The Evolution of Genomic Imprinting

zyxwvutsrqpo zyxwvutsrqpon zyxwvutsrq zyxw zyxwvutsrq zyxwvutsrq Copyright 0 1996 hy the Genetics Society of America T h e Evolution of Genomic Imprinting Atsushi Mochizuki,Yasuhiko Takeda and Yoh Iwasa Department of Biology, Faculty of Science, Kyushu University, Fukuoka 812-81, Japan Manuscript received November 24, 1995 Accepted for publication July SO, 1996 ABSTRACT In some mammalian genes, the paternally and maternally derived alleles are expressed differently: this phenomenon is called genomic imprinting. Herewe study the evolution of imprinting using multivariate quantitative genetic models to examine the feasibility of the genetic conflict hypothesis. This hypothesis explains the observed imprinting patterns as an evolutionary outcome of the conflict between the paternal and maternal alleles. We consider the expression of a zygotic gene, which codes for an embryonic growth factor affecting the amount of maternal resources obtained through the placenta. We assume that the gene produces thegrowth factor in two different amountsdepending on its parental origin. Weshow that genomic imprinting evolves easily if females have some probability of multiple partners. This is in conflict with the observation that not all genes controlling placental development are imprinted and that imprintingin some genes is not conserved between mice and humans.We show however that deleterious mutationsin the coding regionof the gene create selection against imprinting. S EVERAL mammalian embryonic genes are known to be subject to genomic imprinting. Only either the paternally or maternally derived allele is actively expressed while the other stays silent (PETERSON and SAPIENZA 1993). Some of these imprinted genes code for an embryonic growthfactor o r its inhibitor and therefore affectplacentaldevelopment, while others are related to suckling and swallowing behaviors, appetite, and attracting maternal attention. Ithas been suggested that imprinted genes arelikely to be involved in controlling the amount of resources supplied by the mother (HAIGand GRAHAM1991; MOORE and HAIC 1991 ) . Genomic imprinting was first noted from thegrowth retardation and death of embryos that possessed the maternal duplication and paternal deficiency ( o r paternal duplication and maternal deficiency) of the whole genome, or of a single chromosome (uniparental disomy) . FERGUSON-SMITH et al. ( 1991 ) incorporated cells with paternal duplication of the distal chromosome 7 intochimerasandfound a considerable growth enhancement in theembryos. In contrast, inembryos with maternal duplication of the distal chromosome 7, both the maternal alleles of the Zgf2 (insulin-like growth factor type-2 ) gene were repressed. The I g f 2gene in this region was suspected to be imprinted. Using gene targeting, DECHIARAut al. (1991 ) demonstrated that the paternally inheriteddisrupted Zgf2 gene causes growth deficiency butthe maternally inherited disrupted gene does not. Nuclease protection and in situ hybridization analyses of the transcripts from the wild- type and mutated alleles indicated that only the paternal allele is expressed in embryos, while the maternal allele is silent. An exception is found in the choroid plexus and leptomeninges, where both the maternal and paternal Zgf2 genes are expressed. Similarly, only the paternally derived alleles are expressed for several genes on human chromosome 15, which are in turn related to behavioral abnormalities, such as Prader-Willi syndrome and Angelmansyndrome ( OZCELIK et al. 1992; SUTCLIFF et nl. 1994; WEVKICK et al. 1994). In contrast, there are genesin which only the maternally derived copy is actively expressed. For example, Zgf2r (insulin-like growth factor type-2 receptor) is expressed only from the maternal chromosome in mice ( BARLOWet nl. 1991 ) . This has been suspected to be an inhibitor of Zgf 2 ( HAIGand GRAHAM 1991 ) . BARTOLOMEI et al. (1991) showed that in the mouse the HI 9 gene is imprinted, with the active copy derived from the mother. HI 9 RNA has tumor-suppressor activity ( HAO et nl. 1993), but it is, however, not translated. Some genes controlling the cell cycle are also imprinted in the mouse: an inhibitorof the cyclin / CDK complex, is transcribed only from the maternal allele ( HATADAand MUKAI1995), while an activator gene, CDC25, is transcribed only from the paternal allele in some tissues (PUSS et nl. 1996) . Control of differential genetic expression is known to be based on the methylation pattern of the genes ( SASAKI et nl. 1993). CHAILLET et nl. (1991) studied methylation of transgenes and found that both maternally and paternally inherited methylation patterns are erased in primordial germcells and thatdistinctive patterns emerge during germ cell maturation. The methylation pattern found in sperm undergoes further modification during embryogenesis. SASAKIet al. ( 1991 ) zyxwvutsrqp zy zyxwvutsrqponm Cmrsponding author; Atsushi Mochizuki, Department of Biology, Faculty of Science, Kyushu University, Fukuoka 812-81, .Japan. E-mail: [email protected] Gcnetirs 144: 12%-I295 (Novcmhel-, 1996) zyxwvuts zyxwvut zyxwvutsrqponm zyxwvutsrq zyxwvutsr 1284 A. Mochizuki, Y . Takeda and Y . Iwasa examined whether the parental-origin-dependent differential methylation observed in transgenes reflects the genomic imprinting of endogenous genes, and confirmed that the methylation patternsare established before early prophase I during spermatogenesis. UEDA et al. (1992) showed that parental-specific adult patterns of transgene methylation are established during gametogenesis. LI et al. ( 1993) examined expression of three imprinted genes ( I gf2, Igf2r, H19) in mutant mice that are deficient in DNA methyltransferase activity. Both maternal and paternal genes of H19 are expressed, but both genes are repressed for Igf2 and Igf 2r. Recent studies have revealed that the genetically imprinted genes, Mash-2, Ins-2, Igf2, and H19, are clustered into a chromosomal domain, implying that imprinting may be regulated in a regional matter (EDEN and CEDAR1995). In a similar vein, a group of genomically imprinted human genes ( S A W N , PAR-5, PARI, IPW) is located within a single region of severalhundred kb ( OZCELIK et al. 1992;WEVRICK et al. 1994) andis suspected to be controlled by an imprinting-controlling element ( REIS et al. 1994; SUTCLIFF et al. 1994; BUITING et al. 1995). Genomic imprinting of a particular gene may differ between species, between individuals of the same species, and between tissues of the same individuals. ZGF2 is monoallelically expressed in various tissues but not in the adult human liver. The human IGF2R gene, which is imprinted in mice, was found to be expressed from both alleles ( KALSCHEUER et al. 1993; OGAWA et al. 1993) .JINNO et al. ( 1994) demonstrated that W T l can undergo tissue-specific imprinting. Furthermore, they found monoallelic expression of W T l in some placentas but not others, suggesting genetic polymorphism in imprinting within the human population. To explain the observed patterns of genomic imprinting, DAVIDHAIC and his colleagues proposed a genetic conflict hypothesis, which explains genomic imprinting as an outcome of evolution based on natural selection [HAIC and WESTOBY 1989, 1991; HAIC and GRAHAM 1991; see VARMUZAand M A N N (1994) for an