Dirichlet mixtures provide an elegant formalism for constructing and evaluating protein multiple sequence alignments. Their use requires the inference of Dirichlet mixture priors from curated sets of accurately aligned sequences. This article addresses two questions relevant to such inference: of how many components should a Dirichlet mixture consist, and how may a maximum-likelihood mixture be derived from a given data set. To apply the Minimum Description Length principle to the first question, we extend an analytic formula for the complexity of a Dirichlet model to Dirichlet mixtures by informal argument. We apply a Gibbs-sampling based approach to the second question. Using artificial data generated by a Dirichlet mixture, we demonstrate that our methods are able to approximate well the true theory, when it exists. We apply our methods as well to real data, and infer Dirichlet mixtures that describe the data better than does a mixture derived using previous approaches.
algorithms; combinatorics; linear programming; machine learning; statistics
A model is a set of possible theories for describing a set of data. When the data are used to select a maximum-likelihood theory, an important question is how many effectively independent theories the model contains; the log of this number is called the model's complexity. The Dirichlet model is the set of all Dirichlet distributions, which are probability densities over the space of multinomials. A Dirichlet distribution may be used to describe multiple-alignment data, consisting of n columns of letters, with c letters in each column. We here derive, in the limit of large n and c, a closed-form expression for the complexity of the Dirichlet model applied to such data. For small c, we derive as well a minor correction to this formula, which is easily calculated by Monte Carlo simulation. Although our results are confined to the Dirichlet model, they may cast light as well on the complexity of Dirichlet mixture models, which have been applied fruitfully to the study of protein multiple sequence alignments.
alignment; computational molecular biology; dynamic programming; multiple alignment; sequence analysis
Dirichlet mixture priors provide a Bayesian formalism for scoring alignments of protein profiles to individual sequences, which can be generalized to constructing scores for multiple-alignment columns. A Dirichlet mixture is a probability distribution over multinomial space, each of whose components can be thought of as modeling a type of protein position. Applied to the simplest case of pairwise sequence alignment, a Dirichlet mixture is equivalent to an implied symmetric substitution matrix. For alphabets of even size L, Dirichlet mixtures with L/2 components and symmetric substitution matrices have an identical number of free parameters. Although this suggests the possibility of a one-to-one mapping between the two formalisms, we show that there are some symmetric matrices no Dirichlet mixture can imply, and others implied by many distinct Dirichlet mixtures. Dirichlet mixtures are derived empirically from curated sets of multiple alignments. They imply “background” amino acid frequencies characteristic of these sets, and should thus be non-optimal for comparing proteins with non-standard composition. Given a mixture Θ, we seek an adjusted Θ′ that implies the desired composition, but that minimizes an appropriate relative–entropy–based distance function. To render the problem tractable, we fix the mixture parameter as well as the sum of the Dirichlet parameters for each component, allowing only its center of mass to vary. This linearizes the constraints on the remaining parameters. An approach to finding Θ′ may be based on small consecutive parameter adjustments. The relative entropy of two Dirichlet distributions separated by a small change in their parameter values implies a quadratic cost function for such changes. For a small change in implied background frequencies, this function can be minimized using the Lagrange-Newton method. We have implemented this method, and can compositionally adjust to good precision a 20-component Dirichlet mixture prior for proteins in under half a second on a standard workstation.
algorithms; combinatorics; linear programming; machine learning; statistics
The Dirichlet process is used to model probability distributions that are mixtures of an unknown number of components. Amino acid frequencies at homologous positions within related proteins have been fruitfully modeled by Dirichlet mixtures, and we use the Dirichlet process to derive such mixtures with an unbounded number of components. This application of the method requires several technical innovations to sample an unbounded number of Dirichlet-mixture components. The resulting Dirichlet mixtures model multiple-alignment data substantially better than do previously derived ones. They consist of over 500 components, in contrast to fewer than 40 previously, and provide a novel perspective on the structure of proteins. Individual protein positions should be seen not as falling into one of several categories, but rather as arrayed near probability ridges winding through amino acid multinomial space.
