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Epistasis Can Facilitate the Evolution of Reproductive Isolation by Peak Shifts: a Two-Locus Two-Allele Model

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Epistasis Can Facilitate the Evolution of Reproductive Isolation by Peak Shifts: a Two-Locus Two-Allele Model Copyright 0 1994 by the Genetics Society of America Epistasis Can Facilitate the Evolution of Reproductive Isolation by Peak Shifts: A Two-Locus Two-Allele Model Andreas Wagner,* Giinter P. Wagner* and Philippe Similiont “Yale University, Department of Biology, Center for Computational Ecology, New Haven, Connecticut 06520-8104 and Yale University, Department for Applied Physics, Becton Center, New Haven, Connecticut 06520-2157 Manuscript received October 13, 1993 Accepted for publication July 8, 1994 ABSTRACT The influence of epistasis on the evolution of reproductive isolation by peak shifts is studied in a two-locus two-allele model of a quantitative genetic character under stabilizing selection. Epistasis is in- troduced by a simple multiplicative term in the function that maps gene effects onto genotypic values. In the model with only additive effects on the trait, the probability of a peak shift and the amount of reproductive isolation are always inverselyrelated, i. e., the higherthe peak shift rate, thelower the amount of reproductive isolation caused by the peak shift. With epistatic characters there is no consistent rela- tionship between these two values. Interestingly, there are cases where both transition rates as well as the amount of reproductive isolation are increased relative to the additive model. This effect has two main causes: a shift in the location of the transition point, and the hybrids between the two alternative optimal genotypes have lower average fitnessin the epistatic case. A review of the empirical literature shows that the fitness relations resulting in higher peak shift rates and more reproductive isolation are qualitatively the same as those observed for genes causing hybrid inferiority. OR sexually reproducing populations, speciation between alternative optimal genotypes aswell as the F consists in the acquisition of genetic differences amount of reproductive isolationbetween those that cause reproductive isolation of one such population genotypes. from other populations. Of all the possible modes of Considerable effort has been devoted to obtaining es- speciation genetic differentiation of isolated popula- timates of the peak-shift rates by modeling specific evo- tions seems to be prevalent in animals [MAYR(1942) ; see lutionary scenarios (BARTONand CHARLESWORTH 1984; also COYNE(1992) for a recentreview]. In this report, we BARTONand ROUHANI1987; LANDE 1985). In principle, consider a scenario in which a small population gets two kinds of models can be distinguished: In “pheno- separated from the main population without undergo- typic” models distinct phenotypes can occupylocal ing a founder event or several crash/flush cycles. We maxima in the fitness landscape. In “genotypic”models, further assume that genetic differentiation is mainly on the other hand,different combinations of genes may caused by random drift, i.e., there is no difference in lead to the same, optimal phenotype. The model studied selection on the main population and the daughter below is an example for the latter kind of models. population. In this case, genetic differences are either Assumptions about modes of interactions between neutral and therefore irrelevant for speciation, or they gene effects are essential for genotypic models. The constitute transitions among adaptively equivalent geno- most elementary of these assumptions is that geneshave types. In the latter case, the population has to cross an linear (additive) effects on the genotypic valueof a adaptive valley (a “peak shift” occurs) (WRIGHT1952). quantitative character. Peak-shift scenarios under the as- Such a transition event will cause a reproductive barrier sumption of purely additive gene effects have been ex- between the source population and its isolate. This is tensively studied (LANDE 1985;FOLEY 1987;BURGER 1988; because not all of the possible hybrid and back cross CHARLESWORTHand ROUHANI1988; BARTON1989a). phenotypes can be adaptively optimal, otherwise the dif- BARTON(1989a), for example, showed that peak shifts ference would be neutral. can occur at an appreciable rate, although the amount Genetic analysis suggests that initial stages of specia- of reproductive isolation obtained perpeak shift is small. tion may be caused by a small number (mostly pairs) of The overall rate of accumulation of reproductive isola- interacting loci (ORR1987, 1989;ORR and COYNE1989). tion has been estimated to be of theorder of the Therefore, peak shifts among alternative genotypes are mutation rate. likely to involve onlyfew genes, if genetic differentiation To what extent the slow pace of evolution predicted is due to drift. We model this process in the framework by additive models carries over to epistatic models is of a two-locus two-allele model. We will ask how the ge- unclear [for areview, see