Assumptions of the models for haploid and diploid selection

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For a bi-allelic locus, the model for haploid Natural Selection is:

$$frac{dp}{dt} = frac{pW_A}{pW_A + (1-p)W_B}$$

, where $p$ is the frequency of the allele $A$, which relative fitness is $W_A$. $W_B$ is the fitness of the allele $B$, which frequency is $1-p$. $frac{dp}{dt}$ is the change in allele frequency through time.

The same model for diploid selection is:

$$frac{dp}{dt} = frac{p^2W_{AA} + p(1-p)W_{AB}}{p^2W_{AA} + 2p(1-p)W_{AB} + (1-p)^2W_{BB}}$$

, where $W_{AB}$ is the fitness of the diploid genotype carrying the allele $X$ on one chromosome and the allele $Y$ on the other chromosome. We define $W_{AB} = W_{BA}$ and therefore we use only the symbol $W_{AB}$ in the above calculations. Note: This equation assumes Hardy-Weinberg equilibrium.

Those equations can be found in any introductory courses in population genetics. For example, here is a source where you can find this same equation for diploid selection. They just express $q$ instead of $1-p$.

My questions are:

• Do these models assume constant population size? Why?
• Do these models assume threshold selection? Why?

Thank you!

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Unless I am missing something the equations you have posted are incorrect. The second equation should be:

$$p' = frac{p^2 W_{AA} + p(1-p)W_{AB}}{p^2W_{AA}+ 2p(1-p)W_{AB}+(1-p)W_{BB}} $$

$p'$ is not interpreted as $frac{dp}{dt}.$

$p'$ is the frequency of allele A in the next generation. If zygotic frequencies via random mating are

$p^2(AA) + 2pq(Aa)+q^2(aa) = 1$

then all of AA individuals and 1/2 the Aa gametes bear the A allele. So we can find $p$ in the next generation as

$p^2+pq = p^2+p(1-p) = p^2+p-p^2 = p$ which shows that in a Hardy-Weinberg population the frequencies are stable.

The assumptions here are those underlying the HW model, which includes infinite population size (hence constant).

Wright-Fisher Model

The Wright–Fisher Model

The Wright–Fisher (WF) model describes a population with discrete, nonoverlapping generations. In each generation the entire population is replaced by the offspring from the previous generation. Parents are chosen via random sampling with replacement. In a haploid population of constant size N, the probability that an allele present in i individuals will be present in j individuals in the next generation is then given by the binomial sampling probability

The transition probabilities P_ij define a discrete-time Markov process on the state space of allele frequencies x(t)=i(t)/N. Expected allele frequencies remain constant across generations, whereas the variance per generation is Var[x]=x(1−x)/N. The probability that an allele eventually becomes fixed is simply its initial frequency. In particular, the fixation probability of a new mutation present in a single copy is 1/N. It is straightforward to extend the WF model to diploid systems by exchanging N→2N, as well as to nonconstant population sizes by taking larger or smaller samples in each generation.

Several alternatives to the WF model have been proposed for describing random genetic drift, notably the Moran and the Cannings models ( Ewens, 2004 ). The Cannings model is a generalization of the WF model that can incorporate arbitrary variance in allele frequency between generations, which can span a wide range in biological populations. The Moran model is a continuous-time model with overlapping generations. Additional levels of biological complexity that one may wish to incorporate include different sexes, age structure, demography, and population substructure. The study of drift in such models is often facilitated by the concept of an effective population size, which specifies the population size of an idealized WF model that approximates evolutionary patterns in the real scenario ( Charlesworth, 2009 ).

Assumptions of the models for haploid and diploid selection - Biology

What are the equilibria of the following equations?

=1 and =0 are the only equilibria.

In the continuous model, dp/dt = (r A -r a ) p (1-p)=0 only when p or 1-p equal zero (or r A =r a ), as in the discrete model.

and and to be greater than zero.

If the denominator is negative, then the numerators must both be negative for and to be greater than zero.

This implies that there are only two cases in which the polymorphic equilibrium is valid:

W Aa > W aa and W Aa > W AA

W AA = 1, W Aa = 1/2, W aa = 1/4:

Keywords: antagonistic selection evolutionary theory haploid selection pollen competition sperm competition.

Conflict of interest statement

The authors declare no conflict of interest.

Figures

Evolution of the selective arena…

Evolution of the selective arena experienced by haploids. ( A ) When the…

Evolution of maternal control when…

Evolution of maternal control when fathers provision. Paternal control selects for reduced haploid…

Discussion

New sex determination systems are typically expected to spread when they equalise the sex ratio and/or when they increase linkage with loci that experience sex differences in selection [33, 34]. In accordance with the latter mechanism, we find that sex differences in selection at the haploid stage can favour cis- or trans-GSD transitions that tighten sex linkage (Conclusion 3A and 3B). Contrary to this expectation, however, we find that trans-GSD transitions can be favoured that loosen linkage with the sex-determining locus, either when linkage is initially tight (Conclusions 1 and 2, Figs 2 and 3) or when there is haploid selection (Conclusion 3C, Figs 4 and 5). Furthermore, we show that the spread of new sex determination systems is not dominated by selection to balance the sex ratio (Conclusions 4 and 5, Fig 5).