alternative hypothesis]. They noted a conflict of interest between thepaternal and maternal alleles of a growth factor genewithin an embryo. Consider the simplest case in which the mother mates with a large number of males, and all the offspring have different fathers. Then the paternally derived copyof a growth factor gene would try to gain as much resources as possible from the mother and thereby maximize the survivorship and growth rate of the embryo, without considering the survivorship of other embryos that have different fathers. In contrast, the maternal gene may benefit by reducing the resource demand because there is a 50% chance of copy of the gene existing in each of the other maternally related sibs. Even if the probability of a female accepting multiple mates islow, and hence most offspring are likely to have acommonfather, there is still some difference between the two alleles in terms of the “optimal” amount of maternal resources. HAIG (1992) demonstrated differences in the optimal amountforthe maternally derived gene in the offspring, for thepaternally derived gene in the offspring, and for the gene in the mother. This hypothesis explains why the paternal allele tends to be active and the maternal allele inactive for embryonic growth factor genes. In contrast, for genes coding for inhibitors of growth factors or genes that work to reduce the resource demand in general, the maternal allele tends to be active and the paternal one inactive. In this paper, we develop a mathematical model of the evolution of genomic imprinting based on the genetic conflict hypothesis, to examine whether this idea presented as a verbal argument is supported in terms of rigorous genetic models of evolution. The model predicts the evolution of a regulatory region that controls the imprinted expression patterns for growth factor genes that affect the amount of nutrients obtained from themother. Assuming thatthe “primary imprinting signal” is gven, the model predicts that even with a small probability of the female accepting multiple mates, it is always very likely for the genes controlling maternal resource supply to evolve to an extreme asymmetry in gene expression. No matter howsmall the probability of the female accepting multiple males, the evolutionary outcome is alwaysan extreme asymmetry in gene expression between maternally and paternally derived genes. This result, however, does not apparently fit with the observed patterns in that not all the genesaffecting nutrient demand are imprinted and some genes are imprinted in mice but not in humans. We then discussseveral modified versionsof the model, incorporating processes that favor the absence of genomic imprinting. In particularwe study the possibility that recessive deleterious mutations may be present in the coding region of the gene. We demonstrate that whether genomic imprinting should evolve is affected by the frequency of deleterious mutations. Two equilibria, one with genomic imprinting and the other without imprinting, can be simultaneously stable. zyxwvu THE MODEL We consider the level of expression of a gene coding for an embryonic growth factor that determines the demand for maternal resources obtained through the placenta. An embryo has a pair of genes: one from the father and the other from the mother. Let x and y be quantitative traits indicating the amount of a growth factor produced by the paternal geneand by the maternal gene, respectively. These are eitherpositive or zero, as they indicate gene expression levels. We assume that their sum, x + y, is proportional to the amount of resources or care received from the mother. Here, we Genomic of Evolution Imprinting z zy 1285 vestigation, but it is suspected that differential methylation during gametogenesis plays a role ( CHAILLET et al. 1991; SASAKI et al. 1991;UEDAet al. 1992; LI et al. 1993). Here we simply assume that some mechanisms exist that give a reliable primary signal that indicates parental origin, and we concentrate on the evolution of the quantitative level of expression of a growth factor gene with this signal. We assume that a genecan produce the growth factor at two different levels in each generation, depending on its parental origin. This property of the growth factor gene is represented as a pair of values (x,y ) , where x is the amount of growth factor produced by the gene if it is paternally transmitted (ie., given the paternal signal), and y is the production by the same gene if it is maternally transmitted. In a single fertilized egg, there are two copies (paternal and maternal),each of which has a pair of values. Let ( xp,y!,) be the paternally derived allele, and (x,, y m ) be the maternally derived allele. By definition, the former produces ,!x and the latter produces ym, and hence the embryo should produce xr ym growth factor in total, which is proportional to the amountof resources acquired through the placenta. The survivorship of the embryo is then W ( xp zyxwvutsrq zyxwvuts zyxwvuts zyxwvu zyxwvutsrqpon zyxwvutsr zyxwvutsrq z 0 1 Total growthfactor 2 x+y FIGURE1.-Survivorship of an embryo as a function of the amount of growth factor production z = x + y. W( z) = wo ( z - a ) / ( 1 + Pz’) for z 2 a , but is zero for z < a. Parameters are as follows: a = 0.5, p = 0.5 and wo = 1. This is a function used for computation in all the numerical results in this paper, unless otherwise stated. Survivorship peaks when z = 2. Since an offspring with growth factor production lower than a dies, only evolution for region z 2 a is considered. examine the evolutionary change in x and y, in particular, the potential evolution toward asymmetric expression from an initial symmetry. We treat the physiological response of the motherto the growth factor as given, and do not consider the evolution of the mother’s response itself. The survivorship of the embryo increases with maternal resource supply, and hence it is a function of x y, denoted by W ( x y ) . An example of a survivorship function is illustrated in Figure 1. It is zero ( o r very small) below a certain threshold thenincreases quickly with x y, and finally saturates at a high levels. There may also be a peak after which it may decline with x y due to any harmful effects of producing too much growth factor. This curve gwes the survivorship of the offspring as a function of the amount of resources it receives from the mother. The embryonic growth factor production (x + y ) that maximizes survivorshipis however not optimal, because the embryo shares the common pool of maternal resources with its sibs, either simultaneously or over the mother’s reproductive lifetime. Hence reducing its resource demand will increase the number and the survivorship of sibs that are genetically related to the embryo. The level of growth factor production favored by natural selection should therefore be lower than the level that achieves the maximum survivorship of the embryo. The optimal growth factor production for the maternally derived gene is still lower than that for the paternally derived gene, because the former is more likely to be present in sibs than the latter, if the mother has accepted multiple males. Therefore, there is a conflict of interest between the alleles even though they are both present in the same cells of the embryo. The primary imprinting mechanism