alignment; computational molecular biology; dynamic programming; multiple alignment; sequence analysis
Motivation: Pairwise protein sequence alignments are generally evaluated using scores defined as the sum of substitution scores for aligning amino acids to one another, and gap scores for aligning runs of amino acids in one sequence to null characters inserted into the other. Protein profiles may be abstracted from multiple alignments of protein sequences, and substitution and gap scores have been generalized to the alignment of such profiles either to single sequences or to other profiles. Although there is widespread agreement on the general form substitution scores should take for profile-sequence alignment, little consensus has been reached on how best to construct profile–profile substitution scores, and a large number of these scoring systems have been proposed. Here, we assess a variety of such substitution scores. For this evaluation, given a gold standard set of multiple alignments, we calculate the probability that a profile column yields a higher substitution score when aligned to a related than to an unrelated column. We also generalize this measure to sets of two or three adjacent columns. This simple approach has the advantages that it does not depend primarily upon the gold-standard alignment columns with the weakest empirical support, and that it does not need to fit gap and offset costs for use with each substitution score studied.
Results: A simple symmetrization of mean profile-sequence scores usually performed the best. These were followed closely by several specific scoring systems constructed using a variety of rationales.
Supplementary Information: Supplementary data are available at Bioinformatics online.
Almost all protein database search methods use amino acid substitution matrices for scoring, optimizing, and assessing the statistical significance of sequence alignments. Much care and effort has therefore gone into constructing substitution matrices, and the quality of search results can depend strongly upon the choice of the proper matrix. A long-standing problem has been the comparison of sequences with biased amino acid compositions, for which standard substitution matrices are not optimal. To address this problem, we have recently developed a general procedure for transforming a standard matrix into one appropriate for the comparison of two sequences with arbitrary, and possibly differing compositions. Such adjusted matrices yield, on average, improved alignments and alignment scores when applied to the comparison of proteins with markedly biased compositions.
Here we review the application of compositionally adjusted matrices and consider whether they may also be applied fruitfully to general purpose protein sequence database searches, in which related sequence pairs do not necessarily have strong compositional biases. Although it is not advisable to apply compositional adjustment indiscriminately, we describe several simple criteria under which invoking such adjustment is on average beneficial. In a typical database search, at least one of these criteria is satisfied by over half the related sequence pairs. Compositional substitution matrix adjustment is now available in NCBI's protein-protein version of BLAST.
substitution matrices; compositional adjustment; protein database searches; BLAST; BLOSUM
BLAST is a commonly-used software package for comparing a query sequence to a database of known sequences; in this study, we focus on protein sequences. Position-specific-iterated BLAST (PSI-BLAST) iteratively searches a protein sequence database, using the matches in round i to construct a position-specific score matrix (PSSM) for searching the database in round i + 1. Biegert and Söding developed Context-sensitive BLAST (CS-BLAST), which combines information from searching the sequence database with information derived from a library of short protein profiles to achieve better homology detection than PSI-BLAST, which builds its PSSMs from scratch.
We describe a new method, called domain enhanced lookup time accelerated BLAST (DELTA-BLAST), which searches a database of pre-constructed PSSMs before searching a protein-sequence database, to yield better homology detection. For its PSSMs, DELTA-BLAST employs a subset of NCBI’s Conserved Domain Database (CDD). On a test set derived from ASTRAL, with one round of searching, DELTA-BLAST achieves a ROC5000 of 0.270 vs. 0.116 for CS-BLAST. The performance advantage diminishes in iterated searches, but DELTA-BLAST continues to achieve better ROC scores than CS-BLAST.
DELTA-BLAST is a useful program for the detection of remote protein homologs. It is available under the “Protein BLAST” link at http://blast.ncbi.nlm.nih.gov.
This article was reviewed by Arcady Mushegian, Nick V. Grishin, and Frank Eisenhaber.
