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PubMed2003

A comparison of physical mapping algorithms based on the maximum likelihood model.

MOTIVATION: Physical mapping of chromosomes using the maximum likelihood (ML) model is a problem of high computational complexity entailing both discrete optimization to recover the optimal probe order as well as continuous optimization to recover the optimal inter-probe spacings. In this paper, two versions of the genetic algorithm (GA) are proposed, one with heuristic crossover and deterministic replacement and the other with heuristic crossover and stochastic replacement, for the physical mapping problem under the maximum likelihood model. The genetic algorithms are compared with two other discrete optimization approaches, namely simulated annealing (SA) and large-step Markov chains (LSMC), in terms of solution quality and runtime efficiency. RESULTS: The physical mapping algorithms based on the GA, SA and LSMC have been tested using synthetic datasets and real datasets derived from cosmid libraries of the fungus Neurospora crassa. The GA, especially the version with heuristic crossover and stochastic replacement, is shown to consistently outperform the SA-based and LSMC-based physical mapping algorithms in terms of runtime and final solution quality. Experimental results on real datasets and simulated datasets are presented. Further improvements to the GA in the context of physical mapping under the maximum likelihood model are proposed. AVAILABILITY: The software is available upon request from the first author.

Algorithms↗

PubMed2005

Comparison of Bayesian and maximum-likelihood inference of population genetic parameters.

UNLABELLED: Comparison of the performance and accuracy of different inference methods, such as maximum likelihood (ML) and Bayesian inference, is difficult because the inference methods are implemented in different programs, often written by different authors. Both methods were implemented in the program MIGRATE, that estimates population genetic parameters, such as population sizes and migration rates, using coalescence theory. Both inference methods use the same Markov chain Monte Carlo algorithm and differ from each other in only two aspects: parameter proposal distribution and maximization of the likelihood function. Using simulated datasets, the Bayesian method generally fares better than the ML approach in accuracy and coverage, although for some values the two approaches are equal in performance. MOTIVATION: The Markov chain Monte Carlo-based ML framework can fail on sparse data and can deliver non-conservative support intervals. A Bayesian framework with appropriate prior distribution is able to remedy some of these problems. RESULTS: The program MIGRATE was extended to allow not only for ML(-) maximum likelihood estimation of population genetics parameters but also for using a Bayesian framework. Comparisons between the Bayesian approach and the ML approach are facilitated because both modes estimate the same parameters under the same population model and assumptions.

Bayes Theorem↗

PubMed2004

Maximum likelihood estimation for Cox's regression model under nested case-control sampling.

Nested case-control sampling is designed to reduce the costs of large cohort studies. It is important to estimate the parameters of interest as efficiently as possible. We present a new maximum likelihood estimator (MLE) for nested case-control sampling in the context of Cox's proportional hazards model. The MLE is computed by the EM-algorithm, which is easy to implement in the proportional hazards setting. Standard errors are estimated by a numerical profile likelihood approach based on EM aided differentiation. The work was motivated by a nested case-control study that hypothesized that insulin-like growth factor I was associated with ischemic heart disease. The study was based on a population of 3784 Danes and 231 cases of ischemic heart disease where controls were matched on age and gender. We illustrate the use of the MLE for these data and show how the maximum likelihood framework can be used to obtain information additional to the relative risk estimates of covariates.

Adult↗

PubMed1996

Maximum likelihood analysis of rare binary traits under different modes of inheritance.

Maximum likelihood methodology was applied to determine the mode of inheritance of rare binary traits with data structures typical for swine populations. The genetic models considered included a monogenic, a digenic, a polygenic, and three mixed polygenic and major gene models. The main emphasis was on the detection of major genes acting on a polygenic background. Deterministic algorithms were employed to integrate and maximize likelihoods. A simulation study was conducted to evaluate model selection and parameter estimation. Three designs were simulated that differed in the number of sires/number of dams within sires (10/10, 30/30, 100/30). Major gene effects of at least one SD of the liability were detected with satisfactory power under the mixed model of inheritance, except for the smallest design. Parameter estimates were empirically unbiased with acceptable standard errors, except for the smallest design, and allowed to distinguish clearly between the genetic models. Distributions of the likelihood ratio statistic were evaluated empirically, because asymptotic theory did not hold. For each simulation model, the Average Information Criterion was computed for all models of analysis. The model with the smallest value was chosen as the best model and was equal to the true model in almost every case studied.

