Copyright0 1992 by the Genetics Society of America
Population Genetic Modelsof Genomic Imprinting
Gavin P. Pearce and Hami...
900 G. P. Pearce andH. G. Spencer
Viabilityparametersused in the models
form of genomic imprinting is mo...
Models of Genomic Imprinting 901
printed with probability 1-a.Thus a = 0 in model 1.
As before, we can assume that thealle...
902 G. P.Pearceand H. G. Spencer
T o see this, suppose that w l l > w12and, without loss
of generality, that w11 3 w22. Th...
Models of Genomic Imprinting 903
therefore, by (21), we require 1.41 <w20< 2.34 and
so let wz0 = 1.5. This gives w;”l = 0....
904 G. P. Pearceand H. G. Spencer
the female population after selection, the genotypic
[AIAI]/= prnpfwlf/Wf,
Models of GenomicImprinting 905
pk = pfwl/zm (49)
6,= pfw1 + qfw2 = 6f. (50)
0.0I0 20 40 60 eo 100
No. Gle...
906 G. P. Pearce and H. G. Spencer
most general form such differential gene expression
means that reciprocal heterozygotes...
Models of Genomic Imprinting 907
alent to a non-imprinting system with it, a property
we callpseudoheterosis.
When the int...
of 9

Population Genetic Models of Genomic Imprinting

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Transcripts - Population Genetic Models of Genomic Imprinting

  • 1. Copyright0 1992 by the Genetics Society of America Population Genetic Modelsof Genomic Imprinting Gavin P. Pearce and Hamish G. Spencer’ Department of Mathematics and Statistics, University of Waikato, Hamilton,New Zealand Manuscript receivedJune 27, 1991 Accepted for publication November 22, 1991 ABSTRACT The phenomenon of genomic imprintinghas recently excited much interest among experimental biologists. The populationgeneticconsequences of imprinting,however, haveremainedlargely unexplored. Several population genetic modelsare presented and the following conclusions drawn: (i) systems with genomic imprinting neednot behave similarly to otherwise identicalsystems without imprinting; (ii) nevertheless,many of the models investigatedcan be shown to be formally equivalent to models without imprinting;(iii) consequently, imprinting often cannotbe discovered by following allele frequency changesor examining equilibrium values; (iv) the formal equivalencesfail to preserve some well known properties. For example, for populations incorporating genomic imprinting, param- eter values exist that cause these populationsto behave like populations without imprinting, butwith heterozygote advantage, even though no such advantage is present in these imprinting populations. We call this last phenomenon“pseudoheterosis.”The imprinting systems that fail to be formally equivalent to nonimprinting systems are those in which males and females are not equivalent, i.e., two-sex viability systems and sex-chromosome inactivation. MANY observations have been made of the com- plementary roles of maternally and paternally derived alleles in the life and development of orga- nisms. As early as the 1920s, while working with the dipteran Sciara, METZ(1938) discovered that during development a chromosome from the paternal parent may function quite differently from its maternal hom- olog in contrasttothe usual Mendelianequivalent action. In 1971 SHARMANconcluded from his exper- iments with kangaroos that the modeof dosage com- pensation ofthe X-linked genes seemedto be paternal X inactivation, in contrast to the random X-inactiva- tion seen in eutherian mammals (SHARMAN1971). By the mid-1980s genomic imprinting becamethe subject of many moreexperimentsand scientific interest. Experiments conductedwith mice in which transgenes had been inserted revealed that theexpression of the transgenedepended on the sex of the parent from which it was inherited (HADCHOUELet al. 1987; REIK et al. 1987; SAPIENZAet al. 1987; SWAIN,STEWART and LEDER1987). Paternally derived alleles were ex- pressed in appropriate tissues, whereasmaternally derived alleles werenot.Nevertheless, males who inherited the transgene from their mothers (andwho thus did not express it), passed on thetransgene into offspring in which it was expressed. The pattern of inactivation of the transgene was thus readjusted at each generation. The inactivation of the maternally derived transgenesappeared to correspond toits level of methylation (HADCHOUELet al. 1987; REIKet al. 1987; SAPIENZAet al. 1987; SWAIN,STEWARTand 56, Dunedin, New Zealand. Genetics 130: 899-907 (April, 1992) ’ Current address:Departmentof Zoology,Universityof Otago, P.O. Box LEDER1987) (reviews in MONK 1987; MARX 1988; SOLTER1988; HALL1990). The maternally inherited transgene was inactivated by theattachment of an increased number of methyl groups.Paternallyde- rivedtransgenes, in contrast,werefoundtohavea lowlevel