A genealogical approach to variable-population-size models in population genetics

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

Motivation: A fundamental problem in population genetics, which being also of importance to forensic science, is to compute the match probability (MP) that two individuals randomly chosen from a population have identical alleles at a collection of loci. At present, 11–13 unlinked autosomal microsatellite loci are typed for forensic use. In a finite population, the genealogical relationships of individuals can create statistical non-independence of alleles at unlinked loci. However, the so-called product rule, which is used in courts in the USA, computes the MP for multiple unlinked loci by assuming statistical independence, multiplying the one-locus MPs at those loci. Analytically testing the accuracy of the product rule for more than five loci has hitherto remained an open problem.Results: In this article, we adopt a flexible graphical framework to compute multi-locus MPs analytically. We consider two standard models of random mating, namely the Wright–Fisher (WF) and Moran models. We succeed in computing haplotypic MPs for up to 10 loci in the WF model, and up to 13 loci in the Moran model. For a finite population and a large number of loci, we show that the MPs predicted by the product rule are highly sensitive to mutation rates in the range of interest, while the true MPs computed using our graphical framework are not. Furthermore, we show that the WF and Moran models may produce drastically different MPs for a finite population, and that this difference grows with the number of loci and mutation rates. Although the two models converge to the same coalescent or diffusion limit, in which the population size approaches infinity, we demonstrate that, when multiple loci are considered, the rate of convergence in the Moran model is significantly slower than that in the WF model.Availability: A C++ implementation of the algorithms discussed in this article is available at http://www.cs.berkeley.edu/∼yss/software.html.Contact: yss@eecs.berkeley.edu

The discrete time mathematical models of evolution (the discrete time Eigen model, the Moran model, and the Wright-Fisher model) have many applications in complex biological systems. The discrete time Eigen model rather realistically describes the serial passage experiments in biology. Nevertheless, the dynamics of the discrete time Eigen model is solved in this paper. The 90% of results in population genetics are connected with the diffusion approximation of the Wright-Fisher and Moran models. We considered the discrete time Eigen model of asexual virus evolution and the Wright-Fisher model from population genetics. We look at the logarithm of probabilities and apply the Hamilton-Jacobi equation for the models. We define exact dynamics for the population distribution for the discrete time Eigen model. For the Wright-Fisher model, we express the exact steady state solution and fixation probability via the solution of some nonlocal equation then give the series expansion of the solution via degrees of selection and mutation rates. The diffusion theories result in the zeroth order approximation in our approach. The numeric confirms that our method works in the case of strong selection, whereas the diffusion method fails there. Although the diffusion method is exact for the mean first arrival time, it provides incorrect approximation for the dynamics of the tail of distribution.

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.

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.

Moran (7) has shown that some models in population genetics lead to a bilinear recurrence relation which, for a finite number of alleles, has a globally stable equilibrium. When the possible alleles are infinite in number, non-trivial problems of existence and stability of the equilibrium arise, which Moran has resolved in special cases. In this paper a powerful sufficient condition is established for the existence of a globally stable equilibrium, and its consequences are explored for cases of genetical interest. A more speculative final section describes a variant of Moran's model which may possibly have some relevance to the assessment of experimental evidence for or against selective neutrality.

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

Entropy and the arrow of time in population dynamics

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.

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.

Because host-plant chemistry is dynamic, chemical defenses are dynamic, and senescence in plants causes many proteins to decompose after flowering, laboratory and field studies on transgenic crops performed over only part of a plant generation or part of a season may not provide sufficient data to evaluate strategies for resistance management. As an example, we focused on the recent introduction of transgenic corn to control European corn borer, Ostrinia nubilalis (Hubner) (Lepidoptera: Pyralidae). We gathered data from literature on the life cycle of European corn borer and corn crop maturation to study the role that phenological relationships may play in resistance evolution. In addition, we simulated hypothetical titer declines resulting in increased survival in European corn borer populations infesting transgenic corn using a model of population dynamics and genetics. The relationship between European corn borer hatching period and corn maturation varies greatly from site to site and year to year. The peak of the hatching period in the late summer generation of the European corn borer occurs at or after the average midpoint of the dough stage of corn. The last larvae tend to hatch after the dough stage is past and after the midpoint of the dent stage. In simulations where 5% of a region is planted with nontransgenic corn in separate refuge plots and 95% is planted with transgenic corn, complete loss of titer as a result of senescence produces resistance at the 3% resistance-allele level after 5–42 yr, which is less than the 83 yr predicted by the standard model for resistance development when the transgenic corn loses no titer as a result of senescence. We hypothesize that genetically engineered toxins will often decrease in leaf and stem titer as crops reach maturation. The insects feeding and surviving on a crop during its senescence may have important consequences for the population genetics of the breakdown of host plant resistance.

This article describes new models in population genetics that extend the neutral Wright–Fisher model by including strong selection and mutation. Fixation times are studied in the limit of small mutation probabilities within the framework of Markov chains with rare transitions. The main result outlines the role of the discrete geometry of the fitness landscape and provides a mean for estimating the expected number of generations for an individual with better fitness value to appear. Some connections to evolutionary algorithms are discussed as well.

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

Hardy–Weinberg Law