Genomic Imprinting Leads to Less Selectively Maintained Polymorphism on X Chromosomes

Abstract

Population-genetic models are developed to investigate the consequences of viability selection at a diallelic X-linked locus subject to genomic imprinting. Under complete paternal-X inactivation, a stable polymorphism is possible under the same conditions as for paternal-autosome inactivation with differential selection on males and females. A necessary but not sufficient condition is that there is sexual conflict, with selection acting in opposite directions in males and females. In contrast, models of complete maternal-X inactivation never admit a stable polymorphism and alleles will either be fixed or lost from the population. Models of complete paternal-X inactivation are more complex than corresponding models of maternal-X inactivation, as inactivation of paternally derived X chromosomes in females screens these chromosomes from selection for a generation. We also demonstrate that polymorphism is possible for incomplete X inactivation, but that the parameter conditions are more restrictive than for complete paternal-X inactivation. Finally, we investigate the effects of recurrent mutation in our models and show that deleterious alleles in mutation–selection balance at imprinted X-linked loci are at frequencies rather similar to those with corresponding selection pressures and mutation rates at unimprinted loci. Overall, our results add to the reasons for expecting less selectively maintained allelic variation on X chromosomes.

A gene is said to be imprinted when its level of expression is dependent on the parent from which it was inherited. Imprinted loci are characterized by the reduced or nonexpression of either the paternally or maternally derived allele at a particular developmental stage or in a specific tissue type (Reik and Walter 2001). Complete inactivation of an imprinted gene results in functional haploidy, with only one of the two copies of a gene expressed. A well-characterized example of an imprinted gene is insulin-like growth factor II in mice (Igf2) and humans (IGF2)—in most tissues the maternally derived copy of the gene is inactive and only the paternally derived copy is expressed (DeChiara et al. 1991; Giannoukakis et al. 1993). To date, 250 imprinted genes have been identified in mammals, including 73 in humans, and many of these genes are thought to be involved in traits such as growth and development (Morison et al. 2005).

A number of studies have demonstrated that genomic imprinting and imprinting-like phenomena may also occur at X-linked loci. It is worth noting that X chromosome inactivation in marsupials and in some tissues of other mammals is formally equivalent to standard single-locus genomic imprinting, since expression or nonexpression of an allele at an X-linked locus is dependent on the parent from which it was inherited. In addition to the well-known examples of paternal-X inactivation in extraembryonic membranes of some eutherian females, including mice, rats, and cattle (Takagi and Sasaki 1975; Wake et al. 1976; West et al. 1977; Harper et al. 1982; Xue et al. 2002; Wagschal and Feil 2006), in preimplantation rodent embryos (Latham 2005) and somatic tissues of kangaroos and other marsupials (Sharman 1971; Graves 1996, Johnston et al. 2002), there is also evidence of such phenomena in the somatic tissues of eutherian mammals. Preferential inactivation of the maternally derived X chromosome (rather than the usual pattern of random inactivation; Graves 1996) has been observed in the somatic cells of female mice (Cattanach and Perez 1970; Falconer and Isaacson 1972; Falconer et al. 1982). More circumstantially, Hunt (1991) demonstrated that the viability of female mice with only one X chromosome was lower in mice whose X chromosome was paternally derived, suggesting that a number of genes relating to fitness may be expressed exclusively from the maternally inherited X chromosome. In addition, mice inheriting only a paternal X chromosome show developmental retardation in early organogenesis, whereas mice with a maternally inherited X chromosome develop normally (Thornhill and Burgoyne 1993; Jamieson et al. 1998). Kay et al. (1994) established that the Xist gene, which influences X inactivation and is located on the long arm of the X chromosome, is imprinted in mice. Female embryos initially express only the paternal Xist allele, but later allele expression is random with respect to parental origin.

Studies of Turner syndrome in humans, a disorder of females in which all or part of one X chromosome is deleted (Jacobs et al. 1990), provide growing evidence of imprinting at one or more loci on the human X chromosome (Knickmeyer 2012). For example, in a group with only one X chromosome, social adjustment was significantly better in individuals whose chromosome was paternally derived (Skuse et al. 1997), suggesting that a locus (or loci) for social cognition may be inactivated on maternally derived X chromosomes. Further, Bishop et al. (2000) found a significant difference in patterns of memory function between individuals whose only X chromosome was paternally derived and those whose X chromosome was maternally derived. This result suggests that there are one or more imprinted genes on the X chromosome that affect memory function (but see Ross et al. 2006).

Six putative imprinted genes or gene clusters on the X chromosome are currently listed in the Catalogue of Imprinted Genes (Morison et al. 2005), with evidence for both paternal and maternal inactivation. While complete paternal X inactivation results in hemizygosity for both males and females, maternal X inactivation in both male and female offspring means that males lack expression of the gene, whereas maternal X inactivation in females allows only hemizygosity for both males (expression of the maternal allele) and females (expression of the paternal allele). A number of genes are completely maternally inactivated (for example, MAP7D2 is expressed from the paternally derived alleles in adult human females and has very low levels of expression in males, suggesting inactivation of the maternal allele in both sexes; Niida and Yachie 2011). There is currently no evidence of female-limited maternal inactivation of genes on the X chromosome. Nevertheless, it is predicted that sexually dimorphic imprinting may play an important role in the resolution of intralocus sexual conflict (Day and Bonduriansky 2004; Bonduriansky and Chenoweth 2009).