ofmethylation. The state of methylation thus depends on thesex of the parent fromwhich the gene came and most importantly this stateis reconsti- tuted in each generation, depending on thesex of the individual passing onthe allele. Morerecently,ge- nomic imprinting has been used to describe the dif- ferentialexpressionofgeneticmaterialwhereboth alleles are expressed, but at differenttimes, in differ- ent tissues, or at different levels, depending on their parental origin [see SOLTER(1 988) andHALL(1 990) for reviews]. Again there is evidence that the imprint- ing occurs by methylation atthe molecular level (MARX1988; SOLTER1988; HALL1990). Genomicimprintingthus conflicts with normal Mendeliangenetics in thatalthough all alleles are passed on to the next generation, their parentalorigin affects their expression. Thus in contrasttoother violations of the tenetsof Mendelian genetics such as meiotic drive, it is the expression not the inheritance that is altered. Belowwe investigate some of the consequences of thesedeviations in somestandard population genetic models. In particular,we examine the effects of the inactivation of an allele (or chro- mosome) onthe dynamics of allele frequencies in various standard models. Last, we study models of differential gene expression in which the phenotype of the individual depends on the quantity of expres- sion of alleles of maternal and paternal origin. This
  • 2. 900 G. P. Pearce andH. G. Spencer TABLE 1 Viabilityparametersused in the models Phenotype form of genomic imprinting is more complicated,but in its simplest forms is identical to theinactivation we model below. MODELS Model 0. Standard Mendelian inheritance:In or- der tofacilitate comparisons between our models and to introduce our terminology, we first review the standard one-locus two-allele viability selection model [see, e.g., CROWand KIMURA (1970) or HARTL (1980)].We label the two allelesA1 and A2 and suppose they are atfrequencies p and q, respectively (and so p +q = 1). Unless there areviability differences between males and females (asin model 2, below), p (and q) will be the same in both sexes after one generation, regardless of any initial differences.Let thethree genotypes AIAI,A1A2 and A2A2 have viabilities wf1, wf2 and w&, respectively. (Table 1 shows the vi- abilities of the various genotypes in the models we construct.) The frequency ofAl in the next generation is then given by p‘ = (p2wf1+p q w f 2 ) h (1) zlr = pzw;”,+ 2pqwf2 + 42w2*2. (2) where rir is the population mean fitness given by Such a dynamic affords upto threeequilibria where Ap =p‘ -p = 0. The equilibrium p = 1 always exists, islocally stable if wfl >w& and is globally stable if wf2 > wz2 as well. Similarly the p = 0 equilibrium alwaysexists,is locallystable if wZ2>wfzand is globally stable if W X >wflas well. An internal equilibrium also exists when either wT2 > wfl and w& (in which case it is globally stable) or wfz< wfl and ~ $ 2(when it is unstable). By globally stable we mean the system iterates to the equilibrium for all initial p E (0,l)and by locally stable,for all p sufficiently close tothe equilibrium value. When wT2 > wfl and w&, wesay there is heterozygote advantage or heterosis and the system maintains both alleles in the population. Model 1. Completeautosomalinactivation: We now introduce autosomalinactivation into model 0 by supposing that the maternally derived alleles are not expressed in an individual at all. The fitness of an individual receiving an A1 allele from its father (an AI- individual) is w1 and the fitness of an individual receiving an A2 allele from its father (an AS- individ- ual) is w2. With random mating,the (preselection) zygotes in thenextgeneration have the following phenotypic frequencies: [AI-]= p2 +pq = p [AZ-] = q2 +pq = q. We may assume that p is the same for both males and females because an individual’s sex does not affect its own viability but that of its offspring (of both sexes). Following selection, the genotypic frequencies are (4) [AIAI]= P2w1/G [AIAZ]= (pqwl + qPw2)b (5) [ A d z ]= q2w2/5, where ZZI = pwl + qw2, Thus p2w1 + $$Q(Wl + w2) p’ = p2w1 +Pq(w1 + w2) + q2w2’ (6) which is the same formula as foranonimprinting system where the fitnesses of AIAI, A1A2and A2A2are respectively wfl = w l ,wf2 = (wl +w2)/2and w& = w2. (We use w*’s to denote viabilitiesof nonimprinting systems, throughout.) By applying the well-known re- sults of model 0, we find thatthe only solutions to the equilibrium equation p’ = p are trivially p = 0 and p = 1. The internal equilibrium,$, does notexist in this case. The stability of the p = 0 equilibrium may also be derived from model 0. The conditions for global stability, w& > wf2 > wfl