## Abstract

Frequency-independent selection is generally considered as a force that acts to reduce the genetic variation in evolving populations, yet rigorous arguments for this idea are scarce. When selection fluctuates in time, it is unclear whether frequency-independent selection may maintain genetic polymorphism without invoking additional mechanisms. We show that constant frequency-independent selection with arbitrary epistasis on a well-mixed haploid population eliminates genetic variation if we assume linkage equilibrium between alleles. To this end, we introduce the notion of frequency-independent selection at the level of alleles, which is sufficient to prove our claim and contains the notion of frequency-independent selection on haploids. When selection and recombination are weak but of the same order, there may be strong linkage disequilibrium; numerical calculations show that stable equilibria are highly unlikely. Using the example of a diallelic two-locus model, we then demonstrate that frequency-independent selection that fluctuates in time can maintain stable polymorphism if linkage disequilibrium changes its sign periodically. We put our findings in the context of results from the existing literature and point out those scenarios in which the possible role of frequency-independent selection in maintaining genetic variation remains unclear.

- frequency-independent selection
- genetic variation
- fluctuating selection
- linkage disequilibrium
- epistasis

THERE is a general understanding in the population genetics community that constant frequency-independent selection on a well-mixed haploid population eliminates genetic variation. Selection is called frequency independent if the fitness of a genotype does not depend on the frequencies of the other genotypes in the population. If constant selection acts on a haploid population, each haploid genotype (haplotype) has an externally determined fitness value. Hence, in the absence of recombination and mutation, it is clear that the haplotype with the highest fitness replaces the other haplotypes (Bürger 2000, Chap. I.9). With multiple recombining loci and epistasis between alleles, it is much less evident that frequency-independent selection on all haplotypes must reduce the population to a single haplotype. Even more, it remains an open task to identify and characterize the conditions under which frequency-independent selection which fluctuates in time may maintain genetic variation.

The matter of debate concerns an idealized population, under the sole influence of selection and recombination. Understanding the action of natural selection is crucial and justifies the analysis of pure selection models (*cf*. Bürger 2000, and references therein). We therefore neglect direct sources of genetic variation (mutation); spatial heterogeneities in selection that may protect alleles in spatial refuges (Strobeck 1979); and factors generating frequency-dependent fitness, *e.g.*, fluctuating selection in combination with overlapping generations (Ellner and Sasaki 1996), or with explicit population regulation (Dean 2005) and other ecological mechanisms. Similarly, a recent study (Gulisija *et al.* 2016) showed that the presence of a modifier of phenotypic plasticity may shelter alleles at selected loci from adverse conditions (termed a “genomic storage effect”) and thus lead to a protected polymorphism under temporally varying conditions. Here, however, we focus directly on fitness as the only trait being modulated. Furthermore, we consider haploid populations, since the random combination of gametes into diploid genotypes may induce frequency-dependent selection on alleles, even if the fitnesses of the diploid genotypes are frequency independent: with heterozygote advantage, even constant selection on diploids is capable of maintaining genetic variation. Similar mechanisms apply for diploid systems under fluctuating selection, which can lead to effective heterozygote advantage in terms of geometric mean fitnesses over time (Haldane and Jayakar 1963; Hoekstra 1975; Nagylaki 1975).

Mathematically precise arguments for the erosion of genetic variation under constant frequency-independent selection are scarce. Various publications cover special cases of fixed numbers of loci and alleles (*e.g.*, appendix S5 by Bank *et al.* 2012). For arbitrary numbers of loci and alleles, Kirzhner and Lyubich (1997) showed that constant selection on haploids leads to genetic monomorphism in two extreme cases: strong linkage between loci allowing for epistasis between genes, and arbitrary linkage between loci, but assuming that contributions of individual alleles to fitness are additive. More recently, Mehta *et al.* (2015) showed that the discrete-replicator dynamics of coordination games always converges to pure Nash equilibria. It follows as a special case that constant haploid selection does not support stable polymorphism under the assumption of linkage equilibrium between loci. However, their intricate proof is cast in the framework of game theory and computer science, and thus may not be accessible to readers with a background in evolutionary biology.

Under temporally fluctuating, frequency-independent selection on haploids, Kirzhner *et al.* (1994) showed numerically that stable polymorphism can be maintained with two diallelic loci, at least for a very narrow range of parameters. Their exemplary gene-frequency dynamics with periodic fluctuations exhibit cyclically converging trajectories, whose period is typically much longer than the period of the selection coefficients. Their examples require strong selection such that gene frequencies change drastically between successive generations. Thus, the parameters needed for maintaining polymorphism may not be representative for natural populations, where frequency changes due to selection are typically small and selection is weaker than recombination, such that the latter efficiently breaks up associations between loci (Nagylaki 1974, 1993). In particular, it has not yet been shown if fluctuating selection may maintain genetic variation in the continuous-time limit, where selection and recombination are both weak.