To date, there has been limited investigation of the population genetics of sex-linked imprinting. Most current theoretical models focus on the evolutionary origin of X-linked imprinting (for example, Iwasa and Pomiankowski 2001; Spencer et al. 2004, Haig 2006; Seymour and Pomiankowski 2006). This article expands upon two population-genetic models for imprinting at a sex-linked locus, namely Cooper (1976), who derived equilibria under viability selection for complete paternal-X inactivation, and Seymour and Pomiankowski (2006), who, although focusing on the evolution of imprinting in the context of two models where selection favors different levels of gene expression in males and females, also derived equilibria and local stability conditions for a general model where each of the six possible genotypes may have a different fitness (see Table 1). Selection at X-linked loci is of general interest, for example, because sex chromosomes are thought to be particularly important in the generation of sexual dimorphism through sexually antagonistic selection (Mank 2009; Connallon and Clark 2010). Cooper’s model of viability selection is extended to apply to two models of maternal-X inactivation, and the equilibrium conditions are derived. The models are extended to explicitly encompass incomplete inactivation; that is, imprinting is treated as a quantitative phenomenon, and the equilibrium allele frequencies, feasibility, and local stability conditions are shown to be equivalent to those of Seymour and Pomiankowski (2006). There is considerable evidence that parent-of-origin inactivation of alleles on the X chromosome is dependent on the developmental stage or tissue type (see, for example, the Catalogue of Imprinted Genes, Morison et al. 2005), and polymorphism in imprinting status (albeit often with restrictive conditions) has been predicted by a number of theoretical models (for example, Spencer et al. 2004; Haig 2006; Seymour and Pomiankowski 2006). In this context the effect of imprinting on viability selection can be treated as incomplete and quantitative in nature. In addition to the selection models, mutation–selection balance at an imprinted, sex-linked locus is also investigated and, for all systems, the feasibility and stability of equilibrium allele frequencies is determined. Finally, in the discussion, we contrast our selection and mutation–selection models for imprinting on the X chromosome with dominance on the X chromosome and with imprinting at an autosomal locus.

Table 1 . Frequencies and viabilities for the genotypes at a diallelic X chromosome locus, $(s,t\le 1,\hspace{0.17em}0\le k_1,k_2,h\le 1,\hspace{0.17em}{w}_i>0)$ .

	Females				Males
Genotype:	A 1 A 1	A 1 A 2	A 2 A 1	A 2 A 2	A 1	A 2
Genotypic frequencies	$p_{\text{f}}{p}_{\text{m}}$	$p_{\text{f}}(1-p_{\text{m}})$	$(1-p_{\text{f}})p_{\text{m}}$	$(1-p_{\text{f}})(1-p_{\text{m}})$	$p_{\text{f}}$	$(1-p_{\text{f}})$
Viability: complete paternal inactivation (see Cooper 1976)	1 − s	1 − s	1	1	1 − t	1
Viability: complete maternal inactivation in female offspring only	1 − s	1	1 − s	1	1 − t	1
Viability: complete maternal inactivation in both male and female offspring	1 − s	1	1 − s	1	1	1
Viability: incomplete inactivation	1 − s	1 − k 2 s	1 − k 1 s	1	1 − t	1
Viability: dominance, no imprinting	1 − s	1 − hs	1 − hs	1	1 − t	1
Viability: general case (see Seymour and Pomiankowski 2006)	w 1	w 2	w 3	w 4	w 5	w 6

The Imprinting Models

We examine the frequencies of two alleles at a single X-linked locus subject to complete inactivation in viability selection and mutation–selection models. We make the standard Hardy–Weinberg assumptions of no genetic drift, no migration, separate generations, and, where appropriate, no mutation.

Consider two alleles, A ₁ and A ₂, at a single locus on the X chromosome. Let $p_{\text{f}}$ and $p_{\text{m}}$ represent the frequency of A ₁ in the gametes of females (f) and males (m). In females, the homozygotes A ₁ A ₁ and A ₂ A ₂ have phenotypes A ₁ and A ₂ respectively, while the phenotype of the heterozygotes is dependent on the genotype of the expressed X chromosome. Writing the maternally derived allele first, paternal inactivation means that A ₁ A ₂ heterozygotes are phenotypically A ₁ and A ₂ A ₁ heterozygotes are phenotypically A ₂, and vice versa under maternal inactivation (see Table 1).

In the following we present the main results from the selection and selection–mutation models of X-linked imprinting. Full mathematical details are provided in supporting information, File S1.

Selection models

Model 1A. Complete paternal X inactivation:

Following Cooper (1976), let us assume that the paternally derived alleles are inactivated and that the A ₁ phenotype has viability (1 − s) in females and (1 − t) in males, $(s,t\le 1)$(Table 1). This framework allows the possibility of sexual conflict; when s and t differ the viability of the A ₁ allele is different in the two sexes.

As Cooper (1976) showed, the recursion equations for allele frequencies are

$$\begin{array}{l}{p^{\prime }}_{\text{f}}=0.5\left[1+p_{\text{m}}-\frac{1-p_{\text{f}}}{1-p_{\text{f}}s}\right]\\ {p^{\prime }}_{\text{m}}=1-\frac{1-p_{\text{f}}}{1-p_{\text{f}}t}.\end{array}$$ (1)

This system affords three equilibrium allele frequencies (i.e., frequencies at which ${p^{\prime }}_{\text{f}}=p_{\text{f}}$ and ${p^{\prime }}_{\text{m}}=p_{\text{m}}$), ${\widehat{p}}_{\text{f}}={\widehat{p}}_{\text{m}}=0$ and ${\widehat{p}}_{\text{f}}={\widehat{p}}_{\text{m}}=1$, corresponding respectively to loss and fixation of the A ₁ allele in both sexes, and

$${\widehat{p}}_{\text{f}}=\frac{s+t}{2st}\text{,}\hspace{0.17em}{\widehat{p}}_{\text{m}}=\frac{(1-t)(s+t)}{t(s-t)}$$. (2)

This polymorphic equilibrium is biologically feasible (i.e., $0<{\widehat{p}}_{\text{f}},{\widehat{p}}_{\text{m}}<1$) provided s + t < 0 and s + t − 2st > 0.

Standard local stability analysis (examining the leading eigenvalue of the system of equations linearized around the equilibrium; see Edelstein-Keshet 1988) shows that the polymorphic equilibrium is locally stable whenever it is feasible and, indeed, whenever both fixation equilibria are unstable (Table 2).

Table 2. Feasibility and stability conditions for equilibria under complete imprinting at a diallelic X chromosome locus, $(s,t\le 1)$ .

