The Evolution of X-Linked Genomic Imprinting

Corresponding author: Yoh Iwasa, Department of Biology, Kyushu University, Fukuoka 812-8581, Japan., yiwasscb{at}mbox.nc.kyushu-u.ac.jp (E-mail)

Communicating editor: C.-I WU

	ABSTRACT

TOP ABSTRACT THE MODEL EVOLUTIONARY EQUILIBRIUM DISCUSSION APPENDIX A APPENDIX B APPENDIX C LITERATURE CITED

We develop a quantitative genetic model to investigate the evolution of X-imprinting. The model compares two forces that select for X-imprinting: genomic conflict caused by polygamy and sex-specific selection. Genomic conflict can only explain small reductions in maternal X gene expression and cannot explain silencing of the maternal X. In contrast, sex-specific selection can cause extreme differences in gene expression, in either direction (lowered maternal or paternal gene expression), even to the point of gene silencing of either the maternal or paternal copy. These conclusions assume that the Y chromosome lacks genetic activity. The presence of an active Y homologue makes imprinting resemble the autosomal pattern, with active paternal alleles (X- and Y-linked) and silenced maternal alleles. This outcome is likely to be restricted as Y-linked alleles are subject to the accumulation of deleterious mutations. Experimental evidence concerning X-imprinting in mouse and human is interpreted in the light of these predictions and is shown to be far more easily explained by sex-specific selection.

IN a recent article (IWASA and POMIANKOWSKI 1999 ), we proposed the novel hypothesis that X-linked imprinting has evolved to control sex-specific gene expression in early embryos. Unlike autosomes, X chromosomes are inherited in an asymmetrical fashion. The maternal X chromosomes are equally likely to be inherited by female or male offspring, whereas the paternal X is always passed to female offspring and never to male offspring. So imprinting of X-linked genes naturally results in sexually dimorphic gene expression.

This hypothesis predicts that imprinting of X-linked genes is likely to evolve when selection favors different levels of embryonic gene expression in the two sexes. Early expression is predicted to be affected because circulating sex hormones are not available as a signal of sex until development is reasonably advanced (IWASA and POMIANKOWSKI 1999 ). Greater gene expression in males relative to females can be achieved by X^p imprinting (silencing/reducing paternal X activity). In contrast, greater gene expression in females relative to males can be achieved by X^m imprinting (silencing/reducing maternal X activity).

These predictions are consistent with the patterns of gene expressions inferred from the phenotypes of XO genotypes in mice and humans (summary in IWASA and POMIANKOWSKI 1999 ). In the mouse, X^pO female embryos (10.5 days post coitum) were found to be much smaller than similar age X^mO and X^mX^p female embryos (THORNHILL and BURGOYNE 1993 ; BURGOYNE et al. 1995 ). This observation indicates that the paternally derived X^p is imprinted and codes for a slower developmental rate and consequently smaller embryo size. As male embryos (X^mY) do not inherit the X^p, they show growth acceleration relative to female embryos (X^mX^p) from the same broods.

The reverse pattern is seen in humans. In this case the X-linked gene(s) subject to imprinting affect cognitive performance. XO humans (like mice) are female and can be either X^mO or X^pO (both suffer from Turner's syndrome; SKUSE et al. 1997 ). The origin of the single X has no effect on physical phenotype or general IQ. However, X^mO individuals lack social awareness, flexibility, and responsiveness, and scored much lower in formal tests of social cognitive skills compared to X^pO individuals (SKUSE et al. 1997 ). These observations are consistent with maternal imprinting of an X-linked gene for social aptitude, probably through effects on early brain development (SKUSE et al. 1997 ). When the same test was applied to age-matched normal children, X^mX^p girls showed higher social skills than X^mY boys, the difference being similar to that seen between X^pO and X^mO females.

