The Evolution of One- and Two-Locus Systems

Assuming age-independent fertilities and mortalities and random mating, continuous-time models for a monoecious population are investigated for weak selection. A single locus with multiple alleles and two alleles at each of two loci are considered. A slow-selection analysis of diallelic and multiallelic two-locus models with discrete nonoverlapping generations is also presented. The selective differences may be functions of genotypic frequencies, but their rate of change due to their explicit dependence on time (if any) must be at most of the second order in s, (i.e., O( s²)), where s is the intensity of natural selection. Then, after several generations have elapsed, in the continuous time models the time-derivative of the deviations from Hardy-Weinberg proportions is of O(s²), and in the two-locus models the rate of change of the linkage disequilibrium is of O(s²). It follows that, if the rate of change of the genotypic fitnesses is smaller than second order in s (i.e., o(s²)), then to O(s²) the rate of change of the mean fitness of the population is equal to the genic variance. For a fixed value of s, however, no matter how small, the genic variance may occasionally be smaller in absolute value than the (possibly negative) lower order terms in the change in fitness, and hence the mean fitness may decrease. This happens if the allelic frequencies are changing extremely slowly, and hence occurs often very close to equilibrium. Some new expressions are derived for the change in mean fitness. It is shown that, with an error of O( s), the genotypic frequencies evolve as if the population were in Hardy-Weinberg proportions and linkage equilibrium. Thus, at least for the deterministic behavior of one and two loci, deviations from random combination appear to have very little evolutionary significance.     

Colony-level selection in the social insects: Single locus additive and nonadditive models

Genetic models of colony-level selection applicable to diploids (termites) and haplodiploids (social Hymenoptera) are analysed. In the Additive model colony fitnesses are just the arithmetic average of the contribution of the worker genotypes. In the Nonadditive model the fitness of the heterogenotypic colonies (those comprised of more than one worker genotype) may be altered due to interaction between the different worker genotypes. This is modelled by multiplying the additive fitness by the variable, e_i. With additive selection the same equilibrium gene frequency occurs in diploids and in haplodiploids with both once and twice mated queens. In haplodiploids if selection is nonadditive and strong, up to three polymorphic equilibria can exist; however, only a maximum of two are possible with weak selection. Multiple mating by queens increases the number of equilibria possible. Worker-produced males alter the conditions for the existence of a polymorphic equilibrium, and shift the male and female equilibrium gene frequencies.     

Male-female interactions in Drosophila melanogaster: model with one-locus and three-alleles

We develop a fertility model of fitness that is general in that it does not assume that the fitnesses of the mating combinations are symmetrical or that they are additive or multilicative (i. e., that they can be inferred from fitnesses of the two genotypes involved in a mating). %he model considers one locus with three alleles. An experimental test with Drosophila rnelanogaster confirms that the fitnesses of the mating types depart from both additivity (or multiplicativity) and symmetry although this last property is of no consequence for the development of analytical models). urnerical simulations yield the same, or very nearly the same, equilibrium freuencies as the analytical model, independently of whether or not Hardy-Weinberg equilibrium Trequencies are assumed at the beginning of each selection cycle.     

Natural selection under population regulation

The dynamical behavior of multi-allele, one-locus systems is analyzed under population regulation. Weak selection is assumed. It is shown that for sufficiently large times, t, the nth time derivative of the population number N(t) is of order n}+1 in the selection coefficients. These order relations imply there is an asymptotic “quasi-equilibrium” in which population size and mean fitness change slowly relative to changes in gene frequencies. Consistent with the results of other authors, in quasi-equilibrium the mean fitness is second-order in the selection coefficients. In an effort to understand dynamic behavior beyond the immediate neighborhood of equilibrium, the topology of mean fitness surfaces is explored. In general, population regulation leads to regions of decreasing mean fitness in which there are important changes in gene frequencies. To illustrate this and other related phenomena, I analyze models in which there is logarithmic population control, and in which genotypic fitnesses W_i(x) are linear in the allele frequencies x. Exact solutions for mean fitness W(x) are obtained for two- and three-allele systems with symmetric fertilities and mortalities.     

