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The Effect of Background Selection at a Single Locus on Weakly Selected Genet. Res., Camb.(1999), 73, pp. 133–146. With 6 figures. Printed in the United Kingdom # 1999 Cambridge University Press 133 The effect of background selection at a single locus on weakly selected, partially linked variants " # # WOLFGANG STEPHAN *, BRIAN CHARLESWORTH GILEAN MVEAN " Department of Biology, Uniersity of Rochester, Rochester, NY 14627-0211, USA # Institute of Cell, Animal and Population Biology, Uniersity of Edinburgh, Edinburgh EH9 3JT, UK (Receied 18 May 1998 and in reised form 18 September 1998) Summary Previous work has shown that genetic diversity at a neutral locus is affected by background selection due to recurrent deleterious mutations as though the effective population size Ne is reduced by a factor that is calculable from genetic parameters such as mutation rates, selection coefficients, and the rates of recombination between sites subject to selection and the neutral locus. Given that silent changes at third coding positions are often subject to weak selection pressures, it is important to develop similar quantitative predictions of the effects of background selection on variation and evolution at weakly selected sites. A diffusion approximation is derived that describes the effects of the presence of a single locus subject to mutation and strongly deleterious selection on variation and evolution at a partially linked, weakly selected locus. The results are validated by computer simulations using the Ito pseudo-sampling method. We show that both nucleotide site diversity and rates of molecular evolution at a weakly selected locus are affected by background selection as though Ne is reduced in the same way as for a neutral locus. Heuristic arguments are presented as to why the change in Ne for the neutral case also applies with weak selection. As in the case of a neutral locus, the number of segregating sites in the population is poorly predicted from the change in Ne. The potential significance of the results in relation to the effects of recombinational environment on molecular variation and evolution is discussed. regions of reduced recombination, suggesting that 1. Introduction selection at weakly selected sites is less effective when There has recently been a great deal of theoretical recombination is infrequent (Kliman & Hey, 1993). work on variation and evolution at loci that are These observations can be explained by (i) ‘selective closely linked to sites that are the targets of selection. sweeps’ of favourable mutations, which result in the This work has largely been motivated by the ob- fixation of adjacent chromosomal regions (Maynard servation of reduced DNA variability in natural Smith & Haigh, 1974; Kaplan et al., 1989; Stephan et populations of Drosophila at loci situated in regions al., 1992; Stephan, 1995), (ii) ‘background selection’, where genetic recombination is relatively infrequent, in which neutral or nearly neutral variants are lost as compared with regions where it occurs at normal a result of linkage to strongly deleterious mutant frequencies (reviewed by Aguade! & Langley, 1994; alleles that are destined to be rapidly eliminated from Aquadro et al., 1994; Moriyama & Powell, 1996). the population (Charlesworth et al., 1993, 1995; Similar patterns have recently been detected in Mus Charlesworth, 1994, 1996; Hudson, 1994; Hudson & (Nachman, 1997), Aegilops (Dvorak et al., 1998) and Kaplan, 1994, 1995; Nordborg et al., 1996), (iii) Lycopersicon (Stephan & Langley, 1998). In addition, temporally varying selection pressures, which can the level of codon bias in D. melanogaster is lower in cause linked variants to be brought close to loss or fixation (Gillespie, 1994, 1997; Barton, 1995). In this paper we will be concerned solely with the effects of background selection. Previous work on this * Corresponding author. Telephone: j1 (716) 275 009. Fax: j1 (716) 275 2070. e-mail: stephan!troi.cc.rochester.edu. problem has concentrated on the properties of neutral Downloaded from https://www.cambridge.org/core. IP address: 170.106.40.139, on 29 Sep 2021 at 23:05:34, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0016672399003705 W. Stephan et al. 134 variants linked to loci under selection. The reduction that of the mutant allele a by q. Furthermore, we in genetic diversity at neutral sites predicted by the assume that q ' 1; thus, selection acts primarily background selection model can be largely understood against heterozygous carriers of the mutant allele. Let in terms of a reduction in effective population size (Ne) u be the mutation rate from A to a, and t be the caused by the fitness effects of the loci under selection product of the dominance coefficient and the (Charlesworth et al., 1993), and useful formulae for coefficient of selection against mutant