The role of recombination in the modifier theory of autosomal segregation distortion

The effects of recombination on the equilibrium structures of two-locus systems of autosomal segregation distortion are studied. Exact conditions pertaining to the stability of polymorphic equilibria maintaining multiple distorters at the segregation-determination locus as well as their resistance to the invasion of mutant distorters are given. Evolutionary patterns of autosomal meiotic drive and the status of Mendelian segregation are reexamined.     

Population models of genomic imprinting. I. Differential viability in the sexes and the analogy with genetic dominance

Many single-locus, two-allele selection models of genomic imprinting have been shown to reduce formally to one-locus Mendelian models with a modified parameter for genetic dominance. One exception is the model where selection at the imprinted locus affects the sexes differently. We present two models of maternal inactivation with differential viability in the sexes, one with complete inactivation, and the other with a partial penetrance for inactivation. We show that, provided dominance relations at the imprintable locus are the same in both sexes, a globally stable polymorphism exists for a range of viabilities that is independent of the penetrance of imprinting. The conditions for a polymorphism are the same as in previous models with differential viability in the sexes but without imprinting and in a model of the paternal X-inactivation system in marsupials. The model with incomplete inactivation is used to illustrate the analogy between imprinting and dominance by comparing equilibrium bifurcation plots for fixed values of dominance and penetrance. We also derive a single expression for the dominance parameter that leaves the frequency and stability of equilibria unchanged for all levels of inactivation. Although an imprinting model with sex differences does not formally reduce to a nonimprinting scheme, close theoretical parallels clearly exist.     

Natural Selection and Y-Linked Polymorphism

Several population genetic models allowing natural selection to act on Y-linked polymorphism are examined. The first incorporates pleiotropic effects of a Y-linked locus, such that viability, segregation distortion, fecundity and sexual selection can all be determined by one locus. It is shown that no set of selection parameters can maintain a stable Y-linked polymorphism. Interaction with the X chromosome is allowed in the next model, with viabilities determined by both X- and Y-linked factors. This model allows four fixation equilibria, two equilibria with X polymorphism and a unique point with both X- and Y-linked polymorphism. Stability analysis shows that the complete polymorphism is never stable. When X- and Y-linked loci influence meiotic drive, it is possible to have all fixation equilibria simultaneously unstable, and yet there is no stable interior equilibrium. Only when viability and meiotic drive are jointly affected by both X- and Y-linked genes can a Y-linked polymorphism be maintained. Unusual dynamics, including stable limit cycles, are generated by this model. Numerical simulations show that only a very small portion of the parameter space admits Y polymorphism, a result that is relevant to the interpretation of levels of Y-DNA sequence variation in natural populations.     

The maintenance (or not) of polygenic variation by soft selection in heterogeneous environments

On the basis of single-locus models, spatial heterogeneity of the environment coupled with strong population regulation within each habitat (soft selection) is considered an important mechanism maintaining genetic variation. We studied the capacity of soft selection to maintain polygenic variation for a trait determined by several additive loci, selected in opposite directions in two habitats connected by dispersal. We found three main types of stable equilibria. Extreme equilibria are characterized by extreme specialization to one habitat and loss of polymorphism. They are analogous to monomorphic equilibria in singe-locus models and are favored by similar factors: high dispersal, weak selection, and low marginal average fitness of intermediate genotypes. At the remaining two types of equilibria the population mean is intermediate but variance is very different. At fully polymorphic equilibria all loci are polymorphic, whereas at low-variance equilibria at most one locus remains polymorphic. For most parameters only one type of equilibrium is stable; the transition between the domains of fully polymorphic and low-variance equilibria is typically sharp. Low-variance equilibria are favored by high marginal average fitness of intermediate genotypes, in contrast to single-locus models, in which marginal overdominance is particularly favorable for maintenance of polymorphism. The capacity of soft selection to maintain polygenic variation is thus more limited than extrapolation from single-locus models would suggest, in particular if dispersal is high and selection weak. This is because in a polygenic model, variance can evolve independently of the mean, whereas in the single-locus two-allele case, selection for an intermediate mean automatically leads to maintenance of polymorphism.     