Equilibrium	Paternal X inactivation	Maternal X inactivation, female offspring only	Maternal X inactivation, female and male offspring
$p_{\text{f}}=p_{\text{m}}=0$ (fixation of A 2 allele)	s + t > 0	s + t − st > 0	s > 0
$p_{\text{f}}=\left(s+t\right)/2st$, $p_{\text{m}}=(1-t)(s+t)/\left(t(s-t)\right)$(polymorphism)	s + t < 0 and s + t − 2st > 0	No internal equilibrium	No internal equilibrium
$p_{\text{f}}=p_{\text{m}}=1$ (fixation of A 1 allele)	s + t − 2st < 0	s + t − st < 0	s < 0

Moreover, since the entries of the Jacobian matrix, $\partial ({p^{\prime }}_{\text{f}},{p^{\prime }}_{\text{m}})/\partial (p_{\text{f}},p_{\text{m}})$, are nonnegative (see Karlin 1972), the polymorphic equilibrium is globally stable whenever it is locally stable. Thus, if s + t < 0 and s + t − 2st > 0 so that both fixation equilibria are unstable, allele frequencies will always converge to the polymorphic equilibrium, an outcome that requires that s and t are of opposite sign. Nevertheless, this condition (st < 0) is not sufficient (see Figure 1) and, indeed, when selection is weak in one sex, there is a limited range that the other selection coefficient may take for the polymorphic equilibrium to exist and be stable (Cooper 1976). Unless selection for the A ₁ allele is markedly different in the two sexes, it is unlikely that a polymorphic equilibrium would be observed.

Figure 1 .

Plot of the stability and feasibility conditions for the three equilibria possible for X-linked complete paternal inactivation: (1) $p_{\text{f}}=p_{\text{m}}=0$, requiring s + t > 0, (2) $p_{\text{f}}=\left(s+t\right)/2st,\hspace{0.17em}{p}_{\text{m}}=(1-t)(s+t)/\left(t(s-t)\right)$, requiring s + t < 0 and s + t − 2st > 0, and (3) $p_{\text{f}}=p_{\text{m}}=1$, requiring s + t − 2st < 0. Note that s + t = 0 and s + t − 2st = 0 represent the border between dark and light shading and the border between light shading and no shading, respectively.

Model 1B. Complete maternal X inactivation, females only:

We assume that the maternally derived alleles are inactivated (but only in female offspring) and again that the A ₁ phenotype has viability (1 − s) in females and (1 − t) in males, $(s,t\le 1)$(Table 1).

The recursion equations for allele frequencies are now

$$\begin{array}{l}{p^{\prime }}_{\text{f}}=0.5\left[1+p_{\text{f}}-\frac{1-p_{\text{m}}}{1-p_{\text{m}}s}\right]\\ {p^{\prime }}_{\text{m}}=1-\frac{1-p_{\text{f}}}{1-p_{\text{f}}t}.\end{array}$$ (3)

There are now just two equilibrium allele frequencies, $p_{\text{f}}=p_{\text{m}}=0$ and $p_{\text{f}}=p_{\text{m}}=1$, corresponding to loss and fixation of the A ₁ allele in both sexes. The above procedure of Karlin (1972) shows one of these to be both locally and globally stable depending on the sign of s + t − st (Table 2).

Model 1C. Complete maternal X inactivation, males and females:

We assume that the maternally derived alleles are inactivated in both male and female offspring and that the A ₁ phenotype has viability (1 − s) in females and there is no selection in males $(s\le 1)$ (Table 1).

The recursion equations for allele frequencies are now

$$\begin{array}{l}{p^{\prime }}_{\text{f}}=0.5\left[1+p_{\text{f}}-\frac{1-p_{\text{m}}}{1-p_{\text{m}}s}\right]\\ {p^{\prime }}_{\text{m}}=p_{\text{f}}.\end{array}$$ (4)

As expected given the absence of sexual conflict to maintain a polymorphic equilibrium, there are again just two equilibrium allele frequencies, $p_{\text{f}}=p_{\text{m}}=0$ and $p_{\text{f}}=p_{\text{m}}=1$, corresponding to loss and fixation of the A ₁ allele in both sexes. The above procedure of Karlin (1972) shows one of these to be both locally and globally stable depending on the sign of s (Table 2).

Model 1D. Incomplete inactivation:

A more general treatment is to let the expression of maternally and paternally derived X alleles differ, such that the viability of reciprocal heterozygotes is not equal and is intermediate between the viability of the two homozygotes. We define the viabilities as (1 − s) for the A ₁ A ₁ genotype, (1 − k ₂ s) for the A ₁ A ₂ genotype, (1 − k ₁ s) for the A ₂ A ₁ genotype, and 1 for the A ₂ A ₂ genotype in females. The parameters k ₁ and k ₂ take values between 0 and 1. When k ₁ = k ₂ there is no imprinting, when k ₁ = 1 − k ₂ there is no dominance, and when k ₁ = k ₂ = 0.5 there is no imprinting or dominance. We assume that there is some expression of the X chromosome allele in males, such that the viability is (1 − t) for the A ₁ genotype and 1 for the A ₂ genotype (Table 1).

