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Inference from Clines Stabilized by Frequency-Dependent Selection

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Inference from Clines Stabilized by Frequency-Dependent Selection Copyright 0 1989 by the Genetics Societyof America Inference From Clines Stabilized by Frequency-Dependent Selection James Mallet* and Nick Barton? *Department of Entomology, Mississippi State University, Mississippi State, Mississippi 39762, and ?Galton Laboratory, Department of Genetics and Biometry, University College, London hW12HE, England Manuscript received November 7, 1988 Accepted for publication April 28, 1989 ABSTRACT Frequency-dependent selection against rare forms can maintain clines. For weak selection, s, in simple linear models offre uenc -dependence, single locus clines are stabilizedwith a maximum slope of between &/& u and ?/s/ 12 6, where u is the dispersal distance. These clinesare similar to those maintained by heterozygote disadvantage.Using computer simulations, the weak-selection analytical results are extended to higher selection pressures with up to three unlinked genes. Graphs are used to display the effect of selection, migration, dominance, and number of loci on cline widths, speeds of cline movements, two-way gametic correlations(“linkage disequilibria”), and heterozygote deficits. The effects of changing theorder of reproduction, migration, and selection,are also briefly explored. Epistasis can also maintain tension zones. We show that epistatic selection is similar in its effects to frequency-dependent selection, except thatthe disequilibria produced in the zone will be higher for a given level of selection. If selection consists of a mixture of frequency-dependence and epistasis,as is likely in nature, the error made in estimating selectionis usually less than twofold. From the graphs, selection and migration can be estimated using knowledge of the dominance and number of genes, of gene frequences andof gametic correlations froma hybrid zone. ENE frequency clines can bestabilized by various ing balance can be divided into four phases. (1) Gene G forms of natural selection. Theoreticians have frequencies drift against the force of selection in a mainly investigated cases where geographic variation local group of demes [or, in a continuous model, a in selection pressures (HALDANE1948; FISHER1950; local group of “neighborhoods” (WRIGHT 1969)J SLATKIN1973, 1975; MAY,ENDLER and MCMURTRIE linked by migration. (2) Once a critical, unstable equi- 1975; ENDLER1977) or heterozygotedisadvantage librium has been passed because of drift, natural se- (BAZYKIN1969; BARTON1979) maintain clines. Clines lection continues the process of pushing these local can also be stabilized purely by epistasis (BAZYKIN demes (or neighborhoods) towards a new stable equi- 1973), and by frequency-dependent selection if the librium. (3)Provided a critical number of demes (or rare form is at a disadvantage (MALLET1986a). Het- neighborhoods) have reached the new equilibrium, erozygote disadvantage, epistasis, and frequency-de- this equilibrium will be protected from the old form pendent selection can all maintain“tension zones” by a set of clines which constitute a stable, though (HEWITT1988). Tension zones differ from ecologi- mobile, tension zone around the new forms. (4) Such cally determined clines in that they are not rooted to zones can move out to spread the new equilibrium or environmentalgradients, but tend to move until collapse inwards to cause the loss of the new form, trapped by dispersal barriers or areas of low popula- depending on the fitness of the new form, as well as tion density (BAZYKIN1969; BARTON1979). on variation in population density and dispersal (BAR- Tension zones connectgroups of populations at TON 1979; ROUHANIand BARTON1987). This fourth differentstable genetic equilibria (often adaptive process of selection across a tension zone has the same peaks); thus they form an important component of effect as Wright’s “interdemic selection.” In a demic WRIGHT’S(1932) “shifting balance” model forthe model with low migration between demes, the entire evolution of new stable equilibria. Of course, natural tension zone is essentially between a pair of adjacent tension zones can form by secondary contact between demes (SLATKIN1981 ; WALSH 1982; LANDE 1985). already-diverged populations;but if divergence by the Innature, more or less continuous tension zones shifting balance does occur in parapatry, a tension spanning many demes are common (BARTONand zone will be present between the nascent equilibrium HEWITT 1985; HEWITT 1988);the shifting balance and the equilibrium from which it evolved. The shift- under these conditionshas been modeled by ROUHANI and BARTON(1987). The potential for rapid, deter- The publication costs of this article were partly defrayed by the payment ministic cline movement to preserve