A mixability theory for the role of sex in evolution

Abstract

The question of what role sex plays in evolution is still open despite decades of research. It has often been assumed that sex should facilitate the increase in fitness. Hence, the fact that it may break down highly favorable genetic combinations has been seen as a problem. Here, we consider an alternative approach. We define a measure that represents the ability of alleles to perform well across different combinations and, using numerical iterations within a classical population-genetic framework, show that selection in the presence of sex favors this ability in a highly robust manner. We also show that the mechanism responsible for this effect has been out of the purview of previous theory, because it operates during the evolutionary transient, and that the breaking down of favorable genetic combinations is an integral part of it. Implications of these results and more to evolutionary theory are discussed.

Theories on the role of sex in evolution have often been led by the assumption that sex should facilitate the increase in population mean fitness, w̄ (1). At the same time, it has been recognized that sex may break down highly favorable combinations of genes, which impedes the increase in fitness (2, 3). Thus, in a prominent review of sex theory, Barton and Charlesworth (4) wrote to the effect that the breaking down of highly favorable gene combinations has been one of the most obvious difficulties in understanding sex.

Here, we examine the role of sex in evolution from a different angle. We develop a measure, M̄, which represents the genomewide ability of alleles to perform well across different combinations. Using numerical iterations within a classical population–genetic framework, we find that sex favors the increase in M̄ in a highly robust manner. Furthermore, we expose the mechanism underlying this effect and find that it operates during the evolutionary transient, which has been studied relatively little. We also find that the breaking down of highly favorable gene combinations is an integral part of this mechanism. Therefore, if the role of sex involves selection not for the best combinations of genes, as would be registered by w̄, but for genes that are favorable in many different combinations, as is registered by M̄, then the breaking down of highly favorable combinations does not necessarily pose a problem.

A precedent to our work can be found in the work of Crow and Kimura (5), who suggested briefly, based on intuition, that sex favored “good mixers.” However, they did not develop this intuition, and in fact considered the breaking down of highly favorable gene combinations to be a disadvantage of sex. In line with their choice of words, we call M̄ the “average mixability” of alleles in the genome.

Theoretical treatments normally assume that genes make separate contributions to fitness. For example, in the case of two loci, the fitness w_ij of a haploid genotype consisting of allele i at the first locus and allele j at the second locus may be represented as w_ij = 1 + s_i + t_j + ε_ij, where s_i and t_j are the separate, additive contributions of alleles i and j, respectively and ε_ij is an interaction or “epistasis” term. In our models, there is no a priori assumption of separate contributions to fitness. However, sex favors the increase in M̄ and, as a result of that increase, as we shall see, a situation emerges where separate-contribution terms could become natural and important elements in the description of genotypic fitnesses. Formulations incorporating separate-contribution terms have long been used in population genetics (6), but how sex relates to them has not been the primary focus.

Our results may help to bridge between sex theory and modularity theory (7–11). In recent simulations, Misevic et al. (10) found that, as compared with asex, sex favored a genomic organization where genetic elements that coded for the same trait were physically closer to each other and farther from other such elements. In their interpretation, this organization was modular (10, 11) and, accordingly, the groups of linked elements featured in it were modules. However, the ideal module in this sense is one whose elements are absolutely linked, and is also a particularly good mixer in the sense that it is transmitted as a whole and is therefore more likely to maintain its individual contribution to fitness across different backgrounds. Hence, the theory to be given here may help to explain these simulation results on a conceptual level.

Throughout this article, and in the tradition of evolutionary theory, we use the words “gene” and “allele” to refer not only to a genetic sequence that codes for a protein or a set of proteins related by alternative splicing, but to any contiguous stretch of the genome that may be represented by a locus in our models and informed by our results (12).

Theory and Results

Consider the haploid 2-locus fitness landscape in Fig. 1. Although genotype A₁B₃ has maximum fitness, there is a sense in which allele A₂ performs best overall among the A alleles across different genetic contexts (same for B₂). Let P_ij,t be the frequency of genotype A_iB_j at generation t, r be the recombination rate (0 ≤ r ≤ $\frac{1}{2}$), and w_ij be the fitness of A_iB_j, with w¯_t = Σ_i,j P_ij,t w_ij and p̃_ij,t = P_ij,t w_ij/w¯_t. The discrete time evolutionary dynamics of a large panmictic population without mutation can be written as:

Iterating this equation numerically from many different initial genotypic frequencies we find that, for r = 0 (asex), genotype A₁B₃ outcompetes all other genotypes, whereas for many values of r with 0 < r ≤ $\frac{1}{2}$ (sex), alleles A₂ and B₂ most often outcompete all other alleles (Fig. 2).

