Density-dependent selection incorporating intraspecific competition. II. A diploid model.

A diploid model is introduced and analyzed in which intraspecific competition is incorporated within the context of density-regulated selection. It is assumed that each genotype has a unique carrying capacity corresponding to the equilibrium population size when only that type is present. Each genotypic fitness at a single diallelic autosomal locus is a decreasing function of a distinctive effective population size perceived as a result of intraspecific competition. The resulting fitnesses are both density and frequency dependent with selective advantage determined by a balance between genotypic carrying capacity and sensitivity to intraspecific competition. A major finding is that intergenotypic interactions may allow genetic variation to be more easily maintained than in the corresponding model of purely density-dependent selection. In addition, numerical study confirms the possible existence of multiple interior equilibria and that neither overdominance in fitness nor carrying capacity is necessary for stability. The magnitude of the equilibrium population size and optimization principles are also discussed.

Frequency-dependent selection: the high potential for permanent genetic variation in the diallelic, pairwise interaction model.

A detailed analytic and numerical study is made of the potential for permanent genetic variation in frequency-dependent models based on pairwise interactions among genotypes at a single diallelic locus. The full equilibrium structure and qualitative gene-frequency dynamics are derived analytically for a symmetric model, in which pairwise fitnesses are chiefly determined by the genetic similarity of the individuals involved. This is supplemented by an extensive numerical investigation of the general model, the symmetric model, and nine other special cases. Together the results show that there is a high potential for permanent genetic diversity in the pairwise interaction model, and provide insight into the extent to which various forms of genotypic interactions enhance or reduce this potential. Technically, although two stable polymorphic equilibria are possible, the increased likelihood of maintaining both alleles, and the poor performance of protected polymorphism conditions as a measure of this likelihood, are primarily due to a greater variety and frequency of equilibrium patterns with one stable polymorphic equilibrium, in conjunction with a disproportionately large domain of attraction for stable internal equilibria.

Sampling theory for cytonuclear disequilibria.

We examine the statistical properties of cytonuclear disequilibria within a system including one diploid nuclear locus and one haploid cytoplasmic locus, each with two alleles. The results provide practical guidelines for the design and interpretation of cytonuclear surveys seeking to utilize the novel evolutionary information recorded in the observed pattern of cytonuclear associations. Important applications include population studies of nuclear allozymes in conjunction with genes from mitochondria, chloroplasts, or cytoplasmically inherited microorganisms. Our attention focuses on the allelic and genotypic disequilibria, which respectively measure the nonrandom associations between the cytotypes and the nuclear alleles and genotypes. We first derive the maximum likelihood estimators and their approximate large sample variances for each disequilibrium measure. These are each in turn used to set up an asymptotic test of the null hypothesis of no disequilibrium. We then calculate the minimum sample sizes required to detect the disequilibria under specified alternate hypotheses. The work also incorporates the deviation from Hardy-Weinberg equilibrium at the nuclear locus, which can significantly affect the results. The practical utility of this new sampling theory is illustrated through applications to two nuclear-mitochondrial data sets.

Definition and properties of disequilibria within nuclear-mitochondrial-chloroplast and other nuclear-dicytoplasmic systems.

We define and determine the interrelationships among five sets of disequilibrium parameters that measure two- and three-locus nonrandom associations in nuclear-dicytoplasmic systems. These assume a diploid nuclear locus and two haploid cytoplasmic loci, with special reference to nuclear-mitochondrial-chloroplast systems. Three sets of two-locus disequilibria measure the association between haplotypes at the two cytoplasmic loci (DMC) and associations between each cytoplasmic locus and nuclear alleles or genotypes (DM, D1M, D2M, D3M; DC, D1C, D2C, D3C). In addition, we present two classes of higher-order disequilibria that measure nonrandom allelic or genotypic associations involving all three loci. The first class quantifies associations between the nuclear locus and the two cytoplasmic loci taken jointly (DA/MC, DAA/MC, DAa/MC, Daa/MC, etc.), whereas the second measures only those associations remaining after all two-locus associations have been taken into account (DA/M/C, DAA/M/C, DAa/M/C, Daa/M/C). Based on combinations of these five sets of measures, we suggest a variety of parameterizations of three-locus, nuclear-dicytoplasmic systems. The dynamics of these disequilibria are then investigated under models of random and mixed mating, either with both cytoplasmic genomes inherited through the same parent or through opposite parents. Except for associations between the cytoplasmic haplotypes, which are constant when the two cytoplasmic genomes are inherited through the same parent, all disequilibria ultimately decay to zero. These randomizations do not necessarily occur monotonically, however, and in some cases are preceded by an initial increase in magnitude or sign change. For both inheritance patterns, the asymptotic decay rates are steadily retarded by increasing levels of self-fertilization. This behavior contrasts with that in the extreme case of complete selfing, for which only the heterozygote disequilibria always decay to zero. For all models considered, the dynamics of the two-locus cytonuclear subsystems are solely a function of the mating system, whereas the dynamical behavior and sign patterns of the cytoplasmic and three-locus disequilibria also depend strongly on the mode of cytoplasmic inheritance.

