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.     

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.     

A chip off the old block: a model for the evolution of genomic imprinting via selection for parental similarity

A consequence of genomic imprinting is that offspring are more similar to one parent than to the other, depending on which parent's genes are inactivated in those offspring. We hypothesize that genomic imprinting may have evolved at some loci because of selection to be similar to the parent of one sex or the other. We construct and analyze an evolutionary-genetic model of a two-locus two-deme system, in which one locus codes for a character under local selection and the second locus is a potential cis-acting modifier of imprinting. A proportion of males only migrate between demes every generation, and prebreeding males are less fit, on average, than females. We examine the conditions in which an imprinting modifier allele can invade a population fixed for a nonimprinting modifier allele and vice versa. We find that the conditions under which the imprinting modifier invades are biologically restrictive (high migration rates and high values of recombination between the two loci) and thus this hypothesis is unlikely to explain the evolution of imprinting. Our modeling also shows that, as with several other hypotheses, polymorphism of imprinting status may evolve under certain circumstances, a feature not predicted by verbal accounts.     

Inbreeding depression and the evolution of dispersal rates: a multilocus model

Inbreeding depression is one of the possible reasons organisms disperse. In this article, we present a two-locus model for the evolution of dispersal in the presence of inbreeding depression. The first locus codes for a modifier of the migration rate, while the second locus is a selected locus generating inbreeding depression. We express the change in frequency of the migration modifier as a function of allele frequencies and genetic associations and then use a quasi-equilibrium assumption to express genetic associations as functions of allele frequencies. Our model disentangles two effects of inbreeding depression: it gives an advantage to migrant individuals because their offspring are on average less homozygous, but it also decreases the degree of population structure, thus decreasing the strength of kin selection for dispersal. We then extend our model to include an infinite number of selected loci. When the cost of dispersal is not too high, the model predictions are confirmed by multilocus simulation results and show that inbreeding depression can have a substantial effect on the dispersal rate. For high costs of dispersal, we observe discrepancies between the model and the simulations, probably caused by associations among selected loci, which are neglected in the analysis.     

ASSORTATIVE MATE CHOICE AND DOMINANCE MODIFICATION: ALTERNATIVE WAYS OF REMOVING HETEROZYGOTE DISADVANTAGE

In genetic polymorphisms of two alleles, heterozygous individuals may contribute to the next generation on average more or fewer descendants than the homozygotes. Two different evolutionary responses that remove a disadvantageous heterozygote phenotype from the population are the evolution of strictly assortative mate choice, and that of a modifier making one of the two alleles completely dominant. We derive invasion fitness of mutants introducing dominance or assortative mate choice in a randomly mating population with a genetic polymorphism for an ecological trait. Mutations with small effects as well as mutants introducing complete dominance or perfect assorting are considered. Using adaptive dynamics techniques, we are able to calculate the ratio of fitness gradients for the effects of a dominance modifier and a mate choice locus, near evolutionary branching points. With equal resident allele frequencies, selection for mate choice is always stronger. Dominance is more strongly selected than assortative mating when the resident (common) alleles have very unequal frequencies at equilibrium. With female mate choice the difference in frequencies where dominance is more strongly selected is smaller than when mutants of both sexes can choose without costs. A symmetric resource-competition model illustrates the results.     

The Evolution of the Y Chromosome with X-Y Recombination

A theoretical population genetic model is developed to explore the consequences of X-Y recombination in the evolution of sex chromosome polymorphism. The model incorporates one sex-determining locus and one locus subject to natural selection. Both loci have two alleles, and the rate of classical meiotic recombination between the loci is r. The alleles at the sex-determining locus specify whether the chromosome is X or Y, and the alleles at the selected locus are arbitrarily labeled A and a. Natural selection is modeled as a process of differential viabilities. The system can be expressed in terms of three recurrence equations, one for the frequency of A on the X-bearing gametes produced by females, one for each of the frequency of A on the X- and Y-bearing gametes produced by males. Several special cases are examined, including X chromosome dominance and symmetric selection. Unusual equilibria are found with the two sexes having very different allele frequencies at the selected locus. A significant finding is that the allowance of recombination results in a much greater opportunity for polymorphism of the Y chromosome. Tighter linkage results in a greater likelihood for equilibria with a large difference between the sex chromosomes in allele frequency.     

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.     

