Infinite Sites Model Of Mutation Research Articles

Efficiently Summarizing Relationships in Large Samples: A General Duality Between Statistics of Genealogies and Genomes.

As a genetic mutation is passed down across generations, it distinguishes those genomes that have inherited it from those that have not, providing a glimpse of the genealogical tree relating the genomes to each other at that site. Statistical summaries of genetic variation therefore also describe the underlying genealogies. We use this correspondence to define a general framework that efficiently computes single-site population genetic statistics using the succinct tree sequence encoding of genealogies and genome sequence. The general approach accumulates sample weights within the genealogical tree at each position on the genome, which are then combined using a summary function; different statistics result from different choices of weight and function. Results can be reported in three ways: by site, which corresponds to statistics calculated as usual from genome sequence; by branch, which gives the expected value of the dual site statistic under the infinite sites model of mutation, and by node, which summarizes the contribution of each ancestor to these statistics. We use the framework to implement many currently defined statistics of genome sequence (making the statistics’ relationship to the underlying genealogical trees concrete and explicit), as well as the corresponding branch statistics of tree shape. We evaluate computational performance using simulated data, and show that calculating statistics from tree sequences using this general framework is several orders of magnitude more efficient than optimized matrix-based methods in terms of both run time and memory requirements. We also explore how well the duality between site and branch statistics holds in practice on trees inferred from the 1000 Genomes Project data set, and discuss ways in which deviations may encode interesting biological signals.

An accurate approximation for the expected site frequency spectrum in a Galton–Watson process under an infinite sites mutation model

If viruses or other pathogens infect a single host, the outcome of infection often hinges on the fate of the initial invaders. The initial basic reproduction number R0, the expected number of cells infected by a single infected cell, helps determine whether the initial viruses can establish a successful beachhead. To determine R0, the Kingman coalescent or continuous-time birth-and-death process can be used to infer the rate of exponential growth in an historical population. Given M sequences sampled in the present, the two models can make the inference from the site frequency spectrum (SFS), the count of mutations that appear in exactly k sequences (k=1,2,…,M). In the case of viruses, however, if R0 is large and an infected cell bursts while propagating virus, the two models are suspect, because they are Markovian with only binary branching. Accordingly, this article develops an approximation for the SFS of a discrete-time branching process with synchronous generations (i.e., a Galton–Watson process). When evaluated in simulations with an asynchronous, non-Markovian model (a Bellman–Harris process) with parameters intended to mimic the bursting viral reproduction of HIV, the approximation proved superior to approximations derived from the Kingman coalescent or continuous-time birth-and-death process. This article demonstrates that in analogy to methods in human genetics, the SFS of viral sequences sampled well after latent infection can remain informative about the initial R0. Thus, it suggests the utility of analyzing the SFS of sequences derived from patient and animal trials of viral therapies, because in some cases, the initial R0 may be able to indicate subtle therapeutic progress, even in the absence of statistically significant differences in the infection of treatment and control groups.

Coalescent: an open-science framework for importance sampling in coalescent theory.

Background. In coalescent theory, computer programs often use importance sampling to calculate likelihoods and other statistical quantities. An importance sampling scheme can exploit human intuition to improve statistical efficiency of computations, but unfortunately, in the absence of general computer frameworks on importance sampling, researchers often struggle to translate new sampling schemes computationally or benchmark against different schemes, in a manner that is reliable and maintainable. Moreover, most studies use computer programs lacking a convenient user interface or the flexibility to meet the current demands of open science. In particular, current computer frameworks can only evaluate the efficiency of a single importance sampling scheme or compare the efficiencies of different schemes in an ad hoc manner.Results. We have designed a general framework (http://coalescent.sourceforge.net; language: Java; License: GPLv3) for importance sampling that computes likelihoods under the standard neutral coalescent model of a single, well-mixed population of constant size over time following infinite sites model of mutation. The framework models the necessary core concepts, comes integrated with several data sets of varying size, implements the standard competing proposals, and integrates tightly with our previous framework for calculating exact probabilities. For a given dataset, it computes the likelihood and provides the maximum likelihood estimate of the mutation parameter. Well-known benchmarks in the coalescent literature validate the accuracy of the framework. The framework provides an intuitive user interface with minimal clutter. For performance, the framework switches automatically to modern multicore hardware, if available. It runs on three major platforms (Windows, Mac and Linux). Extensive tests and coverage make the framework reliable and maintainable.Conclusions. In coalescent theory, many studies of computational efficiency consider only effective sample size. Here, we evaluate proposals in the coalescent literature, to discover that the order of efficiency among the three importance sampling schemes changes when one considers running time as well as effective sample size. We also describe a computational technique called “just-in-time delegation” available to improve the trade-off between running time and precision by constructing improved importance sampling schemes from existing ones. Thus, our systems approach is a potential solution to the “28 programs problem” highlighted by Felsenstein, because it provides the flexibility to include or exclude various features of similar coalescent models or importance sampling schemes.

On the genealogy of asexual diploids.