Comparative effects of pollen and seed migration on the cytonuclear structure of plant populations. I. Maternal cytoplasmic inheritance.

We explicitly solve and analyze a series of deterministic continent-island models to delimit the effects of pollen and seed migration on cytonuclear frequencies and disequilibria in random-mating, mixed-mating and self-fertilized populations. Given the critical assumption of maternal cytoplasmic inheritance, five major findings are (i) nonzero cytonuclear disequilibria will be maintained in the island population if and only if at least some migration occurs each generation through seeds with nonrandom cytonuclear associations; (ii) immigrant seeds with no cytonuclear disequilibria can strongly affect the genetic structure of the island population by generating significant and long-lasting transient associations; (iii) with all else being equal, substantially greater admixture disequilibria are generally found with higher rates of seed migration into, or higher levels of self-fertilization within, the island population (with the possible exception of the heterozygote disequilibrium); (iv) pollen migration can either enhance or reduce the cytonuclear disequilibria caused by seed migration, or that due to mixed-mating in the absence of seed migration, but the effect is usually small and appears primarily to make a noticeable difference in predominantly outcrossing populations; and (v) pollen migration alone cannot generate even transient disequilibria de novo in populations with completely random associations. This same basic behavior is exhibited as long as there is some random outcrossing in the island population. Self-fertilized populations represent a special case, however, in that they are necessarily closed to pollen migration, and nonzero disequilibria can be maintained even in the absence of seed migration. All of these general results hold whether the population is censused as adults or as seeds, but the ability to detect nonrandom cytonuclear associations can depend strongly on the life stage censused in populations with a significant level of random outcrossing. We suggest how these models might be used for the estimation of seed and pollen migration.

Comparative effects of pollen and seed migration on the cytonuclear structure of plant populations. II. Paternal cytoplasmic inheritance.

We continue our study of the effects of pollen and seed migration on the cytonuclear structure of mixed-mating plant populations by analyzing two deterministic continent-island models under the critical assumption of paternal cytoplasmic inheritance. The major results of this study that contrast with our previous conclusions based on maternal cytoplasmic inheritance are (i) pollen gene flow can significantly affect the cytonuclear structure of the island population, and in particular can help to generate cytonuclear disequilibria that greatly exceed the magnitude of those that would be produced by seed migration or mixed mating alone; (ii) with simultaneous pollen and seed migration, nonzero cytonuclear disequilibria will be maintained not only when there is disequilibrium in the immigrant pollen or seeds, but also through a variety of intermigrant admixture effects when the two pools of immigrants differ appropriately in their cytonuclear compositions; (iii) either immigrant pollen or immigrant seeds can generate disequilibria de novo in populations with initially random cytonuclear associations, but pollen migration alone generally produces lower levels of disequilibrium than does comparable seed migration, especially at high levels of self-fertilization when the overall fraction of immigrant pollen is low; (iv) the equilibrium state of the island population will be influenced by the rate of pollen gene flow whenever there is either allelic disequilibrium in the immigrant pollen or simultaneous seed migration coupled with different cytoplasmic or nuclear allele frequencies in immigrant pollen and seeds or nonzero allelic disequilibrium in either immigrant pool. The estimation of pollen migration should therefore be facilitated with paternal cytoplasmic inheritance relative to the case of maternal cytoplasmic inheritance. These basic conclusions hold whether the population is censused as seeds or as adults, but with simultaneous pollen and seed migration, the relationship between census time and the ability to detect nonrandom cytonuclear associations is complex. When migration is through pollen alone, however, the cytonuclear structure of the island population is independent of the life stage censused.