On the status of mean fitness maximization as a governing principle in evolution

Mean fitness is not always maximized under natural selection. In particular, in two locus models, recombination and epistasis may combine to prevent the operation of a maximizing priciple for mean fitness. If inversion phenomena are considered, however, there exist fully polymorphic equilibria which maximize the mean fitness and moreover the initial progress of an inversion can proceed if and only if it gives rise to an increase in mean fitness. While not applicable to all models, the principle of mean fitness maximization is still useful heuristically.     

Phenogenetic drift and the evolution of genotype-phenotype relationships

Maintenance of a stable two-locus polymorphism is analyzed statistically by fitting a logistic regression with a quadratic function of genotypic fitnesses to the probability for a fitness set to maintain a polymorphism. The regression is fitted using a data set containing information on stable equilibria maintained by 32,00 randomly generated fitness sets with three recombination values (0. 005, 0.05, 0.5). Fitted logistic regressions discriminate with 88 to 90% accuracy between fitness sets maintaining and not maintaining a stable internal equilibrium, which implies the existence of a fitness structure (balance of fitnesses) maintaining a two-locus polymorphism. Aspects of the balance of fitnesses revealed by logistic regressions are discussed. It is demonstrated that logistic regression also discriminates between types of a stable polymorphism: globally stable polymorphism, several simultaneously stable polymorphisms, and stable equilibria in addition to a polymorphic one, which implies that different balances of fitnesses are responsible for the maintenance of different types of polymorphism.     

Identification and characterization of nuclear microsatellites in Mediterranean cedars (Cedrus sp.)

We developed primers for the amplification of six polymorphic nuclear microsatellites in Mediterranean Cedrus taxa. Microsatellites originated from two Cedrus atlantica genomic libraries enriched for TC (four markers) and TG (two markers) motifs. No distortion from Mendelian segregation was observed. There were up to eight alleles per locus and mean expected heterozygosity was 0.75. Four microsatellites also amplified in the Himalayan cedar. These markers should be helpful for species identification, diversity studies, parentage analysis and genome mapping and to monitor biodiversity in endangered Mediterranean Cedrus forests.     

The multiple-locus symmetric fertility model

Selection due to variation in the fecundity among matings of genotypes with respect to many loci each with two alleles is studied. The fitness of a mating depends only on the genotypic distinction between homozygote and heterozygote at each locus in the two individuals, and differences among loci are allowed. This symmetric fertility model is therefore a generalization of the multiple-locus symmetric viability model. The phenomena seen in the two-locus symmetric fertility model generalize—e.g., the possibility of joint stability of equilibria with linkage equilibrium and with linkage disequilibrium, and the existence of different types of totally polymorphic equilibria with the gametic proportions in linkage equilibrium. The central equilibrium with genotypic frequencies in Hardy-Weinberg proportions and gametic frequencies in Robbins proportions exists for all symmetric fertility models. For some symmetric fertility regimes additional equilibria exist with gametic frequencies in linkage equilibrium and with genotypic frequencies in Hardy-Weinberg proportions at all except one locus. These equilibria may exist in the dioecious symmetric viability model, and then they will be locally stable. For free recombination the stable equilibria show linkage equilibrium, but several of these with different numbers of polymorphic loci may be stable simultaneously.     

Selection for Linkage Modification II. a Recombination Balance for Neutral Modifiers

A stable polymorphic equilibrium may be established at a selectively-neutral gene locus which controls the extent of recombination between two other selected loci. The condition for the existence of the stable polymorphism is analogous to heterozygous advantage. The heterozygote at the modifying locus should produce a recombination fraction allowing the greatest linkage disequilibrium. In the models treated this has the effect of producing the highest mean fitness. The relationship of these findings to general problems of coadaptation is discussed.     