give w2 > wl. Similarly, the p = 1equilibrium is globally stable if and only if w1 > w2. The behavior of this model can also bededuced fromthe observation that, since viabilities depend solely on thepaternal gamete, selection can be viewed as acting on thegametes, and so the model is formally equivalent to a haploid one of gametic selection. As is well known, deterministicconstant viability haploid models cannot maintain polymorphism without mu- tation or structured populations, and so the system will iterate to fix the fitter of the two gametes: A1 if w1>W P ,A2 if w2 >w1. One way this model can be generalized is to remove the restriction that only the maternal alleles are inac- tivated. We therefore introduce a parameter, a,the probability that the paternal allele is inactivated, re- quiring,therefore,thatthematernal allele is im-
  • 3. Models of Genomic Imprinting 901 printed with probability 1-a.Thus a = 0 in model 1. As before, we can assume that theallele frequencies are equal in both sexes. Eight different zygotic phe- notypes are possible and their frequencies are [A,-] = p2(1 - a) where the unexpressed allele is a maternal A1 [-AI] = p2a where the unexpressed allele is a paternal A1 [A,-] = pq(1 - 4 where the unexpressed allele is a maternal A2 [-A21 = Pqa where the unexpressed allele is a paternalA1 [&-I = qp(1 - 4 where the unexpressed allele is a maternalA I [-AI] = qpa where the unexpressed allele is a paternalA2 [AZ-] = q2(1 - a) where the unexpressed allele is a maternalA2 and [-A21 = q2a where the unexpressed allele is a paternal A2. After viability selection we obtain p’ =lp2(( 1 -a)w1+ awl)++q(( 1-a)w1+ aw2) +iqp((1- +awl )I/$ which is the same equation as before. In otherwords, provided one allele is always imprinted, the dynamics of the system are unaffected by which sex’salleles are imprinted. Model 2. Partialautosomalinactivation: Let us now consider the case where inactivation occurs in only some individuals. That is, we introduce viability selection into CHAKRABORTY’S(1989) model. CHAK- RABORTY assumed aHardy-Weinbergpopulation (with no selection), but with a constant parameter, 8, equal to the probability that the maternally derived allele is unexpressed (0 d I9 d 1). The same I9 value applied to both Al and A2 alleles, and the paternally derived alleles were assumed to always be expressed (although these assumptions were shown to be easily modified). Model 1 thus has an implicit I9 value of 1; model 0 one of 0. There areseveral possible interpre- tations of 19.For example, 0 may be envisaged as the proportion of females in the population who pass on imprinted alleles (e.g., if inactivation were tempera- ture sensitive), the rest having standardinheritance patterns; it can beregarded as theproportionof imprinted eggs that (all) females pass on (e.g., if the phenomenon were dependent on the age of the fe- male); or it may be some combination of these two possibilities. Nevertheless, we are notyet aware of any reports which have demonstrated such intermediate values of I9 in living organisms. For a particular allele we consider 0 to he fixed (butsee model 4,below). If both alleles are expressed then in the next gen- eration,the zygotes have the following phenotypic frequencies (CHAKRABORTY1989): [ A I A I ]= p2(1- 0) [AI&] = 2pq(l - 6) (7) [A2A2] = q2(1- 0). If the maternally inherited allele is unexpressed then in the next generation, thezygotes have the following phenotypic frequencies (CHAKRABORTY1989): [AI-]= p219 where the unexpressed allele is A1 [AI-] = pqI9 where the unexpressed allele is A2 [Az-] = pq0 where the unexpressed allele is A I [Az-] = 4% where the unexpressed allele is AS. (8) If we let w1 be the viability of an A1 A1or A I- individual, w12 that of an AlA2individual and w22 that of an A2A2 or A2- individual, we obtain the postselec- tion frequencies [ A I A I ]= p2wll/G [ A d z ] = [ ~ P ~ w z+ +q(wll + wzz - 2~1z)]/W (9) [AzAz] = q2wz2/W, where the standardized mean fitness is w = p2w11 + 2pqw12 + q2w22 + Opq(wl1 + w22 - 2w12). Therefore +q2w22 This shows the same formula as for a non-imprinting systeminwhich wfl = w11, w& = wZ2 and wT2 = If the fitness of a heterozygotein which both alleles are expressed isless than-the average homozygote fitness (in the imprinting system), then in the equiva- lentnonimprinting system the heterozygote fitness will be greater by the amountof iB(w11 + wZ2- 2 ~ 1 2 ) . There is a limit to this increase, however: if w12 is less than w11 or w22, then wf2 can not be simultaneously larger than W E and w&. So if heterozygote advantage is not presentin the imprintingsystem then it will not be exhibited in the equivalent nonimprinting system. w12 + + w 1 1 + w22 - 2w12).