The exact test for cytonuclear disequilibria.

We extend the analysis of the statistical properties of cytonuclear disequilibria in two major ways. First, we develop the asymptotic sampling theory for the nonrandom associations between the alleles at a haploid cytoplasmic locus and the alleles and genotypes at a diploid nuclear locus, when there are an arbitrary number of alleles at each marker. This includes the derivation of the maximum likelihood estimators and their sampling variances for each disequilibrium measure, together with simple tests of the null hypothesis of no disequilibrium. In addition to these new asymptotic tests, we provide the first implementation of Fisher's exact test for the genotypic cytonuclear disequilibria and some approximations of the exact test. We also outline an exact test for allelic cytonuclear disequilibria in multiallelic systems. An exact test should be used for data sets when either the marginal frequencies are extreme or the sample size is small. The utility of this new sampling theory is illustrated through applications to recent nuclear-mtDNA and nuclear-cpDNA data sets. The results also apply to population surveys of nuclear loci in conjunction with markers in cytoplasmically inherited microorganisms.

Effects of differential selection in the sexes on cytonuclear polymorphism and disequilibria.

We develop a series of models that examine the effects of differential selection between the sexes on cytonuclear polymorphism and disequilibria. A detailed analysis is provided for populations under constant fertility or viability selection censused at life stages without frequency differences in the sexes. We show analytically that cytonuclear disequilibria can be generated de novo if the cytoplasmic and nuclear loci each affect female fitness and there is no nonmultiplicative fitness interaction between them. While computer simulations demonstrate that the majority of disequilibria produced by random selection are transient and small in magnitude, measurable permanent disequilibria can result from selective differences both within and between the two sexes. We derive analytic conditions for a protected cytonuclear polymorphism and use numerical simulations to quantitate the likelihood of obtaining permanent nuclear, cytoplasmic, and cytonuclear variation under various patterns of selection. The numerical analysis identifies special selection regimes more likely to generate disequilibria and maintain cytonuclear polymorphism and reveals a direct correlation to the strength of selection. As a byproduct, our models also provide the first decomposition of the different parental contributions to cytonuclear dynamics and the analytic conditions under which selection can cause cytoplasmic frequency changes or a cytonuclear hitchhiking effect.

Cytonuclear theory for haplodiploid species and X-linked genes. I. Hardy-Weinberg dynamics and continent-island, hybrid zone models.

We develop models that describe the cytonuclear structure for either a cytoplasmic and nuclear marker in a haplodiploid species or a cytoplasmic and X-linked marker in a diploid species. Sex-specific disequilibrium statistics that summarize nonrandom cytonuclear associations in such systems are defined, and their basic Hardy-Weinberg dynamics and admixture formulae are delimited. We focus on the context of hybrid zones and develop continent-island models whereby individuals from two genetically differentiated source populations migrate into and mate within a single zone of admixture. We examine the effects of differential migration of the sexes, assortative mating by pure type females, and census time (relative to mating and migration), as well as special cases of random mating and migration subsumed under the general models. We show that pure type individuals and nonzero cytonuclear disequilibria can be maintained within a hybrid zone if there is continued migration from both source populations, and that females generally have a greater influence over these cytonuclear variables than males. The resulting theoretical framework can be used to estimate the rates of assortative mating and sex-specific gene flow in hybrid zones and other zones of admixture involving haplodiploid or sex-linked cytonuclear data.

The effects of pollen and seed migration on nuclear-dicytoplasmic systems. I. Nonrandom associations and equilibrium structure with both maternal and paternal cytoplasmic inheritance.