Fig. 1.

A haploid 2-locus fitness landscape with 3 alleles per locus. Alleles A₁, A₂, and A₃ in locus A and B₁, B₂, and B₃ in locus B give 9 genotypes A_iB_j with fitnesses w_ij, represented by the heights of the bars (w₁₁ = 0.840, w₁₂ = 1.046, w₁₃ = 1.100, w₂₁ = 1.040, w₂₂ = 1.060, w₂₃ = 1.000, w₃₁ = 1.020, w₃₂ = 1.050, w₃₃ = 0.820).

Fig. 2.

Typical evolutionary dynamics of the haploid 2-locus example. Here, we start with equal genotypic frequencies (1/9 for each of 9 genotypes) at generation 0 and calculate the genotypic frequencies for generations 1−500 based on Eq. 1 for both r = $\frac{1}{2}$ (sexual, solid lines) and r = 0 (asexual, dotted lines). Alleles A₁, A₂, A₃; B₁, B₂, B₃, and w_ij are as in Fig. 1.

To formalize this observation, let M_i be the average fitness of allele i, i.e.:

where G is the set of all genotypes with ordering (i.e., for the case of diploid genomes, symmetric genotypes are considered distinct), w_g is the fitness of genotype g ∈ G, and i_g is the number of times that allele i appears in genotype g (assuming 0 or 1 times in haploid genomes and 0, 1, or 2 times in diploid genomes). For example, in Fig. 1, M_A2 = (w₂₁ + w₂₂ + w₂₃)/3 and, with the values of w_ij specified, M_A2 > M_A1, M_A3 and M_B2 > M_B1, M_B3. Now, M_i can be taken to represent the ability of allele i to perform well across different combinations; and in the given example, whereas asex favors the most fit genotype (A₁B₃), sex favors the alleles (A₂, B₂) that are highest for this ability, henceforth called “mixability.”

As another example, consider the diploid 1-locus model with genotypes A_iA_j having fitnesses w_ij = w_ji, with w_ij for all i;j ≥ i taking the same values as the corresponding w_ij in the previous example [supporting information (SI) SI 1 in SI Text]. Iterating the appropriate equation (see Methods), again we see that sex favors the alleles of highest mixability (SI 1 in SI Text and Figs. S1 and S2), this time via segregation rather than recombination. In these examples the alleles of highest mixability fix, but the selection for alleles of high mixability occurs during the evolutionary transient rather than near equilibrium.

To obtain a unified picture of many different cases from 3 fundamental models of sex: haploid 2-locus (recombination), diploid 1-locus (segregation), and diploid 2-locus (both), we use a robust means of comparison between sex and asex based on a populationwide measure of mixability, M̄_t, which we define as: M̄_t = (Σ_ℓ∈L Σ_i∈ℓ P_i,t M_i)/〈L〉, where L is the set of loci, 〈L〉 is the cardinality of that set, at each locus ℓ ∈ L there is a set of alleles indexed by i, and P_i,t is the frequency of allele i at generation t. For example, in the haploid 2-locus model, M̄_t = (Σ_i P_i,t M_i + Σ_j P_j,t M_j)/2. M̄, which is the average mixability in the population, can be contrasted with a fundamental measure in population genetics, namely w̄, average (genotypic) fitness.

We iterated the three models (see Methods) in four cases, namely 2, 3, 4, or 5 alleles per locus, three r values (in the sex cases of the 2-locus models; r ∈ {0.5, 0.2, 0.05}) and three values of a parameter (f) that determined the range of fitness values (see Methods). In each condition (one model, one allele number, one r, and one f value), we chose a random set of n fitness matrices, {W¹, … , Wⁿ} and a random set of n initial genotypic frequency matrices, {P¹, … , Pⁿ}, n = 25 (see Methods), and for each of the n² pairs of P and W matrices, iterated the corresponding equations for a certain number of generations (e.g., 2¹⁵ ≈32,000 generations) both with and without sex; each such iteration was called a “trial.” At a representative set of generations (e.g., {0, 1, 2, 4 … , 2¹⁵}), we recorded the percentage of trials in which the sexual population had a higher M̄ than the asexual one (% M̄^sex > M̄^asex) and vice versa (% M̄^asex > M̄^sex) and the percentage of trials in which the asexual population had a higher w̄ than the sexual one (% w̄^asex > w̄^sex) and vice versa (% w̄^sex > w̄^asex). These pairs of percentages do not necessarily add up to 100 because frequency differences between sex and asex below a very small threshold were taken to mean that these measures were essentially the same (see Methods). Hence we call the difference (easily inferred by eye) between the percentage of trials in which one population was higher in M̄ and the percentage of trials in which the other population was higher in M̄ the “advantage” to the former population in M̄ (and similarly for w̄).

