Inference from Clines Stabilized by Frequency-Dependent Selection

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 asbeing 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 atone 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 more fre- Frequency-Dependent Selection Frequency-Dependent 969 where the individual selection coefficients, 2s, inwarn- ing color dependon the memorability of that pattern's genotype. The three selection coefficients represent the actual reduction in fitness due to rarity for each genotype. If sAa > sAA and sAa > sau, heterozygotes have a disadvantage-this may occur with warning color if the hybrid phenotype is less memorable. In warning color, if sAA < sa, the A pattern is more memorable 0.24 / than a, and interdemic selection between two color patterns might be expectedto move clines so as to fix A over thewhole range [see BARTON(1 979) forsimilar ~I'I'.~I'I.I~I.I'~'~'I' -10 -5 0 5 10 movement of clines with heterozygous disadvantage Distance and selection against one homozygote]. The simplest FIGURE1 .--Single locus clines at equilibrium under weak selec- case is to assume thatthere are no differences in tion. Curve A, a codominant locus, Equation 3. Equation 6 is similar: memorability of the phenotypes (sAA = sAa = sa, = s); selection is approximately as effective in the center, but only 1/2 as then effective at the edge: the cline will be as steep. Curve B, a dominant locus, equation (8). Distance is measured in standardized s(p)= 2spq[(1 - h)p(2p2 - 1) + hq(i - 2q2)]. (5) units of 0/4Js. If the locus A is codominant, h = 1/2, and quently, and so has a lower averagefitness. The model in (2) is unrealistic because the fitness Q) = SPdP' + Q2)@ - 9). (6) of heterozygotes is expected to depend on the fre- Again S(p) approximates that for heterozygous dis- quencies of similar and different phenotypes, rather advantage,differing only by afactor of (p' + q2), than being a simple average of the homozygote fit- which always lies between 1 (at the edge of the cline) nesses. The possibility of dominance also needs to be and 1/2 (in thecenter). The cline shape therefore incorporated. With warning color, heterozygotes can differs little from that in (3) (Figure lA), except that beremembered as aseparate phenotype, although the maximum gradient obtained by integration (dp/ "generalization" between heterozygotes and homozy- dx &/ma) is dmas great.This is confirmed gotes (the tendency for predators to confuse similar in the simulations below for low s. In contrast, domi- phenotypes) is likely. Predator generalization between nance (h = 1) gives: all three genotypes may also occur, but will not cause differentialselection, and so is ignored. We use a linear frequency-dependent model for three reasons: A cline set up with this selection regime will move a linear model can be viewed as a regression through because of the interdemic selective asymmetry due to the actual frequency-dependent function; the actual dominance (MALLET 1986a). It is hardto find the nonlinear selection function is not known; and linear shape of a moving cline because the velocity cannot, and nonlinear functions produce similar clines (MAL- in general,be found analytically. We can find the LET 1986a). Selection is assumed not to affect popu- approximate cline shape by supposing that there is a lation density, so that only genotype frequencies need selective asymmetry which just suffices to prevent to be modeled.This is likely for a butterflywith strong movement. In nature also, movement may often be density-dependent factors controlling population size prevented by a dispersal or selection gradient which in the larval stage, as is apparently true in Heliconius will produce similar effects. The necessary asymmetry (GILBERTand SMILEY1978; MALLET 1984). In the is given by: simplest case, selection reducesthe frequency of a genotype in proportion to its phenotypic rarity (1 - S(p) = 2spq2 (4/5 - 2~'). (8) j), j), where f is the frequency of that genotype and of similar phenotypes. For example, W,.,,4 = 1 - 2s[l - Setting dp/dt = 0 in (1) gives for this model: fAA - h ~AQ]= 1 - 2SEQ + (1 - h)fAQ], where the 9 dominance of A (equivalent to predatorgeneralization