is still under in- + + + + + + Ym) . Nowwe consider the fitness of the (x, y ) allele, defined as the expected numberof copies in the following generation for each gene. We calculate the fitness as the multiplication rate from the adultstage in a generation to the adultstage in the next generation (not from the embryo to the embryo, as explained below). It depends on thesex of the gene carrier.Let +*( x, y; % 7) be the fitness for an ( x,y ) -allele in a male. It therefore depends on the population mean traits ( F, 7). Similarly, we consider r#JJ( x, y; % 7) , as the fitness of the (x, y ) allele in a female’s body. We call r#Jm and r#JJ male and female fitness, respectively, of the allele (x,y) . In the initial population, the two values may be equal ( % = 9 , indicating theabsence of genomic imprinting. In time, natural selection working on the regulatory region may cause some difference in the gene expression between thepaternal and maternal copies. We model this process using multi-variate quantitative genetics. The heritable values of x and y of individuals in the population areassumed to be sharply concentrated around their means( %and7) . One-generation changes in the population averages of two quantitative traits are given by the product of a genetic variance matrix and a selection gradient vector (see IWASA et al. 1991) : zyxwv G, and Gy are additive genetic variances for x and y, respectively, and B is the additive genetic covariance between them. These are determined by the balance of several different forces, such as mutation and stabilizing 1286 zyxwvutsrqp zyxwvutsr zyxwvutsr zyxwvutsr zyxwvuts zyxwvutsrqp zyxwvutsr A. Mochizuki, Y. Takeda and Y. Iwasa selection, pleiotropy and assortative mating, but here we simply treat them as constant, and assume that the genetic variance-covariance matrix is not degenerate (having an inverse) . This is acceptable ifwe concentrate on the evolutionary equilibrium rather than the transient. P5 and /?, are selection gradients with respect to the expression of maternally derived and paternally derived genes in the embryo: d l dx 2 Px= - - ( 1nc,bm+ and where the partial derivative is estimated at the population averages ( (x, y ) (E, 7) ) . These indicate the direction and magnitude of natural selection working on x and y, respectively. The factor ‘/2 indicates sexlimitation, i.e., the allele is a paternally derived gene in half of all the generations, and a maternally derived gene in the other half. Justification of the evolutionary dynamics Equation 1, a and b, is given by IWASA et al. (1991). The assumption of weak selection is used for the derivation. The formulation of Equation 1 has been used in computing evolutionary equilibrium and evolutionary limit cycle for male sexual ornaments and female matingpreferences (e.g., POMIANKOWSKIand IWASA 1993; IWASA and POMIANKOWSKI 1995). In Equation 1, a and b, the effect of socialinteractions (in this case competition for maternal resources between sibs) is incorporated in computing the fitness, and defined as the expected number of copies in the following generation. This is called the “neighbormodulated fitness” method ( HAMILTON 1964). Alternatively, we mayconsider thecost and benefit of an embryonic geneproducingdifferentamounts of growth factor, by separating the direct effect on its own survivorship and theindirect effect on the number and survivorship of its sibs, weighted by relatedness (HAMILTON 1964). Thelatter approachis based on inclusive fitness, and is veryuseful because it gives a clear intuitive understanding of the evolutionary stability. However the inclusive fitness approach is justified only for the invadability of rare mutants to the equilibrium, and not for calculating continual change in quantitative characters. In this paper we need to trace the evolutionary trajectory of two quantitative traits and hence we adopt the straightforward quantitative genetic formulation Equation l. To specify the dynamics in Equation 1, we need to compute the fitness functions + m ( x, y; % 7) and +f( x, y; E, 7)from the survivorship function of the offspring W( z ) . In this step we must specify how embryos of the same mother compete for maternalresources and how they are related genetically with each other. In the following, we examine a model in which the mother produces a large number of offspring simultaneously, the + offspring share theresources, the total amount of which is limited. We consider the fitness of a single gene of type (x, y ) in a reproductive female’s body. The expected number of copies in the next generation is the expected number of offspring produced by the female, multiplied by the probability that the copy of the allele is transmitted to each offspring, and also by the survivorship of the embryo carrying the allele. The survivorship depends on the alternative gene from her mate in the embryo. We adopt two assumptions. First, a reproductive female has a limited amount of resources T that can be used by her offspring. The amount of resource allocated to each embryo is proportional to the growth factor produced by the embryo with a proportionality coefficient a. Hence the total number of offspring is T divided by the average resource demand per embryo. Second, thefemale may mate with a single male or with two males, the latter occurringwith probability g, which we call the female polygamy rate. When a female mates with two mates, the males are equally likely to sire her offspring. Under these assumptions, we calculate the fitness of an allele ( x, y ) in a populationin which breeding values are concentrated sharply around ( % 7). According to the calculations in APPENDIX A , the fitness of allele (x, y ) in a female is +f( x, y; % 7)= T +++-) 1 - W( F + y). (2a) This can be interpreted intuitively as the product of three factors: the first factor of the RHS is the expected number of offspring, which is the total resource T divided by the average amount of resource demand per offspring. The average resource demand per offspring is calculated from the growth factor production by the allele concerned ( y ) , the average production by alternative alleles in the female (7),and theaverage production by the allele from her mate (x). The second factor ‘/2 is the probability that the allele concerned will be transmitted to each offspring. The third factor is the survivorship of an embryo, which is a function of the total growth factor production, x + y, the sum of the products of paternal and maternal alleles. From the assumption of random mating, there is no correlation between the geneticvalues of the allele from the female and that from her mate. Absence of inbreeding also assures that the two homologous alleles in the same female are not correlated. For the purpose of evaluating the selection gradients in Equation 1, ie., the partial derivative of fitness with respect to the breeding values of the alleles of concern ( x or y ) , we can set all the other breeding values to population averages (see APPENDIX A , for details). Note that female fitness Equation 2a is independent of the female polygamy rate g. Next, we consider +m( x, y; g Y), the male fitness of zyx zyxwvut zyxwvutsrq zyxwvutsr zyxwvu zyxwvutsr zy zyxwvuts zyxwvutsrq zyxwv zyxwvutsr Evolution of Genomic Imprinting 1287 A an (x, y)-allele in the population with mean traits ( % 7 ) .Let M be the expected mean number of females mating with the male. Mmay depend on the population size, the number of males and other variables, but it is assumed independent of x or y. The probability of an allele in a male being transmitted to each embryo depends on the number of males that a female accepts. If the female accepts a single male (with probability ( 1 - g) ) , the probability is but if the female accepts two males [with probability g (see APPENDIX A for the reason of another factor 2 ) ] , the probability is The average resource demand per embryo also depends on the numberof malesthat afemale accepts. If the female accepts a single male, the average resource demand is 7 ( x E ) / 2, but if the female accepts two males, it is p (x 3 E ) /4. The survivorship of the embryo carrying the ( x , y)-allele is W ( x + 7). We can derive the fitness of the allele ( x , y ) in a male as + zyxwvuts zyxwvutsr + + + c .