco - U.S.A.

2 2 Sasha Gimelfarb Sasha died in San Francisco last May. This presentation is dedicated to him.

3 3 Pyrenestes ostrinus Food specializations feeds mostly on large, hard seeds (Scleria verrucosa) feeds mostly on soft seeds (S. mikawana, S. boivinii) large-billed morph small-billed morph frequency distribution of lower mandible width (mm) (Smith T.B., 1987, 1993)

4 4 Predatorial tactics and food types Haematopus ostralegus (oistercatcher) worm-feeder hammerer stabbers hammerers worm-feeders

5 5 Sympatric speciation ? Phylloscopus trochilus sibilatrix Regulus regulus ignicapillus Parus major caeruleus

6 6 Questions about discrete adaptive polymorphisms æ Can they be generated and maintained stably throughout evolution by frequency dependent disruptive selection? æ What prevents appearance of intermediate variation between extreme forms, given multiple alleles and loci? æ Under what conditions such polymorphisms lead to sympatric speciation?

7 7 The Point of View of “Long-Term Evolution” æ Adaptive evolution of a population is viewed as a succession of – transient – equilibrium states produced by the short term demographic dynamics. æ Transit along this succession is driven by mutation, causing the appearance of new types in a resident population. Natural selection determines whether such mutations are eliminated or become established (invasion). æ A long term equilibrium is therefore a state, either monomorphic or polymorphic, that cannot be invaded by any mutation.

8 8 Frequency dependent disruptive selection æ A continuous trait (strategy) with values in [-1,1] æ Random pairwise contests where y against x gets payoff v(y,x) = 1+  y 2 –(  +  )xy+  x 2, 0 <  <  ; v(x,x)  1 æ Evolutionary singularity at x° = 0 ("PEAST" – see Christiansen, 1991 – or "Branching Point") æ Fitness of y in a population of mean m and variance s 2 is w(y,m,s 2 ) = E{v(y,x)|y} = 1+  y 2 –(  +  )my+  (m 2 +s 2 ) æ Mean fitness is therefore W = 1 + (  )s 2 and is maximized in a population containing only the two extreme types -1 and 1 at equal frequencies, in which s 2 = 1. æ This fitness representation approximates any specific ecological situation causing disruptive selection, as long as trait values in the population are close to x°

9 9 Summary Primary trait controlled by two (non additive) loci: æ simulation of long term dynamics æ outline of possible polymorphic long term equilibria æ coevolution of modifiers: assortment recombination variability of expression æ likelihood that such modifiers actually evolve

10 10 The genetic model æ Loci A and B with alleles A 1, …, A m and B 1, …, B n (m, n ≤ 20), and recombination rate r B 1 B 1 B 1 B 2 …B n B n A 1 A 1 x 11 11 x 11 12 …x 11 nn A 1 A 2 x 12 11 x 12 12 …x 12 nn ………x ij kl … A m A m x mm 11 x mm 12 …x mm nn maximum difference is ≤ 1 maximum difference is ≤ 1 æ A phenotypic table, X={x ij kl }, assigns trait values to genotypes {A i A j B k B l } such that x ij kl  [-1,1] and each locus has a “limited effect”:

11 11 Simulation of Long Term Evolution 1. Start from a monomorphic resident population of phenotype x, taken at random in [-1,1] 3. This population, with phenotypic table {X,Y}, is iterated according to the exact, deterministic two-locus recurrence equations (short term dynamics), under frequency dependent disruptive selection and random mating, until equilibrium is reached. B 1 B 1 …B n B n B*B 1 …B*B n B*B* A 1 A 1 x 11 11 …x 11 nn y 11 *1 …y 11 *n y 11 ** …………………… A m A m x mm 11 …x mm nn y mm *1 …y mm *n y mm ** 2. A mutant allele A* or B* is introduced at low frequency in the current resident population. The mutation produces an extension, Y, of the resident phenotypic table X, e.g. long term dynamics

12 12 Two types of mutations Pattern mutations. Each element of Y is chosen at random in [-1,1], subject to the limited effect constraint. Scale mutations. (i) Choose at random a resident allele of the mutating locus, e.g., A 1. (ii) Mutant trait values, {y *j kl, y ** kl }, result from a linear transformation of the trait values induced by A 1, {x 1j kl }: mean{y *j kl : kl } = mean{x 1j kl : kl } + d, d ~ [-0.5,0.5] ;  j y *j kl - mean{y *j kl : kl } = c [ x 1j kl - mean {x 1j kl : kl } ], c ~ [0,1] or [1,1.5] with equal probability ;  j mean{y ** kl : kl } = mean{y *1 kl : kl } + d y ** kl - mean{y ** kl : kl } = c [ y *1 kl - mean{y *1 kl : kl } ]

13 13 24 parameter sets considered: i.e., all combinations of  = 0.1, 0.9  = 0.1, 0.5, 1 r = 0.01, 0.1, 0.25, 0.5 Parameter Values

14 14 Results - 50% Pattern and 50% Scale Mutations r = 0.01r = 0.5 phenotypic variance number of mutation events  = 0.1  = 0.1  = 0.9  = 1