On the one hand, sex ratio biases caused by haploid selection can facilitate trans-GSD transitions or transitions from GSD to ESD [42]. For instance, alleles favoured by haploid selection in males often become associated with the Y allele, which leads to an ancestral male-biased zygotic sex ratio. This male bias increases the potential for a neo-W or ESD allele to invade (Table 2), equalising the sex ratio (e.g., see Fig 5B for related examples, see [42]). On the other hand, sex ratio selection can be overwhelmed by additional selective effects, preventing a neo-W or ESD allele from invading, even if it would balance the sex ratio (e.g., when selection also acts in opposite directions in male and female diploids, Table 3). Indeed, transitions between sex-determining systems can generate stronger sex ratio biases (e.g., Fig 5A and step 1 in [43]). In one of our key results, we find that with weak selection, there is no difference in conditions allowing XY-to-ZW and ZW-to-XY transitions (Conclusion 4), even when haploid selection always acts in the same sex (e.g., males). That is, the sex ratio bias created by male haploid selection facilitates the spread of a neo-W allele into an XY system to the same degree that male haploid selection drives the spread of a neo-Y into a ZW system with a 1:1 sex ratio (Fig 5).

Because both fisherian selection to equalise the sex ratio and the benefits of hitchhiking with driven alleles can facilitate transitions among sex chromosome systems, we predict that haploid selection should increase the lability of sex determination systems. Even in animal and plant species that have much larger and more conspicuous diploid phases than haploid phases, many loci have been shown to experience haploid selection through gamete competition and/or meiotic drive [38–41, 51–56], which can generate biased sex ratios [57–64]. In animals, a relatively small proportion of all genes are thought to be expressed and selected during competition in animal sperm [39, 65, 66]. Nevertheless, expression in the gamete is not required for haploid selection if the fitness of a gamete depends on its ability to condense DNA [67]. Furthermore, expression during gamete production often underlies systems of meiotic drive [68–70], which may be a common form of haploid selection in animals [71]. Recent studies have demonstrated that sperm competition, even within a single ejaculate, can alter haploid allele frequencies and increase offspring fitness [72, 73]. In plants, competition among gametophytes may be particularly important. It is estimated that 60%–70% of all genes are expressed in the male gametophyte, and these genes exhibit stronger signatures of selection than randomly chosen genes [74–76]. Furthermore, artificial selection pressures applied to male gametophytes are known to cause a response to selection (e.g., [77–80]).

Linking haploid expression with the evolution of sex-determination, a recent transcriptome analysis in Rumex shows that pollen-biased expression (relative to expression in flower buds or leaves) is enhanced among XY-linked genes, compared to autosomal genes or compared to hemizygous genes that are only linked to the X [81]. In addition, Y-linked genes are overexpressed relative to X-linked genes in pollen (but not in flower buds or leaves). This suggests that the spread of neo-Y chromosomes in this clade could have been favoured through linkage with haploid-selected genes rather than those under sexually antagonistic selection.

Frequent turnovers driven by haploid selection may help to explain the relative rarity of heteromorphic sex chromosomes in plants. If haploid selection is strong, but selective differences between male and female diploids are weak, we specifically predict that trans-GSD transitions are favoured more strongly than cis-GSD transitions, with transitions to ESD intermediate (e.g., with , we have Eq 3). Among the relatively few dioecious clades in which multiple species have well-characterised sex chromosomes [6], trans-GSD transitions have been inferred in Silene otites [15] and in Salicaceae [16, 17]. Assuming that transitions from dioecy to hermaphroditism (equal parental investment in male and female gametes) are favoured in a similar manner to the ESD examined here (equal probability of zygotes developing as males or females), our results suggest that competition among haploid pollen could drive transitions between dioecy and hermaphroditism, which are frequent in plants [82, 83]. To further examine this link, future theory could also include inbreeding, which is an important consideration during transitions between dioecy and hermaphroditism [84]. Future empirical studies could look for evidence of haploid selection acting on former sex chromosomes in hermaphroditic species (e.g., a study such as [81] on ancestral, rather than derived, sex chromosomes).

