A nonparametric approach to analyzing large human populations.

There are many statistical techniques that require the assumption that the population being studied is normally distributed--regression analysis, multivariate analysis, time series analysis, and so on. Unfortunately, as the development of survey sampling has long acknowledged, large human populations are usually stratified into several different subpopulations. Since the boundaries between the strata are somewhat blurred, they are not independent, so the overall distribution of the population tends to be multimodal rather than normal. In this paper, a technique is developed to find these multimodal techniques using nonparametric density estimation. Its effectiveness is demonstrated by means of an example.     

Evolutionary stability for large populations

We present a revision of Maynard Smith's evolutionary stability criteria for populations which are very large (though technically finite) and of unknown size. We call this the large population ESS, as distinct from Maynard Smith's infinite population ESS and Schaffer's finite population ESS. Building on Schaffer's finite population model, we define the large population ESS as a strategy which cannot be invaded by any finite number of mutants, as long as the population size is sufficiently large. The large population ESS is not equivalent to the infinite population ESS: we give examples of games in which a large population ESS exists but an infinite population ESS does not, and vice versa. Our main contribution is a simple set of two criteria for a large population ESS, which are similar (but not identical) to those originally proposed by Maynard Smith for infinite populations.     

Spatial ecology of large herbivore populations

Models of the dynamics of large herbivore populations represent density feedbacks on the population growth rate either directly or indirectly through interactions with vegetation resources. Neither approach incorporates the spatial heterogeneity that is an essential feature of most natural environments, and modifies the population dynamics generated. This is especially true for large herbivores exploiting food resources that are rooted in space but temporally variable in quantity and quality both seasonally and annually. In this review I explore how environmental variation at different spatiotemporal scales influences the abundance of herbivore populations controlled via resources, predators or social mechanisms. Changes in abundance can be spatially disparate and dependent on different resource components at different stages of the seasonal cycle, including buffer resources restricting population crashes in extremely adverse years. GPS telemetry enables movement responses generating spatial patterns to be documented in fine spatiotemporal detail, including migration and dispersal. Models incorporating spatial heterogeneity either implicitly or explicitly are outlined, exemplifying how herbivores cope with temporal variability by exploiting spatial variability in resources and conditions. Global human dominance is generating widened climatic variation while opportunities for herbivore movements are becoming constricted. Theoretical population ecologists need to shift their focus from the workings of demographic structure towards effects of changing environmental contexts, in order to project the likely trajectories of large herbivore populations through the Anthropocene.     

Equilibrium distributions for deleterious genes in large stationary populations.

In a large population of constant size, there is a unique equilibrium distribution for every deleterious autosomal dominant or deleterious X-linked gene. The purpose of this paper is to determine the mean vector and covariance matrix for such an equilibrium distribution. The theory of branching processes with immigration provides the framework for our investigation. Autosomal dominants can be treated using single-type branching processes; X-linked genes, using two-type branching processes. Application is made to Huntington's chorea and Becker's muscular dystrophy.     

Changes in large herbivore populations across large areas of Tanzania

We collated aerial census data collected during the late 1980s to early 2000s for large herbivore populations in eight large census zones in Tanzania. Of the ungulate populations that showed significant changes in densities at the start versus end of this decade, most declined; very few populations increased significantly. Thomson's gazelle, Grant's gazelle, hartebeest, reedbuck, roan antelope, sable antelope, warthog and zebra, for example, declined in over 50% of the zones where they were surveyed. Interestingly, small‐bodied species fared particularly poorly in many census zones, whereas elephant and giraffe generally fared well across the country. Most populations of all herbivores declined in some portions of the country (e.g. Burigi‐Biharamulo, Katavi, Greater Ruaha and Tarangire census zones). These surveys suggest that, even in a country renowned for its protected areas and conservation commitment, some large herbivore populations need more conservation attention in order to remain stable.     

Cultural barriers associated with large gene frequency differences among Italian populations.

