A model for genetic relationship in areally continuous plant populations

Representations are based on plant populations, continuously distributed over their habitats according to specified density patterns. Migration of genetic material takes place via pollen and seed dispersal. Monoecious plants with arbitrary rates of self-fertilization and dioecious plants are considered. The model was constructed with the intention of determining coefficients of inbreeding and kinship for all locations within the seed population after its dispersal over the habitat, assuming the respective genetic relationships of the parental generation to be known. To display the consequences of single components hidden in the general result, the following specifications have been treated: finite population size combined with random dispersal of seed, equilibrium states for hypothetically infinite population size with “limited” dispersal of pollen and seed, random dispersal of pollen, and random dispersal of seed.     

Selection in plant populations of effectively infinite size II. Protectedness of a biallelic polymorphism

The biologically important problem of protectedness of genetic polymorphisms in monoecious plant populations exhibiting genotypically determined variation in rates of self-fertilization and sexually asymmetrical fertilities has hitherto escaped exact, analytical treatment for the reason that appropriate mathematical techniques relying on allelic frequencies do not seem to exist. For the particular case of one locus and two alleles it was possible to develop such a technique which provides conditions of high precision for protectedness of an allele. A comparison of the results with those already known from models that appear to be specializations of the present model showed that some of the earlier conclusions can be generalized, while others have to be handled with great care or should even be rejected. Above all, this concerns the role of self-fertilization, which is frequently considered to counteract the establishment of genetic polymorphisms. However, it turned out that increasing the heterozygote selfing rate also increases protectedness for both alleles in all situations. Moreover, even if the amount of self-fertilization is the same for all genotypes, asymmetry in the production of ovules and pollen, which is more the rule than an exception, may imply protectedness only for comparatively large selfing rates. The probably most outstanding finding is that, depending on the ovule and pollen fertilities, protectedness may be realized only within small ranges of selfing rates, and these ranges may vary from arbitrarily low to arbitrarily high rates. On the other hand, if the ovule fertilities show strong overdominance for the heterozygote—more precisely, if the heterozygote produces more than twice as many ovules as either of the homozygotes—both alleles are protected irrespective of the pollen fertilities and rates of self-fertilization; this generalizes earlier results obtained for more specific models.     

Partitioning of trait variation among communities: measures of apportionment and differentiation based on binary sampling

The notion that trait variation is partitioned among communities essentially rests on the supposition that the total variation in the metacommunity exceeds the variation within the communities due to differences between their trait distributions (the partitioning principle). Two elementary perspectives of partitioning can be distinguished: apportionment (members of the same community tend to hold the same trait state) and differentiation (members of different communities tend to hold different trait states). While the apportionment perspective reaches its extreme when each community is monomorphic (fixation), the differentiation perspective does so if communities share no trait states. Even though both perspectives can be shown to be involved in the analysis of community dynamics, their assessment is still almost completely limited to the apportionment perspective (chiefly G _ST and its relatives). To overcome this limitation, methods of quantifying both partitioning perspectives are developed for qualitatively and quantitatively varying traits, where variation is represented by the differences in type resulting from sampling two individuals. It is shown that the validity of the partitioning principle and the design of corresponding measures crucially depend on proper specification of the modes of sampling and specification of differences between types that follow a concavity principle. This approach allows comparison between measures of apportionment and measures of differentiation. Such comparisons enable conclusions about the share that random (drift) and directional (selection, migration) processes have in the partitioning of variation among communities. One of the more far-reaching conclusions is that effects of forces apportioning variation outperform the effects of forces of differentiation, if the average similarity within communities exceeds the average difference between them.     

The Analysis of Association Between Traits When Differences Between Trait States Matter

Because of their elementary significance in almost all fields of science, measures of association between two variables or traits are abundant and multiform. One aspect of association that is of considerable interest, especially in population genetics and ecology, seems to be widely ignored. This aspect concerns association between complex traits that show variable and arbitrarily defined state differences. Among such traits are genetic characters controlled by many and potentially polyploid loci, species characteristics, and environmental variables, all of which may be mutually and asymmetrically associated. A concept of directed association of one trait with another is developed here that relies solely on difference measures between the states of a trait. Associations are considered at three levels: between individual states of two variables, between an individual state of one variable and the totality of the other variable, and between two variables. Relations to known concepts of association are identified. In particular, measures at the latter two levels turn out to be interpretable as measures of differentiation. Examples are given for areas of application (search for functional relationships, distribution of variation over populations, genomic associations, spatiogenetic structure).     