x' - x'o = d(iqq 4 between Aa and AA) is given by 0 5 h 5 1. The j- 0.5 dl - q)J[d4 + 5911 complete model is then: where distance is now measured in standardized units of x' = a/2 Js. This can be integrated numerically to give the cline in Figure 1B. The cline is asymmetrical: on the side where dominants are common, most recessive alleles will be found in heterozygotes, and so will escape selection, giving a long tail; on the recessive 970 J. Mallet and N. Barton side of the cline on the other hand, selection weeds loci: see MALLET (1986a,1989) and MALLET and out the rare dominant alleles even when in heterozy- BARTON(1989)l. Therefore we ran simulations with gotes, giving an abrupt tail. HALDANE (1 948)showed selection pressures, 2s,of 0.05,0.1, 0.2, 0.3, 0.5, that similar asymmetries are expected with dominance 0.75, 0.9 and 0.99. Migration levels, u2,used for each in clines responding to environmental discontinuities. selection pressure were 10, 10, 15, 15, 30, 30, 30 and Because thereare only two phenotypes, we might 30, respectively. The fitness of a genotype is assumed expect the gradient in the center of the cline to be to be the product of the separate fitnesses at each similar to that of (3): this is confirmed in the simula- locus. Such multiplicative fitnesses can be justified if tions. selection occursindependently on each locus: the probability of survival is then equal to a product of CLINESWITH ARBITRARILY STRONG the separate probabilities of escape due toeach locus. SELECTION; 1-3 LOCI Additive fitnesses could have been used, but there is Methods: In the above results S(p) was calculated very little difference between these modes with weak assuming that =: 1,and that genotypicfrequencies selection. are in Hardy-Weinbergequilibrium. Neither holds In reality, selection will of course be much more under strong selection. Analytic results also become complex. Consider warning color: a numberof param- difficult with morethan one locus. We therefore eters will characterize phenotypic selection, the fre- performed computer simulations of clines on a linear quency-dependence will be nonlinear, and epistasis, array of demes to investigate how selection affects the which is modeled below for some special cases, will maximum gradient, speedof movement, heterozygote produce strong interactions between the fitnesses at deficiency within demes (F), andlinkage disequilibria individual loci (e.g., MALLET 1989). Inference from these simplified models will therefore give an “effec- (R = D/J[p~q,p~q~],where D = fAB - PAP, and fAB is the frequency of AB gametes). tive” selection pressure that is dependent on this ideal- The simulations employ the exact linear frequency- ized model situation. dependent selection given by (4), with either complete For the most part, we assume that genotypes or dominance (h = 1, also used by MALLET[ 1986a1) or phenotypes have equalfrequency-dependence (in codominance (h = 112). We modeled a one-dimen- warning color, the phenotypes would then be equally sional cline in a series of demes connected by migra- memorable) so that a single per locus selection pres- tion. In MALLET (1 986a), “stepping-stone” (KIMURA sure, 2s, characterizes all phenotypes within and be- and WEISS 1964) migration between adjacent demes tween different loci. We repeated the simulations for was used. However, this causes inaccuracies when s/u one-locus clines, and for hybrid zones involving two = 1 orgreater because the clines become grossly or threeloci, with various combinations of dominance discontinuous. Here we use binomial migration,a for each locus (h = 1/2 or 1). The loci are assumed to generalization of the stepping-stone model, with the be unlinked throughout. Measurement of p, F, R, and “number of trials” = 2n demes. The proportion arriv- thirdorder D [as defined by SLATKIN(1972) and ing in a deme i demes away from the source is: BARTON(1 982)] took place immediately before reproduction in each generation. The order of processes (‘n)! a(n-t) (1 - was normally assumed to be: reproduction (consisting (n - i)! (n + i)! of mating, offspring production,and growth to adult- For Heliconius this seems For symmetric migration as used here, LY = 1 - LY = hood)-migration-selection. 