Pleiotropy As a Factor Maintaining Genetic Variation in Quantitative Characters Under Stabilizing Selection

Pleiotropy as a factor maintaining genetic variation in quantitative characters under stabilizing selection

A. GIMELFARB Department of Ecology and Evolution, University of Chicago, Chicago, IL 60637, USA (Received 11 July 1995 and in revised form 16 January 1996)

Summary A model of pleiotropy with TV diallelic loci contributing additively to N quantitative traits and stabilizing selection acting on each of the traits is considered. Every locus has a major contribution to one trait and a minor contribution to the rest of them, while every trait is controlled by one major locus and N—\ minor loci. It is demonstrated that a stable equilibrium with the allelic frequency equal to 0-5 in all A' loci can be maintained in such a model for a wide range of parameters. Such a ' totally polymorphic' equilibrium is maintained for practically any strength of selection and any recombination, if the relative contribution by a minor locus to a trait is less than 20 % of the contribution by a major locus. The dynamic behaviour of the model is shown to be quite complex with a possibility under sufficiently strong selection of multiple stable equilibria and positive linkage disequilibria between loci. It is also suggested that pleiotropy among loci controlling traits experiencing direct selection can be responsible for apparent selection on neutral traits also controlled by these loci.

mutation-selection balance models, it remains un- 1. Introduction certain whether this mechanism can account for the The maintenance of genetic variation under natural amounts of genetic variation in quantitative characters selection in multilocus genetic systems controlling that are observed in natural populations (Barton & quantitative traits is an important and long-standing Turelli, 1989; Keightley & Hill, 1990). problem in population biology. Given that phenotypic In addition to models of natural selection acting variation is practically always reduced by stabilizing directly on the character of interest, models have been selection (Shnoll & Kondrashov, 1993), what bio- proposed in which genetic variation for a given trait is logical mechanisms can counterbalance the effect of a result of polymorphisms in the loci controlling the selection and be responsible for the maintenance of trait, but the forces maintaining the polymorphisms genetic variation in natural populations? For some are not related to the trait itself. The polymorphisms time the field of quantitative genetics was dominated can be maintained due to either overdominance for by polygenic models assuming that a quantitative fitness (Robertson, 1956; Gillespie, 1984; Barton, character is controlled by many loci contributing 1990), or selection against unconditionally deleterious additively and equally to the trait. It has been mutations (Barton, 1990; Keightley & Hill, 1990; demonstrated for such models that, if selection is Kondrashov & Turelli, 1992), or epistatic viability weak relative to recombination, no genetic variation selection (Gavrilets & De Jong, 1993). Thus, the (except for a possible polymorphism in just one locus) character itself is neutral, but differences in fitness can be maintained under stabilizing selection without between individuals with differing values of the trait an input of new variation (Wright, 1935; Robertson, give an appearance of stabilizing selection on the trait. 1956; Lewontin, 1964; Barton, 1986). It is important, however, that, while there is ex- Models have been proposed in which mutations perimental evidence of unconditionally deleterious provide new variation, and genetic variation in a mutations, overdominance for fitness or the particular population is maintained by the balance between forms of epistasis for viability are just postulated in stabilizing selection and mutations (Lande, 1975; the above models, but no known developmental or Turelli, 1984; Barton, 1986; Slatkin, 1987; Burger, hereditary mechanisms have been suggested which 1988; Bulmer, 1989). In spite of an intensive study of would produce these phenomena. On the other hand,

