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Genetics, Vol. 170, 1345-1357, July 2005, Copyright © 2005
doi:10.1534/genetics.104.036889
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Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, Massachusetts 02138
1 Corresponding author: St. John's College, Oxford OX1 3JP, United Kingdom.
E-mail: francisco.ubeda{at}sjc.ox.ac.uk
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ABSTRACT |
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 TOP ABSTRACT MODELSTABILITY ANALYSISDISCUSSIONAPPENDIX AAPPENDIX BACKNOWLEDGEMENTSLITERATURE CITED |
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Models addressing the evolution of fair segregation have considered a primary locus (with alleles A1 and A2) undergoing viability selection and a modifier locus that determines the segregation ratio at the primary locus. If the two loci are linked, modifiers that change the segregation ratio at the primary locus are able to invade a population undergoing Mendelian segregation (PROUT et al. 1973; HARTL 1975; LIBERMAN 1976). The intuitive reason for this result is that a modifier that confers a segregation advantage on allele A1 will be favored by natural selection because it comes to be preferentially associated with A1 and thus shares in that allele's segregation advantage. By contrast, a modifier that confers a segregation disadvantage on A1 (i.e., segregation advantage on A2) will become preferentially associated with A2. The introduction of either kind of modifier by itself would destabilize Mendelian segregation.
ESHEL (1985) proposed an elegant solution to this conundrum. He showed that if the modifier locus is unlinked to the primary locus, then natural selection disfavors modifier alleles that take the segregation ratio away from Mendelian expectations but favors alleles that bring the segregation ratio closer to Mendelian expectations. Therefore, Mendelian segregation has the property of evolutionary genetic stability (ESHEL 1996) with respect to unlinked modifiers. Furthermore, an increase in recombination between the main and modifier locus would be favored by natural selection until they become unlinked and segregation distortion is eliminated (HAIG and GRAFEN 1991). Taken together, these results seem to explain the ubiquity of fair segregation in diploid organisms with multiple chromosomes by invoking mutual policing between genes over deviations from fair segregation. Fair segregation is maintained because most loci in the genome, and hence the majority of potential modifiers of the segregation ratio, are unlinked to any particular locus. The intuitive explanation for ESHEL's (1985) result is that an unlinked modifier conferring a segregation advantage on A1 is not preferentially associated with this allele, thus sharing in A1 segregation advantage as much as in A2 segregation disadvantage. Alleles at an unlinked modifier locus can gain no direct advantage from segregation distortion at the primary locus. Therefore, such alleles should favor whatever segregation ratio maximizes population mean fitness, which in Eshel's model is Mendelian segregation.
Brief reflection, however, reveals that Mendelian segregation does not maximize mean fitness at a locus subject to heterozygote advantage because this segregation scheme always produces some offspring with the less-fit homozygous genotypes. Rather, mean fitness is maximized by what we call an all-and-none segregation scheme in which one of the alleles is transmitted to all sperm (or microspores) and no eggs (or megaspores) or vice versa (ÚBEDA and HAIG 2004). Under such a segregation scheme, all adults will be maximally fit heterozygotes. This possibility was considered neither by ESHEL (1985) nor by earlier models of the evolution of the segregation ratio because these models made the simplifying assumption that segregation was the same in males and females. Assuming that a modifier of segregation has equal effects in spermatogenesis (or microsporogenesis) and oogenesis (or macrosporogenesis) is far from being realistic, however. A detailed review of genetic systems in which segregation distortion has been reported fails to provide a single case with identical segregation in males and females (see Figure 1 and references therein). This comes as no surprise since mechanisms underlying male and female gametogenesis are extremely different (PARDO-MANUEL DE VILLENA and SAPIENZA 2001). Thus, it is difficult to posit a modifier of segregation having identical effects in the two processes.
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First we introduce a two-locus model for the interaction between viability selection and segregation distortion. Then, we carry out stability analysis of the parameter space for sex-specific segregation with a focus on Mendelian and all-and-none segregation. Finally, we analyze the particular case of permanent translocation heterozygotes and discuss some possible explanations for the scarcity of all-and-none segregation and the ubiquity of Mendelian segregation.