  • 4. 902 G. P.Pearceand H. G. Spencer T o see this, suppose that w l l > w12and, without loss of generality, that w11 3 w22. Then wf2 = w12(1- e) + + q W l l + w22) s w12(1 - e) + ow11 < w l l ( l - e) + ewll = w11 (12) = W f l , a.e., w;E2 < WT1. If heterozygoteadvantage exists in theimprinting system, however, it can beabsent in the equivalent nonimprinting system. That is,if w12 > w11 and w22 then wI2 > (w11 + w22)/2 and w;"2 < w12, and with certain parameter values we have w;I; < wTl or w& so thatthe equivalent nonimprinting system exhibits no heterozygoteadvantage. Consequently no stable polymorphic equilibrium will be present in either system. An example is: 0 = 0.9, w11 = 0.95, w12 = 1.0 and w22 = 0.85, giving wrl = 0.95, w& = 0.85, and w;"2 = 0.91 < wTl. The only solutions to p' = p are (from model 0), p = 0, p = 1 and (if it exists) In order that 0 < < 1 we require 0 # 1 and either (a) w h > w;FI and w&, which gives w12 > (2 - q w l l - ew22 2(1 - 8) and (2 - qw22 - ow11 w12> 2(1 - e) ' or (b)w h <w h and wfr, which reverses theselast two inequalities. In the first case is stable, in the second unstable. T o see the effect of the level of imprinting on the internal equilibrium, we examine afi/dO, which (when fi is stable) has the same sign as w11 - w22. Thus increasing the level of imprinting increases the equi- librium frequency of the allele of the fitter homozy- gote. Model 3: Generalized autosomalinactivation: We now generalize model 2 so that the fitnesses of indi- viduals with an unexpressed allele are not necessarily equal tothe fitnesses of the homozygotes forthe expressed allele. As before, let w l l , w12and w22 be the viabilities of AlAl, A1A2 and A2A2 individuals respec- tively, but suppose imprinted individuals AI- and AS- have respective viabilities wl0 and wz0.Proceeding as before, we obtain p2[(1- e)wll + flyl0] p' = + p q w - e)Wl2 + ? m l 0 + wz0)1 p - 8)Wll + ewlo] + 2Pq[(l - qw12 + vqw10 + w20)] + q2[(1 - qw22 + OWZO] (15) This equation for p' shows equivalence to a nonim- printing system where W E = (1 - 0)Wll + ow10 w;I; = (1 - O)w,, + +d(WlO + w20) (16) w'& = (1 - d)W22 + ew20. It is easily seen that when (wl0 +wz0)/2 > w12,wf2 is larger than w12. Thus, it is possible for an imprinting system to exhibit the dynamics of anonimprinting system with heterozygote advantage, even though no heterozygote advantage actually exists. T o construct such an example, let us assume, without loss of gen- erality, that w11 > w12 > w22. We require that w& > w11, a.e., * ' (1 - o ) ~ ~ ~ + + e ( w ~ ~ + ~ ~ ~ ) > ( 1- e ) ~ ~ ~ + e ~ ~ ~(17) which gives (1 - fl) w20 > 2 -( ~ 1 1- ~ 1 2 )+ w10 (18) 8 and also that wT2 >w&, i.e., (1 - o ) W 1 2 + $ e ( ~ l o + ~ 2 0 ) > ( 1- O ) W ~ ~ + B W ~ ~(19) which gives (1 - 6)w20 < 2 ~ (w12 - w22) + w10. (20) e Thus, 2-(1 - 6) e (w11 - w12) + w10 <w20 < 2 7(1 - 6) (w12 - w22) + w10 (21) and so T(W11 + w22) < w12. (22) 1 If w22 > w12 > w l l then similar expressions can be found which must be met for heterozygote advantage to be mimicked. Thus if (21) holds and w11 > w12 > wZ2 thenapparent heterozygoteadvantage will be shown in the dynamically equivalent nonimprinting system. Wecall this property pseudoheterosis. As a numerical example consider the following: let w11 = 0.9, wZ2= 0.1 and 0 = 0.3, and therefore, by (22),we require w12 > 0.5, say w12 = 0.6. Let w10 = 0.01
  • 5. Models of Genomic Imprinting 903 therefore, by (21), we require 1.41 <w20< 2.34 and so let wz0 = 1.5. This gives w;”l = 0.6330, w?z = 0.64665, and w& = 0.5200. Thus we have w11>W I Z > wZ2,whereas w;”2> wfl > w& so that there is het- erozygote advantage in the nonimprintingsystem but clearly none exists in theimprinting system. (Of course,the viabilities in thisexample may not be particularly realistic: w11 >w22, but w10 <W Z O ,but see model 5 , below.) The stability analysisof the polymorphic equilib- rium in this model is constructed in a similar manner to that of model 2. The polymorphic equilibrium is feasible and stable if and only if wg2, w?1 < wf2 which gives (21)and hence (22). The effect of 8on # is similar to the effect in model 2:d#/d8 has the same sign asw10-W Z O .That is, greater penetrance of imprinting increases the equilibrium frequency of the allele of the fitter hemizygote. Model 4. Imprinting us. Mendelizingalleles: A natural question to ask about genomic imprinting is how thephenomenonoriginated. Our nextmodel, therefore, looks at how imprinting affects an allele’s ability to entera non-imprinting population. Suppose we have one allele AI that is neverimprintedand another A2 that is imprintable, with probability 8. As in model 2, suppose that Ai-’s have the same viabilities as A,Ai’s, wii(i = 1 and 2). The interative equation for p is thus p’ = [ P 2 W l l + & q W l l + $ p q m + +(I - ~)pqw12]/zs) (23) = [p2wll+ pq(Wl2+ $qwll - w12))l/zlt zs, = p2wll + 2pq[w12+ $8(Wl1- w12)1+ q2Wz2. (24) where This behaves as model 0 where wrl = wll,w& = wZ2and w h = w12+ 28(w11-w12).Like model 2, this system cannot display pseudoheterosis,for