Here we develop the verbal arguments of IWASA and POMIANKOWSKI 1999 , using a quantitative genetic model for the evolution of X-linked imprinting. This model allows genomic imprinting to evolve as a continuous change in the level of gene expression (MOCHIZUKI et al. 1996 ). As well as providing a formal theoretical treatment, we aim to extend the analysis presented in the previous article. In particular, we explicitly consider the conflict hypothesis (HAIG and GRAHAM 1991 ; MOORE and HAIG 1991 ; HURST 1997 ) as another force creating patterns of X-linked imprinting. The conflict hypothesis proposes that paternally expressed alleles demand greater nutrient resources from the mother because they have lower relatedness among sibs than do maternally inherited alleles. The strength of this effect varies with the degree of polygamy. The conflict hypothesis was originally developed to explain patterns of autosomal imprinting, but it should apply equally to X-linked genes. In our model, we make a quantitative assessment of how X-linked imprinting is affected by sex-specific selection and conflict due to different degrees of relatedness among sibs.

In our model we consider models where there is dosage compensation or where the imprinted X-linked gene escapes from dosage compensation, and we also analyze models in which an active Y-linked homologue of the imprinted X-linked gene is present or absent. However, we do not directly model the evolution of dosage compensation. Nor do we model forces such as deleterious mutation pressure that are important in the evolution of Y-linked genes (GRAVES and SCHMIDT 1992 ; CHARLESWORTH 1996 ). The importance of this is raised in the DISCUSSION.

	THE MODEL

TOP ABSTRACT THE MODEL EVOLUTIONARY EQUILIBRIUM DISCUSSION APPENDIX A APPENDIX B APPENDIX C LITERATURE CITED

Evolutionary dynamics:
We study the evolution of gene expression of an X-linked gene. The gene involved has embryonic expression that affects the supply of maternal resources. It is assumed to be subject to genomic imprinting and thus has two different levels of expression, depending on parental origin. We use a quantitative genetic model that was used previously to analyze autosomal genomic imprinting (MOCHIZUKI et al. 1996 ). The basic underlying assumption is weak selection; that is, there is only a small change in fitness over the range of the trait (IWASA et al. 1991 ). Unlike other formalism of quantitative genetics (e.g., LANDE 1976 ), it does not require normality in the distribution of genetic values.

The regulatory part of the gene is represented by (p, m) in which p is the expression of the allele when inherited from the father and m is the expression of the allele when inherited from the mother. There is a corresponding copy of the gene on the Y chromosome. The expression of the Y copy is denoted by y. If the Y copy is inactive, y = 0.

A female embryo receives two X chromosomes, one from the egg (p₁, m₁) and the other from the sperm (p₂, m₂). In female mammals, most X-linked genes undergo dosage compensation by random X inactivation (GRAVES and SCHMIDT 1992 ). The maternal X is silenced in one-half of the cells, and the paternal X is silenced in the other one-half. Assuming random X inactivation, the gene expression in female offspring is z = . In contrast, a male embryo receives an X only from the egg (p₁, m₁). This X is active in all cells. In addition, the male receives a Y chromosome, so the gene expression in male embryos is z = m₁ + y.

Due to the asymmetric inheritance of the sex chromosomes (IWASA and POMIANKOWSKI 1999 ), there are different selective pressures on m, p, and y. Selection on m has two components dependent on whether the offspring is male or female, whereas selection on p has only a single component as it is always inherited by a female. Likewise, selection on y has a single component as it is always inherited by a male.

Selection occurs in both sexes, so we need to define sex-specific fitness functions (Fig 1). We calculate the evolutionary change in m and p in each sex (see Appendix A). For an X-linked gene possessed by a female in the current generation, we calculate the expected number of copies in females in the following generation (i.e., her daughters) and the expected number of copies in males in the following generation (i.e., her sons). We call these the "mother-to-daughter" fitness (denote by _I) and the "mother-to-son" fitness (denote by _II), respectively. For an X-linked gene possessed by a male in the current generation, we calculate the expected number of copies in females in the following generation (i.e., his daughters), which is the "father-to-daughter" fitness (denoted by _III). There is no "father-to-son" fitness for X-linked genes as paternal X-linked genes are only passed on to daughters, never to sons. The father-to-son fitness applies to Y-linked genes (denoted by ), which are passed exclusively through the male lineage.

