Mutation-Selection Balance at a Modifier-of-Imprinting Locus

We propose a pair of population genetic models for a modifier-of-imprinting locus for which different genotypes imprint different proportions of an imprintable target locus in their gametes. The two models examine the situations in which imprinting is advantageous or disadvantageous, and we discuss three cases for which the modifier is respectively partially dominant, dominant, or recessive. The models predict the stable equilibrium frequencies of the mutant modifier and functionally diploid individuals in a large population in terms of up to four parameters: the mutation rate at the modifier locus, V; the selection coefficient against the disadvantageous phenotype, s; the proportion of unimprinted eggs produced by homozygotes for the mutant modifier, θ, and, in the partially dominant models, the dominance parameter, k. The equilibrium frequency of the mutant phenotypes is shown to be approximately twice that of standard Mendelian models: 2V/s or 4V/s when the modifier is recessive or dominant, respectively. Mathematical equivalences between these and nonimprinting models are noted.     

Mutation-Selection Balance under Genomic Imprinting at an Autosomal Locus

I model the effect of genomic imprinting on the equilibrium allele frequencies at an autosomal diallelic locus subject to viability selection and mutation. The population size is assumed to be very large; male and female mutation rates may be unequal. Different models examine cases of the inactivation of one gene (with both complete and partial penetrance) and of differential expression of genes according to the parent of origin. In the simplest cases the frequency of the deleterious allele is approximately twice that of a dominant nonimprinting mutant, but considerably less than that of a recessive nonimprinting mutant. Under imprinting, selection and unequal mutation rates interact: other things being equal, male-biased mutation leads to lower mutant frequencies under maternal imprinting and higher frequencies under paternal imprinting. I also model cases where just one allele is imprintable (and the other not). These models allow us to predict the frequency of a failure to imprint in a normally imprinting system, as well as the frequency of imprinting at a standard nonimprinting locus.     

Mutation modification with multiplicative fertility selection

Two diallelic loci in an infinite panmictic population of diploid individuals are modelled. The A/a locus is subject to unidirectional mutation and either multiplicative fertility selection or, equivalently, sex-asymmetric viability selection. The M/m locus acts as a selectively neutral modifier of the mutation rate at A/a. The loci recombine at rate R. If the M/m locus is initially monomorphic, and the A/a locus has reached equilibrium, the fate of a new modifier allele is found to depend not just on its relative effect on mutation but also upon the linkage, R. Each initial equilibrium may be characterized by a critical value of the recombination rate, R*. If 0 less than R* less than 0.5, a sufficiently small "down" modifier of the mutation rate will invade the population when R less than R* whereas a sufficiently small "up" modifier will succeed when R greater than R*. If R* less than 0 or R* greater than 0.5, only mutation reduction may occur. Numerical analysis of 56,000 sample equilibria indicates that mutation rates may be increased, but only when the selection regime is such that the A/a locus would remain polymorphic in the absence of mutation.     

Dynamics of the classical genetic model for the evolution of dominance

Using a generalized method of Ljapunov functions, the dynamics of the classical genetic model for the evolution of dominance is studied. The model is treated as a two locus two allele system of a primary and a modifying locus with selection, mutation, and recombination. Its behavior may be described either by a system of four differential equations or by a system of four difference equations. In particular, it is proved that under very general conditiones on the five parameters involved, in both cases the well-known fixed point for the mutation-selection balance at the primary locus when the modifier is completely selected is globally asymptotically stable. If, however, the unmodified heterozygote is completely recessive or underdominant, the modifier is only selected if at the beginning of the evolution its frequency and that of the favorable primary allele is not extremely low. Otherwise, it may happen that the favorable primary allele becomes extinct.     

