Sharp Asymptotics for Fixation Times in Stochastic Population Genetics Models at Low Mutation Probabilities

Many of our theories for the generation and maintenance of diversity in nature depend on the existence of specialist biotic interactions which, in host–pathogen systems, also shape cross-species disease emergence. As such, niche breadth evolution, especially in host–parasite systems, remains a central focus in ecology and evolution. The predominant explanation for the existence of specialization in the literature is that niche breadth is constrained by trade-offs, such that a generalist is less fit on any particular environment than a given specialist. This trade-off theory has been used to predict niche breadth (co)evolution in both population genetics and eco-evolutionary models, with the different modelling methods providing separate, complementary insights. However, trade-offs may be far from universal, so population genetics theory has also proposed alternate mechanisms for costly generalism, including mutation accumulation. However, these mechanisms have yet to be integrated into eco-evolutionary models in order to understand how the mechanism of costly generalism alters the biological and ecological circumstances predicted to maintain specialism. In this review, we outline how population genetics and eco-evolutionary models based on trade-offs have provided insights for parasite niche breadth evolution and argue that the population genetics-derived mutation accumulation theory needs to be better integrated into eco-evolutionary theory.

Stochastic models in population genetics which lead to diffusion equations are considered. A geometric formula for the asymptotic expansions of the fundamental solutions of these equations is presented. Specifically, the random genetic drift process of one-locus theory and the Ohta–Kimura model of two-locus di-allelic systems with linkage are studied. Agreement with the work of Keller and Voronka for the two-allele one-locus case is obtained. For the general n-allele problem, the formulas obtained here are apparently new.

P opulation genetics relies heavily on mathematical modeling to make quantitative predictions about the behavior of genes in populations. These models are based on the principles of classic Mendelian gene inheritance; the Hardy-Weinberg equilibrium law, which predicts the gene frequency in a population; and Darwin's theory of natural selection. PopGene version 1 is a simulation program that explores the parameters that affect the behavior of alternative forms of a gene (alleles) in a population. The program window is composed of three smaller windows specifying the parameters used, the calculated allele frequencies through each generation, and the graphical output of the calculations. PopGene provides simulations for both basic and more advanced models in population genetics. The simplest model demonstrates the principles of the Hardy-Weinberg equilibrium law for the case of a single gene with two alleles. Increasingly complex examples based on the assumptions of the Hardy-Weinberg equilibrium law demonstrate the effect of multiple alleles of a gene and multiple gene loci on the genotype of all the individuals in a population. Default parameters for each model are built into the program, but users may also enter their own values. The simulation results, parameters, and graphs may be saved or printed. Up to 15 simulations may be run for a particular model with the results plotted on a single graph for easy comparison. Natural selection and inbreeding can be studied with the program. Natural selection of a gene with two alleles occurs when the fitness of each genotype is not equal. The program permits users to examine the outcomes of several scenarios—selection of an allele that exhibits dominance over another, alleles that combine to produce an intermediate (semidominant) phenotype, an allele that is recessive to another, or selection against a recessive allele because of its lethality. The model for inbreeding explores the consequence of nonrandom mating between individuals on the genotype frequencies of that population. Mating pairings are included to demonstrate self-fertilization (the most extreme form of nonrandom mating), as well as matings between parent and offspring; matings between siblings, first cousins, and second cousins; and random mating. PopGene also provides simulations for genetic drift, mutation, and migration—factors that contribute to evolutionary changes in a population that have long-term effects and are more difficult to study. In summary, PopGene is a valuable tool to use in conjunction with traditional textbook-based teaching of an introductory population genetics course. It provides students with illustrations of the principles they have learned. The program is straightforward, easy to use, and is available for Macintosh and Windows platforms. No preprinted manual is available for PopGene; however, a concise and well-written one is supplied as a PDF file that is readable with Adobe Acrobat. It covers the specifics of using the program and provides a step-by-step tutorial. A brief summary of population genetics, mathematical modeling, and the value and limitations of computer simulations is also discussed. The system requires a PC with a 486 or Pentium processor that uses Windows 3.1, 95, 98, or NT with 8 MB RAM or any Macintosh with 4 MB RAM, a 1.4-MB disk drive, and a hard drive. PopGene Trinity Software, Inc. Plymouth, NH. $95.00; $39.95, college students. 800-352-1282

A modified Wright–Fisher model that incorporates Ne: A variant of the standard model with increased biological realism and reduced computational complexity

A general exchangeable model is introduced to study gene survival in populations whose size changes without density dependence. Necessary and sufficient conditions for the occurrence of fixation (that is the proportion of one of the types tending to 1 with probability 1) are obtained. These are then applied to the Wright–Fisher model, the Moran model, and conditioned branching-process models. For the Wright–Fisher model it is shown that certain fixation is equivalent to certain extinction of one of the types, but that this is not the case for the Moran model.