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while the balance between deleterious mutations and multicharacter systems, and have concluded that, if selection can explain the maintenance of polymorp- selection is weak relative to recombination, the number hisms in mutating loci, it cannot explain the evolution of loci maintaining a stable polymorphism cannot be of phenotypes, and, hence, different biological mech- greater than the number of characters they control. A anisms are supposed to be responsible for the stable polymorphism can, however, be maintained in maintenance of genetic variation and for phenotypic more loci than the number of traits they control, if evolution, yet there is no evidence of this. Also, while selection is sufficiently strong (Gavrilets & Hastings, it is conceivable that some biological traits are neutral 1994a). Indeed, a model of several loci contributing with respect to natural selection, there must also be additively to two traits with half the loci contributing traits (e.g. obvious adaptations) experiencing direct, similarly to both traits and the other half contributing and possibly quite strong, natural selection. Models antagonistically was shown to maintain a stable are needed, therefore, that can explain the main- polymorphism in as many as six (Gimelfarb, 1992) tenance of genetic variation under selection acting and eight (Gavrilets & Hastings, 1994a) loci. directly on quantitative traits. Even if as many loci are maintained polymorphic as It has become quite evident by now that genetic the number of traits they control, pleiotropy can still variation can be maintained in a multilocus system be an important factor responsible for the maintenance controlling a quantitative trait, even if it is under of genetic variation in natural populations, since the direct stabilizing selection, if some of the assumptions number of pleiotropically related traits can be po- of the polygenic model are relaxed. This was shown tentially quite large. In this paper, we shall investigate for unequal effects of loci (Gale & Kearsey, 1968; the maintenance of polymorphisms in a model of Kearsey & Gale, 1968; Nagylaki, 1989; Gavrilets & pleiotropy with N loci controlling N quantitative Hastings, 1993, 1994a, b), for selection strong relative traits. The model is inspired by experimental evidence to recombination (Gavrilets & Hastings, 1993, 1994a, that not all loci controlling a trait contribute equally b; Gimelfarb, 1995), for dominance (Lewontin, 1964) to the trait (Edwards et al 1987; Paterson et al. 1990; and for epistasis (Gimelfarb, 1989). Mackay et al. 1992). The majority of quantitative In spite of the scarcity of direct experimental traits whose genetic basis has been investigated appear evidence of pleiotropy among quantitative traits, it is to be ' under control of a few major genes supported supposed to be ubiquitous based on the widespread by numerous genes with smaller effects' (Shrimpton & genetic correlations between different traits (Falconer, Robertson, 1988). 1989), on the complexity and interconnection of biochemical processes underlying the development of 2. The model quantitative traits (Wright, 1968; Barton, 1990) and on the general philosophical consequence of an It is assumed that N diallelic loci contribute additively organism being 'a unity of an infinite number of traits to N quantitative characters. Each locus has a major and finite number of genes' (Serebrovsky, 1973). effect on one of the characters and a minor effect on Recently, evidence of pleiotropy among quantitative the rest of them. On the other hand, each character is traits has been obtained experimentally based on controlled by one major locus and by TV—1 minor spontaneous and P-element induced mutations loci. The contribution to any trait by one of the alleles (Mackay et al. 1992; Santiago et al. 1992; Clark et al. in each locus is zero. The contribution by the other 1995). The effect of pleiotropy on genetic correlations allele depends on whether the locus is major or minor between quantitative traits has been studied quite for the trait. If Lt is a minor locus for a trait Xs{i =t= f), extensively, particularly in mutation-selection balance its contribution is assumed to be a fraction, bip of the models (Lande, 1980; Turelli, 1985; Wagner, 1989; contribution to the trait by the major locus, L}, i.e. Slatkin & Frank, 1990). Relatively little work has major and minor loci contribute to a trait in a been done, however, investigating pleiotropy among proportion \:bir It is also assumed without loss of loci controlling several quantitative traits as a potential generality that the minimum and the maximum of any factor directly responsible for the maintenance of trait are 0 and 1, respectively. Hence, the actual allelic genetic variation in multilocus systems under natural contributions to a trait X} are