We determine the nuclear-dicytoplasmic effects of unidirectional gene flow via pollen and seeds upon a mixed-mating plant population, focusing on nuclear-mitochondrial-chloroplast systems where mitochondria are inherited maternally and chloroplasts paternally, as in many conifers. After first delineating the general effects of admixture (via seeds or individuals) on the nonrandom associations in such systems, we derive the full dicytonuclear equilibrium structure, including when disequilibria may be indicators of gene flow. Substantial levels of permanent two- and three-locus disequilibria can be generated in adults by (i) nonzero disequilibria in the migrant pools or (ii) intermigrant admixture effects via different chloroplast frequencies in migrant pollen and seeds. Additionally, three-locus disequilibria can be generated by higher-order intermigrant effects such as different chloroplast frequencies in migrant pollen and seeds coupled with nuclear-mitochondrial disequilibria in migrant seeds, or different nuclear frequencies in migrant pollen and seeds coupled with mitochondrial-chloroplast disequilibria in migrant seeds. Further insight is provided by considering special cases with seed or pollen migration alone, complete random mating or selfing, or migrant pollen and seeds lacking disequilibria or intermigrant admixture effects. The results complete the theoretical foundation for a new method for estimating pollen and seed migration using joint cytonuclear or dicytonuclear data.

The effects of pollen and seed migration on nuclear-dicytoplasmic systems. II. A new method for estimating plant gene flow from joint nuclear-cytoplasmic data.

A new maximum-likelihood method is developed for estimating unidirectional pollen and seed flow in mixed-mating plant populations from counts of joint nuclear-cytoplasmic genotypes. Data may include multiple unlinked nuclear markers with a single maternally or paternally inherited cytoplasmic marker, or with two cytoplasmic markers inherited through opposite parents, as in many conifer species. Migration rate estimates are based on fitting the equilibrium genotype frequencies under continent-island models of plant gene flow to the data. Detailed analysis of their equilibrium structures indicates when each of the three nuclear-cytoplasmic systems allows gene flow estimation and shows that, in general, it is easier to estimate seed than pollen migration. Three-locus nuclear-dicytoplasmic data only increase the conditions allowing seed migration estimates; however, the additional dicytonuclear disequilibria allow more accurate estimates of both forms of gene flow. Estimates and their confidence limits for simulated data sets confirm that two-locus data with paternal cytoplasmic inheritance provide better estimates than those with maternal inheritance, while three-locus dicytonuclear data with three modes of inheritance generally provide the most reliable estimates for both types of gene flow. Similar results are obtained for hybrid zones receiving pollen and seed flow from two source populations. An estimation program is available upon request.

Effects of differential selection in the sexes on cytonuclear dynamics. Life stages with sex differences.

We extend our investigation of cytonuclear selection by determining when differential selection between the sexes will generate allele frequency changes or cytonuclear disequilibria in populations with constant viability selection and an adult census. We demonstrate analytically that there can be a cytonuclear hitchhiking effect upon a selectively neutral marker in either sex provided the other marker is selected in that sex and there is allelic disequilibrium between the loci in females. Cytonuclear disequilibria are generated de novo in both sexes when both loci affect fitness in females and there is a nonmultiplicative fitness interaction between them. Similar fitness interactions in males generate male disequilibria only. Through numerical analyses, we investigate the potential magnitude of such disequilibria, their qualitative dynamics, the expected frequency of detectable disequilibria under particular patterns or strengths of selection, and the possible disequilibrium sign patterns resulting from selection. These adult/viability results subsume those for populations with a gamete census and either constant fertility or viability selection. Although previous work suggests that the disequilibria generated by cytonuclear selection may be difficult to detect experimentally, this study shows that cytonuclear disequilibria at life stages with sex differences can be useful markers of the presence and strength of selection.

Dynamics of the Linkage Disequilibrium Function under Models of Gene-Frequency Hitchhiking.

The dynamic behavior of the linkage disequilibrium (D) between a neutral and a selected locus is analyzed for a variety of deterministic selection models. The time-dependent behavior of D is governed by the gene frequency at the selected locus (p) and by the selection (s) and recombination (r) parameters. Thomson (1977) showed numerically that D may increase under certain initial conditions. We give exact conditions for D to increase in time, which require that the selection intensity exceed the recombination fraction (s > r) and that p be near zero or one. We conclude from this result that gene frequency hitchhiking is most likely to be important when a new favorable mutant enters a population. We also show that, for what can be a wide range of gene frequencies, D will decay at a faster rate than the neutral rate. Consequently, the hitchhiking effect may quickly diminish as the selected gene becomes more common.-The method of analysis allows a complete qualitative description of the dynamics of D as a function of s and r. Two major findings concern the range of gene frequencies at the selected locus for which D either increases over time or decays at a faster rate than under neutrality. For all models considered, the region where D increases (i) first enlarges then shrinks as selection intensifies, and (ii) steadily shrinks as r increases. In contrast, the region of accelerated decay constantly enlarges as the selection intensity increases. This region will either shrink or enlarge as r increases, depending upon the form of selection in force.