Towards a theory of the evolution of modifier genes

The main findings of a study of the evolution of modifier gene frequencies in models of deterministic population genetics are presented. A wide variety of random mating systems are subject to selection with modifiers operating, in different cases, on mutation rates, migration between subpopulations, and linkage between other loci. In all these instances, the modifier frequencies evolve in such a way as to maximize the mean fitness of the population at equilibrium. This is remarkable since, the modifier genes are selectively neutral in the sense that they do not affect the fitness of their individual carriers. In nonrandom mating systems, the mean fitness concept is not well-defined, and there does not appear to be such a simple principle governing the evolution of modifier frequencies. In assortative mating systems, modifiers favoring reduced assortment propensities tend to increase. In contrast, for selfing-outcrossing systems, modifiers favoring increased selfing tend to increase.     

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.     

Nuclear androdioecy and gynodioecy

We formulate two single-locus Mendelian models, one for androdioecy and the other one for gynodioecy, each with 3 parameters: t the male (female) fertility rate of males (females) to hermaphrodites, s the fraction of the progeny derived from selfing; and g the fitness of inbreeders. Each model is expressed as a transformation of a 3 dimensional zygotic algebra, which we interpret as a rational map of the projective plane. We then study the dynamics for the evolution of each reproductive system; and compare our results with similar published models. In this process, we introduce a general concept of fitness and list some of its properties, obtaining a relative measure of population growth, computable as an eigenvalue of a mixed mating transformation for a population in equilibrium. Our results concur with previous models of the evolution of androdioecy and gynodioecy regarding the threshold values above which the sexual polymophism is stable, although the previous models assume constant the fraction of ovules from hermaphrodites that are self pollinated, while we assume constant the fraction of the progeny derived from selfing. A stable androdioecy requires more stringent conditions than a stable gynodioecy if the amount of pollen used for selfing is negligible in comparison with the total amount of pollen produced by hermaphrodites. Otherwise, both models are identical. We show explicitly that the genotype fitnesses depend linearly on their frequencies. Simulations show that any population not at equilibrium always converges to the equilibrium point of higher fitness. However, at intermediate steps, the fitness function occasionally decreases.     

Development of SCAR Markers to Determine the Mating Types of Lepista nuda Protoplast Monokaryons

Lepista nuda (Bull. ex Fr.) Cooke belongs to Tricholomataceae and is an edible fungus with both economic and medical value. Mycelia were isolated from the fruiting bodies of L. nuda and were used to prepare the protoplast monokaryons. One hundred and fifteen monokaryons were obtained and their mating types were determined using somatic incompatibility tests. Protoplast monokaryons segregated into either the A₁B₁ or the A₂B₂ mating types. Inter-simple sequence repeats and sequence-related amplified polymorphism fingerprinting were used to analyse the mating types of these protoplast monokaryons and 16 sequence-characterised amplified region primers were developed to efficiently differentiate between the monokaryon mating types. Multiplex PCR analyses were also established. The data presented here outline a method for the precise and rapid identification of protoplast monokaryon mating types, which has the promise to shorten the period required for conventional crossbreeding.     

An application of kinship process to the gene frequencies: linkage disequilibrium due to random drift in mendelian genetics with reversible mutation and in molecular genetics.

Selection at the colony level in social Hymenoptera with colonies containing single, once-mated queens is examined under a simple two-allele model. The condition for balanced polymorphism is $\text{X>}\text{2V}{}^2\text{(V + 1)}$, where V is the fitness of colonies with all workers homozygous and X that of colonies with both heterozygous and homozygous workers, relative to the fitness of colonies with all workers heterozygous. For certain fitness combinations satisfying the above relationship and characterized by values of V and X much lower than one, iteration reveals the development of stable limit cycles of allele frequencies rather than convergence to an equilibrium point. Addition of a third allele, or overlap between generations, eliminates these cycles. Queen-level overdominance is sufficient but not necessary for balanced polymorphism when V < 1, is both sufficient and necessary when V = 1, and is necessary but not sufficient when V > 1. Colony-level selection is a potentially powerful force maintaining genetic variation in populations of social insects, but does not imply correspondence between queen and worker genotype frequencies.     