selection. Rose (1982) has shown that a stable a} = 0-5/B} (major locus), (1 a) polymorphism can be maintained in two loci having /? = 0-5b /B (minor locus), (1 b) antagonistic pleiotropic effects on components of 0 t] } fitness. Gimelfarb (1986) proposed a model of two loci where Bs = 1 +'Llc^jbkj. Notice that aj + 'Lks¥]/2kJ = 05 contributing additively to two traits with the contri- for any / Consequently, the genotypic value of a butions by one locus being the same to both traits but 'total heterozygote' (all AHoci heterozygous) is 0-5 for with antagonistic contributions to the traits by the any trait. other locus. Such a model maintains a stable poly- Each trait, Xs{j = \,2,...,N), is assumed, inde- morphism in both loci for any recombination and any pendently of other traits, to be under stabilizing strength of stabilizing selection on each character. selection with a quadratic fitness function: Hastings & Horn (1989, 1990) investigated multilocus \-Q£X,-0F, (2)

and the fitnesso f an individual with a set of characters 0 for ir 4= j, a N x N model becomes a trivial model of {X,} is N independent traits each controlled by one (major) locus. Since each trait experiences selection favouring (3) the heterozygote, there is total overdominance for n fitness (the fitness of an individual is an increasing function of the number of heterozygous loci in the Random mating is assumed, and the effect of individual's genotype) in such a model. The over- environment is neglected. dominance will, obviously, be preserved if contri- The number of parameters {btp Q} and 0}) is very butions by minor loci are small. It can be said, large, making it impossible to analyse a general therefore, that aNxN model of pleiotropy with small model. We shall, therefore, consider in detail a contributions by minor loci produces total over- simplified model which assumes symmetry in the dominance for fitness. This is not necessarily true, effects of loci (btj = b for any / =|= j, hence a.} = a, fi(j = however, if the contributions by minor loci are not /?) and symmetry in selection (Q; = Q and 6} = 0). It very small. It follows from Appendix B that in order will be demonstrated, however, that the symmetry for total overdominance for fitness to exist in a assumptions are not necessary, and genetic poly- symmetric model, the relative contribution by a minor morphisms can be maintained in models that are not locus must be below the following limits: 0-268, 0121, symmetric. If 0 = 0-5 (the optimum phenotype is that 0-079, 0-058 and 0025 for N = 2, 3, 4, 5 and 10, of a total heterozygote), the value of Q cannot, respectively. It is seen that for N > 2, these limits are obviously, exceed four in order for fitness of any below the limits required for the stability of a phenotype to be non-negative, and, hence, Q = 4 polymorphic equilibrium. Hence, polymorphisms in represents the strongest possible quadratic selection NxN pleiotropic models can be maintained without on a trait. total overdominance for fitness.

3. Weak selection analysis A complete dynamical analysis is not possible even for 4. General analysis a symmetric NxN model. It is possible, however, to A symmetric 2x2 model under selection with 0 = 0-5 analyse its 'weak selection' approximation, assuming represents a special case of the ' symmetric viability' selection sufficiently weak relative to recombination, model (e.g. Karlin & Feldman, 1970) whose para- so that linkage disequilibrium between loci can be meters are as follows: neglected. An analysis of an TV x N model under such selection is presented in Appendix A. It is demonstrated there that, if selection is weak, a symmetric (5) model with btj = b, 0t = 0, Qj = Q, has an equilibrium with allelic frequency a = 2(0-5-2z)2e-(0-5-2z)4g2,

0-0* \+(N-\)b2 where z = 0-5/(1 +b). Such a model always has a where 0* = (4) 1-2(9*' 4[\+(N-l)b]2 symmetric (allelic frequency 0-5 in both loci) equilibrium. It may also have asymmetric equilibria, but at each locus. It is seen that, if the optimum phenotype, we shall not be concerned with them here. The linkage 6 — 0-5, the equilibrium allelic frequency, p = 0-5. disequilibrium at a symmetric equilibrium is described Generally, in order for 0