Estimation of mitotic stability in conidial fungi: a theoretical framework.

Mitotic stability refers to the probability that genetic elements are transmitted to both daughters during mitosis. This is of practical importance in molecular genetics because autonomous cloning vectors should be transmitted at high frequency during mitosis. In filamentous coencytic fungi it is difficult to quantify mitotic stability because a fluctuation test is not feasible. We show how to get around this problem by formulating a general model of the transmission of nuclear genetic elements through the course of conidiogenesis. We derive formulas by two different methods for the expected proportion of conidiospores that retain the element as a function of its mitotic stability and the number of generations of spore production. An important by-product yields the exact probability distributions for the number of conidiospores retaining elements at each stage of conidiophore development. We outline, and illustrate through specific numerical examples, how to use these formulas to estimate mitotic stability. Although we use Aspergillus nidulans as our biological paradigm, the same general framework can be extended to other fungal species, and possibly to less closely related systems as well.

Genetic diversity at a single locus under viability selection and facultative apomixis: equilibrium structure and deviations from Hardy-Weinberg frequencies.

We extensively analyze the maintenance of genetic variation and deviations from Hardy-Weinberg frequencies at a diallelic locus under mixed mating with apomixis and constant viability selection. Analytical proofs show that: (1) at most one polymorphic equilibrium exists, (2) polymorphism requires overdominant or underdominant selection, and (3) a simple, modified overdominance condition is sufficient to maintain genetic variation. In numerical analyses, only overdominant polymorphic equilibria are stable, and these are stable whenever they exist, which happens for approximately 78% of random fitness and mating parameters. The potential for maintaining both alleles increases with increasing apomixis or outcrossing and decreasing selfing. Simulations also indicate that equilibrium levels of heterozygosity will often be statistically indistinguishable from Hardy-Weinberg frequencies and that adults, not seeds, should usually be censused to maximize detecting deviations. Furthermore, although both censuses more often have an excess rather than a deficit of heterozygotes, analytical sign analyses of the fixation indices prove that, overall, adults are more likely to have an excess and seeds a deficit at equilibrium.

Three-locus systems impose additional constraints on pairwise disequilibria.

Combinations of allele frequencies and pairwise linkage disequilibrium terms, each of which is permissible at the two-locus level, may not always be permissible at the three-locus level. These additional constraints on the possible maximum and minimum values for the pairwise disequilibrium terms are formally determined and numerically analyzed. In some cases, the three-locus constraints on a pairwise disequilibrium (D) may be equivalent to the usual two-locus constraints, while in others, the positive or negative range may be restricted. This can result in situations where the allowable values of D are limited to only positive or only negative values up to the extreme case where there is only a single admissible value. No additional restrictions are placed on pairwise disequilibrium values when four loci are considered, other than those imposed by the three-way combinations containing the two loci of interest. A new measure of normalized pairwise linkage disequilibrium, allowing for the three-locus constraints, is defined and illustrated by an application to data from the human histocompatibility antigen (HLA) system. An analogous normalized three-way disequilibrium measure is also formulated.

Definition and properties of disequilibrium statistics for associations between nuclear and cytoplasmic genotypes.

We define and establish the interrelationships of four components of statistical association between a diploid nuclear gene and a uniparentally transmitted, haploid cytoplasmic gene: an allelic (gametic) disequilibrium (D), which measures associations between alleles at the two loci; and three genotypic disequilibria (D1, D2, D3), which measure associations between two cytotypes and the three respective nuclear backgrounds. We also consider an alternative set of measures, including D and the residual disequilibrium (d). The dynamics of these disequilibria are then examined under three conventional models of the mating system: (1) random mating; (2a) assortative mating without dominance (the "mixed-mating model"); and (2b) assortative mating with dominance ("O'Donald's model"). The trajectories of gametic disequilibria are similar to those for pairs of unlinked nuclear loci. The dynamics of genotypic disequilibria exhibit a variety of behaviors depending on the model and the initial conditions. Procedures for statistical estimation of cytonuclear disequilibria are developed and applied to several real and hypothetical data sets. Special attention is paid to the biological interpretations of various categories of allelic and genotypic disequilibria in hybrid zones. Genetic systems for which these statistics might be appropriate include nuclear genotype frequencies in conjunction with those for mitochondrial DNA, chloroplast DNA, or cytoplasmically inherited microorganisms.