Algorithms↗

PubMed1999

Using maximum likelihood to estimate population size from temporal changes in allele frequencies.

We develop a maximum-likelihood framework for using temporal changes in allele frequencies to estimate the number of breeding individuals in a population. We use simulations to compare the performance of this estimator to an F-statistic estimator of variance effective population size. The maximum-likelihood estimator had a lower variance and smaller bias. Taking advantage of the likelihood framework, we extend the model to include exponential growth and show that temporal allele frequency data from three or more sampling events can be used to test for population growth.

Alleles↗

PubMed2000

Monte Carlo evaluation of the likelihood for N(e) from temporally spaced samples.

A population's effective size is an important quantity for conservation and management. The effective size may be estimated from the change of allele frequencies observed in temporally spaced genetic samples taken from the population. Though moment-based estimators exist, recently Williamson and Slatkin demonstrated the advantages of a maximum-likelihood approach that they applied to data on diallelic genetic markers. Their computational methods, however, do not extend to data on multiallelic markers, because in such cases exact evaluation of the likelihood is impossible, requiring an intractable sum over latent variables. We present a Monte Carlo approach to compute the likelihood with data on multiallelic markers. So as to be computationally efficient, our approach relies on an importance-sampling distribution constructed by a forward-backward method. We describe the Monte Carlo formulation and the importance-sampling function and then demonstrate their use on both simulated and real datasets.

Algorithms↗

PubMed2002

Estimation of effective population size of HIV-1 within a host: a pseudomaximum-likelihood approach.

Using pseudomaximum-likelihood approaches to phylogenetic inference and coalescent theory, we develop a computationally tractable method of estimating effective population size from serially sampled viral data. We show that the variance of the maximum-likelihood estimator of effective population size depends on the serial sampling design only because internal node times on a coalescent genealogy can be better estimated with some designs than with others. Given the internal node times and the number of sequences sampled, the variance of the maximum-likelihood estimator is independent of the serial sampling design. We then estimate the effective size of the HIV-1 population within nine hosts. If we assume that the mutation rate is 2.5 x 10(-5) substitutions/generation and is the same in all patients, estimated generation lengths vary from 0.73 to 2.43 days/generation and the mean (1.47) is similar to the generation lengths estimated by other researchers. If we assume that generation length is 1.47 days and is the same in all patients, mutation rate estimates vary from 1.52 x 10(-5) to 5.02 x 10(-5). Our results indicate that effective viral population size and evolutionary rate per year are negatively correlated among HIV-1 patients.

Bayes Theorem↗

PubMed2002

A general likelihood approach to trait-based multipoint linkage analysis in large groups of half-sibs and super sisters.

The idea of trait-based linkage analysis in half-sibs is extended by comparing the frequency of parental marker haplotypes in animals with different phenotypes. This article first presents the likelihood of observing different classes of paternal haplotypes in a half-sib family, where only family members of a certain phenotype (e.g., affected) are genotyped and are fully informative. The likelihood function is then generalized to multiple phenotypic categories. A linear predictor allows for discontinuous as well as for continuous phenotypes and other explanatory variables. Finally, how to incorporate not fully informative offspring and how to analyze super sister families are shown. Maximum-likelihood estimates of all parameters can be found by a Newton-Raphson algorithm, which mimics an iteratively weighted least-squares procedure. The method allows for any multilocus feasible mapping function and, among others, for situations with selective or nonselective genotyping, single or multiple traits, and continuous or categorical traits. No parameters are required to describe the mode of inheritance and the method copes with virtually any family size. Fields of applications are therefore mapping experiments in species with a high reproductive capacity, such as cattle, pigs, horses, honey bees, trees, and fish.

Animals↗

PubMed2003

Maximum-likelihood and markov chain monte carlo approaches to estimate inbreeding and effective size from allele frequency changes.