7/ •" / (b) /' / / /' i 1 Ji i I if I fi it 1 j / /

0-2 0-4 0-6 0-8 10

0-2 0-4 0-6 0-8 10 0 0-2 0-6 0-8 10 Relative allelic contribution Fig. 1. Areas of existence of stable totally polymorphic equilibria on the plane of selection strength and relative allelic contribution for a given recombination rate, r (stable equilibria exist in areas above a corresponding curve), (a), 2x2; (b), 3x3; (c) 4x4, (d), 5x5. , r = 0-05; , r = 01; , r = 0-3; , r = 05.

equilibrium was classified as stable or unstable depending on whether the perturbed system returned to it or not. Gimelfarb (1989) argued that it is (6) impossible to misclassify an unstable equilibrium as stable by using this method, although it is possible in principle to miss a locally stable equilibrium with a very small domain of attraction. Computer iterations where, g,, g2, g3 are gametes, and pk, pk+1 are the for any given set of parameters were initiated from a frequencies of gametes in generations k and k+\. The random point in the vicinity of the centre of the fitness, w{gx,g^), of a genotype is obtained by simplex of gametic frequencies. The centre is the point computing the value of each of the N characters for at which frequencies of all gametes are the same, i.e. the genotype and substituting the values into formulas the allelic frequency is 0-5 in each locus and the loci (2) and (3). The function ff(g3\gi,g2) in (6) is the are in complete linkage equilibrium. If {x,} denotes probability that a gamete produced by an individual coordinates of the centre of a simplex, a starting point with the genotype (gj,^2) will be g3. This function is was chosen with random coordinates {_)>,} such that, determined by the pattern of recombination between besides the usual constraints defining the simplex: the loci in a model. It is assumed throughout the paper I,. yj = 1 and yt ^ 0, they also satisfied an additional that recombination is the same between any pair of constraint: adjacent loci and there is no interference. For a given set of parameters, system (6) was iterated on a computer starting from an initial (7) distribution of gametes, and when the distance, i.e. a starting point was located at a distance p from the centre. It is not a good idea to choose the same p between gametic distributions in two consecutive for models with different numbers of loci, since a generations became less than 10~12, it was assumed given distance may be regarded as 'small' in one that an equilibrium had been reached. To test the simplex but as 'large' in a simplex of a higher local stability of the equilibrium, the equilibrium dimension. For example, a sphere of radius 0-05 lies gametic frequencies were perturbed by adding to completely within the two-locus simplex, but the same each of them its own very small (of the order 10~5) sphere 'sticks out' of the five-locus simplex. For this random number, and then normalizing them so that reason, the distance for a starting point was chosen as their sum was unity. Iterations were repeated starting /9 = /9*/100, where p* is the maximum radius of a with the perturbed gametic frequencies, and an sphere completely contained within the AMocus