We consider a well-mixed haploid population with nonoverlapping generations and frequency-independent selection on multiple loci, each with any number of alleles. We allow arbitrary epistasis between alleles. This includes nonlinear selection on polygenic traits, where epistasis in fitness may emerge even if the underlying genes contribute additively to the trait. First, we present a simple yet rigorous proof that, at linkage equilibrium, no genetic variation can be maintained by constant frequency-independent selection alone. We note that the result does not follow from frequency-independent selection at the level of genotypes as defined above, since frequency dependence in the fitnesses of alleles may be induced, *e.g.*, in diploid organisms. Instead, we provide a formal criterion for frequency-independent selection at the allelic level that is sufficient for proving the loss of genetic polymorphism. Our argument applies to haploid populations when the alleles are at or close to linkage equilibrium, which is the case when selection is weak relative to recombination (Nagylaki 1993). Next, we show numerically that stable equilibria are highly unlikely when selection and recombination are weak but of the same order, so that linkage may be strong. Finally, we provide an example of weak and periodically fluctuating frequency-independent selection and recombination between two loci that maintains genetic variation at a stable equilibrium. We find that linkage disequilibrium is required for stable polymorphism, though its magnitude can be small. We conclude by summarizing and discussing our results, and by laying out the remaining open questions concerning the conundrum of frequency-independent selection and genetic variation.

## Model

We assume that there are *L* genetic loci. Locus *i* has possible allelic variants and we write for the total number of alleles across all loci. Denote the *k*-th allele at locus *i* by and the frequency of allele by Furthermore, we write for the vector of allele frequencies at locus *i* and collect the allele frequencies at all loci in a vector

The mean fitness of the population is obtained by averaging the fitnesses of the genotypes over their frequencies. Similarly, the marginal fitness of an allele is defined as the mean fitness of all genotypes containing that allele. Consequently, the mean fitness of the population can be seen as an average over the marginal fitnesses of alleles at any given locus,(1a)Note that the choice of locus *i* to average across does not play a role here, *i.e.*, the sum evaluates to the same value for any locus *i*.

We say that selection on haploids is frequency independent (at the level of alleles) if the marginal fitness of an allele does not depend on the frequencies of alleles at the same locus. That is, for every choice of alleles *j* and *k* on any locus *i*, we have(1b)This definition encompasses the case of frequency-independent selection on haploids, where every haplotype has a fixed fitness value (Appendix A; Figure A1). However, its scope of application is broader, since it does not require a certain ploidy level and it allows for certain kinds of frequency-dependent interactions between loci.

Importantly, the marginal fitnesses of alleles at locus *i* generally depend on the allele frequencies at the other loci, leading to nontrivial interactions between alleles. From Equations 1a and 1b, we may derive the familiar identity(2)(*cf*. Wright 1937). Hence, the derivative of mean fitness with respect to an allele yields the marginal fitness of that allele under frequency-independent selection.

### Data availability

The authors state that all data necessary for confirming the conclusions presented in the article are represented fully within the article.

## Results

### Constant frequency-independent selection at linkage equilibrium eliminates genetic variation

Assume that the *L* loci are at linkage equilibrium at all times. This is a restrictive assumption, but is valid in the limit where recombination breaks up linkage disequilibria between loci quickly relative to the rate of selection. At linkage equilibrium, the frequency of each haplotype is the product of the frequencies of its constituent alleles. Consequently, it suffices to follow the dynamics of allele frequencies instead of those of the genotypes. We consider the standard selection dynamics from population genetics in discrete time (generations), neglecting all other evolutionary forces (*e.g.*, mutation), and only describing the effect of selection. In particular, there is no genetic drift, *i.e.*, the population is practically infinite and the dynamics are deterministic. The frequency of allele *k* on locus *i* in the next generation is calculated from the frequencies in the previous generation as(3)To prove that constant frequency-independent selection at linkage equilibrium eliminates genetic variation, we consider the dynamics (3) and assume frequency-independent selection at the level of alleles as defined in Equation 1b. Our argument consists of three parts. First, the selection dynamics converge to the set of its equilibrium points, *i.e.*, there is no complicated dynamical behavior. This result has been established for a wide range of (mostly diploid) selection models, *e.g.*, for additivity (Lyubich 1992), weak epistasis, and weak selection (Nagylaki *et al.* 1999). It is a consequence of the mean fitness increasing steadily under selection, which (assuming linkage equilibrium) is implied by Fisher’s fundamental theorem of natural selection (Fisher 1930) and by Wright’s selection gradient formula (Wright 1937). Note that fitness may decrease when recombination breaks up linkage disequilibria (Bürger 2000, Chap. II.1). For our specific setting with linkage equilibrium, however, a commonly used argument that applies the inequality of Baum and Eagon (1967) (a method noted by Nagylaki 1977) shows that all trajectories of the dynamics (3) converge to the set of their equilibria (Appendix B). Since the haploid selection dynamics can be seen as a special case of the diploid dynamics (*cf*. Bürger 2000, p. 30), the proof may be extended to loose linkage, analogous to Nagylaki *et al.* (1999).

