The maintenance of phenotypic and genetic variation in threshold traits by frequency-dependent selection. D. A. Roff. Dcprtmcnt oj' Biology, McGill University, ...
0 Rirkhauser Verlag, Rasel, 1998 J. evol. biol. I I (1998) 1010-061x 98’040513
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1Journal of Evolutionary
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The maintenance of phenotypic and genetic variation threshold traits by frequency-dependent selection
Biology
in
D. A. Roff Dcprtmcnt oj’ Biology, McGill University, 1205 Dr. Ptwfield Ace., Montred, Quebec, Cundu, H3A lBI, e-mail: ~roj~~biol.lun.r~zcgill.ca Ktly u~or&: Threshold traits; mutation; heritability; quency-dependent selection; dimorphism.
additive genetic variance; fre-
Abstract Many traits are phenotypically dimorphic but determined by the action of many loci, the phenotype being a result of a threshold of sensitivity. Quantitative genetic analysis has shown that generally there is considerable additive genetic variation for the trait, the average heritability being 0.52. In numerous casesthreshold traits have been shown, or are assumed, to be under frequency-dependent selection; examples include satellite-territorial behaviour, sex-determination, wing dimorphism and trophic dimorphism. In this paper I investigate the potential for frequency-dependent selection to maintain both phenotypic and additive genetic variation in threshold traits. The qualitative results are robust to the particular form of the frequency-dependent selection function. The equilibrium proportion is more or less independent of population size but the heritability increases with population size, typically approaching its maximal value at a population size of 5000, when the mutation rate is 10P4. A tenfold decrease in the mutation rate requires an approximate doubling of the population size before an asymptotic value is approached. Thus frequency-dependent selection can account for both the existence of two morphs in a population and the observed levels of heritability. It is also shown, both via simulation and theory, that the quantitative genetic model and a simple phenotypic analysis predict the same equilibrium morph proportion. Introduction There is a large class of traits that are expressed as two or a few discrete phenotypes (Wright, 1968; Falconer, 1989; Roff, 1996). Examples include cyclomorphosis, paedomorphosis and paedogenesis, “weaponry” (enlarged mandibles, cerci, head structures etc) in insects, trophic dimorphisms in amphibians and fish, 513
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(semelparity vs wing dimorphism in insects, diapause, modes of reproduction iteroparity), and mating behaviours (e.g. satellites vs territorial males: for a review see Roff, 1996). Though these traits are phenotypically discrete, their inheritance may be polygcnic, the particular manifestation of the trait being a function of a threshold of sensitivity (Falconer, 1989; Roff, 1996). According to the threshold model a continuously varying character underlies the expression of the trait, individuals with values lying above the threshold being of one type, whereas individuals lying below the threshold are the other. The underlying trait, termed the liability, is assumed to be inherited in the usual polygenic manner. Although many threshold traits arc clearly closely connected to fitness (e.g. life cycle alternatives such as paedomorphosis, or mating types displaying alternative morphological structures and/or behaviours), there frequently exist significant proportions of each morph in a population. To address the question of why two morphs are maintained in the population the mathematical approach known as game theory has generally been employed (Gross and Charnov, 1980; Maynard while this technique can Smith, 1982; Gross, 1985; Pfennig, 1992). However, address the question of the equilibrium proportions it cannot predict the population trajectory if the population is displaced from its equilibrium, nor can it address the question of genetic variation for the alternative morphs. All three issues can be analysed using quantitative genetic methods. Such an approach is of particular importance for an understanding of the evolution of alternative morphs because there is typically considerable additive genetic variation for the trait; for example, the mean heritability of the 18 cstimatcs for dimorphic traits reported in Table 1 of Roff (1996) is 0.52. The central assumption of game theoretic models is that selection is frequencydependent (Maynard Smith, 1982). In the present context examples in which frequency-dependent selection has been or can be assumed include satellite-territorial dichotomies (Gross, 1982, 1985, 199 1; Myers, 1986; Hutchings and Myers, 1994; these typically also include morphological differences), dimorphisms in male weaponry (Gadgil, 1972), wing dimorphism and male mating success (Hamilton, 1979; Tsuji et al., 1994) some cases of trophic dimorphism (e.g., cannabalistic vs herbivorous morphs, Pfennig, 1992), and sex determination (Charnov and Bull, variation can also result in 1977; Deeming and Ferguson, 1989). Spatio-temporal frequency-dependent selection on wing dimorphism when viewed from a meta-population perspective (Roff, 1994). Frequency-dependent selection can readily maintain genetic variation at a single locus (Wright, 1948; Haldane and Jayakar, 1963; Clarke and O’Donald, 1964; Clarke, 1974; Anderson, 1969; Cockerham ct al., 1972) but the situation is much less clear for a polygenic trait (Bulmer, 1985), though Mani et al. (1990) demonstrated, using a simulation model, that it can be important in some instances. The potential of frequency-dependent selection maintaining genetic variation in a threshold trait has not been studied: given the high levels of genetic variation in threshold traits and the generality of this type of selection on such traits, the potential importance of frequency-dcpcndcnt selection on the maintenance of genetic variation warrants investigation. Two papers have addressed the question of equilibrium phenotypic frequencies for frequency dependent selection acting on
Frequency-dependent
selection
xid
threshold
traits
