A genome-wide departure from the standard neutral model in natural populations of Drosophila

We analyze nucleotide polymorphism data for a large number of loci in areas of normal to high recombination in Drosophila melanogaster and D. simulans (24 and 16 loci, respectively). We find a genome-wide, systematic departure from the neutral expectation for a panmictic population at equilibrium in natural populations of both species. The distribution of sequence-based estimates of 2Nc across loci is inconsistent with the assumptions of the standard neutral theory, given the observed levels of nucleotide diversity and accepted values for recombination and mutation rates. Under these assumptions, most estimates of 2Nc are severalfold too low; in other words, both species exhibit greater intralocus linkage disequilibrium than expected. Variation in recombination or mutation rates is not sufficient to account for the excess of linkage disequilibrium. While an equilibrium island model does not seem to account for the data, more complicated forms of population structure may. A proper test of alternative demographic models will require loci to be sampled in a more consistent fashion.     

Linkage and the Maintenance of Heritable Variation by Mutation-Selection Balance

The role of linkage in influencing heritable variation maintained through a balance between mutation and stabilizing selection is investigated for two different models. In both cases one trait is considered and the interactions within and between loci are assumed to be additive. Contrary to most earlier investigations of this problem no a priori assumptions on the distribution of genotypic values are imposed. For a deterministic two-locus two-allele model with recombination and mutation, related to the symmetric viability model, a complete nonlinear analysis is performed. It is shown that, depending on the recombination rate, multiple stable equilibria may coexist. The equilibrium genetic and genic variances are calculated. For a polygenic trait in a finite population with a possible continuum of allelic effects a simulation study is performed. In both models the equilibrium genetic and genic variances are roughly equal to the house-of-cards prediction or its finite population counterpart as long as the recombination rate is not extremely low. However, negative linkage disequilibrium builds up. If the loci are very closely linked the equilibrium additive genetic variance is slightly lower than the house-of-cards prediction, but the genic variance is much higher. Depending on whether the parameters are in favor of the house-of-cards or the Gaussian approximation, different behavior of the genetic system occurs with respect to linkage.     

Recombination,selection, and the evolution of tandem gene arrays

Multigene families—immunity genes or sensory receptors, for instance—are often subject to diversifying selection. Allelic diversity may be favored not only through balancing or frequency-dependent selection at individual loci but also by associating different alleles in multicopy gene families. Using a combination of analytical calculations and simulations, we explored a population genetic model of epistatic selection and unequal recombination, where a trade-off exists between the benefit of allelic diversity and the cost of copy abundance. Starting from the neutral case, where we showed that gene copy number is Gamma distributed at equilibrium, we derived also the mean and shape of the limiting distribution under selection. Considering a more general model, which includes variable population size and population substructure, we explored by simulations mean fitness and some summary statistics of the copy number distribution. We determined the relative effects of selection, recombination, and demographic parameters in maintaining allelic diversity and shaping the mean fitness of a population. One way to control the variance of copy number is by lowering the rate of unequal recombination. Indeed, when encoding recombination by a rate modifier locus, we observe exactly this prediction. Finally, we analyzed the empirical copy number distribution of 3 genes in human and estimated recombination and selection parameters of our model.     

Multilocus selection in subdivided populations I. Convergence properties for weak or strong migration

The dynamics and equilibrium structure of a deterministic population-genetic model of migration and selection acting on multiple multiallelic loci is studied. A large population of diploid individuals is distributed over finitely many demes connected by migration. Generations are discrete and nonoverlapping, migration is irreducible and aperiodic, all pairwise recombination rates are positive, and selection may vary across demes. It is proved that, in the absence of selection, all trajectories converge at a geometric rate to a manifold on which global linkage equilibrium holds and allele frequencies are identical across demes. Various limiting cases are derived in which one or more of the three evolutionary forces, selection, migration, and recombination, are weak relative to the others. Two are particularly interesting. If migration and recombination are strong relative to selection, the dynamics can be conceived as a perturbation of the so-called weak-selection limit, a simple dynamical system for suitably averaged allele frequencies. Under nondegeneracy assumptions on this weak-selection limit which are generic, every equilibrium of the full dynamics is a perturbation of an equilibrium of the weak-selection limit and has the same stability properties. The number of equilibria is the same in both systems, equilibria in the full (perturbed) system are in quasi-linkage equilibrium, and differences among allele frequencies across demes are small. If migration is weak relative to recombination and epistasis is also weak, then every equilibrium is a perturbation of an equilibrium of the corresponding system without migration, has the same stability properties, and is in quasi-linkage equilibrium. In both cases, every trajectory converges to an equilibrium, thus no cycling or complicated dynamics can occur.      