Second, we establish that every polymorphic equilibrium of the selection dynamics is generically unstable. A polymorphic equilibrium of Equation 3 is a vector of only nonzero entries that satisfy From standard linear stability analysis, the stability of a polymorphic equilibrium is determined by the eigenvalues of the Jacobian matrix (see Appendix C for details). If all eigenvalues of have a modulus less than one, the equilibrium is asymptotically stable; if the modulus of a single eigenvalue is more than one, it is unstable. In the case when the modulus of the largest eigenvalue equals one, further analysis would be needed. Neglecting this degenerate case, a simple dimensionality argument shows that any polymorphic equilibria of the dynamics (3) under frequency-independent selection, Equation 1b, are unstable (Theorem C1 in Appendix C).

Due to the first two parts, all trajectories converge to the boundary of the state space of the dynamics (3). Hence, the selection dynamics cannot maintain all alleles in the population; some of them are lost, some loci may become monomorphic, and the set of possible genotypes is reduced. Thus, third, we repeat the argument for the remaining genotypes by restricting the system to the loci that are still polymorphic, thereby iteratively eliminating alleles from the population. This procedure ends when all loci have become monomorphic, *i.e.*, when only a single genotype is left. Therefore, in the absence of mechanisms creating variation, constant frequency-independent selection eventually deprives the population of all its genetic variation.

### Constant frequency-independent selection with weak selection and recombination rates eliminates variation

We have shown that when selection is much weaker than recombination, so that the population is close to linkage equilibrium, constant frequency-independent selection cannot maintain variation. At the opposite extreme, when selection is much stronger than recombination, we know that the fittest haplotype will fix. In the intermediate case, where epistasis and recombination are of the same order, can constant frequency-independent selection maintain variation? To simplify matters, we assume that selection and recombination are weak, but of the same order, so that we can assume continuous time.

Appendix D gives an expression for the Jacobian matrix of the continuous-time dynamics in terms of the equilibrium genotype frequencies and the selection coefficients for each haplotype. Crucially, the equilibrium frequencies uniquely determine the selection coefficients, which are proportional to the recombination rates. Necessarily, linkage equilibrium implies neutrality: both epistasis and directional selection on alleles must be zero. Linkage disequilibrium requires epistasis, and directional selection must be set to the value that keeps allele frequencies constant.

We have not found a way to show that the Jacobian matrix necessarily and in general implies instability. However, we have calculated eigenvalues numerically for random genotype frequencies, for up to six loci. Table A1 summarizes the leading eigenvalue, and the number of positive eigenvalues, for loci; calculations become prohibitively slow for >6 loci, since they involve calculating a matrix. Results are based on drawing genotype frequencies uniformly from the space of genotype frequencies. This typically implies strong linkage disequilibrium, and hence epistasis of the same order as the recombination rate. Moreover, if some genotype is by chance very rare, the corresponding selection coefficient is very negative. The leading eigenvalue *λ* is typically of the same order as the recombination rate between adjacent loci, implying that variation is lost over the same timescale as recombination. However, with two loci, the leading eigenvalue may by chance be extremely small, in which case variation would be lost extremely slowly. These results show that stable equilibria are at best rare.

### Weak, periodically fluctuating, frequency-independent selection may maintain stable polymorphism in a two-locus model

#### Specification and intuition:

We now consider two linked loci with two alleles each. The recombination rate between the two loci is For convenience, we denote the alleles at the first locus by and and at the second locus by and Thus, we have four possible genotypes with frequencies and We assume a symmetric fitness model, in which the alleles *A* and *B* provide an additive selective advantage and there is an epistatic coefficient *ɛ* in genotypes consisting of an upper- and a lower-case allele, see Table 1 and Table 2.

Instead of following the four genotype frequencies, we may describe the temporal dynamics of selection by the frequencies and of the alleles *A* and *B*, and the linkage disequilibriumDue to the symmetry of the parameters, however, it is useful to consider the mean *P* and difference *δ* of the upper-case allele frequencies, *i.e.*, we write and We assume that selection and recombination are both weak, *i.e.*, we consider the limit where *s*, *ϵ*, and *r* go to zero simultaneously. Then, the dynamics can be approximated by a set of differential equations in continuous time that may be written as (*cf*. *Formulation of the Dynamics* in Appendix E)(4a)(4b)(4c)Note that the use of all three parameters, *s*, *ϵ*, and *r*, is redundant in this limit since we may always absorb any one of them into the time variable *t* by proper rescaling. For clarity, however, we explicitly denominate all parameters. Consequently, the dynamics remain unchanged if we replace and for any

To further exploit the symmetry in the fitness coefficients we imposed above, we look for equilibria around where the frequencies of alleles *A* and *B* are approximately equal. If additionally the alleles are at linkage equilibrium (), the dynamics of *P* is inverted in time if we replace *s* and *ϵ* by and Thus, fixing *s* and *ɛ* and periodically applying Equation 4a with parameters and for *τ* time units each leads to a trajectory that traces itself back to its initial point. This trivially maintains polymorphism, yet in a degenerate way: perturbations tip the balance of forces and lead to the fixation of one of the alleles at each locus. However, as we see below, including the dynamics of linkage disequilibrium *D* may stabilize trajectories and thus lead to protected polymorphism.