112, and the variance of parent-offspring distances, approximately valid: adults eclose, immediately dis- u2,in simply n. This migration is approximately Gaus- perse, andthen set uphome ranges (MALLET sian for large n. Beyond each edge of the deme array 1986b,c), andmany vertebrate species will be similar. there were assumed to be n imaginary demes with With strong selection the order of processes matters genotypic frequencies identical to those of the edge [see also MANI (1985)l so we have also investigated deme. These imaginary demes acted as sources for another possibility: reproduction-selection-migra- the immigrants that arrived within the deme array. tion. For afew special caseswe have also demonstrated We checked for approximate continuity by doubling the effect of removing disequilibria at meiosis, and of n: if the resulting cline widths did not scale linearly combining epistasis with frequency-dependent selec- with u, we used a still higher level of migration. Edge tion. effects were eliminated similarly; if the cline shape Epistasis,which is likelyin many naturalhybrid was affected by changing the total number of demes, zones (e.g., in Heliconius, see MALLET[ 19891) could the demes were increased until no further effect was be modelled in a wide variety of ways, but the main noticed. feature needed here is an interaction between genes Strong selection on warning color genes is likely in such that the fitness of pure phenotypes is greater. the field [each gene may consist of many tightly linked For simplicity, we suppose that the two pure warning Frequency-Dependent Selection 97 1

1’u0.5 1.Oh0.5 5 0.3 E 0.2 -30 -20 -10 0 10 20 30 0.1 0.0 0.0 0.5 2 0.0 0.5 1.0 0.0 0.5 1 .O E1 I 0.0 0.0 0.5 0.0 1.0 FIGURE2.-Maximum slope (standardized by multipl ing by a) in simulated tension zones plotted against selection, /-(2s). (A) 1 locus; (B) 2 loci; (C, D) 3 loci; (A)-(C) all have the standard order reproduction-migration-selection; (D) has an alternative order of reproduction-selection-migration. The lines represent all dominant (open squares), and all codominant (filled squares) clines. In -20 .10 0 10 30 (C)the mean slope (dashed line) of clines in a zone with 1 codominant locus and 2 dominant loci is also plotted. The low s approxi- D (ABC) mations for dominant genes (J[s/S], upper dotted line), and codominant genes (J[s/12], lower dotted line) are also shownin each figure. U $ -0.01 color patterns are equally fit, but that any hybridization with theother pattern reduces fitness by an 4.02 -30 -20 -10 0 30 amount e for each whole phenotypic element changed, - and e/2 for each heterozygous elementat codominant loci. Suppose a mixture of one codominant locus, A, n I and two completely dominant loci, B and C interact in a tension zone (as occurs in Heliconius). Then AA B- C- and aa bb cc are both pure patterns having maximum fitness 1; AA B- cc, AA bb C-, AA bb cc, aa B- C-, aa B- cc and aa bb C- are one stepaway from pure, with fitness 1 - e; Aa B- C- and Aa bb cc are Demes half a step away, and have fitness 1 - e/2; Aa B- cc FIGURE3.-Simulated three-locus tension zone consisting of one codominant locus (A) and two completely dominant loci (B, C). and AA bb C- are 1% steps away, and have fitness (1 - Such a hybrid zone occurs in Heliconius (MALLETand BARTON e)(l - e/2). Notethat a result of this conceptually 1989). The clines are stabilized by frequency-dependent selection, simple model of “epistasis” is an automatic heterozy- 2s = 0.5 per locus, and migration, u2 = 15 demes per generation. gote disadvantage of e/2 for codominant loci: their The figure shows variation along a demic continuum of all the heterozygotes are always deviants from the pure pat- parameters needed to reconstruct the genotypic distributions. Only the central section of the simulation is shown out of a total of 150 tern. Simulations have been run for a few cases using demes modeled. The simulation has been running for 250 genera- various mixtures of this epistatic model and the fre- tions since contact between fixed populations, has stabilized in shape quency-dependent model, with the two types of fit- and is moving to the left