The recursion equations for allele frequencies are now

$$\begin{array}{l}{p^{\prime }}_{\text{f}}=0.5\left[1-\frac{1-p_{\text{f}}-p_{\text{m}}-p_{\text{f}}{p}_{\text{m}}s}{1-s(k_1{p}_{\text{m}}(1-p_{\text{f}})+k_2{p}_{\text{f}}(1-p_{\text{m}})+p_{\text{f}}{p}_{\text{m}})}\right]\\ {p^{\prime }}_{\text{m}}=1-\frac{1-p_{\text{f}}}{1-t{p}_{\text{f}}}.\end{array}$$ (5)

This system affords three equilibrium allele frequencies, ${\widehat{p}}_{\text{f}}={\widehat{p}}_{\text{m}}=0$ and ${\widehat{p}}_{\text{f}}={\widehat{p}}_{\text{m}}=1$, and the polymorphic equilibrium

$$\begin{array}{c}{\widehat{p}}_{\text{f}}=\frac{k_{2}s+t+k_{1}s(1-t)}{2s((1-t)(k_1-1)+k_2)},\\ {\widehat{p}}_{\text{m}}=\frac{(1-t)(s(k_1+k_2)+t-k_{1}st)}{2s(-1+k_1+k_2)+st(2-k_2-3k_1)+t^2(-1+k_{1}s)},\end{array}$$ (6)

which is equivalent to (2) when k ₁ = 0 and k ₂ = 1. The polymorphic equilibrium is feasible provided s((−2 + k ₁)(1 − t) + k ₂) − t < 0 and s(−k ₁ − k ₂ + k ₁ t) − t > 0. Seymour and Pomiankowski (2006) derived the recursion equations (see their Equation 2), polymorphic equilibrium (see their Equation A1), and local stability conditions (see their Appendix A) for the general case where all six genotypes (four in females and two in males, see Table 1) have separate viabilities; our recursion equations and equilibria above are, as expected, equivalent when the appropriate viabilities are substituted.

Mutation–selection models

We now incorporate recurrent mutation into models 1A–1D and investigate its effect on the above selectively maintained fixations. In particular, we investigate mutation from A ₁ to A ₂ when A ₁ would otherwise be fixed by selection and from A ₂ to A ₁ when A ₂ would otherwise be fixed by selection.

Model 2A(i). Complete paternal X inactivation, mutation from A ₂ to A ₁:

Suppose A ₂ mutates to A ₁ at a rate μ. The recursion equations are now

$$\begin{array}{l}{p}_{\text{f}}^{\prime }=p_{\text{f}}^s(1-\mu )+\mu \\ p_{\text{m}}^{\prime }=p_{\text{m}}^s(1-\mu )+\mu ,\end{array}$$

in which $p_{\text{f}}^s$ and $p_{\text{m}}^s$ are $p_{\text{f}}^{\prime }$ and $p_{\text{m}}^{\prime }$ of (1), respectively.

The equilibria of the system are given by

$${\widehat{p}}_{\text{f}}=\mu \left[\frac{3}{s+t}\right]\text{and}{\widehat{p}}_{\text{m}}=\mu \left[\frac{3+s-2t}{s+t}\right]$$, (7)

$${\widehat{p}}_{\text{f}}=\frac{s+t}{2st}+\mu \left[\frac{2s^2+t^2-3st}{2st(s+t)}\right]$$

and

$${\widehat{p}}_{\text{m}}=\frac{(1-t)(s+t)}{t(s-t)}+\mu \left[\frac{3s^2-t^2-2s^{2}t+4s{t}^2-4st}{t(s+t)(s-t)}\right]$$

or

$${\widehat{p}}_{\text{f}}={\widehat{p}}_{\text{m}}=1$$

with terms in μ² and above ignored.

The equilibrium of biological interest, (7), occurs when the fixation of A ₂ is stable in Model 1 (i.e., when s + t > 0). Feasibility additionally requires $\mu <\left(s+t\right)/3$ whenever $t<s/2$ and $\mu <\left(s+t\right)/\left(3+s-2t\right)$ whenever $t>s/2.$ By ignoring terms in μ² and above, approximate stability conditions for this equilibrium are s + t > 0 and

$$\mu <\frac{A(s+t)(A-s-3)}{A(s-1)(5s-t)+3s+10s^2-5s^3-15t-10st+s^{2}t+16t^2}$$,

where $A=\sqrt{9-2s+s^2-8t}$. As expected, the mutation–selection balance given by (7) is feasible and stable when the pure selection A ₂ fixation is feasible and stable, and mutation is sufficiently rare. Further, equilibria are globally stable wherever they are locally stable, provided μ is not large.

Model 2A(ii). Complete paternal X inactivation, mutation from A ₁ to A ₂:

We now consider the perturbation of the A ₁ fixation by mutation. Let ν be the mutation rate from A ₁ to A ₂. The recursions are now

$$\begin{array}{l}{p}_{\text{f}}^{\prime }=p_{\text{f}}^s(1-v)\\ p_{\text{m}}^{\prime }=p_{\text{m}}^s(1-v).\end{array}$$

Again, this system admits three equilibria, but the only one of biological interest is

$${\widehat{p}}_{\text{f}}=1-v\left[\frac{3(s-1)(t-1)}{2st-s-t}\right]\hspace{0.17em}\text{and}\hspace{0.17em}{\widehat{p}}_{\text{m}}=1-v\left[\frac{3+2st-4s-t}{2st-s-t}\right].$$ (8)

This equilibrium is feasible provided s + t − 2st < 0 (which ensures that the A ₁ fixation is feasible and stable under pure selection), and $v<\left(2st-s-t\right)/\left(3(s-1)(t-1)\right)$ whenever $t>s/\left(2-s\right)$ and $v<\left(2st-s-t\right)/\left(3-4s+2st-t\right)$ whenever $t<s/\left(2-s\right)$. By ignoring terms in ν² and above, approximate stability conditions for this equilibrium are s + t − 2st < 0 and

$$\begin{array}{l}v<\left(-2{(2st-s-t)}^2\right)/[3s+2s^2+3t-15st+2s^{2}t+t^2+4s{t}^2\\ -\frac{(t-1)(-8s^3(1+t)+t(9+t)+s^2(4+38t)+s(9+t(4t-49)))}{\sqrt{(t-1)(-9+16s-8s^2+t)}}]\end{array}$$

Provided mutation is rare, equilibria are globally stable wherever they are locally stable. Thus, the mutation–selection balance given by (8) is feasible and stable when the selection-only fixation of A ₂ is feasible and stable and mutation is sufficiently rare.