Establishment of allelic two-locus polymorphisms

An exact analysis of necessary and sufficient conditions for the establishment and protectedness of biallelic two-locus polymorphisms is developed for the classical model with constant, sexually symmetric fitnesses and random association of the successful gametes. To demonstrate application of the results to common model types, the model of symmetric viabilities depending on the degree of heterozygosity only is chosen as a paradigm. It is pointed out that a unique locally stable internal equilibrium may exist even though all marginal equilibria (including the fixation states) are locally attractive. This example is quoted as an indication of the priority that analyses of protectedness deserve over analyses of local stability or instability of internal equilibria. Further applications of broader appeal concern the role that recombination plays in protecting polymorphisms. Probably the most interesting finding is that with increasing recombination frequency the chances for protectedness of a polymorphism generally decline. Yet, if a certain hierarchic ordering of the fitnesses with respect to the degree of heterozygosity is realized, the polymorphism is protected for arbitrary amounts of recombination. If recombination is rare, heterozygote advantage is not a universal precondition for persistence of polymorphisms. This phenomenon is utilized to derive conditions under which deleterious recessive mutants can be maintained in a population.     

Single-locus gametophytic incompatibility: the symmetric equilibrium is globally asymptotically stable

The deterministic dynamics of the classical single-locus multiple-allele model of gametophytic incompatibility is analyzed with the intention to prove the conjecture that the symmetric state (uniform distribution of genotypes) is the only polymorphic equilibrium and that this equilibrium is globally asymptotically stable in the interior of the frequency simplex. It is shown that the minimum allelic frequency increases strictly over the generations as long as a uniform allelic distribution is not realized. Hence, the minimum allelic frequency is a Ljapunov function for the invariant set of genotypic frequencies characterized by a uniform allelic distribution. Within this set, the uniform genotypic distribution is approached in an exponential fashion, which proves the assertion. An evolutionary optimization rule associated with the global convergence to the symmetric state is implied by the fact that at this state the overall amount of pollen elimination resulting from incompatible crosses is minimized.     

Pyrolysis-field ionization mass spectrometry of agricultural soils and humic substances: Effect of cropping systems and influence of the mineral matrix

Whole soil samples, extracted humic substances, the corresponding fulvic (FA) and humic acids (HA) and the extraction residues (humins) from long-term, agricultural test plots were investigated by in-source pyrolysis-field ionization mass spectrometry (Py-FIMS). For the soils distinct differences in the chemical composition of the organic matter in differently managed fields were observed. The FI mass spectra of the extracted humic substances gave complementary chemical information, as they cover a larger mass range compared to the whole soil spectra. The chemical, structural information of the conventional alkaline extraction residues was demonstrated by Py-FIMS spectra to be similar to that of the related soil samples. Influences of mineral matrix to organic matter ratios were studied on mixtures of extracted humic substances with defined mineral components such as quartz, basalt, iron oxide (Fe₂O₃), Ca-montmorillonite, kaolinite and illite. It was shown that in these mixtures the number of mass signals detected and the covered mass range decreased, when organic carbon concentrations (C_org) in this synthetic mineral matrix dropped below 2% (w/w). Limitations in the direct application of Py-FIMS might arise in the case of natural soil samples with C_org concentrations below 0.5% (w/w), high contents of swelling clay minerals and iron oxides. ei]{gnR}{fnMerckx}     

Deterministic single-locus density-dependent selection

Density-regulated selection is considered for a single, multiallelic gene locus and separated generations. Characteristics resulting from the basic assumption that the average population fitness decreases with increasing density are derived. Under this assumption, it proves to be necessary to distinguish between regions of allelic frequencies which imply limited population growth, unlimited growth, or ultimate extinction when the population stays in the respective region. Particular attention is given to the investigation of the region of limited growth and the carrying capacity function defined on it. Relationships between and the average fitness (adaptive surface) in the non-density dependent model are explained.Besides stability properties of equilibrium points, more general characteristics concerning the asymptotic behavior of population trajectories are treated. In this context, the problems of sudden loss of alleles and of population extinction as a result of large fluctuations in density are discussed.