Gene I. Res., Camb. (1989), 54, pp. 59-77 With 8 text-figures Printed in Great Britain 59 The divergence of a polygenic system subject to stabilizing selection, mutation and drift NICK BARTON Department of Genetics and Biometry, University College London, 4 Stephenson Way, London NWI 2HE (Received 4 December 1987 and in revised form 26 January 1989) Summary Polygenic variation can be maintained by a balance between mutation and stabilizing selection. When the alleles responsible for variation are rare, many classes of equilibria may be stable. The rate at which drift causes shifts between equilibria is investigated by integrating the gene frequency distribution W2NY\{pqYNl''1. This integral can be found exactly, by numerical integration, or can be approximated by assuming that the full distribution of allele frequencies is approximately Gaussian. These methods are checked against simulations. Over a wide range of population sizes, drift will keep the population near an equilibrium which minimizes the genetic variance and the deviation from the selective optimum. Shifts between equilibria in this class occur at an appreciable rate if the product of population size and selection on each locus is small (Nsa2 < 10). The Gaussian approximation is accurate even when the underlying distribution is strongly skewed. Reproductive isolation evolves as populations shift to new combinations of alleles: however, this process is slow, approaching the neutral rate (x/i) in small populations. possibility is that mutation alone is responsible for the 1. Introduction high heritabilities which are found in nature (Lande, How is polygenic variation maintained, and how does 1975). If this is so, one might hope that the genetic it evolve? How does a population separate into two variance could be predicted from measurable incompatible species? How can a population search quantities, such as the total mutation rate, the effective through the vast space of possibilities to find better number of loci, and the strength of selection. adapted combinations of alleles? How is selection on Unfortunately, predictions depend on the distribution the whole organism reflected in the evolution of of allelic effects at individual loci: if a large number of individual genes? Because many genes interact to alleles segregate, giving an approximately Gaussian produce the observable phenotype, these questions distribution, then the variance is proportional to the have proved hard to approach, either empirically or square root of the ratio between mutation rate and theoretically. selection pressure, whereas if variation depends on In this paper, I will analyse the effects of sampling rare alleles with large effects, it is directly pro- drift on polygenic variation which is maintained by a portional to this ratio (Turelli, 1984). In general, the balance between mutation and stabilizing selection. A evolution of the mean and variance of a character character is determined by the sum of effects of a large (which are usually all that can be measured) depends number of genes, each of which segregates for two on the higher moments of the underlying distribution alleles. Stabilizing selection tends to reduce variation, of allelic effects: only by making strong and unrealistic and is strong enough that each gene is close to assumptions can a closed set of equations be obtained fixation. This model was first proposed by Wright (Barton & Turelli, 1987). Even when all the loci are (1935); it has since been elaborated by Latter (1960), close to fixation, as in the case considered here, the Bulmer (1972, 1980), Kimura (1981), Barton (1986), outcome may still be unpredictable. Barton (1986) Hastings (1987), and Burger el al. (1988). Despite its showed that many stable equilibria can coexist: simplicity, it is relevant to each of the issues raised the genetic variance can range from that given by the above. 'rare allele' approximation up to that given by Since most quantitative characters must be subject the Gaussian approximation. So, the phenotypic state to stabilizing selection, at least in the long run, some depends on how populations move between different other force must maintain variability. The simplest equilibria. Downloaded from https://www.cambridge.org/core. IP address: 170.106.33.14, on 25 Sep 2021 at 11:07:38, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0016672300028378 N. Barton 60 Since many different combinations of alleles may be is assumed to be completely heritable: environmental able to satisfy the constraints imposed by selection, variation would merely weaken selection, and would many distinct stable equilibria will generally be have no qualitative effect. Individual fitness follows a available. Wright (1932; Provine, 1986) illustrated this Gaussian curve centred on some optimum z0, and point using the 'adaptive landscape': a graph of mean with variance \/s. fitness plotted against the possible