Infinite Sites Model Research Articles

Maximum Likelihood Estimation of Population Divergence Times and Population Phylogenies under the Infinite Sites Model

In this paper, a maximum likelihood estimator of population divergence time based on the infinite sites model is developed. It is demonstrated how this estimator may be applied to obtain maximum likelihood estimates of the topology of population phylogenies. This approach addresses several classical problems occurring in the inference of the phylogenetic relationship of populations, most notably the problem of shared ancestral polymorphisms. The method is applied to previously published data sets of human African populations and of Caribbean hawksbill turtles.

Estimation of the amount of DNA polymorphism when the neutral mutation rate varies among sites.

Knowing the amount of DNA polymorphism is essential to understand the mechanism of maintaining DNA polymorphism in a natural population. The amount of DNA polymorphism can be measured by the average number of nucleotide differences per site (pi), the proportion of segregating (polymorphic) site (s) and the minimum number of mutations per site (s*). Since the latter two quantities depend on the sample size, theta is often used as a measure of the amount of DNA polymorphism, where theta = 4Nmu, N is the effective population size and mu is the neutral mutation rate per site per generation. It is known that theta estimated from pi, s and s* under the infinite site model can be biased when the mutation rate varies among sites. We have therefore developed new methods for estimating theta under the finite site model. Using computer simulations, it has been shown that the new methods give almost unbiased estimates even when the mutation rate varies among sites substantially. Furthermore, we have also developed new statistics for testing neutrality by modifying Tajima's D statistic. Computer simulations suggest that the new test statistics can be used even when the mutation rate varies among sites.

Model dependency of error thresholds: the role of fitness functions and contrasts between the finite and infinite sites models

Based on a deterministic mutation–selection model the concept of error thresholds is critically examined. It has often been argued that genetic information – for instance, an advantageous allele – can be selectively maintained in a population only if the mutation rate is below a certain limit, the error threshold, which is inversely related to the genome size. Here, I will show that such an inverse relationship strongly depends on the fitness model. To produce the error threshold, as given by Eigen (1971), requires that the fitness model is an extreme form of diminishing epistasis. The error threshold, in a strict sense, vanishes as epistasis changes from diminishing to synergistic. In the latter case even the usual definition of error thresholds becomes ambiguous. Initially, a finite sites model has been used to describe error thresholds. However, they can also be defined within the framework of the infinite sites model. I study both models in parallel and compare their properties as far as error thresholds are concerned. It is concluded that error thresholds possibly play a much less important role in molecular evolution than has often been assumed in the past.

Statistical properties of a DNA sample under the finite-sites model.

Statistical properties of a DNA sample from a random-mating population of constant size are studied under the finite-sites model. It is assumed that there is no migration and no recombination occurs within the locus. A Markov process model is used for nucleotide substitution, allowing for multiple substitutions at a single site. The evolutionary rates among sites are treated as either constant or variable. The general likelihood calculation using numerical integration involves intensive computation and is feasible for three or four sequences only, it may be used for validating approximate algorithms. Methods are developed to approximate the probability distribution of the number of segregating sites in a random sample of n sequences, with either constant or variable substitution rates across sites. Calculations using parameter estimates obtained for human D-loop mitochondrial DNAs show that among-site rate variation has a major effect on the distribution of the number of segregating sites; the distribution under the finite-sites model with variable rates among sites is quite different from that under the infinite-sites model.

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.

The effects of variable mutation rates across sites on the phylogenetic estimation of effective population size or mutation rate of DNA sequences.

Multiple hits at some sites of human mitochondrial DNA sequences suggest that the commonly assumed infinite-sites model can be violated. Under the neutral Wright-Fisher model without recombination and population subdivision, we investigated, by computer simulations, the effect of multiple hits on the estimation of the essential parameter theta = 4Nmu by FU's UPBLUE procedure. We found that with moderate mutation rate heterogeneity, UPBLUE performs very well in terms of unbiasness and efficiency. Under extreme mutation rate heterogeneity, if sample size is reasonably large (e.g., > 60), UPBLUE is still very satisfactory; otherwise we developed a new correction equation. Given knowledge of the degree of mutation rate heterogeneity, the performance of UPBLUE with the new correction equation was tested to be fairly satisfactory: there is almost no bias and the sampling variance is only slightly higher than the theoretical minimum variance. Thus, with an appropriate correction, UPBL.UE is relatively robust to the multiple hits. In genealogies reconstructed by UPGMA, we found that the total length of branches directly linked to the tips is underestimated, and those far away tend to be overestimated, while the total length of all branches is not biased.