Sex-specific viability, sex linkage and dominance in genomic imprinting

Genomic imprinting is a phenomenon by which the expression of an allele at a locus depends on the parent of origin. Two different two-locus evolutionary models are presented in which a second locus modifies the imprinting status of the primary locus, which is under differential selection in males and females. In the first model, a modifier allele that imprints the primary locus invades the population when the average dominance coefficient among females and males is >12 and selection is weak. The condition for invasion is always heavily contingent upon the extent of dominance. Imprinting is more likely in the sex experiencing weaker selection only under some parameter regimes, whereas imprinting by either sex is equally likely under other regimes. The second model shows that a modifier allele that induces imprinting will increase when imprinting has a direct selective advantage. The results are not qualitatively dependent on whether the modifier locus is autosomal or X linked.     

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.     

Population genetics of modifiers of meiotic drive. I. The solution of a special case and some general implications

This is a study of the formal population genetics of a two locus model where the alleles at one locus are subject to meiotic drive and zygotic selection and the only effect of the other locus is the modification of drive intensity. A complete analytic solution is obtained for a biologically reasonable special case. It is then argued, partially with the aid of computer analysis, that with moderate relaxation of assumptions of the special case, the conclusions derived from that case still hold. These conclusions are that if there is linkage a stable two locus polymorphism can result. There is permanent linkage disequilibrium with the loosing allele at the drive locus in coupling with the suppressor allele at the modifier locus, and the driven allele coupled with the modifier allele which enhances drive. It is suggested that this result explains how the SD system in Drosophila maintains its integrity in natural populations.     

On the Evolution of Genetic Incompatibility Systems. VI. a Three-Locus Modifier Model for the Origin of Gametophytic Self-Incompatibility

Recent genetic analyses have demonstrated that self-incompatibility in flowering plants derives from the coordinated expression of a system of loci. To address the selective mechanisms through which a genetic system of this kind evolves, I present a three-locus model for the origin of gametophytic self-incompatibility. Conventional models assume that a single locus encodes all physiological effects associated with self-incompatibility and that the viability of offspring depends only on whether they were derived by selfing or outcrossing. My model explicitly represents the genetic determination of offspring viability by a locus subject to symmetrically overdominant selection. Initially, the level of expression of the proto-S locus is insufficient to induce self-incompatibility. Weak gametophytic self-incompatibility arises upon the introduction of a rare allele at an unlinked modifier locus which enhances the expression of the proto-S locus. While conventional models predict that the origin of self-incompatibility requires at least two- to threefold levels of inbreeding depression, I find that the comparatively low levels of inbreeding depression generated by a single overdominant locus can ensure the invasion of an enhancer of self-incompatibility under sufficiently high rates of receipt of self-pollen. Associations among components of the incompatibility system promote the origin of self-incompatibility. Enhancement of heterozygosity at the initially neutral proto-S locus improves offspring viability through associative overdominance. Further, the modifier that enhances the expression of self-incompatibility develops a direct association with heterozygosity at the overdominant viability locus. These results suggest that the evolutionary processes by which incompatibility systems originate may differ significantly from those associated with their breakdown. The genetic mechanism explored here may apply to the evolution of other systems that restrict reproduction, including maternal-fetal incompatibility in mammals.     

The Hill-Robertson effect and the evolution of recombination

In finite populations, genetic drift generates interference between selected loci, causing advantageous alleles to be found more often on different chromosomes than on the same chromosome, which reduces the rate of adaptation. This "Hill-Robertson effect" generates indirect selection to increase recombination rates. We present a new method to quantify the strength of this selection. Our model represents a new beneficial allele (A) entering a population as a single copy, while another beneficial allele (B) is sweeping at another locus. A third locus affects the recombination rate between selected loci. Using a branching process model, we calculate the probability distribution of the number of copies of A on the different genetic backgrounds, after it is established but while it is still rare. Then, we use a deterministic model to express the change in frequency of the recombination modifier, due to hitchhiking, as A goes to fixation. We show that this method can give good estimates of selection for recombination. Moreover, it shows that recombination is selected through two different effects: it increases the fixation probability of new alleles, and it accelerates selective sweeps. The relative importance of these two effects depends on the relative times of occurrence of the beneficial alleles.     