How multiple mating by females affects sexual selection

Multiple mating by females is widely thought to encourage post-mating sexual selection and enhance female fitness. We show that whether polyandrous mating has these effects depends on two conditions. Condition 1 is the pattern of sperm utilization by females; specifically, whether, among females, male mating number, m (i.e. the number of times a male mates with one or more females) covaries with male offspring number, o. Polyandrous mating enhances sexual selection only when males who are successful at multiple mating also sire most or all of each of their mates'' offspring, i.e. only when Cov_♂(m,o), is positive. Condition 2 is the pattern of female reproductive life-history; specifically, whether female mating number, m, covaries with female offspring number, o. Only semelparity does not erode sexual selection, whereas iteroparity (i.e. when Cov_♀(m,o), is positive) always increases the variance in offspring numbers among females, which always decreases the intensity of sexual selection on males. To document the covariance between mating number and offspring number for each sex, it is necessary to assign progeny to all parents, as well as identify mating and non-mating individuals. To document significant fitness gains by females through iteroparity, it is necessary to determine the relative magnitudes of male as well as female contributions to the total variance in relative fitness. We show how such data can be collected, how often they are collected, and we explain the circumstances in which selection favouring multiple mating by females can be strong or weak.     

Fitness of Allozyme Variants in DROSOPHILA PSEUDOOBSCURA. I. Selection at the PGM-1 and ME-2 Loci

We have studied in Drosophila pseudoobscura the effect of allozyme variation on seven fitness components: female fecundity, egg hatchability, egg-to-adult survival under near-optimal and under competitive conditions, rate of development under near-optimal and under competitive conditions, and mating capacity of males. Three genotypes at each of two loci, Pgm-1 and Me-2, have been studied in various combinations. These two loci are highly polymorphic in natural populations of D. pseudoobscura. Statistically significant differences involving one or more genotypes exist for all components of fitness. No single genotype is best for all fitness components; rather the relative fitnesses of genotypes are reversed when different parameters are considered, or when they are studied in different environmental conditions. Also, the average egg-to-adult survival and rate of development are better when different genotypes are reared together than when they occur in pure culture. Four different modes of selection have been uncovered by our experiments. These forms of selection may account for the persistence of the two allozyme polymorphisms in nature, and for previously observed seasonal fluctuations of the allelic frequencies in natural populations.     

Mate choice for non-additive genetic benefits: a resolution to the lek paradox

In promiscuous mating systems, females often show a consistent preference to mate with one or a few males, presumably to acquire heritable genetic benefits for their offspring. However, strong directional selection should deplete additive genetic variation in fitness and consequently any benefit to expressing the preference by females (referred to as the lek paradox). Here, we provide a novel resolution that examines non-additive genetic benefits, such as overdominance or inbreeding, as a source of genetic variation. Focusing on the inbreeding coefficient f and overdominance effects, we use dynamic models to show that (1) f can be inherited from sire to offspring, (2) populations with females that express a mating preferences for outbred males (low f) maintain higher genetic variation than populations with females that mate randomly, and (3) preference alleles for outbred males can invade populations even when the alleles are associated with a fecundity cost. We show that non-additive genetic variation due to overdominance can be converted to additive genetic variation and becomes “heritable” when the frequencies of alternative homozygous genotypes at fitness loci deviate from equality. Unlike previous models that assume an infinite population size, we now show that genetic drift in finite populations can lead to the necessary deviations in the frequencies of homozygous genotypes. We also show that the “heritability of f,” and hence the benefit to a mating preference for non-additive genetic benefits, is highest in small populations and populations in which a smaller number of loci contribute to fitness via overdominance. Our model contributes to the solution of the lek paradox.     