and spread new of page charges. This article must thereforebe hereby marked “aduertisment” in accordance with 18 U.S.C. $1734 solely to indicate this fact. equilibria has not generally been appreciated. This Genetics 122: 967-976 (August, 1989) 968 J. Mallet and N. Barton kind of selection across continuous tension zones, as (1986) for examples of the method applied to other well as interdemic selection, makes the shifting bal- clines]. ance much more likely than is often assumed, in the same way that individual selection allows the spread CLINES AT ASINGLE LOCUS UNDER WEAK of rare advantageous mutations within a population SELECTION (WRIGHT1980). The study of tension zones is there- When selection is weak, gene flow may be approxi- fore important for understanding genetic divergence, mated by diffusion at a rate u2, so that over a linear includingchromosomal evolution (BAZYKIN 1959) dimension x, the gene frequencyp of a locus A changes and some kinds of speciation (WRIGHT 1982;BARTON with time t as follows (HALDANE1948; NAGYLAKI and CHARLESWORTH 1984). 1975): Warning and mimetic colors of insects often differ greatly between regions. Stable clines (tension zones) in warning pattern can bemaintained by density- dependent predator attacks: when a form is rare, it is where u2 is the variance in parent-offspring distance selected against because predators do notrecognize it along some axis, and S (p)is the effect of selection. A as being unpalatable. This predator behavior gives simple case of a diploid frequency-dependent S (p)has rise to frequency-dependent selection which causes intermediate heterozygote fitness: Mullerian mimicry betweenunpalatable species, as well as lack of color polymorphism within unpalatable species (MALLET and SINGER1987; ENDLER1988). There have been no previous theoretical analyses of clines maintained by frequency-dependent selection, although such theory would be useful for analyzing many hybrid zones between mimetic and warningly w,,= (w,, + w,n)/2 = 1 - s I colored insect taxa: there arethousands of such hybrid where W is fitness, f is the frequency of genotypes, zones known in neotropicalbutterflies alone (e.g., and 0 5 s 5 0.5 is a constant determiningthe strength BROWN, SHEPPARDand TURNER1974; BROWN1979); of frequency-dependence. The selection coefficient of similar clines are found in bumblebees (OWEN and 0 5 2s 5 1 was chosen to make the similarity between PLOWRICHT1980; P. H. WILLIAMSpersonal commu- frequency-dependent selection and heterozygous dis- nication) and soldier beetles (MCLAIN1988) among advantage self-evident, as explained below. At low many other insects. Of course, other mechanisms of selection pressures, mean fitness z 1 and these fit- selection could also be involved: however, the pres- nesses lead to: ence of tight Mullerian mimicry between many of the species is good evidence that frequency-dependence is operating within these species. Frequency-depend- This is identical to S(p) for a heterozygous disadvan- ent selection is likely to stabilize a variety of other tage of s (BAZYKIN 1969).Thus the homozygote fit- clines: for example fronts between strains of bacteria nesses in the frequency-dependent model play little with and without the ability to produceallelochemicals role in the dynamics under the diffusion approxima- (LEVIN 1988),and clines between races having assor- tion, which applies when selection is weak. The differ- tative mating (MOORE 1979) or sexual selection pro- ences between heterozygous disadvantage andfre- vided that the rare form is at a disadvantage (LANDE quency-dependent selection enter only through the s. 1982; MOORE 1987; JOHNSON, MURRAYand CLARKE mean fitness, which will be close to 1for small 1987). There is also a close relationship, demonstrated Equation 1 can then be integrated to give, at equilib- below, between clines maintained by frequency-de- rium: pendent selection and thosemaintained, like many pz-1 { + tanh [" - 20) ""1 j chromosomal clines, by heterozygous disadvantage.2a 2 We here analyze tractable casesof clines at one locus with weak frequency-dependent selection, and where X" is the point at which p = 0.5 (BAZYKIN1969; use computer simulations to investigate more complex BARTON1979). Figure 1Ashows this integral. The cases of strong selection on one, two and three loci. maximum gradientfor this frequency-dependent The immediate purpose of this work is to provide model (dp/dx z &/au) is of course again the same models for analyzing hybrid zones in Heliconius (Lep- as that for heterozygous disadvantage. Heterozygous idoptera): these models enable one to infer bothselec- disadvantage can bethought of as leading to fre- tion and gene flow without recourse to field experi- quency-dependent selection at the haploid level: the ments [see BARTON(1982) and SZYMURA BARTONand rarer gene encounters the opposite allele
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