To obtain an intuitive understanding of how stability may come about, assume that it suffices to consider stability in the direction of *P* only. Then, the stability of a trajectory of *P* is given by the sign (negative sign implies stability) of(5)where the expectation is taken across a period of selection of length Note that besides the variables *P*, *D*, and *δ*, the parameters *s* and *ϵ* are also functions of time. We conclude from this expression that including the dynamics of *D* explicitly may stabilize trajectories of *P* if linkage disequilibrium *D* and epistasis *ϵ* correlate positively (). Equation 4c indicates that the latter may be the case at least for small *D*, when

#### Stable polymorphism in the limit of rapid fluctuations:

We assume that selection fluctuates through the course of two seasons of equal length *τ*. In the first season, the intensities of selection and epistasis are given by constant *s* and *ɛ*, see Table 2. In the second season, selection and epistasis simultaneously change their signs, yet maintain their moduli. We further assume the consistency of this assumption with our results will be affirmed below. As a consequence, we may neglect terms containing in Equation 4. Thus, the equations for *P* and *D* become independent of *δ*.

Finally, we assume quasi-linkage equilibrium, *i.e.*, that is at its equilibrium given *P* at all times. This quasi-steady-state assumption is only approximate, but the absolute error in Equations 4a and 4b is likely to be small, at least if *D* is small itself. By letting the recombination rate *r* be large relative to *s* and *ϵ*, the latter is satisfied and we may neglect terms containing From Equation 4c, we then have(6)for Hence, in particular, *D* correlates positively with *ϵ*.

We now iterate the two seasons as described in Table 3 with a very short duration, each, so that we may approximate the solution to the differential Equation 4 by a single step of the Euler method. For season 1 and the equation for *P*, this means

where Analogously, the expression for season 2 is To apply the two seasons in sequence, we thus need to calculate Neglecting terms of order this yields(7)Equating this to zero produces a polymorphic equilibrium at given that

Assuming that *D* is determined by Equation 6, we thus have an equilibrium at and (the latter is evident from Equation 4b). Since for small *δ* the equation for *P* is independent of *δ* (see above), the Jacobian matrix at this equilibrium becomes triangular, hence its eigenvalues that determine stability can be read from its diagonal. Thus, stability of *P* and *δ* can be assessed separately. For *P*, linearizing the right-hand side of Equation 7 yields a negative eigenvalue:(8)Deriving the dynamics for similarly to Equation 7 and taking the corresponding derivative shows that the second eigenvalue is negative as well: Thus, we have a stable polymorphism at and In particular, the above assumption is justified. Hence, with two seasons alternating rapidly, the selection dynamics converge to a polymorphic state where the alleles *A* and *B* are approximately equally abundant.

#### Stable oscillations in the symmetric case:

The previous section shows that genetic variation can be maintained in the limit of rapid alternations of the two seasons, assuming quasi-linkage equilibrium. In this section, we numerically demonstrate that protected polymorphism is possible for the full dynamics, Equation 4, with a finite duration of seasons.

Assume that both seasons last for time units, and that *s*, *ϵ*, and *r* in the two seasons are given by Table 3. This conforms to the symmetric situation of the previous section. We iterate the dynamics (4) through multiple seasons for various initial conditions. Initially, *δ* rapidly decays and converges to The variables *P* and *D* enter periodic oscillations around the point and (see Figure 1), as predicted by the above analysis. Their limit may be a global attractor of the system, since it is approached from all initial conditions we tested, see *Stability of a Periodic Trajectory Under Fluctuating Selection* in Appendix E. The correlation between *ϵ* and *D* across one cycle of selection can be calculated to be about This is found to be the main contribution to the negativity of Equation 5, indicating that these correlations secure the stability of the trajectory.

The stability of the equilibrium trajectory can be confirmed numerically by linearizing a suitable Poincaré map, see *Stability of a Periodic Trajectory Under Fluctuating Selection* in Appendix E for details. Figure 2 shows the leading eigenvalue of this linearization as a function of the recombination rate *r*. Its value is a measure of the stability of the limit trajectory; the further it is below one, the larger the perturbations the system can sustain without the trajectory losing its stability. Interestingly, there is an intermediate value of *r* for which stability is greatest. Since the variation in linkage disequilibrium *D* scales with *r* (larger *r* keeps *D* closer to zero), this implies that intermediate linkage disequilibrium is most likely to lead to stable polymorphism. When recombination dominates, the alleles remain at linkage equilibrium and polymorphism is maintained in the degenerate way discussed above for (the leading eigenvalue equals one). Without recombination (), we observe that the dynamics (4) similarly become symmetric with respect to replacing and the same logic applies. The latter is in concordance with the literature about fluctuating selection on a single haploid locus which predicts the fixation of the allele with the highest geometric mean fitness over time (Felsenstein 1976). Thus, intermediate linkage disequilibrium is required to maintain stable polymorphism in our example.