View larger version (15K): In this window In a new window Download PPT slide   Figure 1. The transmission of X and Y chromosomes is asymmetric. While Xm is inherited by both sexes of offspring, only females inherit Xp and only males inherit Y. Selection is calculated by the fitness functions (I, II, III, and ) associated with each pattern of inheritance.

Summing across the sexes (with a two-thirds female to one-third male weighting for X-linked genes), we have a simple result for the per generation change in the mean traits (under the weak selection assumption),

(1a)

in which G_i is the genetic variation and ß_i the selection gradient acting on trait i (i = m, p, y). Using the definitions of fitness above, the selection gradients are

(1b)

The details of these derivations are given in APPENDIX A.

To model multiple mating by females, we assume that a fraction g of females accept two males as mates (polygamy). The two males are assumed to be unrelated and each is assumed to contribute equally to the progeny of the female. The remaining 1 - g females mate with a single male (monogamy). We call g the female polygamy rate. The model is not supposed to reflect the details of mammalian mating systems in the wild, which are far more complex. It is adopted for its simple representation of multiple mating.

Fitness:
To further specify the evolutionary dynamics, we need to consider how selection is generated. The amount of resources allocated to an embryo is proportional to its gene expression, z. The survivorship of an embryo is taken to be an increasing function of gene expression and is different in male and female embryos W_mal(z) and W_fem(z). The mother has limited resources, T, that can be used for reproduction. So, as the average resource demand per embryo increases, the total number of embryos produced declines.

These two forces are the main components of the fitness functions. First consider the mother-to-daughter fitness _I. To model the trade-off between gene expression and number of offspring produced, we use a resource division model (MOCHIZUKI et al. 1996 ),

(2a)

The first factor, N, is the expected number of offspring,

(2b)

The denominator of N is the average embryo gene expression. This is determined by the sex ratio of offspring s, the fraction of male offspring. The focal allele (p, m) is equally represented in male and female offspring (following Mendelian inheritance). Its homologue is a random sample from the population, which has mean trait values (, ). a is the conversion coefficient from gene expression into resources. The second factor in Equation 2a, (1 - s), is the fraction of female offspring. The third factor of ¹/₂ is the relatedness, indicating the probability of an X-linked allele in the mother being transmitted to her daughter. The final factor is the survivorship of daughters that carry the focal allele (p, m). The mother-to-son fitness is calculated in a similar way,

(3)

where the number of offspring N is given by Equation 2b. The fraction of males is s, and their survivorship is W_mal(m + ).

To define the other two fitness functions (_III and ), we need to consider a third selective force. Females mate with a variable number of males throughout their reproductive life. Here we consider the simple case in which 1 - g females mate with a single male, while g females mate with two males, each having the same probability of fathering offspring. First we note that female polygamy does not affect the transmission of the female's genes to the next generation. Hence neither _I nor _II are affected by g. In contrast, the father-to-daughter fitness is

(4a)

M is the expected number of females mating with the male. (1 - g)/(1 + g) is the fraction of matings with a monogamous female, and 2g/(1 + g) is the fraction of matings with a polygamous female. N_mono and N_poly are the expected numbers of offspring produced by monogamous and polygamous females, respectively,

Once again, N_mono and N_poly reflect the average gene expression of embryos, which varies with female polygamy. From the perspective of the focal paternal allele (p, m), if the female is polygamous, one-half of her offspring will be sired by another male. This reduces the relatedness by ¹/₂, indicating the probability of an X-linked allele in the father being transmitted to his daughter if the female is polygamous. In a similar way, the male-to-son fitness is likewise affected by the degree of polygamy,

(5a)

with

(5b)

(5c)

	EVOLUTIONARY EQUILIBRIUM

TOP ABSTRACT THE MODEL EVOLUTIONARY EQUILIBRIUM DISCUSSION APPENDIX A APPENDIX B APPENDIX C LITERATURE CITED

Selection gradients:
The selection gradients can now be calculated by substituting the fitness terms above into Equation 1aEquation 1b. Setting the mean gene expression in a female embryo to Z_f = and that in a male embryo to Z_m = + , the selection gradient with respect to p is

(6a)

In a similar way,

(6b)

(6c)

To calculate the terms on the right-hand side of Equation 6aEquation 6b HREF="#FD6c">Equation 6c, we need to specify the functional form of the fitness functions. For example,

(7)

The graphs of these functions have S-shaped curves (Fig 2). a_f and a_m give the rate of increase and b_f and b_m give the asymptotic values of the curves.