simplex. For 3 x 3, 4 x 4 and 5x5 model, the distance Table 1. Percentage of trajectories converging to an is equal to 00013, 00006 and 00003, respectively. equilibrium with linkage disequilibrium D under Fig. 1 shows regions of existence of stable equilibria selection of strength Q in a 2x2 model with relative in N x N models on the plane of relative allelic allelic contribution, b = 01, and recombination, r = contribution, b, and strength of selection, S, for four 0-05 (all equilibria are symmetric) values of recombination, r. Only 'totally polymorphic', i.e. with allelic frequency 05 at each locus, D equilibria are considered. Such equilibria are, obviously, symmetric for a 2 x 2 model, but not for a 01 -0010 100 model with N > 2, since linkage disequilibria generally 31 -0198 100 differ between different pairs of loci. For a given value 3-2 -0-201 81 of r, a locally stable totally polymorphic equilibrium 0147 19 exists in the area above the corresponding curve (left 40 -0-215 59 side of the graph), whereas no such equilibrium exists 0-208 41 in the area below the corresponding curve (right side of the graph). It should be remembered that a stable equilibrium in Fig. 1 is approached from a random Table 2. Percentage of trajectories converging to an point in a vicinity of the centre of the simplex of equilibrium with linkage disequilibria Dfj under gametic frequencies. Other totally polymorphic equil- selection of strength Q in a 3x3 model with relative ibria, to which trajectories originating in different allelic contribution, b = 01, and recombination, r = parts of the simplex converge, may in principle exist. 005 (all equilibria are totally polymorphic with the It is seen in Fig. 1 that totally polymorphic stable allelic frequency 0-5 in each locus) equilibria exist in Nx N models for a wide range of parameters, and, if the contribution by a minor locus Q % is less than 0-2 of that by a major locus, there is a totally polymorphic stable equilibrium for any model 01 -0008 -0004 -0008 100 with practically any strength of selection and any 1-5 -0072 -0032 -0072 100 recombination, which is in accordance with the 2-3 -0091 -0005 -0091 100 analysis of weak selection discussed previously. It can 2-4 -0098 0003 -0098 100 also be noticed that the effect of recombination 2-6 -0115 0026 -0115 100 diminishes with increased dimensionality of the model. 2-7 -0125 0041 -0-125 70 In a 5 x 5 model, the existence and stability of totally 0083 -0084 -0171 15 polymorphic equilibria are mostly determined by the -0171 0084 0-083 15 strength of selection, whereas recombination has very 3-5 -0182 0129 -0182 32 little effect, unless linkage is quite tight (r = 005). 0178 -0151 -0-204 32 It is not possible to investigate completely the -0-204 -0151 0-178 32 0186 0142 0186 4 dynamics of Nx N models. However, a good idea about dynamic properties of a model with a given set of parameters can be obtained by employing a computer procedure introduced by Karlin & Carmelli quite a large number of parameter sets in 2 x 2 and (1975) to investigate two-locus symmetric viability 3x3 models, but to fewer parameter sets in models of models, and used by Gimelfarb (1995) to investigate a higher dimension (computing time becomes prohi- two- and three-locus models of a quantitative trait bitively long for the latter models). The general under strong stabilizing selection. A large number of picture emerging from this is that the dynamic trajectories of system (6) are generated starting from properties of NxN models are determined, not points randomly chosen in the total simplex of gametic surprisingly, by the relative strength of recombination frequencies, and equilibria are observed to which the and selection. If r > 0-05, i.e. linkage is not very tight, trajectories converge. The proportion of trajectories a totally polymorphic stable equilibrium, if it exists, converging to a particular equilibrium can be regarded seems to be the only equilibrium with non-zero allelic as a measure of the size of the domain of attraction to frequencies in all loci. It is apparently globally stable the equilibrium. Obviously, only equilibria that are under selection that is only slightly stronger than the stable and have domains of attraction that are not minimum required to maintain a stable equilibrium in extremely small can be discovered by this method, but a model with given b and r (the minimum strength of on the other hand only such equilibria are of biological selection is represented in Fig. 1 by the vertical significance. If all the randomly initiated trajectories coordinate of the point on the curve for a given r, converge to one equilibrium, we shall call such an whose horizontal coordinate is b). equilibrium 'apparently' globally stable. If, however, linkage is tight (r = 005), dynamic This procedure (with 500 random trajectories properties of NxN models are more complex, generated for each set of parameters) was applied to particularly for small contributions by minor loci.