Analysis of geographic variation for eight red cell markers in Italy shows significant spatial structure for most alleles. Effective population sizes estimated from FST values at these loci are much smaller than those predicted from data on consanguineous marriage, suggesting the presence of factors (presumably barriers) that have reduced gene flow and enhanced the evolutionary weight of genetic drift. Most regions of sharp gene frequency change correspond to geographic and linguistic barriers. Two allele frequencies are significantly correlated with measures of linguistic differentiation but not with indexes describing broad religious and social attitudes. The similarity between patterns of genetic and linguistic variation in Italy, also observed in a previous study, suggests that in specific areas linguistic diversity has acted as a biological barrier constraining mating, dispersal, or both. There is no evidence for a similar role of other extent cultural barriers.     

Stochastic selection in large and small populations

We describe two models of stochastic variation in selection intensity. In both models the arithmetic mean fitness of all genotypes is equal; in both models the geometric mean fitness of the heterozygous genotype is greater than that of both homozygous genotypes. In one model the correlation between the fitnesses of the homozygous genotypes is +1; in the other it is −1. We show that the expected time to absorption of an allele in a finite population is significantly retarded for all initial gene frequencies in the former model. The expected time to absorption of an allele in the latter model is retarded only at extreme initial gene frequencies; at intermediate initial gene frequencies the expected time to absorption is accelerated. We conclude that the criterion for polymorphism based on the geometric mean of the heterozygote being greater than that of both homozygotes provides only limited information about the fate of gene frequency.     

The speed of adaptation in large asexual populations

In large asexual populations, beneficial mutations have to compete with each other for fixation. Here, I derive explicit analytic expressions for the rate of substitution and the mean beneficial effect of fixed mutations, under the assumptions that the population size N is large, that the mean effect of new beneficial mutations is smaller than the mean effect of new deleterious mutations, and that new beneficial mutations are exponentially distributed. As N increases, the rate of substitution approaches a constant, which is equal to the mean effect of new beneficial mutations. The mean effect of fixed mutations continues to grow logarithmically with N. The speed of adaptation, measured as the change of log fitness over time, also grows logarithmically with N for moderately large N, and it grows double-logarithmically for extremely large N. Moreover, I derive a simple formula that determines whether at given N beneficial mutations are expected to compete with each other or go to fixation independently. Finally, I verify all results with numerical simulations.     

Persistence times of populations with large random fluctuations

The growth of populations which undergo large random fluctuations can be modelled with stochastic differential equations involving Poisson processes. The problem of determining the persistence time is that of finding the time of first passage to some small critical population size. We consider in detail a simple model of logistic growth with additive Poisson disasters of fixed magnitude. The expectation and variability of the persistence time are obtained as solutions of singular differential-difference equations. The dependence of the persistence time of a colonizing species on the parameters of the model is discussed. The model may also be viewed as random harvesting with fixed quotas and a comparison is made between the mean extinction time and those for deterministic models.     

Molecular hyperdiversity and evolution in very large populations

The genomic density of sequence polymorphisms critically affects the sensitivity of inferences about ongoing sequence evolution, function and demographic history. Most animal and plant genomes have relatively low densities of polymorphisms, but some species are hyperdiverse with neutral nucleotide heterozygosity exceeding 5%. Eukaryotes with extremely large populations, mimicking bacterial and viral populations, present novel opportunities for studying molecular evolution in sexually reproducing taxa with complex development. In particular, hyperdiverse species can help answer controversial questions about the evolution of genome complexity, the limits of natural selection, modes of adaptation and subtleties of the mutation process. However, such systems have some inherent complications and here we identify topics in need of theoretical developments. Close relatives of the model organisms Caenorhabditis elegans and Drosophila melanogaster provide known examples of hyperdiverse eukaryotes, encouraging functional dissection of resulting molecular evolutionary patterns. We recommend how best to exploit hyperdiverse populations for analysis, for example, in quantifying the impact of noncrossover recombination in genomes and for determining the identity and micro‐evolutionary selective pressures on noncoding regulatory elements.     