ata constant speed of -0.20 demes nesses combined multiplicatively. (-0.051 dispersal distances, u) per generation. At lower selection Cline shape: We have seen that the maximum gra- pressures such a zone would separate into a stationary A cline and a pair of rapidly moving clines, B and C. Top to bottom: gene dient of a single locus cline under weak selection is frequencies; two-way disequilibria, R,; three-way disequilibrium proportional to d&. When selection is increased, the DAB(;(defined as in SLATKIN1972); and heterozygote deficits, F,,,. diffusion approximation breaks down and the cline steepens (Figure 2A) because second order terms in s This is partly due to second order terms in the mul- become important. As expected from the arguments tiplicative fitnesses; in addition, gametic correlations above,dominance steepens the clines because the (“linkagedisequilibria”) are generatedboth by the diversity of phenotypes decreases. selection model, and by migration (see below). These An example of a simulated tension zone involving disequilibria tendto steepen the cline still further three loci is shown in Figure 3. The shape of each because selection on one locus will act on associated cline in a multilocus hybrid zone is also affected by loci (SLATKIN1975; BARTON1983). This explanation the number of loci involved. As before, themaximum can be tested by artificially removing the disequilibria slope increases linearly with at low selection at meiosis in each simulated generation. The slopes of pressures (Figure 2, B and C). At high selection pres- such clines are considerably reduced compared with sures, the approximation breaks down, and the slope those for the same loci in tension zones where dise- increases even more steeply than for a single locus. quilibria are allowed to accumulate naturally, though 972 J. Mallet and N. Barton

TABLE 1 Effect of removing disequilibriaat meiosis

u X Max. (dpldx) Speedlo Max. R Max. F Min. F Three loci, normal: A 0.550 0.053 ABIAC 0.604 A 0.315 A 0.000 B,C 0.597 BC 0.620 B,C 0.394 B,C -0.079 Mean 0.591 Mean 0.609 Mean 0.368 Three loci, disequilibrium removed: A 0.410 0.059 ABIAC 0.278 A 0.190 A 0.000 B,C 0.488 BC 0.325 B,C 0.345 B,C -0.085 Mean 0.462 Mean 0.293 Mean 0.294 One locus, codominant: 0.260 0.000 0.075 0.000 One locus, dominant: 0.337 0.083 0.283 0.083 0.337 -0.103

Results of simulations of three- and one-locus tension zones with2s = 0.75 and u’ = 30, using the standard order of processes: reproduction-migration-selection. In the three-locus results, A is codominant, B and C are completely dominant, as in Figure 3. they still remain steeper than in the equivalent single 0.1 5 locus clines (Table 1); the remainder of the increase in slope is due to the non-additive epistasis inherent in multiplicative fitnesses and to the disequilibria pro- 0.10 ‘0 duced by migration within each generation. In the Q) center of multilocus hybrid zones, disequilibria cause n0) loci to be selected as though they were a linked block, 0.05 leading to a sharp stepin gene frequency;at theedges, each locus is selected on its own, giving long tails of introgression on either side of the central step. This 0.00 will also be true at low s, provided there are many 0.0 0.5 1.o loci. Here, the effect is only appreciable at high s because the small number of unlinked loci are other- FIGURE4.-Cline movement due to dominance. Curves show wise “weakly coupled” (see BARTON1983). movement speeds of tension zones (measured in units of dispersal As in Figure 1, dominant clines are asymmetric with distances, u, per generation) plotted against selection pressure, long tails of recessive alleles on the side of the cline m).Open symbols and solidlines: circles, 1 dominant locus; where dominants are most common (Figure 3). Due diamonds, 2 dominant loci; squares, 3 dominant loci; triangles and dashed lines, 1 codominant and 2 dominant loci. All are for the to disequilibria with dominant loci, codominant clines standard order (reproduction-migration-selection) except for the in mixed tension zones also become slightly asymmet- upper curve (filled squares) which is for 3 dominant ge- with the ric (Figure 3). Alone, they are completely symmetrical