Abstract Given molecular genetic data from diploid individuals that, at present, reproduce mostly or exclusively asexually without recombination, an important problem in evolutionary biology is detecting evidence of past sexual reproduction (i.e., meiosis and mating) and recombination (both meiotic and mitotic). However, currently there is a lack of computational tools for carrying out such a study. In this article, we formulate a new problem of reconstructing diploid genealogies under the assumption of no sexual reproduction or recombination, with the ultimate goal being to devise genealogy-based tools for testing deviation from these assumptions. We first consider the infinite-sites model of mutation and develop linear-time algorithms to test the existence of an asexual diploid genealogy compatible with the infinite-sites model of mutation, and to construct one if it exists. In this ideal case, our chance of detecting signatures of past sexual reproduction is maximized. Then, we relax the infinite-sites assumption and develop an integer linear programming formulation to reconstruct asexual diploid genealogies with the minimum number of homoplasy (back or recurrent mutation) events. If this number is substantially larger than that expected for typical asexual organisms, then it may suggest that sexual reproduction or recombination may have played an important role in the evolutionary history. We apply our algorithms on simulated data sets with sizes of biological interest.

Accurate Prediction of Genetic Values for Complex Traits by Whole-Genome Resequencing

Whole-genome resequencing technology has improved rapidly during recent years and is expected to improve further such that the sequencing of an entire human genome sequence for $1000 is within reach. Our main aim here is to use whole-genome sequence data for the prediction of genetic values of individuals for complex traits and to explore the accuracy of such predictions. This is relevant for the fields of plant and animal breeding and, in human genetics, for the prediction of an individual's risk for complex diseases. Here, population history and genomic architectures were simulated under the Wright-Fisher population and infinite-sites mutation model, and prediction of genetic value was by the genomic selection approach, where a Bayesian nonlinear model was used to predict the effects of individual SNPs. The Bayesian model assumed a priori that only few SNPs are causative, i.e., have an effect different from zero. When using whole-genome sequence data, accuracies of prediction of genetic value were >40% increased relative to the use of dense approximately 30K SNP chips. At equal high density, the inclusion of the causative mutations yielded an extra increase of accuracy of 2.5-3.7%. Predictions of genetic value remained accurate even when the training and evaluation data were 10 generations apart. Best linear unbiased prediction (BLUP) of SNP effects does not take full advantage of the genome sequence data, and nonlinear predictions, such as the Bayesian method used here, are needed to achieve maximum accuracy. On the basis of theoretical work, the results could be extended to more realistic genome and population sizes.

The number of recombination events in a sample history: conflict graph and lower bounds

We consider the following problem: Given a set of binary sequences, determine lower bounds on the minimum number of recombinations required to explain the history of the sample, under the infinite-sites model of mutation. The problem has implications for finding recombination hotspots and for the Ancestral Recombination Graph reconstruction problem. Hudson and Kaplan gave a lower bound based on the four-gamete test. In practice, their bound Rm often greatly underestimates the minimum number of recombinations. The problem was recently revisited by Myers and Griffiths, who introduced two new lower bounds Rh and Rs which are provably better, and also yield good bounds in practice. However, the worst-case complexities of their procedures for computing Rh and Rs are exponential and super-exponential, respectively. In this paper, we show that the number of nontrivial connected components, Rc, in the conflict graph for a given set of sequences, computable in time O(nm2), is also a lower bound on the minimum number of recombination events. We show that in many cases, Rc is a better bound than Rh. The conflict graph was used by Gusfield et al. to obtain a polynomial time algorithm for the galled tree problem, which is a special case of the Ancestral Recombination Graph (ARG) reconstruction problem. Our results also offer some insight into the structural properties of this graph and are of interest for the general Ancestral Recombination Graph reconstruction problem.

The Coalescent in an Island Model of Population Subdivision with Variation among Demes

A simple genealogical structure is found for a general finite island model of population subdivision. The model allows for variation in the sizes of demes, in contributions to the migrant pool, and in the fraction of each deme that is replaced by migrants every generation. The ancestry of a sample of non-recombining DNA sequences has a simple structure when the sample size is much smaller than the total number of demes in the population. This allows an expression for the probability distribution of the number of segregating sites in the sample to be derived under the infinite-sites mutation model. It also yields easily computed estimators of the migration parameter for each deme in a multi-deme sample. The genealogical process is such that the lineages ancestral to the sample tend to accumulate in demes with low migration rates and/or which contribute disproportionately to the migrant pool. In addition, common ancestor or coalescent events tend to occur in demes of small size. This provides a framework for understanding the determinants of the effective size of the population, and leads to an expression for the probability that the root of a genealogy occurs in a particular geographic region, or among a particular set of demes.

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.

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).

The structure of genealogies and the distribution of fixed differences between DNA sequence samples from natural populations.

When two samples of DNA sequences are compared, one way in which they may differ is in the presence of fixed differences, which are defined as sites at which all of the sequences in one sample are different from all of the sequences in a second sample. The probability distribution of the number of fixed differences is developed. The theory employs Wright-Fisher genealogies and the infinite sites mutation model. For the case when both samples are drawn randomly from the same population it is found that genealogies permitting fixed differences are very unlikely. Thus the mere presence of fixed differences between samples is statistically significant, even for small samples. The theory is extended to samples from populations that have been separated for some time. The relationship between a simple Poisson distribution of mutations and the distribution of fixed differences is described as a function of the time since populations have been isolated. It is shown how these results may contribute to improved tests of recent balancing or directional selection.