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MODEL |
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TOPABSTRACTMODELSTABILITY ANALYSISDISCUSSIONAPPENDIX AAPPENDIX BACKNOWLEDGEMENTSLITERATURE CITED |
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Alleles A1 and A2 determine the viability of their carrier. Let the viability parameters corresponding to genotypes A1A1, A1A2, A2A1, A2A2 be v11, v12, v21, v22, where paternally inherited alleles are listed first. Viability parameters are arranged in a four-by-four matrix, V, with elements Vij that are matrices themselves,
 | (1) |
Alleles M1 and M2 determine the segregation ratio of alleles at the A locus. Let the segregation ratio of A1 corresponding to genotypes M1M1, M1M2, M2M1, M2M2 be k11, k12, k21, k22 in males and 
11, 12, 21, 22 in females, with kij = kji, ij = ji, and 0 
kij, ij 1. The segregation ratio of A2 corresponding to genotype MiMj is 1 – kij in males and 1 – ij in females. Segregation ratios in males are arranged in a matrix, Sm, with elements Smij that are matrices themselves,
 | (2) |
 | (3) |
In a single generation, there are four possible transmission paths for one haplotype: from male to male, from male to female, from female to male, and from female to female. Each path relates to a fitness matrix that results from multiplying the viability of the transmitting individual and the segregation ratio of that particular haplotype: Wmm = V 
Sm, Wmf = V Sf, Wfm = VT Sm, Wff = VT Sf. The symbol represents the Schur product of two matrices, which is another matrix with elements V S = Vij Sij = vmnsmn. Following PROUT et al. (1973), LIBERMAN (1976), and ESHEL (1985) we assume no pleiotropic effect of the modifier locus over the fitness locus. Let the frequency of haplotypes A1M1, A2M1, A1M2, A2M2 be x1, x2, x3, x4 in sperm and y1, y2, y3, y4 in eggs, with 0 xi, yj 1, and 
ixi = jyj = 1. Given an initial distribution of haplotype frequencies, random union of gametes results in individuals whose chances of reproducing are determined by the viability of each genotype. Prior to the formation of a new gamete pool, recombination and segregation take place. We assume independent assortment between A and M because this is the most favorable case for Mendelian segregation (ESHEL 1985).The frequency of each haplotype in the next generation is
 | (4a) |
 | (4b) |
im and if represent the linkage disequilibrium function for haplotype i in males and females, respectively. These are 
and 
, with d = v12 (x1y4 – x3y2) + v21(x4y1 – x2y3). The symbol · represents the inner product of two vectors, which is the number x · y = ixiyi. The normalizing factor in (4) is the population mean fitness, which is equal in the two sexes,
 | (5) |
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STABILITY ANALYSIS |
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TOPABSTRACTMODELSTABILITY ANALYSISDISCUSSIONAPPENDIX AAPPENDIX BACKNOWLEDGEMENTSLITERATURE CITED |
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, 
) given fixation of M1 at the modifier locus. This equilibrium is stable in the sense that it remains unaltered over small perturbations of the frequency of A1 and A2 (short-term stability) with M1 fixed but it may not be stable to the introduction of new alleles (long-term stability) (ESHEL 1996). To explore the long-term stability of equilibrium (, ) to the introduction of M2 we simplify our notation by using (k, ) to refer to the segregation scheme (k11, 11) of the common M1M1 homozygotes, and (k+1, +1) to refer to the segregation scheme (k12, 12) of rare M1M2 heterozygotes.
Methods:
To study the long-term stability of a particular segregation scheme it is necessary to have a polymorphic equilibrium at the main locus; otherwise modifiers have no effect over segregation and their fate is determined by drift instead of selection. For this reason, we start by considering a short-term stable equilibrium (, ) polymorphic at the main locus. Hence inequalities kv12 + v21 > v22 and (1 – )v12 + (1 – k)v21 > v11 derived in ÚBEDA and HAIG (2004) must be satisfied.
Let matrix G be the gradient matrix of system (4) evaluated at equilibrium (, ). Matrix G is a block matrix that contains submatrix L (see APPENDIX A). If the leading eigenvalue of matrix L > 1, 
(L) > 1, modifier M2 introduced in a population at equilibrium (, ) increases in frequency at a geometric rate. However, if (L) = 1 nothing can be concluded about the long-term stability of (, ). We deal with this problem using the method suggested by LESSARD (1989). In his work LESSARD (1989) defines the term Q from a generic gradient matrix and its second derivatives, concluding that a particular equilibrium shows long-term instability when Q is positive. Therefore, if the term Q derived from our gradient matrix G (see APPENDIX A) is positive, Q(G) > 0, modifier M2 introduced in a population at equilibrium (, ) increases at an arithmetic rate. To summarize, allele M2 will be favored by natural selection when rare whenever (L) > 1 or whenever (L) = 1 and Q(G) > 0. If this is the case, segregation scheme (k, ) does not show evolutionary genetic stability (EGS) and can be invaded by segregation scheme (k+1, +1).