if wll > w12, wTl - w& = (1 - $8)(wll - w12)> 0 and so wfl > wT2.Alternatively, if w l l < w12< wZ2,then w& The internalequilibrium, a, exists and is stable provided w& >wTl and w&, i.e., w12>w l l and (2wZ2 - 8w11)/(2 - 8). Putting 8 = 0 recovers model 0 (as expected).Examining d$/d8 reveals thatjust as in model 2 the level of imprinting increases the equilib- rium frequency of the allele of the fitter homozygote. The p = 1 equilibriumwhere A1 is fixed will be stable if w?] > w t , ie., if w11 >w12,the same condition as without imprinting (model0).In a finite population, however, deviations fromdeterministicbehavior mean that the A2 allele may still not invade evenif w12 > ~ 1 1 ,the probability of success being an increasing function of w t - w ? ~= (1 - T B ) ( W ~ ~- wll).Thus in- creasing the probability of imprinting decreases the 1 <w12 <w22 = w& 1 chances that an imprintable allele will invade a finite population of nonimprintablealleles. The invadability of apopulation of imprintable alleles by annonimprintable allele (AI) dependson the stability of the p = 0 equilibrium. Such an invasion will be successful if and only if w& < w h , i.e., if w22 C wI2+i8(wll-wI2).Thus for fixedviabilities,a larger value of 8 increases the chance ofsuccess only if imprinting of the A2 allele in an A1A2 heterozygote increases the viability. Model 5. Generalized imprintingus. Mendelizing alleles: Model 4 can be generalized in the same way model 3 generalizes model 2, introducing parameters w10 and wz0.We obtain a system for which and so and w2*2= w22 + O(W20 - w22). The value of # and the conditions for its existence and stability may be calculated as previously. We can see from equations (26) that pseudoheterosis is possi- ble, e.g., with 8 = 0.8, wll = w10 = w20 = 1, w12 = 1.1, wZ2= 1.2. (This example certainly appears more re- alistic than that in model 2.) The condition for the imprintingalleleA2 to invade becomes 2(Wl2 - w11) > e(w12 - WIO) (27) which means that a greater level of imprinting favors invasion when A1- individuals are fitter than AIA2’s. Since W & - W ~ I= ~ 1 2- W ~ I- ~B(w12- w ~ o ) , (28) if wlo is close to wll, we preserve our model 4 result that larger8’sreduce theprobability of A2 successfully invading a finite population (for given w’s). If wlo is ratherlargerthan w I 1 ,however, then this result is reversed. (The deterministic model may no longer admit such an invasion, however, if wl0is very large.) Model 6. Autosomal inactivation with two-sex vi- abilities: The next system we consider differs from model 1 in only one feature: viability selection affects the sexes differently. The following set ofviability fitnesses are assumed: the fitness of male Al”s is wlm, that of female A,”s wlf, that of male A2”s w2, and that of female Ay’s w2p The first model is thus the special case in which wlm= wlf(=w l ) and wZm= W2f (= w2). Following our previous procedure, we obtain, in 1
  • 6. 904 G. P. Pearceand H. G. Spencer the female population after selection, the genotypic frequencies: [AIAI]/= prnpfwlf/Wf, [ A I A ~ ] ~= (PrnqfWlf + qrnPfw2f)/Wf (29) and [A2A2]f = qrnqfw2~/5~, where W/= PmW, + q r n ~ 2 ~ . (30) Similarly, in the male population following selection the genotypic frequencies are: [AIAI], = p m p f w l m / ~ m and (31) [AIAz]~= (pmqfwlm+ qrnP~w~rn)/~rn [A2A2]m = qmqfWzm/Wm where Wrn = p m w l m + qmw2m. (32) Now, the frequencies of the A I allele in the female and male populations of this generation (denoted by pj and p: respectively) are: pj = [AIAI]~+ $[AIAP~ (33) = [PmPfWlf + i(PmqfW1f + ~ , P ~ W V ) I / W ~ and PA = FrnPfWlrn + ivrnqfwlrn + qrnpf~2rn)l/Wrn- (34) Unfortunately this system is not formally equivalent to the well-known two-sex viability scheme of OWEN (1953), nor to a fertility selection scheme (BODMER 1965). The difference is a consequence of the differ- ent viabilitiesof the reciprocal heterozygotes. We therefore analyze the system in more detail. Let us first suppose that neither wl, nor w1f is zero, so that we may divide (33) by wlfand (34) by wl, and write wf = w2f/wlf and w, = w2,/wlm. We can see from (33) and (34) that if W, = wf then pj = PA, otherwise pi # PA. Thus theprevious result that allele frequen- cies are equal inmales and females doesnothold when there are viability differences between the two sexes. Now and Apm = PA - p m (36) -- pm(2qm qJ) + qm(pf - 2pm)wm 2(pm + qrnwm) At equilibrium Apf = Ap, = 0. Thus, the numerators of (35)and (36) will equal zero. Adding the numera- tors gives: 2Pmqm(l - wm) + pfqm(wm - wf) = 0. (37) This equality holds if either:-(a) q, = 0 (ie., p, = 1) which on substituting into Ap, = 0 gives qf = 0 (ie., PJ = 1) and so the population is fixed for theA1 allele in both sexes; or (b)qm# 0 which gives If p, = 0 then pf must also be equal to zeroand so the A2 allele is fixed in both sex systems. Ifp, # 0 then: Substituting (38)and (39) into Apf = 0 gives: As amis an allele frequency it must lie between zero and one, and so by enforcing this range on (40) we see that for 0 <$, either