-0 fn Ea Y X W 1 r 1 0.5 X W( x + 7).(2b) f / yi . .r\ T =MU z B fn i L .1 1 In APPENDIX A , we also derived the fitnesses for the case 0 0.5 1 X in which females may accept more than two males. Substituting Equation1 by Equations 2a and 2b gives Expression of paternal allele us the evolutionary dynamics of the average traits ( % FIGURE2.-Evolutionary trajectories of the basic model. 7). Figure 2 illustrates a typical trajectory of the evoluThe two axes are population average of paternal allele exprestionary dynamics of ( E 7). Starting from any initial sion xand maternal allele expression ( A ) Female polygamy state the population quickly converges to a line of x rate is large ( g = 0.5). Other parameters are G, = G, = y = constant and thenmoves towarda biased expression 0.2, B = 0, and W (z ) given by Figure 1. (The dynamics are with x increasing and p decreasing, finally converging independent of T, A, or M ) . The location of ( z 7)for every other generation is indicated. ( B ) Parameters are the same to the state in which only the paternalallele is expressed as in A exceptfemale polygamy rate is small ( g = 0.05). ( X > 0, 7= O ) , thus implying extreme imprinting of There is a globally stable equilibrium on the faxis in which the growth factor gene. When the population reaches the maternal allele is silent. This implies that stronggenomic this state, the mean trait should stop decreasing further,imprinting should evolve and that only the paternal copy of since a negative value of Xor7is notbiologically meanthe gene is expressed. The speed of the convergence to the final equilibrium depends on the female polygamy rate g. ingful. This behavior can be understood as follows: Suppose that the total growth factor production optimal for the paternal allele is expressed and the maternal allele is paternal allele X + p = ,z is a little larger than that silent. optimal for the maternalallele z + p = z(, z, > z, ) , two In APPENDIX A , we show that this conclusion is quite parallel lines on the ( % y) -plane are formed. Theinitial general: there is no equilibrium satisfying 3 > 0 and transient is the quick convergence of the average traits > 0, provided that thereis some probability of a female ( g 7)to the region between these two lines, in which accepting multiple males ( g > 0 ) . When g = 0, the two the total growth factor production is smaller than the lines where there are no selection gradients ( px = p, paternal allele optimum but larger than the maternal = 0 ) coincide exactly with each other, producing a line allele optimum. Subsequently E increases and p deof equilibria. creases very slowly but the population average traits ( % If there is only a small probability of females ac7)stay in the region between two lines. The asymmetry cepting multiple mates, we might expect that an interin gene expression increases and the final outcome is mediate degree of imprinting could evolve, so that two on theboundary of the positive orthant, where only the alleles are expressed atdifferent levels. The present + 1288 zyxwvutsrqp zyxwvutsr zyxwvuts zyxwvu zyxw zyx zyxwvuts A. Mochizuki. Y. Takeda and Y. Iwasa model does not support this conjecture. There is no possibility of a positive equilibrium in which both the paternal and maternal genes are expressed. If the degree of female polygamy g is small, the evolutionary movement toward genomic imprinting is slow,but even a slight possibility of multiple mating by the female is enough to make the evolution of genomic imprinting a certainty (Figure 2 ) . Complete monogamy, g = 0 , is found rarely if at all in nature with the study of the mating systems of mammals revealing that some degree of sperm competition is common (DEWSBURY 1984; HARW and HARCOURT 1984; SMITH1984). We can therefore conclude that the model predicts that genomic imprinting evolves easily in genes that affect the embryo’s growth. However, contrary to this prediction,thereare growth factors that affect the formation of the placenta that are not imprinted. For example, the Igf 1 gene is imprinted in neither mice norhumans despite its marked effect on the development of the embryo ( LIU et al. 1993). In addition, Igf2r in the placenta, which is a suspected inhibitor of Igf 2 ( HAIG and GRAHAM 1991), is maternally imprinted in the mouse but not in humans ( KALSCHEUER et al. 1993; OGAWA et al. 1993). These findings are in conflict with the simple and general prediction of the model that all the genes affecting the amountof nutrients from the mothershould evolve strong differential expression. Hencethere must be some additional processes that inhibit the evolution of strong genomic imprinting. EFFECT OF DELETERIOUSMUTATIONS These discrepancies can be explained by additional processes that favor equal expression of both genes, thereby discouraging the evolution of genomic imprinting. Here we examine the most plausible candidate; the presence of deleterious mutation in the coding region of the gene. The pointis that thereis a mutation-selection balance at all loci that creates a selection against imprinting. Intuitively speaking, the advantage of not being imprinted, i.e., having biallelic expression, is the same as the advantage of being diploid. Since recurrent recessive deleterious mutations are thought to be the most important process favoring diploidy (e.g., PERROTet al. 1991) , they are a promising candidate to explain the disadvantage of genomic imprinting. Hence, we consider deleterious mutations in the structural gene of the growth factor that may affect the evolution of the regulatory region. In the population, such deleterious nonfunctional mutations occur every generation and are eliminated by natural selection. They are maintained at a frequency determined by the balance between mutation and selection. We assume that linkage between the regulatory region and the structural gene is tight, and we consider only the regulatory region linked with a wild-type (functional) structural gene. Regulatory regions linked with a defective structural gene are neglected, since they will be eliminated from the population. However, the evolution of a regulatory region linked with a wild-type gene can still be affected by the existence of mutations in the population because they may be associated within the same embryo. Let f be the frequency of deleterious (or null)mutation of the structural