order reproduction-selection-migration. Cline movement due to (Figure 1). advantage of one of the homozygotes also shows an approximately Cline movement: Clines maintained by heterozy- linear dependence on o& (BARTON1979). gous disadvantage tend to move with constant speed between 0.4 and 0.5. We do not analyse here cases of if one homozygote is fitter than another (BAZYKIN a selective difference between homozygotes [fora 1969; BARTON1979); this applies also to the similar treatment applicable to codominant frequency-de- clines with frequency-dependent selection on codom- pendence, see BARTON(1979)]; such movement like inant loci. In contrast, dominant clines move even in that due to dominance, is at a speed approximately the absence of differential selection of phenotypes proportional to u & (Figure 4). (Figures 3 and 4), because dominance introduces a Gameticcorrelations: Gene flow across a set of selective asymmetry onthe genotypes (MALLET selected clines will produce correlations both within 1986a).This cline movement due to dominance is loci-heterozygote deficiencies, F-and between loci- little affected by the numbers of dominant loci; how- linkage, or gametic, disequilibria, measured by the ever, if one of the loci is codominant, the movement correlation coefficient R (BARTON1982, 1983; Szy- is considerably slowed. Clines differing in dominance MURA and BARTON 1986). If gene flowis kept tend to move apart below some critical s; at higher s, constant and selection is increased, the cline steepens, disequilibria and epistasis cause the clines to be at- and migration moves genotypes across a sharper gra- tracted together (Figures 3 and 4) (SLATKIN1975). dient, causing an increase in disequilibrium. In con- For instance, where one codominant and two domi- trast, increasing gene flowwhile keeping selection nant loci interact in a tension zone, the critical 2s is constant rescales the cline width without altering R. Frequency-Dependent Selection 973

1 I 1.o 1 .o 1 .o

0.5 -5 i; = 0.5 I 0.5 0.5 a U m (A B

0.0 0.0 0.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 0.0 0.5 1.0 2s 2s FIGURE6.-Heterozygote deficit. The maximum heterozygote FIGURE5.-Disequilibria. The maximum gametic correlation R deficit F is plotted against 2s. Dominant loci, opensymbols; codom- is plotted against the per locus selection pressure 2s. Lines represent inant loci, closed symbols. Left, 1 locus; center, 2 loci; right, 3 loci; values from simulations; all loci dominant (open squares), all loci dashed line indicates 1 codominant and 2 dominant loci. The codominant (closed squares), mean disequilibria when 1 locus is standard order (reproduction-migration-selection) is plotted in codominant, and 2 others are completelydominant (dashed lines in square symbols; the orderreproduction-selection-migration is plot- center graph). Dottedlines represent predictions based on Equation ted in circles. 9 and theactual dp/dx produced in the simulations (Figure 2). Left, 2 loci; center, 3 loci, using the standard order (reproduction- zero to positive values, switch to negative values, and migration-selection); right, 3 loci using the order reproduction- then back to zero as we move across the clines (Figure selection-migration. 3), as expected (BARTON1982). Thus the strength of naturalselection in a hybridzone Positive correlations between alleles within a locus can be measured simply by comparing the disequili- are also expected because of gene flow. However, brium in the center of the zone with that produced in since Hardy-Weinberg equilibrium (Fls = 0) results models of a particular selective process. after one round of random mating, within-locus cor- With a number of loci under weak selection and relations, F,,,, are not expected to equilibrate at as with no epistasis, the disequilibrium produced will be: high a level in the center of a cline as between-locus correlations, R; simulations confirm this (Figure 6). Selection itself may cause deviations from Hardy- Weinberg. This is especially true for dominant loci. As we have already seen, a longtail of recessive alleles (BARTON1982, 1983) whereTAB is the recombination is produced in the edge the cline where the reces- rate