15 15 Phenotypic Variances at the End  = 0.1  = 0.9  = 0.1  = 0.5  = 1  = 0.1  = 0.5  = 1 phenotypic variance

16 16 Phenotypic Distribution at the End mean of 120 runs: 5 replicates for each of the 24 parameter sets intervals of trait value mean frequency

17 17 Maximal Phenotypic Tables 123456 019831911 Distribution of the number of alleles per locus no. alleles frequency æ In 88 cases out of 120, the final population is – either exactly or approximately – at a state consisting of 2 alleles per locus, with a phenotypic table that contains only the trait values {-1, 0, 1}, a class of tables that we may call maximal. type I51 type II33 type III3 type IV1 complex32 Distribution of table types 011 -100 100 0-1-1 101 0-10 101 0-10 101 type III type IV type I type II

18 18 Coevolution of modifiers 1. Modifiers of mating behavior, increasing – or decreasing – strength of assortative mating with respect to the primary trait (possibly leading to sympatric speciation). 2. Modifiers of linkage, increasing or decreasing the rate of recombination, r, between the A and B loci (possibly resulting in the formation of a “supergene”). 3. Modifiers of expression of the primary trait, increasing – or decreasing – the non genetic variability of the trait (parameter ).

19 19 The simulation procedure 1. With a given probability (e.g., 10%), a mutation affects a locus of the modifier phenotype instead of the primary trait. 2. With equal probabilities this mutation increases or decreases, of a small fixed amount, , the trait value, m, of the modifier in the current population. 3. The dominant eigenvalue, , of the linearized recurrence equations for the short term dynamics of the (rare) mutants is evaluated. If > 1 the mutant invades and the trait value of the modifier is changed to m' = m ± . Otherwise the modifier remains unchanged.

20 20 Model of Assortative Mating (a particular case of Gavrilets & Boake, 1998) æ Given that a female of (primary) trait value x has encountered a male of trait value y, the probability that she accepts to mate with this male is  (x,y) = exp{-S(x-y) 2 }, S = strength of assortment, 0 ≤ S < ∞. æ When a male is rejected, the female searches for an other male, till one is found that is suitable. Hence, each female is certain to mate and there is no cost of assortment to females.

21 21 Results for Assortment

22 22 Results for Recombination  = 0.1,  = 1, r = 0.5

23 23 Variable expression of the primary trait Distribution of trait value given genotypic value X X1 large X1 = 0 æ Suppose that the trait value, x, is the sum of a genotypic value, X, and a random error due to various disturbances  Assume that x has Beta distribution over [-1,1], with mean X and variance (1-X 2 )/(1+ ) æ The mean payoff to individuals of genotypic value Y, when opponent has genotypic value X, is æ Hence the genotypic value is subject to disruptive selection and tends to converge to the polymorphic ESS of maximal dispersion

24 24 Results for Variability of Expression  = 0.1  = 0.1 r = 0.5 N=exp{- } 0 < N ≤ 1

25 25 Comparison of invasion eigenvalues Small mutational step: at least 100 steps required to cover entire range assortment:0 ≤ S ≤ 1 recombination:0.5 ≤ r ≤ 0 variability:0 < N ≤ 1

26 26

27 27 Comparison of invasion eigenvalues Large mutational step: at least 50 steps required to cover entire range assortment:0 ≤ S ≤ 1 recombination:0.5 ≤ r ≤ 0 variability:0 < N ≤ 1

28 28

29 29 Effect of strength of selection

30 30 Conclusions I æ Disruptive selection for a continuous trait creates adaptive polymorphisms that are intrinsically stable in the long term. In the meanwhile it raises the variance, and hence the mean fitness of the population. æ Only if the primary trait is controlled by a single locus maximal variance is reached and the polimorphism consists of two discrete and extreme forms ("branching"). æ If several loci are involved, polymorphism takes instead the form of a continuous distribution, the variance of which increases with increasing strength of selection and decreasing recombination, but stays far below its maximum level. æ In these cases there remains selective pressure to increase phenotypic variance. Such pressure could in principle produce discrete morphs by acting on appropriate modifiers that might be available to coevolve with the primary trait.

31 31 Conclusions II æ Modifiers that could have such effect in all circumstances are (1) increase of variability of expression of the primary loci, and (2) reduction of recombination. In case (1) the genetic contribution to polimorphism is likely to be moderate or even negligible. In case (2) the support of the polymorphism is fully genetic and would appear as a “super-gene”. æ Also modifiers of assortment could have the same effect, leading eventually to sympatric speciation, but only if strength of selection is high. Otherwise, assortment might not even invade random mating, or if invading it might precipitate destruction of polymorphism. æ However, none of these modifier effects is likely to actually evolve. Any modifier is most likely to carry with itself some intrinsic cost, inducing a negative density independent selective component that, even if quite small, can easily overcome the tiny frequency dependent fitness advantage carrried by the modifier. æ In this respect, of the three types of modifiers considered, modifiers of assortment are the most unlikely to evolve, while modifiers of variability are the least unlikely.