w, + wf> wfw, + 1 and w, + wf > 2wfw, (41) or w, + wf < wfw, + 1 and w, + wf < 2wf w, (42) and for$, < 1 either w, +wJ> 2 when (41) holds or w, + wf < 2 when (42) holds. So for 0 < $, < 1 we have that either w, + wf> max (2, wfw, + 1, 2wfw,) (43) or w, + wf< min (2, wfw, + 1, 2wfw,). (44) Let us now consider the cases when wl, or wlf is zero. If they are bothzero thenclearlypfand pmhalve every generation and thesole equilibrium is pf= p, = 0. Now, if wlm= 0, but w1f# 0 then (34) reduces to 1 PA = Fpm* (45) Now the only equilibrium value for (45) is pm= 0, which on substitution into (33) gives pf = ipf and so the equilibrium value is pf = 0. Similar arguments show that when wlf = 0, the only equilibrium is p, = Figure 1 illustrates the changes in the AI allele frequency over time within a system constructed on the assumptions of model 6. In this system we have that 2w,wf = 2.0 <w, + wf = 2.5 and 2 < w, + wf= 2.5 (refertoEquation 43). Thus the necessary ine- qualities for apolymorphism hold and we see that the internal equilibrium is reached in both sexes. For the male population: pf = 0. $m = Wf + w, - 2WfW, 2(wm + wf - wfwm - 1) = 0.500(40)
  • 7. Models of GenomicImprinting 905 and pk = pfwl/zm (49) where 6,= pfw1 + qfw2 = 6f. (50) O-.iI 0.0I0 20 40 60 eo 100 No. Gleneratiom FIGURE1.-The frequency of the AI allele over successivegen- erations in model 6. w,,, = 0.5 and w,= 2.0. Initiallyp. = 0.01 and p, = 0.99. Males = solid line. Females = dotted line. and for the female population: Note that,if the sex ratio is even, the total population has more A2 alleles than Ai's, even thoughwmwf= 1.O, because the maternally derived alleles (more likely to be A2’s) are hidden from selection in the following generation’s males. Model 7. X chromosomeimprinting: So far we have considered only imprinting involving the inacti- vation of autosomal alleles. Consider now the case in which the paternal X chromosome(or parts of it) are inactivated in female offspring. That is, only the ma- ternal chromosome is expressed throughout an indi- vidual’s soma [see MONK( 1987) for a review of this phenomenon]. This situation differs fromthe classical X-inactivation in eutherian mammals which occurs after the zygote has undergone several cell divisions. Within each cell of the zygote, inactivation is random with respect to theparental originof the chromosome. In the classical case, therefore, cell lineages express only one X chromosome but the organism generally has bothX chromosomes expressed. Continuing our formulation, assuming that males are the heterogametic sex, let A1 males and -A1 fe- males have viability wl,and A2 males and -A2 females have viability w2. If the frequency of AI in the male population is p, and that in the female population is pf we obtain pi = [ P m p f W l + i ( P m q f W 2 + q m p f w 1 ) ] / ~ f (47) where $f = p m p f w1 + p m q f UP + q m p f w1 + q m q f WP (48) = pfwl + QfW2 Again our system is different from standard Men- delian models; in particular, its dynamics are different from thoseof a sex-linked locus with viabilityselection (MANDEL1959). Clearly pf = p, = 0 and pf = pm= 1 are equilibria. T o discover whether any internal equilibria, ( f i f , fi,), are possible, we can follow MANDEL’S(1959) treat- ment, defining Rf = pr/qf and R, = p m / q m - From (49) and (50) we immediately get that at equilibrium, R m = R ~ w ~ / w z (51) which yields (excluding irrelevant cases, e.g., Rf = 0 and w1= wp) Rf = - W ~ / W I < 0. (52) Obviously we have a nonsensical value for Rf and so we can conclude there are nopolymorphic equilibria possible. We can also explore the consequences of random X-inactivation by introducing P, the probability that the paternal X chromosome is imprinted. Thus the maternal X chromosomeis imprinted with probability 1 - B and we have so far considered /3 = 1. Following our procedure in model 1 we obtain P ; = ~ m p f w l ( P w l + ( 1 - P ) w l ) +&mqf (PW +(1-P)W 1) (53) +&mPAPwl+ (1-P ) w z ) ] / ~ ~ prnpfW1+ +[pme(PW +( 1 -P ) w ~ > -- +q m p f (awl +(1-P)w2)3 PrnpfWl + P m e ( P W : , +( 1 -P)wl) +q m p f (Pwl+ (1-P)w) +q m e ~ 2 and p: = Pfw1 P f W l + qfw2 (49) In contrast to random autosomal inactivation, ran- dom sex-chromosome activation does affect the allelic dynamics, but only those in the females. Using MAN- DEL’S technique again,we can showthat for an internal equilibrium we obtain the contradiction Rf <0and so no polymorphic equilibria exist. Thus, although the dynamicsare altered by random inactivation, the equi- libria are not. Model 8. Differential gene expression: Most re- cently, genomic imprinting has been used to describe the differential expression of alleles(orchromosomes) depending on whether theallele (orchromosome) was paternally or maternally inherited (HALL1990). In its