View larger version (15K): In this window In a new window Download PPT slide   Figure 2. Sex-specific survivorship functions for male and female embryos are functions of gene expression. Equations for Wmal(Zm) and Wfem(Zf) are given in the text. In this example, the rate of increase in survivorship and fitness asymptote are higher for males (am = 1, af = 0.2, bm = 1.8, bf = 1).

Inactive Y copy:
We first focus on the case in which the Y copy is inactive (i.e., = 0 and Z_m = ). At equilibrium and ß_p = 0 and ß_m = 0,

(8)

Rearranging and assuming that the sex ratio is even (s = 0.5),

(9)

The ratio of male to female gene expression is

(10)

This result nicely encapsulates the two forces operating on genomic imprinting. First, it shows that the male-to-female ratio of gene expression decreases with female polygamy g. This is so because paternally inherited genes tend to be resource demanding (MOORE and HAIG 1991 ), so female embryos are selected to have higher gene expression than males because only females inherit the paternal X. A second factor, sex-specific selection, is also important and can work in either direction. If there is selection for greater gene expression in males, a_m > a_f, this acts to increase the Z_m/Z_f ratio, whereas the reverse occurs if a_f > a_m.

Gene silencing:
The interaction between polygamy and sex-specific selection determines whether genomic imprinting causes greater gene expression from X^p or X^m. We can define the equilibrium ratio of paternal to maternal gene expression by substituting Z_f = and Z_m = into Equation 9,

(11)

The two forces that affect imprinting can be examined by exclusion. The effect of polygamy can be seen by setting a_f = a_m (no sex-specific selection),

(12)

When females mate with only one male (g = 0), paternal and maternal X gene expressions are equal. As the probability of polygamy g increases, so does the relative expression of the paternal allele. This does not lead to silencing of the maternal copy. In the present model, the maximum rate of polygamy is g = 1 (all females mate with two males), which generates about twice as large as . This differs from autosomal gene imprinting, which leads to maternal silencing for any g > 0 (MOCHIZUKI et al. 1996 ). The intuitive reason for this difference is that, if = 0, there is no gene expression in males. For most slowly changing selection functions, selection for some gene expression in the male embryo will prevent maternal gene silencing.

The effect of sex-specific selection can be investigated in a similar way by excluding the effect of polygamy. Making all matings monogamous (g = 0),

(13)

In the absence of sex-specific selection (a_m = a_f), paternal and maternal X gene expressions are equal. If selection on males is stronger than selection on females (a_m > a_f), maternal gene expression is greater than paternal gene expression. Paternal gene silencing is achieved if this selection difference is reasonably strong (a_m > 4a_f). At this equilibrium, male gene expression is constrained to be no more than twice female gene expression (as Z_m = m and Z_f = ).

If selection operates more strongly on females (a_f > a_m), paternal gene expression evolves to be greater than maternal gene expression, but the rate of increase is lower than with stronger selection on males. However, even if a_f >> a_m, some weak gene expression is expected from the maternal copy. Maternal gene silencing occurs only when male fitness is independent of gene expression and a_m = 0.

Active Y copy:
We now allow evolution of y, the quantitative expression of the Y copy. The existence of an active homologue on the Y chromosome changes the evolutionary outcome, making it similar to that seen with autosomal imprinting.

First consider the case in which g = 0 and females are monogamous. Given the dynamics in Equation 1aEquation 1b, equilibrium requires that the three selection differentials in Equation 6aEquation 6b HREF="#FD6c">Equation 6c must equal 0. Among these three, only two are independent. So for g = 0, there is a line of equilibria in three-dimensional space (, , , see Appendix B). The two equations specifying Z_f and Z_m are the same as Equation 9. But now there are many combinations of positive values of , , and that satisfy Z_f = and Z_m = + . The dynamics cause convergence of the population mean values to a point on the line of equilibria, but the dynamics then are neutrally stable. This result holds even if there is sex-specific selection. If a_m a_f, this changes the location of the line of equilibrium in three-dimensional space (see Equation B1).