We used analytical expressions for (L) when we were able to derive these, but used numerical analysis to draw conclusions when we were unable to derive an analytical solution. In our numerical analyses, for each combination of k and considered we explored all combinations of v11, v12, v21, v22 in the range [0.1, 1.9] separated at intervals 0.225. For each set of viability parameters yielding a short-term stable polymorphic equilibrium we explore all combinations of k+1 and +1 in the range [0.02, 0.98] separated at intervals 0.08. We then calculated (L) and use this value to classify the parameter sets. This routine was implemented in Matlab 5.3 (MATHWORKS 1991).
Results:
Mendelian segregation:
Consider a population undergoing Mendelian segregation (k, ) = (1/2, 1/2). Whenever 1/2(v12 + v21) > v11, v22 the corresponding equilibrium (, )(1/2,1/2) shows short-term stability. The simplifying assumption v11 = v22 allows us to get a tractable expression for (L) and Q(G).
The leading eigenvalue of L evaluated at equilibrium (, )(1/2,1/2) is
 | (6) |
+1)(v12 – v21) and f2 = v11 + v12 + v21 + v22. Simple algebra (see APPENDIX B) shows that (L(1/2,1/2)) > 1 when
 | (7) |
Whenever reciprocal heterozygotes have the same fitness f1 = 0 and (L(1/2,1/2)) = 1. Term Q(G) evaluated at equilibrium (, )(1/2,1/2) is
 | (8) |
+1), g2 = 2(k+1 – +1), and g3 = (1 – 2k+1)(1 – 2+1).
The sign of Q(G) depends on the relative viabilities of homozygotes and heterozygotes. The two extreme cases are lethal homozygotes (v11/v12 = 0) and equal viability of both homozygote and heterozygote classes (v11/v12 = 1). Taking limits in Q(G) for each of these cases we get
 | (9a) |
 | (9b) |
In our systematic exploration of the parameter space we find 293,384 combinations of v11, v12, v21, v22 yielding a short-term stable polymorphic equilibrium. In the 119,496 cases in which (k+1 – +1)(v12 – v21) < 0, eigen-value (L(1/2,1/2)) > 1. In the 119,496 cases in which (k+1 – +1)(v12 – v21) > 0, eigenvalue (L(1/2,1/2)) < 1. Finally, in the 54,392 cases in which (k+1 – +1)(v12 – v21) = 0 (due to either k+1 = +1 or v12 = v21), (L(1/2,1/2)) takes the unit value. Hence, in principle, our analytical results can be extended to the case of differential viability of homozygote classes.
All-and-none segregation:
Consider a population with segregation (k, ) = (1, 0), one of the two possible forms of all-and-none segregation. Whenever v12 > v11, v22 the corresponding equilibrium (, )(1,0) shows short-term stability. The simplifying assumption v11 = v22 allows us to get a tractable expression for (L).
The leading eigenvalue of L evaluated at equilibrium (, )(1,0) is
 | (10) |
+1)v12 and f4 = v11 + v12. Simple algebra (see APPENDIX B) shows that (L(1,0)) > 1 when
 | (11) |
, )(1,0) is a sufficient condition for its long-term stability. Whether the reciprocal heterozygotes take the same or a different value does not affect the value of (L(1,0)).
Again we resort to numerical analysis to determine whether our analytical results can be extended to the more general case of differential viability of homozygote classes. In our systematic exploration of the parameter space we find 310,284 combinations of v11, v12, v21, v22 yielding a short-term stable polymorphic equilibrium. We failed to find a single case in which (L(1,0)) < 1. This allows us to extend our analytical results to the case of differential viability of homozygote classes.
Other segregation schemes:
We used numerical analysis to investigate the long-term stability of all combinations of k and in the range [0.02, 0.98] separated at intervals 0.08. We failed to find a single case in which (k, ) could not be invaded by some (k+1, +1).