Genetic draft and quasi-neutrality in large facultatively sexual populations

Large populations may contain numerous simultaneously segregating polymorphisms subject to natural selection. Since selection acts on individuals whose fitness depends on many loci, different loci affect each other’s dynamics. This leads to stochastic fluctuations of allele frequencies above and beyond genetic drift—an effect known as genetic draft. Since recombination disrupts associations between alleles, draft is strong when recombination is rare. Here, we study a facultatively outcrossing population in a regime where the frequency of outcrossing and recombination, r, is small compared to the characteristic scale of fitness differences σ. In this regime, fit genotypes expand clonally, leading to large fluctuations in the number of recombinant offspring genotypes. The power law tail in the distribution of the latter makes it impossible to capture the dynamics of draft by an effective neutral model. Instead, we find that the fixation time of a neutral allele increases only slowly with the population size but depends sensitively on the ratio r/σ. The efficacy of selection is reduced dramatically and alleles behave “quasi-neutrally” even for Ns≫1, provided that |s| < s_c, where s_c depends strongly on r/σ, but only weakly on population size N. In addition, the anomalous fluctuations due to draft change the spectrum of (quasi)-neutral alleles from f(ν) ∼ ν⁻¹, corresponding to drift, to ∼ ν⁻². Finally, draft accelerates the rate of two-step adaptations through deleterious intermediates.THE genetic diversity of a population is determined by mutation, selection, recombination, and genetic drift, i.e., the stochasticity inherent in reproduction. Understanding how genetic diversity depends on these elements of evolutionary dynamics is central to population genetics, since it allows us to make inferences about the past history and to predict how rapidly populations can adapt.Population genetic inference focuses on the diversity at putatively neutral sites and assumes that the history of these sites is described by the neutral “coalescent” (Kingman 1982). Coalescent theory models the genealogy of asexual organisms or nonrecombining segments of a genome by positing that lineages merge at random, backward in time, due to common ancestry. Under this assumption, the mean time to the most recent common ancestor, T_C, of the extant N individuals, is 2N generations. The coalescence timescale is very important, since the genetic diversity of the population is given by the number of mutations that occur in all lineages descending from the most recent common ancestor of the population. Genetic diversity is therefore controlled by T_C and hence, under the assumption of neutral evolution, proportional to N. [Coalescent theory has been extended to weak selection (Krone and Neuhauser 1997) and recombination (Hudson 1983; Griffiths and Marjoram 1996).]However, the prediction that neutral genetic diversity is proportional to N is at odds with observations: Population sizes of different organisms differ by many orders of magnitude, while genetic variation among organisms is fairly constant (Lewontin 1974). To resolve this “paradox of variation”, Maynard Smith and Haigh (1974) suggested that selection acting on linked loci might reduce diversity at a neutral locus. Rapid fixation of a novel mutation at a linked locus will perturb the allele frequencies. These perturbations can bring alleles to fixation and, more generally, reduce the coalescence time and hence the average genetic diversity (Kaplan et al. 1989; Barton 1998; Gillespie 2001). Such “hitchhiking” of neutral alleles on linked selected loci will dominate over genetic drift in large populations. Since hitchhiking leads to larger perturbations for more closely linked loci, one expects genetic variation to correlate with the recombination rate, as is indeed observed in Drosophila (Begun and Aquadro 1992; Stephan and Mitchell 1992; Andolfatto and Przeworski 2001; Sella et al. 2009).A related effect was described earlier by Hill and Robertson (1966), who studied the reduction in the fixation probability of a novel beneficial mutation because of selection acting at a linked locus. This effect is commonly known as Hill–Robertson interference (Felsenstein 1974). Hitchhiking and Hill–Robertson interference are different aspects of the same phenomenon, one focusing on the effects of linked selection on genetic diversity and the other on the efficacy of selection. While hitchhiking and Hill–Robertson effects are primarily associated with positive selection for novel alleles, purifying selection against deleterious mutations also affects genetic diversity. The elimination of (neutral) alleles linked deleterious mutations is known as background selection. The lower the recombination rates are, the larger is the target for linked deleterious mutations, resulting in stronger background selection (Charlesworth et al. 1993; Hudson and Kaplan 1995; Nordborg et al. 1996).Most models used to study Hill–Robertson and hitchhiking effects between positively selected mutations consider only two loci. Deleterious mutations, however, are expected to be much more