Dominance and the potential for seasonally balanced polymorphism

Genic dominance is a key component of fitness in diploid genotypes. Modelers exploring the conditions for balanced polymorphism under seasonal selection have argued that a reversal of dominance (where the fitness regime cyclically alternates the direction of dominance between a pair of alleles) is a powerful stabilizer of biallelic variation across a broad space of selection intensities. An alternative genetic mechanism, cumulative overdominance (in which the fitness regime maintains a constant direction of dominance), has been argued to preferentially stabilize alleles characterized by strong selection intensities, while requiring an implausibly strict parity under weak selection. Previous analytical conclusions were typically made under the assumption of symmetries for the dominance parameters. Here I investigate generalized dominance schemes for a bivoltine population in order to compare the proportional contribution of these genetic mechanisms to the stabilization of selective polymorphism. In particular, I derive the potential for polymorphism (a measure of the total parameter space conferring stability) for the generalized sex-independent model in four parameters.


Introduc.on
Interest in cyclic regimes of natural selec=on has renewed in recent years, especially with the discovery of hundreds to thousands of allelic oscilla=ons in temperate Drosophila melanogaster, where biallelic sites spread across the genome exhibited seasonal changes of allele frequency of ~5-20% (Bergland et al. 2014, Machado et al. 2021).Results from the determinis=c theory of cyclical selec=on in diploid bivol=ne and mul=vol=ne organisms (2 or 3+ genera=ons per year, respec=vely) include the condi=ons for protected polymorphism at a single-locus (sensu Prout 1968), with models assuming fitnesses to be either sex-independent (Haldane and Jayakar 1963, Bertram and Masel 2019, Ewing 1977, 1979, Hoekstra 1975, Nagylaki 1975, Dempster 1955, Gillespie and Langley 1974, Frank and Slatkin 1990) or sex-dependent (Reinhold 1999,2000, Yamamichi and Hoso 2017); see Johnson et al. (2023) for a recent review of theory and data on fluctua=ng selec=on.Notably, Haldane and Jayakar (1963) derived the condi=ons for balanced polymorphism in the general sex-independent model: the geometric mean fitnesses for each of the homozygous genotypes must be less than that of the heterozygote if polymorphism is to establish in the popula=on.More recent work on the topic focuses on the role of seasonspecific dominance (a component of fitness, along with season-specific selec=on coefficients) in stabilizing varia=on (Bertram and Masel 2019;see Wicmann et al. 2017 for mul=locus results).Bertram and Masel (2019) examined the poten=al for biallelic polymorphism owing to two gene=c mechanisms (reversal of dominance, cumula=ve overdominance) and two ecological mechanisms (boom-bust demography, non-overlapping genera=ons).In their examina=on of the gene=c mechanisms, they derived the condi=ons for protected polymorphism and compared the sizes of the stability-yielding parameter regions when ploced for par=cular values of the dominance coefficient (h ! ) in season 1 and the dominance coefficient (h " ) in season 2 (Table 1).Four parameters being present, they simplified macers by assuming either: (i) equality of dominance parameters (h != h " , which corresponds to symmetric reversal of dominance) or (ii) that the season 1 dominance is the complement of the season 2 dominance (h != 1 − h " , cumula=ve overdominance; Dempster 1955).The pacern that emerges from plofng given dominance values under these assump=ons is that (1) the stability region for cumula=ve overdominance tends to the line s != s " as the selec=on coefficients become small, and (2) dominance reversal, at least for h !, h " less than and not too close to one-half, is far more permissive of unequal selec=on coefficients, with broad intervals of s !, s " even for weak selec=on.Indeed, for complete beneficial dominance reversal (h != h " = 0), any combina=on of selec=on coefficients yields stable polymorphism.The authors conclude that cumula=ve overdominance preferen=ally stabilizes large-effect alleles, and hence may be responsible for the Bergland et al. (2014) result of many large oscilla=ons.While dominance reversal also stabilizes large-effect alleles, the permissive inclusion of small-effect alleles would result in a gene=c architecture of seasonal fitness (assuming diminishing returns epistasis, Wicmann et al. 2017) that is composed predominantly of small oscilla=ons.Dominance reversal thus suffers from a "weak allele problem", they conclude, and therefore is an implausible mechanism for the strong genome-wide seasonal evolu=on of allele frequencies.Superior explana=ons incorporate a mechanism for the preferen=al exclusion of small-effect alleles, as occurs for cumula=ve overdominance and the ecological mechanisms.
Pufng aside the mul=locus and ecological aspects, here I focus on a broader comparison of the two gene=c mechanisms in the single-locus model to see if the general pacern just described holds for viola=ons of the simplifying assump=ons (i) and (ii) above, or in other words, for generalized dominance.I ask: what is the propor=onal contribu=on of each class of dominance scheme (reversal of dominance, cumula=ve overdominance) to the establishment of seasonally balanced polymorphism?
For the generalized sex-independent model in four parameters (s !, h !, s " , h " ), I show that a complete treatment for weak selec=on is possible based on geometric reasoning about the size of parameter space conferring stability, this size being the 'poten=al for polymorphism' (Asmussen andBasnayake 1990, Trocer andSpencer 2007).