Modifiers of mutation rate: a general reduction principle

A deterministic two-locus population genetic model with random mating is studied. The first locus, with two alleles, is subject to mutation and arbitrary viability selection. The second locus, with an arbitrary number of alleles, controls the mutation at the first locus. A class of viability-analogous Hardy-Weinberg equilibria is analyzed in which the selected gene and the modifier locus are in linkage equilibrium. It is shown that at these equilibria a reduction principle for the success of new mutation-modifying alleles is valid. A new allele at the modifier locus succeeds if its marginal average mutation rate is less than the mean mutation rate of the resident modifier allele evaluated at the equilibrium. Internal stability properties of these equilibria are also described.     

On the evolution of genetic incompatibility systems. IV. Modification of response to an existing antigen polymorphism under partial selfing

A 2-locus model of the evolution of self-incompatibility in a population practicing partial selfing is presented. An allele is introduced at a modifier locus which influences the strength of the rejection reaction expressed by the style in response to antigens recognized in pollen. Two causes of inbreeding depression are investigated. First, offspring viability depends solely on the source (self or non-self) of the fertilizing pollen. Second, offspring viability declines with the expression of recessive deleterious alleles, segregating at a third (disease) locus, which exhibit an imperfect association with antigen alleles. Evolutionary changes occurring at the disease locus are not considered in this study. The condition under which a modifier allele that intensifies the incompatibility reaction increases when rare depends upon the number of antigens, the frequency of recessive deleterious alleles at the disease locus, and the level of association between the antigen locus and the disease locus. It is the improvement of viability among offspring derived by outcrossing, rather than the prevention of self-fertilization, that may represent the primary evolutionary function of genetic incompatibility systems.     

Gene inactivation as a mechanism for the expression of recessive phenotypes.

A series of Chinese hamster ovary cell hybrids were constructed which were heterozygous at the emtB and chr loci. These loci encode two recessive drug-resistance genes (emetine resistance and chromate resistance, respectively) located on a structurally hemizygous region on the long arm of chromosome 2. These heterozygous hybrids therefore exhibit wild-type sensitivity to both emetine and chromate. Drug-resistant variants were then selected in medium containing either emetine or chromate, and the mechanism of reexpression of the recessive drug-resistant allele was determined by karyotypic analysis of the resultant colonies. In previous studies at these loci we have determined that segregation of the recessive phenotype occurs primarily by (1) the loss of the chromosome 2 carrying the wild-type, drug-sensitive, allele, (2) deletion of the long arm of chromosome 2, or (3) loss of one chromosome 2 followed by duplication of the remaining homologue. However, a small proportion of segregants have also been detected which may have arisen by the mechanisms of de novo gene inactivation or mutation. In this report, hybrids are described which were constructed to allow selection for the retention of the chromosome carrying the wild-type allele and which therefore optimize isolation of these rare segregants. We demonstrate by karyotypic analysis, mutation frequency analysis, and microcell-mediated chromosome transfer that these rare segregants occur primarily by gene inactivation. We also demonstrate a dramatic increase in the proportion of segregants occurring by gene inactivation in two of these hybrids as compared with those previously reported, indicating that this mechanism may be an important mode of phenotype segregation in diploid cells and, therefore, in the development of cancers--such as the childhood tumors retinoblastoma and Wilms tumor--resulting from recessive alleles     

Parental origin of de novo constitutional deletions of chromosomal band 11p13.

One-half of all cases of Wilms tumor (WT), a childhood kidney tumor, show loss of heterozygosity at chromosomal band 11p13 loci, suggesting that mutation of one allele and subsequent mutation or loss of the homologous allele are important events in the development of these tumors. The previously reported nonrandom loss of maternal alleles in these tumors implied that the primary mutation occurred on the paternally derived chromosome and that it was "unmasked" by loss of the normal maternal allele. This, in turn, suggests that the paternally derived allele is more mutable than the maternal one. To investigate whether germinal mutations are seen with equal frequency in maternally versus paternally inherited chromosomes, we determined the parental origin of the de novo germinal 11p13 deletions in eight children by typing lymphocyte DNA from these children and from their parents for 11p13 RFLPs. In seven of the eight cases, the de novo deletion was of paternal origin. The one case of maternal origin was unremarkable in terms of the size or extent of the 11p13 deletion, and the child did develop WT. Transmission of 11p13 deletions by both maternal and paternal carriers of balanced translocations has been reported, although maternal inheritance predominates. These data, in addition to the general preponderance of paternally derived, de novo mutations at other loci, suggest that the increased frequency of paternal deletions we observed is due to an increased germinal mutation rate in males.     