Multiple totally polymorphic stable equilibria may any recombination and any strength of selection. exist simultaneously under sufficiently strong selection Since contributions by loci to traits are additive, the in such a case. For example, Tables 1 and 2 show the genotypic variance of any trait is equal to the additive percentage of trajectories (out of 500 initiated ran- genetic variance of the trait, and it can be utilized by domly) converging to totally polymorphic equilibria natural selection in the process of phenotypic evol- in 2 x 2 and 3x3 models with r = 005 and b = 01. ution. The 2x2 model has only one stable symmetric A question can be raised concerning symmetries in equilibrium with negative linkage disequilibrium, the TVx N models analysed so far: it is assumed that which is apparently globally stable, under selection the relative contributions of all minor loci to any trait with Q < 3-1. If, however, Q > 3-2, two stable sym- are the same, and that any trait experiences the same metric equilibria exist simultaneously, one with nega- selection. Are the models robust to deviations from tive and the other with positive linkage disequilibrium. these assumptions? The answer is 'yes'. It is easy to As for the 3x3 model, it has only one stable totally see from the analysis in Appendix A, that a stable polymorphic equilibrium, which is apparently globally totally polymorphic equilibrium may exist under weak stable, under selection with Q ^ 26, whereas three selection, even in a model that is not symmetric. Also, and even four stable totally polymorphic equilibria consider a 5x5 model with parameters shown in can exist simultaneously under stronger selection. It is Table 3. Even though the model is clearly not also interesting that, while linkage disequilibria be- symmetric with respect to either contributions by tween adjacent loci, D12 and £>23, in the 3x3 model minor loci or selection, there is a stable equilibrium are always negative under selection with Q ^ 2-6, and with a polymorphism in all five loci (equilibrium their absolute value increases with stronger selection, allelic frequencies are shown in the bottom row of the the behaviour of linkage disequilibrium between the table), and it is apparently globally stable. It is distant loci, Dl3, is more complex. It is negative and its important to notice that the equilibrium mean value absolute value increases with stronger selection until of a trait, m}, deviates from the selection optimum, 0}, Q remains less than 1-5. If, however, Q > 1-5, its which means, as discussed by Gavrilets & Hastings absolute value decreases until it reaches zero, after (1993), that each character will exhibit a directional which Dl3 becomes positive and increases with component of selection even in a population at increasing strength of selection. If selection is suffi- equilibrium. It can also be noted that the allelic ciently strong {Q ^ 2-7), linkage disequilibrium can be contributions by minor loci in Table 3 are not of the positive between any two loci, including a possibility same sign. This means that minor loci do not have to of all three linkage disequilibria being positive. act in the same direction on all traits for a stable Existence of stable equilibria with positive linkage polymorphism to be maintained. Actually, Table 3 is disequilibrium was first discovered by Gavrilets & only one example of a N x TV model with parameters Hastings (19946) in a two-locus model of an additive chosen randomly from the following intervals: —0-2 quantitative trait under strong stabilizing selection. s= bi} ^ 0-2; 0-3 < g, < 1-0; 0-4 < 0, < 0-6. Five hun- dred such models were generated for each N = 2, 3, 4, 5, and a stable equilibrium with a polymorphism at all N loci existed for 96 % of 2 x 2 models, 95 % of 3 x 3 5. Discussion and conclusions models, 81% of 4x4 models and 56% of 5 x 5 The results demonstrate that Nx N models of pleio- models. tropy can maintain a stable equilibrium with the Another potential concern about the models dis- allelic frequency 0-5 in all N loci for a wide range of cussed here is the assumption that every locus parameters. If the relative contribution by a minor contributes to all N characters, and, consequently, locus is less than 20% of that by a major locus, a every character must be genetically correlated with all 'total polymorphism' is maintained for practically other characters, which may not be true in reality.

Table 3.5x5 model without symmetry: relative contributions by locus Lt to character Xp parameters of

selection (0;. and Qj) on character Xp the equilibrium mean, mp and variance, vp of character Xp the equilibrium allelic frequencies, p,. Recombination pattern, r = {0-5, 04, 0-3, 0-2}

100 013 -0-20 -019 017 0-49 0-32 0-47 0047 -014 1-00 -013 018 -011 0-41 0-48 0-33 0041 018 010 100 -014 009 0-56 0-36 0-61 0048 015 -012 008 100 012 0-60 0-44 0-70 0048 011 -010 -018 009 100 0-48 0-40 0-55 0059 0-56 0-24 0-71 0-74 0-55