Maximum-likelihood and Bayesian (MCMC algorithm) estimates of the increase of the Wright-Malécot inbreeding coefficient, F(t), between two temporally spaced samples, were developed from the Dirichlet approximation of allelic frequency distribution (model MD) and from the admixture of the Dirichlet approximation and the probabilities of fixation and loss of alleles (model MDL). Their accuracy was tested using computer simulations in which F(t) = 10% or less. The maximum-likelihood method based on the model MDL was found to be the best estimate of F(t) provided that initial frequencies are known exactly. When founder frequencies are estimated from a limited set of founder animals, only the estimates based on the model MD can be used for the moment. In this case no method was found to be the best in all situations investigated. The likelihood and Bayesian approaches give better results than the classical F-statistics when markers exhibiting a low polymorphism (such as the SNP markers) are used. Concerning the estimations of the effective population size all the new estimates presented here were found to be better than the F-statistics classically used.

Gene Frequency↗

PubMed2003

Comparison of Bayesian and maximum likelihood bootstrap measures of phylogenetic reliability.

Owing to the exponential growth of genome databases, phylogenetic trees are now widely used to test a variety of evolutionary hypotheses. Nevertheless, computation time burden limits the application of methods such as maximum likelihood nonparametric bootstrap to assess reliability of evolutionary trees. As an alternative, the much faster Bayesian inference of phylogeny, which expresses branch support as posterior probabilities, has been introduced. However, marked discrepancies exist between nonparametric bootstrap proportions and Bayesian posterior probabilities, leading to difficulties in the interpretation of sometimes strongly conflicting results. As an attempt to reconcile these two indices of node reliability, we apply the nonparametric bootstrap resampling procedure to the Bayesian approach. The correlation between posterior probabilities, bootstrap maximum likelihood percentages, and bootstrapped posterior probabilities was studied for eight highly diverse empirical data sets and were also investigated using experimental simulation. Our results show that the relation between posterior probabilities and bootstrapped maximum likelihood percentages is highly variable but that very strong correlations always exist when Bayesian node support is estimated on bootstrapped character matrices. Moreover, simulations corroborate empirical observations in suggesting that, being more conservative, the bootstrap approach might be less prone to strongly supporting a false phylogenetic hypothesis. Thus, apparent conflicts in topology recovered by the Bayesian approach were reduced after bootstrapping. Both posterior probabilities and bootstrap supports are of great interest to phylogeny as potential upper and lower bounds of node reliability, but they are surely not interchangeable and cannot be directly compared.

Animals↗

PubMed2003

The effect of positive selection on a sexual reproduction gene in Thalassiosira weissflogii (Bacillariophyta): results obtained from maximum-likelihood and parsimony-based methods.

Maximum-Likelihood-based and parsimony-based methods were used to test for potential effects of positive selection on the sexually induced gene 1 (Sig1) in Thalassiosira weissflogii. The Sig proteins are thought to play a role in mediating sperm-egg recognition during the sexual reproduction phase. The results obtained from parsimony-based analyses showed that none of the amino acid sites were influenced by positive selection. Maximum-likelihood analyses indicated that positive selection was affecting a maximum of seven and a minimum of four amino acid sites in the polypeptide derived from Sig1. It was concluded that the results obtained from the maximum-likelihood-based method are more reliable than those obtained from the parsimony-based approach. This is apparently the first study that has shown that reproductive proteins in unicellular eukaryotes are influenced by positive selection.

Algorithms↗

PubMed2001

Maximum-likelihood divergence date estimates based on rRNA gene sequences suggest two scenarios of Trypanosoma cruzi intraspecific evolution.