Model 2B(i). Complete maternal X inactivation, females only, mutation from A ₂ to A ₁:

Again, suppose A ₂ mutates to A ₁ at a rate μ. The recursion equations are

$$\begin{array}{l}{p}_{\text{f}}^{\prime }=p_{\text{f}}^s(1-\mu )+\mu \\ p_{\text{m}}^{\prime }=p_{\text{m}}^s(1-\mu )+\mu \end{array}$$

in which $p_{\text{f}}^s$ and $p_{\text{m}}^s$ are $p_{\text{f}}^{\prime }$ and $p_{\text{m}}^{\prime }$ of (3), respectively. Under maternal-X inactivation, two equilibria are possible. The equilibrium of biological interest is

$${\widehat{p}}_{\text{f}}=\mu \left[\frac{3-s}{s+t-st}\right]\hspace{0.17em}\text{and}\hspace{0.17em}{\widehat{p}}_{\text{m}}=\mu \left[\frac{3-2t}{s+t-st}\right]$$

and represents the perturbation from fixation of the A ₂ allele by mutation. It is feasible provided s + t − st > 0 (and so the fixation of A ₂ under selection alone is stable), and $\mu <\left(s+t-st\right)/\left(3-2t\right)$ when $t>s/2$ and $\mu <\left(s+t-st\right)/\left(3-s\right)$ when $t<s/2$ (which means that mutation must be sufficiently rare). In addition, the local and global stability conditions require that

$$\mu <\frac{B(3-B)}{16(1-s)(1-t)-B-1}$$,

where $B=\sqrt{1+8(1-s)(1-t)}$, ignoring terms in μ² and above.

Model 2B(ii). Complete maternal X inactivation, females only, mutation from A ₁ to A ₂:

Finally, we incorporate a mutation rate ν from A ₁ to A ₂. The recursion equations are

$$\begin{array}{l}{p}_{\text{f}}^{\prime }=p_{\text{f}}^s(1-v)\\ p_{\text{m}}^{\prime }=p_{\text{m}}^s(1-v)\end{array}$$

and the equilibrium of biological relevance can be shown to be

$${\widehat{p}}_{\text{f}}=1-v\left[\frac{(2s-3)(t-1)}{st-s-t}\right]\hspace{0.17em}\text{and}\hspace{0.17em}{\widehat{p}}_{\text{f}}=1-v\left[\frac{(s-1)(t-3)}{st-s-t}\right].$$

This equilibrium, close to the selection only fixation of A ₁, is feasible provided st − s − t > 0 and $v<\left(st-s-t\right)/\left((2s-3)(t-1)\right)$ when $t>s/\left(2-s\right)$ and $v<\left(st-s-t\right)/\left((s-1)(t-3)\right)$ when $t<s/\left(2-s\right)$. Local and global stability conditions require that

$$v<\frac{C(3(1-s)(1-t)-C)}{(1-s)(1-t)(16-(s-1)(t-1)-C)}$$,

where $C=\sqrt{(1-s)(1-t)(8+(1-s)(1-t))}$, ignoring terms in ν² and above.

Model 2C(i). Complete maternal X inactivation, males and females, mutation from A ₂ to A ₁:

Again, suppose A ₂ mutates to A ₁ at a rate μ. The recursion equations are

$$\begin{array}{l}{p}_{\text{f}}^{\prime }=p_{\text{f}}^s(1-\mu )+\mu \\ p_{\text{m}}^{\prime }=p_{\text{m}}^s(1-\mu )+\mu ,\end{array}$$

in which $p_{\text{f}}^s$ and $p_{\text{m}}^s$ are $p_{\text{f}}^{\prime }$ and $p_{\text{m}}^{\prime }$ of (4), respectively. Under maternal-X inactivation of both males and females, two equilibria are possible. The equilibrium of biological interest is

$${\widehat{p}}_{\text{f}}=\mu \left[\frac{3-s}{s}\right]\hspace{0.17em}\text{and}\hspace{0.17em}{\widehat{p}}_{\text{m}}=\mu \left[\frac{3}{s}\right]$$

and represents the perturbation from fixation of the A ₂ allele by mutation. It is feasible provided that s > 0 (and so the fixation of A ₂ under selection alone is stable) and $\mu <s/\left(3-s\right)$ (which means that mutation must be sufficiently rare). In addition, the local and global stability conditions require that

$$\mu <\frac{D(3-D)}{15-16s-D}$$,

where $D=\sqrt{9-8s}$, ignoring terms in μ² and above.

Model 2C(ii). Complete maternal X inactivation, males and females, mutation from A ₁ to A ₂:

Finally, we incorporate a mutation rate ν from A ₁ to A ₂. The recursion equations are

$$\begin{array}{l}{p}_{\text{f}}^{\prime }=p_{\text{f}}^s(1-v)\\ p_{\text{m}}^{\prime }=p_{\text{m}}^s(1-v)\end{array}$$

and the equilibrium of biological relevance can be shown to be

$${\widehat{p}}_{\text{f}}=1-v\left[\frac{2s-3}{s}\right]\hspace{0.17em}\text{and}\hspace{0.17em}{\widehat{p}}_{\text{m}}=1-v\left[\frac{3(s-1)}{s}\right]$$

This equilibrium, close to the selection-only fixation of A ₁, is feasible provided that s < 0 and $v<s/\left(2s-3\right)$. Local and global stability conditions require that

$$v<\frac{E(3(1-s)-E)}{(1-s)(15+s-E)}$$,

where $E=\sqrt{(1-s)(9-s)}$, ignoring terms in ν² and above.

Model 2D(i). Incomplete imprinting, mutation from A ₂ to A ₁:

Again, suppose A ₂ mutates to A ₁ at a rate μ. The recursion equations are

$$\begin{array}{l}{p}_{\text{f}}^{\prime }=p_{\text{f}}^s(1-\mu )+\mu \\ p_{\text{m}}^{\prime }=p_{\text{m}}^s(1-\mu )+\mu ,\end{array}$$

in which $p_{\text{f}}^s$ and $p_{\text{m}}^s$ are $p_{\text{f}}^{\prime }$ and $p_{\text{m}}^{\prime }$ of (5), respectively. Under maternal-X inactivation of both males and females, two equilibria are possible. The equilibrium of biological interest is

$${\widehat{p}}_{\text{f}}=\mu \left[\frac{3-k_{1}s}{s(k_1+k_2)+t-k_{1}st}\right]$$

and

$${\widehat{p}}_{\text{m}}=\mu \left[\frac{3+k_{2}s-2t}{s(k_1+k_2)+t-k_{1}st}\right]$$

and represents the perturbation from fixation of the A ₂ allele by mutation.