  • 8. 906 G. P. Pearce and H. G. Spencer most general form such differential gene expression means that reciprocal heterozygotes havedistinguish- able phenotypes and hence possibly different viabili- ties. We can modify the Mendelian model 0 to this situation by simply requiringthat AlA2 individuals have viability w12 whereas A2A1 individuals have via- bility w ~ l(which is not necessarily equal to ~ 1 2 ) .The recurrence equation for the frequencyof the A1 allele is thus which is the same formula as a nonimprinting system in which w?, = w11 (55) and w& = w22. That is, differentialgene expression gives identical dynamics to a standardsystem inwhich heterozygotes have viability equal to thearithmetic mean of the reciprocal imprinted heterozygote viabilities. The de- struction of the symmetry of the viability matrix W (with entry wq in row i column j ) thus does not lead to novel behavior of the system. Indeed, this result is known in another context, thatof the parallel between the game theoreticapproach to ESS (evolutionarily stable strategy) theory and standard one-locus Men- delian genetics [see CANNINCSand VICKERS(1989) for a recent example]. The viability matrices of the latter are payoff matrices of the former.(Allpayoff matrices are not viability matrices-even asymmetric viability matrices-however.) Returning to Equations 55 we see that in addition to the p = 0 and p = 1 equilibria,the internal equilib- rium w12 + WPl - 2w22 = 2(w12 + WPl - w11 - w22) will exist provided i(w12+ wpl) > w l l and w22. Thus for the dynamics of Mendelian heterosis to be mim- icked at least one of the heterozygotes must have the highest viability. Nevertheless, a kind of pseudohet- erosis is possible where one heterozygote has low viability,e.g.,w l l = w22 = 1.0, w12 = 0.95,w2I = 1.1. The above formulationallows the easy examination of special cases of interest. For example, suppose the paternal and maternal allelescontribute to theoverall phenotype in the ratio 4:(1 - 4). Under allelic selec- tion, letting the Ai allele contribute wi to the viabil- ity, the viability of an AiAj individual is wq = 4wi + (1 - 4)wj (i,j 1, 2). Thus w;"]= w1, w T ~= ~ ( w I + we) and w& = wp which, like model 1, is formally haploid selection, affording no polymorphic equilibria. TABLE 2 Equivalencesbetween nonimprinting and imprinting models Model 0 parameter Model w ;, :2 w:2 1 WI f(Wl +WP) w 2 2 WII (1 - e ) ~ ,++e(wII+wZ2) wqq 3 (1 - e ) ~ ,+8wla(i - 0)wI2+&wl0 +wsa) (1 - o)wz2+&uzo 4 WII (1 -$e)wIz++&ull W22 5 WII (1 - $B)Wl2 ++&ulO (1 - qwss +OW2,l 8 W I I f(Wl* + WZI) w 2 2 DISCUSSION The most important of our conclusions is that the majority of our models incorporatingimprinting could be shown to be formally equivalent to models in which there was no imprinting. That is, the allele frequencies in a particular imprintingsystem changed in exactly the same manner as those in certain corre- spondingnonimprinting systems. (The equivalences are summarized in Table 2.)Consequently, the equi- libria in the imprinting system also corresponded to those found in these certain nonimprinting systems. Forexample,a systeminwhichall the maternally derived alleles were inactivated was shown to behave as a haploid selection model. Completeinactivation of an autosomal allele thus leads to monomorphism. This equivalence between imprinting and nonimprinting systems has the important consequence that imprint- ing can never be detected by simply following allele frequency changes.(Of course, imprintingof this sort can easily be detected by setting upappropriate crosses and examiningoffspringgenotypepropor- tions.) This resultmirrors several other discoveries that show how little can be deduced about a genetic population from analyzing the changes in allele fre- quencies. These discoveries include the equivalence of viability selection models and some fertility selec- tion models (FELDMAN, LIBERMANand CHRISTIANSEN 1983), the equivalence of constant viability selection models and some frequency dependent selection models (DENNISTONand CROW 1990), and theequiv- alence of BODMER'Smultiplicative fertility selection system and OWEN'S (1953)two-sex viability selection system (BODMER1965). The formal equivalences do not,unfortunately, pre- serve some of the properties of the viabilities in the respective models. For example, an imprintingsystem with heterozygote advantagecould behave like a non- imprinting system without such advantage. Conse- quently, under genomic imprinting, heterozygote ad- vantage is not a sufficient condition for the mainte- nance of a diallelic polymorphism. Conversely, heterozygote advantage is not a necessary condition: an imprintingsystem without it can be formally equiv-