between loci A and B. With low s, the maximum of gradient is - k m,and so disequilibriumshould sives are rare. Here, recessive homozygotes will be peak in the center of two identical clines, where: produced from a large reservoirof heterozygotes, but they will bestrongly selected against,producing a negative FlS. By a similar argument, we expect selection to produce a positive Fly in the other tail, where dominant phenotypes are rare, adding to that pro- Simulations show that disequilibria do peak in the duced by gene flow. The effect of selection on F,,s is centers of clines (Figure 3), and dependalmost linearly thus expected to be much lower on codominant loci on selection pressure (Figure 5). When some loci are than on dominant loci. In the simulations, F,, is about codominant and others arecompletely dominant, the three times as large in dominant clines than it is in average disequilibria are intermediatebetween the case where all loci are dominant and the case where codominant clines (Figure 3 and 6). In comparison, K all are codominant (Figure 5), as might be expected. (Figure 5) differs less than twofold between simula- The frequency-dependent selection used here causes tions with codominant and dominant loci, about as a higher constant of proportionality than expected expected on the basis of differences in slope alone (Figure 2). from Equation 9b (ie., RAB/s> 4k2/rAB). This is for two reasons. Multiplicative fitnesses generate some Epistasis: The assumption of multiplicative fit- nonadditive epistasis, which causes extra disequilibria. nesses implies that selection acts independentlyon Adding more loci then increases the disequilibrium each locus. In nature,color patterns under frequency- further, because the additional selection imposed by dependent selection may often be produced by epi- more disequilibria reduces cline widths. statically interacting loci: for example, in Heliconius Theory predicts that higher order disequilibria will warningpatterns (SHEPPARDet al. 1985; MALLET also be produced in multilocus clines (BARTON1983). 1989), pure patterns arelikely to be more memorable For instance, third order disequilibria change from than hybrid patterns. Atension zone could potentially 974 J. Mallet and N. Barton

TABLE 2

Effect of varying proportionsof epistasis (e) and frequency-depemdence (29)

a X max. dpl 2s e ds Speedlo Max. R Max. F Min. F 0.5 0 0.051A 0.250 ABIAC 0.356 A 0.116 A 0.000 B,C 0.314 BC 0.42 1 B,C 0.226 B,C -0.054 Mean 0.292 Mean 0.378 Mean 0.190 0.5 0.1 0.056A 0.356 ABIAC 0.493 A 0.223 A 0.000 B,C 0.368 BC 0.514 B,C 0.278 B,C -0.058 Mean 0.364 Mean 0.500 Mean 0.256 0.4 0.1 0.4 0.053A 0.278 ABIAC 0.417 A 0.169 A 0.000 B,C 0.292 BC 0.443 B,C 0.221 B,C -0.049 Mean 0.287 Mean 0.425 Mean 0.203 0.25 0.25 0.25 0.057A 0.274 ABIAC 0.475 A 0.226 A 0.000 B,C 0.261 BC 0.491 B,C 0.219 B,C -0.046 Mean 0.266 Mean 0.48 1 Mean 0.221 0 0.5 0.062A 0.253 ABIAC 0.588 A 0.322 A 0.000 B,C 0.226 BC 0.613 B,C 0.241 B,C -0.049 Mean 0.235 Mean 0.597 Mean 0.190

Results of simulations of three-locus tension zones with u2 = 30, using the standard order of processes: reproduction-migration-selection. Locus A is codominant, loci B and C are completely dominant,as in Figure 3. be stabilized by such epistasis alone (BAZYKIN 1973), speed is similar, but does not tail off at high s (Figure and we here model one such case. 4); disequilibria, R, are reduced andbecome closer to Table 2 shows the effect of varying combinations the small s approximation of equation 9 (Figure 5); of epistasic and frequency-dependent selection on a and within-locus correlations, F, are reducedalso (Fig- three-locus tension zone. If the clines are maintained ure 6). Many of these effects result because selection purely by epistasis (2s = 0, e = 0.5), the shape and acts on genotypes in Hardy-Weinbergequilibrium, speed are little different from the pure frequency- rather than ongenotypes that have just been disturbed dependent model (2s = 0.5, e = 0). The heterozygous by migration. Additional effects are due to thepartial deficit F is increased for codominant loci because the reduction of disequilibrium by reproduction before simple phenotype model of epistasis used here