Evolution of dominance under frequency-dependent intraspecific competition

A population-genetic analysis is performed of a two-locus two-allele model, in which the primary locus has a major effect on a quantitative trait that is under frequency-dependent disruptive selection caused by intraspecific competition for a continuum of resources. The modifier locus determines the degree of dominance at the trait level. We establish the conditions when a modifier allele can invade and when it becomes fixed if sufficiently frequent. In general, these are not equivalent because an unstable internal equilibrium may exist and the condition for successful invasion of the modifier is more restrictive than that for eventual fixation from already high frequency. However, successful invasion implies global fixation, i.e., fixation from any initial condition. Modifiers of large effect can become fixed, and also invade, in a wider parameter range than modifiers of small effect. We also study modifiers with a direct, frequency-independent deleterious fitness effect. We show that they can invade if they induce a sufficiently high level of dominance and if disruptive selection on the ecological trait is strong enough. For deleterious modifiers, successful invasion no longer implies global fixation because they can become stuck at an intermediate frequency due to a stable internal equilibrium. Although the conditions for invasion and for fixation if sufficiently frequent are independent of the linkage relation between the two loci, the rate of spread depends strongly on it. The present study provides further support to the view that evolution of dominance may be an efficient mechanism to remove unfit heterozygotes that are maintained by balancing selection. It also demonstrates that an invasion analysis of mutants of very small effect is insufficient to obtain a full understanding of the evolutionary dynamics under frequency-dependent selection.     

Selection, Generalized Transmission and the Evolution of Modifier Genes. I. the Reduction Principle

Modifier gene models are used to explore the evolution of features of organisms, such as the genetic system, that are not directly involved in the determination of fitness. Recent work has shown that a general "reduction principle" holds in models of selectively neutral modifiers of recombination, mutation, and migration. Here we present a framework for models of modifier genes that shows these reduction results to be part of a more general theory, for which recombination and mutation are special cases. The deterministic forces that affect the genetic composition of a population can be partitioned into two categories: selection and transmission. Selection includes differential viabilities, fertilities, and mating success. Imperfect transmission occurs as a result of such phenomena as recombination, mutation and migration, meiosis, gene conversion, and meiotic drive. Selectively neutral modifier genes affect transmission, and a neutral modifier gene can evolve only by generating association with selected genes whose transmission it affects. We show that, in randomly mating populations at equilibrium, imperfect transmission of selected genes allows a variance in their marginal fitnesses to be maintained. This variance in the marginal fitnesses of selected genes is what drives the evolution of neutral modifier genes. Populations with a variance in marginal fitnesses at equilibrium are always subject to invasion by modifier genes that bring about perfect transmission of the selected genes. It is also found, within certain constraints, that for modifier genes producing what we call "linear variation" in the transmission processes, a new modifier allele can invade a population at equilibrium if it reduces the level of imperfect transmission acting on the selected genes, and will be expelled if it increases the level of imperfect transmission. Moreover, the strength of the induced selection on the modifier gene is shown to range up to the order of the departure of the genetic system from perfect transmission.     

On the evolution of dominance modifiers I. A nonlinear analysis

A well known mathematical model of evolution of dominance is subjected to a nonlinear analysis. For the case where the primary locus and the modifying locus are completely linked (r = 0) a global Ljapunov function is given. This proves that selection of dominance modifiers entirely due to their modifying effect is possible. This result is also extended to small recombination fractions r by using the method of Ljapunov functions in a more sophisticated way. For r = 0 and μ = 0 a lower bound for the success of selection of the modifier is given. Furthermore, the influence of the dominance relations between the alleles (measured by the parameters h and k) is investigated. Finally it is shown that differential and difference equations lead to the same results (which need not be the case in general). As a by-product we obtain a new equilibrium point in the classical one locus selection-mutation model.     

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.     

