Evolution of Amphimixis and Recombination Under Fluctuating Selection in One and Many Traits

Evolution of amphimixis and recombination under fluctuating selection in one and many traits

ALEXEY S. KONDRASHOV* AND LEV YU. YAMPOLSKY Section of Ecology and Systematics, Cornell University, Ithaca, NY 14853, USA (Received 6 June 1995 and in revised form 14 May 1996)

Summary Both stabilizing and directional selection acting on one or many quantitative traits usually reduce the genetic variance in a polymorphic population. Amphimixis and recombination restore the variance, pushing it closer to its value under linkage equilibrium. They thus increase the response of the population to fluctuating selection and decrease the genetic load when the mean phenotype is far from optimum. Amphimixis can have a short-term advantage over apomixis if selection fluctuates frequently and widely, so that every genotype often has a low fitness. Such selection causes high genetic variance due to frequent allele substitutions, and a high load even with amphimixis. Recombination in an amphimictic population is maintained only if selection is usually strong and effectively directional. A modifier allele causing free recombination can have a significant advantage only if fluctuations of selection are such that the load is substantial. With smaller fluctuations, an intermediate recombination rate can be established, either due to fixation of alleles that cause such a rate or due to the stable coexistence of alleles causing high and low recombination. If many traits simultaneously are under fluctuating selection, amphimixis and recombination can be maintained when selection associated with individual traits is weaker and the changes in their mean values are smaller than with a single trait. Still, the range of changes of the fitness optima in each trait must be at least of the order of the trait's standard deviation, and the total load must be high.

1. Introduction Amphimixis and recombination can be beneficial in an infinite population under a changing environment because they destroy linkage disequilibria created by epistatic selection (see Kondrashov, 1993). The corresponding Environmental Deterministic (ED) hypotheses can be subdivided into two subclasses. Non-responsiveness hypotheses claim that destruction of disequilibria is beneficial because it reduces the response of a population to frequent and relatively small fluctuations of selection, while better responsiveness hypotheses postulate that this destruction is beneficial because it increases the response to rarer and wider fluctuations, or to invariant directional Trait selection (Barton, 1995). 'Narrowing' selection (Shnol & Kondrashov, 1993) Fig. 1. Under selection with fluctuating fitness function amphimixis can be advantageous only if every genotype acting on a quantitative trait reduces the genetic sometimes has a low fitness. With all-or-nothing variance below its linkage equilibrium level by creating truncation selection, this is the case when there is at least a pair of fitness functions, w^x) and w2(.v), each • Corresponding author. Telephone: +1(607) 254-4221. Fax: describing selection at some moments of time, which do + 1(607) 255-8088. e-mail: [email protected]. not overlap.

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Fig. 2. The average deviation of the mean value (ji) of the trait from m (a) and from o (b), and the genetic load corresponding to the geometric mean fitness (c) with w = 1, S = 1-5, and r = 20 (circles), 80 (squares), and 320 (diamonds) in the amphimictic (filled markers) and apomictic (open markers) populations as functions of d (compare with figure 3 in KY-96).

repulsion linkage disequilibria, while amphimixis and mean reaches the fitness optimum. Under invariant recombination move it closer to this level by destroying directional selection strong enough to provide a short- them and thus increase the resulting changes in the term (Maynard Smith, 1978) advantage for trait mean value (Mather, 1943; Fraser, 1957; amphimixis, the higher variance is beneficial but the Felsenstein, 1965; Eshel & Feldman, 1970; Crow, mean value of the trait changes rapidly and 1992; Charlesworth, 1993). Both directional selection irreversibly. Apparently this is not common in nature, (if the increase in fitness caused by two beneficial although if many traits are selected simultaneously a alleles in the same genotype is less than the product of short-term advantage may be consistent with slow their individual effects, i.e. if they interact sub- evolution in each of them (Maynard Smith, 1988, multiplicatively or with diminishing returns epistasis) p. 62; Crow, 1992). and stabilizing selection are usually narrowing (see Thus, the better responsiveness hypotheses are Shnol & Kondrashov, 1993). more plausible under fluctuating selection (Mather, Under invariant stabilizing selection, the higher 1943;Treisman, 1976; Maynard Smith, 1980, 1988; variance becomes deleterious after the population Korol & Preygel 1989; Crow, 1992; Charlesworth,