Regions of Stable Equilibria for Models of Differential Selection in the Two Sexes under Random Mating

The equilibrium structure of models of differential selection in the sexes is investigated. It is shown that opposing additive selection leads to stable polymorphic equilibria for only a restricted set of selection intensities, and that for weak selection the selection intensities must be of approximately the same magnitude in the sexes. General models of opposing directional selection, with arbitrary dominance, are investigated by considering simultaneously the stability properties of the trivial equilibria and the curve along which multiple roots appear. Numerical calculations lead us to infer that the average degree of dominance determines the equilibrium characteristics of models of opposing selection. It appears that if the favored alleles are, on the average, recessive, there may be multiple polymorphic equilibria, whereas only a single polymorphic equilibrium can occur when the favored alleles are, on the average, dominant. The principle that the average degree of dominance controls equilibrium behavior is then extended to models allowing directional selection in one sex with overdominance in the other sex, by showing that polymorphism is maintained if and only if the average fitness in heterozygotes exceeds one.     

ESS gene expression of X-linked imprinted genes subject to sexual selection

We define ESS (Evolutionary Stable Strategy) conditions for the evolution of genomic imprinting at an X-linked locus. The system analysed is designed for mammalian imprinting in which X-linked genes typically undergo random X-inactivation and lack Y-linked homologues. We consider two models that map cellular gene expression to fitness in females subject to random X-inactivation. In the first model, female fitness is simply a function of the average gene expression across all cells. In the second model, each cell contributes independently to fitness, and female fitness is assessed as the average of these contributions across all cells. In both models, imprinting readily evolves when sexual selection favours different levels of gene expression in the two sexes. Imprinting is beneficial as it improves adaptation in both sexes. There are limits to the improvement in adaptation when sexual selection is strong and favours greater gene expression in males (the heterogametic sex). We also consider the consequences of an active Y-linked homologue on the evolution of imprinting. Our analysis suggests that restrictive conditions apply for the evolution of polymorphic ESSs at an X-linked imprinted loci.     

Polymorphism at two loci through selection for linear metric deviation.

A model of phenotypic stabilising selection in which the fitness of an individual depends solely on its phenotype, and not directly on its genetic constitution, is explored algebraically for a system of two linked loci of unequal effect. It is found that selection for metric deviation gives rise to polymorphic gametefrequency equilibria for a variety of fitness regimes. Stability of non-trivial equilibria occurs for a wide range of parameter sets. Stability is facilitated by close linkage and inequality between gene effects. It is suggested that, in general genetic variation may be maintained under stabilising selection when the fitness of double heterozygotes exceeds that of the phenotypically intermediate homozygotes.     

On the relation between the instability of ESS in discrete dynamics and segregation distortion

There is a simple correspondence between discrete dynamical systems associated with evolutionary game dynamics and general locus multiallele selection models with non-Mendelian segregation. When interpreted properly the payoff matrix has two components, a fitness matrix component and a segregation matrix component. The presence of segregation distortion which corresponds to a non-symmetric payoff matrix, is a source of instability. With non-symmetric payoff an ESS does not usually correspond to a stable equilibrium. It is always externally stable but does not necessarily have an internally stable equilibrium.     

Inheritance of allozyme variants in bishop pine (Pinus muricuta D. Don)

Isozyme phenotypes are described for 45 structural loci and 1 modifier locus in bishop pine (Pinus muricata D. Don,) and segregation data are presented for a subset of 31 polymorphic loci from 19 enzyme systems. All polymorphic loci had alleles that segregated within single-locus Mendelian expectations, although one pair of alleles at each of three loci showed significant segregation distortion. The consistency of resolution and segregation at many loci in bishop pine makes electrophoretic analyses feasible for many purposes in this species.     

Gene frequency patterns in the levene subdivided population model

In the context of a multideme population structure subject to selection, migration, and mating forces, it is desired to ascertain the stable evolutionary outcomes for various classes of selection regimes. These include selection patterns as (i) a mosaic of local directed selection effects, (ii) overdominance throughout with varying intensities of the local heterozygote advantage, (iii) varying degrees of underdominance throughout, (iv) a mixed underdominant-overdominant regime. An accounting of the nature of the equilibrium configurations in the Levene population subdivision model was done with respect to the above classes of selection regimes. In particular, it is established that multiple polymorphic equilibria do not arise for selection structures (i) and (ii), while for the mixed underdominant-overdominant selection form (iv) with appropriate parameter ranges there can exist two stable internal equilibria. A surprising finding is that the number and/or character of the equilibria is not changed by increased population division beyond that of two habitats, while there is a significant difference in the equilibrium possibilities for a one- as against a two-deme population. These results appear to be a special limiting feature of the Levene population subdivision formulation.     

