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Quantitative Genetic Models for the Balance Between Migration and Stabilizing Selection

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Quantitative Genetic Models for the Balance Between Migration and Stabilizing Selection Genet. Res., Camb. (2000), 76, pp. 285–293. With 3 figures. Printed in the United Kingdom # 2000 Cambridge University Press 285 Quantitative genetic models for the balance between migration and stabilizing selection JARLE TUFTO* Department of Mathematical Sciences – Lade section, Norwegian Uniersity of Science and Technology, 7491 Trondheim, Norway (Receied 15 July 1999 and in reised form 10Noember 1999) Summary The evolution of a quantitative trait subject to stabilizing selection and immigration, with the immigrants deviating from the local optimum, is considered under a number of different models of the underlying genetic basis of the trait. By comparing exact predictions under the infinitesimal model obtained using numerical methods with predictions of a simplified approximate model based on ignoring linkage disequilibrium, the increase in the expressed genetic variance as a result of linkage disequilibrium generated by migration is shown to be relatively small and negligible, provided that the genetic variance relative to the squared deviation of immigrants from the local optimum is sufficiently large or selection and migration is sufficiently weak. Deviation from normality is shown to be less important by comparing predictions of the infinitesimal model with a model presupposing normality. For a more realistic symmetric model, involving a finite number of loci only, no linkage and equal effects and frequencies across loci, additional changes in the genetic variance arise as a result of changes in underlying allele frequencies. Again, provided that the genetic variance relative to the squared deviation of the immigrants from the local optimum is small, the difference between the predictions of infinitesimal and the symmetric model are small unless the number of loci is very small. However, if the genetic variance relative to the squared deviation of the immigrants from the local optimum is large, or if selection and migration are strong, both linkage disequilibrium and changes in the genetic variance as a result of changes in underlying allele frequencies become important. (e.g. Felsenstein, 1977; Slatkin, 1978; Barton, 1999) is 1. Introduction available. Most species are divided into a number of smaller In general, exact predictions about the evolution of subpopulations connected by migration. Migration, quantitative traits assuming Mendelian inheritance in general, tends to homogenize a population and can cannot be made unless a large amount of detailed prevent adaptation of subpopulations to local en- information about the genetic basis of the trait is vironmental conditions. This is a relatively well available (Barton & Turelli, 1989). For a trait understood phenomenon when differences in fitness determined by n diallelic loci, there are 2n different are caused by genes at a single locus only (e.g. haplotypes, which, assuming random mating, implies Haldane, 1930; Slatkin, 1973, 1985; Nagylaki & that general analytic treatment of the effect of Lucier, 1979). Most traits, however, are quantitative, evolutionary forces such as selection, migration and that is, influenced by genes at many loci (Lande, reproduction seldom is possible. For only about 10 1982).Quantitative genetic theory involving both loci, the number of haplotypes also becomes so large selection and migration, despite being an important that numerical analysis becomes infeasible. Similarly, problem, is more sparse, although some theory while not increasing exponentially if one assumes focusing on geographic variation in quantitative traits additivity, the number of parameters to be estimated between demes (Bulmer, 1980, p. 180) and in clines in order to make prediction based on explicit multilocus models would typically require very large * Tel: j47 73591888. Fax: j47 73591038. e-mail: jarlet! math.ntnu.no amounts of data. Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.234, on 24 Sep 2021 at 15:40:31, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0016672300004742 J. Tufto 286 Quantitative genetic evolutionary theory therefore as follows. In each generation, a proportion m of the has to be based on models relying on different degrees population is replaced by immigrants with mean of simplifying assumptions. The infinitesimal model breeding value z" and variance V". For simplicity, it is (Fisher, 1918; Bulmer, 1980) can be seen as a way of assumed that there is no linkage disequilibrium within overcoming the difficulties of explicit multilocus the immigrants. This will be approximately true if the models by assuming that the number of loci effectively immigrants originate from a single panmictic popu- is infinite and the allelic effects at individual loci are lation, deviating from the optimum as a result of, for additive and small. This approach reduces all genetic example, artificial selection. Migration is followed details to one parameter: the genetic variance at by stabilizing selection with fitness of individuals linkage disequilibrium VLE. Further simplifications of with phenotype P l ZjE equal to W!