Dynamics of Fst for the island model

F(st) is a measure of genetic differentiation in a subdivided population. Sewall Wright observed that F(st)=1/1+2Nm in a haploid diallelic infinite island model, where N is the effective population size of each deme and m is the migration rate. In demonstrating this result, Wright relied on the infinite size of the population. Natural populations are not infinite and therefore they change over time due to genetic drift. In a finite population, F(st) becomes a random variable that evolves over time. In this work we ask, given an initial population state, what are the dynamics of the mean and variance of F(st) under the finite island model? In application both of these quantities are critical in the evaluation of F(st) data. We show that after a time of order N generations the mean of F(st) is slightly biased below 1/1+2Nm. Further we show that the variance of F(st) is of order 1/d where d is the number of demes in the population. We introduce several new mathematical techniques to analyze coalescent genealogies in a dynamic setting.     

The distribution of <Emphasis Type="Italic">F</Emphasis><Subscript><Emphasis Type="Italic">st</Emphasis></Subscript> and other genetic statistics for a class of population structure models

We examine genetic statistics used in the study of structured populations. In a 1999 paper, Wakeley observed that the coalescent process associated with the finite island model can be decomposed into a scattering phase and a collecting phase. This decomposition becomes exact in the large population limit with the coalescent at the end of the scattering phase converging to the Ewens sampling formula and the coalescent during the collecting phase converging to the Kingman coalescent. In this paper we introduce a class of limiting models, which we refer to as G/KC models, that generalize Wakeley’s decomposition. G in G/KC represents a completely general limit for the scattering phase, while KC represents a Kingman coalescent limit for the collecting phase. We show that both the island and two-dimensional stepping stone models converge to G/KC models in the large population limit. We then derive the distribution of the statistic F _st for all G/KC models under a large sample limit for the cases of strong or weak mutation, thereby deriving the large population, large sample limiting distribution of F _st for the island and two-dimensional stepping stone models as a special case of a general formula. Our methods allow us to take the large population and large sample limits simultaneously. In the context of large population, large sample limits, we show that the variance of F _st in the presence of weak mutation collapses as O(\frac1logd){O(\frac{1}{\log d})} where d is the number of demes sampled. Further, we show that this O(\frac1logd){O(\frac{1}{\log d})} is caused by a heavy tail in the distribution of F _st . Our analysis of F _st can be extended to an entire class of genetic statistics, and we use our approach to examine homozygosity measures. Our analysis uses coalescent based methods.     

Neutral two-locus multiple allele models with recombination

General formulae for the homozygosity and variance of linkage disequilibrium are derived for neutral, stationary, two-locus multiple allele models where there is a symmetric type of mutation at each locus. Particular cases examined are K allele models, the infinite alleles model, and the stepwise mutation model. The two-locus infinite allele model is examined at the molecular level and a joint probability generating function is found for the number of heterozygous sites at each locus in two randomly chosen gametes.     

Variogram analysis of the spatial genetic structure of continuous populations using multilocus microsatellite data

A geostatistical perspective on spatial genetic structure may explain methodological issues of quantifying spatial genetic structure and suggest new approaches to addressing them. We use a variogram approach to (i) derive a spatial partitioning of molecular variance, gene diversity, and genotypic diversity for microsatellite data under the infinite allele model (IAM) and the stepwise mutation model (SMM), (ii) develop a weighting of sampling units to reflect ploidy levels or multiple sampling of genets, and (iii) show how variograms summarize the spatial genetic structure within a population under isolation-by-distance. The methods are illustrated with data from a population of the epiphytic lichen Lobaria pulmonaria, using six microsatellite markers. Variogram-based analysis not only avoids bias due to the underestimation of population variance in the presence of spatial autocorrelation, but also provides estimates of population genetic diversity and the degree and extent of spatial genetic structure accounting for autocorrelation.     

Properties of a neutral allele model with intragenic recombination

An infinite-site neutral allele model with crossing-over possible at any of an infinite number of sites is studied. A formula for the variance of the number of segregating sites in a sample of gametes is obtained. An approximate expression for the expected homozygosity is also derived. Simulation results are presented to indicate the accuracy of the approximations. The results concerning the number of segregating sites and the expected homozygosity indicate that a two-locus model and the infinite-site model behave similarly for 4Nu less than or equal to 2 and r less than or equal to 5u, where N is the population size, u is the neutral mutation rate, and r is the recombination rate. Simulations of a two-locus model and a four-locus model were also carried out to determine the effect of intragenic recombination on the homozygosity test of Watterson (Genetics 85, 789-814; 88, 405-417) and on the number of unique alleles in a sample. The results indicate that for 4Nu less than or equal to 2 and r less than or equal to 10u, the effect of recombination is quite small.     