Wolbachia-Host Interactions: Host Mating Patterns Affect Wolbachia Density Dynamics

Wolbachia are maternally inherited intracellular bacteria that infect a wide range of arthropods and cause an array of effects on host reproduction, fitness and mating behavior. Although our understanding of the Wolbachia-associated effects on hosts is rapidly expanding, our knowledge of the host factors that mediate Wolbachia dynamics is rudimentary. Here, we explore the interactions between Wolbachia and its host, the two-spotted spider mite Tetranychus urticae Koch. Our results indicate that Wolbachia induces strong cytoplasmic incompatibility (CI), increases host fecundity, but has no effects on the longevity of females and the mating competitiveness of males in T. urticae. Most importantly, host mating pattern was found to affect Wolbachia density dynamics during host aging. Mating of an uninfected mite of either sex with an infected mite attenuates the Wolbachia density in the infected mite. According to the results of Wolbachia localization, this finding may be associated with the tropism of Wolbachia for the reproductive tissue in adult spider mites. Our findings describe a new interaction between Wolbachia and their hosts.     

Inbreeding and incompatibility in Trichogramma nr. brassicae: evidence and implications for quality control

Inbreeding effects and incompatibility relationships were examined in strains of the egg parasitoid Trichogramma nr brassicae (Hymenoptera: Trichogrammatidae) from southeastern Australia. Crosses between strains provided weak evidence of incompatibility in a few cases. However sex ratio in crosses within strains tended to be more female-biased than in crosses between strains. Inbreeding was imposed for four generations (F>0.59) of sib mating. The fitness of inbred strains was compared to that of outbred strains generated by crossing the inbred strains. No effects of inbreeding were found for any of the four female traits examined (fecundity, body length, head width and hind tibia length), indicating that T. nr. brassicae is not subjected to inbreeding depression. Inbreeding effects were also not found for male mating success as expected for the haploid sex. There were differences among strains for all traits apart from fecundity, indicating heritable variation. Strain differences for fitness measures were uncorrelated with wasp size. The potential use of inbreeding in the quality control of Trichogramma for mass-release is discussed. Inbreeding may be a useful tool in minimising the effects of laboratory adaptation, thereby extending the useful life of a strain.     

Trait fitness is not a propensity,but fitness variation is

The propensity interpretation of fitness draws on the propensity interpretation of probability, but advocates of the former have not attended sufficiently to problems with the latter. The causal power of C to bring about E is not well-represented by the conditional probability Pr(E|C). Since the viability fitness of trait T is the conditional probability Pr(organism O survives to adulthood|O has T), the viability fitness of the trait does not represent the degree to which having the trait causally promotes surviving. The same point holds for fertility fitness. This failure of trait fitness to capture causal role can also be seen in the fact that coextensive traits must have the same fitness values even if one of them promotes survival and the other is neutral or deleterious. Although the fitness of a trait does not represent the trait’s causal power to promote survival and reproduction, variation in fitness in a population causally promotes change in trait frequencies; in this sense, fitness variation is a population-level propensity.     

Asymmetric Assortative Mating Behaviour Reflects Incomplete Pre‐zygotic Isolation in the Nasonia Species Complex

Preference of con‐ over heterospecific mates leading to assortative mating can substantially contribute to pre‐zygotic reproductive isolation and prevent fitness losses if post‐zygotic hybridization barriers already exist. The jewel wasp genus Nasonia displays quite strong and well‐studied post‐zygotic reproductive isolation due to a ubiquitous Wolbachia infection causing cytoplasmic incompatibility between different species. Pre‐zygotic isolation, however, has received far less research attention in this model organism, especially concerning the mechanisms and criteria of mate choice. In the present study, we analysed mate rejection and mate acceptance rates in cross‐comparisons between all four Nasonia species. We put emphasis on observing which sex is more likely to interrupt interspecific matings and how discriminatory behaviour varies across the different species in all possible combinations. We found an asymmetric distribution of assortative mating among the four Nasonia species that appears to be highly influenced by the respective combinations of sex and species. Females appeared to be the main discriminators against heterospecific mating partners, but interestingly, we could also detect mate discrimination and rejection behaviour in males, a widely neglected factor in research on mating behaviour in general and on Nasonia in particular. Moreover, the asymmetry in the assortative mating behaviour was partially reflective of sym‐ or allopatric distributions of natural Nasonia populations.     