(P) l exp # the infinitesimal model, based on assuming normality (ks!(Pkz!) \2) such that the mean fitness of indi- or by ignoring the dynamics of the genetic variance, viduals with breeding value Z is proportional to W(Z) # are sometimes made in order to enable analytic l exp (ks(Zkz!) \2) where 1\s l 1\s!jVE. Without treatment of models of more specific evolutionary any loss of generality, we can let z! l 0 and z" l 1, situations. This is done, for example, in theory for the which is essentially equivalent to rescaling the model evolution of plasticity (Via & Lande, 1985; van in terms of z- by the deviation of the immigrants from Tienderen, 1997), theory of source-sink dynamics the local optimum z"kz!, remembering the VLE and s (Kirkpatrick & Barton, 1997) and in some non-spatial are now measured on this new dimensionless scale. models (e.g. Lande et al., 1997). An important question is therefore how well these different approximations 2. The infinitesimal model predict the evolutionary dynamics relative to more realistic models assuming finite number of loci, The assumption of the infinitesimal model that the deviations from normality and linkage disequilibrium. trait is affected by genes at an infinite number of loci Here I consider different approaches to modelling with infinitesimal and additive effects implies that the evolution of a population subject to local allele frequency changes will be infinitesimally small stabilizing selection with one-way immigration of so that there will be no change in the genetic variance individuals deviating from the local phenotypic of the genotypic values at linkage disequilibrium, VLE. optimum. This situation arises, for example, in models In addition, except for some patterns of strong linkage of source-sink dynamics (Holt & Gomulkiewicz, disequilibrium (Dawson, 1997), the distribution of 1997a, b) and in conservation genetics in the context genotypic values among offspring, conditional on the of reintroduction and intentional or unintentional genotypic values X and Y of selected parents, will supplementation of a wild population with individuals always be normal with expectation equal to the mid- kept in captivity (Hindar et al., 1991). In general, both parental value (XjY)\2 and variance equal to one stabilizing selection and migration will generate half the genetic variance at linkage equilibrium VLE linkage disequilibrium and consequently changes in (Bulmer, 1980; Turelli & Barton, 1994). The full the genetic variance. Unlike stabilizing Gaussian unconditional offspring distribution after repro- selection, however, migration will make the distri- duction, however, will depend on the distribution of bution of genotypic values depart from normality (e.g. parental genotypic values in the previous generation, Grant & Grant, 1994). More importantly, migration and does not need to be normal. will also pull the genotypic mean away from the local optimum, and thereby change the underlying allele (i) Exact Fourier transform method frequencies. The complexities of this situation make it Based on these general results, the joint effects of a good test of the various approaches to modelling migration, stabilizing selection and recombination on quantitative traits that have been proposed in the the distribution genotypic values can be found literature. numerically using the method of Turelli & Barton In the first part of the paper, exact predictions for (1994). Let ψ(z) be the initial distribution of the trait. the infinitesimal model obtained using numerical Migration changes this distribution to methods are compared with less realistic and simpler approaches based on ignoring departures from nor- ψh(z) l (1km)ψ(z)jmψ"(z), (1) mality and ignoring linkage disequilibrium altogether. where ψ"(z) is the (normal) distribution of the trait In the latter case a simple analytic result is available. among the immigrants. After selection the distribution In the second part of the paper, exact results for the of z is infinitesimal model are compared with a more realistic W(z)ψh(z) model involving a finite number of loci only and more ψd(z) l . (2) !W(z)ψ (z) specific details about the underlying genetic basis of h the trait. The life cycle is completed by reproduction, which is The details of the general evolutionary situation are equivalent to taking the mean of two randomly Downloaded from https://www.cambridge.org/core. IP address: 170.106.35.234, on 24 Sep 2021 at 15:40:31, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0016672300004742 Quantitatie genetic models for migration and stabilizing selection 287 After migration After selection After reproduction 4 2 0 4 2 0 4 2 0 4 2 0 Generations 1–7 4 2 0 4 2 0 4 2 0 –0·4 0 0·4 0·8 1·2 –0·4 0 0·4 0·8 1·2 –0·4 0 0·4 0·8 1·2 Fig. 1. Iterations of the infinitesimal mode for 7 generations (rows 1–7) with z! l 0, z" l 1, s l 1, m l 0n2, and VLE l 0n005, that is, immigrants deviating 14 genetic standard deviations from the optimum.
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