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threshold traits: the first, by Hazel et al. (1990) uses R quantitative genetic model, whereas the second, by Gross and Repka (1997) adopts a non-Mendelian mechanism of inheritance, which makes interpretation of the results suspect. Both models assume that the amount of variation is fixed and hence do not address the issue of the maintenance of genetic variation, the central focus of the present paper. The purpose of the present analysis is to test the hypotheses that frequency-dependent selection acting on a threshold trait can maintain both morphs in the population and also maintain genetic variation as high as observed. These questions are addressed using both theory and simulation modelling. The simulation
model
The simulation model follows that used in an earlier analysis of the effect of directional selection on genetic variance (Roff, 1998). The underlying continuously distributed character (liability) is assumed to be determined by II (= 10 unless otherwise stated) unlinked loci, each with two alleles contributing either 0 or 1 to the phenotypic value. Dominance and epistasis are assumed to be absent. The initial distribution of allelic values is obtained by randomly assigning each as 0 or I, thereby giving an average per locus frequency of 0.5. Thus the additive genetic variance, assuming Hardy-Weinberg equilibrium and linkage equilibrium in the unselected population, is equal to 0.5t7. The phenotypic value of an individual is obtained as the summed contribution of all loci plus a random normal deviate distributed with zero mean and variance I$. The environmental variance, V,:, is determined from the heritability in the unselected population and the relationship /I’ = V, ,‘( V,[ + I’,_), where /I’ is set in most simulations at 0.5 to correspond to the average value observed for threshold traits (see introduction). In the founding population both morphs are set at equal frequency, obtained by setting the threshold value equal to 12: the effect of varying the threshold is also investigated. Each generation consists initially of N individuals (N= 25, 50, 100, 250, 500, 2000, 5000). The selection coefficient of the two morphs (hereafter referred to a morph A for individuals with liabilities below the threshold and morph B for those with liabilities above the threshold) is determined from their proportions in the population. For each individual a random number between 0 and I is generated and compared to the relevant selection coefficient; if the random number falls below the selection coefficient the individual is retained, otherwise it is discarded (i.e. does not contribute to the next generation). To generate the next generation, N pairs are chosen with replacement from the selected population, each pair contributing one offspring. Each allele in the offspring has the same probability, ,L(, of mutating to the alternate allele. I used two values of 11; ,LL= IO-” and 1~= 10 ‘. The former rate is at the upper end of published estimates, but Turelli (1984) has suggested that these rata are probably too low. Although frequency-dependent selection has been well documented (see introduction) the functional form of the relationship has not been studied: therefore, I chose two arbitary fmictions, one based on that used by Mani ct al. (1990) and one suggested by Maynard Smith (1982). The former function is
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where J’, is the selection coefficient of morph A, .f’ is the proportion of the population that is morph A, and a, /j are constants which determine the shape and strength of selection. The selection coefficient of morph B is determined in three to one in which separate ways: 1) Model 1; sg = 1 - .s,#. This model corresponds fitness is determined by some resource that is partitioned between the two morphs. Alternately, it describes the situation in which a predator takes a constant proportion of individuals from the population. 2) Model 2; sg is calculated by substituting the proportion of morph B into equation 1 (i.e. replaceJ’with 1 -.f’). In this model the only important characteristic of the morph is its frequency. For models 1 and 2 I use five different combinations of x and fi, producing functions ranging from highly concave through linear to highly convex (Fig. 1). 3) Model 3; the value of p varies between morphs. Unlike model 2, in this case the fitness of each morph depends both upon its frequency and characteristics specific to the particular morph. For simplicity, for both morphs x is set equal to 1, .SA=
1-f P/If+ 1 -J
S”=/j
R(1 i/,+f
(2)
Two combinations of /Is are used: ljA = 4, ljR = 2, and BB = 0.1. The second function is that used in the “Hawk-Dove” model (Maynard Smith, 1982, pp. 11-12) s/l = tm - B1+(1 -.m .sn = $x( 1 -J‘). Because the functional placed in the “Model used: a= 1, jj’=4.
A theoretical
form differs 3” category.
(3) between the two morphs, the above model is A single combination of parameter values is
approach
For an infinite population all loci will have the same allelic frequency, p; therefore, the selection intensity will be the same at all loci and we can apply standard single locus theory (Crow and Kimura, 1970, p. 258). Considering a single focal locus, the contributions to the phenotypic value of three possible genotypes, labelled AA, Aa and au, are 2, 1, and 0, respectively. The fitnesses of these three genotypes are, respectively, w/f,
= P,s, + (1 - P,)s,,
w,4,, = p,s,‘l + (1 - p, bn,
w,,‘,= p,,s,+ (1- P,,).~,,
(4)
Frequency-dependent
selection
and threshold
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traits
Fig. 1. Frequency-dependent fitness functions used in the present study. Top panel: Functions used in models 1 and 2. Reading from top to bottom curves, parameter combinations are x = 2, ,0 = 0. I; a = 2, /J’ = 0.5; CL= I, /J = I; 3~= 0.9, /I = 5; CY= 0.9, [I’ = 200. Bottom panel: Functions used in model 3 (morph indicated by A or B). Solid lines; /j’,, = 4, ljB = 2. Dashed lines; PA = I, /J’~ = 0.1. Dotted lines; 3~= I, p=4.
where P, is the probability that the phenotypic value of the particular genotype lies i units below the threshold value, T, before the contribution of the focal locus is considered. The phenotypic value set by the n - 1 “non-focal” loci and the environmental effect is distributed as a random normal with mean p = 2(n - 1)~ and variance (T’ = 2(n - l)p( 1 -p)+ V,, where V, is determined as described in the preceding section. Therefore,
(5) Because we are concerned with equilibrium conditions, linkage disequilibrium is ignored in the above formulation. At equilibrium we have (Crow and Kimura, 1970, p. 259, equation 6.22)
tP2W,4A+p(l -P)W,,,)(l P=P2w4A+2Pu
-P)W,,,f(l
-p) -P)2W