Frequency- and density-dependent selection on a quantitative character

The equilibrium distribution of a quantitative character subject to frequency- and density-dependent selection is found under different assumptions about the genetical basis of the character that lead to a normal distribution in a population. Three types of models are considered: (1) one-locus models, in which a single locus has an additive effect on the character, (2) continuous genotype models, in which one locus or several loci contribute additively to a character, and there is an effectively infinite range of values of the genotypic contributions from each locus, and (3) correlation models, in which the mean and variance of the character can change only through selection at modifier loci. It is shown that the second and third models lead to the same equilibrium values of the total population size and the mean and variance of the character. One-locus models lead to different equilibrium values because of constraints on the relationship between the mean and variance imposed by the assumptions of those models.——The main conclusion is that, at the equilibrium reached under frequency- and density-dependent selection, the distribution of a normally distributed quantitative character does not depend on the underlying genetic model as long as the model imposes no constraints on the mean and variance.     

Evolution of Bacterial Transformation: Is Sex with Dead Cells Ever Better than No Sex at All?

Computer simulations of bacterial transformation are used to show that, under a wide range of biologically reasonable assumptions, transforming populations undergoing deleterious mutation and selection have a higher mean fitness at equilibrium than asexual populations. The source of transforming DNA, the amount of DNA taken up by each transforming cell, and the relationship between number of mutations and cell viability (the fitness function) are important factors. When the DNA source is living cells, transformation resembles meiotic sex. When the DNA source is cells killed by selection against mutations, transformation increases the average number of mutations per genome but can nevertheless increase the mean fitness of the population at equilibrium. In a model of regulated transformation, in which the most fit cells of a transforming population do not transform, transforming populations are always fitter at equilibrium than asexual populations. These results show that transformation can reduce mutation load.     

Expected Shannon Entropy and Shannon Differentiation between Subpopulations for Neutral Genes under the Finite Island Model

Shannon entropy H and related measures are increasingly used in molecular ecology and population genetics because (1) unlike measures based on heterozygosity or allele number, these measures weigh alleles in proportion to their population fraction, thus capturing a previously-ignored aspect of allele frequency distributions that may be important in many applications; (2) these measures connect directly to the rich predictive mathematics of information theory; (3) Shannon entropy is completely additive and has an explicitly hierarchical nature; and (4) Shannon entropy-based differentiation measures obey strong monotonicity properties that heterozygosity-based measures lack. We derive simple new expressions for the expected values of the Shannon entropy of the equilibrium allele distribution at a neutral locus in a single isolated population under two models of mutation: the infinite allele model and the stepwise mutation model. Surprisingly, this complex stochastic system for each model has an entropy expressable as a simple combination of well-known mathematical functions. Moreover, entropy- and heterozygosity-based measures for each model are linked by simple relationships that are shown by simulations to be approximately valid even far from equilibrium. We also identify a bridge between the two models of mutation. We apply our approach to subdivided populations which follow the finite island model, obtaining the Shannon entropy of the equilibrium allele distributions of the subpopulations and of the total population. We also derive the expected mutual information and normalized mutual information (“Shannon differentiation”) between subpopulations at equilibrium, and identify the model parameters that determine them. We apply our measures to data from the common starling (Sturnus vulgaris) in Australia. Our measures provide a test for neutrality that is robust to violations of equilibrium assumptions, as verified on real world data from starlings.     

Age-specific life history tactics in organisms with determinate growth: Optimal models for non-optimal behavior?

