Cross-validated maximum likelihood enhances crystallographic simulated annealing refinement

Recently, the target function for crystallographic refinement has been improved through a maximum likelihood analysis, which makes proper allowance for the effects of data quality, model errors, and incompleteness. The maximum likelihood target reduces the significance of false local minima during the refinement process, but it does not completely eliminate them, necessitating the use of stochastic optimization methods such as simulated annealing for poor initial models. It is shown that the combination of maximum likelihood with cross-validation, which reduces overfitting, and simulated annealing by torsion angle molecular dynamics, which simplifies the conformational search problem, results in a major improvement of the radius of convergence of refinement and the accuracy of the refined structure. Torsion angle molecular dynamics and the maximum likelihood target function interact synergistically, the combination of both methods being significantly more powerful than each method individually. This is demonstrated in realistic test cases at two typical minimum Bragg spacings (dmin = 2.0 and 2.8 Å, respectively), illustrating the broad applicability of the combined method. In an application to the refinement of a new crystal structure, the combined method automatically corrected a mistraced loop in a poor initial model, moving the backbone by 4 Å.

Using rare mutations to estimate population divergence times: A maximum likelihood approach

In this paper we propose a method to estimate by maximum likelihood the divergence time between two populations, specifically designed for the analysis of nonrecurrent rare mutations. Given the rapidly growing amount of data, rare disease mutations affecting humans seem the most suitable candidates for this method. The estimator RD, and its conditional version RDc, were derived, assuming that the population dynamics of rare alleles can be described by using a birth–death process approximation and that each mutation arose before the split of a common ancestral population into the two diverging populations. The RD estimator seems more suitable for large sample sizes and few alleles, whose age can be approximated, whereas the RDc estimator appears preferable when this is not the case. When applied to three cystic fibrosis mutations, the estimator RD could not exclude a very recent time of divergence among three Mediterranean populations. On the other hand, the divergence time between these populations and the Danish population was estimated to be, on the average, 4,500 or 15,000 years, assuming or not a selective advantage for cystic fibrosis carriers, respectively. Confidence intervals are large, however, and can probably be reduced only by analyzing more alleles or loci.

Maximum likelihood estimation of a migration matrix and effective population sizes in n subpopulations by using a coalescent approach

A maximum likelihood estimator based on the coalescent for unequal migration rates and different subpopulation sizes is developed. The method uses a Markov chain Monte Carlo approach to investigate possible genealogies with branch lengths and with migration events. Properties of the new method are shown by using simulated data from a four-population n-island model and a source–sink population model. Our estimation method as coded in migrate is tested against genetree; both programs deliver a very similar likelihood surface. The algorithm converges to the estimates fairly quickly, even when the Markov chain is started from unfavorable parameters. The method was used to estimate gene flow in the Nile valley by using mtDNA data from three human populations.

A maximum likelihood algorithm for genome mapping of cytogenetic loci from meiotic configuration data.

Frequencies of meiotic configurations in cytogenetic stocks are dependent on chiasma frequencies in segments defined by centromeres, breakpoints, and telomeres. The expectation maximization algorithm is proposed as a general method to perform maximum likelihood estimations of the chiasma frequencies in the intervals between such locations. The estimates can be translated via mapping functions into genetic maps of cytogenetic landmarks. One set of observational data was analyzed to exemplify application of these methods, results of which were largely concordant with other comparable data. The method was also tested by Monte Carlo simulation of frequencies of meiotic configurations from a monotelodisomic translocation heterozygote, assuming six different sample sizes. The estimate averages were always close to the values given initially to the parameters. The maximum likelihood estimation procedures can be extended readily to other kinds of cytogenetic stocks and allow the pooling of diverse cytogenetic data to collectively estimate lengths of segments, arms, and chromosomes.

A stochastic approximation algorithm with Markov chain Monte-Carlo method for incomplete data estimation problems

We propose a general procedure for solving incomplete data estimation problems. The procedure can be used to find the maximum likelihood estimate or to solve estimating equations in difficult cases such as estimation with the censored or truncated regression model, the nonlinear structural measurement error model, and the random effects model. The procedure is based on the general principle of stochastic approximation and the Markov chain Monte-Carlo method. Applying the theory on adaptive algorithms, we derive conditions under which the proposed procedure converges. Simulation studies also indicate that the proposed procedure consistently converges to the maximum likelihood estimate for the structural measurement error logistic regression model.

