Infinite Sites Model Research Articles

Statistical properties of the number of recombination events in the history of a sample of DNA sequences.

Some statistical properties of samples of DNA sequences are studied under an infinite-site neutral model with recombination. The two quantities of interest are R, the number of recombination events in the history of a sample of sequences, and RM, the number of recombination events that can be parsimoniously inferred from a sample of sequences. Formulas are derived for the mean and variance of R. In contrast to R, RM can be determined from the sample. Since no formulas are known for the mean and variance of RM, they are estimated with Monte Carlo simulations. It is found that RM is often much less than R, therefore, the number of recombination events may be greatly under-estimated in a parsimonious reconstruction of the history of a sample. The statistic RM can be used to estimate the product of the recombination rate and the population size or, if the recombination rate is known, to estimate the population size. To illustrate this, DNA sequences from the Adh region of Drosophila melanogaster are used to estimate the effective population size of this species.

Lines-of-descent and genealogical processes, and their applications in population genetics models

This paper reviews a variety of results for genealogical (or line-of-descent) processes that arise in connection with the theory of some classical selectively neutral haploid population genetics models. While some new results and derivations are included, the principal aim of the paper is to demonstrate the central importance and simplicity of genealogical Markov chains in this theory. Considerable attention is given to ‘diffusion time scale’ approximations of such genealogical processes. A wide variety of results pertinent to (diffusion approximations of) the classical multi-allele single-locus WrightFisher model and its relatives are unified by this approach. Other examples where the genealogical process plays an explicit role (for example, the infinite-sites models) are discussed.

Estimation of the neutral mutation rate in a finite population from DNA sequence data

The sampling theory for the infinite site model taking into account the phylogenetic relationship between the alleles is developed for those cases in which two or three alleles are observed in the sample. From this theory a maximum likelihood estimate of θ = 4 Nμ can be obtained. Unlike the maximum likelihood estimate of θ based on the infinite allele model or the number of segregating sites, this estimate of θ is a function of the frequencies of the alleles. This method is used to estimate θ for mitochondrial DNA in Drosophila melanogaster and D. virilis from data obtained by Shah and Langley (1979. Nature (London) 281, 696–699) using restriction endonucleases.

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 4 Nu ⩽ 2 and r ⩽ 5 u, 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 4 Nu ⩽ 2 and r ⩽ 10 u, the effect of recombination is quite small.

The distribution of nucleotide site differences between two finite sequences

A simple method to obtain the distribution of site differences between two randomly chosen cistrons in a finite population is shown. It is assumed that the number of sites and the number of alleles at each site are finite and that no recombination occurs between sites. The mean and variance of the percent difference and the number of site differences between the two sequences are shown to be simple functions of two variables. The differences between a finite-site model an infinite-site model are discussed. The results are applied to the β-globin polymorphism in man.

The number of heterozygous loci between two randomly chosen completely linked sequences of loci in two subdivided population models

The stationary probability distribution of the number of heterozygous loci in two randomly chosen sequences of completely linked infinite alleles loci, with mutation at each locus, is found in the island model for within and between islands. Results for an infinite site model are found as a limit. A single charge state locus is also studied in the island model and distributions found for the charge difference between two genes. Similar results are derived for a stepping stone model.

Transient distribution of the number of segregating sites in a neutral infinite-sites model with no recombination

The transient distribution of the number of segregating sites in a sample from a large population of 2Ngenes is found. Segregating sites are split into those in common with the sites segregating in the initial population and those segregating due to new mutations. The waiting time distribution for a population to lose all of its initial sites is also studied. A neutral infinite-sites model with no recombination is used. This paper extends the work of Watterson (1975), from the stationary case; and of Li (1977) from the transient distribution in a sample of 2.

Transient distribution of the number of segregating sites in a neutral infinite-sites model with no recombination

The transient distribution of the number of segregating sites in a sample from a large population of 2Ngenes is found. Segregating sites are split into those in common with the sites segregating in the initial population and those segregating due to new mutations. The waiting time distribution for a population to lose all of its initial sites is also studied. A neutral infinite-sites model with no recombination is used. This paper extends the work of Watterson (1975), from the stationary case; and of Li (1977) from the transient distribution in a sample of 2.