  • 9. Models of Genomic Imprinting 907 alent to a non-imprinting system with it, a property we callpseudoheterosis. When the internal equilibrium, $, exists (as it will when the equivalentnonimprintingmodelexhibits heterosis) its value dependsonthepenetrance of imprinting, 8. The larger the value of 8 the closer fi moves to fixation of the fitter type (AJi and/or Ai-). Model 4 examines diallelic systemsin which only one of the alleles is imprinted. Again this model is formally equivalent to anonimprintingmodel. The effect of imprinting on anallele attempting to invade a finitenonimprintingpopulation is toreducethe invasion’s chance of success. Imprinting reduces the advantage the heterozygote might have over the un- imprinted homozygote since many of the heterozy- gotes are selectively undifferentiated from the homo- zygotes. In model 5 (in which imprinted heterozygotes and homozygotes are not necessarily identical) a sim- ilar result holds if AJ, and Ai- individuals are pheno- typically similar. In this lattermodel, however, the conditions for asuccessful invasion in the determinis- tic case depend on 8, unlike those in model 4. The exceptions to the ruleof equivalence between imprinting and non-imprintingsystemsarise when the sexes are different,either in their viabilities, or in their levelof ploidy. In nonimprinting models, e.g., OWEN’S(1953) two-sex viability system, or BODMER’S (1965) fertility selection scheme, reciprocal heterozy- gotes had the same fitnesses. Imprinting contravenes this principle and even the sex-symmetry property of many fertility selection schemes (FELDMAN,LIBERMAN and CHRISTIANSEN1983) does not apply.Our two-sex viability model has apolymorphicequilibriumpro- vided certainrestrictions onthe viabilities apply, whereas with X-inactivationno polymorphic equilibria exist, even when the inactivation is random with re- spect to parental origin.This nonequivalence between imprinting and nonimprintingsystems reveals the im- portance of the synonymity of reciprocal heterozy- gotes to standard Mendelian models. When viabilities from these latter models are represented in a matrix, this synonymity is manifested in the symmetry of the matrix. Such a matrixalso corresponds to a symmetric payoff matrix of ESS (evolutionarily stable strategy) theory [see HINES(1987) for a review]. ESS theory, however, does not require that the payoff matrix be symmetric and, in general, it is not. Thus, ourimprint- ing models can also be expectedto be moregenerally equivalent to various models in game theory. Of course, the models we have constructed are idealized inmany respects, e.g., in having constant viabilities and in ignoringgeneticdrift.Moreover, some (but by no means all) of the peculiar behaviors requireparameter values that might beconsidered unlikely. Nevertheless, the models do serve to illus- trate thatgenomic imprinting systems need not act at all like otherwise identical non-imprinting ones. We thank LISABROOKS,RAYLITTLERand an anonymous re- viewer for helpful comments on earlier versionsof the manuscript, andJANET SMITHfor assistance with typing. LITERATURECITED BODMER,W. F., 1965 Differential fertility in population genetics models. Genetics 51: 41 1-424. CANNINGS,C., and G. T. VICKERS,1989 Patterns and invasions of evolutionarily stable strategies. Appl. Math. Camp. 3 2 227- 253. CHAKRABORTY,R., 1989 Can molecular imprinting explain het- erozygote deficiencyand hybrid vigor?Genetics 122 713-7 17. CROW,J. F., and M. KIMURA,1970 An IntroductiontoPopulation Genetics Theory. Harper & Row, New York. DENNISTON,D., and J. F. CROW,1990 Alternative fitness models with the same allele frequency dynamics. Genetics 125 201- 205. FELDMAN,M. W., U. L I B E R M A N ~ ~ ~F. B. CHRISTIANSEN,1983 On some models of fertility selection. Genetics 105: 1003-1010. HADCHOUEL,M., H. FARZA,D. SIMON,P. TIoLLAIsand C.POURCEL, 1987 Maternal inhibition of hepatitis B surface antigen gene expression in transgenic mice correlates with de novo methyl- ation. Nature 329 454-456. HALL,J. G., 1990 Genomic imprinting: review and relevance to human diseases. Am.J. Hum. Genet. 46: 857-873. HARTL,D. L., 1980 Principles of Population Genetics. Sinauer, Sunderland, Mass. HINES,W. G. S., 1987 Evolutionarilystable strategies-a review of basic theory. Theor. Popul. Biol. 31: 195-272. MANDEL,S. P. H., 1959 Stable equilibrium of a sex-linked locus. Nature 183: 1347-1 348. MARX,J. L., 1988 A parent’s sex may affect gene expression. Science 239: 352-353. METZ,J. L., 1938 Chromosome behavior, inheritance and sex determination in Sciara. Am. Nat. 72: 485-520. MONK,M., 1987 Memoriesof the mother and father. Nature 328: OWEN,A. R. G., 1953 A geneticalsystemadmitting of twodistinct stable equilibria under natural selection. Heredity 7: 97-102. REIK,W., A. COLLICK,M. L. NORRIS,S. C.BARTONand M. A. SURANI,1987 Genomic imprinting determines methylation of parental allelesin transgenic mice. Nature 328 248-251. SAPIENZA,C., A. C. PETERSON,J. ROSSANT and R. BALLING, 1987 Degree of methylation of transgenes is dependenton gamete of origin. Nature 328: 251-254. SHARMAN,G. B., 1971 Late DNA replication in the paternally derived X chromosome of female kangaroos. Nature 230: SOLTER,D., 1988 Differential imprinting and expression of ma- ternal and paternal genomes. Annu. Rev. Genet. 22: 127-1415, SWAIN,J. L., T. A. STEWARTand P. LEDER,1987 Parental legacy determines methylation and expression of an autosomal trans- gene: a molecular mechanism for parental imprinting. Cell 50: 719-727. 203-204. 231-233. Communicating editor: A. G. CLARK

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