These results hold for female monogamy (g = 0). If females have some positive probability of accepting a second male (g > 0), then the line of equilibria disappears. The only stable equilibrium occurs when the paternally inherited alleles are active ( > 0, > 0) and the maternally inherited allele is silenced ( = 0, see Appendix B for proof). At this equilibrium, ß_p = ß_y = 0, so substituting = 0 and s = 0.5 (one-to-one sex ratio),

(14)

In the absence of sex-specific selection (a_m = a_f), both male and female embryos increase gene expression with the polygamy rate g. The increase is the same in both sexes because polygamy reduces the relatedness of X^p and Y-linked genes among sibs at an identical rate (unless the sex ratio is not 1:1).

The single stable equilibrium with > 0, = 0, and > 0 holds irrespective of sex-specific selection. Both with a_m > a_f and a_m < a_f, the maternal X is silenced. This happens because p evolves to fulfill selection on female expression and y evolves to fulfill selection on male expression. Sex-specific selection only changes the relative expression of these two genes,

(15)

This ratio is independent of g and depends entirely on the relative strength of selection on the two sexes.

In many cases where there is an active Y-linked homologue, the X-linked genes in the female do not undergo dosage compensation by X inactivation (DISTECHE 1995 ). All the previous calculations remain the same except that the fitness functions and selection gradients change to reflect the lack of X inactivation in females (see Appendix C). This does not change the results in (14) and (15), except that Z_f = rather than Z_f = .

Although the results for an active Y copy may look strange, they are equivalent to those found with autosomal imprinting (MOORE and HAIG 1991 ). An imprinted autosomal locus has expression levels and when maternally or paternally inherited, respectively (MOCHIZUKI et al. 1996 ). Under female monogamy, there is no genetic conflict and natural selection works only on the sum + , the expression level in females and males. The optimal expression, Z = + , defines a line of equilibria in two-dimensional space (, ). This line disappears when g > 0. Polygamy causes a conflict between paternal and maternal copies, resulting in the evolution of higher and lower , ending with silencing of the maternally inherited allele (MOCHIZUKI et al. 1996 ).

	DISCUSSION

TOP ABSTRACT THE MODEL EVOLUTIONARY EQUILIBRIUM DISCUSSION APPENDIX A APPENDIX B APPENDIX C LITERATURE CITED

In this article we formalized the analysis of X chromosome imprinting presented by IWASA and POMIANKOWSKI 1999 . We used a quantitative genetic model to allow continuous change in the level of gene expression (IWASA et al. 1991 ; MOCHIZUKI et al. 1996 ). An alternative modeling strategy is to let imprinting evolve in an all-or-nothing fashion using a major gene model (ANDERSON and SPENCER 1999 ). The modeling assumes that gene expression in the embryo affects the supply of resources received by the embryo from the mother. The modeling allows us to place a number of verbal predictions on a more secure footing. It also permits a quantitative comparison of sex-specific selection and genomic conflict as causes of X-linked imprinting patterns. Evolutionary outcomes differ depending on the lack or presence of an active Y-linked copy, so we discuss these in turn.

Inactive Y copy:
Most X-linked genes in human and mouse lack Y-linked homologues (GRAVES and SCHMIDT 1992 ). These genes are generally subject to X inactivation, which standardizes dosage across the sexes (GRAVES and SCHMIDT 1992 ). To reflect this, we first consider the evolution of X-linked imprinting when the gene involved lacks a Y-linked homologue and undergoes random X inactivation.

The conflict hypothesis predicts that genomic imprinting arises because of differential selection on paternal and maternal copies, which is generated by polygamy (HAIG and GRAHAM 1991 ; MOORE and HAIG 1991 ). If a female mates with several males throughout her reproductive life, the average relatedness of paternally inherited genes among sibs will be less than that of maternally inherited genes. This logic applies equally to X-linked genes as to autosomal genes. It predicts that, under polygamy, selection will cause the evolution of higher , the gene expression from X^p (paternally inherited X), than , the gene expression from X^m (maternally inherited X).