A model for the evolution and control of generative apomixis

A model is presented for the evolution and control of generative apomixis—a collective term for apomixis in animals and diplosporous apomixis in flowering plants. Its development takes into account data obtained from studies of apomictic-like processes in sexual organisms and in non-apomictic parthenogens, as well as data obtained from studies of generative apomicts. This approach provides insights into the evolution and control of generative apomixis that cannot be obtained from studies of generative apomicts alone. It is argued that the control of the avoidance of meiotic reduction during egg production in generative apomicts resides at a single locus, the identity of which can vary between lineages. This variation accounts for the observed variation between taxa in the pattern of avoidance of meiotic reduction. The affected locus contains a wild-type allele that codes for meiotic reduction and excess copies of a mutant allele that codes for its avoidance. The dominance relationship between these is determined by their ratio and by the environment. Environmental differences between female generative cells and somatic cells are such that the phenotypic expression of the mutant allele is favoured in the former, while that of the wild-type allele is favoured in the latter. This is important, for the locus is also involved in the control of mitosis which would be disrupted by the expression of the mutant allele in somatic cells. The requirement to maintain a viable pattern of growth and development explains why the wild-type allele is retained by generative apomicts, and this in turn explains why the ability to produce meiotically reduced eggs is retained by facultative forms and why it appears to be suppressed in, rather than absent from, obligate forms. The requirement for excess copies of the mutant allele in generative cells explains why generative apomicts are typically polyploid, as this condition provides a simple and effective means of generating the correct balance of mutant and wild-type alleles. Environmental effects can also lead to the dominance relationship between wild-type and mutant alleles varying between generative cells. In plants, this can lead to the apomixis gene being expressed, and thus to meiotic reduction being avoided, in only some ovules. Meiotically reduced, as well as meiotically unreduced, eggs are produced when this occurs. If compatible and viable pollen is available the meiotically reduced eggs may be fertilized, resulting in these organisms reproducing as facultative apomicts. It is argued that the control and evolution of parthenogenesis in generative apomicts varies between taxa. In some, the parthenogenetic initiation of embryos may result from the acquisition of a parthenogenesis gene or genes; but there is no reason to believe that this is either a general or a common requirement. Indeed, in some it may be an ancestral trait, these apomicts differing from their sexual ancestors in the ability to mature, rather than in the ability to initiate, embryos from unfertilized eggs; or it may result from physiological or developmental changes induced, for example, by polyploidization, hybridization, or the avoidance of meiotic reduction. In some plants it may be induced by pollination (without fertilization) or by the activity of a developing endosperm. Although it is argued that most generatively apomictic lineages may have acquired this form of reproduction relatively easily, by the acquisition of a mutation at a single locus, it is argued that newly initiated lineages may often be reproductively inefficient. These will begin to accumulate mutations that improve the efficiency of apomictic reproduction. Thus several loci may be involved in the control of generative apomixis in established lineages, even though only a single locus was involved in its initiation in these lineages. Care must be taken to distinguish between these initiator and modifier genes when considering the evolution of generative apomixis. Finally, it is argued that although generatively apomictic lineages have easily acquired this form of reproduction, its evolution in some taxa may be so difficult, requiring the acquisition of mutations simultaneously at two or more loci, that these may never acquire it. Thus, evidence obtained from taxa that have successfully made the transition from sexual reproduction to generative apomixis that its evolution was straightforward should not be used as evidence that its evolution will always be relatively easily achieved. Its uneven taxonomic distribution indicates that it is much more easily evolved by some taxonomic groups than by others.     

Coevolution of self-fertilization and inbreeding depression. II. Symmetric overdominance in viability

We describe the evolutionary dynamics of a modifier of selfing coevolving with a locus subject to symmetric overdominance in viability under general levels of reduction in pollination success as a consequence of self-fertilization (pollen discounting). Simple models of the evolution of breeding systems that represent inbreeding depression as a constant parameter do not admit the possibility of stable mixed mating systems involving both inbreeding and random mating. Contrary to this expectation, we find that coevolution between a modifier of selfing and a single overdominant locus situated anywhere in the genome can generate evolutionarily attracting mixed mating systems. Two forms of association between the modifier locus and the viability locus promote the evolution of outcrossing. The favored heterozygous genotype at the viability locus develops positive associations with modifier alleles that enhance outcrossing and with the heterozygous genotype at the modifier locus. Associations between outcrossing and high viability evolve immediately upon the introduction of a rare modifier allele, even in the absence of linkage.     

The Evolutionary Reduction Principle for Linear Variation in Genetic Transmission

The evolution of genetic systems has been analyzed through the use of modifier gene models, in which a neutral gene is posited to control the transmission of other genes under selection. Analysis of modifier gene models has found the manifestations of an “evolutionary reduction principle”: in a population near equilibrium, a new modifier allele that scales equally all transition probabilities between different genotypes under selection can invade if and only if it reduces the transition probabilities. Analytical results on the reduction principle have always required some set of constraints for tractability: limitations to one or two selected loci, two alleles per locus, specific selection regimes or weak selection, specific genetic processes being modified, extreme or infinitesimal effects of the modifier allele, or tight linkage between modifier and selected loci. Here, I prove the reduction principle in the absence of any of these constraints, confirming a twenty-year-old conjecture. The proof is obtained by a wider application of Karlin’s Theorem 5.2 (Karlin in Evolutionary biology, vol. 14, pp. 61–204, Plenum, New York, 1982) and its extension to ML-matrices, substochastic matrices, and reducible matrices. Dedicated to my doctoral advisor Marc Feldman on his 65th birthday, and to the memory of Marc’s doctoral advisor, Sam Karlin, who each laid the foundations necessary for these results; and to my mother Elizabeth Lee and to the memory of my father Roger Altenberg, who together laid the foundation necessary for me.