Mitochondrial mismatch analysis is insensitive to the mutational process.

Mismatch distributions are histograms showing the pattern of nucleotide (or restriction) site differences between pairs of individuals in a sample. They can be used to test hypotheses about the history of population size and subdivision (if selective neutrality is assumed) or about selection (if a constant population size is assumed). Previous work has assumed that mutations never strike the same site twice, an assumption that is called the model of infinite sites. Fortunately, the results are surprisingly robust even when this assumption is violated. We show here that (1) confidence regions inferred using the infinite-sites model differ little from those inferred using a model of finite sites with uniform site-specific mutation rates, and (2) even when site-specific mutation rates follow a gamma distribution, confidence regions are little changed until the gamma shape parameter falls well below its plausible range, to roughly 0.01. In addition, we evaluate and reject the proposition that mismatch waves are produced by pooling data from several subdivisions of a structured population.

FIXATION PROBABILITIES OF SELFING RATE MODIFIERS IN SIMULATIONS WITH SEVERAL DELETERIOUS ALLELES WITH LINKAGE.

The fixation rates of selfing rate modifiers were found by stochastic simulation in an infinite site model, including effects of several deleterious alleles with variable effects, which were randomly distributed in the genome without assuming any pollen discounting. Previous results on the evolution of selfing obtained by more precise methods were in this study further validated, and it was concluded that the effect of genetic associations on the evolution of mating systems is small except in the case of full pollen discounting. Furthermore, attention was given to the uneven distribution of the genetic load in the population, and the accompanying large among-genome variation in fixation rates. This among-genome variation will be of significance for the evolution of mating systems.

The amount of DNA polymorphism maintained in a finite population when the neutral mutation rate varies among sites.

The expectations of the average number of nucleotide differences per site (pi), the proportion of segregating site (s), the minimum number of mutations per site (s*) and some other quantities were derived under the finite site models with and without rate variation among sites, where the finite site models include Jukes and Cantor's model, the equal-input model and Kimura's model. As a model of rate variation, the gamma distribution was used. The results indicate that if distribution parameter alpha is small, the effect of rate variation on these quantities are substantial, so that the estimates of theta based on the infinite site model are substantially underestimated, where theta = 4Nv, N is the effective population size and v is the mutation rate per site per generation. New methods for estimating theta are also presented, which are based on the finite site models with and without rate variation. Using these methods, underestimation can be corrected.

The Amount of Genetic Variation at the DNA Level Maintained in a Finite Population

Abstract The amount of DNA polymorphism can be estimated either from the average number of pairwise nucleotide differences per site (π) or from the proportion of segregating (polymorphic) site (s) among a sample of DNA sequences. Under a number of assumptions, including panmictic population and neutrality, it is known that the expectations of π and s are given by E(π) =θ and E(s) = a1(n)θ, where θ= 4Nv, a1(n) = 1 + 1/2 +…+ 1/(n‐1), N is the effective population size, v is the mutation rate per site per generation, and n is the sample size. Therefore, θ can be estimated by π or s/a1(n). These assumptions, however, are not always correct. In this paper, using a simple non‐random sampling model, I have examined the effect of non‐random sampling on the estimates of the amount of DNA polymorphism. The results indicate that the effect of non‐random sampling on the proportion of segregating site is substantially large whereas the effect of non‐random sampling on the average number of nucleotide differences per site is negligibly small unless non‐randomness is extremely large. Using a finite site model with and without rate variation, I have also examined the effect of rate variation among sites on the estimates of the amount of DNA polymorphism. The results indicate that if the neutral mutation rate varies among sites substantially, the estimates of θ based on the infinite site model are substantially underestimated. New methods for estimating θ are also presented.