With autosomal imprinting, even the smallest degree of polygamy leads to silencing of the maternal copy, with gene expression only from the paternal copy (MOCHIZUKI et al. 1996 ). However, silencing of maternal alleles is not an outcome with X-linkage (see Equation 12). Due to the asymmetric inheritance of the X chromosomes (Fig 1), silencing of X^m leads to a complete absence of gene expression in male offspring. This acts as a check on the evolution of maternal silencing ( = 0). In our model, the most extreme difference in gene expression generated by polygamy was about twice as large as . Although the quantitative value of this result is contingent on our model of polygamy and selection, other models are likely to give similar outcomes. The general point is that genomic conflict does not result in extreme differences in gene expression between X^m and X^p and cannot explain X^m silencing when quantitative variation in gene expression is possible.

This outcome is in contrast to that with sex-specific differences in selection, which can cause extreme differences in gene expression and even gene silencing. If selection favors higher gene expression in males than in females, this results in higher expression from , the maternal copy, than from , the paternal copy (see Equation 13). This "reversed" pattern of imprinting cannot easily be explained by the conflict hypothesis (IWASA and POMIANKOWSKI 1999 ). If the difference in sex-specific selection is moderately large (a_m > 4a_f), the X^p gene is silenced.

When selection favors higher gene expression in females than in males, higher gene expression is predicted from than from . The degree of imprinting reflects the difference in sex-specific selection. If selection is considerably stronger on females, then the ratio of to can be very large. Ultimately, gene silencing of can occur, but only if selection favors no gene expression in males. The difference is expression between and does not increase as quickly when selection favors higher expression in females (compared to selection favoring higher expression in males). This is a reflection of the asymmetric inheritance of the X chromosomes. Imprinting of X^p reduces gene expression in females only, whereas imprinting of X^m reduces gene expression both in males and to a lesser extent in females.

Active Y copy:
The conclusions above assume that the Y chromosome lacks genetic activity. This assumption must have been invalid prior to the degeneration of the mammalian Y chromosome. It also remains invalid for a number of X-linked genes that retain Y-linked homologues (JEGALIAN and PAGE 1998 ). With these limitations in mind, we modeled the evolution of X-imprinting when a fully active Y-linked homologue also contributes to gene expression.

When there is an active Y copy, the evolutionary outcome of X-imprinting more closely resembles patterns seen with autosome imprinting. If there is complete monogamy and no sex-specific selection, there are multiple equilibria. Many combinations of , , and can satisfy the gene expression of males and females. Formally there is a line of equilibria in three-dimensional space (, , ). A similar set of multiple equilibria are seen with autosomal imprinting (MOCHIZUKI et al. 1996 ) but limited to two dimensions (, ).

If there is any degree of polygamy, the line of equilibrium breaks down to a single stable equilibrium. This occurs when the paternally inherited alleles are active ( > 0, > 0) and the maternally inherited allele is silenced ( = 0). As the paternal X is inherited solely by daughters and the paternal Y solely by sons, these two genes can evolve to independent states that satisfy selection on the two sexes of offspring.

The single equilibrium with maternal gene silencing also holds when there is sex-specific selection. This merely changes the exact equilibrium values of and . A similar adjustment needs to be made for X-linked genes that escape X inactivation. These tend to have Y-linked homologues. This will cause a doubling of the paternal X gene expression (as this is no longer subject to dosage compensation), but it does not alter the existence or stability of the single equilibrium.

The results above assume that Y-linked genes evolve under the same conditions as X-linked or autosomal genes. This is not likely to be the case as most Y-linked genes are subject to the build up of deleterious mutations because they do not undergo regular recombination (GRAVES and SCHMIDT 1992 ; CHARLESWORTH 1996 ). The analysis presented above relates to the special condition in which the population of Y-linked genes is not subject to deleterious mutation. The effect of deleterious mutations was analyzed by MOCHIZUKI et al. 1996 for autosomal imprinting. Their conclusion was that deleterious mutation pressure acts to stop the evolution of gene silencing by imprinting, as monoallelic expression leads to severe fitness loss when the single expressed gene carries an inactivating mutation.

