Equilibrium Values of Measures of Population Subdivision for Stepwise Mutation Processes

Expected values of WRIGHT's F-statistics are functions of probabilities of identity in state. These values may be quite different under an infinite allele model and under stepwise mutation processes such as those occurring at microsatellite loci. However, a relationship between the probability of identity in state in stepwise mutation models and the distribution of coalescence times can be deduced from the relationship between probabilities of identity by descent and the distribution of coalescence times. The values of F(IS) and F(ST) can be computed using this property. Examination of the conditional probability of identity in state given some coalescence time and of the distribution of coalescence times are also useful for explaining the properties of F(IS) and F(ST) at high mutation rate loci, as shown here in an island model of population structure.     

On the Genealogy of a Rare Allele

The gene genealogy is derived for a rare allele that is descended from a mutant ancestor that arose at a fixed time in the past. Following Thompson (1976,Amer. J. Human Genet.28, 442–452), the fractional linear branching process is used as a model of the demography of a rare allele. The model does not require the total population size to be constant or the mutant class to be neutral; so long as individuals in the class are selectively equivalent, the class as a whole may have a selective advantage, or disadvantage, relative to other alleles in the population. An exact result is given for the joint probability distribution of the coalescence times among a sample of alleles descended from the mutant. A method is described for rapidly simulating these coalescence times. The relationship between the genealogical structure of a discrete generation branching process and a continuous generation birth–death process is elucidated. The theory may be applied to the problem of estimating the ages of rare nonrecurrent mutations.     

PREDICTING NUCLEAR GENE COALESCENCE FROM MITOCHONDRIAL DATA: THE THREE-TIMES RULE

Abstract.— Coalescence theory predicts when genetic drift at nuclear loci will result in fixation of sequence differences to produce monophyletic gene trees. However, the theory is difficult to apply to particular taxa because it hinges on genetically effective population size, which is generally unknown. Neutral theory also predicts that evolution of monophyly will be four times slower in nuclear than in mitochondrial genes primarily because genetic drift is slower at nuclear loci. Variation in mitochondrial DNA (mtDNA) within and between species has been studied extensively, but can these mtDNA data be used to predict coalescence in nuclear loci? Comparison of neutral theories of coalescence of mitochondrial and nuclear loci suggests a simple rule of thumb. The “three‐times rule” states that, on average, most nuclear loci will be monophyletic when the branch length leading to the mtDNA sequences of a species is three times longer than the average mtDNA sequence diversity observed within that species. A test using mitochondrial and nuclear intron data from seven species of whales and dolphins suggests general agreement with predictions of the three‐times rule. We define the coalescence ratio as the mitochondrial branch length for a species divided by intraspecific mtDNA diversity. We show that species with high coalescence ratios show nuclear monophyly, whereas species with low ratios have polyphyletic nuclear gene trees. As expected, species with intermediate coalescence ratios show a variety of patterns. Especially at very high or low coalescence ratios, the three‐times rule predicts nuclear gene patterns that can help detect the action of selection. The three‐times rule may be useful as an empirical benchmark for evaluating evolutionary processes occurring at multiple loci.     

Structured Coalescent Processes on Different Time Scales

It is demonstrated that the structured coalescent model can readily be extended to include phenomena such as partial selfing and background selection through the use of an approximation based on separation of time scales. A model that includes these phenomena, as well as geographic subdivision and linkage to a polymorphism maintained either by local adaptation or by balancing selection, is derived, and the expected coalescence time for a pair of genes is calculated. It is found that background selection reduces coalescence times within subpopulations and allelic classes, leading to a high degree of apparent differentiation. Extremely high levels of subpopulation differentiation are also expected for regions of the genome surrounding loci important in local adaptation. These regions will be wider the stronger the local selection, and the higher the selfing rate.     

Maximum-likelihood estimation of coalescence times in genealogical trees

We develop a method for maximum-likelihood estimation of coalescence times in genealogical trees, based on population genetics data. For this purpose, a Viterbi-type algorithm is constructed to maximize the joint likelihood of the coalescence times. Marginal confidence intervals for the coalescence times based on the profile likelihoods are also computed. Our method of finding MLEs and calculating C.I.'s appears to be more accurate than alternative numerical maximization methods, and maximum-likelihood inference appears to be more accurate than other existing model-free approaches to estimating coalescent times. We demonstrate the method on two different data sets: human Y chromosome DNA data and fungus DNA data.     

Inbreeding coefficients and coalescence times.