Model and Results
Consider a bivol=ne popula=on with discrete, non-overlapping genera=ons that is subject to a two-season selec=on regime.Without loss of generality, suppose that the season 1 selec=on coefficient is greater than or equal to that of season 2. Following Haldane and Jayakar (1963), the condi=ons for a protected biallelic polymorphism are given by the annual fitnesses conforming to the inequality W(Aa) > W(AA) , W(aa), or in terms of the parameters {W(AA, spring This reduces to the Stable Polymorphism (SP) condi=ons: . These inequali=es imply the following: Lemma 1: For weak selec=on coefficients, the SP condi=ons cover the unit square of dominance parameters in the form of a right triangle  with ver=ces at (0, 0), (0, 1), and (α, 0).(Figure 1).
Proof: (0,0) is a vertex since the SP condi=ons include the origin (h != h " = 0).(0, 1) is a vertex since the permissible h " parameter interval is [0,1).(α, 0) is a vertex since the SP condi=ons cross the h !axis when h " = 0 at: The ver=ces (α, 0) and (0, 1) are connected by a line if we ignore products of selec=on coefficients.That is, the upper bound of permissible h !(h !,+,-) conforms to: Corollary 1: For weak selec=on coefficients,  does not intersect the upper right quadrant of the unit square for which < h " ≤ 1 (the region associated with deleterious reversal of dominance).
Proof: Ignoring second-order terms, .∎ Lemma 2a: For weak selec=on coefficients and s2 ≤ s1 < 2s2,  can be subdivided into three regions:   ,   , and .  is a triangle similar to  with ver=ces at 80, Proof: (1) (2) is formed by drawing a midline through  parallel to the h1 axis.By the midpoint theorem, the base of   has length one-half of the base of , and is equal to is the propor=onal contribu=on of the Cumula=ve Overdominance stability condi=ons to the overall poten=al for polymorphism P().

Discussion
The propor=on of parameter space that is stabilized owing to a constant direc=on of dominance was derived for the sex-independent seasonal selec=on model: 25% to 50% of all stabilityconferring dominance schemes maintain a constant direc=on of dominance, with parity of selec=on coefficients tending towards the 50% figure and disparity tending towards the 25% figure.

Generality and rela-on to other models of antagonis-c pleiotropy
While the proofs presented here invoke weak selec=on, selec=on coefficients may range up to, say, 20% without significantly distor=ng the results.This owes to the fact that the curvature of the triangular stable regions remains negligible unless selec=on is rather intense (Fig 1c ); deleterious dominance reversal contributes to the poten=al for polymorphism under these more extreme scenarios of fitness varia=on.Note that all results apply just as well to mul=vol=ne popula=ons that have an equal number of genera=ons per season, the geometric mean fitness criterion being iden=cal to that of the bivol=ne model (Eqn.1).
Supposing a single-locus model with a constant selec=on regime, antagonis=c pleiotropy in which fitness components interact mul=plica=vely (e.g. an allele affec=ng both reproduc=on and viability, Hedrick 1999) is characterized by the same geometric mean fitness criterion as analyzed here, and so the conclusions carry over.For models of antagonis=c pleiotropy in which the harmonic mean fitness of the heterozygotes must exceed that of the homozygotes to establish polymorphism (mul=ple-niche selec=on, Levene 1953), the conclusions apply without altera=on under weak selec=on assump=ons for the two-niche case; this owes to that fact that the iden=cal h !,+,-approxima=on is obtained upon ignoring products of selec=on coefficients.For antagonis=c models in which the criterion is that the arithme=c fitness of the heterozygotes must exceed that of the homozygotes, the weak selec=on approxima=on obtained herein is general (that is, the h !,+,-approxima=on is exact).