General discussion:
As in our previous article (IWASA and POMIANKOWSKI 1999 ), we can use the results here to compare the ability of conflict and sex-specific selection to explain patterns of X-linked genomic imprinting. Both these forces can potentially contribute to X-imprinting.

For the majority of X-linked genes there are no Y-linked homologues (GRAVES and SCHMIDT 1992 ). In this case, the conflict hypothesis can explain weak imprinting of X^m ( > ), but cannot explain silencing of X^m ( = 0), or the reverse pattern of X^p imprinting ( < ). In contrast, selection for sex-specific gene expression can potentially explain imprinting of X^m, reverse imprinting of X^p, and silencing of either allele. When there is an active Y-linked copy, the conflict hypothesis predicts silencing of X^m ( = 0), with male gene expression controlled by the Y copy and female expression controlled by the paternal X.

How do these predictions fare in the face of the available data? X-imprinting has so far been detected in human and mouse. Unfortunately the genes involved have not been mapped, so the interpretation of data remains tentative. In humans, X-imprinting is thought to contribute to the development of social cognitive skills and memory (SKUSE et al. 1997 ; BISHOP et al. 2000 ). The maternal X is thought to be silenced, the paternal X remains active, and no Y effect is present ( = 0, > 0, = 0). This creates greater gene expression in females, which is consistent with selection for greater female gene expression, a_f > a_m (IWASA and POMIANKOWSKI 1999 ; SKUSE 1999 ). Maternal X silencing is not predicted by the conflict hypothesis. At best, part of the greater paternal X expression might be attributed to selection due to conflict. However, the gene(s) involved does not seem to be involved in fetal-maternal resource accrual; rather it appears to regulate differential growth of regions in the brain (BISHOP et al. 2000 ). So there appears to be little room for conflict in this case.

The second example of X-imprinting affects early developmental arrest and growth rates in the mouse (THORNHILL and BURGOYNE 1993 ; BURGOYNE et al. 1995 ). Early in development, the paternal X is thought to be silenced, the maternal X remains active, and a Y effect is present ( > 0, = 0, > 0). This pattern does not directly fit either hypothesis. The pattern cannot be explained by the conflict hypothesis, which predicts greater paternal gene expression. However, the pattern is consistent with selection for greater gene expression in males, a_m > a_f, if we assume that the Y effect is a small fixed effect. If this is the case, the Y provides insufficient expression for the male and so needs to be augmented by the maternal X.

The few data available on X-imprinting appear to be far more easily explained by sex-specific selection than by the conflict hypothesis. The data suggest that selection generated by polygamy for greater paternal gene expression is easily outweighed by selection for sexual dimorphism. However, our interpretations remain tentative until more detailed genetic evidence becomes available.

	ACKNOWLEDGMENTS

This work was supported in part by a Grant-in-Aid from the Ministry of Education, Science, Sports and Culture of Japan (Y.I.), a CREST project (Y.I.), and a Royal Society Fellowship (A.P.).

Manuscript received March 9, 2001; Accepted for publication May 3, 2001.

	APPENDIX A

TOP ABSTRACT THE MODEL EVOLUTIONARY EQUILIBRIUM DISCUSSION APPENDIX A APPENDIX B APPENDIX C LITERATURE CITED

We develop a quantitative genetic model originally developed to study sexual selection (IWASA et al. 1991 ; IWASA and POMIANKOWSKI 1994 ) and more recently applied to study autosomal imprinting (MOCHIZUKI et al. 1996 ). The model assumes weak selection (small changes in fitness over the range of the trait), a constant variance-covariance matrix, and similar values of p and m in the two sexes. These assumptions will be violated when there is strong selection, when selection leads to changes in the variance-covariance matrix, or when values of p and m are significantly different in the two sexes. These extra factors will alter the evolutionary trajectory but should not make major differences in the location of equilibria. They deserve to be studied further in major gene models (e.g., ANDERSON and SPENCER 1999 ).