gene at equilibrium. Although f may change in evolution, here we treat f as a constant for the sake of clarity of argument. If the gene is imprinted in the population (x > 0 and y = 0 ) , paternal deleterious mutants produce a phenotype deficient in growth factor, though the maternal deleterious mutant is normal. In contrast, if the gene is not imprinted (x = y ) , deleterious mutations (both maternal and paternal) produce a phenotype with growth factor production at a lower rate (half of the normal phenotype). Considering the homeostatic adjustment of develop ment, reducing the amount of growth factor would be less serious to embryonic development than the complete absence of growth factor, which is likely to be lethal. The possibility of being associated with deleterious genes may discourage the evolution of genomic imprinting. This conjecture may, however, depend on the shape of survivorshipas a function of growth factor production W ( z ) andneeds to be examined using quantitative mathematical models. In the model, a female may mate with a single male or two males as her mates, with probability 1 - g and g, respectively. In computing the fitness of an (x, y ) allele in a reproductive female, we need to separate cases according to whether she and her mate ( s ) carry a mutant or wild-type a copy. Sincethe mutanthomozygotes are lethal ( W( 0 ) = 0 ) , the adult population is composed of wild-type homozygotes ( + / + ) and heterozygotes ( + / - ) only. We denote the frequency of the mutant genein the population by f . Then the fraction of wild-type homozygotes ( + / + ) is 1 - 2 f and that of heterozygotes ( + / - ) is 2 f , for both males and females, A random mating assumption allows us to use these fractions in computing the combination of the genotypes of a female’s mates. On the other hand, the female herself is either heterozygous or homozygous, with probability (1 - 2 f ) / ( 1 - f ) and f / (1 - f ) , respectively (further explanation is in APPENDIX B ) . By computing the fitness of the (x, y)-allele for different cases and by averaging them with the probability of these cases, we have the female fitness function 4,. A male fitness function 4m, the expected number of copies of the ( x, y ) -allele linked with a wild-type structural gene in males is also derived in APPENDIX B. The results are straightforward but are too bulky to show in the text. By replacing these in Equation 1, we have the evolutionary dynamics of ( g 7). The evolutionary outcome is as follows: ( 1) no imprinting when f is large, or ( 2 ) full imprinting when zy zyx zy Evolution of Genomic Imprinting zyxwz 1289 lmminted / 7 zyxwvu zyxwvutsrqponmlkjih / / A t I 0.5 0.25 / 1 - X imprinted L 0.003 0.006 f gene frequencyof deleterious mutations in the population zyxwvutsrqpon zyxwvutsrqp zyxwvutsrqp zyxwvuts Expression of paternal allele FIGL‘RE 3.-Evolutionary trajectorieswith deleterious recessive mutations of the structural gene. Frequency of deleterious mutations is J = 0.02. Other parameters are the same as in Figure 2R. At the evolutionary equilibrium, both paternal and maternal alleles are expressed ( X > 0, -y > 0 ) , implying the absence of strong genomic imprinting. f is small, o r ( 3 ) an intermediate degreeof imprinting when f is intermediate. Figure 3 shows the evolutionary trajectories of the model in case 1. Even for polygamy rate .g > 0 , if f is significantly large, there is a stable equilibrium in the middle of the positive orthant ( K > 0 and f > 0 ) . In this particular case, maternal and paternal genes are expresseda at similar level, indicating the absence of genomic imprinting. A large polygamy rate g . causes a strong conflict of interest behveen hvo growth factor alleleswith different parental origin andresults in a strong asymmetry in the expression levels. Ontheotherhand,genomic imprinting is discouraged by deleterious mutations with frequency f in the population. In Figure4, the relative magnitude of production of maternal alleles to that of paternal alleles (-y/ r) is illustrated as a contour map over hvo parameters: the frequency of deleterious mutations f and female polygamy rate g. The basic model discussed in the previoussection corresponds to the points on thevertical axis ( f = 0 ) . In the shaded region nearthe vertical axis, stronggenomicimprinting should evolve (-y= 0 ) . As the frequency of mutants in the population fincreases and exceeds a line on this plane, the maternal alleles becomes expressed, though at a level lower than the paternal allele. As the ratio of f to .g increases, the difference between maternal and paternal alleles becomes smaller, reducing the degree of genomic imprinting. FIGL‘RE4.-The ratio of the expression of the maternal allele to the paternal allele at evolutionary equilibrium .y/ R Horizontal axis is the frequency of tlcletcrious mutations in the population J , vertical axis, female polygamy rate g. The resource division model with sunivorship crlnre in Figure 1 is assumed. imprinting, by considering the risacting regulatory region controlling the level of expression dependent on parental origin. MTe observed that asymmetric expression of the hvo alleles evolves very easily from the initial symmetry, if there is even a small chance of a female mating with more than one male. By slight modification, we can show that the gene coding for inhibitors should evolve genomic imprinting in the reverse direction with the paternal allelebecomingsilent, as explained in APPENDIX (;. This confirms the genetic conflict hypothesis, previously stated verbally by DAVID HAIC and his colleagues ( HAIC;and M’ESTOBY 1989, 1991; HNC and GRAI-IAM 1991; MOOREand HAK:1991 ) . ” h a t is notable in the basic model is that theevolutionary outcome is always an extremeasymmetry even if the probability of female polygamy is small. In this paper, we examine thc case of a “resource division model” in which the mother prodwes a large number of offspring simultaneously and the offspring share the resources. I n many mammals, however, a single offspring is gestated at a time, and the production of the next offspring becomes possible only when the mother stopsinvesting in the“current” offspring, which is apparently in conflict with the assumptions o f the resource division model. In a separate paper, we examinethe“sequentialcaremodel” in which offspring are born andreceive maternal care one ata time (Y. IWASA, A. MO(;HIZLTKI and Y. TAKEDA, unpublished results). Thesurvivorship ofan ofkpringincreases with the length of time that the mother invests her care t o that offspring. A longer periodof care for eachindividual reduces the total number of ofl-yxing produced in the mother’s lifetime. The result of this model is zyxwvuts DISCUSSION We examined mathematical models of the evolution of differential expression of the paternally and the maternally derived allelesof growth factor genes, genomic 1290 zyxwvutsrqp A. Mochizuki.and Y. Takeda qualitatively the same as the present model.If the probability that a female accepts multiple mates is positive, the expression levelof the gene of a growth factor should evolve to show extreme asymmetry.We will show this result in another paper. The ease of the evolution of genomic imprinting predicted by the basic model is not totally consistent with the observation. Some genes known to