#### Stable oscillations in asymmetric cases:

Our analysis of the leading eigenvalue confirms stability in two ways. First, the periodic trajectory is dynamically stable with respect to perturbations of the variables, *e.g.*, due to genetic drift. Second, the model is structurally stable because of its continuous dependence on the model parameters. This implies that a stable periodic trajectory continues to exist under deviations from our reference parameters, even if they break the symmetry between the two seasons we imposed earlier. In particular, this concerns variation in the selection or epistatic coefficients of one or both seasons (*e.g.*, a systematic bias such that the mean selection coefficient is nonzero), random perturbations to *s* and *ɛ* between seasons, and variation in the durations of seasons.

For example, consider two seasons as described in Table 4. Note that we chose a higher recombination rate of to afford reasonably large perturbations of the parameters relative to the previous example (compare Table 3). For initial conditions close to and the dynamics converge to a stable trajectory as shown in Figure 3. This trajectory can be shown to be locally stable by the same techniques we used in the symmetric case. However, stability is not global; initial conditions with appreciable converge toward the monomorphic equilibria given by and Note that rescaling time during one of the seasons corresponds to rescaling *s*, *ϵ*, and *r* in this season. Thus, this example also shows that stable polymorphism is possible if the two seasons do not have the same duration.

Stable polymorphism can also be maintained if the symmetry between the two loci is perturbed, *i.e.*, if the two genotypes and have different fitness values. This can be parameterized by assuming two epistatic coefficients, for these genotypes. In *An Asymmetric Example of Stable Polymorphism* in Appendix E (Table A2; Table A3), we provide a numerical example for stable polymorphism under this asymmetry with substantial differences between and

## Discussion

When discussing the inability of frequency-independent selection to maintain genetic variation, population geneticists implicitly refer to constant frequency-independent selection at the allelic level or on haploid populations, rather than to the broader definition of frequency-independent selection at the level of genotypes. For example, consider constant overdominance (heterozygote advantage) at a single diploid locus with alleles *A* and *a*. In this selection regime, the fitnesses of the three possible genotypes and are typically scaled as 1, and respectively, where and These values are independent of the genotype frequencies, so that selection is frequency independent at the level of genotypes. Nevertheless, there is genetic polymorphism at equilibrium under selection; under the dynamics (3), the *A* allele is maintained at a frequency of The fact that genetic variation is maintained in this example is not surprising, since with random mating it is simply not possible to have a population that consists solely of heterozygotes (unless both homozygotes are nonviable). One may thus argue that it is not selection that maintains genetic variation, but rather, the random pairing of gametes to form diploid organisms. However, since the latter cannot be separated from the dynamics of diploid genotypes, such a discussion is futile. In contrast, the dynamics of allele frequencies, Equation 3, does not involve a pairing process. Accordingly, the notion of frequency-independent selection at the allelic level, as introduced by Equation 1b, is sufficient to prove the erosion of genetic variation, and contains the notion of frequency-independent selection on haploids.

In *Constant frequency-independent selection at linkage equilibrium eliminates genetic variation*, we show that constant frequency-independent selection on a set of haploid loci at linkage equilibrium eliminates genetic variation. However, it remains unclear if genetic variation may be stably maintained when recombination and selection are comparable. The case of complete linkage between loci is trivial, and Kirzhner and Lyubich (1997) showed that no stable polymorphism is possible with recombination and additive selection. Our argument covers the complementary case of linkage equilibrium, yet allows for arbitrary epistasis between alleles. For two diallelic loci under constant selection, a complete analysis was performed by Bank *et al.* (2012), yet the interaction of recombination and epistasis in multi-locus systems is still not well understood.

Trivially, fluctuating selection may maintain polymorphism indefinitely if the direction of selection on two alleles at a single locus changes whenever either allele frequency falls below a threshold value. However, this example is artificial, since the pattern of fluctuations depends on the frequencies of the alleles. In the time-varying framework, frequency-independent selection requires that the change of the fitness functions be independent of the allele frequencies. Temporally fluctuating, frequency-independent selection is capable of maintaining genetic polymorphism, as demonstrated by Kirzhner *et al.* (1994). Their examples require strong selection such that gene frequencies undergo large changes between successive generations, which may be considered unrealistic in practice. In *Weak*, *periodically fluctuating*, *frequency-independent selection may maintain stable polymorphism in a two-locus model*, we provide an example of two diallelic loci under weak periodically fluctuating selection and recombination that maintains polymorphism in a stable periodic trajectory. Crucially, including linkage disequilibrium *D* explicitly is necessary to stabilize the trajectory; under complete linkage or at linkage equilibrium, the effect vanishes.

Heuristically, we may conclude from Equation 5 that the dynamics of *D* may lead to a stable trajectory if they correlate positively with epistasis. This is the case if we approximate *D* by assuming quasi-linkage equilibrium at all times, when *D* has the sign of *ϵ* (see Equation 6). Fluctuating epistasis is thus a key element in the maintenance of genetic polymorphism in our example. Patterns of positive correlation between linkage disequilibrium and epistasis emerge naturally under quasi-linkage equilibrium. Thus, they may be of general importance in maintaining stable polymorphism under frequency-independent selection.