To calculate changes in the mean values , , and , we have to follow inheritance and selection through the maternal and paternal lineages. We introduced the following notation. For generation t, let (_t, _t) be the average over all females, (^'_t, ^'_t) be the average over all males, and _t be the average of y in males. In addition, (^*_t, ^*_t) is the average over all maternal X chromosomes transmitted to daughters and (^'*_t, ^'*_t) is the average over all paternal X chromosomes transmitted to daughters.

These averages determine the values of , , and in the next generation via four fitness functions. To illustrate this, we concentrate on change in p (we can derive the same set of equations for m). The mean female X expression in the next generation is

(A1)

(derived as described in the Appendix of IWASA et al. 1991 ). Equation A1 indicates that the change is equal to the product of additive genetic variance and the selection gradient. The mean male X expression in the next generation is

(A2)

Combining these two equations and taking into account the pattern of inheritance of the X chromosomes,

(A3)

Under the weak selection assumption, the difference in mean trait value of X chromosomes in females and in males ( and ') is small; hence,

(A4a)

In a similar way,

(A4b)

(A4c)

Now we note that the mother-to-daughter fitness _I and the mother-to-son fitness _II are functions of m but independent of p. In contrast, the father-to-daughter fitness _III is independent of m. Hence we have Equation 1aEquation 1b in the text. Note, in addition, that we assume that the genetic covariance between m and p is much smaller than the genetic variances G_m and G_p. Although a nonzero genetic covariance will change the dynamics, it will not affect the location or stability of equilibria.

	APPENDIX B

TOP ABSTRACT THE MODEL EVOLUTIONARY EQUILIBRIUM DISCUSSION APPENDIX A APPENDIX B APPENDIX C LITERATURE CITED

When there is an active Y-linked copy and no polygamy (g = 0), there is a line of equilibria. The three equations ß_m = 0, ß_p = 0, and ß_y = 0 given in (6) are not independent. Given Z_f = and Z_m = + as in (9), and setting g = 0, the line of equilibria in the (, , ) space is

(B1)

This line collapses when there is some degree of polygamy (g > 0). To investigate the location and stability of the equilibria, we note that ß_p = 0 holds if > 0 at equilibrium. If instead = 0 at equilibrium, ß_p < 0 must hold. In a similar way, ß_m = 0 and > 0, or ß_m < 0 and = 0; likewise ß_y = 0 and > 0, or ß_y < 0 and < 0. To find which of these relationships hold, we note from (6) that

(B2)

Now we can consider possible equilibria:

1. At the equilibrium > 0, > 0, > 0, it must be the case that ß_p = ß_m = ß_y = 0, which is inconsistent with (B2). Hence there is no equilibrium of this type.

2. At the equilibrium with = 0, > 0, > 0, it must be the case that ß_m = ß_y = 0. (B2) requires ß_p > 0, which implies that the equilibrium is unstable against increases in .

3. At the equilibrium with > 0, > 0, = 0, it must be the case that ß_p = ß_m = 0. (B2) requires ß_y > 0, which implies that this equilibrium is unstable against increases in . However, this equilibrium is possible if the expression of y is perfectly silenced by some additional mechanism, such as the accumulation of deleterious mutations (see DISCUSSION).

4. At the equilibrium with > 0, = 0, > 0, it must be the case that ß_p = ß_y = 0. (B2) requires ß_m < 0, so this equilibrium is stable.

Hence with female polygamy (g > 0), we can conclude that the only stable equilibrium is > 0, = 0, > 0.

	APPENDIX C

TOP ABSTRACT THE MODEL EVOLUTIONARY EQUILIBRIUM DISCUSSION APPENDIX A APPENDIX B APPENDIX C LITERATURE CITED

When there is no random X inactivation, or the gene concerned is not subject to dosage compensation, the gene expression in female is Z_f = m + p, instead of Z_f = . All the calculations in the text remain the same, except for the fitness functions, which change to reflect the lack of X inactivation,

(C1)

and the selection gradients are

(C2)

	LITERATURE CITED

TOP ABSTRACT THE MODEL EVOLUTIONARY EQUILIBRIUM DISCUSSION APPENDIX A APPENDIX B APPENDIX C LITERATURE CITED

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