Model 2D(ii). Incomplete imprinting, mutation from A ₁ to A ₂:

Finally, we incorporate a mutation rate ν from A ₁ to A ₂. The recursion equations are

$$\begin{array}{l}{p}_{\text{f}}^{\prime }=p_{\text{f}}^s(1-v)\\ p_{\text{m}}^{\prime }=p_{\text{m}}^s(1-v)\end{array}$$

and the equilibrium of biological relevance can be shown to be

$${\widehat{p}}_{\text{f}}=1-v\left[\frac{(-3+s(2+k_2))(1-t)}{s(2-k_1-k_2)+t-st(2-k_1)}\right]$$

and

$${\widehat{p}}_{\text{m}}=1-v\left[\frac{-3+s(4-k_1)+t-st(2-k_1)}{s(2-k_1-k_2)+t-st(2-k_1)}\right]$$.

Discussion

Our modeling reveals a critical difference between the effects of viability selection on complete paternal vs. maternal inactivation of X-linked loci: only the former can lead to a selectively maintained polymorphism. This difference arises because of the sexual asymmetry in the number of X chromosomes. Under paternal-X inactivation, the X chromosome derived from the father is hidden from selection for a generation when it is inactivated in his daughters. With maternal-X inactivation, by contrast, the paternal X is exposed to selection in daughters and the maternal X in sons, so that selection is effectively acting on a haploid system. We note that maintenance of allelic variation by viability selection on another naturally haploid system, Y chromosome loci, is also impossible (Clark 1987). Nevertheless, even under paternal inactivation, the conditions for the maintenance of polymorphism are restrictive. Not only must the selection coefficients act in opposing directions in the two sexes—that is, there is sexual conflict, such that the A ₁ allele is beneficial in one sex but costly in the other—they must be of similar magnitude, especially when weak (Cooper 1976; see Figure 1). Intuitively, the maintenance of polymorphism requires not only that there is sexual conflict, but that the harm (i.e., reduction in viability) in one sex is of similar magnitude to the benefit (i.e., increase in viability) in the other. Imbalance in the strength of selection between sexes means that either the A ₁ or A ₂ allele will be driven to fixation, as the costs to one sex outweigh the benefits to the other (or vice versa) (Rice 1984).

It is revealing to compare the equilibria under complete paternal-X inactivation with those under complete maternal autosomal inactivation with two sex viabilities (Pearce and Spencer 1992; Anderson and Spencer 1999). Following the convention of female and male viabilities being (1 − s) and (1 − t), respectively, we find that their three equilibria are equivalent to the three equilibria above, with both equilibrium frequencies for males and females and the selection coefficients s and t swapped (Table 3 and File S2). Further, the stability conditions for the maintenance of a stable internal polymorphism are the same, whereas conditions for the stability of each of the fixation equilibria are swapped (results not shown). This swapping is necessitated because the previous results concern maternal inactivation, whereas the X-linked polymorphism occurs only under paternal inactivation. Indeed, by deriving the equivalent paternal autosomal inactivation recursion equations to those in Pearce and Spencer (1992) and Anderson and Spencer (1999), we find that the equilibria (Table 3 and File S2) and stability conditions (results not shown) are identical to those under paternal-X inactivation.

Table 3. Comparison of equilibria for X-linked and autosomal imprinting with two sex viabilities, $(s,t\le 1)$ .

X-linked paternal inactivation	${\widehat{p}}_{\text{f}}={\widehat{p}}_{\text{m}}=0$, ${\widehat{p}}_{\text{f}}={\widehat{p}}_{\text{m}}=1$ and ${\widehat{p}}_{\text{f}}=\left(s+t\right)/2st$, ${\widehat{p}}_{\text{m}}=(1-t)(s+t)/(t(s-t))$
Autosomal maternal inactivation	${\widehat{p}}_{\text{f}}={\stackrel{}{p}}_{\text{m}}=0$, ${\widehat{p}}_{\text{f}}={\widehat{p}}_{\text{m}}=1$ and ${\widehat{p}}_{\text{f}}=(1-s)(s+t)/(s(t-s))$, ${\widehat{p}}_{\text{m}}=\left(s+t\right)/2st$
Autosomal paternal inactivation	${\widehat{p}}_{\text{f}}={\widehat{p}}_{\text{m}}=0$, ${\widehat{p}}_{\text{f}}={\widehat{p}}_{\text{m}}=1$ and ${\widehat{p}}_{\text{f}}=\left(s+t\right)/2st$, ${\widehat{p}}_{\text{m}}=(1-t)(s+t)/(t(s-t))$

This equivalence between autosomal and the X chromosome two-sex viability imprinting models contrasts with the lack of correspondence between nonimprinting models for differential male and female selection at autosomal and sex-linked loci (Owen 1953; Bennett 1957, 1958). Under standard Mendelian expression, two alleles in both males and females are exposed to selection on an autosome whereas two alleles in females and only one allele in males are exposed to selection at an X-linked locus. In both autosomal and sex-linked imprinting models, however, a single active chromosome is exposed to selection in both males and females. Nevertheless, although viability selection can maintain genetic variation at an unimprinted X-linked locus (Bennett 1957, 1958), the likelihood of such maintenance has been considered low, because, as in our imprinted case, the conditions, especially when selection is weak, are restrictive (Bennett 1958).