Invasion by extremes: population spread with variation in dispersal and reproduction

For populations having dispersal described by fat-tailed kernels (kernels with tails that are not exponentially bounded), asymptotic population spread rates cannot be estimated by traditional models because these models predict continually accelerating (asymptotically infinite) invasion. The impossible predictions come from the fact that the fat-tailed kernels fitted to dispersal data have a quality (nondiscrete individuals and, thus, no moment-generating function) that never applies to data. Real organisms produce finite (and random) numbers of offspring; thus, an empirical moment-generating function can always be determined. Using an alternative method to estimate spread rates in terms of extreme dispersal events, we show that finite estimates can be derived for fat-tailed kernels, and we demonstrate how variable reproduction modifies these rates. Whereas the traditional models define spread rate as the speed of an advancing front describing the expected density of individuals, our alternative definition for spread rate is the expected velocity for the location of the furthest-forward individual in the population. The asymptotic wave speed for a constant net reproductive rate R0 is approximated as (1/T)(piuR)/2)(1/2) m yr(-1), where T is generation time, and u is a distance parameter (m2) of Clark et al.'s 2Dt model having shape parameter p = 1. From fitted dispersal kernels with fat tails and infinite variance, we derive finite rates of spread and a simple method for numerical estimation. Fitted kernels, with infinite variance, yield distributions of rates of spread that are asymptotically normal and, thus, have finite moments. Variable reproduction can profoundly affect rates of spread. By incorporating the variance in reproduction that results from variable life span, we estimate much lower rates than predicted by the standard approach, which assumes a constant net reproductive rate. Using basic life-history data for trees, we show these estimated rates to be lower than expected from previous analytical models and as interpreted from paleorecords of forest spread at the end of the Pleistocene. Our results suggest reexamination of past rates of spread and the potential for future response to climate change.     

Statistics for Microsatellite Variation Based on Coalescence

The stepwise mutation model, which was at one time chiefly of interest in studying the evolution of protein charge-states, has recently undergone a resurgence of interest with the new popularity of microsatellites as phylogenetic markers. In this paper we describe a method which makes it possible to transfer many population genetics results from the standard infinite sites model to the stepwise mutation model. We study in detail the properties of pairwise differences in microsatellite repeat number between randomly chosen alleles. We show that the problem of finding the expected squared distance between two individuals and finding the variance of the squared distance can be reduced for a wide range of population models to finding the mean and mean square coalescence times. In many cases the distributions of coalescence times have already been studied for infinite site problems. In this study we show how to calculate these quantities for several population models. We also calculate the variance in mean squared pairwise distance (an estimator of mutation rate × population size) for samples of arbitrary size and show that this variance does not approach zero as the sample size increases. We can also use our method to study alleles at linked microsatellite loci. We suggest a metric which quantifies the level of association between loci—effectively a measure of linkage disequilibrium. It is shown that there can be linkage disequilibrium between partially linked loci at mutation–drift equilibrium.     

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.     

Estimation of effective population size and migration rate from one- and two-locus identity measures

Standard methods for inferring demographic parameters from genetic data are based mainly on one-locus theory. However, the association of genes at different loci (e.g., two-locus identity disequilibrium) may also contain some information about demographic parameters of populations. In this article, we define one- and two-locus parameters of population structure as functions of one- and two-locus probabilities for the identity in state of genes. Since these parameters are known functions of demographic parameters in an infinite island model, we develop moment-based estimators of effective population size and immigration rate from one- and two-locus parameters. We evaluate this method through simulation. Although variance and bias may be quite large, increasing the number of loci on which the estimates are derived improves the method. We simulate an infinite allele model and a K allele model of mutation. Bias and variance are smaller with increasing numbers of alleles per locus. This is, to our knowledge, the first attempt of a joint estimation of local effective population size and immigration rate.     

