The lattice models of neutral multi-alleles in population genetics theory

The aim of this article is to study lattice models of neutral multi-alleles including Ohta-Kimura's step-wise mutation model. We shall show an outline of the construction of a unique strongly continuous non-negative semi-group associated with the infinite dimensional generator and show a general and straightforward method of obtaining the time dependent and equilibrium solutions of all polynomial moments of the gene frequencies. We shall discuss the spectrum of the diffusion processes and as an application we obtain all higher moments of the homozygosity.     

Stochastic theory of population genetics

Stochastic models of population genetics are studied with special reference to the biological interest. Mathematical methods are described for treating some simple models and their modifications aimed at the problems of the molecular evolution. Unified theory for treating different quantities is extensively developed and applied to some typical problems of current interest in genetics. Mathematical methods for treating geographically structured populations are given. Approximation formulae and their accuracy are discussed. Some criteria are given for a structured population to behave almost like a panmictic population of the same total size. Some quantities are shown to be independent of the geographical structure and their dynamics are described.     

Stability of discrete one-dimensional population models

We give conditions for local and global stability of discrete one-dimensional population models. We give a new test for local stability when the derivative is −1. We give several sufficient conditions for global stability. We use these conditions to show that local and global stability coincide for the usual models from the literature and even for slightly more complicated models. We give population models, which are in some sense the simplest models, for which local and global stability do not coincide.     

Separable models for age-structured population genetics

This paper is concerned with the applications of nonlinear age-dependent dynamics to population genetics. Age-structured models are formulated for a single autosomal locus with an arbitrary number of alleles. The following cases are considered: a) haploid populations with selection and mutation; b) monoecious diploid populations with or without mutation reproducing by self-fertilization or by two types of random mating. The diploid models do not deal with selection. For these cases the genic and genotypic frequencies evolve towards time-persistent forms, whether the total population size tends towards exponential growth or not.     

Spatially explicit Bayesian clustering models in population genetics

This article reviews recent developments in Bayesian algorithms that explicitly include geographical information in the inference of population structure. Current models substantially differ in their prior distributions and background assumptions, falling into two broad categories: models with or without admixture. To aid users of this new generation of spatially explicit programs, we clarify the assumptions underlying the models, and we test these models in situations where their assumptions are not met. We show that models without admixture are not robust to the inclusion of admixed individuals in the sample, thus providing an incorrect assessment of population genetic structure in many cases. In contrast, admixture models are robust to an absence of admixture in the sample. We also give statistical and conceptual reasons why data should be explored using spatially explicit models that include admixture.     

Quasispecies theory in the context of population genetics

Background  

A number of recent papers have cast doubt on the applicability of the quasispecies concept to virus evolution, and have argued that population genetics is a more appropriate framework to describe virus evolution than quasispecies theory.     

Interactive use of computer models in teaching population genetics

A technique using a computer with a graphical display unit to teach students the effects of genetic drift, selection and migration is described. Both diallelic and triallelic loci are discussed. The Fokker-Planck equation is used as the mathematical model of the genetic system, and its validity as an approximation in this context is demonstrated by an investigation into selection at the ABO locus. An appendix contains a derivation from the Fokker-Planck equation of the formula used in the paper for the gene frequency distribution at a multiallelic locus in equilibrium under selection and migration.     

Convergence of one-dimensional diffusion processes to a jump process related to population genetics

A conjecture on the convergence of diffusion models in population genetics to a simple Markov chain model is proved. The notion of bi-generalized diffusion processes and their limit theorems are used systematically to prove the conjecture. Three limits; strong selection-weak mutation limit, moderate selection-weak mutation limit, weak selection-weak mutation limit are considered for typical diffusion models in population genetics.Supported in part by Research Grant from the Ministry of Education, Science and Culture of JapanSupported by the Air Force Office of Scientific Research Contract No. F49620 85C 0144     

Some population genetic models combining artificial and natural selection pressures: I. One-locus theory

A series of one-locus genetic models involving the combined effects of artificial and natural selection forces are analyzed. Other factors incorporated in this work include the influence of the imposed or inherent mating system, the relevance of timing in application of the two types of selection forces, the importance of multiallelism and dominance relations.     