The effects of assortative mating and migration on cytonuclear associations in hybrid zones.

We examine the influence of nonrandom mating and immigration on the evolutionary dynamics of cytonuclear associations in hybrid zones. Recursion equations for allelic and genotypic cytonuclear disequilibria were generated under models of (1) migration alone, assuming hybrid zone matings are random with respect to cytonuclear genotype; and (2) migration in conjunction with refined epistatic mating, in which females of the pure parental species preferentially mate with conspecific males. Major results are as follows: (a) even the slightest migration removes the dependency of the final outcome on initial conditions, producing a unique equilibrium in which both pure parental genotypes are maintained in the hybrid zone; (b) in contrast to nuclear genes, the dynamics of cytoplasmic allele frequencies appear robust to changes in the assumed mating system, yet are particularly sensitive to gene flow; (c) continued immigration can generate permanent cytonuclear disequilibria, whether mating is random or assortative; and (d) the order of population censusing (before versus after reproduction by immigrants) can have a dramatic effect on the magnitude but not the pattern of cytonuclear disequilibria. Using the maximum likelihood method, the parameter space of migration rates and assortative mating rates was examined for best fit to observed cytonuclear disequilibria data in a hybrid population of Hyla tree frogs. An epistatic mating model with a total immigration rate of about 32% per generation produces equilibrium gene frequencies and cytonuclear disequilibria consistent with the empirical observations.

Symbiont survival and host-symbiont disequilibria under differential vertical transmission.

Interspecific genetic interactions in host-symbiont systems raise intriguing coevolutionary questions and may influence the effectiveness of public health and management policies. Here we present an analytical and numerical investigation of the effects of host genetic heterogeneity in the rate of vertical transmission of a symbiont. We consider the baseline case with a monomorphic symbiont and a single diallelic locus in its diploid host, where vertical transmission is the sole force. Our analysis introduces interspecific disequilibria to quantify nonrandom associations between host genotypes and alleles and symbiont presence/absence. The transient and equilibrium behavior is examined in simulations with randomly generated initial conditions and transmission parameters. Compared to the case where vertical transmission rates are uniform across host genotypes, differential transmission (i) increases average symbiont survival from 50% to almost 60%, (ii) dramatically reduces the minimum average transmission rate for symbiont survival from 0.5 to 0.008, and (iii) readily creates permanent host-symbiont disequilibria de novo, whereas uniform transmission can neither create nor maintain such associations. On average, heterozygotes are slightly more likely to carry and maintain the symbiont in the population and are more randomly associated with the symbiont. Results show that simple evolutionary forces can create substantial nonrandom associations between two species.

Detection of deleterious genotypes in multigenerational studies. II. Theoretical and experimental dynamics with selfing and selection.

A mathematical model was developed to help interpret genotype and allele frequency dynamics in selfing populations, with or without apomixis. Our analysis provided explicit time-dependent solutions for the frequencies at diallelic loci in diploid populations under any combination of fertility, viability, and gametic selection through meiotic drive. With no outcrossing, allelic variation is always maintained under gametic selection alone, but with any fertility or viability differences, variation will ordinarily be maintained if and only if the net fitness (fertility x viability) of heterozygotes exceeds that of both homozygotes by a substantial margin. Under pure selfing and Mendelian segregation, heterozygotes must have a twofold fitness advantage; the level of overdominance necessary to preserve genetic diversity declines with apomixis, and increases with segregation distortion if this occurs equally and independently in male and female gametes. A case study was made of the Arabidopsis act2-1 actin mutant over multiple generations initiated from a heterozygous plant. The observed genotypic frequency dynamics were consistent with those predicted by our model for a deleterious, incompletely recessive mutant in either fertility or viability. The theoretical framework developed here should be very useful in dissecting the form(s) and strength of selection on diploid genotypes in populations with negligible levels of outcrossing.