Typical evolutionary dynamics are given for the 2-locus diploid model in Fig. 3 and for other cases in SI 2 in SI Text and Figs. S3–S8. The main result is that whereas the asexual population has a decisive advantage in w̄, the sexual population has a decisive transient advantage in M̄. This result holds remarkably well under all conditions investigated and in all 3 models (SI 2 in SI Text and Figs. S3–S8).

Fig. 3.

M̄ and w̄ comparisons between sex and asex for the 2-locus diploid model without position effects (see Methods), for r = 0.5, f = 0.5. See Theory and Results for how the parameters were chosen.

A Verbal Explanation of the Results

That w̄ is maximized under asex is clear, because in that case the geometric growth of genotypic frequencies according to their fitnesses leads to the eventual extinction of all but the most fit genotypes. But how does sex favor mixability? Consider the haploid 2-locus model in Eq. 1. The second term in this equation shows that recombination and segregation of nonhomologous chromosomes act to reduce the variance in the frequencies of genotypes that carry a given allele. This reduction of variance shifts the growth rate of this allele in the long term from being associated with the performance of the most fit genotype that carries it toward being associated with how well it performs on average with its various genetic partners. Consequently, alleles of the same gene compete with each other based on how well they perform on average rather than how well they perform in any one specific combination. Examination of the diploid 1-locus equation shows that segregation of homologous chromosomes has a similar effect (SI 3.1 in SI Text).

Why the reduction in variance has this effect can be understood intuitively by analogy to finance. To diversify a portfolio (13) among various investments and thus average its growth rate over time, an investor must keep rebalancing the portfolio at regular intervals (14). Otherwise, the portfolio will soon be biased toward the investments that had the best individual returns so far. The investments here correspond to the genotypes that share an allele, the portfolio corresponds to that allele, and by analogy, it is the persistent rehomogenization by sex of frequencies of genotypes carrying each allele that shifts the focus of natural selection over the generations from the individual performance of genotypes to the average performance of alleles.

Because this average performance of alleles can only be registered on the multigenerational time scale, it is essentially different from the average performance of alleles discussed by Fisher (15). Indeed, population genetics normally focuses on frequency changes that occur in one generation only and on the equilibria that could be calculated from them, and the effect just mentioned is not amenable to this approach (SI 3.2 in SI Text).

We can now see the benefit of M_i, which is an average of genotypic fitnesses unweighted by genotypic frequencies, as compared with a familiar measure such as

which is an average of genotypic fitnesses weighted by the genotypic frequencies at generation t, P_g,t (in the diploid 1-locus model, φ_i,t is related to the marginal fitness). Because of the multigenerational nature of the effect, the “instantaneous” frequencies P_g,t, which refer to generation t only, would make for inappropriate weights for the fitnesses w_g. This benefit, however, comes with a cost. By noting that M_i = φ_i when all P_g,t are equal, we see that M_i is fundamentally a proxy that becomes less informative the further the genotypic frequencies are far from uniformly distributed. Thus, as genetic variation is lost during the numerical iterations because of selection, and the genotypic frequency distribution becomes less uniform, the M_i proxy loses power, and the advantage to sex in mixability as seen through M_i decreases.

We can now see why the advantage to sex in M̄ peaks in the transients and is <100%. On the one hand, either the decrease in the power of M_i begins immediately or M_i is inaccurate from the beginning (SI 3.3 in SI Text). On the other hand, it takes multiple generations for the advantage to sex in M̄ to accumulate (SI 3.4 in SI Text). The superposition of these two effects cuts the observed advantage to sex in mixability on both the left and right to create the peak mentioned above. Importantly, this is a limitation on our ability to observe selection for mixability through M̄, not a limitation on the selection for mixability itself.