However, this assumption is also not necessary, and a for apparent selection on 'neutral' traits. Indeed, total polymorphism can be maintained even more consider a N x N model, and let there be some other easily in a model of N loci controlling N traits with traits which do not experience direct selection but are some of the loci contributing to fewer than N controlled by the N loci (or some of them). If characters (an extreme and trivial case is that of every parameters of the Nx N model are such that a total locus contributing to a different trait). Consider, for polymorphism is maintained, there will be apparent example, a model of five loci controlling five quan- stabilizing selection on any 'neutral' trait to which titative traits with bHi+l) = b{i = 1-4), bbX = b and b(j these loci contribute. The apparent selection can be = 0 otherwise, i.e. each locus contributes to only two quite strong if the number of traits experiencing direct traits, and any two traits have either one or no loci in selection is large, even if selection on each individual common. It follows from Fig. 1 d that the minimum trait is weak. strength of selection necessary to maintain a total polymorphism in a 5 x 5 model with b = 0-3 and r = Appendix A 0-5 is Q= 1-5. On the other hand, a totally stable polymorphism is maintained in the discussed model The following analysis was suggested by Nick Barton. with the same parameters b and r under selection with Given that the loci controlling traits are at linkage Q as small as 001. Under such weak selection, linkage equilibrium, i.e. the distribution of genotypes in the disequilibria are very close to zero and, consequently, loci are independent, the mean and the variance of a only characters having a locus in common are character X} are genetically correlated (e.g. character X is genetically l (Al) correlated with characters X2 and Xh, while its genetic correlation with other traits is practically zero). The model of antagonistic pleiotropy for fitness (A 2) components by Rose (1982) predicts a negative genetic k correlation between the traits. Yet, many experimental where pk is the frequency of the non-zero allele in estimates of the genetic correlation between fitness locus Lk(gk = 1 —pk) and fijk is the actual contribution components are either zero or positive (Rose et ai, by locus Lk to character X}(fi1} = a,). According to 1987). On the other hand, the genetic correlation equations (2) and (3) in the text, the fitness of an between characters in models discussed here can be individual is either positive or negative or zero, depending on the signs and values of allelic contributions and on the pattern of pleiotropy. i Assuming that selection is sufficiently weak that terms A question may be asked whether it is actually 2 pleiotropy that is responsible for the maintenance of a in (A 3) of the order Q and higher can be neglected, total polymorphism in Nx N models. Would not the fitness of an individual is approximately stabilizing selection acting on just one of the characters 2 produce a similar result? Indeed, a stable poly- W = 1 - 2 Qj{Xi - 6>3.) . (A 4) i morphism in several loci with unequal contributions to a quantitative trait can be maintained under Consequently, the average fitness in a population is stabilizing selection on the trait, provided selection is (A 5) sufficiently strong relative to recombination (Gavrilets & Hastings 1993, 1994a, b; Gimelfarb, 1995). If, The dynamics of allelic frequencies under weak however, selection is relatively weak, it is in fact selection is described approximately by the following pleiotropy among traits, each experiencing selection, system of differential equations (Wright, 1935; Barton, that is responsible for the maintenance of a total 1986): polymorphism in N x N models. Consider, for ex- dp, dW ample, a 5 x 5 model with btl = b = 0-2 and r = 0-5 (A 6) between any pair of loci. If selection acts only on one of the traits, no totally polymorphic equilibrium can Given (A 5), (A 1) and (A 2), be maintained under stabilizing selection of any d strength (including the strongest possible stabilizing P * (A 7) selection which culls all individuals except those _ = Ap-c, whose phenotype is exactly 6). Yet, as Fig. \d where p is a vector whose elements are allelic demonstrates, a stable totally polymorphic equilib- frequencies, whereas A and c are a matrix and a vector rium exists for the same values of b and r even under with elements extremely weak selection, if selection acts on each of the five traits. Pleiotropy among characters experiencing stabilizing selection can also be a mechanism responsible (for i*./), {A 8 a)