Analysis of genome sequences of 159 isolates of Plasmodium falciparum from Senegal yields an extraordinarily high proportion (26.85%) of protein-coding genes with the ratio of nonsynonymous to synonymous polymorphism greater than one. This proportion is much greater than observed in other organisms. Also unusual is that the site-frequency spectra of synonymous and nonsynonymous polymorphisms are virtually indistinguishable. We hypothesized that the complicated life cycle of malaria parasites might lead to qualitatively different population genetics from that predicted from the classical Wright-Fisher (WF) model, which assumes a single random-mating population with a finite and constant population size in an organism with nonoverlapping generations. This paper summarizes simulation studies of random genetic drift and selection in malaria parasites that take into account their unusual life history. Our results show that random genetic drift in the malaria life cycle is more pronounced than under the WF model. Paradoxically, the efficiency of purifying selection in the malaria life cycle is also greater than under WF, and the relative efficiency of positive selection varies according to conditions. Additionally, the site-frequency spectrum under neutrality is also more skewed toward low-frequency alleles than expected with WF. These results highlight the importance of considering the malaria life cycle when applying existing population genetic tools based on the WF model. The same caveat applies to other species with similarly complex life cycles.

ASYMPTOTIC BEHAVIOR OF A CLASS OF DISCRETE-TIME MODELS IN POPULATION GENETICS

Let $\{X_n\}$ be a Markov chain on a bounded set in $R^d$ with $E_x(X_1) = f_N(x) = x + \beta_N h_N(x)$, where $x_0$ is a stable fixed point of $f_N(x) = x$, and $\operatorname{Cov}_x(X_1) \approx \sigma^2(x)/N$ in various senses. Let $D$ be an open set containing $x_0$, and assume $h_N(x) \rightarrow h(x)$ uniformly in $D$ and either $\beta_N \equiv 1$ or $\beta_N \rightarrow 0, \beta_N \gg \sqrt{\log N/N}$. Then, assuming various regularity conditions and $X_0 \in D$, the time the process takes to exit from $D$ is logarithmically equivalent in probability to $e^{VN\beta_N}$, where $V > 0$ is the solution of a variational problem of Freidlin-Wentzell type $\lbrack \text{if} \beta_N \rightarrow 0 \text{and} d = 1, V = \inf\{2 \int^y_{x_0}\sigma^{-2}(u)|h(u) du|: y \in \partial D\} \rbrack$. These results apply to the Wright-Fisher model in population genetics, where $\{X_n\}$ represent gene frequencies and the average effect of forces such as selection and mutation are much stronger than effects due to finite population size.

Hardy–Weinberg Law

In this chapter, I elaborate the population genetics perspective introduced in chapter 1 and adopted in the subsequent chapters. I submit more justification here for the position that there are indeed heuristic advantages in approaching language evolution on the model of population genetics, assuming that a language is a species but not an organism. However, I also argue that the linguistic species need not be a clone of any biological species, despite the fact that it shares several properties with the parasitic species. In fact, the proposed population genetics of language evolution is more than an analog of population genetics, although its heuristics has been very much inspired by the latter. A linguistic species must be defined on its own independent terms and its evolutionary properties hypothesized according to its own combination of ontological characteristics. These account for both the similarities and differences which it displays with its closest kin in biological evolution. The basic assumption is that there are general evolutionary principles which apply similarly to the linguistic and biological species. However, there are species-specific principles which distinguish them from each other, based in part on whether a species is of the Darwinian or Lamarckian kind, on whether traits are transmitted sexually or asexually, horizontally and/or vertically, on whether the default quality of copying in trait inheritance is with or without modification, etc. One cluster of factors that plays an important role in any theory of evolution is “ecology.”

This paper considers a semilinear elliptic equation which arises in a selection-migration model in population genetics, involving two alleles $A_1$ and $A_2$ such that $A_1$ is at an advantage over $A_2$ in certain subregions and at a disadvantage in others. The system is studied on all of $R^n$ and is assumed to possess radial symmetry. Existence and asymptotic properties of solutions of the corresponding ordinary differential equation are investigated and, by using shooting method type arguments, results are obtained on the bifurcation of solutions from the trivial solutions corresponding to the cases where $A_1$ or $A_2$ is extinct. The nature of the results obtained varies according to whether $A_1$ or $A_2$ has an overall advantage.

This thesis was scanned from the print manuscript for digital preservation and is copyright the author. Researchers can access this thesis by asking their local university, institution or public library to make a request on their behalf. Monash staff and postgraduate students can use the link in the References field.

Simulating Probability Distributions in the Coalescent

In the present article, we investigate the effects of dormancy on an abstract popula- tion genetic level. We first provide a short review of seed bank models in population genetics, and the role of dormancy for the interplay of evolutionary forces in gen- eral, before we discuss two recent paradigmatic models, referring to spontaneous resp. simultaneous switching of individuals between the active and the dormant state. We show that both mechanisms give rise to non-trivial mathematical objects, namely the (continuous) seed bank diffusion and the seed bank diffusion with jumps, as well as their dual processes, the seed bank coalescent and the seed bank coalescent with simultaneous switching.

A model is developed describing the operation of selection upon broods of full sibs. Fitness is regarded as a function of individual genotype, the genetic composition of the brood of which the individual is a member, and/or the genotypes of the individual's parents. The model is formulated such that the fundamental equations describing brood selection at a single locus in a randomly-mating population are identical to the equations arising from a frequency-dependent model in population genetics. Hence, the mathematical analysis of brood selection is equivalent to that of frequency-dependent selection. In particular, brood selection results in multiple equilibria that in general do not maximize average fitness. Several specific examples are discussed, including social selection against a genetic disease, reproductive compensation, the evolution of aposematic coloration, and the evolution of altruistic or selfish brood behavior.