control the embryonic developmentare notimprinted (Lru et al. 1993), and there are genes that are imprinted in the mouse but not inhumans (KALSCHEUER et al. 1993; OGAWA et al. 1993) . To explain why there are only a few genes imprinted in mammals, we considered models that incorporated potential processes that favor equal expression of the maternal and paternal alleles in the embryo. We examined one candidate, the effect of deleterious mutation in the coding region of the growth factor gene. Depending on the valuesof the polygamy rate and the frequency of mutation in the population, the model predicts either the absence or the presence of genomic imprinting. In a separate paper (Y. IWASA, A. MOCHIZUKIand Y. TAKEDA, unpublished results), we examined two other candidate processes that potentially favor the absence of genomic imprinting. First, a mistake in labeling the parental origin of one allele would make both genes in an embryo stay silent or both genes be active, resulting in deficient or overproduction of growth factor. Hence it is possible that this may disfavor the evolution of genomic imprinting.However, mistakesoccur tooinfrequently ( REIK et al. 1995) to be important. Second, the possibility of differential expression causing additional energy and time expenditure that may reduce the fitness by a small amount therefore favors the absence of genomic imprinting. The models considered are qualitatively the same as the model with the deleterious mutations examined in this paper. Depending on parameter values, the models predict genomic imprinting may or may not evolve. The analysis in this paper suggests that whether an embryonic growth factor gene evolves to become imprinted or not may be determined by the balance between the polygamy rate and the frequency of deleterious mutations of the structural genein the population. Imprinting is more likely to evolve in a population with a high polygamy rate. Since the polygamy rate is the same for different genes of the same species, the variation between genes in terms of imprinting should be explained by the difference in the frequency of deleterious mutations perstructuralgene,thelatterdetermined by the mutation-selection balance. Hence a gene including moreof functional sites should have a higher mutant rate per gene andis less likely to be imprinted than genes with fewer functional sites. Recent study of the molecular mechanism of imprinting has revealed that imprintedgenes tend to form Y. Iwasa clusters that are located in “imprinted chromosomal domains”, and that many genes tend to be controlled by a single “imprinting control element” ( REIS et al. 1994; SUTCLIFFE et al. 1994; BUITINCet al. 1995; EDEN and CEDAR1995). This suggests that reliable information on the parental origin of a chromosome is not always available, and once it is available then many genes may in fact evolve to show genomic imprinting. If this interpretation is correct, the conclusion of the basic model in the present paper may be justified. The question then iswhy are imprinted chromosomal regions rare. The hypothesis of the cost of imprinting including all the molecular machinery for imprinting may be worth more careful examination. In this paper, in examining the evolutionary change in the level of gene expression, we assumed that a primary signalfor parental origin is given. Underlying this is the assumption that the degree of expression should be determined by the nucleotide sequenceof the regulatory region of the growth factor genes. The primary imprinting of the gene should be given during the gametogenesis. If the pattern of imprinting for a gene is determined by the parental molecular machinery rather than the DNA sequence that is to become imprinted, we need to examineamodelin which the parental diploidgenotype would determine the pattern of imprinting for their gametes. The predicted evolutionary outcome should be different from the analysis in the present paper due to parent-offspring conflict (TRIVEKS 1974; QUELLER 1994). Another form of involvement of the parental genome is the reaction of the mother to the amount of growth factor produced in each embryo, which is certainly an important theme of future theoretical investigation (see HAIG 1993) . In the currentpaper we concentrateonthe embryo’s growth factor production by simply assuming that the mother would invest more to embryos producing more growth factor. We considered the evolution of cis-acting regulatory regions of growth factor genes and succeeded in explaining patternsof extreme asymmetry ofgene expression depending on the parental origin. This in effect explains genomic imprinting as “voluntary” regulation of maternally derived and paternally derived alleles. MOOREet al. ( 1995) and MOORE and REIK ( 1996) suggested additional mechanisms by which an allele suppresses the opponentallele directly: Some substances in the oocyte cytoplasmmay modify the paternally derived allele after fertilization to control the expression level of the allele, which can be counteracted by the sperm through the production of cytosolic factors. They also suggest that such postzygotic reprogramming may be a reason for the nonimprinting of genes that affect the growth of the embryo. If a different survivorship function W (z ) is adopted, some of the conclusions can be changed for the case with deleterious mutations favoring the absence of im- zyxwvut zyxwvutsrq zyxwvutsr z zyxwvut Evolution of Genomic Imprinting A 0 1 2 zy B I two stable equilibria. Hence the model predicts that, depending on evolutionary history, two species having similar ecology and mating system may show extremely different levelsof genomic imprinting. This may explain the difference inimprintingpattern between mice and humans ( KALSCHEUER et al. 1993; OGAWA et al. 1993) . However the range of parameter values f and g that allow the evolution of bistability as shown in Figure 5B is quite small. In addition to the genetic conflict hypothesis, several alternative hypotheses have been proposed to explain the pattern of genomic imprinting. For example, VARMUZA and MANN (1994) proposedthatimprinting might be a device that protects female mammals from thepotential ravagesof ovarian trophoblast disease caused by the spontaneous startof the development of unfertilized eggs. If the maternal allele is silent (7= O ) , the risk of ovarian trophoblast disease would be reduced, to the advantage of both parents, thus favoring the reduction of maternal gene expression. This, together with subsequent compensatory evolution of paternal gene expression, would produce the evolution of genomic imprinting, even in the absence of a conflict of interest between paternal and maternal alleles. This hypothesis can be studied by a similar theoretical framework to the present paper, butwe would like to pursue this project in a separate paper. To examine thefeasibility of several different hypotheses proposed to explain the phenomena of genomic imprinting, to establish when and how each mechanism works, and to predict quantitatively the relative importance of the alternative hypotheses, we need to develop theoretical studies for each,just as we have in this paper for the genetic conflict hypothesis. zyxwvu Total growth factorx+y * 1291 / J . . zyxwv zyxwvutsrqponml zyxwvutsrqpo zyxwvuts - X Expression of paternal allele FIGURE 5.