An analysis of the fixation probability of a mutant on special classes of non-directed graphs

There is a growing interest in the study of evolutionary dynamics on populations with some non-homogeneous structure. In this paper we follow the model of Lieberman et al. (Lieberman et al. 2005 Nature 433, 312–316) of evolutionary dynamics on a graph. We investigate the case of non-directed equally weighted graphs and find solutions for the fixation probability of a single mutant in two classes of simple graphs. We further demonstrate that finding similar solutions on graphs outside these classes is far more complex. Finally, we investigate our chosen classes numerically and discuss a number of features of the graphs; for example, we find the fixation probabilities for different initial starting positions and observe that average fixation probabilities are always increased for advantageous mutants as compared with those of unstructured populations

Clines with partial panmixia across a geographical barrier in an environmental pocket

In a geographically structured population, partial global panmixia can be regarded as the limiting case of long-distance migration. On the entire line with homogeneous, isotropic migration, an environmental pocket is bounded by a geographical barrier, which need not be symmetric. For slow evolution, a continuous approximation of the exact, discrete model for the gene frequency at a diallelic locus at equilibrium, where denotes position and the barrier is at , is formulated and investigated. This model incorporates viability selection, local adult migration, adult partial panmixia, and the barrier. The gene frequency and its derivatives are discontinuous at the barrier unless the latter is symmetric, in which case only is discontinuous. A cline exists only if the scaled rate of partial panmixia ; several qualitative results also are proved. Formulas that determine in a step-environment when dominance is absent are derived. The maximal gene frequency in the cline satisfies . A cline exists if and only if and the radius of the pocket exceeds the minimal radius , for which a simple, explicit formula is deduced. Given numerical solutions for and , an explicit formula is proved for in ; whereas in , an elliptic integral for must be numerically inverted. The minimal radius for maintenance of a cline in an isotropic, bidimensional pocket is also examined

Modelling the spread of Wolbachia in spatially heterogeneous environments

The endosymbiont Wolbachia infects a large number of insect species and is capable of rapid spread when introduced into a novel host population. The bacteria spread by manipulating their hosts' reproduction, and their dynamics are influenced by the demographic structure of the host population and patterns of contact between individuals. Reaction–diffusion models of the spatial spread of Wolbachia provide a simple analytical description of their spatial dynamics but do not account for significant details of host population dynamics. We develop a metapopulation model describing the spatial dynamics of Wolbachia in an age-structured host insect population regulated by juvenile density-dependent competition. The model produces similar dynamics to the reaction–diffusion model in the limiting case where the host's habitat quality is spatially homogeneous and Wolbachia has a small effect on host fitness. When habitat quality varies spatially, Wolbachia spread is usually much slower, and the conditions necessary for local invasion are strongly affected by immigration of insects from surrounding regions. Spread is most difficult when variation in habitat quality is spatially correlated. The results show that spatial variation in the density-dependent competition experienced by juvenile host insects can strongly affect the spread of Wolbachia infections, which is important to the use of Wolbachia to control insect vectors of human disease and other pests

Evolutionary Dynamics on Small-Order Graphs

Abstract. We study the stochastic birth-death model for structured finite populations popularized by Lieberman et al. [Lieberman, E., Hauert, C., Nowak, M.A., 2005. Evolutionary dynamics on graphs. Nature 433, 312-316]. We consider all possible connected undirected graphs of orders three through eight. For each graph, using the Monte Carlo Markov Chain simulations, we determine the fixation probability of a mutant introduced at every possible vertex. We show that the fixation probability depends on the vertex and on the graph. A randomly placed mutant has the highest chances of fixation in a star graph, closely followed by star-like graphs. The fixation probability was lowest for regular and almost regular graphs. We also find that within a fixed graph, the fixation probability of a mutant has a negative correlation with the degree of the starting vertex. 1

Evolutionary Epidemiology of Drug-Resistance in Space

The spread of drug-resistant parasites erodes the efficacy of therapeutic treatments against many infectious diseases and is a major threat of the 21st century. The evolution of drug-resistance depends, among other things, on how the treatments are administered at the population level. “Resistance management” consists of finding optimal treatment strategies that both reduce the consequence of an infection at the individual host level, and limit the spread of drug-resistance in the pathogen population. Several studies have focused on the effect of mixing different treatments, or of alternating them in time. Here, we analyze another strategy, where the use of the drug varies spatially: there are places where no one receives any treatment. We find that such a spatial heterogeneity can totally prevent the rise of drug-resistance, provided that the size of treated patches is below a critical threshold. The range of parasite dispersal, the relative costs and benefits of being drug-resistant compared to being drug-sensitive, and the duration of an infection with drug-resistant parasites are the main factors determining the value of this threshold. Our analysis thus provides some general guidance regarding the optimal spatial use of drugs to prevent or limit the evolution of drug-resistance

A Model-Based Analysis of GC-Biased Gene Conversion in the Human and Chimpanzee Genomes