Among organisms with determinate growth, optimization models predict that reproductive effort should increase as individuals approach old age, but the assumptions of these models may be inappropriate because the senescence that generates the necessary selective pressure may be not itself be optimal. Population genetics models were constructed to examine whether genes for age-specific changes in reproductive effort could invade a population in which senescence was maintained at equilibrium levels by a balance between mutation and selection. In asexually reproducing organisms, it was found that strategies of increasing reproductive effort could not normally invade the population. In sexually reproducing organisms, however, recombination was found to be important and genes for age-specific changes in effort could spread in the population under most circumstances.     

The Sampling Distribution of Linkage Disequilibrium

The probabilities of obtaining particular samples of gametes with two completely linked loci are derived. It is assumed that the population consists of N diploid, randomly mating individuals, that each of the two loci mutate according to the infinite allele model at a rate µ and that the population is at equilibrium. When 4Nµ is small, the most probable samples of gametes are those that segregate only two alleles at either locus. The probabilities of various samples of gametes are discussed. The results show that most samples with completely linked loci have either a very small or a very large association between the alleles of each locus. This causes the distribution of linkage disequilibrium to be skewed and the distribution of the correlation coefficient to be bimodal. The correlation coefficient is commonly used as a test statistic with a chi square distribution and yet has a bimodal distribution when the loci are completely linked. Thus, such a test is not likely to be accurate unless the rate of recombination between the loci and/or the effective population size are sufficiently large enough so that the loci can be treated as unlinked.     

The evolution of function-valued traits for conditional cooperation

In this paper we study the evolution of function-valued traits for cooperation in environments that display varying degrees of population viscosity. Traits measure an individual's intrinsic propensity to cooperate in a standard bilateral Prisoner's dilemma and can be increasing, decreasing or constant functions of the probability to interact with individuals of ones own genotype. We first analyse adaptation to homogenous environments (with constant degree of viscosity). Comparing environments characterized by different degrees of viscosity, we find that the relation between viscosity and the equilibrium type distribution is not monotone. In fact, it is possible that in fluid populations (no viscosity) there is more cooperation in equilibrium than in populations with intermediate degrees of viscosity. In a second step we analyse heterogenous environments (with varying degrees of viscosity). We find that under very weak assumptions on the distribution of the viscosity parameter strictly increasing functions are always selected and under some parameter constellations they are uniquely so.     

On the determination of recombination rates in intermated recombinant inbred populations

The recurrent intermating of F(2) individuals for some number of generations followed by several generations of inbreeding produces an intermated recombinant inbred (IRI) population. Such populations are currently being developed in the plant-breeding community because linkage associations present in an F(2) population are broken down and a population of fixed inbred lines is also created. The increased levels of recombination enable higher-resolution mapping in IRI populations relative to F(2) populations. Herein we derive relationships, under several limiting assumptions, for determining the expected recombination fraction in IRI populations from the crossover rate per meiosis. These relationships are applicable to situations where the inbreeding component of IRI population development is by either self-fertilization or full-sib mating. Additionally, we show that the derived equations can be solved for the crossover rate per meiosis if the recombination fraction is known for the IRI population. Thus, the observed recombination fraction in any IRI population can be expressed on an F(2) basis. The implications of this work on the expansion of genetic maps in IRI populations and limits for detecting linkage between markers are also considered.     

The Affected Sib Method. I. Statistical Features of the Affected Sib-Pair Method

The distribution of the number of HLA haplotypes shared by sibs affected with the same HLA-linked disease can be used to obtain information on the genetics of the disease. Since the inception of the use of sib-pair methods for the analysis of the HLA-associated diseases, the question has been raised of how to include families with more than two affected sibs in the sib-pair analysis. This paper presents appropriate weighting schemes. A procedure for estimating the frequency of the disease allele in the general population, under the assumptions of single-allele recessive, additive, dominant and intermediate models, with negligible recombination (theta = 0) between the disease-predisposing gene and the HLA region, and no selective disadvantage of the trait, is also given. Cluster-sampling techniques are used in the analysis.     