BARTON(1989b) 1. The discus- netic architecture influences both, the rateof transition sion is mostly framed in terms of the contrast between Genetics 138 533-545 (October, 1994) 534 A. Wagner, G. P.and Wagner P. Similion type I architectures, where many genes have approxi- and Y, with two alleles each: X, and X, at locus X and Yo mately additive effects,and type I1 architectures, where and Y, at locus Y. Contributions to G by the underlying there are few genes with strongly epistatic effects and variables which are influenced by locus X are denoted many “modifier” loci. The results are difficult to gen- by the lowercase letters x,, x,, x, for genotypes &Xo, eralize, since there are countless many ways to model XoX,and XIXI, respectively. An analogous notation is epistatic gene interactions. In the present paperwe con- used for contributions yk+, of the genotypes YkYl at centrate on anepistatic model of a quantitative charac- the locus Y. The genotypic values for the underlying ter under stabilizing selection. The relative strength of variables are arbitrarily chosen to be x, = yo = 3, epistasis can be adjusted by changes in one parameter x, = y1 = 2 and x, = y2 = 1. Additionally, we suppose (see below). Thus, aone parameter class of models with that thereexists a smooth function g( x,y) of the under- different degrees of epistasis in the quantitative char- lyingvariables x and y which determines the geno- acter is defined. This class includes a purely additive typic value G, G := g( x, y). In this paper wewill be case. A member of this class which shows a high rate concerned onlywith two special cases ofsuch a function, of evolution of reproductive isolation is analyzed in alinear (additive) one, and a “mildly” non-linear detail and compared to the additivecase. A compari- (epistatic) one. son of this member to data about genescausing hybrid The linear functiong, is given by g, (x, y ) := e, + cIx + inferiority suggests that it is representative of many cg, c,, c, and c, being arbitrary real numbers. To specify genes involved in theemergence of reproductive the coefficients ci in the above functions, we introduce isolation. the following three assumptions. First, since we want to study the transition between alternative genetic adap- THE MODEL tations, it is assumed that two complementary double We consider a diploid randomly mating population homozygous genotypes have the same genotypic value with overlapping generations thatis either dioecious or zero, i. e., g, ( x,, y,) : = 0 and g, (x,, yo) := 0. Further, we monoeciouswith the same pattern of selection acting on assume that the contributions of the loci are symmetrical, both sexes. Heritable fitness differences areassumed to i.e., 9 = I+ These assumptions specify the additive geno- be due to the variation in one quantitative character P. type phenotype map up to a multiplicative constant: No genotype-environment interaction is assumed to be present, such thatthe genotypic value G andthe g,(x, y) := c (x + y - 4). (44 environmental contribution E are independent: Since fitness onlydepends on the product of st the con- P= G+E. (1) stant c can be chosen arbitrarily withoutloss of generality The environmental effect E is supposed to be normally asc= 1. distributed with mean zero and variance one. As a modelof the epistatic genotype phenotype map x, Selection is supposed to bestabilizing with P = 0 as g2( y) we use a modification of GIMELFARB’S(1989) the optimal phenotype. More precisely, we assign to additive/multiplicative model, which one obtains by each phenotypicvalue Pa fitness value mp(P) , defined adding a multiplicative term to the additive model: as g,(x, y) = g,(s y) + 4x9 97 m,(P) = 1 - SPZ, (2) where where s denotes a measure for the strength of stabi- lizing selection. However, since the environmental ef- v(x, y) = c, xy + e,. fect E is assumed to be superimposed onto G, it is necessary to evaluate the mean fitnessm( G)of a geno- Since we want to maintain that the two complemen- type with genotypic value G. By using (2) and the tary double homozygous genotypes have the same ge- above assumptions about the distribution of E, %( G) notypic value zero in order to have the same optimal becomes genotypes in both models, the interaction termu( x, y) has tobe zero for these two genotypes too. This specifies its coefficients up to a multiplicative con- stant: u( x, y) = b( xy - 3). In the resulting model for the epistatic character = 1 - s - sG2. g,(x, y) := c’(x + y - 4) + b (xy - 3) Because the selection dynamics with overlapping gen- erations is invariant to the addition of a constant to the strength of epistasis is determined by the ratio P = the fitness, we can ignore the term s in %( G). b/c’ . It is thus convenient to rewrite g2(x,y) as Furthermore, we assume that thegenotypic value G of g,(x, y) := c’[(x + y - 4) + P (q- 3)]. (4b) the quantitative character under consideration is deter- mined solely by the genetic composition of two loci X As in the linear model, e‘ only enters the fitness as set*, E pistasis and Speciation 535 Speciation and Epistasis (CROWand K~MURA1970).
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