The phylogenetic relationships of Trypanosoma cruzi strains were inferred using maximum-likelihood from complete 18S rDNA sequences and D7-24Salpha rDNA regions from 20 representative strains of T. cruzi. For this we sequenced the 18S rDNA of 14 strains and the D7-24Salpha rDNA of four strains and aligned them to previously published sequences. Phylogenies inferred from these data sets identified four groups, named Riboclades 1, 2, 3, and 4, and a basal dichotomy that separated Riboclade 1 from Riboclades 2, 3, and 4. Substitution models and other parameters were optimized by hierarchical likelihood tests, and our analysis of the 18S rDNA molecular clock by the likelihood ratio test suggests that a taxa subset encompassing all 2,150 positions in the alignment supports rate constancy among lineages. The present analysis supports the notion that divergence dates of T. cruzi Riboclades can be estimated from 18S rDNA sequences and therefore, we present alternative evolutionary scenarios based on two different views of T. cruzi intraspecific divergence. The first assumes a faster evolutionary rate, which suggests that the divergence between T. cruzi I and II and the extant strains occurred in the Tertiary period (37-18 MYA). The other, which supports the hypothesis that the divergence between T. cruzi I and II occurred in the Cretaceous period (144-65 MYA) and the divergence of the extant strains occurred in the Tertiary period of the Cenozoic era (65-1.8 MYA), is consistent with our previously proposed hypothesis of divergence by geographical isolation and mammalian host coevolution.

Animals↗

PubMed2001

A general empirical model of protein evolution derived from multiple protein families using a maximum-likelihood approach.

Phylogenetic inference from amino acid sequence data uses mainly empirical models of amino acid replacement and is therefore dependent on those models. Two of the more widely used models, the Dayhoff and JTT models, are estimated using similar methods that can utilize large numbers of sequences from many unrelated protein families but are somewhat unsatisfactory because they rely on assumptions that may lead to systematic error and discard a large amount of the information within the sequences. The alternative method of maximum-likelihood estimation may utilize the information in the sequence data more efficiently and suffers from no systematic error, but it has previously been applicable to relatively few sequences related by a single phylogenetic tree. Here, we combine the best attributes of these two methods using an approximate maximum-likelihood method. We implemented this approach to estimate a new model of amino acid replacement from a database of globular protein sequences comprising 3,905 amino acid sequences split into 182 protein families. While the new model has an overall structure similar to those of other commonly used models, there are significant differences. The new model outperforms the Dayhoff and JTT models with respect to maximum-likelihood values for a large majority of the protein families in our database. This suggests that it provides a better overall fit to the evolutionary process in globular proteins and may lead to more accurate phylogenetic tree estimates. Potentially, this matrix, and the methods used to generate it, may also be useful in other areas of research, such as biological sequence database searching, sequence alignment, and protein structure prediction, for which an accurate description of amino acid replacement is required.

Algorithms↗

PubMed2001

A dependent-rates model and an MCMC-based methodology for the maximum-likelihood analysis of sequences with overlapping reading frames.

We present a model and methodology for the maximum-likelihood analysis of pairwise alignments of DNA sequences in which two genes are encoded in overlapping reading frames. In the model for the substitution process, the instantaneous rates of substitution are allowed to depend on the nucleotides occupying the sites in a neighborhood of the site subject to substitution at the instant of the substitution. By defining the neighborhood of a site to extend over all sites in the codons in both reading frames to which a site belongs, constraints imposed by the genetic code in both reading frames can be taken into account. Due to the dependency of the instantaneous rates of substitution on the states at neighboring sites, the transition probability between sequences does not factorize and therefore cannot be obtained directly. We present a Markov chain Monte Carlo procedure for obtaining the ratio of two transition probabilities between two sequences under the model considered, and we describe how maximum-likelihood parameter estimation and likelihood ratio tests can be performed using the procedure. We describe how the expected numbers of different types of substitutions in the shared history of two sequences can be calculated, and we use the described model and methodology in an analysis of a pairwise alignment of two hepatitis B sequences in which two genes are encoded in overlapping frames. Finally, we present an extended model, together with a simpler approximate estimation procedure, and use this to test the adequacy of the former model.

Base Sequence↗

PubMed2001

Quartet-mapping, a generalization of the likelihood-mapping procedure.