Some population genetic models combining artificial and natural selection pressures: II. Two-locus theory

Some population genetic systems of traits controlled principally at two or three main loci subject to the combined effects of artificial and natural selection are studied. A number of genotypic-phenotypic associations are formulated, and both random and selfing mating patterns are examined. The interaction of some forms of epistasis and linkage in these systems are evaluated.     

Sam Karlin and the stochastic theory of evolutionary population genetics

Sam Karlin’s role in the development of the stochastic theory of evolutionary population genetics is outlined, together with his work in developing BLAST theory.     

Some models of population structure and evolution

Models of population structure analyzed include: the uniform model (k populations all exchanging at a constant rate) and the multiuniform model (s clusters of k populations exchanging at different rates within and between clusters). Analogies of the latter model with trees of descent have been shown. The algorithm used allows exact solutions for the equilibrium variances and covariances in a variety of cases, including the circular stepping-stone model.Effects of multiple sources of stabilizing pressures have been shown in the multiuniform and the star model. The latter is of interest in situations of population radiation and of centripetal migration.     

The relation between multilocus population genetics and social evolution theory

Evolution at multiple gene positions is complicated. Direct selection on one gene disturbs the evolutionary dynamics of associated genes. Recent years have seen the development of a multilocus methodology for modeling evolution at arbitrary numbers of gene positions with arbitrary dominance and epistatic relations, mode of inheritance, genetic linkage, and recombination. We show that the approach is conceptually analogous to social evolutionary methodology, which focuses on selection acting on associated individuals. In doing so, we (1) make explicit the links between the multilocus methodology and the foundations of social evolution theory, namely, Price's theorem and Hamilton's rule; (2) relate the multilocus approach to levels-of-selection and neighbor-modulated-fitness approaches in social evolution; (3) highlight the equivalence between genetical hitchhiking and kin selection; (4) demonstrate that the multilocus methodology allows for social evolutionary analyses involving coevolution of multiple traits and genetical associations between nonrelatives, including individuals of different species; (5) show that this methodology helps solve problems of dynamic sufficiency in social evolution theory; (6) form links between invasion criteria in multilocus systems and Hamilton's rule of kin selection; (7) illustrate the generality and exactness of Hamilton's rule, which has previously been described as an approximate, heuristic result.     

Epidemic models with spatial spread due to population migration

A spatial diffusion operator that governs the migration of polymorphic populations is derived and some specific epidemic models are analyzed in the presence of this type of diffusion. Threshold criteria and asymptotic behavior of solutions are derived, and it is shown that spatially heterogeneous steady states can occur in these models.The work of this author was partially supported by the National Science Foundation's Ecosystem Studies Program under Interagency Agreement No. DED80-21024 with the U.S. Department of Energy under contract W-7405-eng-26 with Union Carbide Corporation     

A two loci viability matrix with implications in current population genetics models

Summary Some equilibrium properties of a two loci deterministic system subject to a postulated viability matrix are given. Two conclusions are reached: i) linkage can not stabilize additive equally contributing two-loci systems under optimizing selection. ii) Hybrid protein superiority can not be invoked for the maintenance of polymorphisms shared by duplicated structural loci.     

Soft sweeps II--molecular population genetics of adaptation from recurrent mutation or migration

In the classical model of molecular adaptation, a favored allele derives from a single mutational origin. This ignores that beneficial alleles can enter a population recurrently, either by mutation or migration, during the selective phase. In this case, descendants of several of these independent origins may contribute to the fixation. As a consequence, all ancestral haplotypes that are linked to any of these copies will be retained in the population, affecting the pattern of a selective sweep on linked neutral variation. In this study, we use analytical calculations based on coalescent theory and computer simulations to analyze molecular adaptation from recurrent mutation or migration. Under the assumption of complete linkage, we derive a robust analytical approximation for the number of ancestral haplotypes and their distribution in a sample from the population. We find that so-called "soft sweeps," where multiple ancestral haplotypes appear in a sample, are likely for biologically realistic values of mutation or migration rates.