Another limitation on observability is the lack of mutation in our models. Although selection for increased w̄ under asex cannot continue indefinitely without mutation because of the depletion of genetic variation, selection for increased M̄ under sex is even more sensitive to that depletion, because it requires not only variation in the genetic material that is being selected [as in Fisher's fundamental theorem (15)] but also variation in genetic material whose tolerance is being selected. With respect to these limitations our results are conservative (SI 3.5 in SI Text). In actuality, variation is continually replenished by mutation, selection continues beyond what is shown here, and the difference in mixability between sexual and asexual populations should grow indefinitely (SI 3.6 in SI Text).

Separate Effects on Fitness

As a corollary to the increase in M̄ in our models, sex also gives rise to separate genetic contributions to fitness. To see this, consider a haploid 2-locus model with fitness values w_ij, i ∈ {1, … , n_A}, j ∈ {1, … , n_B} with a mean

Each w_ij could be represented as a deviation from the mean: w_ij = 1 + x_ij, where x_ij can be positive, negative, or zero. However, suppose that the set of fitness values belonging to a certain allele i at the first locus has a high average; that is,

is significantly >1. In this case it is statistically meaningful to represent each w_ij in that set as a deviation from the mean of that set: w_ij = 1 + s′_i + ε′_ij, where s′_i is positive. Furthermore, suppose that allele j at the second locus also has a high M_j, then it is statistically meaningful to use a form such as w_ij = 1 + s_i + t_j + ε_ij.Thus, as the set of alleles of interest in our models becomes restricted through the selection for mixability from the original set of alleles to the subset of alleles that have high M values, it becomes appropriate to describe the fitnesses of genotypes using a form such as the above, which features separate, additive contributions to fitness in the s and t terms.

Note that, first, by using the additive form just mentioned we do not mean to exclude other forms that include separate contributions to fitness, such as w_ij = (1 + s_i)(1 + t_j) + ε_ij [this form is also additive in the sense of Fisher (15), where additivity refers to the exponent]. Second, even if alleles A_i and B_j make separate contributions to fitness, the organism may still have to have some allele at locus A and some allele at locus B to be viable at all and, accordingly, such separability says nothing about the strength of the interaction between loci A and B that is common to all pairs of alleles A_i and B_j.

Discussion

It has long been suggested that sex breaks down highly favorable combinations of genes (2, 3). This has been seen as a problem because of the assumption that sex should facilitate the increase in population mean fitness, w̄ (16, 17). Specific conditions have been sought under which sex might facilitate that increase (1, 4), but these conditions were found to be restrictive (1, 18–20). However, our analysis has shown an advantage to sex in average mixability, M̄, and furthermore it appears that dissociation of gene combinations, in which the breaking down of highly favorable combinations is featured prominently, is needed to cause that advantage. Hence we suggest that the role of sex is to enable selection not for highly favorable specific combinations of genes but for genes that are favorable in many different combinations (SI 3.7 in SI Text).

The mechanism enabling the selection for mixability operates on the multigenerational time scale. It has been out of the purview of previous theory, which often focused on single-generational changes and the equilibria that could be calculated from them. This mechanism shows that, while at any one generation natural selection operates on genotypic fitnesses, over the generations and in the presence of sex, it is particularly efficient not in increasing population mean fitness but in increasing the ability of alleles to perform well across different combinations. Thus, in this mechanism, natural selection and sex operate interdependently and need to be understood in the context of each other.

Our iterations start with random genotypic fitness values that do not depend in any meaningful way on their constituent alleles. In the haploid 2-locus model, for example, the genotypic fitnesses w_ij can be written at this stage as w_ij = 1 + x_ij, where i and j refer to alleles at the first and second locus, respectively, and x_ij can be positive, negative, or zero. Then, during the iterations, in the presence of sex, alleles are selected that have high M values. However, when alleles have high M values it becomes statistically meaningful to represent the fitnesses of genotypes symbolically with forms that feature separate contributions to fitness, such as w_ij = 1 + s_i + t_j + ε_ij, where s_i and t_j represent the additive contributions to fitness of alleles i and j, respectively, and ε_ij is an interaction or “epistasis” term. Thus, we start with genetic elements that have no meaning except for the fact that they recombine and end up with genes that make separate contributions to fitness. Forms such as the above have been used in population genetics since Fisher's reconciliation of Mendelism and biometry (6). But now we see that, within the confines of our framework, sex provides causal justification to them.