where a = 0-5/[l +(JV-l)6]and/? = 0-56/[l +(N-\) b] are the actual contributions by a major and a minor locus, respectively, whereas x denotes the total Provided matrix A is non-singular, allelic frequencies t contribution to character Xt by the identical N— 1 at an equilibrium are obtained as loci. The fitnesso f an individual under weak selection = A1c. (A 9) is described by (A 4). Hence, the fitnesseso f individuals differing only at locus Lt will be For a symmetric model with Qi = Q, 0} = 6, fiu = a. and f} = /? for / 4= j, the elements of matrix A and (j 2 vector c are W(AA) = 1 -Q[(x} + 2cc-0-5) + £ (xk + 2fi-0-5)*\, (B2a) 2 Atj = -8Q[2a/l + (N-2)/1 ] (for / * j), (A 10a) 2 2 W{Aa) = 1 -Q[(x] + a-0-5 ct = 2Q[d-a -(N--i)/3 ]. (A 10b) Consequently, the equilibrium allelic frequencies are (B26) d-d* , where*?* = a2 + (/V-l)^2. (B2c) Pi=P = 1-2(9* (All) Total overdominance for fitness means that for any Substituting the values of a and /? from equations (1 a, genotype at N— 1 loci, a heterozygote at locus L1 has b) in the text, a higher fitness than a homozygote, i.e. the difference in fitness between a heterozygote and a homozygote (A 12) must be positive. Hence the difference

In order for 0 < p < 1, the optimum phenotype must W{Aa)- W{aa) = be within the limits 6* < 6 < 1 -6*. *#; The local stability of the equilibrium (All) is -\)/3\ (B3) determined by the eigenvalues of the matrix A. A must be positive for any values contributed by the NxN matrix with elements R on the main diagonal TV—1 loci, including their maxima. The maximum and elements S elsewhere has one eigenvalue A = 1 of xs is (N—\)p, whereas the maximum of xk is R + S(N-1) and (N-\) eigenvalues A2 = R-S. For a + (N—2)/3. Hence, the following inequality must be the matrix A, satisfied: 2 + 6(N-l)a/3+2(N-\)(N-2)/3 ], 8(/V-l)a/? + (7V-l)(4/V-7)/?2 + a2-0-5 > 0. (B4) (A 13a) -4a/?-(7V-3)/?2]. (A 136) Substituting the expressions for a and /? from equations (1 a, b) of the text yields The first value is always negative, whereas in order for 2 the second value to be negative, and, hence, for the (7V-l)(2/V-5)6 -l)6-l <0 (B5) equilibrium to be locally stable, the expression in It can be shown that the same inequality must hold in square brackets of (A 136) must be positive, i.e. the order for the difference W(Aa) — W(AA) to be positive. following inequality must hold: Hence, total overdominance for fitnessi s present in a l-46-(/V-3)62>0. (A 14) NxN model if This requires that (TV = 2) (B6a) (A 15a) V[3(JV-l)(2/V-3)]-2(/V-l) (A 156) b< (N-l)(2N-5) b <(VN+ l-2)/(N-3) (N>3). (A 15c) I wish to thank Carlo Matessi for his interest and useful discussions, Reinhard Burger for an important suggestion, Appendix B and Thomas Nagylaki for his help. The very valuable contribution by Nick Barton is highly appreciated. I am also Consider individuals with identical genotypes in N— 1 grateful to William Hill for constructive criticism. This work loci, but having different genotypes (AA, Aa or aa) at was supported by National Science Foundation grant BSR- 9006285. one locus, say Ly In a symmetric NxN model, such individuals will have the following values of the characters: References

XjiAA) = x, + 2a, Xk(AA) = xk (k + j) (B 1 a) Barton, N. H. (1986). The maintenance of polygenic X (Aa) = Xj + a, X (Aa) = x (k + j) (B 1 b) variation through a balance between mutation and } k k stabilizing selection. Genetical Research, Cambridge 47, Xjiqa) = x, Xk(aa) = xk (k * j) (B 1 c) 209-216.

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