1993; Korol et al. 1994), either directional or stabilizing (when the population mean is far from the 3. Results optimum, stabilizing selection becomes effectively (i) Amphimixis versus apomixis: one trait directional and the increase in variance is beneficial). A short-term advantage of amphimixis requires that When not stated otherwise, we used the default no phenotype has high fitness for a long time (Fig. 1; parameters from KY-96. Figure 2 presents the data on Treisman, 1976). Maintenance of recombination in an the deviations of the mean value of the trait /* from m amphimictic population may require even more (Fig. Id) and o (Fig. 2b), and on the genetic load (Fig. stringent conditions (Charlesworth, 1993). 2 c). In the amphimictic population the deviation of fi from o is smaller, while its deviation from m is larger, Here we will supplement the recent analytical results so that amphimixis facilities the response to selection, of Charlesworth (1993) and Barton (1995) by in- due to higher genetic variance (see KY-96). Figure 2 is vestigating a computer simulational model of the based on the data from a single run, observed from evolution of amphimixis and recombination under generation 3d during 5d generations. The results of fluctuating selection acting on one or many quan- different runs were very similar (data not reported). titative traits. Not surprisingly, this better responsiveness of amphimixis leads to a lower load when d> S. Amphimicts outcompete apomicts with twofold re- 2. Model productive advantage (Maynard Smith, 1978) if (l-^amP)/[2(l-4po)] > 1, where Lamp and Lapo are The basic model was described in Kondrashov & the values of the load, in terms of geometric mean Yampolsky (1996; hereafter KY-96). To consider the fitness, in the two populations. This requires Lap0 > evolution of recombination, we added a modifier 0-5, which does not happen if there is a phenotype locus R, which does not mutate, does not influence which always has high fitness (a compromise strategy), fitness directly, is unlinked to the selectable loci, and i.e. if d

= m( + dsm{2n[t/T + i/ (1) Fig. 3. The genetic load corresponding to the geometric mean fitness in the amphimictic and apomictic population (compare with equation 1 in KY-96). According to (a) under the same parameters as in Fig. 2 (5 = 1-5, eqn (1), different o deviate maximally from the d = 4-5 and T = 80) but with non-zero environmental t variance, and (b) under the same parameters as in Fig. 2 corresponding mt at different moments, so that the (S = 1-5, T = 80), but with alleles having arbitrary effects distribution of the values of o( does not change much (compared with figures 6 and 7 in KY-96). with time. The THINK C programs are available on request.

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XXXXBBSBI 10 0 3 6 9 12 15 d Fig. 4. Evolution at the R locus under the same conditions as in Fig. 2 with S = 1-5 (o) or S = 30 (6). Fixations of alleles causing more and less recombination are denoted by black and white squares, respectively; their stable coexistence is denoted by crossed squares, their neutral coexistence (selection at R too weak to be detected) is denoted by crosses, and extinction of the population is denoted by dots. Only a pair of alleles, marked above the corresponding pictures, was present in each case. With d = 0, selection at R was always too weak to be detected, although r0 probably won eventually (Charlesworth, 1993).

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all cases, the Lamp required for the protection of recombination, while with d> S recombination can amphimixis was high (at least ~ 03). Of course, be selected for. Free recombination was selected for without the twofold advantage of apomixis, Lamp < when Lamp > ~ 0-3, but still the advantage of Lapo is sufficient for the protection of amphimixis. An amphimixis was often not sufficient to overcome the advantage of amphimixis disappears only when S -4 1 twofold advantage of apomixis (compare Figs. 2 c and (data not reported). 4). Recombination was favoured most with T = 20 If the mutation rate u is not much smaller than N'1, or 40 under S = 1-5, and with r = 40 or 80 under allele fixations are rare and some variability is usually S = 30. With d> S, but not sufficient to favour free present at a locus, while with u < 001A^1 even an recombination, intermediate recombination rates were amphimictic population can run out of variability and established, due to either fixation of alleles causing die out when d^> S. In contrast, with a high u such rates or stable coexistence of two alleles causing (perhaps unrealistic) even an apomictic population a high and a low rate (Fig. 5). can adapt rapidly, due to new mutations. Thus, With S < 0-3 recombination was not favoured, u ~ 0-1 TV"1 (as in Fig. 2) usually leads to the largest because the increase in variance always led to lower advantage of amphimixis (data not presented). Trunc- fitness. The properties of alleles causing three or more ation stabilizing selection, as well as random chiasmata were very similar to those of rm. Dominance fluctuations of the fitness optimum, led to roughly the of an r allele led, as expected,to the decreased efficiency same advantage of amphimixis (data not reported; see of selection when this allele was frequent (data not Charlesworth, 1993). reported). The advantage of amphimixis increased slightly As with the advantage of amphimixis, non-zero with the number of loci that influence the trait (data environmental variance increases the range of o not reported). In the presence of environmental fluctuations required for the advantage of recom- variance the advantage of amphimixis requires even bination. For example, if 5 = 1-5 and d = 6-0, allele rx wider fluctuations of fitness optimum, such that replaced r0 only when crenv < 1-5 (T = 20) or 0'env )> because, with smaller d, the popu- (T = 40), i.e. when rfexceeded ViS + S. With