Incest versus abstinence: reproductive trade‐offs between mate limitation and progeny fitness in a self‐incompatible invasive plant

Plant mating systems represent an evolutionary and ecological trade‐off between reproductive assurance through selfing and maximizing progeny fitness through outbreeding. However, many plants with sporophytic self‐incompatibility systems exhibit dominance interactions at the S‐locus that allow biparental inbreeding, thereby facilitating mating between individuals that share alleles at the S‐locus. We investigated this trade‐off by estimating mate availability and biparental inbreeding depression in wild radish from five different populations across Australia. We found dominance interactions among S‐alleles increased mate availability relative to estimates based on individuals that did not share S‐alleles. Twelve of the sixteen fitness variables were significantly reduced by inbreeding. For all the three life‐history phases evaluated, self‐fertilized offspring suffered a greater than 50% reduction in fitness, while full‐sib and half‐sib offspring suffered a less than 50% reduction in fitness. Theory indicates that fitness costs greater than 50% can result in an evolutionary trajectory toward a stable state of self‐incompatibility (SI). This study suggests that dominance interactions at the S‐locus provide a possible third stable state between SI and SC where biparental inbreeding increases mate availability with relatively minor fitness costs. This strategy allows weeds to establish in new environments while maintaining a functional SI system.     

Frequency-Dependent Selection at the LAP Locus in DROSOPHILA MELANOGASTER

Results of fitness estimates for the Lap locus in Drosophila melanogaster revealed that under crowded media conditions gene frequency equilibrium was maintained by frequency-dependent selection. Evidence was obtained that indicated that mating and egg-to-adult viability were frequency dependent.     

Emergence and loss of assortative mating in sympatric speciation

We have studied an agent model which presents the emergence of sexual barriers through the onset of assortative mating, a condition that might lead to sympatric speciation. In the model, individuals are characterized by two traits, each determined by a single locus A or B. Heterozygotes on A are penalized by introducing an adaptive difference from homozygotes. Two niches are available. Each A homozygote is adapted to one of the niches. The second trait, called the marker trait has no bearing on the fitness. The model includes mating preferences, which are inherited from the mother and subject to random variations. A parameter controlling recombination probabilities of the two loci is also introduced. We study the phase diagram by means of simulations, in the space of parameters (adaptive difference, carrying capacity, recombination probability). Three phases are found, characterized by (i) assortative mating, (ii) extinction of one of the A alleles and (iii) Hardy-Weinberg like equilibrium. We also make perturbations of these phases to see how robust they are. Assortative mating can be gained or lost with changes that present hysteresis loops, showing the resulting equilibrium to have partial memory of the initial state and that the process of going from a polymorphic panmictic phase to a phase where assortative mating acts as sexual barrier can be described as a first-order transition.     

The effects of unidirectional incompatibility on cytonuclear disequilibria in a hybrid zone

Unidirectional incompatibility selection is examined as an alternate mechanism of natural selection to cytoplasmic male sterility (CMS) for generating cytonuclear disequilibria. Differences in the dynamics and equilibrium behavior of cytonuclear disequilibria between these two cytonuclear selection models may allow for statistical tests of CMS vs. unidirectional incompatibility between mating cytotypes. Unlike CMS without migration, unidirectional incompatibility causes the cytoplasmic allele frequency to change over time rather than remain constant, and the nuclear allele frequencies hitchhike on the cytoplasmic frequencies. The decay of disequilibria is also distinctive in the absence of migration. Furthermore, in comparing both models with migration it is seen that the opportunity for internal equilibrium can be two or three times higher in a unidirectional incompatibility vs. CMS model. An example is presented that shows how unidirectional incompatibility can be statistically eliminated as a possible mechanism of cytonuclear selection. This revised version was published online in July 2006 with corrections to the Cover Date.