Conclusion:
Our results are simplest when there is a balanced polymorphism at the primary locus for a fitness scheme in which reciprocal heterozygotes have distinct fitnesses (v12 
v21). In this case, a rare modifier coding for a segregation scheme (k+1, +1) such that (k+1 – +1)(v12 – v21) > 0 can invade a population fixed for Mendelian segregation. Suppose that v12 > v21; then the population can be invaded by any segregation scheme such that A1 is transmitted in greater proportion to sperm than to eggs, i.e., k+1 > +1 (Figure 2a.2). The reason for this instability is straightforward. At the Mendelian equilibrium, A1 has higher fitness when transmitted via sperm than via eggs and A2 has higher fitness when transmitted via eggs than via sperm. Therefore, heterozygotes would gain a reproductive advantage by increasing the frequency of A1 among their sperm or by increasing the frequency of A2 among their eggs. Of particular significance, Mendelian segregation can be invaded by segregation schemes (k+1, 1/2), where k+1 > 1/2 or (1/2, +1), where +1 < 1/2. That is, changes in segregation ratio do not need to be coordinated between the sexes: a successful modifier can change the segregation ratio in spermatogenesis without a change in oogenesis, or the reverse.
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+1) located below the surface Q(G) = 0 will be favored by natural selection over a modifier coding for Mendelian segregation and fixed in the population (Figure 2b.1). Simple observation of surface Q(G) = 0 reveals that successful modifiers must code for a segregation scheme with opposite effects in spermatogenesis and oogenesis. Moreover, the precision with which the segregation advantage in one sex is complemented by a segregation disadvantage in the opposite sex becomes increasingly stringent as there is a progressively smaller advantage of A1A2 heterozygotes over homozygous genotypes.
Under the simplifying assumption that v11 = v22, the two extreme cases are minimum heterozygote advantage, v11/v12 
1, and maximum heterozygote advantage, v11/v12 0. In the first scenario, it is only modifiers with equal, but opposite, effects in males and females, i.e., k+1 + +1 = 1, that can invade a population in which Mendelian segregation is the norm. In the second scenario, it is enough that the modifier has opposite effects in males and females, i.e., (k+1 – 1/2)(+1 – 1/2) < 0, to be favored by natural selection (Figure 2b.3). For example, consider v12 = v21 and v11 = v22 = 0; a Mendelian population can be invaded by a modifier that increases the transmission of A1 to sperm (k+1 > 1/2) but reduces its transmission to eggs (+1 < 1/2). The same population can be invaded by a modifier that reduces the transmission of A1 to sperm (k+1 < 1/2) but increases its transmission to eggs (+1 > 1/2) (Figure 2b.3).
If reciprocal heterozygotes have identical viability, the modifiers that can invade a population fixed for Mendelian segregation must cause coordinated changes in spermatogenesis and oogenesis. This is because A1 and A2 have the same fitness whether transmitted via eggs or sperm when allele frequencies are at the equilibrium determined by Mendelian segregation. Selection is initially weak because, in a panmictic population, the rare eggs produced by the modified segregation scheme gain a fitness advantage only from their even rarer unions with the rare sperm produced by the modified segregation scheme. Modifications need to be coordinated between oogenesis and spermatogenesis because these unions need to produce an increased frequency of heterozygotes whereas some combinations of changes, including unilateral changes in one sex but not in the other, will result in increased production of the less-fit homozygous genotypes.
The intuitive reason why fair segregation shows evolutionary instability is that this segregation scheme does not maximize population mean fitness when sex-specific segregation is allowed (ÚBEDA and HAIG 2004). Hence, those segregation schemes able to bias the offspring production in favor of the fittest heterozygote will be favored by natural selection. The link between fitness and segregation can be clarified by using the concept of genetic load. CROW (1970) defined genetic load as the fraction by which the population mean fitness at equilibrium differs from the fitness of the most viable genotype,
 | (12) |
While the enforcement of Mendelian segregation eliminates drive load it does not affect segregation load (Figure 3). However, if sex-specific segregation is allowed, distorters can modify both types of load (Figure 3). If the net result is a reduction of load, distorters are beneficial to their host genotype and we would expect them to invade a Mendelian population. That is, distorters of Mendelian segregation can be beneficial to their host genotype if they reduce segregation load. This might call into question the use of the adjective "ultraselfish" (CROW 1988) to describe segregation distorters.
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Our results back this intuition. For example, consider the case v12 > v21 in which fair segregation can be invaded by any segregation scheme (k+1, +1) such that k+1 > +1. Numerical evidence suggests that none of these segregation schemes except all-and-none segregation of the type (1, 0) show evolutionary stability (see Figure 4). For another example, consider the case v12 = v21 and v11 = v22 = 0 in which fair segregation can be invaded by any segregation scheme (k+1, +1) such that (k+1 – 1/2)(+1 – 1/2) < 0. Numerical evidence suggests that none of these segregation schemes except all-and-none segregation show evolutionary stability. Furthermore, analytical results demonstrate that all-and-none segregation of the type (1, 0) shows evolutionary stability when v12 > v11, v22 while its symmetric segregation (0, 1) shows evolutionary stability when v21 > v11, v22.