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Fig. 6. The geometric mean genetic load in the amphimictic (filled markers) and apomictic (open markers) population with fluctuating selection acting on 10 traits, T = 80 and S = 1-5 (continuous lines) and S = 3-0 (dotted lines).

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Fig. 7. The geometric mean genetic load (continuous lines) and frequency of the allele rx after IT generations (dotted lines) under fluctuating selection acting on 10 traits with S = 1-5, T = 20 (circles), 80 (squares) and 320 (diamonds), and different values of d. Initially only the alleles and r0 and rx were present.

such d the load is very high even with amphimixis. However, some recombination can be maintained Under shorter and longer periods of selection with a smaller load, although it still must be at least fluctuations the pattern was similar (data not re- ~ 0-5. As with one trait, a stable polymorphism at the ported). R locus can be established with intermediate d (data With random fluctuations in O, Lamp was even not reported). higher under the conditions necessary for the advantage of amphimixis. However, the minimal D required for this advantage was slightly lower than 4. Discussion under periodic fluctuations according to eqn (1), (i) When does fluctuating stabilizing selection because here all position of O inside the multi- maintain amphimixis'! dimensional cube (mi — d^Of^nif + d) are possible (data not reported). Amphimixis may have an advantage when every genotype is sometimes maladapted (Fig. 1). In our model this happens when d> ~ ^/(SP + cr^2) (Figs. (iv) Recombination: many traits 2, 3). The advantage of amphimixis appears when Some data on the outcomes of evolution at the R fluctuations of o are ~ 2 times wider than that locus with fluctuating selection acting on 10 traits are required for increased genetic variance, because this presented in Fig. 7. As with a single trait, an increased advantage depends on simultaneous allele substitu- range of optimum fluctuations favours free recom- tions, while even isolated substitutions increase the bination, while the period most favourable to it is 80. variance (compare Fig. 2c with figure 3 in KY-96). Even with such T, free recombination wins only when Fluctuations must be also frequent enough, thus selection changes so rapidly that the load is very high. causing a substantial load even in an amphimictic

population, because otherwise apomicts can adapt Fluctuating selection acting on many traits may using fresh mutations. However, if fluctuations are favour amphimixis more because in this case the too frequent the advantage of amphimixis disappears, movements of O can be effectively irreversible (e.g. because the population cannot respond to selection. circular), although each ot changes back and forth. Apparently, intermediate frequencies of fluctuation Thus, the trajectory of O may be free of 'returns', are most favourable to amphimixis (Fig. 2 c). The which are unavoidable with n = 1, which favour exact mode of selection and its fluctuations are not apomixis and linkage. However, we did not observe critical. (Figs. 2 c, 6) a better protection of amphimixis with Thus, our results are in a qualitative agreement with n > 1. In contrast, it was protected only when the those of Charlesworth (1993). However, his analytical genetic load was very high even in the amphimictic predictions should be used with caution, because he population. did not take into account the dramatic increase in the variance due to changing selection (KY-96). Although (ii) When does fluctuating stabilizing selection both deterministic and stochastic factors were involved maintain recombination! in our simulations, we believe that, because our populations were very large, the advantage of Under constant selection tight linkage is ultimately amphimixis and recombination we observed was favoured (see Zhivotovsky et al. 1994), although mostly deterministic. recombination can enjoy a temporary advantage if the In nature, the width of the fitness curve S1 usually population is initially far from the selection optimum exceeds a because: (i) the data suggest that for many (Bergman & Feldman, 1990). An allele causing free traits S ~ 10 S (see KY-96). If so, d > Vi^ + cr^2) implies Smith (1988, figure 3), who observed that the allele dp a. Thus, because fluctuations of o cause fluctua- that increases recombination rate won only under tions of [i with only a slightly smaller amplitude (Fig. wide fluctuations of O which caused wide fluctuation 2 a, b), if amphimixis is maintained by fluctuating of (i. Contrary to his opinion (p. 59), the lag load (see selection acting on one quantitative trait, the ranges Maynard Smith, 1978) that corresponds to the of changes of fitness optimum and of the mean value geometric mean fitness is a useful predictor of the of this trait must exceed its standard deviation. evolution of recombination. Under truncation selec- When selection acts on many traits, the compromise tion the immediate advantage of a higher variance strategy is absent when D > S. With the Euclidean requires the load (i.e. the proportion truncated) to be metric D = d\/n, and this requires S < d\/n. With above 0-5. However, the conditions where recom- n -> oo, S < a\/n would cause too high a load, because bination is selected for are less restrictive, because a the proportion of a multivariate Gaussian distribution higher variance makes selection more efficient, so that with the same variance S the range of changes in o(, 2d, must mutational deterministic mechanism, although the exceed 2