Most pairwise and multiple sequence alignment programs seek alignments with optimal scores. Central to defining such scores is selecting a set of substitution scores for aligned amino acids or nucleotides. For local pairwise alignment, substitution scores are implicitly of log-odds form. We now extend the log-odds formalism to multiple alignments, using Bayesian methods to construct “BILD” (“Bayesian Integral Log-odds”) substitution scores from prior distributions describing columns of related letters. This approach has been used previously only to define scores for aligning individual sequences to sequence profiles, but it has much broader applicability. We describe how to calculate BILD scores efficiently, and illustrate their uses in Gibbs sampling optimization procedures, gapped alignment, and the construction of hidden Markov model profiles. BILD scores enable automated selection of optimal motif and domain model widths, and can inform the decision of whether to include a sequence in a multiple alignment, and the selection of insertion and deletion locations. Other applications include the classification of related sequences into subfamilies, and the definition of profile-profile alignment scores. Although a fully realized multiple alignment program must rely upon more than substitution scores, many existing multiple alignment programs can be modified to employ BILD scores. We illustrate how simple BILD score based strategies can enhance the recognition of DNA binding domains, including the Api-AP2 domain in Toxoplasma gondii and Plasmodium falciparum.
Multiple sequence alignment is a fundamental tool of biological research, widely used to identify important regions of DNA or protein molecules, to infer their biological functions, to reconstruct ancestries, and in numerous other applications. The effectiveness and accuracy of sequence comparison programs depends crucially upon the quality of the scoring systems they use to measure sequence similarity. To compare pairs of DNA or protein sequences, the best strategy for constructing similarity measures has long been understood, but there has been a lack of consensus about how to measure similarity among multiple (i.e. more than two) sequences. In this paper, we describe a natural generalization to multiple alignment of the accepted measure of pairwise similarity. A large variety of methods that are used to compare and analyze DNA or protein molecules, or to model protein domain families, could be rendered more sensitive and precise by adopting this similarity measure. We illustrate how our measure can enhance the recognition of important DNA binding domains.
Position specific score matrices (PSSMs) are derived from multiple sequence alignments to aid in the recognition of distant protein sequence relationships. The PSI-BLAST protein database search program derives the column scores of its PSSMs with the aid of pseudocounts, added to the observed amino acid counts in a multiple alignment column. In the absence of theory, the number of pseudocounts used has been a completely empirical parameter. This article argues that the minimum description length principle can motivate the choice of this parameter. Specifically, for realistic alignments, the principle supports the practice of using a number of pseudocounts essentially independent of alignment size. However, it also implies that more highly conserved columns should use fewer pseudocounts, increasing the inter-column contrast of the implied PSSMs. A new method for calculating pseudocounts that significantly improves PSI-BLAST's; retrieval accuracy is now employed by default.
Motivation: The flexibility in gap cost enjoyed by hidden Markov models (HMMs) is expected to afford them better retrieval accuracy than position-specific scoring matrices (PSSMs). We attempt to quantify the effect of more general gap parameters by separately examining the influence of position- and composition-specific gap scores, as well as by comparing the retrieval accuracy of the PSSMs constructed using an iterative procedure to that of the HMMs provided by Pfam and SUPERFAMILY, curated ensembles of multiple alignments.
Results: We found that position-specific gap penalties have an advantage over uniform gap costs. We did not explore optimizing distinct uniform gap costs for each query. For Pfam, PSSMs iteratively constructed from seeds based on HMM consensus sequences perform equivalently to HMMs that were adjusted to have constant gap transition probabilities, albeit with much greater variance. We observed no effect of composition-specific gap costs on retrieval performance. These results suggest possible improvements to the PSI-BLAST protein database search program.
Availability: The scripts for performing evaluations are available upon request from the authors.
TBLASTN is a mode of operation for BLAST that aligns protein sequences to a nucleotide database translated in all six frames. We present the first description of the modern implementation of TBLASTN, focusing on new techniques that were used to implement composition-based statistics for translated nucleotide searches. Composition-based statistics use the composition of the sequences being aligned to generate more accurate E-values, which allows for a more accurate distinction between true and false matches. Until recently, composition-based statistics were available only for protein-protein searches. They are now available as a command line option for recent versions of TBLASTN and as an option for TBLASTN on the NCBI BLAST web server.