Our work confirms the intuition of Crow and Kimura (5) that sex favors good mixers, which they described verbally as alleles that made large additive contributions to fitness. It is also consistent with empirical evidence they provided on this point. They noted that ANOVA of the contribution of different chromosomes to drug resistance that evolved under sexual reproduction in Drosophila showed strong additivity (5, 21), whereas drug resistance that evolved under asexual reproduction in Escherichia coli was diminished by subsequent recombination in a manner that implied strong interactions (22). Along the same line, Malmberg (23) showed that contributions of different genomic regions to drug resistance in bacteriophage T4 showed stronger additivity and weaker interactions in populations evolving under higher recombination rates. Although our work agrees with this evidence, it also suggests that mixability needs to be attended to as a focal issue.

Recently, interest has increased in the evolution of phenotypic robustness to genetic changes (11, 24). It has been suggested that either this robustness evolves to keep a trait at its optimum (24) [directly as a response to genetic variance or as a correlated response to environmental variance (25)] or this robustness is a byproduct of being near an optimum (24), and that conditions for the former may be more pronounced in sexual than in asexual populations (11, 24, 26, 27). Fitness is a phenotypic trait of special interest in this context (24), and indeed, recent simulations found higher fitness robustness to recombinational (28) and mutational (10, 28) genetic changes in sexual than in asexual populations. Because alleles of high mixability maintain high fitness despite recombination, our iterations also show higher fitness robustness to recombinational changes in sexual populations, and on this point they are in agreement with the above. However, our iterations demonstrate an additional route by which this fitness robustness may evolve, because here it evolves as the immediate consequence of sex and natural selection and not as a result of pressure to stay near a fitness optimum.

Importantly, although selection for mixability in our models favored a genetic variant if it maintained high fitness with multiple genetic partners that it encountered through recombination, such a variant may also be more likely to interact well with genetic partners that it has not yet encountered, including newly mutated ones, because those are likely to differ by only a small amount from the ones that it has encountered. This possibility would accord with recent simulations (28), which showed that recombinational fitness robustness led to mutational fitness robustness (28). Now, according to de Visser et al. (24), fitness robustness may enhance evolvability (24), and in particular, the ability of loci to tolerate changes in other loci may allow evolution to advance by local changes rather than necessitate a coordinated change across the genome (24). Accordingly, we conjecture that mixability enhances evolvability.

The possibility that mixability enhances evolvability may provide an answer to the question “what is mixability good for?” It is commonly believed that, among higher organisms, sexual species (obligate and facultative) are more evolvable than asexual ones (29–31), because they are the founders of wide taxa (12, 29) and are vastly more common (4, 32), while obligate asexuals are mostly recent derivations from sexual ancestors (referred to as “evolutionary dead-ends”; but see ref. 33) and are phylogenetically sparse (12, 29). If mixability enhances evolvability, then (because it is favored under sex) it may have been contributing to the proliferation of sexual species and sexuality itself. In addition, the tolerance to genetic changes that mixability entails may explain why the same or similar genes often appear in different combinations within and across species (9). Kirschner and Gerhart attribute this to the components of life being appropriate for “versatility and modification rather than for dedicated single use” (9).

Our analysis shows that sex favors compatibility between alleles. The inverse is that lack of sex allows alleles to develop incompatibility as they evolve. This connects to the Dobzhansky-Muller model (34) of hybrid fitness reduction. In the basic case of this model, a population of diploid genotypes A₁B₁A₁B₁ splits into two separate demes, one in which A₁B₂A₁B₂ evolves and one in which A₂B₁A₂B₁ evolves; and after these demes come back into contact, negative epistatic interactions are said to occur in the hybrid A₁B₁A₂B₂ between alleles A₂ and B₂. The question arises, however, as to why alleles A₂ and B₁ are compatible, alleles A₁ and B₂ are compatible, but alleles A₂ and B₂ are incompatible. The answer, in terms of our theory, is that sex selects for alleles with an additive effect that rises above epistasis. Without sex, epistasis between alleles in nonbreeding individuals is free to increase before resulting in a hybrid. Thus, evidence for the Dobzhansky-Muller effect (35) [and the fact that hybrids are less functional in species that are further along the process of speciation (19)] is consistent with our work.