Gene Genealogies Within Mutant Allelic Classes

A coalescent theory of the gene genealogy within an allelic class that arises by a unique mutational event is developed and analyzed. To interpret this theory it was necessary to expand on existing theory for populations of varying size. Two features of the gene genealogy--the average pairwise distance and the total tree length--within the mutant class and within the nonmutant class are found. An index, I, is proposed that describes the extent to which a genealogy is similar to one from a population of constant size (for which I = 0) or to a star genealogy (for which I = 1). The value of I is positive in growing populations and is generally positive for the gene genealogy for the mutant class. The value of I is negative for a population decreasing in size and for the nonmutant class, if the mutant arose recently. The results are discussed in the context of the infinite sites model of mutation, which is appropriate for nucleotide sequence data, and the generalized stepwise mutation model, which is appropriate for microsatellite loci. The same genealogical methods are used to find the probability of at least one recombination event between the nucleotide that defines an allelic class and a marker at a nearby linked site.

Infinite-allele model and infinite-site model in population genetics

Both the infinite-allele model and infinite-site model have contributed to development of population genetics. Although the former is a model mainly for protein polymorphism and the latter is mainly for DNA polymorphism, these two models are related: the expected heterozygosity and homozygosity can be obtained from the infinite-site model, and the expectation of the amount of DNA polymorphism can be obtained from the infinite-allele model.

The pattern of neutral molecular variation under the background selection model.

Stochastic simulations of the infinite sites model were used to study the behavior of genetic diversity at a neutral locus in a genomic region without recombination, but subject to selection against deleterious alleles maintained by recurrent mutation (background selection). In large populations, the effect of background selection on the number of segregating sites approaches the effect on nucleotide site diversity, i.e., the reduction in genetic variability caused by background selection resembles that caused by a simple reduction in effective population size. We examined, by coalescence-based methods, the power of several tests for the departure from neutral expectation of the frequency spectra of alleles in samples from randomly mating populations (Tajima's, Fu and Li's, and Watterson's tests). All of the tests have low power unless the selection against mutant alleles is extremely weak. In Drosophila, significant Tajima's tests are usually not obtained with empirical data sets from loci in genomic regions with restricted recombination frequencies and that exhibit low genetic diversity. This is consistent with the operation of background selection as opposed to selective sweeps. It remains to be decided whether background selection is sufficient to explain the observed extent of reduction in diversity in regions of restricted recombination.

Estimates of linkage disequilibrium and the recombination parameter determined from segregating nucleotide sites in the alcohol dehydrogenase region of Drosophila pseudoobscura.

The alcohol dehydrogenase (Adh) region of Drosophila pseudoobscura, which includes the two genes Adh and Adh-Dup, was used to examine the pattern and organization of linkage disequilibrium among pairs of segregating nucleotide sites. A collection of 99 strains from the geographic range of D. pseudoobscura were nucleotide-sequenced with polymerase chain reaction-mediated techniques. All pairs of the 359 polymorphic sites in the 3.5-kb Adh region were tested for significant linkage disequilibrium with Fisher's exact test. Of the 74,278 pairwise comparisons of segregating sites, 127 were in significant linkage disequilibrium at the 5% level. The distribution of five linkage disequilibrium estimators D(ij), D2, r(ij), r2 and D(ij) were compared to theoretical distributions. The observed distributions of D(ij), D2, r(ij) and r2 were consistent with the theoretical distribution given an infinite sites model. The observed distribution of D(ij) differed from the theoretical distribution because of an excess of values at -1 and 1. No spatial pattern was observed in the linkage disequilibrium pattern in the Adh region except for two clusters of sites nonrandomly associated in the adult intron and intron 2 of Adh. The magnitude of linkage disequilibrium decreases significantly as nucleotide distance increases, or a distance effect. Adh-Dup had a larger estimate of the recombination parameter, 4Nc, than Adh, where N is the effective population size and c is the recombination rate. A comparison of the mutation and recombination parameters shows that 7-17 recombination events occur for each mutation event. The heterogeneous estimates of the recombination parameter and the inverse relationship between linkage disequilibrium and nucleotide distance are no longer significant when the two clusters of Adh intron sites are excluded from analyses. The most likely explanation for the two clusters of linkage disequilibria is epistatic selection between sites in the cluster to maintain pre-mRNA secondary structure.

Estimation of levels of gene flow from DNA sequence data.