Cycling linkage disequilibrium is also present in a study by Gulisija *et al.* (2016), who demonstrate that stable polymorphism can be maintained in a haploid system with a locus subject to periodic frequency-independent selection and a modifier locus for phenotypic plasticity. They argue that an allele that promotes phenotypic plasticity reduces the effect of selection on linked alleles, and hence may cause a genomic-storage effect by sheltering deleterious alleles. Their model reduces to a dialellic, two-locus selection model with epistasis in which the genotype fitnesses fluctuate periodically; its fitness structure, however, differs from the example we present above. Their study provides further indication that the explicit process of recombination between loci is vital for maintaining genetic variation; in the limit of high recombination rate (*i.e.*, at linkage equilibrium), their proposed mechanism fails. In some cases, dynamically fluctuating linkage disequilibria may indeed give rise to effective negative frequency dependence over time, and thus to stable polymorphism. Further analyses will be necessary to understand the dynamics of linkage disequilibria and their effect on the persistence of genetic variation.

We have shown that constant frequency-independent selection on haploids, or on diploids in the absence of dominance, cannot maintain variation at linkage equilibrium; our numerical results show that constant and weak selection and recombination very likely do not maintain variation, even when linkage disequilibrium is strong. However, we do show, using a two-locus example, that fluctuating epistasis can maintain variation when recombination is of the same order as the strength of epistasis. This works through a general mechanism, whereby a positive correlation between linkage disequilibrium and epistasis helps stabilize the polymorphism. Since epistasis among a set of loci generates linkage disequilibrium among those loci, we expect this mechanism to act as a general stabilizing process.

However, this process is likely to be weak, especially when epistasis is weak (). The stabilizing force is proportional to and since *D* is itself proportional to epistasis, the stabilizing force is of order This makes polymorphisms sensitive to biases in directional selection, these must be weaker than for the polymorphism to be stable. Moreover, if such biases push the equilibrium away from the center of the allele frequency space, the stabilizing force becomes much weaker, since A caveat here is that it is not clear how these arguments scale with the numbers of loci: epistatic coefficients become weaker relative to directional coefficients, but there are many more of them. An extension to many loci would be desirable, though not easy.

Is fluctuating frequency-independent selection likely to maintain variation in nature? We know that selection on traits is typically strong and typically fluctuates across generations (Kingsolver *et al.* 2001); classic examples of selection on discrete loci also involve strong selection which changes through time (Lewontin *et al.* 1981; Cook *et al.* 1986; Lenormand *et al.* 1999). Recent genome-wide surveys of polymorphism in *Drosophila melanogaster* show that large numbers of SNPs rapidly change frequency through the seasons (Bergland *et al.* 2014), though it is not clear just how many causal loci drive this pattern. Over the longer timescale of the fossil record, we see that phenotypes may change rapidly in the short term, and yet much more slowly in the long term (Gingerich 2009). This suggests that selection may typically fluctuate strongly, yet largely cancel in the long term.

Nevertheless, we believe that the mechanism that we identify is unlikely to be significant in explaining the bulk of variation across the genome. Although selection on the organism can be strong, selection on nucleotide sites must typically be weak; indeed, indirect estimates from patterns of molecular variation suggest selection coefficients that are typically small, even if often strong relative to drift ( say; Charlesworth 2015). Most sets of loci are loosely linked (), and so weak epistasis is unlikely to provide much stability. More importantly, as discussed above, systematic bias in directional selection is likely to be much stronger than the stabilizing interaction between epistasis and linkage disequilibrium. In principle, frequency-dependent selection gives a much more robust mechanism that can give an advantage to rare alleles, and thereby maintain variation. Even here, though, many of the proposed mechanisms require either strong selection or a delicate balancing of parameters, *e.g.*, the Levene model (Levene 1953) or, more generally, Maynard Smith and Hoekstra (1980). Thus, after half a century of debate, it remains unclear that balancing selection can account for widespread variation across the genome.

## Acknowledgments

We thank Katarína Bod’ová, Reinhard Bürger, Josef Hofbauer, Meike Wittmann, and two anonymous reviewers for valuable suggestions and stimulating ideas. This project has received funding from the European Union’s Seventh Framework Programme for research, technological development, and demonstration under grant agreement 618091: Speed of Adaptation in Population Genetics and Evolutionary Computation.

## Appendix A: Frequency Independence at the Allelic Level

A haplotype (where ) is a list of alleles that sit at their respective loci. Under the assumption of linkage equilibrium between loci, the frequency of any haplotype is given by the product of the frequencies of the alleles it consists of,We denote the fitness of the haplotype by the constant Note that this does not imply anything about epistatic interactions between the individual alleles.

The mean fitness of the population is obtained by averaging the haplotype fitnesses over the haplotype frequencies. Then, the mean fitness is a homogeneous polynomial of degree *L*, whose positive coefficients are the fitness values of the haplotypes,

Similarly, the marginal fitness of an allele can be written as(A1b)From the mean and marginal fitnesses under constant haploid selection, Equations A1a and A1b, it is straightforward to see that the marginal fitness of an allele is independent of the allele frequency at its locus, Equation 1b.