Coevolution of self-fertilization and inbreeding depression. III. Homozygous lethal mutations at multiple loci.

We study the evolution of the rate of self-fertilization in response to deleterious mutations at multiple loci. Although partial selfing induces associations among loci even in the absence of linkage, associations among mutations at different loci are of a smaller order of magnitude than the mutation rate. Genotypes that carry homozygous lethal mutations in heterozygous form at i loci occur in frequencies of the order (Ti) mu i, in which T denotes the number of viability loci and mu the mutation rate. While associations between mutations at different loci remain small even under inbreeding, each viability locus develops an association with the modifier of the rate of self-fertilization that substantially affects the evolution of the breeding system. Positive associations between enhancers of selfing and haplotypes carrying multiple wild-type alleles and positive associations in heterozygosity between the modifier locus and the viability loci promote evolutionary increases in the rate of self-fertilization.     

A model of migration modification

Two subpopulations whose different sizes are in a constant ratio interact via migration. The fitness of the diploid organisms is determined by two alleles at a single locus and by the niche the organism is in. The rates of migration depend upon two neutral modifier genes at a second locus. The second modifying allele is introduced into an equilibrium where the first modifying allele is fixed, and where the other two alleles are already polymorphic. It is shown that the new migration modifier is selected for when it reduces migration. The similarity between this result and some recombination modifier models is noted.     

Curly bare (cub), a new mouse mutation on chromosome 11 causing skin and hair abnormalities,and a modifier gene (mcub) on chromosome 5

In the outcrossing of a new recessive mouse mutation causing hair loss, a new wavy-coated phenotype appeared. The two distinct phenotypes were shown to be alternative manifestations of the same gene mutation and attributable to a single modifier locus. The new mutation, curly bare (cub), was mapped to distal Chr 11 and the modifier (mcub) was mapped to Chr 5. When homozygous for the recessive mcub allele, cub/cub mice appear hairless. A single copy of the dominant Mcub allele confers a full, curly coat in cub/cub mice. Reciprocal transfer of full-thickness skin grafts between mutant and control animals showed that the skin phenotype was tissue autonomous. The hairless cub/cub mcub/mcub mice show normal contact sensitivity responses to oxazolone. The similarity of the wavy coat phenotype to those of Tgfa and Egfr mutations and the map positions of cub and mcub suggest candidate genes that interact in the EGF receptor signal transduction pathway.     

Oculocutaneous albinism (OCA2) in sub-Saharan Africa: distribution of the common 2.7-kb P gene deletion mutation

Oculocutaneous albinism (OCA2) is the most common autosomal recessive disorder in the South African Negroid population, occurring with a prevalence of 1/3900 individuals. The OCA2 locus, P, has been mapped to chromosome 15q11–q13 and a 2.7-kb interstitial deletion has been found to be the common mutation in Africa. This study reports the detection of the deletion allele in OCA2-affected individuals from the southern African, Zambian and Central African Republic (CAR) Negroid populations (0.77, 131/170 OCA2 chromosomes; 0.79, 11/14; 0.33, 4/12, respectively). Normally pigmented individuals from different African countries were also tested. The deletion mutation was found at a frequency of 0.013 (10/780) in the normally pigmented southern African Negroid population and at a lower frequency in individuals from central Africa (0.002; 2/834), including individuals from Zambia, Cameroon, Zaire and the CAR. The study confirms the African origin of this deletion allele. Haplotype analysis suggests that the deletion mutation probably occurred only once and that it arose before the divergence of these African populations, which is estimated to be about 2000– 3000 years ago. The unusually high frequency of OCA2 mutations, in particular the 2.7-kb deletion, suggests some selective agent or genetic drift. Received: 24 September 1996 / Revised: 8 November 1996     