frequent, and background selection models typically consider many mutations with small deleterious effects. A systematic study of the effect of interference between many weakly selected sites in a mutation/selection/drift equilibrium was presented by McVean and Charlesworth (2000), who used computer simulations of a model of codon bias evolution. They showed that linkage-dependent interference between a large number of weakly selected sites has substantial effects on genetic diversity, fixation probability of mutations, and the degree of adaptation measured as the frequency of preferred codons. This and subsequent computational studies reinforced the understanding that the Hill–Roberson effect reduces the effectiveness of selection and made clear that a quantitative understanding of Hill–Robertson effects in multilocus systems requires an analysis that goes beyond two-locus models (Comeron and Kreitman 2002; Seger et al. 2010); see Comeron et al. (2008) for a recent review.It is common to interpret the effect of linked selection in terms of increased variance in offspring number. In this interpretation, linked selection can be thought of as a stochastic force analogous to genetic drift and is often referred to as genetic draft—a term coined by Gillespie (2000). Following Hill and Robertson (1966) and Felsenstein (1974), the increased variance in offspring number is often captured by a reduction in the “effective population” size, N_e, in a neutral model (which means enhanced drift and accelerated coalescence). It has, however, been noted that a rescaled neutral model does not consistently explain all observables (Charlesworth et al. 1993; Braverman et al. 1995; Fay and Wu 2000; McVean and Charlesworth 2000; Barton and Etheridge 2004; Seger et al. 2010) and that different effective population sizes need to be defined depending on the question and timescale of interest (Ewens 2004; Karasov et al. 2010). Furthermore, the dependence of N_e on the actual population size and other relevant parameters is not understood (Wiehe and Stephan 1993; Gillespie 2000; Lynch 2007).Here, we provide analytic results on the effect of draft in a stochastic multilocus evolution model. Instead of a mutation/selection equilibrium considered in McVean and Charlesworth (2000), our focus here is a continuously adapting and facultatively sexual population, like human immunodeficiency virus (HIV) in coevolution with the host’s immune system. Our model and its relation to the biology of HIV are described in more detail below. The scope of the model, however, extends beyond HIV and is equally applicable to scenarios where background selection is dominant. Many important and well-studied organisms such as influenza, yeast, and plants are facultatively sexual. Rice, for example, is a partly selfing species and strong selection has acted during its domestication (Caicedo et al. 2007). While dominance effects can render the selfing of diploid organisms more complicated than facultatively sexual propagation of haploid organisms (Charlesworth et al. 1991; Kelly and Williamson 2000), our analysis still provides a null model on top of which dominance effects can be investigated.Using computer simulations of an adapting population, we first demonstrate how quantities such as the coalescence time, the fixation probability of beneficial or deleterious mutations, and the allele frequency spectra depend on the rate of outcrossing relative to selection. We also show that our in silico observations cannot be described by a neutral model with a reduced effective population size. This is because single genotypes can, through clonal expansion, give rise to a wildly fluctuating number of recombinant genotypes. The distribution is so broad that its variance diverges, which in turn makes an effectively neutral diffusion limit impossible. To provide an analytic understanding of the simulation results, we use a branching process model that allows us to study the stochastic dynamics of novel mutations (neutral, beneficial, and deleterious) as they spread through the population. We calculate fixation probabilities and the typical time to fixation, T_fix (and more generally, the probability of attaining n copies after time T), for a new mutant allele, making explicit the dependence on the rate of recombination, fitness variance, and the population size. An important consequence of genetic draft is a qualitatively different frequency spectrum of rare alleles, which we also calculate analytically. Finally, we show that empirical HIV allele frequency spectra are in agreement with our theoretical prediction, confirming the relevance of our model to the dynamics of HIV adaptation.     

Evolution favors protein mutational robustness in sufficiently large populations

Background  

An important question is whether evolution favors properties such as mutational robustness or evolvability that do not directly benefit any individual, but can influence the course of future evolution. Functionally similar proteins can differ substantially in their robustness to mutations and capacity to evolve new functions, but it has remained unclear whether any of these differences might be due to evolutionary selection for these properties.