This paper describes the relationship between probabilities of identity by descent and the distribution of coalescence times. By using the relationship between coalescence times and identity probabilities, it is possible to extend existing results for inbreeding coefficients in regular systems of mating to find the distribution of coalescence times and the mean coalescence times. It is also possible to express Sewall Wright's FST as the ratio of average coalescence times of different pairs of genes. That simplifies the analysis of models of subdivided populations because the average coalescence time can be found by computing separately the time it takes for two genes to enter a single subpopulation and time it takes for two genes in the same subpopulation to coalesce. The first time depends only on the migration matrix and the second time depends only on the total number of individuals in the population. This approach is used to find FST in the finite island model and in one- and two-dimensional stepping-stone models. It is also used to find the rate of approach of FST to its equilibrium value. These results are discussed in terms of different measures of genetic distance. It is proposed that, for the purposes of describing the amount of gene flow among local populations, the effective migration rate between pairs of local populations, M, which is the migration rate that would be estimated for those two populations if they were actually in an island model, provides a simple and useful measure of genetic similarity that can be defined for either allozyme or DNA sequence data.     

THE EFFECT OF COLLECTIVE DISPERSAL ON THE GENETIC STRUCTURE OF A SUBDIVIDED POPULATION

Correlated dispersal paths between two or more individuals are widespread across many taxa. The population genetic implications of this collective dispersal have received relatively little attention. Here we develop two‐sample coalescent theory that incorporates collective dispersal in a finite island model to predict expected coalescence times, genetic diversities, and F‐statistics. We show that collective dispersal reduces mixing in the system, which decreases expected coalescence times and increases F_ST. The effects are strongest in systems with high migration rates. Collective dispersal breaks the invariance of within‐deme coalescence times to migration rate, whatever the deme size. It can also cause F_ST to increase with migration rate because the ratio of within‐ to between‐deme coalescence times can decrease as migration rate approaches unity. This effect is most biologically relevant when deme size is small. We find qualitatively similar results for diploid and gametic dispersal. We also demonstrate with simulations and analytical theory the strong similarity between the effects of collective dispersal and anisotropic dispersal. These findings have implications for our understanding of the balance between drift–migration–mutation in models of neutral evolution. This has applied consequences for the interpretation of genetic structure (e.g., chaotic genetic patchiness) and estimation of migration rates from genetic data.     

The coalescent in a continuous,finite, linear population

In this article we present a model for analyzing patterns of genetic diversity in a continuous, finite, linear habitat with restricted gene flow. The distribution of coalescent times and locations is derived for a pair of sequences sampled from arbitrary locations along the habitat. The results for mean time to coalescence are compared to simulated data. As expected, mean time to common ancestry increases with the distance separating the two sequences. Additionally, this mean time is greater near the center of the habitat than near the ends. In the distant past, lineages that have not undergone coalescence are more likely to have been at opposite ends of the population range, whereas coalescent events in the distant past are biased toward the center. All of these effects are more pronounced when gene flow is more limited. The pattern of pairwise nucleotide differences predicted by the model is compared to data collected from sardine populations. The sardine data are used to illustrate how demographic parameters can be estimated using the model.     

Linkage disequilibrium under skewed offspring distribution among individuals in a population

Correlations in coalescence times between two loci are derived under selectively neutral population models in which the offspring of an individual can number on the order of the population size. The correlations depend on the rates of recombination and random drift and are shown to be functions of the parameters controlling the size and frequency of these large reproduction events. Since a prediction of linkage disequilibrium can be written in terms of correlations in coalescence times, it follows that the prediction of linkage disequilibrium is a function not only of the rate of recombination but also of the reproduction parameters. Low linkage disequilibrium is predicted if the offspring of a single individual frequently replace almost the entire population. However, high linkage disequilibrium can be predicted if the offspring of a single individual replace an intermediate fraction of the population. In some cases the model reproduces the standard Wright-Fisher predictions. Contrary to common intuition, high linkage disequilibrium can be predicted despite frequent recombination, and low linkage disequilibrium under infrequent recombination. Simulations support the analytical results but show that the variance of linkage disequilibrium is very large.     

From gene trees to species trees II: species tree inference by minimizing deep coalescence events

When gene copies are sampled from various species, the resulting gene tree might disagree with the containing species tree. The primary causes of gene tree and species tree discord include incomplete lineage sorting, horizontal gene transfer, and gene duplication and loss. Each of these events yields a different parsimony criterion for inferring the (containing) species tree from gene trees. With incomplete lineage sorting, species tree inference is to find the tree minimizing extra gene lineages that had to coexist along species lineages; with gene duplication, it becomes to find the tree minimizing gene duplications and/or losses. In this paper, we present the following results: 1) The deep coalescence cost is equal to the number of gene losses minus two times the gene duplication cost in the reconciliation of a uniquely leaf labeled gene tree and a species tree. The deep coalescence cost can be computed in linear time for any arbitrary gene tree and species tree. 2) The deep coalescence cost is always not less than the gene duplication cost in the reconciliation of an arbitrary gene tree and a species tree. 3) Species tree inference by minimizing deep coalescence events is NP-hard.     