Segregation and the evolution of sex under overdominant selection

The segregation of alleles disrupts genetic associations at overdominant loci, causing a sexual population to experience a lower mean fitness compared to an asexual population. To investigate whether circumstances promoting increased sex exist within a population with heterozygote advantage, a model is constructed that monitors the frequency of alleles at a modifier locus that changes the relative allocation to sexual and asexual reproduction. The frequency of these modifier alleles changes over time as a correlated response to the dynamics at a fitness locus under overdominant selection. Increased sex can be favored in partially sexual populations that inbreed to some extent. This surprising finding results from the fact that inbred populations have an excess of homozygous individuals, for whom sex is always favorable. The conditions promoting increased levels of sex depend on the selection pressure against the homozygotes, the extent of sex and inbreeding in the population, and the dominance of the invading modifier allele.     

Contagious parthenogenesis, automixis, and a sex determination meltdown

Because of the twofold cost of sex, genes conferring asexual reproduction are expected to spread rapidly in sexual populations. However, in reality this simple prediction is often confounded by several complications observed in natural systems. Motivated by recent findings in the Cape honey bee and in the parasitoid wasp Lysiphlebus fabarum, we explore through mathematical models the spread of a recessive, parthenogenesis inducing allele in a haplodiploid population. The focus of these models is on the intricate interactions between the mode of parthenogenesis induction through automixis and complementary sex determination (CSD) systems. These interactions may result in asexual production of diploid male offspring and the spread of the parthenogenesis-inducing allele through these males. We demonstrate that if parthenogenetic females produce a substantial proportion of male offspring, this may prevent the parthenogenesis-inducing allele from spreading. However, this effect is weakened if these diploid males are at least partially fertile. We also predict a degradation of multilocus CSD systems during the spread of parthenogenesis, following which only a single polymorphic CSD locus is maintained. Finally, based on empirical parameter estimates from L. fabarum we predict that male production in parthenogens is unlikely to prevent the eventual loss of sexual reproduction in this system.     

Is there selection in favour of heterozygotes in families with merosin-deficient congenital muscular dystrophy?

Merosin-deficient congenital muscular dystrophy is an autosomal recessive neuromuscular disorder caused by partial or total absence of laminin-2 (merosin) in the skeletal muscle. Affected children have severe weakness, hypotonia at birth, high creatine kinase (CK) levels (more than 10 times normal) and are not able to walk or stand unsupported. Linkage and mutation analysis demonstrated that the gene encoding for the laminin-alpha2 chain, mapped on chromosome 6q22-23, is invariably responsible for this form of congenital muscular dystrophy. We investigated the pattern of inheritance of the haplotypes associated with the mutated allele in 29 informative merosin-deficient families, using tightly linked informative polymorphic microsatellite markers. This allowed us to identify heterozygous individuals from normal homozygotes, who are clinically, pathologically and biochemically indistinguishable. By linkage analysis, we found a statistically significant increase in the number of heterozygous individuals carrying either the paternal or the maternal haplotypes associated with the mutated allele. This could suggest a selection in favour of the alleles carrying mutations at the laminin alpha2-chain locus.     

Studies of a spontaneous lethal mutation at the albino locus in SELH/Bc mice.