The lattice models of neutral multi-alleles in population genetics theory

The aim of this article is to study lattice models of neutral multi-alleles including Ohta-Kimura's step-wise mutation model. We shall show an outline of the construction of a unique strongly continuous non-negative semi-group associated with the infinite dimensional generator and show a general and straightforward method of obtaining the time dependent and equilibrium solutions of all polynomial moments of the gene frequencies. We shall discuss the spectrum of the diffusion processes and as an application we obtain all higher moments of the homozygosity.     

Mutation load and mutation-selection-balance in quantitative genetic traits

Haldane (1937) showed that the reduction of equilibrium mean fitness in an infinite population due to recurrent deleterious mutations depends only on the mutation rate but not on the harmfulness of mutants. His analysis, as well as more recent ones (cf. Crow 1970), ignored back mutation. The purpose of the present paper is to extend these results to arbitrary mutation patterns among alleles and to quantitative genetic traits. We derive first-order approximations for the equilibrium mean fitness (and the mutation load) and determine the order of the error term. For a metric trait under mutation-stabilizing-selection balance our result differs qualitatively from that of Crow and Kimura (1964), whose analysis is based on a Gaussian assumption. Our general approach also yields a mathematical proof that the variance under the usual mutation-stabilizing-selection model is, to first order, µ/s (the house-of-cards approximation) as µ/s tends to zero. This holds for arbitrary mutant distributions and does not require that the population mean coincide with the optimum. We show how the mutant distribution determines the order of the error term, and thus the accuracy of the house-of-cards approximation. Upper and lower bounds to the equilibrium variance are derived that deviate only to second order as µ/s tends to zero. The multilocus case is treated under the assumption of global linkage equilibrium.     

Mismatch repair blocks expansions of interrupted trinucleotide repeats in yeast

Disease-causing expansions of trinucleotide repeats (TNRs) can occur very frequently. In contrast, expansions are rare if the TNR is interrupted (imperfect). The molecular mechanism stabilizing interrupted alleles and thereby preventing disease has been elusive. We show that mismatch repair is the major stabilizing force for interrupted TNRs in Saccharomyces cerevisiae. Interrupted alleles expand much more often when mismatch repair is blocked by mutation or by poorly corrected mispairs. These results suggest that interruptions lead to mismatched expansion precursors. In normal cells, expansions are prevented in trans by mismatch repair, which coexcises the mismatches plus the aberrant, TNR-mediated secondary structure that otherwise resists removal. This study indicates a novel role for mismatch repair in mutation avoidance and, potentially, in disease prevention.     

DNA dinucleotide evolution in humans: fitting theory to facts

We examine length distributions of approximately 6000 human dinucleotide microsatellite loci, representing chromosomes 1-22, from the GDB database. Under the stepwise mutation model, results from theory and simulation are compared with the empirical data. In both constant and expanding population scenarios, a simple single-step model with parameters chosen to account for the observed variance of microsatellite lengths produces results inconsistent with the observed heterozygosity and the dispersion of length skewness. Complicating the model by allowing a variable mutation rate accounts for the homozygosity, and introducing a small probability of a large mutation step accounts for the dispersion in skewnesses. We discuss these results in light of the long-term evolution of microsatellites.     

F-statistics under alternation of sexual and asexual reproduction: a model and data from schistosomes (platyhelminth parasites)

Accurate inferences on population genetics data require a sound underlying theoretical null model. Nearly nothing is known about the gene dynamics of organisms with complex life cycles precluding any biological interpretation of population genetics parameters. In this article, we used an infinite island model to derive the expectations of those parameters for the life cycle of a dioecious organism obligatorily alternating sexual and asexual reproductions as it is the case for schistosomes (plathyhelminth parasites). This model allowed us to investigate the effects of the degree of mixing among individuals coming from different subpopulations at each new generation (represented in the model by the migration rates before and after clonal reproductions) and the variance in the reproductive success of individuals during the clonal phase. We also consider the effects of different migration rates and degrees of clonal reproductive skew between male and female individuals. Results show that the variance in the reproductive success of clones is very important in shaping the distribution of the genetic variability both within and among subpopulations. Thus, higher variance in the reproductive success of clones generates heterozygous excesses within subpopulations and also increases genetic differentiation between them. Migration occurring before and after asexual reproduction has different effects on the patterns of F(IS) and F(ST). When males and females display different degrees of reproductive skew or migration rates, we observe differences in their respective population genetic structure. While results of the model apply to any organism alternating sexual and clonal reproductions (e.g. all parasitic trematodes, many plants, and all aphididae), we finally confront some of these theoretical expectations to empirical data from Schistosoma mansoni infecting Rattus rattus in Guadeloupe.     