We compare the utility of two methods for estimating the average levels of gene flow from DNA sequence data. One method is based on estimating FST from frequencies at polymorphic sites, treating each site as a separate locus. The other method is based on computing the minimum number of migration events consistent with the gene tree inferred from their sequences. We compared the performance of these two methods on data that were generated by a computer simulation program that assumed the infinite sites model of mutation and that assumed an island model of migration. We found that in general when there is no recombination, the cladistic method performed better than FST while the reverse was true for rates of recombination similar to those found in eukaryotic nuclear genes, although FST performed better for all recombination rates for very low levels of migration (Nm = 0.1).

A multi-dimensional coalescent process applied to multi-allelic selection models and migration models

For a sample of two genes from a population divided into an arbitrary number of allele classes, a general mathematical framework is developed to address the expectation and variance of the time of the most recent common ancestor. Depending on the meaning of allele classes and the manner in which genes can change among them, this framework can be applied to a diversity of population genetic models. By adoption of the infinite sites model, the effect on heterozygosity is modelled for balancing selection among allele classes, mutation between allele classes, migration among populations, and gene conversion between loci. Most results are described for a continuous time approximation to a discrete generation model. It is also shown how the discrete generation model can be used to study the hitch-hiking effect of favorable mutations.

Estimating the recombination parameter of a finite population model without selection.

SummaryAn estimator is proposed for the parameter C = 4Nc. where N is the population size andcis the recombination rate. The estimator is appropriate for use with sequence or restriction site data from random samples from within populations. Properties of the estimator are investigated for an infinite-sites neutral model using Monte Carlo simulations. The median and mode of the distribution of the estimator are close to the true value for all parameter values examined, but large data sets are required to obtain reliable estimates.

On the divergence of genes in multigene families

Statistical properties of the amount of divergence of genes in multigene families are studied. The model considered is an infinite-site neutral model with unbiased intrachromosomal conversion, unbiased interchromosomal conversion, and recombination. By considering the time back to the most recent common ancestor of two genes, both the probability of identity and the moments of S, the number of sites that differ between two sampled genes, are obtained. We find that (a) if recombination rates are large or conversion is always interchromosomal, then the expectation of S is 4 Nμn where N is the poulation size, μ is the rate of mutation per generation per gene and n is the number of genes in the gene family, (b) as the conversion rates approach zero, the moments of divergence do not approach the moments of divergence with conversion rates equal to zero, and (c) it is possible for a decrease in the rate of intrachromosomal conversion to result in a higher probability of identity, but a greater mean divergence of the two genes.

DIFFERENTIATION OF A MULTIGENE FAMILY BETWEEN POPULATIONS.

Differentiation of a multigene family between populations is investigated under the forces of intrachromosomal unbiased gene conversion, neutral mutation, recombination, and random drift. The infinite-site model without recombination within the gene is employed, and the average numbers of the different sites between pairs of genes at the same locus and at different loci from two populations that divided some time ago are calculated as functions of time. Since differentiation is a relative concept in which differences between populations are expressed in comparison with those within a population and since there is most likely a reduction in population size after the division of populations, the time-dependent behavior of the corresponding quantities within a population is also investigated using approximate and numerical methods. It is found that intrachromosomal unbiased gene conversion promotes differentiation at the same locus but promotes or retards that between different loci, depending on the parameters in a recently divided population.

On the divergence of alleles in nested subsamples from finite populations.

Within-population variation at the DNA level will rarely be studied by sequencing of loci of randomly chosen individuals. Instead, individuals will usually be chosen for sequencing based on some knowledge of their genotype. Data collected in this way require new sampling theory. Motivated by these observations, we have examined the sampling properties of a finite population model with two mutation processes and with no selection or recombination. One mutation process generates new alleles according to an infinite-alleles model, and the other generates polymorphisms at sites according to an infinite-sites model. A sample of n genes is considered. The stationary distribution of the number of segregating sites in a subsample from one of the allelic classes in the sample conditional on the allelic configuration of the sample is studied. A recursive scheme is developed to compute the moments of this distribution, and it is shown that the distribution is functionally independent of the number of additional alleles in the sample and their respective frequencies in the sample. For the case in which the sample contains only two alleles, the distribution of the number of segregating sites in a subsample containing both alleles conditional on the sample frequencies of the alleles is studied. The results are applied to the analysis of DNA sequences of two alleles found at the Adh locus of Drosophila melanogaster. No significant departure from the neutral model is detected.