The optimization principle in phylogenetic analysis tends to give incorrect topologies when the number of nucleotides or amino acids used is small

In the maximum parsimony (MP) and minimum evolution (ME) methods of phylogenetic inference, evolutionary trees are constructed by searching for the topology that shows the minimum number of mutational changes required (M) and the smallest sum of branch lengths (S), respectively, whereas in the maximum likelihood (ML) method the topology showing the highest maximum likelihood (A) of observing a given data set is chosen. However, the theoretical basis of the optimization principle remains unclear. We therefore examined the relationships of M, S, and A for the MP, ME, and ML trees with those for the true tree by using computer simulation. The results show that M and S are generally greater for the true tree than for the MP and ME trees when the number of nucleotides examined (n) is relatively small, whereas A is generally lower for the true tree than for the ML tree. This finding indicates that the optimization principle tends to give incorrect topologies when n is small. To deal with this disturbing property of the optimization principle, we suggest that more attention should be given to testing the statistical reliability of an estimated tree rather than to finding the optimal tree with excessive efforts. When a reliability test is conducted, simplified MP, ME, and ML algorithms such as the neighbor-joining method generally give conclusions about phylogenetic inference very similar to those obtained by the more extensive tree search algorithms.

Accurate reconstruction of a known HIV-1 transmission history by phylogenetic tree analysis.

Phylogenetic analyses are increasingly used in attempts to clarify transmission patterns of human immunodeficiency virus type 1 (HIV-1), but there is a continuing discussion about their validity because convergent evolution and transmission of minor HIV variants may obscure epidemiological patterns. Here we have studied a unique HIV-1 transmission cluster consisting of nine infected individuals, for whom the time and direction of each virus transmission was exactly known. Most of the transmissions occurred between 1981 and 1983, and a total of 13 blood samples were obtained approximately 2-12 years later. The p17 gag and env V3 regions of the HIV-1 genome were directly sequenced from uncultured lymphocytes. A true phylogenetic tree was constructed based on the knowledge about when the transmissions had occurred and when the samples were obtained. This complex, known HIV-1 transmission history was compared with reconstructed molecular trees, which were calculated from the DNA sequences by several commonly used phylogenetic inference methods [Fitch-Margoliash, neighbor-joining, minimum-evolution, maximum-likelihood, maximum-parsimony, unweighted pair group method using arithmetic averages (UPGMA), and a Fitch-Margoliash method assuming a molecular clock (KITSCH)]. A majority of the reconstructed trees were good estimates of the true phylogeny; 12 of 13 taxa were correctly positioned in the most accurate trees. The choice of gene fragment was found to be more important than the choice of phylogenetic method and substitution model. However, methods that are sensitive to unequal rates of change performed more poorly (such as UPGMA and KITSCH, which assume a constant molecular clock). The rapidly evolving V3 fragment gave better reconstructions than p17, but a combined data set of both p17 and V3 performed best. The accuracy of the phylogenetic methods justifies their use in HIV-1 research and argues against convergent evolution and selective transmission of certain virus variants.

Half tetrad analysis in alfalfa using multiple restriction fragment length polymorphism markers.

A maximum likelihood approach of half tetrad analysis (HTA) based on multiple restriction fragment length polymorphism (RFLP) markers was developed. This procedure estimates the relative frequencies of 2n gametes produced by mechanisms genetically equivalent to first division restitution (FDR) or second division restitution and simultaneously locates the centromere within a linkage group of RFLP marker loci. The method was applied to the diploid alfalfa clone PG-F9 (2n = 2x = 16) previously selected because of its high frequency of 2n egg production. HTA was based on four RFLP loci for which PG-F9 was heterozygous with codominant alleles that were absent in the tetraploid tester. Models including three linked and one unlinked RFLP loci were developed and tested. Results of the HTA showed that PG-F9 produced 6% FDR and 94% second division restitution 2n eggs. Information from a marker locus belonging to one linkage group was used to more precisely locate the centromere on a different linkage group. HTA, together with previous cytological analysis, indicated that in PG-F9, FDR 2n eggs are likely produced by diplospory, a mechanism common among apomictic species. The occurrence of FDR 2n eggs in plant species and their importance for crop evolution and breeding is discussed together with the potential applicability of multilocus HTA in the study of reproductive mutants.

Calculating the probability of multitaxon evolutionary trees: bootstrappers Gambit.

The reconstruction of multitaxon trees from molecular sequences is confounded by the variety of algorithms and criteria used to evaluate trees, making it difficult to compare the results of different analyses. A global method of multitaxon phylogenetic reconstruction described here, Bootstrappers Gambit, can be used with any four-taxon algorithm, including distance, maximum likelihood, and parsimony methods. It incorporates a Bayesian-Jeffreys'-bootstrap analysis to provide a uniform probability-based criterion for comparing the results from diverse algorithms. To examine the usefulness of the method, the origin of the eukaryotes has been investigated by the analysis of ribosomal small subunit RNA sequences. Three common algorithms (paralinear distances, Jukes-Cantor distances, and Kimura distances) support the eocyte topology, whereas one (maximum parsimony) supports the archaebacterial topology, suggesting that the eocyte prokaryotes are the closest prokaryotic relatives of the eukaryotes.