GC-biased gene conversion (gBGC) is a recombination-associated process that favors the fixation of G/C alleles over A/T alleles. In mammals, gBGC is hypothesized to contribute to variation in GC content, rapidly evolving sequences, and the fixation of deleterious mutations, but its prevalence and general functional consequences remain poorly understood. gBGC is difficult to incorporate into models of molecular evolution and so far has primarily been studied using summary statistics from genomic comparisons. Here, we introduce a new probabilistic model that captures the joint effects of natural selection and gBGC on nucleotide substitution patterns, while allowing for correlations along the genome in these effects. We implemented our model in a computer program, called phastBias, that can accurately detect gBGC tracts about 1 kilobase or longer in simulated sequence alignments. When applied to real primate genome sequences, phastBias predicts gBGC tracts that cover roughly 0.3% of the human and chimpanzee genomes and account for 1.2% of human-chimpanzee nucleotide differences. These tracts fall in clusters, particularly in subtelomeric regions; they are enriched for recombination hotspots and fast-evolving sequences; and they display an ongoing fixation preference for G and C alleles. They are also significantly enriched for disease-associated polymorphisms, suggesting that they contribute to the fixation of deleterious alleles. The gBGC tracts provide a unique window into historical recombination processes along the human and chimpanzee lineages. They supply additional evidence of long-term conservation of megabase-scale recombination rates accompanied by rapid turnover of hotspots. Together, these findings shed new light on the evolutionary, functional, and disease implications of gBGC. The phastBias program and our predicted tracts are freely available. © 2013 Capra et al

Statistical Genetics and Evolution of Quantitative Traits

The distribution and heritability of many traits depends on numerous loci in the genome. In general, the astronomical number of possible genotypes makes the system with large numbers of loci difficult to describe. Multilocus evolution, however, greatly simplifies in the limit of weak selection and frequent recombination. In this limit, populations rapidly reach Quasi-Linkage Equilibrium (QLE) in which the dynamics of the full genotype distribution, including correlations between alleles at different loci, can be parameterized by the allele frequencies. This review provides a simplified exposition of the concept and mathematics of QLE which is central to the statistical description of genotypes in sexual populations. We show how key results of Quantitative Genetics such as the generalized Fisher's "Fundamental Theorem", along with Wright's Adaptive Landscape, emerge within QLE from the dynamics of the genotype distribution. We then discuss under what circumstances QLE is applicable, and what the breakdown of QLE implies for the population structure and the dynamics of selection. Understanding of the fundamental aspects of multilocus evolution obtained through simplified models may be helpful in providing conceptual and computational tools to address the challenges arising in the studies of complex quantitative phenotypes of practical interest.Comment: to appear in Rev.Mod.Phy

Calculating Evolutionary Dynamics in Structured Populations

Evolution is shaping the world around us. At the core of every evolutionary process is a population of reproducing individuals. The outcome of an evolutionary process depends on population structure. Here we provide a general formula for calculating evolutionary dynamics in a wide class of structured populations. This class includes the recently introduced “games in phenotype space” and “evolutionary set theory.” There can be local interactions for determining the relative fitness of individuals, but we require global updating, which means all individuals compete uniformly for reproduction. We study the competition of two strategies in the context of an evolutionary game and determine which strategy is favored in the limit of weak selection. We derive an intuitive formula for the structure coefficient, σ, and provide a method for efficient numerical calculation

Isolation-by-Distance and Outbreeding Depression Are Sufficient to Drive Parapatric Speciation in the Absence of Environmental Influences

A commonly held view in evolutionary biology is that speciation (the emergence of genetically distinct and reproductively incompatible subpopulations) is driven by external environmental constraints, such as localized barriers to dispersal or habitat-based variation in selection pressures. We have developed a spatially explicit model of a biological population to study the emergence of spatial and temporal patterns of genetic diversity in the absence of predetermined subpopulation boundaries. We propose a 2-D cellular automata model showing that an initially homogeneous population might spontaneously subdivide into reproductively incompatible species through sheer isolation-by-distance when the viability of offspring decreases as the genomes of parental gametes become increasingly different. This simple implementation of the Dobzhansky-Muller model provides the basis for assessing the process and completion of speciation, which is deemed to occur when there is complete postzygotic isolation between two subpopulations. The model shows an inherent tendency toward spatial self-organization, as has been the case with other spatially explicit models of evolution. A well-mixed version of the model exhibits a relatively stable and unimodal distribution of genetic differences as has been shown with previous models. A much more interesting pattern of temporal waves, however, emerges when the dispersal of individuals is limited to short distances. Each wave represents a subset of comparisons between members of emergent subpopulations diverging from one another, and a subset of these divergences proceeds to the point of speciation. The long-term persistence of diverging subpopulations is the essence of speciation in biological populations, so the rhythmic diversity waves that we have observed suggest an inherent disposition for a population experiencing isolation-by-distance to generate new species

Population Genetics of Trypanosoma evansi from Camel in the Sudan

Genetic variation of microsatellite loci is a widely used method for the analysis of population genetic structure of microorganisms. We have investigated genetic variation at 15 microsatellite loci of T. evansi isolated from camels in Sudan and Kenya to evaluate the genetic information partitioned within and between individuals and between sites. We detected a strong signal of isolation by distance across the area sampled. The results also indicate that either, and as expected, T. evansi is purely clonal and structured in small units at very local scales and that there are numerous allelic dropouts in the data, or that this species often sexually recombines without the need of the “normal” definitive host, the tsetse fly or as the recurrent immigration from sexually recombined T. brucei brucei. Though the first hypothesis is the most likely, discriminating between these two incompatible hypotheses will require further studies at much localized scales