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DISCUSSION |
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TOPABSTRACTMODELSTABILITY ANALYSISDISCUSSIONAPPENDIX AAPPENDIX BACKNOWLEDGEMENTSLITERATURE CITED |
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All-and-none segregation is not a theoretical caprice: it is the segregation scheme employed by at least 57 species of flowering plants (in seven genera) that exist as permanent translocation heterozygotes (HOLSINGER and ELLSTRAND 1984). For example, some species of Oenothera are permanent structural heterozygotes for two chromosome complexes, with one set of chromosomes (the 
-complex) transmitted to all megaspores ( = 1), and the other set (the ß-complex) transmitted to all microspores (k = 0) (CLELAND 1972).
CHARLESWORTH (1979) proposed that systems of permanent translocation heterozygosity evolved to fix a beneficial heterozygous genotype in inbred populations. His model assumed heterozygote advantage and obligate self-fertilization. Under these assumptions, any modifier of Mendelian segregation in one sex is neutral, but once there is a bias in segregation of one of the "alleles" to one class of gametes/spores, there is positive selection for modifiers that established the opposite bias in segregation to the other class of gametes/spores.
Our model suggests an alternative path to permanent heterozygosity. If there is differential viability of reciprocal heterozygotes, one allele will have higher fitness at the Mendelian equilibrium when transmitted by sperm/microspores and the other allele will have higher fitness when transmitted by eggs/megaspores. Therefore, modifiers of the segregation ratio in one sex will be favored by selection, even without an opposite bias of the segregation ratio in the other sex (VON WANGENHEIM 1962 provides evidence of genomic imprinting in Oenothera; see interpretation of his results in HAIG and WESTOBY 1991). Unlike Charlesworth's model, our model does not require initial inbreeding. The natural history of permanent translocation heterozygosity does not strongly favor one model or the other, because these species are usually self-fertilizing but with outcrossing relatives (e.g., GRANT 1975, p. 407).
It has not escaped our notice that Mendelian segregation is the rule and all-and-none segregation the rare exception. What processes then could account for the ubiquity of Mendelian segregation? We make four suggestions. There may well be others.
v21). However, such balanced polymorphisms may be rare. In the simplest form of genomic imprinting, an allele is silent when inherited from one parent, but expressed when inherited from the other. If so, the allele inherited from one parent does not affect fitness and each heterozygous genotype has a fitness equal to one of the homozygous genotypes (either v12 = v11 and v21 = v22 or v12 = v22 and v21 = v11). No balanced polymorphism is possible for such fitness schemes (PEARCE and SPENCER 1992). v21 cannot be rejected so simply, however. Imprinted genes are often clustered, with maternally expressed genes tightly linked to paternally expressed genes. Moreover, some imprinted genes are expressed biallelically in most tissues, but have monoallelic expression in some cell types. In such cases, an imprinted haplotype will have effects when it is both maternally and paternally inherited. Thus, the heterozygous genotypes need not be phenotypically equivalent to the homozygous genotypes. The model of this article also assumes that fitnesses are fixed properties of an individual's genotype. However, if an individual's fitness is influenced by the genotypes of other family members, the fitnesses of the different genotypes are frequency dependent. A1A2 heterozygotes may exist in family environments different from those of A2A1 heterozygotes and from that of either homozygous genotype (e.g., in models of sib competition with multiple paternity within litters). In such models, A1A2 and A2A1 heterozygotes may have different fitnesses even at an unimprinted locus. To conclude, the prevailing solution to the evolutionary puzzle of Mendelian segregation (ESHEL 1985) does not apply when sex-specific segregation is allowed. In our model, fair segregation does not show evolutionary genetic stability while all-and-none segregation does. Clearly, the selective forces that maintain Mendelian segregation in most organisms are not fully understood.
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APPENDIX A |
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TOPABSTRACTMODELSTABILITY ANALYSISDISCUSSIONAPPENDIX AAPPENDIX BACKNOWLEDGEMENTSLITERATURE CITED |
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 | (A1) |
Straight differentiation in (4) yields the first-order derivatives
 | (A2a) |
 | (A2b) |
 | (A2c) |
 | (A2d) |
 | (A3a) |
 | (A3b) |
ij is the Kronecker delta; that is, ij = 1 if i = j, and ij = 0 otherwise.