implies selection. Because populations are nearer to linkage heterozygous disadvantage at codominant loci; domi- equilibrium, we expect that clines will not be attracted nant loci have similar F across models. Because epis- together as strongly. This prediction is borne out: for tasis favors particular gene combinations,disequilibria the model of Figure 3 and Table 2 under the order are strongly affected; purely epistatic selection causes reproduction-selection-migration,codominant and correlations, R, in the center of the zone to increase dominant clines always separate for 2s 5 1, compared by about 60% over those produced in a purely phe- with clines under the standard order, in which the notypic model. However, if only a small proportion critical 2s = 0.5. of the selection is due to epistasis (for example 2s = 0.5, e = 0.1, compared with 2s = 0.5, e = 0) the DISCUSSION disequilibria are less strongly affected. Linear frequency-dependent selection on codomi- This analysis is somewhat crude because 2s and e nant genes produces, in the low selection limit, recur- are not really equivalent. However, Table 2 shows in sions that have the same form as those for heterozy- a general way that large amounts of epistasis will lead gote and disadvantage. Dominant clines are similar, to an overestimate of selection if disequilibria in a but differ in that asymmetries of selection caused by natural tension zone were assumed to result solely the dominance produce an assymmetric cline. Domi- from frequency-dependent selection. With these res- nant clines will also move, even in the absence of ervations, it seems that the differencesin the estimates phenotypic selective differences: codominant clines do will usually be less than twofold [see also BARTON not move unless there are such selective differences. (1983), p. 4641. Clines can provide estimates of gene flow, u'l, and Order of processes: The orderof processes makes selection pressure, s, which do not require directfield no difference at low s. The order assumed thus far measures (BARTON1982; SZYMURA and BARTON has been reproduction-migration-selection. If s is high 1986). For biologically possible values (2s Il), these and the orderis instead reproduction-selection-migra- estimates will be correctto within afactor of -2, tion, themaximum gradients are reduced andbecome whatever the exact form of selection. However, clines closer tothe low s a~~proximations(Figure 2); the are often found in which only a few genes or chro- Frequency-Dependent Selection Frequency-Dependent 975 mosomes strongly affect phenotypes, and presumably urements of F will be the only way to make inferences fitness [e.g., Heliconius (MALLET 1986a, 1989; MAL- from single locus clines (e.g. MALLET1986a). LET and BARTON 1989), Bombus (OWEN and Correlations between loci, R, equilibrate at a higher PLOWRICHT1980), and birds (GILL1973, 1980)l. To level than F because they are only partly broken down analyze tension zones, we will therefore often need to by recombination in each generation, will thus be extend theweak selection theory, as is attempted here. easier to measure, and can give an efficient means of These extensions are problematical because, whereas estimating the absolute value of s (Figure 5). Higher a number of models collapse into a single model at order disequilibria (Figure 3) could also be used, but lows s, as we increase s the order of processes as well require greater sample sizes for accurate estimation. as other assumptions beginto matter. (Whetherfitness Finally, it should beborne in mind that the selection differences at such stronglyselected loci were built up inferred by this means depends on a one parameter gradually is not at issue here: provided r E 0.01 or model of frequency-dependent selection. Consider less, intragenic recombinants are selected out more warning color: among the actual parameters of im- rapidly than they are produced by recombination in portance are likely to be the epistasis between loci tension zones, so that they can be ignored for the affecting the phenotype, the unpalatability of the apo- purposes of inference). sematic species, relativenumbers of predators and Analytical theory is most useful