An interesting aspect in the general model of incomplete inactivation is whether a stable polymorphism is possible, and if so, the parameter space available for the polymorphism to be maintained. As shown in the results above, complete inactivation of the maternal X in females does not allow a stable polymorphism, despite the potential for selection to act in opposing directions in the two sexes. This lack of polymorphism is clear when examining the expression for equilibrium allele frequencies under incomplete inactivation in (6); when k ₁ = 1 and k ₂ = 0, the denominator for ${\widehat{p}}_f$ is zero and the equilibrium allele frequency does not exist. This also occurs when k ₂ = (1 − t)(1 − k ₁); assuming t is small (i.e., selection is weak), polymorphism requires that the sum of k ₁ and k ₂ is not too close to 1. Further, by examining the polymorphic equilibrium for incomplete imprinting (6), it can be seen that feasibility and stability conditions are the least restrictive for paternal-X inactivation and become more restrictive as the degree of inactivation tends toward maternal-X inactivation. For example, let t = −0.1. A polymorphic equilibrium is possible for s for a range of 0.017 (0.083 < s < 0.1) when k ₁ = 0 and k ₂ = 1, for a range of 0.011 (0.086 < s < 0.097) when k ₁ = 1/3 and k ₂ = 2/3, and for a range of 0.006 (0.088 < s < 0.094) when k ₁ = 2/3 and k ₂ = 1/3. Thus, everything else being equal, X-linked loci with incomplete inactivation are less likely to be variable than loci with complete paternal inactivation.

The incorporation of recurrent mutation into our viability selection models shows that disadvantageous alleles can be maintained at low mutation–selection balance frequencies. Unsurprisingly, these results require mutation to be rare and local stability of the selective fixation of the advantageous allele in the absence of mutation, conditions that also apply to most other models of mutation–selection balance. Mutation–selection balance at an X-linked locus was first examined by Nagylaki (1977). However, selection coefficients are not directly comparable to s and t in the above analyses, as Nagylaki (1977) derived equilibria in terms of average selection coefficients in males and females. To allow comparison in mutation–selection balance between X-linked loci with and without imprinting, we use Table 1 to derive recursion equations (see File S3) and hence find mutation–selection balance frequencies for h = 0, h = 0.5, and h = 1, where (1 − hs) is the viability of the heterozygous females, and h = 0 implies complete dominance of the A ₂ allele, h = 0.5 implies no dominance, and h = 1 implies complete recessivity of the A ₂ allele. The recursion equations for allele frequencies in the absence of mutation are now

$$\begin{array}{l}{p^{\prime }}_{\text{f}}=0.5\left[1-\frac{1-p_{\text{f}}-p_{\text{m}}+s{p}_{\text{f}}{p}_{\text{m}}}{1-s{p}_{\text{f}}{p}_{\text{m}}-hs(p_{\text{f}}+p_m-2p_{\text{f}}{p}_{\text{m}})}\right]\\ {p^{\prime }}_{\text{m}}=1-\frac{1-p_{\text{f}}}{1-t{p}_{\text{f}}}\end{array}$$

(see also Rice 1984). This system permits three equilibrium allele frequencies,

$${\widehat{p}}_{\text{f}}={\widehat{p}}_{\text{m}}=0\text{,}\hspace{0.17em}{\widehat{p}}_{\text{f}}={\widehat{p}}_{\text{m}}=1$$,

$${\widehat{p}}_{\text{f}}=\frac{-2hs-t+hst}{2s(1-2h+ht-t)},{\widehat{p}}_{\text{m}}=\frac{(t-1)(-2hs+hst-t)}{s(h{(t-2)}^2+2(t-1))-t^2}$$. (9)

The values for the polymorphic equilibria when h = 0, h = 0.5, and h = 1 are

$$(p_{\text{f}},p_{\text{m}})=$$

$$\left(\frac{t}{2s(t-1)},\frac{t(t-1)}{2s-2st+t^2}\right),$$

$$\left(\frac{2s+2t-st}{2st},\frac{(t-1)(st-2s-2t)}{t^2(s-2)}\right)\text{and}$$

$$\left(\frac{2s+t-st}{2s},\frac{(t-1)(st-2s-t)}{2s-2st+s{t}^2-t^2}\right),$$

respectively. In the absence of dominance (h = 0.5) the existence of the polymorphic equilibrium requires st < 0, 2s + 2t − 3st > 0 and 2s + 2t − st < 0. As expected, when k ₁ = k ₂ = h (i.e., there is no imprinting), the polymorphic equilibrium allele frequencies for the incomplete inactivation model (6) are equivalent to (9).

Interestingly, it is not possible to recover the polymorphic equilibrium for the incomplete imprinting model (6) by substituting h = 0.5(k ₁ + k ₂) into (9). In general, ignoring imprinting (i.e., setting h = 0.5(k ₁ + k ₂)) underestimates the frequency of A ₁ when k ₁ < k ₂ and overestimates the frequency of A ₁ when k ₂ < k ₁. For an autosomal imprinted locus, the expected response to selection in one generation is equivalent whether calculated using separate terms for the reciprocal heterozygotes or simply using the heterozygote mean value (Santure and Spencer 2011). The lack of equivalence in the X chromosome system is likely due to the lack of symmetry in the interaction between the male and female allele frequencies (and hence the male selection parameter t) and the k_i imprinting parameters in the allele frequency recursions (5).

Tables 4 and 5 compare mutation–selection balance at an X-linked locus for mutation rates μ (from A ₂ to A ₁, when A ₂ would be fixed by selection alone) and ν (from A ₁ to A ₂, when A ₁ would be fixed by selection alone), respectively. Despite the lack of equivalence of the polymorphic equilibrium when substituting h = 0.5(k ₁ + k ₂) into (9), both Table 4 and Table 5 show an interesting correspondence between the equilibrium frequencies with and without imprinting. For both mutation rates (ignoring terms in μ² and ν² and above), the feasibility condition for mutation–selection balance with no imprinting and h = 0.5 is both the sum of the feasibility conditions under paternal and maternal imprinting, and the sum of the feasibility conditions with no imprinting and h = 0 and h = 1. The feasibility conditions are also in general the denominators of each equilibrium. From comparison with the no imprinting, $h=1/2$ case for both mutation rates, it is clear that the frequency of the deleterious allele is at similar frequency to that at imprinted loci when similar selection pressures apply. The rightmost column in each table gives a numerical example.