New sex-determining alleles have previously been shown to spread when they arise in linkage with loci that experience sex differences in selection because beneficial associations build up between alleles that determine sex and alleles that are favoured in that sex [35–37, 43]. In support of this hypothesis, researchers have identified genes on recently derived sex chromosomes that might be under sexually antagonistic selection [21, 85–87]. However, we show that, if selected loci are tightly linked to the ancestral sex-determining locus, they can drive trans-GSD transitions that reduce sex-linkage (Conclusions 1 and 2), thus widening the range of genomic locations where selection could be driving observed trans-GSD transitions. In addition, we find that polymorphic sex-determining systems (X, Y, and neo-W alleles all segregating) can be maintained when a selected locus is tightly linked to the ancestral sex-determining system (e.g., S9B and S9C Fig), which is not possible with loose linkage [36]. This pair of conclusions applies in cases with or without haploid selection.

Our tight linkage result—in particular, the prediction that invasion can lead to polymorphic sex determination—is consistent with empirical data from species in which new feminising mutations are found segregating with ancestral XY loci. For example, in the platyfish (X. maculatus), X,Y, and W alleles segregate at one locus (or two closely linked loci) near potentially sexually antagonistic genes for pigmentation and sexual maturity [44, 88–90]. Furthermore, several rodent species maintain feminising alleles along with the ancestral X and Y sex determination alleles (reviewed in [91]). In nine Akadon rodent species, it appears that male-determining sry expression is suppressed by an autosomal feminising allele (a neo-W allele), creating XY females [92, 93]. XY females have increased fitness relative to XX females [94]. However, it is not yet clear whether loci linked to the feminising factor or the ancestral Y cause this effect. Most convincingly, in Mus minutoides, females can have XX, XX*, or X * Y genotypes [95]. Previous theory would predict that the dominant X* chromosome (potentially an autosome that has fused with the sex chromosome) harbours female-beneficial alleles, driving its spread. However, XX and XX* females have similar fitness, whereas X * Y female fitness is enhanced [96–98]. Although Y-linkage of female-beneficial alleles is counterintuitive, our model suggests that it can be stably maintained when linkage is initially tight between the sex-determining region and the selected locus, subsequently favouring new feminising mutations, which would be a parsimonious explanation for the spread of feminising alleles in this case.

Our models assume that sex-determining alleles do not experience direct selection except via their associations with sex and selected alleles. However, in some cases, there may be significant degeneration around the sex-limited allele (Y or W) in the ancestral sex-determining region because recessive deleterious mutations and/or deletions accumulate in the surrounding nonrecombining regions [99–102]. During trans-GSD transitions, but not cis-GSD transitions, any recessive deleterious alleles linked to the Y or W are revealed to selection in YY or WW individuals [4]. This phenomenon was studied by van Doorn and Kirkpatrick [36], who found that degeneration can prevent fixation of a neo-W or a neo-Y allele, leading to a mixed sex-determining system in which the ancestral and new sex-determining loci are both segregating. However, they noted that very rare recombination events around the ancestral sex-determining locus can allow the completion of trans-GSD transitions. Degeneration around the Y or W could explain why trans-GSD transitions are not observed to be much more common than cis-GSD transitions despite the fact that our models demonstrate that they are favoured under a wider range of conditions, especially with haploid selection. For example, there are a dozen sex chromosome configurations among dipteran species but only one transition between male and female heterogamety [9], but Y degeneration or absence is also very common among Diptera [9].

In this study, we have only considered new sex-determining alleles of large effect. However, we expect similar selective forces to act on masculinising and feminising alleles of weaker effect. For example, small-effect masculinising and feminising alleles within a threshold model of sex determination can be favoured when linked to loci that experience sexually antagonistic selection [37]. These results echo those for large-effect neo-Y and neo-W alleles [35, 36]. It should be noted, however, that the dynamics of sex-determining alleles with very weak effect will be influenced by genetic drift, which itself has been shown to bias transitions towards epistatically dominant sex-determining systems when there is no direct selection [103].

Selection for pollen competitive ability in mixed-mating systems

Coexpression of genes in plant sporophytes and gametophytes allows correlated gametic and sporophytic selection. Theory predicts that, under outcrossing, an allele conferring greater pollen competitive ability should fix within a population unless antagonistic pleiotropy with the sporophyte stage is strong. However, under strong selfing, pollen competitiveness is immaterial as superior and inferior competitors are deposited on opposite stigmas, producing assortative competition. Because many plant species have mixed-mating systems, selfing should be critical in the spread and maintenance of pollen-expressed genes affecting competitiveness. We present two one-locus, two-allele population genetic models for the evolution of a locus controlling pleiotropic antagonism between pollen competitiveness and diploid fitness. Analytical solutions provide minimum and maximum selfing rates allowing invasion of alleles with greater diploid and haploid fitness, respectively. Further, polymorphism is only maintained when diploid selection is recessive. Fixation of the allele conferring greater pollen competitiveness may be prevented, even with weak sporophytic counterselection, with sufficiently high selfing. Finally, selfing expands and limits the range of haploid–diploid selection coefficients allowing polymorphism, depending on dominance and selfing mode.

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