Recombination and the properties of Tajima's D in the context of approximate-likelihood calculation

I show that Tajima's D, a commonly used summary of the site-frequency spectrum for single-nucleotide polymorphism data, is a biased summary of the site-frequency spectrum. Under neutral models, this bias depends on the population recombination rate. This bias of D in summarizing the data makes inference of demographic parameters sensitive to assumptions about recombination rates.     

The ideal free distribution as an evolutionarily stable strategy

We examine the evolutionary stability of strategies for dispersal in heterogeneous patchy environments or for switching between discrete states (e.g. defended and undefended) in the context of models for population dynamics or species interactions in either continuous or discrete time. There have been a number of theoretical studies that support the view that in spatially heterogeneous but temporally constant environments there will be selection against unconditional, i.e. random, dispersal, but there may be selection for certain types of dispersal that are conditional in the sense that dispersal rates depend on environmental factors. A particular type of dispersal strategy that has been shown to be evolutionarily stable in some settings is balanced dispersal, in which the equilibrium densities of organisms on each patch are the same whether there is dispersal or not. Balanced dispersal leads to a population distribution that is ideal free in the sense that at equilibrium all individuals have the same fitness and there is no net movement of individuals between patches or states. We find that under rather general assumptions about the underlying population dynamics or species interactions, only such ideal free strategies can be evolutionarily stable. Under somewhat more restrictive assumptions (but still in considerable generality), we show that ideal free strategies are indeed evolutionarily stable. Our main mathematical approach is invasibility analysis using methods from the theory of ordinary differential equations and nonnegative matrices. Our analysis unifies and extends previous results on the evolutionary stability of dispersal or state-switching strategies.     

On the Recombination Rate Estimation in the Presence of Population Substructure

As recombination events are not uniformly distributed along the human genome, the estimation of fine-scale recombination maps, e.g. HapMap Project, has been one of the major research endeavors over the last couple of years. For simulation studies, these estimates provide realistic reference scenarios to design future study and to develop novel methodology. To achieve a feasible framework for the estimation of such recombination maps, existing methodology uses sample probabilities for a two-locus model with recombination, with recent advances allowing for computationally fast implementations. In this work, we extend the existing theoretical framework for the recombination rate estimation to the presence of population substructure. We show under which assumptions the existing methodology can still be applied. We illustrate our extension of the methodology by an extensive simulation study.     

Optimal Sequencing Strategies for Surveying Molecular Genetic Diversity

Two commonly used measures of genetic diversity for intraspecies DNA sequence data are based, respectively, on the number of segregating sites, and on the average number of pairwise nucleotide differences. Expressions are derived for their variance in the presence of intragenic recombination for a panmictic population of fixed size that is at neutral equilibrium at the region sequenced. We show that, in contrast to the slow decrease in variance with increasing sample size, if the recombination rate is nonzero, the asymptotic rate of decrease of variance with increasing sequence length, for fixed sample size, is quite rapid. In particular, it is close to that which would be obtained by sequencing independent chromosome regions. The correlation between measures of diversity from linked regions is also examined. For a given total number of bases sequenced in a particular region, optimal sequencing strategies are derived. These typically involve sequencing relatively few (three to 10) long copies of the region. Under optimal strategies, the variances of the two measures are very similar for most parameter values considered. Results concerning optimal sequencing strategies will be sensitive to gross departures from the underlying assumptions, such as population bottlenecks, selective sweeps, and substantial population substructure.     

High intraspecific recombination rate in a native population of Candidatus pelagibacter ubique (SAR11)