A model of electrophoretic variation: Finite populations and finite numbers of mutable sites

In this paper we present a model of electrophoretic variation for a protein with M amino acid positions, each of which can be in the selectively equivalent charge states 0 and 1. At each amino acid position the mutation rate from one charge state to the other is v. We derive the expected variance in charge, for a population of N individuals in generation t, which is α 2(t) = [ 2MNv (1 + 8Nv) ]{1 − exp[−(4v + 1 2N )t]} for an initially monomorphic population. This variance, which is unaffected by intragenic recombination and gene conversion, does not increase linearly with N as does the steady-state variance in charge for the older electrophoretic models. We also derive the probability, P j ( t), that a random gene in a population exceeds another by j charge units. We conclude: (1) that, except for species which have maintained effective population numbers in the millions, the infinite mutable sites model adequately describes “neutral” genetic variation; (2) that very high levels of “neutral” genetic variation can be attained only by genes with a fairly large number of “neutral” amino acid sites in species which have maintained an effective size in the millions for millions of generations; (3) that, because all species surveyed have a mean heterozygosity of less than 25 %, it is likely that most genes have only a few “neutral” covarions.

A model allowing continuous variation in electrophoretic mobility of neutral alleles.

The infinite-sites model with no recombination is extended to include mutations that affect electrophoretic mobility. The model allows the effect of a single-site mutation to have a continuous effect on mobility. Formulae are obtained for the variance of electrophoretic mobility of alleles after an arbitrary lenght of time. A special case of the general model is the case of stepwise production of neutral alleles with an arbitrary number of steps.

Hidden genetic variability within electromorphs in finite populations.

The amount of hidden genetic variability within electromorphs in finite populations is studied by using the infinite site model and stepwise mutation model simultaneously. A formula is developed for the bivariate probability generating function for the number of codon differences and the number of electromorph state differences between two randomly chosen cistrons. Using this formula, the distribution as well as the mean and variance of the number of codon differences between two identical or nonidentical electromorphs are studied. The distribution of the number of codon differences between two randomly chosen identical electromorphs is similar to the geometric distribution but more leptokurtic. Studies are also made on the number of codon differences between two electromorphs chosen at random one from each of two populations which have been separated for an arbitrary number of generations. It is shown that the amount of hidden genetic variability is very large if the product of effective population size and mutation rate is large.

A note on the sampling theory for infinite alleles and infinite sites models

This note considers sampling theory for a selectively neutral locus where it is supposed that the data provide nucleotide sequences for the genes sampled. It thus anticipates that technical advances will soon provide data of this form in volume approaching that currently obtained from electrophoresis. The assumption made on the nature of the data will require us to use, in the terminology of Kimura ( Theor. Pop. Biol. 2, 174–208 (1971) ), the “infinite sites” model of Karlin and McGregor ( Proc. Fifth Berkeley Symp. Math. Statist. Prob. 4, 415–438 (1967) ) rather that the “infinite alleles” model of Kimura and Crow ( Genetics 49, 174–738 (1964) ). We emphasize that these two models refer not to two different real-world circumstances, but rather to two different assumptions concerning our capacity to investigate the real world. We compare our results where appropriate with corresponding sampling theory of Ewens ( Theor. Pop. Biol. 3, 87–112 (1972) ) for the “infinite alleles” model. Note finally that some of our results depend on an assumption of independence of behavior at individual sites; a parallel paper by Watterson (submitted for publication (1974)) assumes no recombination between sites. Real-world behavior will lie between these two assumptions, closer to the situation assumed by Watterson than in this note. Our analysis provides upper bounds for increased efficiency in using complete nucleotide sequences.

Care of the dying.

Analysis of genomic data requires an efficient way to calculate likelihoods across very large numbers of loci. We describe a general method for finding the distribution of genealogies: we allow migration between demes, splitting of demes [as in the isolation-with-migration (IM) model], and recombination between linked loci. These processes are described by a set of linear recursions for the generating function of branch lengths. Under the infinite-sites model, the probability of any configuration of mutations can be found by differentiating this generating function. Such calculations are feasible for small numbers of sampled genomes: as an example, we show how the generating function can be derived explicitly for three genes under the two-deme IM model. This derivation is done automatically, using Mathematica. Given data from a large number of unlinked and nonrecombining blocks of sequence, these results can be used to find maximum-likelihood estimates of model parameters by tabulating the probabilities of all relevant mutational configurations and then multiplying across loci. The feasibility of the method is demonstrated by applying it to simulated data and to a data set previously analyzed by Wang and Hey (2010) consisting of 26,141 loci sampled from Drosophila simulans and D. melanogaster. Our results suggest that such likelihood calculations are scalable to genomic data as long as the numbers of sampled individuals and mutations per sequence block are small.