states of the Provided that selection is weak, the population can population. In all but the simplest cases, this surface be considered to change approximately continuously will contain many local 'adaptive peaks'. The central in time. If selection is also much slower than problem for Wright was how populations can move recombination, linkage disequilibrium can be towards superior peaks, despite their tendency to be neglected (Bulmer, 1980; see Turelli & Barton, 1989, trapped by local irregularities in the surface. He for a general justification). The effects of selection are proposed that sampling drift, or perhaps random then given by the general relation: fluctuations in selection pressures, could knock dp, populations between peaks, and that, for a variety of (1) reasons, populations would tend to accumulate at the dt dp, higher peaks. Here, I consider just one factor which (p,, qt are the frequencies of the + and — alleles at the favours higher peaks: random transitions from low z'th locus). peaks to high tend to be more likely than those in the If the number of loci is large, and selection is weak, opposite direction. In the present model of a quan- then the mean fitness, W, of a population with mean titative character, the question is whether this tendency z and variance v is approximately: is efficient enough that the population will usually be 2 s(z-z ) in the equilibrium with lowest variance, and the log(»f) = - 0 sv (2) highest mean fitness. If this is so, then the simple 'rare "2 alleles' approximation can safely be used to predict where the evolution of the mean and variance. 2 z = Ilac(pi-qi), v = '£l2a. piqi Shifts between alternative equilibria are important I i even if they have no direct phenotypic or adaptive Including mutation at an equal rate /i in each direction: consequences. Even when stabilizing selection is strong enough that the phenotype is confined within narrow limits, shifts between different combinations of alleles (3) may still be possible. This process is of interest from two points of view. First, the rate and pattern of Here, 8 = (z — zo)/x is the deviation from the optimum substitution at individual loci will be affected by relative to the effect of a substitution at a single locus. The equilibria of eqn (3) satisfy a cubic; provided selection (Kimura, 1981, Hastings, 1987, Foley, 1987): 2 is the observed level of selection on the whole organism that S and /i are not too large (\S\ < f, /i < sa /8; consistent with effectively neutral evolution at the Barton, 1986), it has three solutions, denoted by molecular level? Second, two populations may [p,p, F\. The equilibria which can be reached by the eventually come to have very different combinations whole system can be described by the numbers of of alleles, leading to substantial reproductive barriers loci which are at each of these three frequencies, between them. How rapid is this process, relative to [m, v, M] (m+u + M = n). Each combination [m, v,M] other mechanisms of speciation? represents a class of equilibria with the same overall Progress on these questions depends on techniques properties, but includes many different permutations for analysing stochastic systems with many degrees of across loci. freedom. Though such techniques are well developed The deterministic equilibria can be calculated by in other fields, their application to genetical problems solving eqn (3) numerically (Table 1). The nature of requires some innovations. The simple model analysed the stable equilibria depends on the ratio between mutation and selection at each locus. When this is here is interesting in its own right. However, the main 2 2 aim is to develop methods which can deal with extremely small (/i < sa /(n + I) ), mutation is neg- evolution in many dimensions. ligible: one locus can be kept polymorphic by (effectively) heterozygote advantage, whilst the re- mainder contribute negligible variance. At the other 2. Deterministic behaviour extreme, when /i > s<x2/S, selection is negligible, and The deterministic behaviour of the model is de- all loci are kept polymorphic by recurrent mutation. scribed by Barton (1986); this section summarizes the In the intermediate range, sa2/(n + 1 )2 < /t < sa.2/S, necessary results. Consider a single character, z, which all loci are close to fixation for one or other allele (i.e., is determined by the sum of effects of n biallelic loci. all stable equilibria are of the class [m, 0, M]). Individuals are diploid. Substitutions at all loci have However, several different classes of stable equilibria an identical effect a on the character; its value must may exist: approximately 1 +(« —2) y//i/sa2. Though therefore range between — net and +na. The character the phenotypic means of these alternatives are never Downloaded from https://www.cambridge.org/core. IP address: 170.106.33.14, on 25 Sep 2021 at 11:07:38, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms.