Seasonally Fluctuating Selection Can Maintain Polymorphism at Many Loci

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This interest. and M.W.F., of conflict A.O.B., no M.J.W, declare authors and The paper; the on paper. input aspects the gave all wrote P.S.S. on input analyses; gave D.A.P. the and M.W.F., of A.O.B., simulations; and analyses performed oiac”,atog naypriua eeain n fthe 20). 19, of (16, one fittest generation, be over- might particular (“marginal homozygotes 18). any fitness heterozy- and in mean if although 17 geometric stable dominance”), refs. highest is see locus the single but have fitness a gotes 16, at mean (ref. polymorphism geometric fixation diploids, higher In to cannot a go generally have eventually alleles will and two one haploids, because In coexist (16). fit- time mean over geometric their nesses on depend selection fluctuating porally number large a at selection loci. should of organisms fluctuating many seasonally whereas polygenic, experience usually reproduction, such resource are Since survival. in traits abundant in life-history investment investment for with select sea- for may seasons seasons across stressful select example, trade-offs might For life-history supply 15). are ani- (14, there for sons availability tree Often, food between (13). in and changes mals in seasonal within fruiting to synchronized leading and often species, flowering is example, forests com- experi- For tropical predators, usually seasonality. populations of some tropical abundance ence Even the parasites. or in petitors, temporal or of availability, rainfall, type temperature, resource in particular multiple example, with a for seasonality, organisms experience heterogeneity: most year fact, per In generations (12). genotypes and (11) uhrcnrbtos ...... n ...dsge eerh M.J.W. research; designed D.A.P. and P.S.S., M.W.F., A.O.B., M.J.W., contributions: Author owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence whom To [email protected]. or se o ayseisadde o ufrfo rbesof problems from suffer not does models. previous and sat- species be many plausibly for may season- isfied condition by This polymorphism selection. fluctuating multilocus condition ally of general more maintenance a the obtain We for at genome. the polymorphism in maintain loci many simultaneously fluctu- seasonally to empiri- of selection recent potential ating by the reevaluate Inspired rarely. we findings, occurs cal this research that previous argued However, has polymorphism. mecha- maintain powerful to potentially a nism sea- year, is per selection generations fluctuating multiple their sonally have in they seasonality if popula- and, some maintains natural habitats, experience What in organisms observed is: Many variation tions? biology genetic evolutionary abundant in the question key A Significance ihdsrt eeain,teftso eoye ne tem- under genotypes of fates the generations, discrete With a alS Schmidt S. 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EVOLUTION PNAS PLUS Extending these results to the multilocus case is nontrivial, alone is not sufficient to reconcile evidence from population and, so far, only two cases are well-understood: (i) multiplicative genomics and quantitative genetics (38). Thus, we need to recon- selection across loci and (ii) temporally fluctuating selection on sider the potential of temporally fluctuating selection to maintain a fully additive trait. Under multiplicative selection in an infinite multilocus polymorphism. population with free recombination, the allele-frequency dynam- As explained above, the conditions for multilocus polymor- ics at a focal locus are independent of those at other loci. Thus, phism under seasonally fluctuating selection have been examined polymorphism is stable if heterozygotes have the highest geo- mostly in two narrow cases. Here, we examine a more general metric mean fitness, as in the single-locus case. However, devi- class of seasonal selection models with various forms of domi- ations from multiplicative selection appear to be the rule. In nance and epistasis. Using deterministic mathematical analysis particular, beneficial mutations often exhibit diminishing-returns and stochastic simulations, we show that multilocus polymor- epistasis (21–23). Additionally, there is the potential problem of phism is possible if the currently favored allele at any time is genetic load. Genetic load is commonly defined as the differ- sufficiently dominant, with dominance measured by using a scale ence between the population’s average fitness and the fitness of on which contributions across loci are additive. This mechanism, the fittest possible genotype. Lewontin and Hubby (1) noticed which we call “segregation lift,” can maintain polymorphism at that this value can become unsustainably high if there is strong a large number of loci across the genome, is robust to many heterozygote advantage at many loci. This was a conundrum for model perturbations, and does not require single individuals to the neoclassical school, which was worried that with high genetic have too many offspring. Depending on the parameter values, load, single individuals would have to produce an astronomically allele-frequency fluctuations can be large and readily detectable, large number of offspring. Others have dismissed this concern, or subtle and hard to discern. arguing, for example, that selection does not generally act on all loci independently or that only relative fitness differences within Basic Model the population are relevant, not fitness relative to some optimum We consider a diploid, randomly mating population in a season- genotype that might not even exist (24–27). However, debate ally fluctuating environment. While asymmetry in various model continues over whether genetic load should be an important con- parameters will be explored later, we start with a fully symmetric sideration (28, 29). model having a yearly cycle with g generations of winter followed The second previously studied scenario is seasonally fluctuat- by g generations of summer. The genome consists of L unlinked ing selection on a trait to which loci contribute additively (30, loci with two alleles each: one summer-favored and one winter- 31). These models generally assume additivity also within loci, favored allele. For a given multilocus genotype, let ns and nw such that the contribution of heterozygotes is exactly intermedi- be the number of loci homozygous for the summer and winter ate between the contributions of the two homozygotes. Tempo- allele, respectively, and nhet the number of heterozygous loci, rally fluctuating selection can then cause intermediate trait val- with ns + nw + nhet = L. ues to be best in the long run (32), i.e., select against variance In the basic model, loci are interchangeable in their effects in fitness. Effectively, this is stabilizing selection on the temporal (see Stochastic Simulations for a more general model), and the average. As such, it can generally maintain polymorphism at only fitness of a multilocus genotype can be computed as a function one locus (33, 34), or two loci if their effect sizes are sufficiently of ns , nw and nhet . In the simplest case, fitness depends only on different (35) or if they are closely linked (30). The reason is that ns + 0.5 · nhet in summer and nw + 0.5 · nhet in winter, i.e., half with multiple loci and additivity within and between loci, there the number of currently favored allele copies. To allow for domi- are multiple genotypes with intermediate phenotypes. For two nance effects, we generalize this simple scenario and assume that loci, for example, there is the double heterozygote (“heterozy- fitness in summer depends on the summer score gous intermediate”) and the genotype homozygous at both loci, but for alleles with opposite effects (“homozygous intermedi- zs := ns + ds · nhet [1] ate”). These genotypes may all have the same high fitness. How- ever, matings between heterozygous intermediates produce a and fitness in winter depends on the winter score range of different genotypes, some of which are less fit than their z := n + d · n . [2] parents. By contrast, matings between homozygous interme- w w w het diates only produce new homozygous intermediates. Homozy- The parameters ds and dw quantify the dominance of the cur- gote intermediates can therefore go to fixation and eliminate all rently favored allele in summer and winter, not with respect to polymorphism. fitness, but with respect to the seasonal scores zs and zw . Because In summary, multiplicative seasonal selection is a power- we are interested in whether temporally fluctuating selection can ful mechanism to maintain multilocus polymorphism, but the maintain polymorphism in the absence of other stabilizing mech- assumed independence across loci and the associated load call anisms, we only consider values of ds and dw between 0 and 1, into doubt its plausibility. On the other hand, selection on addi- and do not allow values >1, which would correspond to standard tive traits can maintain polymorphism at only a few loci. So heterozygote advantage. far, there has been little need for further exploration because The relationship between the seasonal score z (z = zs in sum- there were no clear empirical examples to challenge the view mer and z = zw in winter), and fitness, w, is given by a mono- that temporal heterogeneity rarely maintains variation. This is tonically increasing fitness function w(z). This function specifies now changing, however, as advances in sequencing technology the strength of selection and accounts for epistasis. With discrete allow detailed studies of genetic variation across time and space. generations, the allele-frequency dynamics at a focal locus are For instance, by sampling the same temperate population of driven by the relative fitnesses of the three possible genotypes at Drosophila melanogaster at several time points, Bergland et al. that locus—for example, the ratio of the fitness of homozygotes (36) detected seasonal allele-frequency fluctuations at hundreds and heterozygotes. We say that there is no epistasis if these ratios of sites in the genome. Many of the SNPs are also shared with and thus the strength of selection are independent of the number African populations of D. melanogaster and some even with the of other loci and their contributions to z. This is the case when sister species Drosophila simulans, indicating that some of them fitness is multiplicative across loci: may be ancient balanced polymorphisms. More generally, recent L L population genomic data appear to suggest that balancing selec- Y X tion contributes more to maintaining genetic variation than pre- w = wi ⇔ ln(w) = ln(wi ), [3] viously assumed (37) and that mutation-selection-drift balance i=1 i=1

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EVOLUTION PNAS PLUS detail below, appears to be the only way in which seasonally ﬂuc- tuating selection can maintain polymorphism under additivity (d = 0.5). Next, we explore whether deviations from additivity (d =6 0.5) can facilitate multilocus polymorphism. Under multiplicative selection, i.e., without epistasis, the conditions for polymorphism at one locus are not affected by the dynamics at other loci. Thus, given the ﬁtness values for individual loci (exp(d) for heterozygotes, exp(1) and exp(0) for currently favored and disfavored homozygotes, respectively, see text below Eq. 3), we can conclude that polymorphism is possible if exp(d)2 > exp(1) · exp(0) ⇔ d > 0.5. [6]

Fig. 3. Potential mechanistic underpinning for beneficial reversal of dom- That is, there must be a reversal of dominance with respect to z, inance. There is antagonistic pleiotropy for two traits, and the seasonal such that at any time the currently favored allele is dominant. scores for winter and summer are computed as weighted averages of traits 1 Now, we explore whether such a beneficial reversal of domi- and 2, with the relative importance of the two traits switching between sea- nance can also maintain polymorphism in the presence of epis- sons. The dashed line indicates the average of the two homozygote traits. If tasis. In each case, a necessary condition for polymorphism is the heterozygotes are closer to the fitter homozygote with respect to both that a population fixed for the fittest possible fully homozygous traits 1 and 2, there is a beneficial reversal of dominance at the level of the genotype can be invaded by mutants. As we have seen above, seasonal score, z. See SI Appendix, Fig. S2 for alternative scenarios. with synergistic epistasis (v 00 > 0), there are two fully homozygous genotypes with maximum fitness, the one with the summer allele at all loci and the one with the winter allele at all loci. In Deterministic Analysis both cases, the resident type has score L in one season and score In this section, we assume that population size is so large that 0 in the other season, whereas mutants differing in one posi- genetic drift does not play a role. We also assume that mutation have scores L − 1 + d and d. For mutants to invade, we tions are rare enough that the allele-frequency dynamics will thus need equilibrate before a new mutation arises at one of the L loci. This v(d) + v(L − 1 + d) > v(L) + v(0). [7] simple deterministic framework allows us to develop an intuitive understanding of the conditions for stable polymorphisms for Using our example class of fitness functions with positive epista- various genotype-to-fitness maps. The intuitions developed here sis, Eq. 5 with q > 1, we thus obtain the condition will then be checked and extended with stochastic simulations in d q + (L − 1 + d)q > Lq . [8] the next section. We will first confirm that the conditions under which season- The critical value of d, dcrit , at which extreme types become inva- ally fluctuating selection can maintain polymorphism are restric- sible, satisfies tive when contributions to the seasonal score z are additive q q dcrit dcrit − 1 within loci (d = ds = dw = 0.5 in Eqs. 1 and 2). Then, zs +zw = L + 1 + = 1. [9] ∗ L L for all possible genotypes, and the mean z over time is z = L/2. The long-term success of a genotype depends on its geometric For q = 2, dcrit takes values 0.707, 0.954, and 0.995, with 1, 10, mean fitness, or, equivalently, on the arithmetic mean of the and 100 loci, respectively. For q > 1 in general, dcrit approaches logarithm of fitness, v(z). Jensen’s inequality or simple geomet- 1 as the number of loci increases. To see this, note first that the ric considerations (Fig. 4) tells us that the arithmetic mean of condition in Eq. 9 is always fulfilled for d = 1 and thus dcrit ≤ 1. v(zs ) and v(zw ) for a given genotype will be smaller than or Thus, as the number of loci, L, goes to infinity, (dcrit − 1)/L equal to v(z ∗) if v 00 < 0 everywhere (Fig. 4A), equal to v(z ∗) becomes small, and we can approximate the second term on if v 00 = 0 (Fig. 4B), and larger than or equal to v(z ∗) if v 00 > 0 the left-hand side of Eq. 9 by a Taylor expansion around 1 (Fig. 4C). to obtain The interannual allele-frequency dynamics (e.g., from summer to summer or from winter to winter) with multiplicative fitness (v 00 = 0) and d = 0.5 are thus neutral. No balancing selection 00 ) z

emerges. With positive epistasis (v > 0), extreme types with ( A B C either only summer or only winter alleles have the highest geo- v metric mean fitness. Therefore, the population ends up in a state where all loci are fixed for the summer allele or all for the winter allele. With negative epistasis (v 00 < 0), the genotypes with Summer the highest geometric mean fitness are those with the same num- Winter ber of summer and winter alleles and thus zs = zw . There are Average always some genotypes heterozygous at one or more loci that fulfill this condition (heterozygous intermediates; Fig. 2). For an 0 L/2 L 0 L/2 L 0 L/2 L Logarithm of fitness, Logarithm of fitness, even number of loci, zs = zw is also true for genotypes homozy- Summer score, zs = L − zw gous for the summer allele at half of the loci and homozygous for the winter allele at the other half (homozygous intermediates). Fig. 4. The logarithm of summer fitness (red) and winter fitness (blue) and When one of the homozygous intermediates fixes in the popu- the average logarithm of fitness (gray) as a function of a genotype’s summer lation, it cannot be invaded by any mutant starting at small fre- score, zs. Assuming d = 0.5, the winter score is zw = L − zs, leading to the mirror symmetry around L/2. If the fitness function is concave on the quency (under the assumptions of the deterministic model; see SI logarithmic scale, intermediate types have the highest average log-fitness Appendix, section S1 for a detailed proof), and all polymorphism (A); if the fitness function is log-linear, then all types have the same average is eliminated. For an odd number of loci, homozygous interme- log-fitness (B); and if the fitness function is convex on a logarithmic scale, diates do not exist, and some polymorphism may be maintained, extreme types have the highest average log-fitness and thus the highest at least at one locus. This case, which we will examine in more geometric mean fitness (C).

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+ 1 + 1 + erae as decreases −1 sapoiaey1i h ubro oiis loci of number the if 1 approximately is d ) + 1 d d − · z crit hs h eutn eesr condition necessary resulting the Thus, . w L = q (L/2 L L 2 2+ (2 + · o eal) hti,freeyindivid- every for is, That details). for 4 twihhmzgu intermediates homozygous which at L/2 n letting and − d ihayexponent any with crit v + 1 − L 00 L swa,adtecniin for conditions the and weak, is (z − + 1 nrae n prahs0.5 approaches and increases L) d = ) 4 1 crit s ohbcuediminishing- because both d + ) + 1 00 L psai rudthe around epistasis L, −y O > = < ot nnt,w can we infinity, to go 1+ (1 w 2+ (2 d L ,tefits pos- fittest the 0), (L/2) 1 + 1 2 o fitness For 1. = 1 z L) 13 5 y ) 1 = L −2 2 h critical the , ihpositive with 2 2 L/2 . sgenerally is ! . decreases 2 . . nboth in [10] [13] [11] [12] aaee Explanation explored ranges the and parameters Parameter model of Overview 1. Table rwne lee o large For allele. winter or parameters, d dominance of different three case for trajectories the frequency For loci. of 6). (Fig. only is rate and mutation loci by of influenced number weakly the with decreases which value, parameter, critical Methods) dominance and the if (Materials emerges selection balancing inde- effective is expected, behavior exponent, This the 5). zero of (Fig. pendent approaches below finally from and neutrality) value, (effective absolute numbers, in odd overall balanc- for of even decreases a negative strength becomes effective at eventually the loci selection S4). increases, ing Fig. loci two of Appendix, or number (SI one the frequency As only intermediate but at strength positive, fluctuate effective time is the become loci, selection of to balancing numbers tend of odd alleles small rare For rarer. that even indicating sim- both the negative, from where is the estimated ulations situations 2), selection balancing (Fig. i.e., of exist strength loci, effective intermediates of heterozygous numbers and even homozygous For 5). within allele (Fig. contributions rare from (d additive a expected arguments, fast loci As theoretical cycle. how yearly above and full the a whether over us frequency in tells increases which S2), Methods section and (Materials selection” balancing of individuals still between that value number population. possible offspring the smallest in in the differences three, for of simu- allows cap supplementary a role run the with also understand lations to better we To set capping, again. is gametes. offspring-number drawn fitness individu- be 10 of its cannot of most it number, parent that at that so a reached contribute 0 has as i.e., individual times generation, an 10 next Once indi- most the each at in model, this drawn als In be load. can genetic of vidual relevance the assess to computa- by muta- approximated larger rates. with well tion populations smaller be manageable may more values tionally rates on the mutation only than depend small smaller processes product genetic rates population the mutation many and here, used larger often rate are [w mutation effects explored. an the mutation dominance than ranges to gives and larger the relative selection 1 strong and scenarios, Table are parameters simulated size. most model simulations population In the supplementary in of run changes overview also seasonal but with constant, size size population lation the and ation probability mutation symmetric model. multiplicative the for simulations Eq. some in run epistasis we synergistic for one the than smaller L N g d µ y o small For . rmnwo,w ilfcso cnro ihlrenumbers large with scenarios on focus will we on, now From strength “effective an estimate we output, simulation the From model “capped” a design we model, basic the to addition In the are simulations stochastic the in parameters Additional 0 = r o odcv omliou polymorphism multilocus to conducive not are .5) ubro loci of Number 1–20 size Population 2g has year a season; per generations of Number e generation per uainpoaiiyprall oy10 parameter Dominance copy allele per probability Mutation xoeto h tesfnto Eq. function fitness the of Exponent d N ahlcsi lotfie ihrfrtesummer the for either fixed almost is locus each , µ eg,rf 5.Tu,lreppltoswith populations large Thus, 45). ref. (e.g., generations 100 y ftefins ucinEq. function fitness the of , d l oiflcut tintermediate at fluctuate loci all , oi i.7soseapeallele- example shows 7 Fig. loci, .Atog aua populations natural Although µ]. N µ egnrlyke popu- keep generally We . 0 e leecp e gener- per copy allele per (z d ) slre hnacertain a than larger is , NSEryEdition Early PNAS and d 4 and − 0.5–4 ag explored Range 1–500 100–10,000 0–1 naddition, In 9. 0.5 IAppendix, SI −6 lo as Also, 4. r much are − | 10 f10 of 5 −4

EVOLUTION PNAS PLUS the fittest individual in the population was on average 1.2 times fitter than the least fit individual and 4.9 times fitter for y = 4 (see SI Appendix, section S4 for a supporting heuristic analysis). When the offspring-number cap is set to three, substantial quantitative differences between uncapped and capped simulations are seen (SI Appendix, Fig. S9). However, 0.5 remains the critical dominance parameter. This result also holds for the multiplicative model (SI Appendix, Fig. S10), but with otherwise larger differences between capped and uncapped model versions, even with an offspring-number cap of 10. Also, the retardation factor for the multiplicative model is often below one, even when the effective strength of balancing selection is positive. The reason appears to be that there is larger variance in fitness under the multiplicative model (SI Appendix, Fig. S11) and that fluctuations are sometimes so large that alleles go to fixation (SI Appendix, Fig. S12). Both of these effects are weakened by offspring number capping, so that the capped multiplicative model behaves more similarly to the diminishing-returns model (SI Appendix,

Fig. 5. Effective strength of balancing selection (be in Eq. 14 in Materi- Fig. S10). als and Methods) in the additive case (d = 0.5) as a function of the num- Additional simulations for the diminishing-returns model sug- ber of loci. Solid lines indicate means and dashed lines indicate means ± gest that the finding of stable multilocus polymorphism for two standard errors. Simulations are always run for successive odd and even d > dcrit ≈ 0.5 still holds under various forms of asymmetry, numbers. N = 1,000, g = 15, µ = 0.0001. e.g., when one season has more generations than the other (SI Appendix, Fig. S13); when the exponent of the fitness function, y, differs between summer and winter (SI Appendix, Fig. S14); and frequency. The critical dominance parameter, dcrit , with 100 loci when there are seasonal changes in population size (SI Appendix, is close to 0.5, independently of the exponent of the fitness func- Fig. S15). To explore the effects of asymmetry in the dominance tion, y (Fig. 8A). For d < 0.5, i.e., if the currently favored allele is parameters, we fixed the winter dominance parameter, dw , at 0.4 recessive, the effective strength of balancing selection is negative and varied the summer dominance parameter, ds . Stable poly- and polymorphism is unstable (Fig. 8A). As d increases beyond morphism then arises for ds > 0.6 (SI Appendix, Fig. S16), sug- 0.5, i.e., as the currently favored allele becomes more dominant, gesting that stable polymorphism requires an arithmetic mean effective balancing selection becomes stronger (Fig. 8A). Both dominance parameter >0.5. Compared with the diminishing- the stabilizing and destabilizing effects increase with increasing returns model, the multiplicative model seems less robust to exponent y (Fig. 8A). asymmetry (SI Appendix, Fig. S17). A tendency for rare alleles to increase in frequency does not With an increasing number of loci under the diminishing- guarantee that the average lifetime of a polymorphism is larger returns model and with d > 0.5, the strength of balancing selec- than under neutrality (42, 46). This is particularly interesting for tion, the retardation factor, and the magnitude and predictabil- fluctuating selection regimes with positive autocorrelation where ity of fluctuations all decrease (Fig. 9). Population size hardly alleles regularly go through periods of low frequency (42). We influences effective strength of balancing selection and effective therefore compute the so-called retardation factor (46), the average lifetime of a polymorphism in the selection scenario relative to the average lifetime under neutrality (see SI Appendix, section S2 for detailed methodology). The results for 100 loci are consis- tent with those for the effective strength of balancing selection: For d > 0.5, polymorphism under segregation lift is lost more slowly than under neutrality (Fig. 8B). To quantify seasonal fluctuations, we compute an effective selection coefficient (Materials and Methods and SI Appendix, section S2). We also compute the predictability of fluctuations as the proportion of seasons over which the allele frequency changes in the expected direction, e.g., where the summer- favored allele increases over a summer season. Both the effective selection coefficient and the predictability of fluctuations have a maximum at intermediate values of d and increase with increasing exponent y of the fitness function Eq. 4 (Fig. 8 C and D). For even higher values of d, fluctuations are not as strong, presum- ably because heterozygotes are fitter, and therefore more copies of the currently disfavored allele enter the next generation. Also, effective strength of balancing selection, effective selection coefficient, and predictability of fluctuations increase with the number of generations per season (SI Appendix, Fig. S6). With an offspring-number cap of 10, the results for the capped model generally match the results for the uncapped model in all Fig. 6. Critical value of the dominance parameter, dcrit , such that the effec- d > 0.5 tive strength of balancing selection (be in Eq. 14 in Materials and Methods), respects, especially for (Fig. 8). It appears that the cap- is positive (stable polymorphism) if d > d , and negative (unstable poly- ping mechanism only rarely takes effect because of relatively nar- crit morphism) if d < dcrit . Symbols represent means across replicates, and lines row distributions of the seasonal score, z, within a generation (SI represent averages ± two standard errors. Since the pattern for odd num- Appendix, Fig. S7), and consequent relatively low variance in fit- bers of loci is more complex (Fig. 5), only even values for the number of loci ness (SI Appendix, Fig. S8). For example, for d = 0.7 and y = 0.5, are included here. N = 1,000, y = 2, g = 15.