The roles of sex, additivity, and epistasis described here tie also to the Fisher–Wright debate, where Fisher (15) advocated additive genetic contributions to fitness with weak epistasis and Wright (36) advocated strong epistasis, because it suggests that both were partly correct. That is, because mating occurs more often within a deme, it reduces epistasis in crosses within a deme; and because it occurs less frequently between demes, it allows epistasis in crosses between demes to increase with the accumulation of mutations. By a similar argument, genetic linkage (5) and the local rates of recombination across the genome may be expected to affect the extent of epistasis.

Finally, our results may speak to a connection between sex and modularity (10, 11). According to Schlosser (37), evolutionary modules are defined partly by the fact that they make relatively context-insensitive (here called “separate”) contributions to fitness. In the presence of sex, the loci in our models acquire this characteristic of evolutionary modules. Furthermore, in simulation experiments with digital organisms, Misevic et al. (10) found that genetic elements that coded for the same trait were physically closer to each other and farther from other such elements in sexual than in asexual organisms. In their interpretation, this result showed that sex favored a modular organization of the genome (10, 11), and accordingly, the groups of linked elements featured in sexual genomes were modules. However, the ideal module in this interpretation is one whose elements are absolutely linked, and this ideal module is also a particularly good mixer in that it is transmitted as a whole and is therefore more likely to maintain its individual contribution to fitness across different backgrounds. Thus, the mechanism of selection for mixability described here may contribute to the understanding of the results of Misevic et al. (10).

Our analysis does not address the origin of sex and does not attempt to relate the evolution of sex to increase or decrease of mean fitness. Instead, we have observed an interesting relationship between the presence of sex and the temporal pattern of mixability. The ways in which mixability and modularity in response to sex relate to evolvability promise to provide an interesting area for future work.

Methods

Three Models.

The haploid 2-locus model is given in Eq. 1. For alleles i∈{1, … , n_A} in locus A and alleles j∈{1, … , n_B} in locus B, M_i=M_i^o=Σ_jw_ij/n_B.

In the diploid 2-locus model, asex case, we keep track of genotypes P_ijkl,t with fitnesses w_ijkl, where i,k∈{1, … , n_A} refer to alleles in locus A and j,l∈{1, … , n_B} refer to alleles in locus B, such that ij constitutes one haplotype and kl constitutes the other haplotype. In the sex case, assuming panmixis, we need only keep track of the haplotype frequencies (i.e., chromosomes), P_ij,t. Position effects entail only w_ijkl=w_klij, whereas lack thereof entails w_ijkl=w_kjil=w_ilkj=w_klij for all i,j,k,l. The next generation frequencies P_ijkl,t+1^asex and P_ij,t+1^sex in the asexual and sexual populations respectively are given by:

where w̄″ is the sum of the right-hand sides of the equations where it appears (for all ijkl in Eq. 2 and for all ij in Eq. 3). Here,

The diploid 1-locus model is obtained from the diploid 2-locus model by setting r=0 and collapsing what was haplotype ij before into allele i and what was haplotype kl before into allele j. Then:

where w¯′_t is defined as w̄″_t was. M_i now takes a form similar to M_i^o, except that now i and j refer to the same set of alleles in the same locus: M_i=M′_i=Σ_jw_ij/n_A (P_ij,t in Eq. 4 differs from P_ij,t in Eq. 1 in the same way).

Iterations.

The values in each fitness matrix were drawn independently at random from a uniform distribution on the interval [1−f,1+f].

To construct the frequency matrices, initial allelic frequencies (e.g., P_i,0, P_j,0 in the haploid 2-locus model) were first drawn independently at random from the uniform distribution over the interval [0,1] and normalized for each locus, then genotypic frequencies were obtained by multiplying the allelic frequencies (e.g., P_ij,0=P_i,0P_j,0).

Comparison.

Because “larger than” comparisons are overly sensitive to the least significant decimal digits, we only compared the sexual and asexual populations' w̄ values when genotypic frequencies, P_g,t, differed beyond a very small threshold (Σ_g∈G (P_g,t^sex−P_g,t^asex)²>10⁻¹⁶), where genotypes of identical fitness were grouped together as one), and only compared their M̄ when both genotypic (previous condition) and allelic frequencies differed beyond a very small threshold (e.g., in the haploid 2-locus model, ((Σ_iP_i,t^sex−P_i,t^asex)²+(Σ_jP_j,t^sex−P_j,t^asex)²)/(n_A+n_B)>10⁻¹⁶); otherwise, the w̄ values (respectively, M̄ values) were taken to be essentially identical.