Let S = Giji, j 
{1, 2}; R = Giji {1, 2}, j {3, 4}; and L = Giji, j {3, 4}. Matrix G has the structure
 | (A4) |
The leading eigenvalue of G is greater than one if either the leading eigenvalue of S or the leading eigenvalue of L is greater than one. Furthermore, the leading eigenvalue of S must be less than one given the short-term stability of (, ). Hence, the long-term stability of equilibrium (, ) is characterized by the leading eigenvalue of L, (L). The full expression of L is
 | (A5) |
 | (A6) |
Hessian matrix and second derivatives:
Hessian matrix H is a matrix with elements that are matrices themselves:
 | (A7a) |
 | (A7b) |
Straight differentiation in (A2) yields the second-order derivatives with respect to xl,
 | (A8a) |
 | (A8b) |
 | (A8c) |
 | (A8d) |
 | (A9a) |
 | (A9b) |
 | (A9c) |
 | (A9d) |
The term Q(G) results from multiplying the left eigenvector of L, elements of H, and pairs of values of the right eigenvector of G as specified in LESSARD (1989).
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APPENDIX B |
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TOPABSTRACTMODELSTABILITY ANALYSISDISCUSSIONAPPENDIX AAPPENDIX BACKNOWLEDGEMENTSLITERATURE CITED |
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| (B1) |
12)(v12 – v21) and f2 = 2v11 + v12 + v21. Hence (L(1/2,1/2)) > 1 whenever
 | (B2) |
 | (B3) |
 | (B4) |
Consequently, the necessary and sufficient condition for the long-term instability of Mendelian segregation is f1 > 0. Substituting f1 for its expression and simplifying, this inequality reads
 | (B5) |
Spectral radius of L(1,0) < 1:
The leading eigenvalue of L(1,0) is
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| (B6) |
12)v12 and f4 = v11 + v12. Hence L(1,0) < 1 whenever
 | (B7) |
 | (B8) |
12 – k12)v12 > v11 and v12 > v11. Given that 3 + 12 – k12 > 1, inequality v12 > v11 is the more restrictive of the two conditions and, therefore, the only one that is relevant. The latter condition is always true if equilibrium (, )(1,0) is short-term stable.
If 4v12 < f3 + f4, we have to reverse the sense of inequality (B7) when squaring:
 | (B9) |
, )(1,0) is short-term stable.
To summarize, condition v12 > v11, v22 is necessary and sufficient to guarantee the long-term stability of the segregation scheme (1, 0). This is the same condition required for the short-term stability of equilibrium (, )(1,0).
Spectral radius of L(0,1) < 1:
Similarly, the leading eigenvalue of L(0,1) is
 | (B10) |
+1 – k+1)v21 and f6 = v11 + v21.
 | (B11) |
 | (B12) |
, )(1,0) is short-term stable. |
ACKNOWLEDGEMENTS |
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TOPABSTRACTMODELSTABILITY ANALYSISDISCUSSIONAPPENDIX AAPPENDIX BACKNOWLEDGEMENTSLITERATURE CITED |
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LITERATURE CITED |
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TOPABSTRACTMODELSTABILITY ANALYSISDISCUSSIONAPPENDIX AAPPENDIX BACKNOWLEDGEMENTSLITERATURE CITED |
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= 5 x 10–3. (a) Nonsymmetric viability of reciprocal heterozygotes. Let v11 = 0.8, v12 = 1.6, v21 = 2, and v22 = 0.8. Our results show that allele M2 coding for segregation scheme (0, 1) in homozygotes fully replaces allele M1. Specifically, haplotypes A2M2 and A1M2 become fixed in sperm and eggs, respectively. (b) Symmetric viability of reciprocal heterozygotes. Let v11 = 0.8, v12 = v21 = 1.8, and v22 = 0.8. Our results show that both (b.1) an allele M2 coding for segregation scheme (0, 1) in homozygotes and (b.2) an alleleM2 coding for segregation scheme (1, 0) in homozygotes fully replace allele M1. While in the former case haplotypes A2M2 and A1M2 become fixed in sperm and eggs, in the latter case they become fixed in eggs and sperm, respectively. In the absence of imprinting the number of generations represented is four times larger than that in its presence. Arrows indicate the sense in which time increases.