homozygotes; this reduction have been developed (Hudson, 1994; then q l u\t (approximately), according to the classi- Hudson & Kaplan, 1994, 1995; Nordborg et al., 1996; cal formula for equilibrium under mutation and Santiago & Caballero, 1998). Charlesworth (1994) selection. The second locus is assumed to be under derived results for the effects of background selection directional selection; furthermore, it is assumed that on the probabilities of fixation of weakly selected dominance is intermediate, so that Bb and BB favourable or deleterious alleles in a non-recombining individuals have fitness 1js and 1j2s, respectively, genome; Peck (1994) studied the effect of background relative to a fitness of 1 for bb. (The latter assumption selection on the fixation probability of a strongly could be generalized to arbitrary degrees of dominance selected favourable allele in a non-recombining without changing the main conclusions of this paper.) genome subject to recurrent deleterious mutations. s may be negative or positive. We assume that QsQ ' t. Barton (1995) has obtained results for the effect of Recombination between the two loci is measured by recurrent deleterious mutations at a single locus on the recombination fraction, rh. the fixation probability of a linked, strongly selected, Since the first locus is assumed to be in equilibrium, favourable mutation. it is convenient to deviate from the standard notation The results of these calculations suggest that of two-locus population genetics theory and to background selection affects fixation probabilities introduce the following variables: x is the frequency through a reduction in Ne, in a way which is virtually of B among chromosomes carrying allele A, and y is identical to its effect on diversity in the neutral case. the frequency of B among chromosomes carrying But no theoretical results on fixation probabilities or allele a. Then the deterministic recurrence equations genetic diversity for the effect of background selection for the changes in x and y are (Nordborg et al., 1996) on partially linked, weakly selected loci are available. 5 (By weak selection, we mean that the product of the ∆x l sx(1kx)kqr(xky), 6 (1) effective population size and the selection coefficient is 1 7 ∆y l sy( ky)jp(rjt)(xky), 8 of the order of one, so that finite population size effects can significantly influence the fate of the alleles where r l rh(1kt). These equations are exact to O(s) in question.) Given the evidence that such weak and O(q). On the boundary, the vector field (defined selection pressures are important in controlling by the right-hand side of (1)) is directed towards the patterns of variation and evolution at synonymous interior of the unit square, except at (0, 0) and (1, 1). sites in Drosophila (Akashi, 1995, 1996; Akashi & In parallel, we consider the corresponding diffusion Schaeffer, 1997), it is clearly important to have a process. On the time scale of generations (measured quantitative theory of the effects of background by τ), the Kolmogorov forward equation for the selection on weakly selected sites. The purpose of the probability densities of x and y is (cf. Nordborg et al., present paper is to present some analytical and 1996) simulation results for the effects of the presence of a c single locus subject to mutation and strongly del- p(x, y, τ) l Lp(x, y, τ), (2) eterious selection on a partially linked, weakly selected cτ locus. We show that both nucleotide site diversity and where rates of molecular evolution at the weakly selected # locus are affected by background selection as though 1 c c N L l # y(1ky)k [sy(1ky)jp(rjt)(xky)] e is reduced in the same way as for a neutral locus. 4Nq cy cy # 1 c c 2. The model and its assumptions j # x(1kx)k [sx(1kx)kqr(xky)]. (3) 4Np cx cx We consider a two-locus, two-allele system, in a population of N diploid, randomly mating individuals. Equation (2), with L defined as in (3), is exact to " We assume that N is sufficiently large and selection is O(N− , s); quadratic and higher-order terms in q are sufficiently strong that the alleles A and a at the first also neglected in the drift operator. To define the locus are in an approximate equilibrium, due to diffusion problem completely, initial and boundary selection against the deleterious mutant allele a. conditions have to be specified. With respect to the Following the notation of Nordborg et al.(1996), the initial value, we assume throughout this paper that a frequency of the wild-type allele A is denoted by p and diffusion process which is at (x, y) at time τ has started Downloaded from https://www.cambridge.org/core. IP address: 170.106.40.139, on 29 Sep 2021 at 23:05:34, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0016672399003705 The effect of background selection on weakly linked sites 135 at (ξ, η) at time 0.
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