Recombination is an important process in microbial evolution. Rates of recombination with extracellular DNA matter because models of microbial population structure are profoundly influenced by the degree to which recombination is occurring within the population. Low rates of recombination may be sufficient to ensure the lateral propagation of genes that have a high selective advantage without disrupting the clonal pattern of inheritance for other genes. High rates of recombination potentially can obscure clonal patterns, leading to linkage equilibrium, and give microbial populations a population genetic structure more akin to sexually interbreeding eukaryotic populations. We examined eight loci from nine strains of candidatus Pelagibacter ubique (SAR11), isolated from a single 2L niskin sample of natural seawater, for evidence of genetic recombination between strains. The Shimodaira-Hasegawa test revealed significant phylogenetic incongruence in seven of the genes, indicating that frequent recombination obscures phylogenetic signals from the linear inheritance of genes in this population. Statistical evidence for intragenic recombination was found for six loci. An informative sites matrix showed extensive evidence for a widespread breakdown of linkage disequilibrium. Although the mechanisms of genetic transfer in native SAR11 populations are unknown, we measured recombination rates, rho, that are much higher than point mutation rates, theta, as a source of genetic diversity in this clade. The eukaryotic model of species sharing a common pool of alleles is more apt for this SAR11 population than a strictly clonal model of inheritance in which allelic diversity is controlled by periodic selection.     

The ideal free distribution as an evolutionarily stable strategy

We examine the evolutionary stability of strategies for dispersal in heterogeneous patchy environments or for switching between discrete states (e.g. defended and undefended) in the context of models for population dynamics or species interactions in either continuous or discrete time. There have been a number of theoretical studies that support the view that in spatially heterogeneous but temporally constant environments there will be selection against unconditional, i.e. random, dispersal, but there may be selection for certain types of dispersal that are conditional in the sense that dispersal rates depend on environmental factors. A particular type of dispersal strategy that has been shown to be evolutionarily stable in some settings is balanced dispersal, in which the equilibrium densities of organisms on each patch are the same whether there is dispersal or not. Balanced dispersal leads to a population distribution that is ideal free in the sense that at equilibrium all individuals have the same fitness and there is no net movement of individuals between patches or states. We find that under rather general assumptions about the underlying population dynamics or species interactions, only such ideal free strategies can be evolutionarily stable. Under somewhat more restrictive assumptions (but still in considerable generality), we show that ideal free strategies are indeed evolutionarily stable. Our main mathematical approach is invasibility analysis using methods from the theory of ordinary differential equations and nonnegative matrices. Our analysis unifies and extends previous results on the evolutionary stability of dispersal or state-switching strategies.     

Some genetic consequences of skewed fecundity distributions in plants

Summary Most plant populations show a skewedrd distribution of fecundity amongst their members, in contrast to the poisson distribution assumed by most population genetical theory. We examine by simulation the consequences of skewed fecundity for plant evolution when combined with sieve selection. In comparison with poisson-based theory, plant populations are likely to show a faster response to selection, especially when the favoured allele is at a low frequency. Selection against a deleterious immigrant allele will also be more effective, reducing its equilibrium frequency in a population. In the special case of heterozygote disadvantage traits will evolve that could not under poisson theory. However, random variation is also higher, giving a 10-plant population an effective population size of about 6.4 under poisson theory. The conclusions are not qualitatively changed by different assumptions on the exact shape of the fecundity distribution, or on heritability, or on reproduction by the smallest plants of the population.     

Birth, growth and death as structuring operators in bacterial population dynamics

A new model is presented that describes microbial population dynamics that emerge from complex interactions among birth, growth and death as oriented, discrete events. Specifically, birth and death act as structuring operators for individual organisms within the population, which become synchronised as age clusters (called cell generations that are structured in age classes) that are born at the same time and die in concert; a pattern very consistent with recent experimental data that show bacterial group death correlates with temporal population dynamics in chemostats operating at carrying capacity. Although the model only assumes “natural death” (i.e., no death from predation or antimicrobial exposure), it indicates that short-term non-linear dynamic behaviour can exist in a bacterial population growing under longer term pseudo-steady-state conditions (a confined dynamic equilibrium). After summarizing traditional assumptions about bacterial aging, simulations of batch, continuous-flow, and bioreactors with recycle are used to show how population dynamics vary as function of hydraulic retention time, microbial kinetics, substrate level, and other factors that cause differential changes in the distribution of living and dead cells within the system. In summary, we show that population structures induced by birth and death (as discrete and delayed events) intrinsically create a non-linear dynamic system, implying that a true steady state can never exist in growing bacterial populations. This conclusion is discussed within the context of process stability in biotechnology.