We evaluate the statistical and retrieval accuracy of the E-values reported by a baseline version of TBLASTN and by two variants that use different types of composition-based statistics. To test the statistical accuracy of TBLASTN, we ran 1000 searches using scrambled proteins from the mouse genome and a database of human chromosomes. To test retrieval accuracy, we modernize and adapt to translated searches a test set previously used to evaluate the retrieval accuracy of protein-protein searches. We show that composition-based statistics greatly improve the statistical accuracy of TBLASTN, at a small cost to the retrieval accuracy.
TBLASTN is widely used, as it is common to wish to compare proteins to chromosomes or to libraries of mRNAs. Composition-based statistics improve the statistical accuracy, and therefore the reliability, of TBLASTN results. The algorithms used by TBLASTN are not widely known, and some of the most important are reported here. The data used to test TBLASTN are available for download and may be useful in other studies of translated search algorithms.
Protein sequence database search programs may be evaluated both for their retrieval accuracy—the ability to separate meaningful from chance similarities—and for the accuracy of their statistical assessments of reported alignments. However, methods for improving statistical accuracy can degrade retrieval accuracy by discarding compositional evidence of sequence relatedness. This evidence may be preserved by combining essentially independent measures of alignment and compositional similarity into a unified measure of sequence similarity. A version of the BLAST protein database search program, modified to employ this new measure, outperforms the baseline program in both retrieval and statistical accuracy on ASTRAL, a SCOP-based test set.
PSI-BLAST is an iterative program to search a database for proteins
with distant similarity to a query sequence. We investigated over
a dozen modifications to the methods used in PSI-BLAST, with the goal
of improving accuracy in finding true positive matches. To evaluate
performance we used a set of 103 queries for which the true positives
in yeast had been annotated by human experts, and a popular measure
of retrieval accuracy (ROC) that can be normalized to take on values
between 0 (worst) and 1 (best). The modifications we consider novel
improve the ROC score from 0.758 ± 0.005
to 0.895 ± 0.003. This does not include
the benefits from four modifications we included in the ‘baseline’ version,
even though they were not implemented in PSI-BLAST version 2.0.
The improvement in accuracy was confirmed on a small second test
set. This test involved analyzing three protein families with curated
lists of true positives from the non-redundant protein database.
The modification that accounts for the majority of the improvement
is the use, for each database sequence, of a position-specific scoring
system tuned to that sequence’s amino acid composition.
The use of composition-based statistics is particularly beneficial
for large-scale automated applications of PSI-BLAST.
The distribution of optimal local alignment scores of random
sequences plays a vital role in evaluating the statistical significance
of sequence alignments. These scores can be well described by an
extreme-value distribution. The distribution’s parameters depend
upon the scoring system employed and the random letter frequencies;
in general they cannot be derived analytically, but must be estimated
by curve fitting. For obtaining accurate parameter estimates, a form
of the recently described ‘island’ method has several
advantages. We describe this method in detail, and use it to investigate
the functional dependence of these parameters on finite-length edge
To characterize gene expression in activated mast cells more comprehensively than heretofore, we surveyed the changes in genetic transcripts by the method of serial analysis of gene expression in the RBL-2H3 line of rat mast cells before and after they were stimulated through their receptors with high affinity for immunoglobulin E (FcεRI). A total of 40,759 transcripts derived from 11,300 genes were analyzed. Among the diverse genes that had not been previously associated with mast cells and that were constitutively expressed were those for the cytokine macrophage migration inhibitory factor neurohormone receptors such as growth hormone- releasing factor and melatonin and components of the exocytotic machinery. In addition, several dozen transcripts were differentially expressed in response to antigen-induced clustering of the FcεRI. Included among these were the genes for preprorelaxin, mitogen-activated protein kinase kinase 3, and the dual specificity protein phosphatase, rVH6. Significantly, the majority of genes differentially expressed in this well-studied model of mast cell activation have not been identified before this analysis.
receptor aggregation; signal transduction; exocytosis; cell differentiation; allergy