Here, we show that frequency-independent selection at the allelic level, as defined by Equation 1b, implies that the mean and marginal fitnesses can be written by Equations A1a and A1b under the assumption of linkage equilibrium. In this sense, constant haploid selection is contained in the notion of frequency-independent selection at the allelic level. Due to its (verbal) definition as the average over genotypes (see the main text), the mean fitness can generally be written as a linear combination of the marginal fitnesses, *i.e.*, Equation 1a holds in general. It thus suffices to show that the mean fitness can be written as a homogeneous polynomial of degree *L* with positive coefficients under the additional assumption of frequency-independent selection at the allelic level as given by Equation 1b.

From Equation 1a and 1b, it follows that(A2)at any locus and for any choice of alleles Furthermore, from Equation 2 we have(A3)since marginal fitnesses are always nonnegative by definition. Thus, in terms of any given locus, the mean fitness is a linear function of the allele frequencies at the locus. Its coefficients are nonnegative functions only of the frequencies of alleles at other loci. For each pair of loci, we may write the mean fitness in two ways as(A4)*cf*. Equation 1a. Taking the derivative with respect to the frequency of any allele *k* at locus *l* yields(A5)The function is constant in the allele frequencies at locus *l*. Choosing (which implies for ) shows that also the derivative is constant in Hence, the marginal fitnesses are linear functions in the allele frequencies at the loci different from locus *i*.

Overall, it follows that is a polynomial with positive coefficients, whose degree is at most *L*. Furthermore, each term of this polynomial contains at most one allele frequency from each locus. If a locus *i* is not represented in any one of the terms, we may multiply it with to homogenize the polynomial. Hence, is completed to degree *L* homogeneously, and our claim is proved.

## Appendix B: Convergence to the Set of Equilibrium Points

From Equation 2, it follows that the dynamics (3) for allele *k* on locus *i* may be written as

Furthermore, is a homogeneous polynomial with positive coefficients, see Appendix A. Thus, the conditions for the inequality of Baum and Eagon (1967) are met. It follows that the mean fitness is strictly increasing along trajectories of under the dynamics (3), remaining constant only at equilibrium,The space of admissible is a compact set, hence the values of are bounded from above. Thus, every trajectory of the dynamics (3) converges to the set of its equilibrium points. This set may be complicated; nevertheless, the existence of other attractors (*e.g.*, periodic orbits, chaotic attractors) can be excluded (*e.g.*, Lyubich 1992, Chap. 9). Further assuming that all equilibria are hyperbolic implies that there are only finitely many equilibria (Nagylaki *et al.* 1999), hence each trajectory of the dynamics converges to an equilibrium; in particular, there is no cycling behavior. The hyperbolicity of equilibria is a generic property (Nagylaki *et al.* 1999, appendix B) which implies the nondegeneracy of equilibria we point out in Theorem C1 below.

## Appendix C: Unstable Polymorphism Under Constant Selection

Because are frequencies, we have for every The dynamics of allele frequencies at locus *i* with alleles evolve on the -dimensional simplex

The joint Equation 3 for the change of allele frequencies is thus a dynamical system on the product of simpliceswhich has degrees of freedom due to the constraints on the being frequencies.

The linearization of the dynamics (3) around an equilibrium is given by the Jacobian This matrix has dimensions and its entry at position is obtained from taking the derivative of the *m*-th function with respect to the *n*-th variable (counting contiguously across loci), and evaluating at For example, the diagonal entries of are given by(C2)where

According to Appendix B, the trajectories of allele frequencies under the dynamics (3) converge to the set of their equilibrium points. These equilibria may be fully polymorphic or on the boundary of the state space Theorem C1 below states that any polymorphic equilibrium is unstable, hence trajectories converge to the latter type of equilibria and at least one allele is lost from the population.

**Theorem C1.** *(Instability of polymorphic equilibria.) Under the assumptions of linkage equilibrium and constant frequency-independent selection*, *any fully polymorphic equilibrium of the dynamics (3) is unstable. This holds for all but a set of degenerate cases of measure zero*, *where all eigenvalues of the Jacobian are equal to 1*.

For the proof of the theorem, we note two simple results:

**Lemma C1.** *The Jacobian at a polymorphic equilibrium of the dynamics (3) has at most nonzero eigenvalues*.

*Proof*. This is the case because the dynamics (3) are a projection on the spacewhich has dimension □

**Lemma C2.** *Consider a polymorphic equilibrium of the dynamics (3) under the same assumptions as in Theorem C1. Then*, *the trace of the Jacobian (i.e.*, *the sum of its diagonal entries) is**Proof*. At any polymorphic equilibrium of the dynamics (3) we have(C3)Using this identity together with Equations 1a and 1b, we calculate the diagonal entries of (Equation C2) asIt follows that the trace of is □*Proof of the theorem*. Consider a polymorphic equilibrium of Equation 3 and its Jacobian Because is a matrix with dimensions it has eigenvalues (counting multiplicities). Due to Lemma C1, at most are nonzero. According to Lemma C2, the trace and hence the sum of eigenvalues of is also Therefore, unless all eigenvalues equal one (the degenerate case), at least one of them has a modulus greater than one. Thus, the polymorphic equilibrium is unstable. □

## Appendix D: Stability of Equilibria in the Continuous-Time Limit

We assume that selection and recombination are weak so that we can approximate time to be continuous. Selection and recombination are of the same order so that there may be strong linkage disequilibria. Since only the ratio of the two processes matters, selection scales relative to the recombination rate. We define the genotype by the list of alleles *X*. There may be multiple alleles per locus; gamete frequencies are and the corresponding fitnesses are Because we assume continuous time, can be positive or negative, and adding a constant to *s* would make no difference.