Asymptotic Distributions of Coalescence Times and Ancestral Lineage Numbers for Populations with Temporally Varying Size

The distributions of coalescence times and ancestral lineage numbers play an essential role in coalescent modeling and ancestral inference. Both exact distributions of coalescence times and ancestral lineage numbers are expressed as the sum of alternating series, and the terms in the series become numerically intractable for large samples. More computationally attractive are their asymptotic distributions, which were derived in Griffiths (1984) for populations with constant size. In this article, we derive the asymptotic distributions of coalescence times and ancestral lineage numbers for populations with temporally varying size. For a sample of size n, denote by T_m the mth coalescent time, when m + 1 lineages coalesce into m lineages, and A_n(t) the number of ancestral lineages at time t back from the current generation. Similar to the results in Griffiths (1984), the number of ancestral lineages, A_n(t), and the coalescence times, T_m, are asymptotically normal, with the mean and variance of these distributions depending on the population size function, N(t). At the very early stage of the coalescent, when t → 0, the number of coalesced lineages n − A_n(t) follows a Poisson distribution, and as m → n, n(n ? 1)T_m/2N(0) follows a gamma distribution. We demonstrate the accuracy of the asymptotic approximations by comparing to both exact distributions and coalescent simulations. Several applications of the theoretical results are also shown: deriving statistics related to the properties of gene genealogies, such as the time to the most recent common ancestor (TMRCA) and the total branch length (TBL) of the genealogy, and deriving the allele frequency spectrum for large genealogies. With the advent of genomic-level sequencing data for large samples, the asymptotic distributions are expected to have wide applications in theoretical and methodological development for population genetic inference.     

Inbreeding and relatedness coefficients: what do they measure?

This paper reviews and discusses what is known about the relationship between identity in state, allele frequency, inbreeding coefficients, and identity by descent in various uses of these terms. Generic definitions of inbreeding coefficients are given, as ratios of differences of probabilities of identity in state. Then some of their properties are derived from an assumption in terms of differences between distributions of coalescence times of different genes. These inbreeding coefficients give an approximate measurement of how much higher the probability of recent coalescence is for some pair of genes relative to another pair. Such a measure is in general not equivalent to identity by descent; rather, it approximates a ratio of differences of probabilities of identity by descent. These results are contrasted with some other formulas relating identity, allele frequency, and inbreeding coefficients. Additional assumptions are necessary to obtain most of them, and some of these assumptions are not always correct, for example when there is localized dispersal. Therefore, definitions based on such formulas are not always well-formulated. By contrast, the generic definitions are both well-formulated and more broadly applicable.     

The effect of selection on genealogies

The coalescent process can describe the effects of selection at linked loci only if selection is so strong that genotype frequencies evolve deterministically. Here, we develop methods proposed by Kaplan, Darden, and Hudson to find the effects of weak selection. We show that the overall effect is given by an extension to Price's equation: the change in properties such as moments of coalescence times is equal to the covariance between those properties and the fitness of the sample of genes. The distribution of coalescence times differs substantially between allelic classes, even in the absence of selection. However, the average coalescence time between randomly chosen genes is insensitive to the current allele frequency and is affected significantly by purifying selection only if deleterious mutations are common and selection is strong (i.e., the product of population size and selection coefficient, Ns>3). Balancing selection increases mean coalescence times, but the effect becomes large only when mutation rates between allelic classes are low and when selection is extremely strong. Our analysis supports previous simulations that show that selection has surprisingly little effect on genealogies. Moreover, small fluctuations in allele frequency due to random drift can greatly reduce any such effects. This will make it difficult to detect the action of selection from neutral variation alone.     

Synchronization of chaotic nonlinear continuous neural networks with time-varying delay

In this paper, the synchronization problem for delayed continuous time nonlinear complex neural networks is considered. The delay dependent state feed back synchronization gain matrix is obtained by considering more general case of time-varying delay. Using Lyapunov stability theory, the sufficient synchronization criteria are derived in terms of Linear Matrix Inequalities (LMIs). By decomposing the delay interval into multiple equidistant subintervals, Lyapunov-Krasovskii functionals (LKFs) are constructed on these intervals. Employing these LKFs, new delay dependent synchronization criteria are proposed in terms of LMIs for two cases with and without derivative of time-varying delay. Numerical examples are illustrated to show the effectiveness of the proposed method.     

An application of the central limit theorem to coalescence times in the structured coalescent model with strong migration

The structured coalescent describes the ancestral relationship among sampled genes from a geographically structured population. The aim of this article is to apply the central limit theorem to functionals of the migration process to study coalescence times and population structure. An application of the law of large numbers to the migration process leads to the strong migration limit for the distributions of coalescence times. The central limit theorem enables us to obtain approximate distributions of coalescence times for strong migration. We show that approximate distributions depend on the population structure. If migration is conservative and strong, we can define a kind of effective population size N _e ^*, with which the entire population approximately behaves like a panmictic population. On the other hand, the approximate distributions for nonconservative migration are qualitatively different from those for conservative migration. And the entire population behaves unlike a panmictic population even though migration is strong.     

Estimating the age of alleles by use of intraallelic variability.

A method is presented for estimating the age of an allele by use of its frequency and the extent of variation among different copies. The method uses the joint distribution of the number of copies in a population sample and the coalescence times of the intraallelic gene genealogy conditioned on the number of copies. The linear birth-death process is used to approximate the dynamics of a rare allele in a finite population. A maximum-likelihood estimate of the age of the allele is obtained by Monte Carlo integration over the coalescence times. The method is applied to two alleles at the cystic fibrosis (CFTR) locus, deltaF508 and G542X, for which intraallelic variability at three intronic microsatellite loci has been examined. Our results indicate that G542X is somewhat older than deltaF508. Although absolute estimates depend on the mutation rates at the microsatellite loci, our results support the hypothesis that deltaF508 arose < 500 generations (approximately 10,000 years) ago.     

Eradicating vector-borne diseases via age-structured culling

We derive appropriate mathematical models to assess the effectiveness of culling as a tool to eradicate vector-borne diseases. The model, focused on the culling strategies determined by the stages during the development of the vector, becomes either a system of autonomous delay differential equations with impulses (in the case where the adult vector is subject to culling) or a system of nonautonomous delay differential equations where the time-varying coefficients are determined by the culling times and rates (in the case where only the immature vector is subject to culling). Sufficient conditions are derived to ensure eradication of the disease, and simulations are provided to compare the effectiveness of larvicides and insecticide sprays for the control of West Nile virus. We show that eradication of vector-borne diseases is possible by culling the vector at either the immature or the mature phase, even though the size of the vector is oscillating and above a certain level.      

Applying the Cox proportional hazards model when the change time of a binary time-varying covariate is interval censored

This paper develops methodology for estimation of the effect of a binary time-varying covariate on failure times when the change time of the covariate is interval censored. The motivating example is a study of cytomegalovirus (CMV) disease in patients with human immunodeficiency virus (HIV) disease. We are interested in determining whether CMV shedding predicts an increased hazard for developing active CMV disease. Since a clinical screening test is needed to detect CMV shedding, the time that shedding begins is only known to lie in an interval bounded by the patient's last negative and first positive tests. In a Cox proportional hazards model with a time-varying covariate for CMV shedding, the partial likelihood depends on the covariate status of every individual in the risk set at each failure time. Due to interval censoring, this is not always known. To solve this problem, we use a Monte Carlo EM algorithm with a Gibbs sampler embedded in the E-step. We generate multiple completed data sets by drawing imputed exact shedding times based on the joint likelihood of the shedding times and event times under the Cox model. The method is evaluated using a simulation study and is applied to the data set described above.     

The effect of a barrier to gene flow on patterns of geographic variation.

Explicit formulae are given for the effects of a barrier to gene flow on random fluctuations in allele frequency; these formulae can also be seen as generating functions for the distribution of coalescence times. The formulae are derived using a continuous diffusion approximation, which is accurate over all but very small spatial scales. The continuous approximation is confirmed by comparison with the exact solution to the stepping stone model. In both one and two spatial dimensions, the variance of fluctuations in allele frequencies increases near the barrier; when the barrier is very strong, the variance doubles. However, the effect on fluctuations close to the barrier is much greater when the population is spread over two spatial dimensions than when it occupies a linear, one-dimensional habitat: barriers of strength comparable with the dispersal range (B approximately equal to sigma) can have an appreciable effect in two dimensions, whereas only barriers with strength comparable with the characteristic scale (B approximately equal to L=sigma/sqrt{2mu}) are significant in one dimension (mu is the rate of mutation or long-range dispersal). Thus, in a two-dimensional population, barriers to gene flow can be detected through their effect on the spatial pattern of genetic marker alleles.