Likelihood-mapping (LM) was suggested as a method of displaying the phylogenetic content of an alignment. However, statistical properties of the method have not been studied. Here we analyze the special case of a four-species tree generated under a range of evolution models and compare the results with those of a natural extension of the likelihood-mapping approach, geometry-mapping (GM), which is based on the method of statistical geometry in sequence space. The methods are compared in their abilities to indicate the correct topology. The performance of both methods in detecting the star topology is especially explored. Our results show that LM tends to reject a star tree more often than GM. When assumptions about the evolutionary model of the maximum-likelihood reconstruction are not matched by the true process of evolution, then LM shows a tendency to favor one tree, whereas GM correctly detects the star tree except for very short outer branch lengths with a statistical significance of >0.95 for all models. LM, on the other hand, reconstructs the correct bifurcating tree with a probability of >0.95 for most branch length combinations even under models with varying substitution rates. The parameter domain for which GM recovers the true tree is much smaller. When the exterior branch lengths are larger than a (analytically derived) threshold value depending on the tree shape (rather than the evolutionary model), GM reconstructs a star tree rather than the true tree. We suggest a combined approach of LM and GM for the evaluation of starlike trees. This approach offers the possibility of testing for significant positive interior branch lengths without extensive statistical and computational efforts.

Computer Simulation↗

PubMed2002

A maximum likelihood method for analyzing pseudogene evolution: implications for silent site evolution in humans and rodents.

We present a new likelihood method for detecting constrained evolution at synonymous sites and other forms of nonneutral evolution in putative pseudogenes. The model is applicable whenever the DNA sequence is available from a protein-coding functional gene, a pseudogene derived from the protein-coding gene, and an orthologous functional copy of the gene. Two nested likelihood ratio tests are developed to test the hypotheses that (1) the putative pseudogene has equal rates of silent and replacement substitutions; and (2) the rate of synonymous substitution in the functional gene equals the rate of substitution in the pseudogene. The method is applied to a data set containing 74 human processed-pseudogene loci, 25 mouse processed-pseudogene loci, and 22 rat processed-pseudogene loci. Using the informatics resources of the Human Genome Project, we localized 67 of the human-pseudogene pairs in the genome and estimated the GC content of a large surrounding genomic region for each. We find that, for pseudogenes deposited in GC regions similar to those of their paralogs, the assumption of equal rates of silent and replacement site evolution in the pseudogene is upheld; in these cases, the rate of silent site evolution in the functional genes is approximately 70% the rate of evolution in the pseudogene. On the other hand, for pseudogenes located in genomic regions of much lower GC than their functional gene, we see a sharp increase in the rate of silent site substitutions, leading to a large rate of rejection for the pseudogene equality likelihood ratio test.

Animals↗

PubMed2000

NJML: a hybrid algorithm for the neighbor-joining and maximum-likelihood methods.

In the reconstruction of a large phylogenetic tree, the most difficult part is usually the problem of how to explore the topology space to find the optimal topology. We have developed a "divide-and-conquer" heuristic algorithm in which an initial neighbor-joining (NJ) tree is divided into subtrees at internal branches having bootstrap values higher than a threshold. The topology search is then conducted by using the maximum-likelihood method to reevaluate all branches with a bootstrap value lower than the threshold while keeping the other branches intact. Extensive simulation showed that our simple method, the neighbor-joining maximum-likelihood (NJML) method, is highly efficient in improving NJ trees. Furthermore, the performance of the NJML method is nearly equal to or better than existing time-consuming heuristic maximum-likelihood methods. Our method is suitable for reconstructing relatively large molecular phylogenetic trees (number of taxa >/= 16).

Algorithms↗

PubMed2002

Using exact Poisson likelihood functions in Bayesian interpretation of counting measurements.

A technique for computing the exact marginalized (integrated) Poisson likelihood function for counting measurement processes involving a background subtraction is described. An empirical Bayesian method for determining the prior probability distribution of background count rates from population data is recommended and would seem to have important practical advantages. The exact marginalized Poisson likelihood function may be used instead of the commonly used Gaussian approximation. Differences occur in some cases of small numbers of measured counts, which are discussed. Optional use of exact likelihood functions in our Bayesian internal dosimetry codes has been implemented using an interpolation-table approach, which means that there is no computation time penalty except for the initial setup of the interpolation tables.

Bayes Theorem↗