aileie or to a polymorphism (Fig. 5). In this case the The main falsifiable prediction of any ED hypothesis load would be lower with free recombination, which is that every genotype must experience frequent provides one more example of the violation of episodes of low fitness. There are no direct data Karlin-McGregor (1972) principle. Such poly- supporting this assertion, but currently it cannot be morphism in a similar ED model was first observed by ruled out. In addition, the ED scenario considered Korol & Preygel (1989) and Korol et al. (1994), while here requires that: Charlesworth (1993, equation 7) concluded that (1) Fluctuating selection is strong, causing a high intermediate ESS recombination rates are possible. A load, and usually effectively directional (even if similar polymorphism was described by Kondrashov formally stabilizing), because most of the time /t (1984, figure 5) in the mutational deterministic model. deviates substantially from o. In contrast, the data apparently suggest that quantitative traits are usually under weak stabilizing selection with ft close to o (Turelli, 1984). (iii) Stabilizing versus directional fluctuating selection (2) The genetic variance is much higher than it Contrary to the opinion of Maynard Smith (1980, would be under invariant selection, because of 1988), fluctuating stabilizing and directional selection frequent allele substitutions (compare figure 3 in KY- maintain amphimixis and recombination by the same 96 with Figs. 2 c and 4). mechanism (Charlesworth, 1993; Barton, 1995). Both (3) The mean values of the traits involved change are usually narrowing (non-narrowing directional at least by ~ a regularly, unless their number is very selection does not maintain amphimixis), and res- high, because the range of the optimum changes must toration of the variance improves the response to be ~ a and 0 otherwise, where a is between populations (see Hamilton et al. 1990), some trait value. Even if the switches between w1 and instead of being imposed by non-biological forces. w2 occur every generation, the amplitude of /i changes Still, any fluctuating selection can be presented as a would be ~ a, while the load exceeds 0-5. If fluctuating fitness function changing with time. The only directional selection acts on very many traits, the difference that biotic interactions could make is to maintenance of amphimixis is consistent with small ensure that selection possesses the parameters most changes in the mean values of individual traits (Crow, advantageous for amphimixis, but it is not clear why 1992). this should be the case. Environmental fluctuations, especially caused by biotic factors, are probably asynchronous in different places. Thus, under widely fluctuating selection we (iv) How plausible is the mechanism ? should find large differences between mean values of Any environmental hypothesis on the maintenance of quantitative traits and, even if these values temporarily amphimixis and recombination depends on con- coincide, between the frequencies of alleles at the loci ditionally beneficial genetic variability (Kondrashov involved, among different populations of the same & Turelli, 1992; Kondrashov, 1993; Judson, 1995). species. We are not aware of any data on geographical Theoretically, several factors can protect such varia- variability that cannot be explained by permanent bility (see KY-96), but we do not know yet how differences between conditions in different locations. common it is. Some of these factors can actually Thus, the ED hypothesis considered here leads to diminish the advantage of amphimixis, by helping several predictions that are not impossible to test apomicts to adapt. New mutations do so by creating experimentally. Before this is done, it is premature to new genotypes, while frequency-dependent selection decide what role is played by fluctuating selection in does so by preserving many clones. the maintenance of amphimixis and recombination.

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