We report a new mutation at the albino locus in SELH/Bc mice. The mutation arose spontaneously in a male mouse that appeared to be a somatic and germ line mosaic for a new albino (c) allele, provisionally named cBc. The mutation is a recessive lethal, causing embryonic death soon after implantation. We have shown that there is no detectable activity of the Mod-2 allele in cis with the mutation and conclude that the mutation is probably a deletion that includes the c locus, the Mod-2 locus, the intervening 2 cM, and at least one locus essential for postimplantation embryonic survival, either proximal to the c locus or distal to the Mod-2 locus. This new mutation is similar to most previously reported spontaneous mutations at the albino locus in that it arose in a somatic and germ line mosaic mutant animal but differs from them in that it is an embryonic lethal when homozygous and is apparently a deletion. SELH/Bc mice appear to have a high mutation rate. This lethal albino mutation that appears to be a postmeiotic deletion should be useful in the search for the mechanism of mutagenesis in SELH/Bc mice. It may also be useful in mapping essential genes in the c-locus region.     

The evolution of intratetrad mating rates

Intratetrad mating, the fusion of gametes formed in a single meiosis, has unusual consequences for genetic diversity, especially in genome regions linked to mating type loci. Here we investigate the fate of modifier alleles that alter the rate of intratetrad mating, under models of heterozygote advantage and of genetic load resulting from recurrent mutation. In both cases, intratetrad mating is favored if the recombination rate between the selected locus and mating type is less than the frequency of lethal recessive alleles at that locus in the population. Positive feedback often accelerates the invasion of modifiers to the intratetrad mating rate. Recombination rate and intratetrad mating rate exert indirect selection on one another, resulting in a cascading decline in outcrossing, even in the absence of any cost of sex. However, under recurrent mutation, alleles for obligate intratetrad mating invade only very slowly, perhaps explaining why outcrossing can persist at low frequencies in a largely intratetrad mating population.     

Beneficial mutations, hitchhiking and the evolution of mutation rates in sexual populations

Natural selection acts in three ways on heritable variation for mutation rates. A modifier allele that increases the mutation rate is (i) disfavored due to association with deleterious mutations, but is also favored due to (ii) association with beneficial mutations and (iii) the reduced costs of lower fidelity replication. When a unique beneficial mutation arises and sweeps to fixation, genetic hitchhiking may cause a substantial change in the frequency of a modifier of mutation rate. In previous studies of the evolution of mutation rates in sexual populations, this effect has been underestimated. This article models the long-term effect of a series of such hitchhiking events and determines the resulting strength of indirect selection on the modifier. This is compared to the indirect selection due to deleterious mutations, when both types of mutations are randomly scattered over a given genetic map. Relative to an asexual population, increased levels of recombination reduce the effects of beneficial mutations more rapidly than those of deleterious mutations. However, the role of beneficial mutations in determining the evolutionarily stable mutation rate may still be significant if the function describing the cost of high-fidelity replication has a shallow gradient.     

Model involving gene inactivation in the generation of autosomal recessive mutants in mammalian cells in culture.

We present evidence for a two-step model for expression of the recessive phenotype at the diploid adenine phosphoribosyl transferase (aprt) locus in Chinese hamster ovary cells. This model proposes a high-frequency event leading to allelic inactivation and a low-frequency event leading to a structural alteration of the APRT protein. Either event can occur first, resulting in two types of heterozygous cells. The proposed model is based on analysis of Chinese hamster ovary presumptive aprt heterozygotes and APRT- mutants, derived by two different laboratories. The major class of heterozygotes (class 1) had approximately 50% parental APRT activity, 50% immunologically precipitable APRT protein, and only wild-type enzyme as based on two-dimensional gel electrophoresis and thermal inactivation studies. We propose that one allele at the aprt locus has been inactivated in these heterozygotes. APRT- mutants derived from any single class 1 heterozygote arose at a low frequency and contained either no immunologically detectable APRT protein or an APRT enzyme which was, in most cases, demonstrably altered. The second class of heterozygotes, consisting of two independent isolates, gave rise to APRT- cells at a high frequency (10(-3) to 10(-5). These heterozygous cell lines had 50% of parental APRT activity and only wild-type spot, or wild-type and an electrophoretic variant spot, on two-dimensional gels. These aprt heterozygotes appear to have arisen by mutation at one allele. APRT- mutants derived from either heterozygote of this class had all lost the wild-type activity, consistent with the proposed model.