The genealogy, site frequency spectrum and ages of two nested mutant alleles

In this paper we consider the genealogy of two nested mutant alleles, assuming the constant-size neutral coalescent model with infinite sites mutation. We study the conditional genealogy and derive explicit formulas for the joint and marginal site frequency spectra for the double, single and zero mutant allele. In addition, we find the mean ages of the two mutations. We show that the age of the youngest mutation does not depend on the frequency of the single mutant allele and that the frequency spectra for the single mutant allele and the zero mutant allele are the same.     

The Inbreeding Effective Population Number and the Expected Homozygosity for an X-Linked Locus

Assuming random mating and discrete nonoverlapping generations, the inbreeding effective population number, (see PDF), is calculated for an X-linked locus. For large populations, the result agrees with the variance effective population number. As an application, the maintenance of genetic variability by the joint action of mutation and random drift is investigated. It is shown that, if every allele mutates at rate u to new types, then the probabilities of identity in state (and hence the expected homozygosity of females) converge to the approximate value (see PDF) at the approximate asymptotic rate (see PDF).     

Ancestral origins of the Machado-Joseph disease mutation: a worldwide haplotype study

Machado-Joseph disease (MJD) is an autosomal dominant neurodegenerative disorder originally described in families of Portuguese-Azorean ancestry. The cloning of the MJD1 gene allowed identification of the disease in many other populations, and MJD is now known to be the most common cause of dominant spinocerebellar ataxia. The hypothesis that its present world distribution could result from the spread of an original founder mutation has been raised, both at historical and molecular levels. In the present study, we tested this hypothesis by linkage-disequilibrium analysis of tightly linked polymorphisms and by haplotype comparison, in 249 families from different countries. We typed five microsatellite markers surrounding the MJD1 locus (D14S1015, D14S995, D14S973, D14S1016, and D14S977), and three intragenic single-base-pair polymorphisms (A(669)TG/G(669)TG, C(987)GG/G(987)GG, and TAA(1118)/TAC(1118)). The results show two different haplotypes, specific to the island of origin, in families of Azorean extraction. In families from mainland Portugal, both Azorean haplotypes can be found. The majority of the non-Portuguese families also share the same intragenic haplotype seen in the families coming from the island of Flores, but at least three other haplotypes were seen. These findings suggest two introductions of the mutation into the Portuguese population. Worldwide, the sharing of one intragenic haplotype by the majority of the families studied implies a founder mutation in MJD.     

Dynamics of stochastic epidemics on heterogeneous networks

Epidemic models currently play a central role in our attempts to understand and control infectious diseases. Here, we derive a model for the diffusion limit of stochastic susceptible-infectious-removed (SIR) epidemic dynamics on a heterogeneous network. Using this, we consider analytically the early asymptotic exponential growth phase of such epidemics, showing how the higher order moments of the network degree distribution enter into the stochastic behaviour of the epidemic. We find that the first three moments of the network degree distribution are needed to specify the variance in disease prevalence fully, meaning that the skewness of the degree distribution affects the variance of the prevalence of infection. We compare these asymptotic results to simulation and find a close agreement for city-sized populations.     

Joint Frequencies of Alleles Determined by Separate Formulations for the Mating and Mutation Systems

A method to calculate joint gene frequencies, which are the probabilities that two neutral genes taken at random from a population have certain allelic states, is developed taking into account the effects of the mating system and the mutation scheme. We assume that the mutation rates are constant in the population and that the mating system does not depend on allelic states. Under either--the condition that mutation rates are symmetric or that the mating unit is large and the mutation rate is small--the general formula is represented by two terms, one for the mating system and the other for the mutation scheme. The term for the mating system is expressed using the coancestry coefficient in the infinite allele model, and the term for the mutation scheme is a function of the eigenvalues and the eigenvectors of the mutation matrix. Several examples are presented as applications of the method, including homozygosity in a stepping-stone model with a symmetric mutation scheme.     

Asymptotic Behavior of Eigen’s Quasispecies Model

We study Eigen’s quasispecies model in the asymptotic regime where the length of the genotypes goes to \(\infty \) and the mutation probability goes to 0. A limiting infinite system of differential equations is obtained. We prove convergence of trajectories, as well as convergence of the equilibrium solutions. We give analogous results for a discrete-time version of Eigen’s model, which coincides with a model proposed by Moran.