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EVOLUTION PNAS PLUS AB between epistasis and strength of selection, future work should also consider more general fitness functions allowing for various combinations of epistasis and selection strength. Whenever there is balancing selection at a large number of loci, genetic load is a potential concern. In the case of segregation lift with diminishing-returns epistasis, however, genetic load does not appear to play an important role. The results for our capped model closely match the results for the original, uncapped model. Apparently, independent segregation at a large number of unlinked loci leads to relatively small variance in seasonal scores within the population and, together with CD the diminishing-returns fitness function, to relatively small variance in fitness. With a much smaller offspring number cap or with multiplicative selection, differences between the capped and uncapped model are more substantial, but even then, balancing

Fig. 9. Influence of population size and the number of seasonally selected loci on effective strength of balancing selection (A; be in Eq. 14, Materials and Methods), retardation factor (B), magnitude of fluctuations (C; se in Eq. 15, Materials and Methods), and predictability of fluctuations (D). Symbols indicate averages across replicates and lines in A, C, and D indicate means ± two standard errors (in C and D, standard errors are too small to be visible). Lines in B simply connect maximum-likelihood estimates obtained jointly from all replicates. Note that in B, some points are missing because the rate of loss of polymorphism was too small to be quantified. d = 0.7, y = 4, g = CD 15, µ = 10−4. Bal. sel., balancing selection; exp., expected.

score. As discussed above, there are several potential mechanisms that can plausibly produce such changes in dominance. Moreover, the required changes are small. Unfortunately, there have been only few relevant empirical studies so far. For instance, in the copepod Eurytemora affinis, there appears to be beneficial reversal of dominance for fitness across salinity conditions (47). In experimental Drosophila populations, changes in dominance for gene expression across environments appear to be common (48). More empirical and theoretical work is required to find out how common changes in dominance are, in particu- EF lar on the relevant scale of the seasonal score. However, even if the required changes in dominance are rare on a per-site basis and the vast majority of polymorphisms are lost under fluctuating selection, there may still be many sites in the genome with appropriate reversal of dominance, and, as we show, those are then the ones that we should see as seasonally fluctuating polymorphisms. In the focal diminishing-returns scenario, the conditions for stable polymorphism via segregation lift are surprisingly robust to changes in the mutation rate (Fig. 6) and to asymmetries in number of generations, strength of selection, or population size between summer and winter (SI Appendix, Figs. S13–S15), appar- Fig. 10. Stability of polymorphism and detectability of allele-frequency ently more so than under the multiplicative model (SI Appendix, fluctuations when parameters vary across loci and seasons. (A) Snapshot Fig. S17). When the dominance parameter differs between sum- of allele-frequency trajectories for stable polymorphisms in one simulation mer and winter, polymorphism is generally stable at those loci run. (B) Average number of stable polymorphisms as a function of the total whose average dominance parameter across seasons is >0.5 (Fig. number of loci for different population sizes. (C and D) As in A and B, 10F and SI Appendix, Fig. S16). Segregation lift is also robust to but only for polymorphisms that are also detectable. (E) Winter effect size, variation in effect sizes and dominance parameters across loci ∆l,w , vs. summer effect size, ∆l,s, for stable and detectable and only sta- (Fig. 10). In reality, the strength of seasonality likely varies in ble polymorphisms. The plot shows pooled results over ten simulation runs with independently drawn parameters. Oval isoclines indicate the shape of space and time, which could make the maintenance of polymor- the original sampling distribution, with 75% of the sampling probability phism by segregation lift even more robust (SI Appendix, Fig. mass inside the outermost isocline. (F) Corresponding dominance parame- S2B and ref. 40). Future work needs to explore whether segrega- ters (see also SI Appendix, Fig. S19). The dominance parameters were orig- tion lift is robust also to linkage between selected loci. Since our inally drawn from a uniform distribution on the unit square. Parameters: diminishing-returns fitness function has a particular relationship y = 4, g = 10, µ = 10−4 and in A, C, E, and FN = 10,000, L = 100.

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EVOLUTION PNAS PLUS ACKNOWLEDGMENTS. For helpful discussion and/or comments on the were performed on Stanford’s FarmShare Cluster and on the Vienna Scien- manuscript, we thank Michael Desai, Joachim Hermisson, Oren Kolodny, tiﬁc Cluster. M.J.W. was supported by fellowships from the Stanford Center Mike McLaren, Richard Nichols, Pleuni Pennings, Jitka Polechova,´ and mem- for Computational Evolutionary and Human Genomics and from the Aus- bers of D.A.P.’s laboratory, as well as two anonymous reviewers. Simulations trian Science Fund (M 1839-B29).

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