Table 4. Comparison of mutation-selection balance with and without imprinting at a sex-linked locus for mutation rate μ from A ₂ to A ₁, $(s,t\le 1,\hspace{0.17em}\mu >0)$ .

	${\widehat{p}}_{\text{f}}$ (females)	${\widehat{p}}_{\text{m}}$ (males)	Feasibility condition	$({\widehat{p}}_{\text{f}},{\widehat{p}}_{\text{m}})$ for s = 0.1, t = 0.2, μ = 10−5
Complete paternal X inactivation	$\mu \left[\frac{3}{s+t}\right]$	$\mu \left[\frac{3+s-2t}{s+t}\right]$	$s+t>0$	(10.0 × 10−5, 9.0 × 10−5)
Complete maternal X inactivation, female offspring only	$\mu \left[\frac{3-s}{s+t-st}\right]$	$\mu \left[\frac{3-2t}{s+t-st}\right]$	$s+t-st>0$	(10.4 × 10−5, 9.3 × 10−5)
Complete maternal X inactivation, male and female offspring	$\mu \left[\frac{3-s}{s}\right]$	$\mu \left[\frac{3}{s}\right]$	$s>0$	(29.0 × 10−5, 30.0 × 10−5)
No imprinting; h = 0 a	$\mu \left[\frac{3}{t}\right]$	$\mu \left[\frac{3-2t}{t}\right]$	$t>0$	(15.0 × 10−5, 13.0 × 10−5)
No imprinting and no dominance; h = 0.5 a	$\mu \left[\frac{6-s}{2s+2t-st}\right]$	$\mu \left[\frac{6+s-4t}{2s+2t-st}\right]$	$2s+2t-st>0$	(10.2 × 10−5, 9.1 × 10−5)
No imprinting; h = 1 a	$\mu \left[\frac{3-s}{2s+t-st}\right]$	$\mu \left[\frac{3+s-2t}{2s+t-st}\right]$	$2s+t-st>0$	(7.6 × 10−5, 7.1 × 10−5)

^aWhere (1 − hs) is the viability of the heterozygous females; see Table 1.

Table 5. Comparision of mutation-selection balance with and without imprinting at a sex-linked locus for mutation rate ν from A ₁ to A ₂, $(s,t\le 1,\hspace{0.17em}v>0)$ .

	${\widehat{p}}_{\text{f}}$ (females)	${\widehat{p}}_{\text{m}}$ (males)	Feasibility condition	(1 − ${\widehat{p}}_f$, 1 −${\widehat{p}}_{\text{m}}$) for s = −0.1, t = −0.2, ν = 10−5
Complete paternal X inactivation	$1-v\left[\frac{3(s-1)(t-1)}{2st-s-t}\right]$	$1-v\left[\frac{2st-4s-t+3}{2st-s-t}\right]$	$2st-s-t>0$	(11.6 × 10−5, 10.7 × 10−5)
Complete maternal X inactivation, female offspring only	$1-v\left[\frac{(2s-3)(t-1)}{st-s-t}\right]$	$1-v\left[\frac{(s-1)(t-3)}{st-s-t}\right]$	$st-s-t>0$	(12.0× 10−5, 11.0 × 10−5)
Complete maternal X inactivation, male and female offspring	$1-v\left[\frac{2s-3}{s}\right]$	$1-v\left[\frac{3(s-1)}{s}\right]$	$s<0$	(32.0× 10−5, 33.0 × 10−5)
No imprinting; h = 0 a	$1-v\left[\frac{(2s-3)(t-1)}{2st-2s-t}\right]$	$1-v\left[\frac{2st-4s-t+3}{2st-2s-t}\right]$	$2st-2s-t>0$	(8.7× 10−5, 8.3 × 10−5)
No imprinting and no dominance; h = 0.5 a	$1-v\left[\frac{5st-5s-6t+6}{3st-2s-2t}-\frac{3st-2s-2t}{2st}\right]$	$1-v\left[\frac{3st-7s-2t+6}{3st-2s-2t}\right]$	$3st-2s-2t>0$	(11.8× 10−5, 10.8 × 10−5)
No imprinting; h = 1 a	$1-v\left[\frac{3(s-1)(t-1)}{st-t}\right]$	$1-v\left[\frac{(s-1)(t-3)}{st-t}\right]$	$st-t>0$	(18.0× 10−5, 16.0 × 10−5)

^aWhere (1 − hs) is the viability of the heterozygous females; see Table 1.

There is an extensive literature regarding the existence and maintenance of variability on the X chromosome. In the context of sexual conflict and over a range of dominance, the X chromosome has less opportunity for polymorphism than the autosomes (Curtsinger 1980; Patten and Haig 2009). However, the speed to fixation of beneficial X-linked compared to autosomal alleles is dependent not only on the degree of dominance, but also factors including the population-effective population size, the effective population size of the X compared to the autosomes, the reproductive variability of males compared to females, and the relevance of standing vs. new mutations (Rice 1984; Charlesworth et al. 1987; Vicoso and Charlesworth 2006; Mank et al. 2009; Connallon and Clark 2010; Connallon et al. 2012), suggesting that these factors are also likely to influence polymorphism on the X chromosome. In addition, the X chromosome is predicted to harbor a large proportion of sexually antagonistic loci (Gibson et al. 2002, although see Connallon and Clark 2010). We have demonstrated that unless there is strong yet balanced sexual conflict, imprinting (either complete or incomplete) is unlikely to be conducive to selectively maintained polymorphism on the X chromosome.

In summary, imprinting of X-linked genes means that the conditions for the selective maintenance of allelic variation are either nonexistent (for complete maternal inactivation) or restrictive, requiring male and female selection coefficients of opposite sign but similar magnitude (for complete paternal inactivation and when imprinting is incomplete). Our results therefore add to the reasons for expecting less variation on this chromosome. Finally, we have shown that the frequency of an allele maintained by mutation–selection balance on the X chromosome is very similar for both a locus without imprinting and an imprinted locus.