The rates of change of genotype frequencies are

where and is the rate at which a pair generates *X*. If we ignore multiple crossovers, this is a sum over the single recombination events, each with rate is the rate at which *X* is broken down, which equals the sum of the recombination rates, Note that this includes recombination events that generate *X* itself, but these cancel with corresponding terms in the sum over Writing this explicitly:(D2)where denotes the frequency of all gametes that match *X* to the left of break point *i*, and similarly for Because the rates of change sum to zero over *X*.

At equilibrium, for all *X*, and the stability is determined by the matrix(D3)where the last term arises from differentiating and is one if and zero otherwise. If we specify the equilibrium frequencies and recombination rates, the is uniquely determined:(D4)Substituting into Equation D3 yields(D5)Note that the sum over *X* is a constant,(D6)which implies that there is a constant eigenvector with eigenvalue This corresponds to perturbations that violate the constraint and therefore can be ignored. The stability of the equilibrium is determined by whether the largest real part of the remaining eigenvalues is positive.

We have not found a way to show analytically that this matrix implies instability, even in the three-locus case. However, it is instructive to examine the completely symmetric case, with two alleles per locus, where all haplotypes are equally common. This corresponds to linkage equilibrium and allele frequencies all equal to With complete symmetry, the eigenvalues can be calculated for up to four loci, and depend in a simple way on the recombination rates; Table A1 shows the 16 eigenvalues for four loci, and suggests the obvious generalization to *n* loci. There are *n* zero eigenvalues, corresponding to neutral change in allele frequency; the question is how these change when the frequencies are perturbed away from linkage equilibrium, and hence away from neutrality. Numerically, one or more of these eigenvalues becomes positive, consistent with our previous result that when selection is much weaker than recombination, so that linkage equilibrium is approached, polymorphism is unstable. However, this does not tell us whether stable equilibria exist when epistasis and linkage equilibria are strong. To investigate that, we calculate eigenvalues of the stability matrix numerically (see the main text).

## Appendix E: Stable Polymorphism at Two Diallelic Loci Under Fluctuating Selection

### Formulation of the Dynamics

Consider two diallelic loci with genotype fitnesses as given in Table 2. Under weak selection and recombination [formally: replacing and letting ], the dynamics of genotype frequencies is(E1a)(E1b)(E1c)(E1d)as first derived by Kimura (1956) (see also Bürger 2000, Chap. II.1). The measure of linkage disequilibrium *D* is given by as in the main text, and Due to the constraint one of these equations is redundant. Rewriting the system (E1) in terms of the allele frequencies and yields(E2a)(E2b)(E2c)where and Introducing and then leads to the system (4).

### Stability of a Periodic Trajectory Under Fluctuating Selection

Using numerical routines of the computer algebra software Mathematica, we simulate fluctuating selection on two diallelic loci by iterating two seasons, as described in the main text in *Weak*, *periodically fluctuating*, *frequency-independent selection may maintain stable polymorphism in a two-locus model*. For all tested initial conditions [two sets of initial conditions were tested: (i) randomly drawn starting points, and (ii) and distributed on a regular grid in the − plane as displayed in Figure A1, and ], the dynamics converge to the periodic trajectory displayed in Figure 1. Through the course of the two seasons, the periodic trajectory interpolates between the approximate valuessee the bold points in Figure 1. These values were calculated numerically as the fixed points of two Poincaré maps, and defined by the values of *P*, *δ*, and *D* at the transition between season 1 and 2, and season 2 and 1 [*i.e.*, at the time points and for respectively],(E3)Numerically calculating the eigenvalues and of the linearization of either Poincaré map at its equilibrium yieldshence the periodic trajectory is asymptotically stable. Repeating the procedure for different recombination rates *r* produces the data presented in Figure 2.

### An Asymmetric Example of Stable Polymorphism

We simulate the dynamics given by Equation 4 using different epistatic coefficients in the fitness values of the two genotypes with mixed upper- and lower-case letters, and The fitness configurations are given by Table A2. We assume the two seasons to be of equal length and the recombination rate to be The specific values of *s*, and in each of the seasons are given in Table A3. For a set of initial conditions (*e.g.*, ), the temporal dynamics of the system converge to a periodic orbit shown in Figure A2. The asymptotic stability of this orbit can be established by the methods outlined above.

## Footnotes

*Communicating editor: G. Coop*

- Received December 20, 2016.
- Accepted July 26, 2017.

- Copyright © 2017 by the Genetics Society of America