Intra- and intergenotypic competition in Drosophila melanogaster: effects of density on larval survival and rate of development

We have examined the effects of density and frequency in the larval competition of Drosophila melanogaster by measuring three fitness components: viability (V), mean development time (MDT) and a combination of these two (E). We have detected (contrary to most published results) non-linear effects of density in single-genotype cultures; in addition, different functions are required to describe the density effects below and above the optimal density. Frequency has also non-linear effects in the two-genotype cultures. Only one polymorphic equilibrium frequency, which is stable, occurs with respect to V; but two polymorphic equilibria, one stable and one unstable, exist with respect to E. The responses in single-genotype cultures do not allow one to predict the outcome of the competition in two-genotype cultures.     

Long-term evolution of multilocus traits

 We analyze monomorphic equilibria of long-term evolution for one or two continuous traits, controlled by an arbitrary number of autosomal loci and subject to constant viability selection. It turns out that fitness maximization always obtains at long term equilibria, but in the case of two traits, linkage determines the precise nature of the fitness measure that is maximized. We then consider local convergence to long term equilibria, for two multilocus traits subject to either constant or frequency dependent selection. From a model of long-term dynamics near an equilibrium we derive a criterion of local long-term stability for 2-dimensional equilibria. It turns out that mutation can be a decisive factor for stability. Received 26 January 1994; received in revised form 26 September 1994     

Stable Equilibria at Two Loci in Populations with Large Selfing Rates

The equilibrium structure of two-locus, two-allele models with very large selfing rates is found using perturbation techniques. For free recombination, r = 1/2, the following results hold. If the heterozygotes do not have at least an approximate 30% advantage in fitness relative to homozygotes, a stable equilibrium with all alleles present is possible only if all of the homozygote fitnesses differ at most by approximately the outcrossing rate, t, and all stable polymorphic equilibria have disequilibrium values, D, that are at most on the order of the outcrossing rate. Once the heterozygote fitnesses are above the threshold, there are stable equilibria possible with D near its maximum possible value. The results show that the observed disequilibria in highly selfed plant populations are not likely to result from selection leading to an equilibrium.     

Some circumstances assuring monomorphism in subdivided populations

It is well known that in a subdivided population subject to soft selection with two alleles at one locus, instability of both fixation states (a “protected polymorphism”) entails at least one stable polymorphic equilibrium. Although stable polymorphic and monomorphic equilibria can coexist in general, a stable fixation state (monomorphic equilibrium) precludes the existence of any polymorphic equilibrium under the circumstances of haploid or submultiplicative diploid viabilities. This provides that a stable monomorphism is robust against random fluctuations in allele frequencies. It also increases the known circumstances where there is a unique globally attracting stable equilibrium, i.e., where allele frequencies are determined by the selection-migration structure independent of the history of the system.     

On the evolutionary stability of Mendelian segregation

We present a model of a primary locus subject to viability selection and an unlinked locus that causes sex-specific modification of the segregation ratio at the primary locus. If there is a balanced polymorphism at the primary locus, a population undergoing Mendelian segregation can be invaded by modifier alleles that cause sex-specific biases in the segregation ratio. Even though this effect is particularly strong if reciprocal heterozygotes at the primary locus have distinct viabilities, as might occur with genomic imprinting, it also applies if reciprocal heterozygotes have equal viabilities. The expected outcome of the evolution of sex-specific segregation distorters is all-and-none segregation schemes in which one allele at the primary locus undergoes complete drive in spermatogenesis and the other allele undergoes complete drive in oogenesis. All-and-none segregation results in a population in which all individuals are maximally fit heterozygotes. Unlinked modifiers that alter the segregation ratio are unable to invade such a population. These results raise questions about the reasons for the ubiquity of Mendelian segregation.