if theorder is aposematic prey, one or more learning parameters of reproduction-selection-migration. This might be par- the predators, and theavailability of alternative foods. tially true if the organism dispersed throughout its The value of s that is inferred will give a good idea of life, and selection occurred relatively early on. The the strength of selection, but not its exact nature. For abolition of F and reduction of R by mating and reasonably large data sets, the accuracy of these esti- recombination just beforeselection make loci virtually mates will be limited by uncertainties as to the model independentfor low numbers of unlinked loci. If ratherthan by sampling error. Nonetheless, these there aremany loci scattered around the genome the measures are likely to be very useful given the inac- average r would be greatly reduced'and disequilibria curacy of most other information on evolution: at the become important, requiring a differentanalysis (BAR- very least they can confirm the results of field exper- TON 1983). iments. Onthe other hand, in many species aburst of We thank the Natural Environmental Research Council for migration is followed by a relatively sedentary period financial support, and N. SANDERSON,M. SLATKIN andan anony- during which selection is likely to occur. Then migra- mous reviewer for comments on the manuscript. tion mainly precedes selection and the increasing deviations from Hardy-Weinberg under stronger selec- LITERATURE CITED tion cause narrower clines whichin turncreate BARTON,N. H.,1979 The dynamics of hybrid zones. Heredity stronger deviation from Hardy-Weinberg: this causes 43: 341-359. a strongly curvilinear relation between and maxi- BARTON,N. H., 1982 The structure of the hybrid zone in Uro- 4 derma bilobatum. Evolution 36863-866. mum gradient. This effectis magnified still further as BARTON,N. H., 1983 Multilocus clines. Evolution 37: 454-471. the number of loci is increased because of a similar BARTON,N. H.,and B. CHARLESWORTH,1984 Genetic revolu- feedback effecton disequilibria (Figure 2). Under this tions, founder effects, and speciation. Annu. Rev. Ecol. Syst. order of processes simulations will berequired to 15: 133-164. BARTON,N. H., and G. M. HEWITT.1985 Analysis of hybrid estimate and s accurately from the cline data. u2 zones. Annu. Rev. Ecol. Syst. 16: 113-148. To make inferences about selection and gene flow BAZYKIN, A. D., 1969Hypothetical mechanism of speciation. Ev- we needmeasurements of at least two parameters olution 23: 685-687. from tension zones. The simplest feature to measure BAZYKIN,A. D., 1973 Population genetic analysis of disruptive is width, defined l/max. (dpldx). This gives the ratio and stabilizing selection. Part 11. Systems of adjacent populations within a continuous area.Genetika 9: 156-166. u/&(Figure 2). One further measurementwill fix the BROWN,K. S., 1979 Ecologza Geograjcae Evoluccio nasFlorestas absolute values of each parameter. One possibility is Neotropicais. Universidade Estadual de Campinas, Campinas, to use heterozygote disadvantage F. However, F de- Brazil. pendsstrongly on details of the model. F is more BROWN, K. S., P. M. SHEPPARDand J. R. G. TURNER, responsive to selection on dominant genes; unfortu- 1974 Quaternary refugia in tropical America: evidence from race formation in Heliconius butterflies. Proc. Roy. SOC.Lond. nately, heterozygote frequencies requiredfor estimat- B 187: 369-378. ing F will usually be difficult to measure unless a locus ENDLER,J. A.,1977 GeographicVariation, Speciation andClines. is codominant in an alternative biochemical test, for Princeton University Press, Princeton, N.J. example. The F at codominantly selected loci are ENDLER,J. A., 1988 Frequency-dependent selection, crypsis, and much lower, and will behard to detect, evenfor aposematism. Phil. Trans. R. SOC. Lond. B319 459-472. FISHER,R. A., 1950 Gene frequencies in a cline determined by relatively high levels of selection (see Figure 6 and selection and diffusion.Biometrics 6: 353-361. LEWONTINand COCKERHAM1959). However, meas- GILBERT,L. E., and J. T. SMXLEY,1978 Determinants of local 976 J. Mallet and N. Barton

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