Implica-ons for modeling
As men=oned, the sizeable contribu=on of cumula=ve overdominance to stabilizing allelic oscilla=ons is apparent when generalizing beyond the strict complement rela=on h != 1 − h " .Indeed, parallel dominance provides, at worst, a quarter of the poten=al for polymorphism.If selec=on coefficients are not so disparate ( near 1), then the two mechanisms have roughly equal poten=als.These conclusions merit a reconsidera=on of both (1) the defini=on of cumula=ve overdominance, and (2) the contribu=on of cumula=ve overdominance to stabilizing oscilla=ons of allele frequencies.
Regarding the defini=onal issue, the cumula=ve overdominance mechanism has been described as " incomplete dominance [reducing] selec=on against rare alleles in the environments they are unsuited to" (Bertram and Masel 2019, p. 884).Strict constancy of magnitude across seasons is unnecessary for dominance to operate along the spirit of this verbal descrip=on, as an overall constancy to direc=on (with changes in magnitude) is a form of incomplete dominance.Iden=fying cumula=ve overdominance as the intersec=on of the stable polymorphism condi=ons with the upper leu and bocom right quadrants of the unit square (called "parallel dominance" in non-temporal models of antagonis=c pleiotropy, Curtsinger et al. 1994), allows for all stability-conferring dominance schemes to be grouped into two classes (dominance reversal and cumula=ve overdominance/parallel dominance) with no regions leu over (addi=vity in one or both seasons may be classified as cases of cumula=ve overdominance by conven=on, as Bertram and Masel 2019 do for the doubly addi=ve case).Each of these classes is a subset of "geometric mean overdominance", which is the condi=on given by Haldane and Jayakar (1963) (or Eqn 1, present ar=cle, as applies for two-season bivol=nism).
Terminology aside, the theorem of Eqn 5 is helpful in clarifying the role of the two gene=c mechanisms in stabilizing seasonal polymorphism.Whereas several authors have concluded that constant dominance requires an exceedingly =ght interval for small selec=on coefficients, the generalized schemes analyzed here lack this restric=ve feature.The theorem makes clear that the argument of Bertram and Masel (2019), that cumula=ve overdominance preferen=ally stabilizes large-effect alleles due to the =ght-interval effect, depends on the dominance coefficients hewing closely to the complement rela=on.Therefore, generalized dominance as a whole tends to suffer from the "weak-allele problem," with exclusion of small-effect alleles occuring only for special restric=ons on dominance.To the extent that muta=ons are "sampled" evenly from the whole of the dominance unit square, gene=c mechanisms fare poorly as compared to ecological factors in preferen=ally stabilizing a system of many high amplitude oscilla=ons.
The mul=locus simula=ons of Wicmann et al. (2017) an=cipated the general conclusion of the analy=cal results presented here.Namely, their Figure 10F reports a sizeable por=on of dominance schemes that are off the complement line which are able to confer stability (in addi=on to a sizeable por=on in the reversal of dominance region).They suggest the criterion (h !+ h " ) 2 ⁄ < 0.5 for predic=ng stable polymorphism, but this clearly fails under asymmetric selec=on coefficients.A sa=sfactory criterion was derived above: h !< α(1 − h " ), keeping in mind the "without loss of generality" assump=on for labeling alleles.
In their study of the mul=plica=ve model of antagonis=c pleiotropy, Curtsinger et al. (1994) asked what percentage of selec=on parameter combina=ons are stabilized by the parallel dominance quadrants, finding approximately 25% of sampled selec=on pairs to be stabilized.This ques=on is dis=nct from that posed in the present ar=cle, namely: for all dominance schemes in SP, what is the propor=on of dominance schemes residing in the parallel dominance Nevertheless, their numerical results clearly indicate the importance of nonreversing quadrants of the dominance unit square to the stability of polymorphism.
Van Dooren ( 2006) examined a single-locus high-dimensional phenotypic model in which allelic dominance is characterized by a vector of trait-specific values.He concluded that antagonis=c pleiotropy and trait-specific dominance are generally required for stabilizing polymorphism.Importantly, he notes that the maximum possible ra=o of stability-conferring parameter space with dominance reversal to that without dominance reversal "appears to be 3."This is consistent with the 3:1 ra=o of P(): P(  ) for α ≤ !" found here.Hence, the conclusions of a simple two-trait model (fitness in two seasons) seem to carry over not only to other classes of antagonis=c pleiotropy, but to higher-dimensional phenotypes.
It remains to be discovered whether modifiers of the gene=c system respond in a qualita=vely different manner depending on which dominance model is opera=ve.
is formed by substrac=ng   and   from .
consists of the interval on h ! between !" and α, and forms an angle  between the line h !,+,-= α(1 − h " ) and the h !-axis.The height of the right triangle so formed is given by 8α − !" 9 tan .As  is also an angle of , it is seen that tan  = Lemma 2b: For weak selec=on coefficients and s1 ≥ 2s2,  is subdivided into the two regions   and , and   is en=rely absent under these more asymmetric condi=ons.  is as described above and  is formed by subtrac=ng   from .Proof: By assump=on, s1 ≥ 2s2, and so α ≤ !" .It follows that the base of  is less than or equal to ! " , and so  never crosses into the bocom right quadrant of the unit square.∎ Defini=on 1. P() is the poten-al for polymorphism, where P():= Area of .Defini=on 2. The SP condi=ons belonging in   and   are the "Cumula=ve Overdominance stability condi=ons".Defini=on 3. The SP condi=ons belonging in  are the "Reversal of Dominance stability condi=ons".Defini=on 4: The ra=o (): = 1(  ) 4 1(  ) 1() ) = Area of   = ?

Figure 1 .Figure 2 .
Figure 1.Stability regions of the dominance unit square.(a) Lefng the season 1 selec=on coefficient equal 0.5% and the season 2 coefficient equal 0.4%, the stable region intersects the unit square to form a right triangle T (shaded region); open circles and the blue dashed line are excluded from the stable region.The quadrants of the unit square divide T into 3 regions (T1, T2, R).(b) The stability region narrows for increasing asymmetry between the s1, s2 parameters, as seen in reduc=on of area for α = 0.25 as compared to α = 0.75.(c) Extreme selec=on parameters (especially as α → 1) induce curvature in the boundary of the stability regions.