- ( A ) Survivorship of an embryo as a function of the amount of growth factor production, W ( z ) = wz2/ (1 + h3), with ~0 = 0.4, h = 0.25. This is slightly different from the one in Figure 1. ( B ) Evolutionarytrajectory with deleterious recessive mutation of structural genes (resource division model). Parameters are as follows: g = 0.1, f = 0.08. There are three equilibria, one stableequilibrium without genomic imprinting ( X > 0, > O ) , another stable equilibriumwith genomic imprinting( X> 0, y= 0 ) ,and an unstable equilibrium lying between these two. The system is bistable, and the evolutionary outcome depends on the initial population. printing. For example, Figure 5B illustrates the evolutionary trajectories when the survivorship function is given by Figure 5A, which is slightly different from the one in Figure 1 used to generate Figures 2 and 3. For a particular parameter values of gand f , we obtain the evolutionary trajectories shown in Figure 5B, in which the system is bistable, having three equilibria, twoof which are locally stable. One stable equilibrium is in the middle of the graph, indicating that both paternal and maternal alleles are expressed. The second stable equilibrium is on the saxis indicating that the maternally derived allele is inactive and that the paternally derived allele is strongly expressed. Between these two stable equilibria, there is an unstable equilibrium. Evolutionary trajectories should be separated into two domains of attraction, each corresponding to one of the We express our sincere thanks to Professor HIROWIUSASAKI, Research Laboratory for Genetic Information, Kyushu University, who kindly introduced us to the problem of genomic imprinting. We also thank following people for their very useful comments: M. B o o r s , C. GODFRAY, D. HAIG,I. KOBAYASHI, HIROTSUGA MATSUDA,HIROVUKI MATSUDA, T. MOORE,W. REIK, A. SASAKI,A. SURANI, K. TAKAHASHI, S. TAKAHASHI, M. UYENOYAMA and T . Y-. This workwas s u p ported in part by a Grant-in-Aidfor Scientific Research by the Ministry of Education, Science, and Culture,Japan. LITERATURECITED BARI.oW, D. P., R. STOGER,B. G. HERRMANN,R SAITO and N. SCHWEIFER, 1991 Themouse insulin-like growthfactortype-2 receptor is imprinted and closely linked to the Tme locus. Nature 349: 84-87. BARTOLOMEI, M. S., S. ZEMEL and S. M. 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ROPERS,1993 The insulin-like growth factor type-2 receptor gene is imprinted in the mouse but not in humans. Nature Genet. 5 74-78. LI, E., C . BEARDand R. JAENISCH, 1993 Role for DNA methylation in genomic imprinting. Nature 366: 362-365. LIU,J.-P., J. BAKER, A. S. PERKINS, E. J. ROBERTSON and A. EFSTRATImIs, 1993 Mice carrying null mutations of the genes encoding insulin-like growth factor 1 ( I g f l ) and type-1 IGF receptor ( I g f I r ) . Cell 75: 59-72. MOORE,T., and D. HAIG,1991 Genomic imprinting in mammalian development: a parental genetic conflict. Trends Genet. 7: 4549. MOORE,T., and W. &IK, 1996 Geneticconflict in early develop ment: parental imprinting in normal and abnormalgrowth. Rev. Reproduction 1: 73-77. MOORE, T., L. D. H U R ~ and T W. REIK, 1995 Geneticconflict and evolution of mammalian X-chromosome inactivation. Dev. Genet. 17: 206-211. OGAWA,O., L.A. MCMOE,R. EGCLES,I. M. MORRISONand A. E. REEVE, 1993 Human insulin-like growth factor type I and type I1 receptors are not imprinted. Hum.Mol. Genet. 2: 2163-2165. OZCEIiK, T., S. LEFF,W. ROBINSON,T. DONLON,M. WNDE et al., 1992 Small nuclear ribonucleoprotein polypeptide N (SNKPN) , an expressed gene in the Prader-Willi syndrome critical region. Nature Genet. 2: 265-269. PETERSON, K.,and C. SAPIENZA, 1993 Imprinting the genome: imprinted genes, imprinting genes, and a hypothesis for theirinteraction. Annu. Rev. Genetics 27: 7-31. PIASS,C., H. SIIIBATA, I. K ” C H E V A , L. MUI.LINS, N. KOTELEWSEVA et al., 1996 RLGSM identification of CDC25M”as an imprinted gene. Nature Genet. (in press). POMIANROWSKI, A,, andY. IWAsA, 1993 Evolution of multiple sexual ornaments by Fisher’s process of sexual selection. Proc. R. Soc. Lond. B. 253: 173-181. PERROT, V., S. RICHERD and M. VALERO, 1991 Transition from haploidy to diploidy. Nature 351: 315-317. QUELLER, D. C., 1994 Male-female conflict andparentuffspring conflict. Am. Nat. 144: S84-S99. REIK, W., K. W. BROWN, H. SCHNEID, Y. LE BOUC,W. BICKMORE and E. R. MAHER, 1995 Imprinting mutations in the Beckwith-Wiedemann syndrome suggested by an altered imprinting pattern in the IGF2-HI9 domain. Human Mol. Genet. 4 2379-2385. REIS, A,, B. DITTRICH, V. GREGER,K. BUITING, and M. L A I A N ~ ~et~ ~ . nl., 1994 Imprinting mutations suggested by abnormal DNA methylation patterns in familial Angelman and Prader-Willi syndromes. Am. J. Hum. Genet. 54: 741-747. SASAIU, H., T. HAMADA, T. UEDA,R. SEKI,T. HIGASHINAKAGAWA el al., 1991 Inherited type of allelic methylation variations in a mouse chromosome region where an integrated transgene shows methylation imprinting. Development 111: 573-581. SASAKI, H., N. D. ALLEN and M. A. SURANI, 1993 DNA methylation and genomic imprinting inmammals, pp. 469-486 in DNA Methylation: Molecular Biology and Biologzcal Sign$cance, edited by J. P. JOST and H. P. SAI.UZ. Birkhauser Verlag, Basel. SMITH,R. L., 1984 Human sperm competition, pp.601-659 in Sperm Competition and the Evolution of Animal Mating Systrms, edited by R. I,. SMITH.Academic Press h c . , New York. SUTCIJFFE, J. S., M. NAKAO,S. CHRISTLAN, K. H. ORSTAVIK, N. TOMMERUP et al., 1994 Deletions of a differentially methylated CpG island at the SNRPN gene define a putative imprinting control region. Nature Genet. 8: 52-58. TRIVERS, R. L., 1974 Parent-offspring conflict. Am. 2001. 14: 249264. UEDA, T.,K. YAMAZAKI, R. SUZUKI, H. FUJIMOTO, H. SASAKI et al., 1992 Parental methylation patterns of a transgenic locus in adult somatic tissues are imprinted duringgametogenesis. Development 116: 831-839. VARMUZA, S., and M. M A N N , 1994 Genomicimprinting-defusing the ovarian time bomb. Trends Genet. 10: 118-123. WEVIUCK, R., J. A. KERNS and U. FRANCKE, 1994 Identification of a novel paternally expressed gene in the Prader-Willi syndrome region. Human Mol. Genet. 3: 1877-1882. zyxwvutsrqp zyxw Communicating editor: L. PARTRIDGE zy zyxwvutsrqpon zy APPENDIX A Derivation of (2a) and (2b): Consider a gene ( x,y ) that is possessed by a reproductive female. Let ( x’, y’) be the alternative allele in the female. She accepts a single malewith probability 1 - g. Let ( xl, y l ) and ( x;, y i ) be the two alleles at the same locus possessed by the male. The female may accept two maleswith probability g. The two genes of the first male are ( x 1 , y l ) and (x;, y;) , those of the second male are ( q ,y2) and (x;, y;). zyxw zy zyx zyxwvutsrqp zyxwvutsr zyxwv zyxwvu zyxwvuts Evolution of Genomic Imprinting 1293 tics similar to that above, we have female fitness +fthe same as ( 2 a ) , and male fitness +m as where E [ 03 implies the operation of the population average with respect to ( x’, y ’ ) , ( ~ 1 yl, ) , ( xi, y i ) , ( x2, B), and ( x;, y 6 ) . Since we assume that the population is concentrated sharply around the mean values, and also that mating is random (no correlation between two homologous alleles of the same individual) , then we can simply replace x‘ = xl = x i = x2 = x; = 5, and y’ = 7, but leave x and y intact, because we need to compute partial derivatives of fitness with respect to x and y. (A1 ) becomes zyxwvutsr T + - I + W( y + x ) . (A4) With pl = 1 - g, p, = g,p, = p4 = - * * = 0 , (A4) becomes ( 2 b ) . Evolutionary equilibrium:At equilibrium, the two selection gradients must be zero: 1 which is (2a) in the text. In computing the male fitness, we consider a gene of type (x, y ) in a reproductive male and consider a female who accepted this male as her mate. Let (q,, yo) and (x;, y;) be the two alleles of the female. Here we note that the probability for a randomly chosen male mates with a female that accepts two males is larger than g, the latter being the probability for a female to accept two males. The former probability (probability for a male to mate with a female that accepts two males) is 2g/ 1 + g,and the probability for a male to mate with a female which accepts one male is 1 - g/ 1 g. Then the fitness for the male, after being multiplied by a factor M , is +,,,=M- X gE[ T g a ( % ( y o + y;) + % ( x + x ’ ) ) - W’(X+ - 7)- W ( x +7) 1 = 0. 2 ( x + 7) (A5b) Both (A5a) and (A5b) specify a line of slope minus 1 ( E + y= constant). If g > 0 , these two lines are parallel, implying that there is no equilibrium. If g = 0, two lines coincide, forming a line of equilibria, and the system is neutrally stable. The total amount of growth factor F + ythat satisfies (A5a) is equal to the paternal optimum and the one satisfjmg (A5b) is thematernal optimum (see also HAIG 1992). APPENDIX B Effect of deleterious mutations on structural genes: Consider a ( x, y) -regulatory region that is linked with a wild-type structural gene. First we consider the case in which it is in a reproductive female. We need to distinguish cases according to ( 1) whether the female accepts a single male or two males, ( 2 ) whether the alternative allele in the same holder iswild type or mutant (denote here by + or - ) , and ( 3 ) whether her mate ( s ) is homozygous wild type or heterozygous (denote by +/ or +/ - ) . There is no possibility of homozygosity of mutants, which are lethal and eliminated before the reproductive stage. Let f be the gene frequency of the mutant at the reproductive stages (instead of the time of fertilization). Then the frequencies of wild-type homozygotes and heterozygotes are 1 - 2fand 2f,respectively. On the other hand, the alternative allele in the same holder for a randomly chosen wild-type allele is also wild type with probability 1 - 2f/ 1 - f,and it is mutant with probability f / 1 - f. zyxwvut zyxwv The denominator, 1 + g of 2g/ 1 + g and 1 - g/ 1 + g can be considered to be neglected because they are a commonfactor. Now by taking the average E [ 0 J with respect to ( x r , y ’ ) , (x2,B),( 4 , y 6 ) , ( ~ 0 yo), , and ( x;, y; ) considering that the breeding values are centered around the population mean, we have (2b) in the text. In a similar way,we can compute the case in which a female accepts n males with probability p,, which sire the offspring of the female equally. Aftersome arithme- + zyxwvu zyxwvut zyxwvuts zyxwvutsrqpo zyxwv zyxw A. Mochizuki, Y. Takeda and Y. Iwasa 1294 The fitness of a female is 4,= ( 1 - g ) [ fitness when she accepts a single male] + g[ fitness when she accepts two males] . From ( B l a ) together with ( B l b ) and ( B l c ) , we have female fitness function 4,. The male fitness +m is zy i zyxwvu (Bla) 4 1,L The fitness when the female mates with a single male is the sum of four terms summarized as zyxwvuts zyxwvut The fitness obtained from mating with a single female is calculated in a similar way as the female fitness function +f, by noting that it differs from female fitness when the female mates with two males. The result is [ fitness when she accepts a single male] I-/ 1 2.f) - = M [ fitness obtained from mating a single female] . f 2f +-1 - f )H+ "+a Y %' 0 0 = + a - il' 21 X-W(y+X)+2f g ) (1 - +ya/ Y+P+? 2 -+- 2 2 2 The fitness when the female accepts two males should be classified into sixcases, according to whether the female herself is homozygous or heterozygous, and to three cases: ( 1) both males are homozygous, ( 2 ) one male is homozygous and the otheris heterozygous, and ( 3 ) both males are heterozygous. mi 1 + -2g a [Fitness when she accepts two males] + f 1 - 2.f' 1 - f I f 1 - 2 f (1 - 2 f ) (1 1 1 - 2f - 2 f ) ___ x +3x 4 1 - 2f (1 - 2 f ) -+2f"1- f 1 - f x 2% +7 4 + 1 - f +P 25 + "1 1 -f a - +4B J x+ 1 - f Y + L F y + -3a . 2 4 2 4 I " 4 +'2 + (2f)2 i -f - r+Y;x 2 2 +-1 - f 4 x+- 2 2 + 1 - 2f (1 - 2 f ) -+ 2 f 1 - f 1-f x+2n 4 7 + -2 2f -q 1 - f + 4 J' 2 z zyxwvutsrq zyxwvu zyxwvutsrqp zyxwvutsr zyxwvutsrqp zyxwvutsr zyxwvut zyxwvuts zyxwvut 1295 Evolution of Genomic Imprinting APPENDIX C A Genomic imprinting evolution €or an inhibitor gene: Here, we consider a gene coding an inhibitor that decreases the growth rate of the embryo. For example, Igf2r in mouse is suspected of being an inhibitor of Igf 2 ( HAIG and G m 1991) . The product of Igf 2 r gene decreases the size of the embryo and is expressed only from the maternal allele ( BARLOWet al. 1991) . We consider the expression levelof aninhibitor gene, the productof which digestsor inactivates embryonic growth factors. If an embryo receives the ( xp,yp) allele from its father and the (x,, y,) -allele from its mother, it produces an quantity of inhibitor xp + ym in total. In contrast to the growth factor, the total quantity of the inhibitor decreases the demand for maternal resource supply and the survivorship of the embryo. The survivorship function of an embryo, denoted by WE( xp + y m ) , is a decreasing function of xp + y,,,. Suppose that the resource allocated by the mother to each offspring is B zyxwvutsrqpo 0.3 where F, is the total amount of growth factor when no inhibitor is present. Then the fitness functions of the resource division model for this gene are 0.1 9 2 Total Inhibitor x+y (C1) a(Fm - ( x p + y m ) ) 1 0 "..-.. .. . . J/ . +.* -. . 4f( x, y; 5 7) 0 0.1 0.3 0.5 Expression of paternal allele r 1 X W ( x + y ) . (C2b) Substituting ( 1 ) by (C2a)and(C2b) givesus the evolutionary dynamics of the average expression levels from the paternal allele and the maternal allele.Suppose that survivorship function is illustrated in Figure 6A. As illustrated by the evolutionary trajectories of (X, y) in Figure 6B, theexpression level from the - X FIGURE6.-(A) Survivorship of an embryo as a function of the amount of growth-inhibitor z = x + y, K (z) = w,, ( F , - z - a ) / ( l +P(F,-z)2),forz<F,-a,butiszerofor z > F,,, - a. Parameters are a = 0.5, p = 0.5, F, = 2 and w,, = 1. It decreases with z. Since an offspring with inhibitor higher than F, - a dies, evolution only for the region z < F, - a is considered. ( B ) Evolutionary trajectories of the resource division model for a growth-inhibitor gene. Female polygamy rate is large ( g = 0.3) . Other parameters are G, = Gy = 0.2, B = 0, and ( z ) is given by A. The dynamics are independent of T, a and M. The locations of ( z 7) for every other generations are indicated. There is a globally stable equilibrium on the Taxis in which the paternal allele is silent. maternal allele becomes very large and the paternal allele become silent as the result of evolution, if there is some probability for the female to accept multiple mates ( g > 0 ) .